null  null
Semi-active Suspension Control
Emanuele Guglielmino • Tudor Sireteanu
Charles W. Stammers • Gheorghe Ghita
Marius Giuclea
Semi-active Suspension
Control
Improved Vehicle Ride and Road Friendliness
123
Emanuele Guglielmino, PhD
Italian Institute of Technology (IIT)
Via Morego, 30
16163 Genoa
Italy
Charles W. Stammers, PhD
Department of Mechanical Engineering
University of Bath
Bath BA2 7AY
UK
Tudor Sireteanu, PhD
Gheorghe Ghita, PhD
Marius Giuclea, PhD
Department of Mathematics
Academy of Economic Studies 6
Piata Romana
010374 Bucharest
Romania
Institute of Solid Mechanics
Romanian Academy
C-tin Mille Street
010141 Bucharest
Romania
ISBN 978-1-84800-230-2
e-ISBN 978-1-84800-231-9
DOI 10.1007/978-1-84800-231-9
British Library Cataloguing in Publication Data
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Library of Congress Control Number: 2008927606
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Preface
The fundamental goals of a car suspension are the isolation of the vehicle from the
road and the improvement of road holding by means of a spring-type element and a
damper.
The inherent limitations of classical suspensions have motivated the
investigation of controlled suspension systems, both semi-active and active. In a
semi-active suspension the damper is generally replaced by a controlled dissipative
element and no energy is introduced into the system. In contrast, an active
suspension requires the use of a fully active actuator, and a significant energy input
is generally required. Due to their higher reliability, lower cost and comparable
performance semi-active suspensions have gained wide acceptance throughout the
automotive engineering community.
This book provides an overview of vehicle ride control employing smart semiactive damping systems. In this context the term smart refers to the ability to
modify the control logic in response to measured vehicle ride and handling
indicators.
The latest developments in vehicle ride control stem from the integration of
diverse engineering disciplines, including classical mechanics, hydraulics,
biomechanics and control engineering as well as software engineering and
analogue and digital electronics.
This book is not intended to be a general-purpose text on vehicle dynamics and
vehicle control systems (traction control, braking control, engine and emissions
control etc.). The focus of this work is on controlled semi-active suspension
systems for ride control and road friendliness using a multidisciplinary
mechatronic approach.
If effective control of a suspension is to be achieved, which makes the most of
the potentialities of a semi-active damping device, it is paramount to understand
the interactions between mechanical, hydraulic and electromagnetic sub-systems.
The book analyses the different facets of the technical problems involved when
designing a novel smart damping system and the technical challenges involved in
its control. The emphasis of this work is not only on modelling and control
algorithm design, but also on the practical aspects of its implementation. It
describes the practical constraints encountered and trade-offs pursued in real-life
viii
Preface
engineering practice when designing and testing a novel smart damping system.
Hence sound mathematical modelling is balanced by large sections on
experimental implementation as well as case studies, where a variety of automotive
applications are described, covering different applications of ride control, namely
semi-active suspensions for a saloon car, seat suspensions for vehicles not
equipped with a primary suspension and control of heavy-vehicle dynamic tyre
loads to reduce road damage and improve handling.
Within the book issues such as road holding, passenger comfort and human
body response to vibration are thoroughly analysed. Appropriate control-oriented
dampers models are described, along with their experimental validation. Vehicle
ride and human body models are illustrated and robust algorithms are designed.
The book is centered around two types of semi-active dampers: friction
dampers and magnetorheological dampers. The former can be viewed as an out-ofthe-box non-conventional damper while the latter can be thought as a conventional
controllable damper (it is used in several cars). Based on these two types of
dampers in the course of the book it is shown how to design a semi-active damping
system (using a friction damper) and how to implement an effective semi-active
control system on a well-established damper (the magnetorheological damper).
The book can be fruitful reading for mechanical engineering students (at both
undergraduate and postgraduate level) interested in vehicle dynamics, electrical
and control engineering students majoring in electromechanical and
electrohydraulic control systems. It should be valuable reading for R&D and
design engineers working in the automotive industry and automotive consultants. It
can be of interest also to engineers, physicists and applied mathematicians working
in the broad area of noise and vibration control, as many concepts can potentially
be applied to other fields of vibration control.
The book is structured as follows:
Chapter 1 is a general introduction to active, semi-active and passive
suspensions and introduces the fundamental concepts of vehicle ride and handling
dynamics.
Chapter 2 focusses on dampers modelling (including hysteresis modelling) and
reviews the main vehicle ride and road surface models.
Chapter 3 analyses the human body response to vibration via appropriate
human body models based on recent studies in the field of biomechanics.
Chapter 4 is dedicated to control algorithms. After a brief qualitative overview
of the fundamentals of modern control theory, the main semi-active suspensions
algorithms are introduced. The focus is on an algorithm known as balance logic,
which is analysed from a mathematical viewpoint. Emphasis is also placed on
robust algorithm design and on techniques to increase the reliability of the systems
(e.g., anti-chattering algorithms).
Chapter 5 details the design of a semi-active suspension system based on a
friction damper.
Chapter 6 illustrates the design of a magnetorheological-based semi-active
suspension.
Chapter 7 offers a comprehensive overview of the applications with a number
of case studies including a friction damper-based suspension unit for a saloon car, a
magnetorheological damper-based seat suspension for vehicles not equipped with
Preface
ix
primary suspensions which uniquely rely on this suspension mounted underneath
the driver seat to provide ride comfort and semi-active suspension for heavy
vehicles where the emphasis is not only on ride comfort but also on road damage
reduction.
Disclaimer
All the experimental work and numerical simulations presented in this book are the
result of academic research carried out at the University of Bath (UK) and at the
Institute of Solid Mechanics of the Romanian Academy (Romania). All devices
described in the book are purely experimental prototypes.
Therefore in no circumstances shall any liability be accepted for any loss or
damage howsoever caused to the fullest possible extent of the law that may result
from the reader's acting upon or using the content contained in the publication.
Acknowledgements
The authors wish to express their gratitude to Springer (London) for the invitation
to write this book. Many thanks to our Editor Oliver Jackson for his guidance
throughout this work.
The book is the fruit of many years of research work. Most of the results
presented were obtained at the Department of Mechanical Engineering, University
of Bath (UK), which we would like to thank.
We are deeply grateful to Professor Kevin Edge, Pro-Vice-Chancellor for
Research of the University of Bath, for his contribution to the hydraulic control of
friction dampers.
We would like to thank the University for providing the technical facilities
and would also like to express our gratitude to the technical staff of the Department
of Mechanical Engineering without whom the manufacture of devices and test rigs
and technical trouble-shooting would not have been possible.
We wish to express our appreciation to the Royal Society of London for
sponsoring a decade of collaboration with the Romanian Academy in Bucharest,
where valuable contributions to this text were made.
Thanks are also due to Dr Georgios Tsampardoukas for kindly providing
valuable material on trucks.
We finally wish to thank all the people whose input, support and
sympathetic understanding was instrumental to the successful completion of such
an undertaking.
Contents
1 Introduction....................................................................................................... 1
1.1 Introduction................................................................................................ 1
1.2 Historical Notes on Suspensions................................................................ 3
1.3 Active and Semi-active Suspensions in the Scientific Literature............... 5
1.4 Comfort in a Vehicle.................................................................................. 7
1.4.1 Comfort Assessment..................................................................... 10
1.5 Introduction to Controlled Dampers ........................................................ 10
1.6 Introduction to Friction Dampers............................................................. 12
1.7 Introduction to MR Dampers ................................................................... 14
2 Dampers and Vehicle Modelling.................................................................... 17
2.1 Introduction.............................................................................................. 17
2.2 Phenomenology of Hysteresis.................................................................. 19
2.3 Damper Hysteresis Modelling ................................................................. 22
2.3.1 Bouc–Wen Model......................................................................... 24
2.3.1.1 Parameter A.................................................................... 24
2.3.1.2 Parameter γ..................................................................... 25
2.3.1.3 Parameter ν .................................................................... 26
2.3.1.4 Parameter n .................................................................... 26
2.4 Bouc–Wen Parameter Identification........................................................ 27
2.5 Vehicle Ride Models................................................................................ 27
2.5.1 Quarter Car Model........................................................................ 29
2.5.2 Half Car Model............................................................................. 31
2.5.3 Full Car Model ............................................................................. 32
2.5.4 Half Truck Model ......................................................................... 36
2.6 Tyre Modelling ........................................................................................ 39
2.7 Road Modelling ....................................................................................... 40
3 Human Body Analysis .................................................................................... 43
3.1 Introduction.............................................................................................. 43
3.2 Human Body Response............................................................................ 44
xiv
Contents
3.3
3.4
3.5
3.6
3.7
3.8
Hysteretic Damping ................................................................................. 44
3.3.1 The Duffing Equation................................................................... 45
3.3.2 Suppression of Jumps ................................................................... 46
Low-frequency Seated Human Model ..................................................... 48
3.4.1 Multi-frequency Input................................................................... 49
Semi-active Control ................................................................................. 51
State Observer.......................................................................................... 51
3.6.1 Luenberger State Observer ........................................................... 51
3.6.2 Simple State Observer .................................................................. 52
3.6.3 Ideal Control................................................................................. 53
Results ..................................................................................................... 54
Seated Human with Head-and-Neck Complex …………. ...................... 57
3.8.1 Driver Seat (Including Cushions) ................................................. 58
3.8.2 Driver Body .................................................................................. 59
3.8.3 Head-and-Neck Complex (HNC) ................................................. 59
3.8.4 Analysis of the Head-and-Neck System ....................................... 60
3.8.5 Head Accelerations During Avoidance Manoeuvre ..................... 64
4 Semi-active Control Algorithms .................................................................... 65
4.1 Introduction.............................................................................................. 65
4.2 PID Controllers ........................................................................................ 67
4.3 Adaptive Control ..................................................................................... 68
4.4 Robust Control…..................................................................................... 69
4.5 Balance, Skyhook and Groundhook ........................................................ 70
4.5.1 Balance Logic............................................................................... 70
4.5.2 Skyhook Logic.............................................................................. 70
4.5.3 Groundhook Logic........................................................................ 70
4.5.4 Displacement-based On–Off Groundhook Logic ......................... 71
4.5.5 Hybrid Skyhook–Groundhook Logic ........................................... 71
4.6 Balance Logic Analysis ........................................................................... 72
4.7 Chattering Reduction Strategies .............................................................. 75
4.8 SA Vibration Control of a 1DOF System with Sequential Dry Friction.. 79
4.8.1 Sequential Damping Characteristics............................................. 81
4.8.2 Free Vibration: Phase Plane Trajectories...................................... 82
4.8.3 Free Vibration: Shock Absorbing Properties................................ 83
4.8.4 Harmonically-Excited Vibration .................................................. 85
4.8.4.1 Time Histories................................................................ 85
4.8.4.2 Amplitude–Frequency Characteristics ........................... 85
4.8.5 Random Vibration ........................................................................ 87
4.8.5.1 Simulation of White Noise Sample Functions ............... 89
4.8.5.2 Numerical Solution of the Equation of Motion.............. 91
4.8.5.3 Numerical Results.......................................................... 92
4.9 Stability of SA Control with Sequential Dry Friction............................. 93
4.10 Quarter Car Response with Sequential Dry Friction................................ 95
5 Friction Dampers ............................................................................................ 99
5.1 Introduction.............................................................................................. 99
Contents
xv
5.2
Friction Force Modelling ......................................................................... 99
5.2.1 Static Friction Models ................................................................ 100
5.2.2 Dynamic Friction Models........................................................... 102
5.2.3 Seven-parameter Friction Model ................................................ 102
5.3 The Damper Electrohydraulic Drive...................................................... 104
5.4 Friction Damper Hydraulic Drive Modelling ........................................ 107
5.4.1 Power Consumption ................................................................... 115
5.4.2 The Feedback Chain ................................................................... 115
5.5 Pilot Implementation of Friction Damper Control................................. 116
5.6 Automotive Friction Damper Design..................................................... 122
5.7 Switched State Feedback Control .......................................................... 126
5.8 Preliminary Simulation Results ............................................................. 129
5.9 Friction Damper Electrohydraulic Drive Assessment............................ 141
5.10 Electrohydraulic Drive Parameters Validation ...................................... 151
5.11 Performance Enhancement of the Friction Damper System .................. 156
5.11.1 Damper Design Modification ..................................................... 157
5.11.2 Hydraulic Drive Optimisation .................................................... 159
5.11.3 Friction Damper Controller Enhancement.................................. 161
6 Magnetorheological Dampers ...................................................................... 165
6.1 Introduction............................................................................................ 165
6.2 Magnetorheological Fluids .................................................................... 165
6.3 MR Fluid Devices.................................................................................. 167
6.3.1 Basic Operating Modes .............................................................. 167
6.3.2 Flow Simulation ......................................................................... 168
6.3.2.1 Pressure-driven Flow Mode with Either Pole Fixed .... 168
6.3.2.2 Direct Shear Mode with Relatively Movable Poles ..... 177
6.3.2.3 Squeeze-film Mode...................................................... 180
6.4 MR Damper Design ............................................................................... 180
6.4.1 Input Data and Choice of the Design Solution ........................... 181
6.4.2 Selection of the Working MR Fluid ........................................... 181
6.4.2.1 MR Fluid Figures of Merit .......................................... 182
6.4.2.2 Choice of the MR Fluid ............................................... 183
6.4.3 Determination of the Optimal Gap Size and Hydraulic Design.. 185
6.4.3.1 Controllable Force and Dynamic Range ...................... 185
6.4.3.2 Parameters of the Hydraulic Circuit............................. 186
6.4.4 Magnetic Circuit Design............................................................. 187
6.5 MRD Modelling and Characteristics Identification ............................... 189
6.5.1 Experimental Data ...................................................................... 190
6.5.2 Parametric Model Simulation ..................................................... 192
6.5.3 Fuzzy-logic-based Model ........................................................... 200
6.5.4 Modelling the Variable Field Strength ....................................... 203
6.5.5 GA-based Method for MR Damper Model Parameters
Identification............................................................................... 209
7 Case Studies................................................................................................... 219
7.1 Introduction............................................................................................ 219
xvi
Contents
7.1.1 Some Aspects of Data Acquisition and Control ......................... 219
Car Dynamics Experimental Analysis ................................................... 221
7.2.1 The Experimental Set-up ............................................................ 221
7.2.2 Post-processing and Measurement Results................................. 224
7.2.3 Suspension Spring and Tyre Tests.............................................. 229
7.3 Passively-Damped Car Validation ......................................................... 230
7.4 Case Study 1: SA Suspension Unit with FD.......................................... 232
7.4.1 Frequency-domain Analysis ....................................................... 233
7.4.2 Time-domain Analysis ............................................................... 234
7.4.3 Semi-active System Validation................................................... 242
7.5 Case Study 2: MR-based SA Seat Suspension....................................... 245
7.5.1 Numerical Results ...................................................................... 248
7.5.2 Conclusions ................................................................................ 250
7.6 Case Study 3: Road Damage Reduction with MRD Truck
Suspension.................................................................................. 251
7.6.1 Introduction ................................................................................ 251
7.6.2 Half Truck and MR Damper Model ........................................... 252
7.6.3 Road Damage Assessment.......................................................... 255
7.6.4 Road Damage Reduction Algorithm .......................................... 255
7.6.5 Time Response ........................................................................... 256
7.6.6 Truck Response on Different Road Profiles ............................... 258
7.6.7 Truck Response to Bump and Pothole........................................ 262
7.6.8 Robustness Analysis ................................................................... 264
7.6.8.1 Trailer Mass Variation ................................................. 266
7.6.8.2 Tyre Stiffness Variation ............................................... 267
7.6.8.3 MRD Response Time................................................... 268
7.7 Conclusions............................................................................................ 270
7.2
References ........................................................................................................... 271
Bibliography........................................................................................................ 283
Authors’ Biographies ......................................................................................... 289
Index .................................................................................................................... 291
1
Introduction
1.1 Introduction
Today’s vehicles rely on a number of electronic control systems. Some of them are
self-contained, stand-alone controllers fulfilling a particular function while others
are co-ordinated by a higher-level supervisory logic. Examples of such vehicle
control systems include braking control, traction control, acceleration control,
lateral stability control, suspension control and so forth. Such systems aim to
enhance ride and handling, safety, driving comfort and driving pleasure. This book
focuses on semi-active suspension control. The thrust of this work is to provide a
comprehensive overview of theoretical and design aspects (including several case
studies) of vehicle semi-active systems based on smart damping devices.
Isolation from the forces transmitted by external excitation is the fundamental
task of any suspension system. The problem of mechanical vibration control is
generally tackled by placing between the source of vibration and the structure to be
protected, suspension systems composed of spring-type elements in parallel with
dissipative elements. Suspensions are employed in mobile applications, such as
terrain vehicles, or in non-mobile applications, such as vibrating machinery or civil
structures. In the case of a vehicle, a classical car suspension aims to achieve
isolation from the road by means of spring-type elements and viscous dampers
(shock absorbers) and contemporarily to improve road holding and handling.
The elastic element of a suspension is constituted by a spring (coil springs but
also air springs and leaf springs), whereas the damping element is typically of the
viscous type. In such a device the damping action is obtained by throttling a
viscous fluid through orifices; depending on the physical properties of the fluid
(mainly its viscosity), the geometry of the orifices and of the damper, a variety of
force versus velocity characteristics can be obtained. This technology is very
reliable and has been used since the beginning of the last century (Bastow, 1993).
However it is possible to achieve a damping effect by other means, as subsequently
discussed.
2
Semi-active Suspension Control
Spring rate and damping are chosen according to comfort, road holding and
handling specifications. A suspension unit ought to be able to reduce chassis
acceleration as well as dynamic tyre force within the constraint of a set working
space. Depending upon the type of vehicle, either the former or the latter criterion
is emphasised. In applications different from automotive ones (e.g., rotating
machinery, vibration mitigation in civil structures) the comfort criterion is not
usually an issue, but other specifications exist, e.g., on the maximum value of some
quantities (displacements, velocities etc.).
Passive suspensions have inherent limitations as a consequence of the trade-off
in the choice of the spring rate and damping characteristics, in order to achieve
acceptable behaviour over the whole range of working frequencies. As is known
from linear systems theory a one-degree-of-freedom (1DOF) spring–mass–damper
system (modelled by a second-order linear differential equation) having high
damping performs well in the vicinity of the resonant frequency and poorly far
from it, whilst a low-damped system behaves conversely (Rao, 1995).
The necessity of compromising between these conflicting requirements has
motivated the investigation of controlled suspension systems, where the elastic and
the damping characteristics are controlled closed-loop. By using an external power
supply and feedback-controlled actuators, controlled suspension systems can be
designed which outperform any passive system.
The external energy needed to generate the required control forces of a smart
suspension is an important issue that must be considered in controller design. The
controllers must be designed so as to achieve an acceptable trade-off between
control effectiveness and energy consumption. From this point of view, the control
strategies can be grouped in two main categories: active and semi-active.
Usually, the active control strategies need a substantial amount of energy to
produce the required control forces. A fully active system can potentially provide
higher performance than its passive counterpart. However in many engineering
applications this goal can be achieved only at the expense of a complex and costly
system, with large energy consumption and non-trivial reliability issues. In
particular when designing an active control system two important aspects must be
taken into consideration: the potential failure of the power source, and the injection
of a large amount of mechanical energy into the structure that has the potential to
destabilise (in the bounded input/bounded output sense) the controlled system.
Hence a careful hazard and failure-modes analysis must be carried out and a failsafe design adopted.
Semi-active control devices offer reliability comparable to that of passive
devices, yet maintaining the versatility and adaptability of fully active systems,
without requiring large power sources. In a semi-active suspension the amount of
damping can be tuned in real time. Hence most semi-active devices produce only a
modulation of the damping forces in the controlled system according to the control
strategy employed. In contrast to active control devices, semi-active control
devices cannot inject mechanical energy into the controlled system and, therefore,
they do not have the potential to destabilise it. Examples of such devices are
variable orifice dampers, controllable friction devices and dampers with
controllable fluids (e.g., electrorheological and magnetorheological fluids).
Introduction
3
Fig. 1.1. Illustration of passive and controlled vehicle suspensions (1 - passive spring, 2 passive damper, 3 - actuator, 4 - controllable damper)
The above discussed control solutions are illustrated schematically in Figure 1.1
for the simplest (1DOF) model of a primary vehicle suspension.
As previously stated, a suspension algorithm is designed to reduce chassis
acceleration as well as dynamic tyre force. Chassis acceleration is related to ride
and comfort, and tyre force to road holding and handling.
Dynamic tyre force reduction results in better handling of the vehicle, as the
cornering force, tractive and braking efforts developed by the tyre are related to
normal load, which can be controlled by semi-active methods. Road holding and
handling performance can be quantified by the consideration of the forces and
moments applied to the chassis and to the tyres.
Comfort is more difficult to quantify and, although standards exist, its
assessment is a controversial issue, because it is an inherently subjective matter. In
Section 1.5 a survey of comfort assessment criteria is presented to shed light on
this matter.
1.2 Historical Notes on Suspensions
Suspensions, as many other vehicle systems, followed relatively closely the
evolution of the transportation technology. For centuries carts were not equipped
with any sort of suspension at all. Only later, in the eigth century, was a primitive
suspension based on an iron chain system developed. Metal springs were first
developed in the 17th century and shortly afterwards leaf springs. Various designs
were developed until the last century, which saw the development of the concept of
suspension based on a spring and a damper.
Early vehicle ride studies date back to the 1920s and 1930s (Lanchester, 1936).
Investigation on handling and steering dynamics followed later in the 1950s as
reported by Milliken WF and Milliken DL (1995) as well as the application of
random vibration theory to vehicle studies. The advent of digital computers with
greater processing power and the development of multi-body vehicle models of
ever increasing complexity has contributed to produce more and more
sophisticated designs.
4
Semi-active Suspension Control
The optimisation of suspensions is achieved not only via a careful design and
tuning of springs and dampers, but also by improving the design of the other
components of the suspensions (e.g., rubber bushes and mountings), so as to better
exploit their damping properties in order to obtain an overall ride improvement,
and through an appropriate design of the suspension geometry (links, arms, levers).
The study of the mechanical design of suspensions and their kinematics is not
within the scope of this book. However a brief overview is here presented. The
literature on the topic is vast (interesting websites on suspension mechanical design
and suspension history are http://www.carbibles.com/suspension_bible.html and
http://www.citroenet.org.uk/miscellaneous/suspension/suspension1.html).
Essentially suspensions can be categorised into two large families: dependent
and independent suspensions, the difference being whether the two suspension
units (on either the front or the rear of the vehicle) are linked or not. Whilst it is
very common to have rear dependent suspension, most front suspensions are of the
independent type. Sometimes suspensions are linked by an anti-roll bar, which is
essentially a torsion spring that helps reduce roll while negotiating a bend.
As far as the kinematics of the suspension is concerned car manufacturers
developed a variety of designs, including the so-called double wishbone system
and the multi-link suspensions (used on the Audi A4 for instance).
It was said that suspension unit is composed of a damper and a spring. Springs
are typically of coil type but leaf springs are still common, particularly on trucks.
A classical design is the so-called MacPherson strut, named after Earle S.
MacPherson who designed it in the 1940s. It is a very compact design where the
damper is mounted within the coil spring. Hydropneumatic suspension is another
type of suspension developed by Citroën, which has worked on controlled
suspensions for many years (Curtis, 1991). Another type of suspension is the
Hydragas suspension employed for instance on the MGF Roadster (Moulton and
Best, 1979a and 1979b; Rideout and Anderson, 2003).
Controlled suspensions (both active and semi-active) have appealed to
automotive engineers for many decades. Semi-active dampers have been developed
by damper manufacturers such as ZF Sachs (ABC -Active Body Control- and CDC
http://www.zf.com/content/en/import/zf_konzern/startseite/f_e/nutzen_fuer_unsere
_kunden/variable_daempfungssysteme/Variable_Daempfungssysteme.html).
At
present many vehicles offer some kind of controlled suspensions. Active
suspensions were first developed for Formula 1 cars: Lotus’s was the first car to be
equipped with an active system in 1983 (Baker, 1984; Milliken, 1987). Besides
racing cars, active systems have been studied and developed for a long time also
for road vehicles (typically saloon cars). Hillebrecht et al. (1992) 15 years ago
discussed the trade-off between customer benefit and technological challenge from
the angle of a car manufacturer. Mercedes have worked for many years on active
suspensions. The Mercedes CL Coupe (Cross, 1999) is equipped with a fully
integrated suspension and traction control. The Citroën BX model was fitted with a
self-leveller system and the Xantia Activa is equipped with active anti-roll bars.
Toyota worked on controlled suspensions, for example in the Toyota Celica
(Yokoya et al., 1990) as well as Volvo (Tiliback and Brood, 1989). Most recently
plenty of high-segment cars are equipped with semi-active suspensions (Mercedes,
Lamborghini and Ferrari vehicles, to name but a few).
Introduction
5
Magnetorheological-based semi-active suspensions are used on a number of
high-segment market cars which employ the Delphi MagneRide™
(http://delphi.com/manufacturers/auto/other/ride/magneride/) system based on
magnetorheological dampers. The system is fitted on a few vehicles including
some Cadillac models (Imaj, Seville, SRX, XLR, STS, DTS), the Chevrolet
Corvette and most recently the Audi TT, the Audi R8 and the Ferrari 599 GTB.
Another interesting type of suspension worth mentioning is the Bose® linear
electromagnetic suspension designed by Dr Amar Bose, which is based on a linear
electric motor and power amplifier instead of a spring and a damper
(http://www.automobilemag.com/features/news/0410_bose_suspension/).
1.3 Active and Semi-active Suspensions in the Scientific
Literature
A vast amount of work on controlled suspension systems is present in the technical
and scientific literature. The first paper dealing with active suspensions dates back
to the 1950s (Federspiel-Labrosse, 1954). One of the first reviews of the state of
the art of controlled suspensions was carried out by Hedrick and Wormely (1975).
Another one was produced in 1983 by Goodall and Kortüm who surveyed the
active suspension technology. A few years later Sharp and Crolla (1987) and
Crolla and Aboul Nour (1988) produced comparative reviews of advantages and
drawbacks of various types of suspensions. Another historical review and also an
attempt to present some design criteria was given by Crolla (1995). A first choice
in the design of a fully active suspension is the type of actuation. The actuator can
be hydraulic, pneumatic or electromagnetic, or a hybrid solution. Williams et al.
(1996) analysed the merits of an oleo-pneumatic actuator, Martins et al. (1999)
proposed a hybrid electromagnetic-controlled suspension. An active suspension
employing a hydraulic actuator, pressure-controlled rather than flow-controlled,
has been proposed by Satoh et al. (1990).
Active suspensions are a challenging field for control engineers. All the main
control techniques developed in the past 30 years have been applied to the problem
of controlling vehicle suspensions. An overview of this research now follows.
At the outset it must be stressed that one of the main problems in the design of
control algorithms is the identification of the vehicle and suspension parameters.
Errors in their knowledge can spoil the performance of the most sophisticated
controllers, designed with very refined mathematical techniques. Majjad (1997)
and Tan and Bradshaw (1997) addressed the problem of the identification of car
suspension parameters. The necessity of trading off among the conflicting
requirements of the suspensions in terms of comfort and road holding led to the use
of optimisation techniques. In 1976 Thompson studied a quarter car model and
employed optimal linear state feedback theory for designing a controlled
suspension; Chalasani (1987) optimised active ride performance using a full car
model. An H∞ algorithm for active suspensions was proposed by Sammier et al.
(2000).
6
Semi-active Suspension Control
The driving conditions in a car change greatly depending upon the road and the
speed. This suggests the necessity of some forms of adaptive control. Hac (1987)
implemented this kind of scheme. Other types of adaptive controls were proposed
over the years. Ramsbottom et al. (1999) and Chantranuwathana and Peng (1999)
studied an adaptive robust control scheme for active suspensions.
Robust control, the control philosophy for dealing with systems with uncertain
parameters, has been investigated by many researchers as well. Mohan and Phadke
(1996) studied a variable structure controller for a quarter car. Park and Kim
(1998) extended this study to a full 7DOF ride model. Sliding mode control was
investigated by Yagtz et al. (1997) and by Kim and Ro (1998). A mixed sliding
mode–fuzzy controller was proposed by Al-Houlu et al. (1999).
Active suspension systems have been studied also for off-road vehicles (Crolla
et al., 1987). Stayner (1988) proposed an active suspension for agricultural
vehicles. Active devices have been investigated also for rail applications as
reported by Goodall et al. (1981).
Semi-active suspensions were firstly introduced in the 1970s (Crosby and
Karnopp, 1973; Karnopp et al., 1974) as an alternative to the costly, highly
complicated and power-demanding active systems. Similar work was performed by
Rakheja and Sankar (1985) and Alanoly and Sankar (1987) in terms of active and
semi-active isolators. A comparative study with passive systems was carried out by
Margolis (1982) and by Ahmadian and Marjoram (1989). The most attractive
feature of that work was that the control strategies were based only upon the
measurement of the relative displacement and velocity. A review can be found in
Crolla (1995).
A control scheme known as skyhook damping, based on the measurement of
the absolute vertical velocity of the body of the car (the aim is to achieve the same
damping force as that produced by a damper connected to an ideal inertial
reference in the sky), was proposed in the 1970s by Karnopp and is still employed
in a number of variations (Alleyne et al., 1993). Yi and Song (1999) proposed an
adaptive version of the skyhook control. Some authors (Chang and Wu, 1997), in
order to improve comfort, designed a suspension based on a biological,
neuromuscular-like control system. Recently Liu et al. (2005) studied four
different semi-active control strategies based on the skyhook and balance control
strategies.
The reduction of the dynamic tyre force is a challenging field. Cole et al.
(1994) did extensive work on it, both theoretical and experimental. Groundhook
control logic was also investigated by Valasek et al. (1998) to reduce dynamic tyre
forces.
As far as the applications of semi-active suspensions are concerned, they have
been envisaged not only for saloon cars but also for other types of vehicles.
Besinger et al. (1991) studied an application of semi-active dampers on trucks. A
skyhook algorithm for train applications was investigated by Ogawa et al. (1999).
Miller and Nobes (1988) studied a semi-active suspension for military tanks. A
study was performed by Margolis and Noble (1991) in order to control heave and
roll motions of large off-road vehicles.
As with active systems, a variety of control schemes have been proposed for
semi-active suspensions: adaptive schemes (Bellizzi and Bouc, 1989), optimal
Introduction
7
control (Tseng and Hedrick, 1994), LQG (Linear Quadratic Gaussian) schemes
(Barak and Hrovat, 1988) as well as robust algorithms (Titli et al., 1993). Crolla
and Abdel Hady (1988) proposed a multivariable controller for a full vehicle
model. Preview control was proposed by Hac (1992) and Hac and Youn (1992),
receding horizon control was investigated by Ursu et al., (1984) and H∞ optimal
control by Moran and Nagai (1992).
An interesting solution has been proposed by Groenewald and Gouws (1996)
who suggested improving ride and handling by using closed-loop control to adjust
the tyre pressure. By controlling tyre pressure it is possible to control wheel-hop
resonance and therefore improve the so-called secondary ride, i.e., the behaviour
close to the wheel-hop resonance (Shaw, 1999) as well as improving the lifespan of
tyres and reducing fuel consumption.
During the last few years there has been a tremendous amount of activity on the
applications of artificial intelligence techniques to suspension systems. Neural
networks and fuzzy logic (Vemuri, 1993; Agarwal, 1997) have attracted the
attention of many researchers in the field (Moran and Nagai, 1994; Watanabe and
Sharp, 1996 and 1999; Ghazi Zadeh et al. 1997; Yoshimura et al., 1997).
This brief survey has shown that a great deal of research has been and is still
being carried out on designing cheap and reliable controlled suspension systems
for vehicles.
1.4 Comfort in a Vehicle
Whilst road holding and handling can be objectively quantified by the analysis of
the dynamic equations of a vehicle, this is not the case with comfort, as it is an
inherently subjective matter. Ride quality, driving pleasure and driving fun are
concerned with passenger comfort and driver feeling in a moving vehicle.
Vibration transmitted to passengers originates from a host of causes, including,
amongst others, road uneveness, aerodynamic forces and engine- and powertraininduced vibration. Road irregularities are indeed the major source of vibration. In a
comfortable vehicle, vibration must stay within some boundaries. In order to
establish these boundaries, it is firstly necessary to assess and quantify how to
measure comfort.
There is no generally accepted method to assess human sensitivity to vibration;
human response is quite subjective and dependent on several factors. Firstly it must
be highlighted that the road forces transmitted to tyres are asymmetrical. In the
occurrence of a bump, vertical upward acceleration can reach several g while if a
pothole is encountered, the vertical downward acceleration cannot be larger than 1
g. This is also a reason why hydraulic dampers are designed with non-symmetrical
characteristics for the bound and rebound strokes.
The human body presents asymmetric reactions to vibration as well: the body
reacts differently if a vertical acceleration of a given magnitude is applied upward
or downward. People can better withstand an increase rather than a decrease in the
gravitational force (as can be experienced in a fast elevator, for instance). Likewise
motions with low roll centre (e.g., rolling and pitching of a ship) are more
troublesome and likely to induce motion sickness than those with a high roll centre.
8
Semi-active Suspension Control
From these considerations it follows that comfort tests carried out with sinusoidal
inputs are not sufficient, but only useful for comparison and benchmarking
purposes. However even if a multi-harmonic input is applied, it is difficult to
weigh the various frequencies to which the body is more, or less, sensitive.
Early studies associated the feeling of comfort with the frequency of vibration
of the walking pace. A review of these early works is provided by Demic (1984).
Thus early car suspensions were designed according to this criterion. Further
studies (Dieckmann, 1958) proved that different frequency bands are
uncomfortable for different organs. Frequencies lower than 1 Hz are related to
symptoms like motion sickness; frequencies in the range 5–6 Hz are troublesome
for the stomach, while frequencies around 20 Hz are pernicious for head and neck.
Several criteria have been proposed over the past 30 years to assess comfort
based on the nature of the vibration. Some of them are general purpose whilst
others are employed in specialist fields such as off-road and military vehicles
(Pollock and Craighead, 1986).
One first criterion, relevant to the automotive field, is Janeway's comfort
criterion (SAE Society of Automotive Engineers, 1965). It relates the comfort to
vertical vibration amplitude and permits to find the largest allowed chassis
displacement for each frequency. In essence it states that within the range 1–6 Hz
the maximum allowable amplitude is the one resulting in a peak jerk value of not
more than 12.6 m/s3. In other words, if X is the maximum allowed displacement
amplitude and ω the angular frequency, Janeway's criterion states that:
Xω 3 = 12.6 .
(1.1)
At higher frequencies the criterion does not refer to jerk but to acceleration and
velocity. It affirms that in the range 6-20 Hz, acceleration peak value should not
exceed 0.33 m/s2, whilst between 20 and 60 Hz the maximum velocity should stay
below 2.7 mm/s. The major limitation of the SAE criterion is that it applies only to
vertical sinusoidal disturbances and it does not give any indication of the situation
in the case of multi-harmonic inputs.
In the same period Steffens (1966) proposed an empirical formula to determine
the amplitude of vibration causing discomfort as a function of the frequency:
X [cm 2 ] = 7.62 ⋅10 − 3 (1 +
125
),
f2
(1.2)
where again vertical displacement (rather than acceleration) is used to assess
comfort.
The most general criterion however is the standard ISO 2631 (1978); it is a
general standard applicable not only to vehicles but to all vibrating environments.
It defines the exposure limits for body vibration in the range 1–80 Hz, defining
limits for reduced comfort, for decreased proficiency and for preservation of
health. A subsequent addendum to the norm also takes into consideration the
frequencies in the particular range 0.1–1 Hz. It relates discomfort to root mean
square (RMS) acceleration as a function of frequency for different exposure times.
Introduction
9
If vibrations are applied in all three directions (foot-to-head, side-to-side, back-tochest), the corresponding boundaries apply for each component. Subsequently to
the ISO standard, the British standard BS 6841 was published in 1987 in address
what were perceived in Britain as the weakest points of the ISO standard and in
1998 Griffin thoroughly reviewed both of them, analysing merits and their weakest
points.
Another criterion proposed is the vibration dose value (Griffin, 1984) which
provides an indication based on the integral of the fourth power of the frequencyweighted acceleration. The vibration dose (VD) value is calculated as:
t
∫
VD = a 4 dt.
(1.3)
0
If the acceleration has components along two or three axes, the total dose value
is the algebraic sum of the values for each axis. A value above 15 m4/s7 is
considered to cause discomfort. This criterion is independent of the type of
waveform; besides the fourth power of weighted acceleration emphasises the peak
value (which is the other important parameter of a waveform —together with the
RMS value— to assess comfort).
Some studies have considered the energy absorbed by the body in a vibrating
environment. An initial study by Zeller (1949) relates the comfort to the maximum
specific kinetic energy absorbed by the body over a period for a sinusoidal input,
and establishes some boundaries for the maximum allowed absorbable energy. The
energy is calculated as E = 0.5v2max/T = a2max/8π2f2. The disturbance considered here
is again sinusoidal, T being its period.
A further integral criterion is known as the absorbed power method (Lee and
Pradko, 1969); it calculates the power absorbed by the body when it experiences
vibration. It has been used by the army to test human tolerance in military vehicles.
The criterion is expressed by:
f
∫
Power = ( A 2 w)df ,
(1.4)
0
the parameter A being the acceleration spectrum along an axis and w a frequency
weighting function. The tolerance limit is taken to be 6 W.
Another criterion, known as the DISC rating value (Leatherwood et al., 1980)
assesses the discomfort with the formula:
DISC = −0.44 + 1.65(DVERT 2 + DLAT 2 ),
(1.5)
where
DVERT = −1.75 + 0.857CFV − 0.102CFV 2 + 0.00346CFV 3 + 33.4GV,
(1.6)
10
Semi-active Suspension Control
CFV being the centre frequency of vertical axis applied vibration band and GV the
peak acceleration within this band (DLAT is defined by an analogous expression).
This survey has shown that the problem of assessing comfort is in some way
quantifiable, but the main issue is the choice of the most suitable comfort criteria
for a particular application.
1.4.1 Comfort Assessment
From the previous survey it is clear that comfort assessment is an ill-posed
problem from a quantitative standpoint. In this work a choice has been made on
how to assess it. For the purpose of this work the assessment will be based on
RMS, peak and jerk values for sinusoidal inputs, on the response to a pseudorandom input and to a bump as well as on the spectral characteristics of the nonlinear acceleration response.
In terms of sinusoidal input, the simplest method to compare passive and semiactive suspension response is through the peak value of chassis accelerations. In a
linear case it is straightforward: for an output displacement expressed by
x(t)=Xsin(2πft), the peak values of the higher-order derivatives (velocity,
acceleration and jerk) are:
x MAX = 2πfX ,
(1.7)
xMAX = 4π 2 f 2 X ,
(1.8)
xMAX = 8π 3 f 3 X .
(1.9)
An assessment based on peak jerk values is a possibility investigated by Hrovat
and Hubbard (1987). Another possible approach to estimate comfort is in the
frequency domain, carrying out a Fourier analysis of the chassis acceleration
amplitude spectrum. Actually this analysis is more appropriate to assess the degree
of non-linearity in terms of harmonic distortion. However, as a rule of thumb, a
large harmonic content may possibly indicate higher peak acceleration and jerk
amplitudes (even if strictly speaking the analysis of the phase spectrum would be
necessary), although there is no precise relationship between spectrum and
comfort.
As far as the bump input is concerned, the relevant quantities to minimise,
comfortwise, are the peak value of the acceleration and the number of oscillations
after the bump.
1.5 Introduction to Controlled Dampers
The kernel of a semi-active system is the controllable damper. Therefore it is of
paramount importance to gain an understanding of the different types of dampers,
Introduction
11
their working principles, and how to model them. Consider a simple spring–mass–
damper system, depicted in Figure 1.2.
Fig. 1.2. 1DOF spring–mass–damper system
A spring is an elastic element which stores potential energy and whose positiondependent characteristic can be expressed by the functional relation F=k(x), x
being the displacement across it. Examples of springs employed in automotive
engineering are coil springs, air springs, leaf springs and torsion bars. Hooke’s law
F=kx, k being the spring stiffness (or spring rate) expressed in N/m, represents the
particular case of a linear spring.
A damper is a device which dissipates energy through an internal mechanism
(e.g., by throttling a viscous flow through an orifice). Typically a damper has a
velocity-dependent characteristic expressed by F = c ( x ), x being the velocity
across it. The particular case F = cx represents an ideal linear viscous damper and
c is the damping coefficient, expressed in Ns/m.
A generic nonlinear semi-active element has a characteristic expressed by the
functional relation F = f ( x , x ) or also F = f ( x , x , x ).
A wide range of dampers exist based on a variety of dissipating mechanisms
(deformation of viscoelastic solids, throttling of fluids, frictional sliding, yielding
of metals, and so forth). The following is a list of some common types of dampers
employed in engineering applications.
•
•
•
viscous dampers
viscoelastic dampers
friction dampers
12
Semi-active Suspension Control
•
•
•
•
•
magnetorheological fluid dampers
electrorheological fluid dampers
shape memory alloy dampers
tuned mass dampers
tuned liquid dampers
Details on the physical principles of these dampers can be found in Soong and
Costantinou (1994). From a black box input–output standpoint, dampers are
characterised by their force versus. velocity and force versus displacement
characteristics. This book will mainly deal with two types of controllable dampers:
friction dampers (FD) and magnetorheological dampers (MRD). The latter semiactive device has become increasingly popular within the automotive engineers
community and hence can be referred to as conventional, whereas the former type
of damper is indeed non-conventional in the automotive world: the concept of dry
frictional damping conflicts with several decades of automotive design culture
centred on the viscous damper.
Both are semi-active control systems since only the dissipative forces are
regulated and the command signals are synthesized by utilising response variables
such as displacements, velocities and accelerations. These smart damping systems
can produce the dissipative forces demanded by the control algorithm if they are
supplied with an appropriate electric signal (voltage or current).
In case of an FD, an actuator (electrohydraulic, electromagnetic or
piezoelectric) regulates the normal force applied to the friction plates. In the case
of an MRD, an electromagnetic circuit is used to modulate the intensity of the
magnetic field applied to the magnetorheological fluid.
By developing sufficiently accurate analytical models to portray the dynamic
behaviour of these smart damping devices, the vehicle and the controller can be
modelled as a dynamic system having as inputs the external excitation (produced
by road unevenness) and the control algorithm command signals.
1.6 Introduction to Friction Dampers
Viscous dampers are the most widely used type of dampers in automotive
engineering. However in principle it is possible to achieve a damping effect by
other means. A possible alternative is frictional damping. Friction in automotive
engineering is usually associated with braking systems or friction-based power
transmission systems (e.g., belt transmissions). However it can also be used in
vibration control.
A friction damper is a device which conceptually is composed of a plate fixed
to a moving mass and a pad pressing against it. A sketch showing the physical
principle of a friction device is depicted in Figure 1.3.
An external normal force Fn is applied to a mass by the pad and consequently,
because of the relative motion between the pad and the plate and of the presence of
friction (which can be represented by a friction coefficient μ or a more elaborated
model), a damping force Fd is produced.
Introduction
13
Frictional damping is one of the oldest techniques of achieving a damping
effect. It is worthwhile remarking that historically early cars were equipped with
friction dampers before the advent of the technology of viscous dampers (Bastow,
1993). Leaf spring suspensions too exploit the damping properties of friction
arising from the sliding motion among leaves when they are bent.
Fig. 1.3. Principle of a friction damper (copyright Elsevier (2003), reproduced from
Guglielmino E, Edge KA, Controlled friction damper for vehicle applications, Control
Engineering Practice, Vol 12, N 4, pp 431–443 and used by permission)
The pure dry (or lubricated) friction characteristic is of no practical use because of
its harshness, but if friction force is modulated by employing modern control
techniques, performance can improve tremendously. Anti-locking brakes (ABS)
are a good example of closed-loop control using friction force.
Controlling friction entails revisiting an early-day damping technology,
improving it in the light of the latest advances in mechatronics and control, and
proving through simulation and experimental studies that a controlled semi-active
friction damper has potentially superior performance to a conventional viscous
damper and could be successfully used in a variety of mobile (and also nonmobile) applications where it can potentially replace a conventional viscous
damper for the purpose of reducing vibration.
The challenge is therefore the control of the friction force so as to obtain
appropriate damping characteristics for an automotive application. If a friction
damper is properly controlled, it can emulate viscous- and spring-type
characteristics or any combination of the two, and it is possible to create any other
type of generalised damping (e.g., proportional to acceleration). It is also
interesting to remark that, if opportunely driven, this device can produce a high
damping at low velocity, unlike a passive viscous damper (where at low velocity
the damping is low) and can clamp a mass if contingencies require it.
14
Semi-active Suspension Control
This device is similar in its physical principle to a controlled brake, even if its
aim is different. In an ABS the target is to prevent slip so as to minimise the
braking distance, whereas in a controlled friction damper the interest is in
minimising displacements, velocities and/or accelerations so as to improve vehicle
ride and handling. The choice of using dry friction as a means of achieving a
damping effect is non-conventional. In most servomechanisms a considerable
amount of effort has been invested in order to identify and compensate for friction
(Åstrom, 1998), which is one of the principal causes of performance deterioration
of control systems (e.g., steady-state errors, limit cycles). Here instead frictional
force is the actual control force in the closed-loop control system.
Applications of frictional damping can be envisaged in various fields of
engineering practice, from automotive applications (controlled suspensions) to
reduction of machinery transmitted loads to supporting structures as well as in
structural engineering.
It is worth noting that non-controlled friction dampers are nowadays largely
employed in the suspensions of freight trains for cost and maintenance reasons
(Sebesan and Hanganu, 1993). In these devices the friction force produced is
proportional to the weight of the train. The cooling is provided by air convection
during the train motion. The Y25 bogie, a common freight bogie, employs noncontrolled dry friction dampers (Bosso et al., 2001). It is clear that a semi-active
controlled suspension could be obtained by retrofitting the existing friction
dampers with an appropriate drive and control. This would greatly improve rail
vehicles’ ride performance and stability (thus potentially allowing trains to travel at
faster speeds) and also reduce mechanical stress and damage to the railway tracks,
with great economical benefit both for train and infrastructure operators. In a study
by O’Neill and Wale (1994) it was pointed out how, in order to reach higher speeds
and have better ride in rail transport, semi-active systems can be a lower-cost
solution than making any modification to the tracks and other infrastructures.
Friction dampers are widely employed in non-mobile applications as well, most
importantly in civil engineering e.g., vibration mitigation in buildings and civil
structures (Nishitani et al., 1999). Friction dampers have been also proposed for
turbine blade vibration control (Sanliturk et al., 1995). Applications to vehicle
driveshaft vibration reduction have also been considered (Wang et al., 1996).
In the course of the book it will be shown that friction force can be electrohydraulically controlled. The damper is force-controlled; in an electrohydraulically
actuated scheme this translates to pressure control, and hence the flow required in
order to set up the working pressure in the hydraulic circuit is negligible, since the
actuator load (the pad) does not move. As a consequence it consumes less energy
compared to a controlled viscous damper. Chapter 5 will deal with FDs in detail.
1.7 Introduction to MR Dampers
A magnetorheological damper (MRD) is not very different from a conventional
viscous damper. The key difference is the magnetorheological (MR) oil and the
presence of a solenoid embedded inside the damper which produces a magnetic
field.
Introduction
15
Fig. 1.4. Principle of a magnetorheological damper (copyright Inderscience (2005),
reproduced from Guglielmino E, Stammers CW, Stancioiu D, Sireteanu T, Conventional
and non-conventional smart damping systems, Int J Vehicle Auton Syst, Vol. 3, N 2/3/4,
used by permission)
An MR oil is a particular type of fluid which contains micron-sized ferromagnetic
particles in suspension. A detailed analysis of MR fluids properties is given in
Agrawal et al. (2001). As a consequence of the polarising magnetic field, particles
tend to form chains, which modifies the value of the oil yield stress. In such a state
rheological properties of oil change and the fluid passes from the liquid state to the
semi-solid state. Hence by controlling the solenoid current, continuously variable
damping can be produced without employing moving parts such as valves or
variable orifices. The energy requirements are extremely low. For control, it is only
necessary to supply the solenoid with a conventional battery.
Likewise it is important to remark that MR fluid rheological properties are
virtually temperature and contamination independent. Therefore MRDs are rugged
and reliable devices, capable of providing excellent performance over a wide
variety of operating conditions.
It is clear that an appropriate control logic is crucial to take full advantage of
the potential offered by an MRD. It will be shown later that similarities exist
between FD and MRD control as their static characteristics are somehow similar.
The main domains of application are automotive and structural. In the latter,
they are employed for earthquake protection and for damping wind-induced
16
Semi-active Suspension Control
oscillations of bridges and flexible structures (Dyke et al., 1996). In the automotive
field they are employed in semi-active suspensions. They are currently present on a
number of high-segment market cars, as detailed in Section 1.2. Chapter 6 is
dedicated to MRDs.
2
Dampers and Vehicle Modelling
2.1 Introduction
The heart of a semi-active suspension is the controllable damper. Its accurate
modelling is crucial for suspension analysis and design. In many practical
applications the damper characteristic exhibits a strong non-linearity, which must
be taken into account in simulation studies in order to obtain realistic results when
investigating system performance.
A damper is identified by its force versus velocity characteristics (damping
characteristics or hydraulic characteristics), which can be expressed by the
functional relation
Fd = f (x ) ,
(2.1)
Fd being the damping force generated and x the velocity across it. Other damping
elements have a more general non-linear characteristics expressed by the functional
relation
Fd = f ( x , x ) .
(2.2)
Such a characteristic is typical of viscoelastic materials, but it could well
represent a controlled semi-active damper, whilst generalised semi-active damping
devices can be made to have a characteristic expressed by
Fd = f ( x , x , x)
(2.3)
with acceleration-dependent damping too.
A certain amount of hysteresis is always present in a damper characteristic,
depending upon its internal dissipation mechanism. As introduced in Chapter 1 a
18
Semi-active Suspension Control
damper can be viewed from an energy standpoint as a device that dissipates energy
through an internal mechanism (e.g., by throttling a viscous flow through an
orifice). In order to fully identify a damper, besides its damping characteristics it is
customary to define also its force versus displacement characteristics (damper
work characteristics), the area of which gives a measure of the energy dissipated
over a complete cycle.
This chapter presents the mathematical techniques necessary to model real
dampers with hysteresis in their characteristics, and subsequently reviews the main
car and truck ride models, developed for suspension studies. The last part of the
chapter deals with road modelling.
Figures 2.1 and 2.2 plot the characteristics of two types of dampers: an ideal
linear viscous damper and an ideal Coulomb friction damper. These are idealised
characteristics as no hysteresis is present in the force versus velocity characteristics
(real dampers always contain a certain amount of hysteresis in their force versus
velocity map).
Fig. 2.1. Linear viscous damper characteristics
Fig. 2.2. Coulomb friction damper characteristics
Dampers and Vehicle Modelling
19
2.2 Phenomenology of Hysteresis
Hysteresis occurs in a variety of physical systems; the most noteworthy examples
are ferromagnetic materials, constituting the core of motors, generators,
transformers and a wide range of other electrical devices. In automotive
applications hysteresis is present not only in damper characteristics but also, for
instance, in tyre characteristics and in all viscoelastic and viscoplastic materials in
general.
From a control systems standpoint, hysteresis must not always be regarded as a
non-linearity hampering controller performance or making its design more
difficult. Under particular circumstances hysteresis can be beneficially exploited in
control systems, namely in on–off control algorithms (Gerdes and Hedrick, 1999)
for reducing chatter which may occur when in a control loop the difference
between setpoint and feedback (i.e., the error) is close to zero. In such systems the
required amount of hysteresis is typically generated within the control software in
microprocessor-based programmable architectures or in hardware by employing
electronic devices or exploiting the inherent hysteresis present in actuator
characteristics (e.g., valves).
Virtually no material or device employed in mechanical and structural systems
is perfectly elastic, and restoring forces generated as a result of deformations are
not perfectly conservative. Likewise internal dissipation within viscous fluids in
dampers result in a hysteretic characteristic.
Fig. 2.3. Typical hysteretic loop for a linear material (copyright Publishing House of the
Romanian Academy (2002), reproduced from Giuclea M, Sireteanu T, Mita AM, Ghita G,
Genetic algorithm for parameter identification of Bouc–Wen model, Rev Roum Sci Techn
Mec Appl, Vol 51, N 2, pp 179–188, used by permission)
A wide variety of micromechanisms contribute to energy dissipation in cyclically
loaded materials and in viscous fluids. This behaviour macroscopically results in a
hysteretic loop when the material or device is subject to a sinusoidal displacement
Q'. However, it is not possible — except in very special cases — to quantitatively
predict the macroscopic hysteretic behaviour starting from physical models of the
microscopic behaviour. Therefore, from an engineering standpoint, an
experimental assessment of the hysteresis loop and the corresponding energy losses
20
Semi-active Suspension Control
is required to characterise a material. Figures 2.3, 2.4 and 2.5 depict typical
hysteretic loops for a linear material, a non-linear hardening material and a nonlinear softening material, respectively.
Fig. 2.4. Typical hysteretic loop for a non-linear hardening material (copyright Publishing
House of the Romanian Academy (2002), reproduced from Giuclea M, Sireteanu T, Mita
AM, Ghita G, Genetic algorithm for parameter identification of Bouc–Wen model, Rev
Roum Sci Techn Mec Appl, Vol 51, N 2, pp 179–188, used by permission)
Fig. 2.5. Typical hysteretic loop for a non-linear softening material (copyright Publishing
House of the Romanian Academy (2002), reproduced from Giuclea M, Sireteanu T, Mita
AM, Ghita G, Genetic algorithm for parameter identification of Bouc–Wen model, Rev
Roum Sci Techn Mec Appl, Vol 51, N 2, pp 179–188, used by permission)
The area enclosed by the loop is a measure of the dissipated energy. The area, and
hence the energy dissipated per cycle, can be calculated by the following contour
integral:
∫
E = Q' dq .
(2.4)
Depending upon the material, the hysteresis loop can be either very thin (and
generally elliptical in shape), resulting in a very small amount of energy dissipation
(Figure 2.6) or larger, hence producing a more significant energy consumption.
Dampers and Vehicle Modelling
21
Thin loops are likely to occur in elastic materials, such as steel, when they are
cyclically loaded within their elastic range. Conversely, materials loaded in their
inelastic range exhibit wider hysteresis loops, as portrayed in Figure 2.7.
Composite materials too dissipate significant amounts of energy.
Fig. 2.6. Deformation in the elastic range (copyright Publishing House of the Romanian
Academy (2002), reproduced from Giuclea M, Sireteanu T, Mita AM, Ghita G, Genetic
algorithm for parameter identification of Bouc–Wen model, Rev Roum Sci Techn Mec
Appl, Vol 51, N 2, pp 179–188, used by permission)
Fig. 2.7. Deformation in the plastic range (copyright Publishing House of the Romanian
Academy (2002), reproduced from Giuclea M, Sireteanu T, Mita AM, Ghita G, Genetic
algorithm for parameter identification of Bouc–Wen model, Rev Roum Sci Techn Mec
Appl, Vol 51, N 2, pp 179–188, used by permission)
A loss function L(E) can therefore be defined for a particular material or damping
device under a sinusoidal load. Measuring the area enclosed by the hysteretic loop,
and dividing it by the cycle period T, the average energy loss per cycle can be
obtained
T
L( E ) =
1
Q' qdt .
T 0
∫
(2.5)
22
Semi-active Suspension Control
2.3 Damper Hysteresis Modelling
The energy dissipation and the impact on control systems performance of
hysteretic materials can be significant and in order to properly assess the system
response the availability of a control-oriented easy-to-handle analytical model of
hysteresis is fundamental.
Hysteresis modelling has been a challenging problem for engineers, physicists
and mathematicians. Several models have been proposed over the years to capture
different classes of hysteretic phenomena using a variety of approaches. Some
models require an in-depth knowledge of specialist areas of mathematics to be
fully appreciated. A survey can be found in Mayergoyz (1991), in Visintin (1994)
and in Sain et al. (1997). Amongst the hysteresis models developed it is worth
citing the Chua–Stromsoe model (Chua and Bass, 1972) and the Hysteron model
proposed by Krasnoel´skii and Pokrovskii (1989). Another model, known as the
Preisach model (Brokate and Visintin, 1989) exists, constructed by superposing the
outputs of a set of hysteretic relays. Other models have been proposed, including
hysteretic biviscous models (Wereley et al., 1998) and polynomial models (Choi et
al., 2001).
A model having an appealing simplicity is the Bouc–Wen model, which has
gained large consensus within the engineering community. A wide variety of
hysteretic shapes can be represented by using this simple differential model
proposed by Bouc (1971) and generalised by Wen (1976). The model is based on a
nonlinear ordinary differential equation which contains a memory variable z,
representing (in the case of a damping system) the hysteretic restoring force, the
position of which is identified by the variable q. The Bouc–Wen equation is
defined as follows (Sain et al., 1997): given T = [t 0 ,t ] ∈ ℜ , the states q(t) , z(t):
T → ℜ , a vector-valued function f : (ℜ m ,ℜ m ,ℜ ,ℜ) → ℜ m and the input u(t):
T → ℜ , the Bouc–Wen model is defined by:
q(t ) = f (q , q , z ,u )
z = −γ q z z
n −1
n
−νq z + Aq
q(t 0 ) = q0 ,
(2.6a)
z (t 0 ) = z 0 ,
(2.6b)
where q is the imposed displacement of the device (or the material deformation).
The quantities γ, ν, A, n are loop parameters defining the shape and the amplitude
of the hysteresis loop.
Figures 2.8, 2.9 and 2.10 depicts three typical hysteretic loops plotted for
different sets of parameters and for an imposed harmonic displacement
q(t)=q0sin(2πft), where f =1 Hz is the frequency and q0 is the amplitude of the
sinusoidal input.
By consideration of the three Figures 2.8, 2.9 and 2.10 it can be seen that the
Bouc–Wen equation can describe both linear and non-linear (softening and
hardening) hysteretic behaviour.
Dampers and Vehicle Modelling
23
z
0.5
q
0.0
-1
0
1
-0.5
Fig. 2.8. Linear hysteretic behaviour, plotted for γ = 0.9, ν = 0, A = 1, n = 1 (copyright
Publishing House of the Romanian Academy (2002), reproduced from Giuclea M, Sireteanu
T, Mita AM, Ghita G, Genetic algorithm for parameter identification of Bouc–Wen model,
Rev Roum Sci Techn Mec Appl, Vol 51, N 2, pp 179–188, used by permission)
0.8
z
0.4
-1.0
-0.5
0.0
0.0
q
0.5
1.0
-0.4
-0.8
Fig. 2.9. Non-linear softening hysteretic behaviour, plotted for γ = 0.75, ν = 0.25, A = 1,
n =1 (copyright Publishing House of the Romanian Academy (2002), reproduced from
Giuclea M, Sireteanu T, Mita AM, Ghita G, Genetic algorithm for parameter identification
of Bouc–Wen model, Rev Roum Sci Techn Mec Appl, Vol 51, N 2, pp 179–188, used by
permission)
24
Semi-active Suspension Control
z
2
1
q
0
-1.0
-0.5
0.0
0.5
1.0
-1
-2
Fig. 2.10. Non-linear hardening hysteretic behaviour (plotted for γ = 0.5, ν = -1.5, A = 1,
n= 1) (copyright Publishing House of the Romanian Academy (2002), reproduced from
Giuclea M, Sireteanu T, Mita AM, Ghita G, Genetic algorithm for parameter identification
of Bouc–Wen model, Rev Roum Sci Techn Mec Appl, Vol 51, N 2, pp 179–188, used by
permission)
2.3.1 Bouc–Wen Model
This section describes how the Bouc–Wen equation coefficients γ, ν, A and n affect
the shape and the amplitude of hysteretic loops. It will be shown how these
parameters define the slope rate and the stiffness (linear, hardening or softening) of
the characteristics and hence the area (and therefore the energy dissipated over a
cycle). The following results are based on a numerical analysis as it is extremely
difficult to define analytical correlations due to the non-linearity of the Bouc–Wen
equation.
2.3.1.1 Parameter A
As it can be evinced from Figure 2.11, the parameter A defines the scale and the
amplitude of the hysteretic curve and the slope of the variation of the stiffness
characteristic. An increase in the parameter A results in a wider hysteresis loop,
and consequently in a larger energy dissipation.
Dampers and Vehicle Modelling
25
z
A=1
A=0.8
0.5
A=0.6
A=0.4
A=0.2
-1.0
-0.5
0.0
0.0
0.5
1.0
q
-0.5
Fig. 2.11. Hysteretic loop dependence on the parameter A (plotted for γ = 0.9, ν = 0.1, n = 1)
(copyright Publishing House of the Romanian Academy (2002), reproduced from Giuclea
M, Sireteanu T, Mita AM, Ghita G, Genetic algorithm for parameter identification of Bouc–
Wen model, Rev Roum Sci Techn Mec Appl, Vol 51, N 2, pp 179–188, used by permission)
2.3.1.2 Parameter γ
The dependence of the hysteretic loops on the parameter γ is portrayed in Figure
2.12 (plotted for ν = 0.1, n = 1, A = 1) and can be summarised by saying that the
area of the loop increases if γ increases from 0 to a value γ0 (between 2 and 3 in the
numerical example of Figure 2.12, but in general dependent on the other
coefficients), and for values of γ larger than γ0 the energy dissipated per cycle
slightly decreases.
z
γ=0
1.0
γ=1
0.5
γ=2
γ=4
0.0
-1.0
-0.5
0.0
0.5
1.0
q
-0.5
-1.0
Fig. 2.12. Hysteretic loop dependence on the parameter γ (plotted for ν = 0.1, n = 1, A = 1)
(copyright Publishing House of the Romanian Academy (2002), reproduced from Giuclea
M, Sireteanu T, Mita AM, Ghita G, Genetic algorithm for parameter identification of Bouc–
Wen model, Rev Roum Sci Techn Mec Appl, Vol 51, N 2, pp 179–188 used by permission)
26
Semi-active Suspension Control
2.3.1.3 Parameter ν
It can be noticed (Figure 2.13) that the parameter ν controls the shape of the
hysteretic curve: ν = 0 corresponds to a linear hysteretic behaviour, ν < 0 produces
a hardening hysteretic behaviour, and ν > 0 results in a softening hysteretic
behaviour.
z
1.5
ν= -1
ν = -0.5
1.0
ν =0
ν =0.5
ν=1
0.5
0.0
-1.0
-0.5
0.0
0.5
1.0
q
-0.5
-1.0
-1.5
Fig. 2.13. Hysteretic loop dependence on the parameter ν (plotted for γ = 0.9, n = 1, A = 1)
(copyright Publishing House of the Romanian Academy (2002), reproduced from Giuclea
M, Sireteanu T, Mita AM, Ghita G, Genetic algorithm for parameter identification of Bouc–
Wen model, Rev Roum Sci Techn Mec Appl, Vol 51, N 2, pp 179–188, used by permission)
2.3.1.4 Parameter n
z
n=4
n=3
n=2
n=1
1.0
0.5
0.0
-1.0
-0.5
0.0
0.5
1.0
q
-0.5
-1.0
Fig. 2.14. Hysteretic loop dependence on the parameter n (plotted for γ = 0.9, ν = 0.1, A = 1)
(copyright Publishing House of the Romanian Academy (2002), reproduced from Giuclea
M, Sireteanu T, Mita AM, Ghita G, Genetic algorithm for parameter identification of Bouc–
Wen model, Rev Roum Sci Techn Mec Appl, Vol 51, N 2, pp 179–188, used by permission)
Dampers and Vehicle Modelling
27
As Figure 2.14 (plotted for γ = 0.9, ν = 0.1, A = 1) shows, the variation is
significant for small values of n (between 1 and 2 in the numerical example of
Figure 2.14), while for larger values (n > 2) its effect is negligible.
2.4 Bouc–Wen Parameters Identification
The Bouc–Wen model has the ability to portray a wide range of hysteretic
behaviour and by an appropriate choice of the equation coefficients both the slope
variation of the stiffness characteristic and the energy dissipated per cycle can be
precisely established. The Bouc–Wen equation can be readily combined with plant
differential equations to yield an overall dynamic model.
The shape of the Bouc–Wen hysteretic loop depends on the four parameters γ,
ν, A and n whose physical meaning has been discussed above. However their
identification is not straightforward as the dependence between z and the set of the
four parameters is strongly nonlinear and not easy to investigate analytically;
furthermore the parameter variation ranges are different. Parameter identification
through least-square-based methods is a possible avenue, but may not be the best
choice. Black-box optimisation methods based on artificial intelligence techniques
such as genetic algorithms (GA) could also be beneficial. Such a method will be
employed in Chapter 6 in the context of MRD parameters identification. The
method is here briefly introduced. The reader interested in furthering the topic can
refer to the textbook of Goldberg (1989).
A GA is a probabilistic search technique inspired from the evolution of species.
Such an optimisation tool has an inherent parallelism and ability to avoid
stagnation in local optima. It starts with a set of potential solutions called
individuals and evolves towards better solutions with respect to an objective
function. Genetic operators are defined (namely crossover, mutation and selection)
and their application drives the solution towards the optimum. The main elements
of a standard GA are genetic representation for potential solutions, an objective
function, genetic operators, characteristic constants such as population size,
probability of applying an operator, number of children and so forth.
The Bouc–Wen coefficients search problem can be stated as finding a set of
parameters γ, ν, A and n such that the Bouc–Wen model given by Equation 2.6
determines a hysteretic curve which is a good approximation of an experimental
one, knowing the imposed displacement q(t) and a set of measured data (qi,
z*i)i=1,…, n corresponding to a complete cycle.
2.5 Vehicle Ride Models
A broad variety of vehicle mathematical models of increasing degree of
complexity has been developed over the years by automotive engineers to provide
reliable models for computer-aided automotive design and vehicle performance
assessement.
28
Semi-active Suspension Control
From a purely mathematical standpoint vehicle models can be categorised as
distributed models (i.e., governed by partial differential equations) and lumped
parameter models (i.e., governed by ordinary differential equations). The former
are mainly of interest to vehicle design rather than control algorithm design.
Distributed models (typically solved numerically with finite-element-based
methods) are widely employed in mechanical, thermal, aerodynamic analyses as
well as crashworthiness analyses. For car dynamics (ride and handling) and control
studies, lumped parameter models are usually employed. They typically aim to
model either ride or handling dynamics or both. In this book only lumped
parameter models will be considered.
A car can be thought as being composed of two main subsystems: the sprung
mass (chassis) and the unsprung masses (wheels, axles and linkages), connected
via a number of elastic and dissipative elements (suspensions, tyres etc.) and
subjected to external inputs coming from the road profile, the steering system and
other external disturbances (e.g., wind gust).
The motion of a vehicle with the nonholonomic constraint of the road has six
degrees of freedom (6DOF), classified as follows:
•
•
•
•
•
•
longitudinal translation (forward and backward motion)
lateral translation (side slip)
vertical translation (bounce or heave)
rotation around the longitudinal axis (roll)
rotation around the transverse axis (pitch)
rotation around the vertical axis (yaw)
Vehicle ride is essentially concerned with car vertical dynamics (bounce, pitch,
roll) whereas handling is concerned with lateral dynamics (side slip, yaw, roll).
Ride models are typically composed of interconnected spring–mass–damper
systems and defined by a set of ordinary differential equations.
The most trivial representation of a vehicle suspension has 1DOF. In this
simple model the chassis (body) is represented by a mass and the suspension unit
by a spring and a damper. Tyre mass and stiffness are neglected as well as any
cross-coupling dynamics.
By incorporating a wheel into the model, a more accurate representation having
2DOF (typically referred to as a quarter car model) can be developed. This model
was (and still is) very popular in the automotive engineering community, especially
before the widespread use of computer simulation, the reason being that the quarter
car model, despite its simplicity, features the main variables of interest to
suspension performance assessment: body acceleration, dynamic tyre force and
suspension working space (Sharp and Hassan, 1986). A merit of the quarter car is
that it permits to evaluate more straightforwardly the effects of modifications in
control parameters because higher-order dynamics and cross-coupling terms with
the other suspension units are not taken into account. A good suspension design
should produce improvement of both vehicle road holding and passenger comfort
(or possibly improvement of one without degradation of the other), although
inherent trade-offs are unavoidable in the design of a passive suspension system.
The quarter car is a 2DOF system having two translational degrees of freedom.
Another classical model can be obtained with only 2DOF: the half vehicle model
Dampers and Vehicle Modelling
29
having a translational degree of freedom and a rotational degree of freedom to
describe, respectively, bounce and pitch motions or, analogously, bounce and roll
motions (in the former case the model is referred to as a bicycle model). Its natural
extension is a 4DOF model, which also includes tyre masses and elasticity. This
model can be employed to study the vehicle pitch (or roll) behaviour. However the
4DOF model cannot take into account the cross-couplings between the right- and
left-hand side of the car (or front and rear in the case of roll motion). These
interactions can be taken into account only by using a 7DOF model (sometimes
referred to as a full car model), which allows to represent bounce, roll and pitch
motions.
The models mentioned above are classical ride models. Higher order ride
models can be developed including further degrees of freedom, e.g., accounting for
seat and engine mounting elasticity. Driver and passengers can be modelled as well
with springs, masses and damping elements. This is particularly important for
accurate human comfort studies. Chapter 3 will deal with this topic in detail.
Analogously to ride vehicle models, also handling models having different
degrees of complexity can be developed. The equivalent handling model of the
quarter car is a linear single track model which describes lateral and yaw dynamic
responses to handling manoeuvres (ignoring the effect of sprung and unsprung
masses).
Models including both ride and handling dynamics are necessary when there is
a need to accurately investigate the interaction between ride and handling (during a
turning manoeuvre, for instance) and to study the limit of handling characteristics
or elements such as anti-roll bars. Multibody techniques allow relatively easy
development of complicated models with many degrees of freedom. In 1991, Zeid
and Chang described a 64DOF model. Models with hundreds of degrees of
freedom have been developed by automotive engineers. Such involved models,
however, despite their sophistication, suffer from two main drawbacks: parameter
uncertainty and long simulation running time. For these reasons they are not
always the best choice for control design. Especially in the early stages of the
design, a less complicated model is preferable, reserving the use of higher-order
models for further refinements and optimisations.
Analogously to cars, several heavy and trailed vehicles (tractor/semi-trailer)
models (Gillespie,1992; Wong, 1993) have been developed to examine their ride
comfort, tractor–trailer interactions, dynamic tyre forces and road damage. A
survey can be found in Jiang et al. (2001). These vehicle models usually include
linear tyre models, linear or nonlinear suspension characteristics, tandem or single
axles. Other truck ride models include suspended tractor cab and driver seat with
linear or non-linear components.
The models described and the simulation results presented in this book are
based on the use of MATLAB® and Simulink® software. In the following sections
the classical vehicle and truck models will be briefly revised.
2.5.1 Quarter Car Model
The quarter car has for a long time been the par excellence model used in
suspension design. It is a very simple model as it can only represent the bounce
30
Semi-active Suspension Control
motion of chassis and wheel without taking into account pitch or roll vibration
modes. However it is very useful for a preliminary design: it is described by the
following system of second-order ordinary differential equations (Figure 2.15; tyre
damping is not shown in the figure):
m1 x1 = −2ξω1 ( x1 − x 2 ) − k s ( x1 − x2 ),
m2 x2 = 2ξω1 ( x1 − x 2 ) + k s ( x1 − x2 ) − k t ( x2 − z 0 ) − ct ( x 2 − z 0 ).
(2.7a)
(2.7b)
If relative displacement is defined as x = x1 − x2 , Equations 2.7a and 2.7b can
be rewritten in a more compact form:
m1 x1 = −2ξω1 x − k s x,
m2 x2 = 2ξω1 x + k s x − k t ( x2 − z 0 ) − ct ( x 2 − z 0 ).
(2.8a)
(2.8b)
Fig. 2.15. Quarter car model (copyright Inderscience (2005) reproduced with minor
modifications from Guglielmino, E, Stammers CW, Stancioiu D and Sireteanu T,
Conventional and non-conventional smart damping systems, Int J Vehicle Auton Syst, Vol.
3, N 2/3/4, pp 216–229, used by permission)
From the analysis of the quarter car model equations some fundamental properties
of the passive suspensions can be analytically evinced and it is possible to quantify
the compromise when reduction of both chassis acceleration, suspension working
space (sometimes referred as rattle space) and tyre deflection is pursued: the
Dampers and Vehicle Modelling
31
quarter car is a dynamic system composed of two interconnected subsystems and
as such is subject to constraint equations, independent of the type of
interconnections. From the analysis of the quarter car model Hedrick and Butsuen
(1988) showed that only three transfer functions can be independently defined and
that invariant points (i.e., values at specified frequencies depending only on kt , m1
and m2 but not on ks) exist at particular frequencies. In particular they showed that
the acceleration transfer function has an invariant point at the wheel-hop
frequency. Similarly, the suspension deflection transfer function has an invariant
point at the rattle space frequency. The trade-off between passenger comfort and
suspension deflection occurs because it is not possible to simultaneously keep both
transfer functions small around the wheel-hop frequency in the low-frequency
range.
2.5.2 Half Car Model
Fig. 2.16. Half car model with 4DOF
Pitch motion can be taken into account with the half vehicle model (also known as
the bicycle model). The governing equations are the following (Figure 2.16), with
α representing pitch:
⎧ mx = −k 2 ( z − bα − z 4 ) − k1 ( z + aα − z3 ) − c2 ( z − bα − z4 ) − c1 ( z + aα − z3 ),
⎪ Jα = k ( z − bα − x )b − k ( x + aα − z )a − c ( z − bα − z )b − c ( z + aα − z )a,
⎪
2
4
1
3
2
4
1
3
(2.9)
⎨
(
)
(
)
(
),
=
+
−
−
−
+
+
−
m
z
k
z
a
α
z
k
z
z
c
z
a
α
z
1
3
1
3
01
3
01
1
3
⎪
⎪⎩
m2 z4 = k 2 ( z − bα − z 4 ) − k 02 ( z 4 − z02 ) + c2 ( z − bα − z4 ),
32
Semi-active Suspension Control
where m is the sprung mass, m1 , m2 the front and rear unsprung masses, J the pitch
inertia, and a and b the distances of the front and rear of the vehicle from its centre
of gravity. Replacing α with the roll angle, pitch inertia J by roll inertia and b by
half-track length this model can be also employed to describe roll motion.
2.5.3 Full Car Model
The 7DOF vehicle ride model (Sireteanu et al., 1981) extends the half car model to
the entire vehicle: 3DOF are used for the sprung mass (bounce, roll and pitch),
while the unsprung masses have 4DOF (1DOF for each tyre), as depicted in Figure
2.17. The governing equations can be written compactly in matrix form (bold
letters denote matrices and vectors):
Mq + P T CPq + P T KPq + Fd = − P T K 0 z 0 − P T C 0 z0 ,
(2.10)
Fig. 2.17. Full car model with 7DOF (copyright ASME (2001), reproduced with minor
modifications from from Guglielmino E, Edge KA, Modelling of an electrohydraulicallyactivated friction damper in a vehicle application, Proc ASME IMECE 2001, New York,
used by permission)
with q ∈ ℜ7, z0 ∈ ℜ4, Fd ∈ ℜ7, M ∈ ℜ7x7, K ∈ ℜ8x8, C ∈ ℜ8x8, K0 ∈ ℜ8x4, C0 ∈ ℜ8x4 and
P ∈ ℜ8x7.
The vertical displacement vector z ∈ ℜ8 is defined as:
z = [ z1 , z 2 , z 3 , z 4 , z 5 , z 6 , z 7 , z8 ]T
(2.11)
Dampers and Vehicle Modelling
33
(the vertical displacements are not all independent).
Let q be the vector of generalised co-ordinates:
q = [q1 , q 2 , q3 , q 4 , q5 , q6 , q7 ]T
(2.12)
with the following choice of co-ordinates:
q1 = z ; q2 = z5 ; q3 = z6 ; q4 = z’’ ; q5 = α ; q6 = β ; q7 = β’’,
(2.13)
z and q being related by the matrix P, dependent upon the vehicle geometry:
z = Pq.
(2.14)
Consider the vertical displacement vector and the matrix P being defined as:
⎡1
⎢
⎢1
⎢1
⎢
⎢1
P = ⎢0
⎢
⎢0
⎢
⎢0
⎢
⎢0
⎣
0
0
0
0
1
0
0
0
0
0
0
1
0 − a d'
0 − a − d'
0 b
d ''
0 b − d ''
0 0
0
0 0
0
0 0 1
0
0
0 0 1
0
0
0 ⎤
⎥
0 ⎥
0 ⎥
⎥
0 ⎥
0 ⎥,
⎥
0 ⎥
E ⎥
2 ⎥⎥
E
− ⎥
2⎦
(2.15)
where a and b are the distances of the front and rear of the vehicle from its centre
of gravity, d’ and d’’ are, respectively, the front and rear half-track lengths and E
the inter-wheel distance.
The road input vector z0 is then defined as:
z 0 = [ z 01 , z 02 , z 03 , z 04 ]T .
(2.16)
Equation 2.10 can be obtained using Lagrangian formalism. The Lagrange
equations, expressed as a function of the kinetic energy are:
d ∂T ∂T
−
= gk
dt ∂q k ∂qk
k=1,…, 7
(2.17)
where T is the total kinetic energy of the system, defined by the quadratic form
34
Semi-active Suspension Control
T=
1 T
q Mq ,
2
(2.18)
M being the mass matrix:
M = diag(m, m1 , m1, m2 , Jα , Jβ , Jβ’’),
(2.19)
where m is the sprung mass, m1 , m2 the front and rear unsprung masses, Jα the
pitch inertia and Jβ and Jβ’’ the roll inertias of sprung mass and rear inter-axis bar.
The right-hand side of Equation 2.17 is defined as:
8
g k = ∑ fi
i =1
∂xi
∂q k
k=1,…, 7.
(2.20)
Defining the vector f of the forces applied to sprung and unsprung masses
f = [ f1 , f 2 , f 3 , f 4 , f 5 , f 6 , f 7 , f 8 ]T .
(2.21)
the forces applied to sprung and unsprung masses are:
f = −[ Kz + Cz + K 0 z 0 + C 0 z0 ] ,
(2.22)
where K is the stiffness matrix:
⎡ k1
⎢ 0
⎢
⎢ 0
⎢
0
K=⎢
⎢− k1
⎢
⎢ 0
⎢ 0
⎢
⎢⎣ 0
0
0
k1
0
0
0
− k1
0
0
0
k2
0
0
0
− k2
0
0
− k1
0
0
0
0
− k1
0
0
0
0
− k2
k2
0
0
0
− k2
0
0
0
k1 + k 01
0
0
0
0
0
k1 + k 01
0
0
0
k 2 + k02
0
and K0 is the unsprung mass stiffness matrix
⎤
0 ⎥⎥
0 ⎥
⎥
− k2 ⎥
0 ⎥
⎥
0 ⎥
0 ⎥
⎥
k 2 + k 02 ⎥⎦
(2.23)
0
Dampers and Vehicle Modelling
⎡ 0
⎢ 0
⎢
⎢ 0
⎢
0
K0 = ⎢
⎢− k 01
⎢
⎢ 0
⎢ 0
⎢
⎣⎢ 0
0
0
0
0
0
0
0
0
0
0
− k 01
0
0
− k 02
0
0
0 ⎤
0 ⎥⎥
0 ⎥
⎥
0 ⎥
.
0 ⎥
⎥
0 ⎥
0 ⎥
⎥
− k 02 ⎦⎥
35
(2.24)
Analogous definitions can be given for matrices C and C0 , the latter being the
tyre damping matrix, usually negligible (tyre damping is not depicted in Figure
2.17).
Combining Equations 2.14 and 2.18 yields:
xk =
7
∑p
ik
qk
k=1,…, 8,
(2.25)
i =1
T=
1 7
∑ mkk q k2 .
2 i =1
(2.26)
Developing (2.17) yields:
gk =
8
∑p
ik
qk = ( P T f )
k=1,…, 7,
(2.27)
k=1,…, 7,
(2.28)
i =1
d ∂T
= ( Mq )k
dt ∂q k
hence
Mq = P T f .
(2.29)
Taking into account Equation 2.22, the governing Equation 2.10 is obtained:
Mq + P T CPq + P T KPq + Fd = − P T K 0 z 0 − P T C 0 z0 .
(2.30)
In the model here described, front suspensions are taken to be independent and
rear suspensions dependent (connected through a rigid axle). However the model
36
Semi-active Suspension Control
can be easily modified to represent independent front suspensions or both front and
rear dependent suspensions, by appropriate choices of the matrices M, K and C.
2.5.4 Half Truck Model
The half truck model is the equivalent of the half vehicle model for an articulated
vehicle, and has seven degrees of freedom. Figure 2.18 depicts a schematic of the
truck model (Tsampardoukas et al., 2007)
Fig. 2.18. Half truck model (copyright Elsevier, reproduced from Tsampardoukas G,
Stammers CW and Guglielmino E, Hybrid balance control of a magnetorheological truck
suspension, accepted for publication in Journal of Sound and Vibration, used by permission)
It is composed of two sprung masses, namely the tractor body (or frame) and the
trailer body, together with three unsprung masses (the three wheels). The tractor
and the trailer are linked through an articulated connection known as the fifth
wheel. Two vibration modes are considered for each sprung unit (heave and pitch)
and one for the unsprung masses (heave).
The governing equations can be readily obtained by consideration of forces and
moments as follows:
Dampers and Vehicle Modelling
37
⎧ m c xc = F sf + F df + F sr + Fc rear − F s 6 − F d 6
⎪ J θ = F l + Fc
sr 2
rear l 2 − F s6 l 4 − F d 6 l 4 − F sf l 1 − F df l 1
⎪ c c
⎪ m t x t = F st + Fc trailer + F s6 + F d6
⎪ ⎨ J t θ t = F st l 6 + Fc trailer l 6 − F s6 l 5 − F d6 l 5
(2.31)
⎪ m x = F − F − F
u1 wf
tf
sf
df
⎪
⎪ m u2 x wr = F tr − F sr − F dr
⎪
⎩ m u3 x wt = F tt − F st − F dt
where
Fsf = k f ( xwf − xc + l1θc )
Fdf = c f (xwf − xc + l1θc )
Fsr = kr (xwr − xc − l2θc )
F = c ( x − x − l θ )
dr
r
wr
c
2 c
Fst = kt (xwt − xt − l6θt )
F = c (x − x − l θ )
dt
t
wt
t
6 t
Fs6 = k5 ( xc + l2θc − xt + l5θ t )
F = c ( x + l θ − x + l θ )
d6
5
c
2 c
t
5 t
Ftf = ktf ( zf − xwf )
Ftr = ktr ( zr − xwr )
Ftt = ktt ( zt − xwt )
rear_ relative_ velocity= ( xc + l2θc − xwr )
trailer_ relative_ velocity= ( xt + l6θt − xwt )
zf = front_ wheel_ input
zr = rear_ wheel_ input
zt = trailer_ wheel_ input
(2.32)
38
Semi-active Suspension Control
The equations can be implemented in Simulink®. Figure 2.19 shows a
schematic of the Simulink® model. The meaning of the symbols in Equations 2.31
and 2.32 are listed in Table 2.1.
Table 2.1. Truck model symbols notation (copyright Elsevier, reproduced from
Tsampardoukas G, Stammers CW and Guglielmino E, Hybrid balance control of a
magnetorheological truck suspension, accepted for publication in Journal of Sound and
Vibration, used by permission)
Fsf
Front tractor suspension spring force
Fdf
Front tractor suspension damping force
Fsr
Rear tractor suspension spring force
Fdr
Rear tractor suspension damping force
Fst
Trailer suspension spring force
Fdt
Trailer suspension damping force
Fs6
Fifth wheel spring force
Fd6
Fifth wheel damping force
xt
Trailer heave
xc
Tractor heave
ϑc
Tractor pitch
ϑt
Trailer pitch
xwf
Front tractor wheel heave
xwr
Rear tractor wheel heave
xwt
Trailer tractor wheel heave
zf
Front tractor road input
zr
Rear tractor road input
zt
Trailer road input
Ftf
Excitation force (front tractor)
Ftr
Excitation force (rear tractor)
Ftt
Excitation force (trailer)
Dampers and Vehicle Modelling
39
Fig. 2.19. Simulink® half truck model
2.6 Tyre Modelling
Tyres are made of rubber, i.e., a viscoelastic material and in ride studies the
vertical tyre stiffness can be approximated by a spring and some damping (either
pure viscous or hysteretic damping, but often negligible). In the vehicle models
described so far, the tyre has been represented as a spring (and a viscous damper
term). This model although quite crude is acceptable for ride analysis. In handling,
braking or traction studies more sophisticated models are required which account
for road–tyre adhesion (both longitudinal and lateral) as well as rolling friction. A
classical model is the so-called Pacejka magic formula (Pacejka and Bakker,
1991). At higher frequencies and for short road obstacles even more sophisticated
models are required.
The majority of workers utilise a point contact model since it is easy to use.
Such a model has at least two defects. Firstly, the point follows the slightest
vertical excursion of the road and hence generates high-frequency inputs which in
practice would not occur as the tyre contact patch bridges or envelops such points.
40
Semi-active Suspension Control
Secondly, such a model cannot generate longitudinal forces although it is
evident that these exist to a greater or lesser degree. Indeed such a model cannot
predict the deceleration of a vehicle on a steady incline.
A more useful model is a radial spring tyre model (Smith, 1977; Bernard et al.,
1981; Crolla et al., 1984). The number of springs needed is not obvious, and
depends on the type of road surface; the authors (unpublished work) have found
360 to be adequate. The stiffness of the springs is adjusted to produce a prescribed
static deflection.
For on-road applications, it is valid to regard the road profile as rigid compared
to the flexible tyre. The chief difficulty is locating the point of contact for each
spring when a pseudo-random road profile is assumed. Such a model generates
longitudinal forces but does not admit enveloping, although Davis (1974) has
extended the model to allow for this. Torsional tyre distortion is not modelled.
2.7 Road Modelling
The representation of the road profile is vital for vehicle dynamic simulations
because it is the main source of excitation. An accurate road model is as important
as a good vehicle model. Sources of vehicle vibration include forces induced by
road surface irregularities as well as aerodynamic forces and vibration that arise
from the rotating mechanical parts of the vehicles (tyres, engine and transmission).
However, the road surface elevation profile (identified by the co-ordinates z 0i , in
the previous models) plays the major role. Road roughness includes any type of
surface irregularities from bump and potholes to small deviations.
The reduction of forces transmitted to the road by moving vehicles (particularly
for heavy vehicles) is also an important issue responsible for road damage. Heavyvehicle suspensions should be designed accounting also for this constraint. The
issue will be dealt in detail in a case study in Chapter 7.
Road inputs can be classified into three types: deterministic road (periodic and
almost-periodic) inputs, random-type inputs and discrete events such as bumps and
potholes.
As far as deterministic inputs are concerned, a variety of periodical waveforms
can be used, such as sine waves, square waves or triangular waves. To a first
approximation the road profile can be assumed to be sinusoidal. Although not
realistic, it is useful in a preliminary study because it readily permits a comparison
of the performance of different suspension designs both in the time and in the
frequency domain (through transmissibility charts, plotted at different frequencies).
A multi-harmonic input which is closer to an actual road profile can be
generated. A possible choice which approximates fairly well a real road profile is a
so-called pseudo-random input (Sayers and Karamihas, 1998; Dukkipati, 2000)
which results from summing several non-commensurately related sine waves (i.e.,
the ratio of all possible pairs of frequencies is not a rational number) of decreasing
amplitude, so as to provide a discrete approximation of a continuous spectrum of a
random input. The trend can be proved to be non-periodic (sometimes referred to
as almost periodic) in spite of being a sum of periodic waveforms (Bendat and
Dampers and Vehicle Modelling
41
Piersol, 1971). To achieve a pseudo-random profile effect it is advisable to select
spatial frequencies of the form
jΔΩ + trascendental term
j=1,…, m.
(2.33)
where j is an integer, ΔΩ is the separation between spatial frequencies and the
added term could be e/1000 or π/1000 for example. The spatial frequency range is
(m-1) ΔΩ. The RMS amplitude at each centre frequency is obtained from the
power spectral law and multiplication by ΔΩ.
Another possible way to generate a realistic multi-harmonic input consists in
making the ratio between frequencies constant and decreasing with amplitude, but
using randomly generated phase angles (between 0 and 360 degrees) for each
component. In this case the resulting waveform is periodical. Simulation results
presented subsequently are based on the latter approach.
Figure 2.20 shows an example of road profile: 20 sine waves with random
phases have been added together in order to create a pseudo-random profile. The
amplitude of the profile is calculated to approximate a smooth highway by using
the spatial frequency data suggested by the Society of Automotive Engineers
(SAE).
Fig. 2.20. Pseudo-random road profile
However, these road profiles are deterministic functions and as a consequence
could not represent a real random pavement.
A stochastic model gives the more realistic representation of a road profile. The
power spectral density (PSD) is the most common way to characterise the road
42
Semi-active Suspension Control
roughness (British standard BS 7853, 1996). A road could well be approximated
(Wong, 1993) by an ergodic process with spectral density expressed by:
S ( Ω ) = CΩ − n ,
(2.34)
where Ω is the spatial frequency, having units of cycles/m (i.e., Ω is the inverse of
wavelength), C is a coefficient dependent on the road roughness, f the frequency in
Hz and n is a rational exponent. It follows that Ω=f/V, where V is the forward
speed of the car, so that:
S( f ) = (
C
) f −n .
V −n
(2.35)
This approach involves an analysis in terms of power spectral densities. For a
linear system the input and output spectral densities Yin(f) and Yout(f) are related
through the transfer function of the system G(f), from the equation:
2
Yout ( f ) = G ( f ) Yin ( f ).
(2.36)
This property in the frequency domain only applies to linear systems and
allows the output spectral density to be readily calculated if the vehicle transfer
function and the input spectral density are known. For a nonlinear system this
property does not hold.
Several techniques (Cebon, 1996) exist to characterise the road roughness by
using spectral densities. For example, the inverse discrete Fourier transform (DFT)
can be used to generate a single road profile. Alternatively a two-dimensional
inverse DFT can be employed to create a pair of correlated road profiles for full
vehicle simulation. Tyre envelopment can be added to the models.
Discrete events such as bumps or potholes cannot be modelled with the
approaches described above and are instead usually modelled using half-sine
waves or also using the Heavyside function.
3
Human Body Analysis
3.1 Introduction
The response of vehicle occupants to road inputs is an important factor in the
design of any vehicle. A suspension ought to provide the greatest possible comfort.
On the other hand, on poor roads and in off-road operation, protection of the
occupants from actual damage is most important. Tractor drivers in particular are
at risk of lower-back injuries and possibly damage to viscera (liver, heart and
brain).
In order to assess the merits of a suspension it is accordingly necessary to
model the seat characteristics, particular those of the cushion. A model for the
occupant(s) is also highly desirable, as will be described below.
The efficiency of isolation of 67 conventional seats and 33 suspension seats has
been reported (Paddan and Griffin, 2002). The measure used was seat effective
amplitude transmissibility (SEAT), which is the frequency-weighted root mean
square acceleration experienced with the seat compared to that experienced with a
rigid seat.
For 25 car seats the SEAT value varied between 57% and 122% with a median
value of 78%. For the 16 trucks tested the SEAT range varied between 44% and
115%, with a mean of 87%. For seven tractors the SEAT value varied between
57% and 118%.
These results indicate the importance of good seat design. A key parameter is
the characteristics of the foam material used for the seat.
Yu and Khameneh (1999) measured the transmissibility of three different types
of foam formulations: two rather similar toluene diisocyanate (TDI) formulations,
and one methylene diphenyl diisocyanate (MDI).
Using a shaped 50 kg load, displacement transmissibility for each foam was
recorded over the range 2.5–6 Hz for inputs of 5 mm and 20 mm. At lowamplitude inputs (5 mm) the natural frequency was around 4 Hz. For all three
foams, the natural frequency for 20 mm inputs was found to be about 0.5 Hz lower
than that for 5 mm. The peak transmissibility was also reduced.
44
Semi-active Suspension Control
It is clear that the foams have a softening spring characteristic with damping
ratio increasing with amplitude.
3.2 Human Body Response
Tests of 12 seated humans to vertical random acceleration in the range 0.2–20 Hz
(Mansfield and Griffin, 2000) indicated that the human body also demonstrates a
softening spring characteristic. The apparent mass resonance frequency fell from
5.4 Hz to 4.2 Hz as the magnitude of vibration increased from 0.25 m/s to 2.5 m/s.
From these tests an equivalent mechanical system was developed (Wei and Griffin,
1998a and 1998b). While Wei and Griffin were careful to point out that no specific
identification with parts of the body can be made, it appears plausible that this
response is that of internal organs (viscera) within the skeleton. Excessive
stretching of the intestinal attachment tissue (the mesentery) could lead to rupture
and internal bleeding. Similarly the liver could be damaged, which implies serious
implications for that vital organ. The visceral mass includes the brain, which
sloshes within the skull. The issue here is that of pressure waves. Rebound of the
brain causes reduced pressure which is thought to be the cause of concussion.
The resonant frequencies reported by Mansfield and Griffin for the human body
are little greater than those found by Yu and Khameneh for mass foam cushions.
The possibility of resonant interaction between cushion and occupant is quite
possible. This may explain why some seats are significantly less comfortable than
others.
Torsional chirp (swept sine) excitation of the wrist of subjects with forearm
supported (Lakie et al., 1984) indicated that the wrist resonant frequency fell
markedly as the magnitude of the oscillatory input was increased. Here is evidence
of the stiffness of tendons decreasing drastically with increasing force and hence
displacement.
In view of the reported experimental work, it is plausible to adopt a spring force
of the form K ( x − ε x 3 ) , where K is the linear stiffness and ε > 0 for both foam
and human body in the low-frequency range.
Hysteric damping is assumed and can be modelled via a complex stiffness
K (1 + iβ ) , where β is the loss factor. This formulation guarantees non-linear
damping which increases with amplitude, as will be shown below. The same form
of damping in the human viscera is assumed as for foam.
3.3 Hysteretic Damping
For sinusoidal response (and the hysteretic model is really only valid for that
condition) adopting the complex response:
x = A eiω t ,
(3.1)
Human Body Analysis
45
x = i ω x ,
(3.2)
x 3 = −i ω 3 x 3 ,
(3.3)
The damping term is hence obtained as:
3
⎛ x
⎛ x ⎞ ⎞
K β ⎜ + ε⎜ ⎟ ⎟ ,
⎜ω
⎝ ω ⎠ ⎟⎠
⎝
(3.4)
The non-linear term produces increased damping when ε > 0.
An effective damping ratio can be obtained as:
ζ eff
2
⎡
⎛ x ⎞ ⎤
ωn β ⎢1 + ε ⎜ ⎟ ⎥
⎝ ω ⎠ ⎦⎥
⎣⎢
=
,
2ω
(3.5)
ωn being the natural frequency of the linear system.
Assuming now that x = C sin(ω t + φ ) , a mean damping of the form
ζ eff =
ωn β (1+ 0.5 ε C 2 )
2ω
(3.6)
can be obtained.
The frequency response of the system can be obtained and is useful in
indicating the general behaviour of the system.
3.3.1 The Duffing Equation
A system governed by an equation of Duffing type (Hagerdon, 1998)
(
)
M x + B x + K x − ε x 3 = F cos(ω t + φ )
(3.7)
can exhibit jumps in amplitude if the damping is sufficiently low or the forcing F
sufficiently large. Such jumps are undesirable, particularly so in the case of the
visceral organs, where injury could result.
The type of behaviour is indicated in Figure 3.2 where the amplitude X of the
response:
x = X cos(ω t ) + higher - order terms
(3.8)
46
Semi-active Suspension Control
is plotted as a function of p =
ω
⎛K⎞
, where ωn = ⎜ ⎟
ωn
⎝M ⎠
0.5
is the natural frequency of
the linear system.
For small F the response is virtually that of the linear system, but for larger
values of F the response curves exhibit a buckled shape. For the top curve in
Figure 3.1, as p is decreased slowly from a large value, when the point A is
reached the response jumps to B. The critical condition is the vertical tangent. The
solution at A is no longer real, while the formerly unreal solution at B becomes
real.
As p is increased slowly again, when the point C is reached the response jumps
to D (for a system in which the spring hardens with amplitude, the curves lean to
the right).
Fig. 3.1. Frequency response, softening spring; p = ω/ωn
3.3.2 Suppression of Jumps
The loss factor β to prevent jumps can be obtained from the condition that no
vertical tangent exists in the frequency response plot.
The equation of motion for a single-degree-of-freedom system has the form:
3
⎛ x
⎛ x ⎞ ⎞
M x + Kβ ⎜ + ε ⎜ ⎟ ⎟ + K ( x − ε x 3 ) = F1sin (ω t ) + F2 cos(ω t ) .
⎜ω
⎝ ω ⎠ ⎟⎠
⎝
(3.9)
It is convenient to non-dimensionalise the equation of motion by setting ωnt =τ.
The equation governing motion becomes
Human Body Analysis
3
⎛ x
⎛ x ⎞ ⎞
x + β ⎜ + ε ⎜⎜ ⎟⎟ ⎟ + x − ε x 3 = ( f1 sin ( p τ ) + f 2 cos( p τ )) ,
⎜p
⎝ p ⎠ ⎟⎠
⎝
(
)
47
(3.10)
where fj=Fj /M, p=ω/ωn and differentiation is now with respect to τ.
Assuming a solution x = C sin ( pτ ) :
(1 − p )C − ε C
2
3
= f1 ,
(3.11)
β C + β eC = f2 ,
(3.12)
e = 0.75ε .
(3.13)
3
where
Then
[(1 − p ) C − e C ] + (β C + β e C )
3 2
2
3 2
= f12 + f 22 = f 2 ,
(3.14)
where f is a chosen input.
∂C
∂p
is infinite, or more usefully, when
Jumps occur when
=0.
∂C
∂p
∂p
=0
Differentiating (3.14) with respect to C, with the condition
∂C
[(
](
)
) (
)(
)
2 1− p2 C − e C 3 1− p2 − 3e C 2 + 2 β C + β e C 3 β + 3 β e C 2 = 0 .
(3.15)
Dividing by 2C and arranging as an equation in p
(
) (
)(
)
(
)(
)
p 4 − 2 p 2 1 − 2 e C 2 + 1 − 3 e C 2 1 − e C 2 + β 2 1 + e C 2 1 + 3e C 2 = 0 .
(3.16)
Jumps are impossible if this equation has no real roots for p2 , i.e., if
(1 − 2 e C ) < (1 − 3 e C ) (1 − e C )+ β (1 + e C )(1 + 3 e C )
2 2
2
2
2
2
2
(3.17)
or
β
2
(1− 2eC ) − (1− 3 e C )(1− e C ) ,
>
(1+ e C )(1+ 3 e C )
2 2
2
2
2
2
(3.18)
48
Semi-active Suspension Control
i.e.,
β2 >
e2 C 4
.
1+ e C2 1+ 3 e C2
(
)(
)
(3.19)
The value of β to suppress jumps is a function of e C 2 . This is a measure of the
magnitude of the nonlinear component of the spring force; see Equation 3.9. When
e C 2 << 1 , to prevent jumps it is necessary that
β > e C2 ,
(3.20)
where e = 0.75 ε .
For e C 2 >> 1 , the required value of β to prevent all jumps is 0.58, greater than
one would expect for human tissue. However, this is an extreme case.
Studies of the human visceral model with β around 0.3 (the level indicated by
experimental work on cushions) suggest that jumps would occur for sinusoidal
oscillations of the viscera in excess of 3 mm in magnitude. On the other hand, the
value of β to prevent jumps of a foam cushion is less than 0.1.
3.4 Low-frequency Seated Human Model
Various detailed models of the human body exist. However, cushion and visceral
natural frequencies are below 6 Hz. Moreover because the amplitude of road
profile fluctuations falls with decreasing wavelength, road inputs experienced by
vehicle occupants are predominantly at low frequency. For realistic vehicle speeds
higher frequency inputs are not important until wheel-hop is experienced at around
12 Hz for cars and nearer 10 Hz for freight vehicles. When traversing rough
ground, drivers instinctively slow, reducing the frequency of input. Hence a simple
human model suitable for low-frequency inputs is adopted here.
The model adopted for the human body is one of those developed by Wei and
Griffin (Wei and Griffin, 1998a). This is shown in the upper part of Figure 3.3. The
non-linear spring Kv models stiffness effects with hysteretic effects providing
damping.
The motion of the visceral mass Mv is denoted by x, and that of the remainder
Mc of the body by y. This second mass could be that of the skeleton. The authors
also produced a two-degree-of-freedom model in order to model response at
frequencies greater than about 8 Hz. Vehicle simulations using this model
indicated no significantly different response for realistic road inputs and vehicle
speeds. Hence the single-degree-of-freedom model is adopted here.
The seat is modelled as a mass Ms and a (non-linear) spring Ks. As indicated
above the damping terms are deduced from a hysteric model.
A seat control force Fc is considered. This would be provided by an actuator
fixed to the vehicle floor beneath the seat.
Human Body Analysis
49
The sprung mass Ms on a spring and damper suspension is modelled since it
acts as a low-pass filter of road inputs. In the frequency range of interest, the
unsprung mass is neglected, as is common with truck models.
The experimental work using vertical inputs indicates a natural frequency of the
human body in the range 4.2–5.4 Hz, depending on the amplitude of input
(Mansfield and Griffin, 2000). This is not far from the resonant frequencies
reported for a loaded foam cushion (Yu and Khameneh, 1999).
3.4.1 Multi-frequency Input
Vehicle-occupant model for low-frequency vibration is depicted in Figure 3.2.
Fig. 3.2. Vehicle and occupant model (copyright Elsevier (1998), reproduced with minor
modifications and with the addition of the bottom part of the figure from Wei and Griffin,
The prediction of seat transmissibility from measures of seat impedance, J Sound Vib,Vol.
214, N 1, used by permission)
Ground input is indicated by zg. The unsprung mass is not modelled as at the low
frequencies considered (below 6 Hz) it follows the road.
Ms represents the sprung mass of the vehicle, restrained by a linear spring and
viscous damper, as is commonly modelled.
The cushion is modelled as a complex spring Kc with loss factor βc; Mc and Mv
represent the two masses of the Wei and Griffin 1DOF model (Wei and Griffin,
1998b), with Kv a non-linear spring with loss factor βv.
50
Semi-active Suspension Control
The equations of motion for an input zg of frequency ω are:
⎛
u 2
⎜⎜1 + ε v 2
ω
x = ωv2 u 1 − ε v u 2 + ωv2 β v u ⎝
ω
(
)
⎞
⎟⎟
⎠,
(3.21)
⎛
w 2
⎜⎜1 + ε c 2
ω
y = − R x + (R + 1) ωc2 w 1 − ε c w 2 + (R + 1) ωc2 β c w ⎝
ω
(
(
)
)
z = ωs2 z g − z + 2 ζ ωs (g − z ) −
⎞
⎟⎟
⎠ + Fc ,
Mc
(3.22)
Fc
Ms
(3.23)
Mv
; ωv is the natural frequency of the linear
Mc
visceral system, ωc is the natural frequency of the total mass Mv + Mc on the linear
cushion, ωs is the natural frequency of the sprung mass on its suspension, zg is the
road surface displacement as experienced by the vehicle.
In the case of vehicles, the actual ground input is nearly always a multifrequency input and the response similarly. The hysteretic damper analysis of
Section 3.3 cannot now be employed.
The damping can be calculated by summing the contributions obtained from
individual frequency inputs (Wettergren, 1997). However, this procedure requires
the amplitude of vibration at each frequency to be known, calling for a continuous
FFT of the response, which would not appear practical and would certainly add
expense to the system. The somewhat less elegant solution of selecting a typical ω
is the strategy adopted here.
The power spectral density for the road is defined over a range [Ω1, Ω2] of the
spatial frequency Ω (cycles/m); the frequency (Hz) experienced by a vehicle
moving at speed V is V Ω.
a j sin 2 π Ω j V t + φ j , where Ωj is a spatial
The excitation has the form
where u = y − x , w = z − y and R =
(
∑
)
frequency (cycles/m), V the vehicle speed and φj a random phase angle; g is the
time
derivative
∑a
j
(
2 π Ω j V cos 2 π Ω j V t + φ j
)
of
the
road
surface
displacement.
This model can be used to obtain the frequency response of the system, be it
passive or controlled.
Human Body Analysis
51
3.5 Semi-active Control
A semi-active device can only dissipate energy. Hence the damper can be on only
when:
Fc, des w < 0
(3.24)
otherwise Fc, des should be set to zero. This can be achieved in a dry friction control
system by separating the plates. For a viscous damper or for a magnetorheological
damper zero force is not possible and the best that can be achieved is to demand
the minimal setting.
The response Fc of the actuator is assumed to be governed by a first-order
system of the form:
Tconst Fc + Fc = Fc,des ,
(3.25)
where Tconst is the time for the error in the response to a step demand to fall to
36%.
The error caused by the non-zero time constant of the actuator can be reduced
by a process of gain compensation. The demanded force is multiplied by a factor
γ > 1, which is found to be effective when the integration time step is less than 20%
of the actuator time constant. For zero error after one time constant,
(
)
−1
γ = 1 − e −1 = 1.58 .
As long as switching decisions are made several times a time constant, there
should be no overshoot; the gain compensation procedure is found to reduce
visceral accelerations by around 10% at vehicle speeds up to 20 m/s.
3.6 State Observer
To achieve comfort, it is necessary to reduce the acceleration x of the viscera.
This could be achieved if the relative internal displacement u and relative velocity
u could be controlled. However, it is not possible in practice to measure these
quantities.
It is therefore necessary to construct a state observer. The observer concept was
first proposed by Luenberger (1964).
3.6.1 Luenberger State Observer
The most general case is the one in which no state variable are measured. This is
termed the full-order observer.
Consider a system with state variables x =[x1 , x2 ,…, xn] which generates an
output y = C x which can be measured.
If the system dynamics are given by
52
Semi-active Suspension Control
x = Ax + Bu ,
(3.26)
where u is a vector of control inputs, an estimate z of x is given by the equation
z = Az + Bu + K ( y − Cz )
(3.27)
and K is the observer gain matrix, the error e = x – z is found to satisfy
e = ( A − KC )e .
(3.28)
The estimate z of x is governed by:
z = ( A − KC ) z + Bu + Ky .
(3.29)
If the eigenvalues of the matrix A – KC are chosen suitably (by an appropriate
choice of K) the error e = x – z should decay rapidly.
The calculation of K involves, among other steps, the formation of a
controllability matrix and an observability matrix, and is not simple. The reader is
referred to Crossley and Porter (1979) or Burns (2001) for details.
In many cases, as in the application considered here, some of the variables
(such as seat relative displacement and relative velocity) can be measured.
However, what cannot be measured are the displacement and velocity of the
viscera relative to the skeleton and the skeleton relative to the seat. In the case
where only some of the state variables need to be estimated, the observer is known
as a reduced-order state observer.
A similar mathematical path is required for the generation of K as in the fullorder observer.
3.6.2 Simple State Observer
The method outlined here (Stammers and Sireteanu, 2004) requires no preprocessing, and is simple enough to be implemented in real time.
The relative acceleration u can be expressed from (3.21) and (3.22) as
⎛ 1
1 ⎞
⎟ u 1 − ev u 2 −
+
u = − K v ⎜⎜
⎟
M
M
c ⎠
⎝ v
(
)
⎛ 1
1 ⎞
⎟⎟ u
+
M
M
c ⎠
⎝ v
ω
β v K v ⎜⎜
⎛
w 2
⎜
+
1
ε
c
⎜
ω2
K w 1 − ε c w2
+ c
+ K c β c w ⎝
Mc
ωMc
(
If Fc is chosen so that
)
⎞
⎟⎟
⎠ + Fc .
Mc
⎛
u 2 ⎞
⎜⎜1 + ε v 2 ⎟⎟
ω ⎠
⎝
(3.30)
Human Body Analysis
⎛
w 2 ⎞
K c β c w ⎜⎜1 + ε c 2 ⎟⎟
ω ⎠ Fc
1− εc w
⎝
+
=0
+
Kc w
Mc
ω Mc
Mc
(
2
)
53
(3.31)
whenever control is possible, u is the response of a damped oscillator and will
decay toward zero. The relative displacement w and relative velocity w of the seat
with respect to the sprung mass can be obtained by the use of an LVDT or a pullwire transducer.
Alternatively the accelerations of the seat and the vehicle floor could be
recorded and the difference integrated and low-pass filtered to obtain the relative
velocity and displacement, although this system might be more expensive than that
needed for the displacement method.
If the system were active and the time constant of the device low enough, the
relative displacement and velocity of the viscera could be driven to zero and no
discomfort would be experienced.
For the semi-active device to be on it is necessary that power be dissipated,
namely that
Fc w < 0 .
(3.32)
Experience with a semi-active control for a random input shows that the
damper can only be on about half of the time.
3.6.3 Ideal Control
The performance of the proposed observer can be assessed by a comparison with
the ideal situation in which the relative displacement u and its derivative are
known.
In this case reference to Equation (3.30) shows that Fc should be chosen to
satisfy
⎛ 1
1 ⎞
⎟ u 1− ε v u2 −
− K v ⎜⎜
+
⎟
M
M
c ⎠
⎝ v
(
)
⎛ 1
1 ⎞ ⎛
u 2 ⎞
⎟⎟ u ⎜⎜1 + ε v 2 ⎟⎟
+
ω ⎠
⎝ Mv Mc ⎠ ⎝
+
ω
β v K v ⎜⎜
⎛
w 2 ⎞
K c β c w ⎜⎜1 + ε c 2 ⎟⎟
2
ω ⎠ Fc
K w 1− ε c w
⎝
+
= 0.
+ c
+
Mc
ωMc
Mc
(
)
(3.33)
54
Semi-active Suspension Control
3.7 Results
The value of εc was estimated from the transmission ratios found experimentally by
Yu and Khameneh (1999). Their work indicated that εc is approximately 400 m-2.
The results of Yu and Khameneh were also used to obtain values of fc and βc.
Values of 4.35 Hz and 0.375 Hz, respectively, were deduced. The work of
Mansfield and Griffin (2000) suggests εv is approximately 5000 m-2.
Figure 3.3 shows the loss factor β required to prevent jumps in the cushion and
viscera as a function of amplitude of response C.
The required value of β for the cushion is sufficiently low that even with
oscillations of 20 mm amplitude, jumps will not in practice occur.
0.7
loss factor to prevent jump
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10
12
14
16
18
20
amplitude (mm)
Fig. 3.3. Loss factor β to prevent jumps as a function of amplitude: viscera (solid line),
cushion (dashed line)
The corresponding results for viscera indicate that, if visceral input oscillations are
kept below 2 mm, a value of βv of only 0.1 or greater is required. If on the other
hand oscillations of up to 4 mm in amplitude occured, βv would need to be greater
than 0.3 (of similar magnitude to that estimated for cushion material) to prevent
jumps. Jumps in the viscera would be physically damaging quite apart from the
effect of the oscillation itself. The value of βv likely to exist is not known but such
predictions appear quite credible.
Human Body Analysis
55
3
transmission ratio
2.5
2
1.5
1
0.5
2.5
3
3.5
4
4.5
5
5.5
input frequency (Hz)
Fig. 3.4. Comparison of predicted (solid curves) and measured cushion transmissibility,
* 5 mm input, + 20 mm input (measured data reproduced from Yu and Khameneh,
Automotive seating foam: subjective dynamic comfort study. Reprinted with permission
from SAE Paper # 1999-01-0588 © 1999 SAE International)
In Figure 3.4 the quoted transmissibility for cushion C (acc. out/acc. in) is
compared with the predicted value for sinusoidal inputs of amplitude (a) 5 mm and
(b) 20 mm; βc = 0.3, εc =100 N/m3 and fn = 4.25 Hz.
Agreement is quite good for the 5 mm inputs, and fair for the 20 mm inputs.
Higher-order stiffness effects could be introduced if required.
1.6
1.4
visceral acc (m/s/s)
1.2
1
0.8
0.6
0.4
0.2
0
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
freq (Hz)
Fig. 3.5. εv = 5000 m-2, εc = 100 m-2, β = 0.3 for both viscera and cushion; viscera fv = 5 Hz.
Solid lines: fc = 4.75 Hz, dashed lines: fc = 4 Hz. Lower curves: 1 mm input, upper curves: 2
mm input
56
Semi-active Suspension Control
With 2 mm seat input (Figure 3.5), a cushion with fc = 4.75 Hz has a peak visceral
acceleration input (upper curves) which is 11% greater than that for fc = 4 Hz.
However, for 1 mm input (lower curves) the increase in peak acceleration is 20%.
Due to nonlinear damping, doubling the input increases the peak response by
only 67%. The shift of peak response with increased magnitude of input is modest
but detectable.
The effect of a simple observer is shown in Figures 3.6 as a frequency response
plot and in Figure 3.7 for a random road input. The observer removes the
resonance at around 3.5 Hz.
1.4
visceral acc (m/s/s)
1.2
1
0.8
0.6
0.4
0.2
0
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
freq (Hz)
Fig. 3.6. εv = 5000 m-2, εc = 100 m-2, βc = βv= 0.4, fc = 4 Hz, fv = 5 Hz. Observer-based seat
control (dotted) versus passive case (solid) for inputs of 1 mm (lower curves) and 2 mm
(upper curves)
0.9
0.8
rms visc acc (m/s/s)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
10
15
20
25
vehicle speed (m/s)
Fig. 3.7. Visceral response passive (dashed), simple observer (solid line); data as above
Human Body Analysis
57
3.8 Seated Human with Head-and-Neck Complex
The seated human model has four subsystems: seat, cushion, driver body and the
head-and-neck complex (HNC). The latter is represented as an inverted double
pendulum.
The entire seated human model, together with the seat mass and under seat
semi-active damper, is presented in Figure 3.8.
Fig. 3.8. Schematic diagram of the seated human model with HNC
The equations of motion for the body and seat are:
(
)
m3 x3 = C3 x1 − x3 + K 3 (x1 − x3 ) = f 3 (t ) ,
(3.34)
m2 x2 = C2 (x1 − x 2 ) + K 2 (x1 − x2 ) = f 2 (t ) ,
(3.35)
58
Semi-active Suspension Control
m1 x1 = Cc (xs − x1 ) + K c (xs − x1 ) − f 2 (t ) − f 3 (t ) ,
(3.36)
mmxs = F (t ) + K (z0 − xs ) − C (xs − x1 ) − (xs − x1 ) + Fbw ⎛⎜⎝t ⎞⎟⎠ + Fbuffer⎛⎜⎝t ⎞⎟⎠ .
(3.37)
Fbw = (k − K s )z − γ z Fbw − β z Fbw ,
(3.38)
C
S
C
⎧⎪k1t (zo − xs − d ) + C1 z
Fbuffer = ⎨
b
1
⎪⎩k
(zo − xs + d ) + k (zo − xs + d )
b
3
3
if (zo − xs ) > d
if (zo − xss ) < −d
(3.39)
The non-linearity of the system is modelled by Equations 3.38 and 3.39 using
the Bouc–Wen method to describe the linkage suspension friction and the end-stop
buffers to protect the system from high-amplitude vibration, respectively.
3.8.1 Driver Seat (Including Cushions)
A typical truck seat is made of a frame that usually contains some sort of
suspension, and foam pads covered with fabric or leather. Cushions are commonly
used in the car industry to protect the human spine and body from vibration due to
road irregularities. The material and the design of foam pads may differ, hence one
cushion may protect the occupant, while another amplifies the input vibration.
Seat cushions generally have nonlinear characteristics (Yu and Khameneh,
1999). Nonlinear hysteretic damping analysis is used here to model such effects.
The model used is shown in Figure 3.9. End-stop buffers are used in order to
protect the system from severe vibration with high amplitude. This system is
modelled as nonlinear stiffness elements in terms of fifth-order polynomial
functions. The coefficients of these polynomials were determined by applying a
least-square curve fit to the measured buffer force–deflection characteristic
(Gunston et al., 2004).
Xm
mm
Xb
Fig. 3.9. Non-linear driver seat model
Human Body Analysis
59
The input to the driver seat is the motion of the vehicle chassis which is combined
heave and pitch. Only the vertical component is considered in the body response
analysis.
3.8.2 Driver Body
The driver model (Figure 3.10) is based on experimental work of Wei and Griffin
and consists of a light frame (m1) and two suspended masses (m2 and m3) each with
a linear spring and damper. The three masses do not represent actual human
organs but are chosen so that the model reproduces the force response of vibrated
subjects.
X3
X2
X1
m3
m2
m1
Fig. 3.10. Driver body model [copyright Elsevier (1998), reproduced with modifications
from Wei and Griffin, The prediction of seat transmissibility from measures of seat
impedance, J Sound Vib,Vol 214, N 1, pp 121–137, used by permission]
3.8.3 Head-and-Neck Complex (HNC)
A two-degree-of-freedom model (Figure 3.8) is used to describe the head-and-neck
system (Fard et al., 2003); a linearised model of the double inverted pendulum is
used to emulate the motion of the head-and-neck complex. The first centre of
rotation is assumed to be very close to the centre of the neck O2, while the second
centre of rotation is situated at O1.
The determination of the viscoelastic parameters is a very difficult task that
requires much experimental work and data with human volunteers. Based on
published work of Fard et al. (2003) these parameters are summarised in Table 3.1.
The head-and-neck complex is attached to the driver body in order to represent the
human seated model including the driver body and the head-and-neck motion. For
vehicle applications the driver body is assumed to vibrate in the vertical direction
only, while the HNC is able to rotate in three dimensions in response to driver
body vertical motion and the vertical, pitch and lateral motion of the vehicle
chassis. The HNC system for 3D analysis is described using the Gibbs–Appel
method.
60
Semi-active Suspension Control
Table 3.1. Head-and-neck complex parameters (copyright Elsevier, reproduced from
Tsampardoukas G, Stammers CW and Guglielmino E, Hybrid balance control of a
magnetorheological truck suspension, accepted for publication in J Sound Vib, used by
permission)
Parameter
Value
L1p
0.042 m
m1p
1.07 kg
J1
0.0012 kgm2
L2p
0.071 m
m2p
4.31 kg
J2
0.0216 kgm2
K1p
15.57 Nm/rad
C1p
0.358 Nms/rad
K2p
10.45 Nm/rad
C2p
0.266 Nms/rad
3.8.4 Analysis of the Head-and-Neck System
The motion of the head-and-neck complex (HNC) due to vertical, pitch and roll
motions of the vehicle chassis is presented using the Gibbs–Appel method
(Blundell and Harty, 2004), an alternative method to Lagrange. With the Gibbs–
Appel method the kinetic energy of the Lagrange method is replaced with the
“energy” of acceleration. The potential energy of the system (Equation 3.46) is
used just as with the Lagrange method. In complicated systems, Gibbs–Appel can
be a simpler tool than Lagrange for the derivation of the equations of motion.
The Appel function A of the system (the acceleration “energy”) in three
dimensions is presented in Equation 3.47. Equations of motion are obtained via
derivation of the total acceleration of each direction as presented by Equations
3.48–3.51.
The external accelerations (Figures 3.11 and 3.12) applied both to the head and
neck due to the motion of the vehicle chassis are given by Equations 3.40–3.42,
taking into account that the pitch and roll chassis accelerations can be analysed into
two components, one vertical and one horizontal. The component in lateral
direction is not illustrated in Figures 3.11 and 3.12.
Human Body Analysis
61
Fig. 3.11. Head accelerations [copyright IMechE (2008), reproduced from Tsampardoukas
G, Stammers CW and Guglielmino E, Semi-active control of a passenger vehicle for
improved ride and handling, accepted for publication in Proceedings of the Institution of
Mechanical Engineers, Part D: Journal of Automobile Engineering, Publisher: Professional
Engineering Publishing, ISSN 0954/4070, Vol 222, D3/2008, pp 325–352, used by
permission]
Fig. 3.12. Neck accelerations [copyright IMechE (2008), reproduced from Tsampardoukas
G, Stammers CW and Guglielmino E, Semi-active control of a passenger vehicle for
improved ride and handling, accepted for publication in Proceedings of the Institution of
Mechanical Engineers, Part D: Journal of Automobile Engineering, Publisher: Professional
Engineering Publishing, ISSN 0954/4070, Vol 222, D3/2008, pp 325–352, used by
permission]
62
Semi-active Suspension Control
The accelerations av, ah and az in the vertical, longitudinal and lateral directions,
respectively are used to form Ax, Ay and Az , the Appel function in terms of axes
fixed in the head.
Ax =
L a v = X c + L 3 ΘC +
ΦC ,
4
(3.40)
ah = hp ΘC ,
(3.41)
a z = hp ΦC ,
(3.42)
(
)
1
cos(Φ) − L Θ
− 2 L Θ 2 sin(Φ) 2
m2 av sin(Θ2 ) − ah cos(Θ2 ) − 2 L1 Θ
1
2
1
1
2
2
1
2 ,
+ m1 av sin(Θ1 ) − ah cos(Θ2 ) − L1 Θ
1
2
(
)
sin(Φ) − L Θ 2 − 2 L Θ 2 cos(Φ)⎞ 2
⎛ av cos(Θ2 ) + ah sin(Θ2 ) + 2 L1 Θ
1
2
2
1
1
⎟
⎜
1
sin(Φ ) − L Φ 2
⎟
Ay = m2 ⎜ + av cos(Φ2roll ) + az sin(Φ2roll ) + 2 L1 Φ
1roll
roll
2
2roll
⎟
⎜
2
2
⎟
⎜ − 2 L1 Φ1roll cos(Φroll )
⎠
⎝
(3.43)
(3.44)
2
⎛ av cos(Θ1 ) + ah sin(Θ1 ) − L1 Θ12 + av cos(Φ1roll ) + az sin(Φ1roll )⎞
1
⎜
⎟
+ m1
,
⎜ − L Φ 2
⎟
2
⎝ 1 1roll
⎠
⎞ 2
⎛ a v sin (Φ2roll ) − a z cos(Φ2roll ) − 2 L1 Φ
1
1roll cos (Φ roll ) − L2 Φ 2 roll ⎟
⎜
Az = m2
⎜ − 2 L Φ 2 sin (Φ )
⎟
2
1
1roll
roll
⎝
⎠
2
1
− a cos(Φ ) ,
+ m1 a v sin (Φ1roll ) − L1 Φ
1roll
1roll
z
2
(
)
(3.45)
Human Body Analysis
PE = m1 L1 g cos(Θ1 ) + m2 g (2 L1 cos(Θ1 ) + L2 cos(Θ2 )) +
(
)
1
2
K 2 (Θ2 − Θ1 )
2
2
1
1
1
C 2 Θ 2 − Θ1 + K1 Θ12 + C1 Θ12 + m1 L1 g cos(Φ1roll )
2
2
2
1
2
+ m2 g 2 L1 cos(Φ1roll ) + L2 cos(Φ2roll ) + K 2 (Φ2roll − Φ1roll )
2
2
1
1
1
+ C 2 Φ 2roll − Φ 1roll + K1 Φ12 + C1Φ 12 ,
2
2
2
+
(
)
(
63
(3.46)
)
A = Ax + Ay + Az ,
(3.47)
The equations of motion are:
∂Ay
∂A
∂ PE
∂ PE
∂Ax
∂ Az
∂ PE
∂ PE
+
+
= 0 ⇔
+
+
+
+
= 0 , (3.48)
∂ Θ1
∂ Θ 1
∂Θ 1
∂ Θ1 ∂ Θ1 ∂ Θ1
∂ Θ 1
∂Θ 1
∂Ay
∂A
∂ PE
∂ PE
∂Ax
∂Az
∂ PE
∂ PE
+
+
= 0 ⇔
+
+
+
+
= 0 , (3.49)
∂Θ 2
∂Θ 2
∂Θ 2
∂Θ 2
∂Θ 2
∂Θ 2
∂Θ 2
∂Θ 2
∂Ay
∂Ax
∂Az
∂A
∂PE
∂PE
∂PE
∂PE
+
+
=0⇔
+
+
+
+
= 0, (3.50)
∂Φ1roll ∂Φ1roll ∂Φ1roll
∂Φ1roll ∂Φ1roll ∂Φ1roll ∂Φ1roll ∂Φ1roll
∂Ay
∂Ax
∂Az
∂A
∂PE
∂PE
∂PE
∂PE
+
+
=0⇔
+
+
+
+
= 0, (3.51)
∂Φ2roll ∂Φ 2roll ∂Φ2roll
∂Φ2roll ∂Φ2roll ∂Φ2roll ∂Φ 2roll ∂Φ2roll
The equations of motion for the head-and-neck complex in three dimensions
are obtained. In matrix form:
M (q ) q + B (q ) q 2 + C q + D q + E (q ) sin (q ) = P (q ) U ,
(3.52)
where
⎡Θ 1 ⎤
⎢
Θ 2 ⎥⎥
q=⎢
,
⎢φ1roll ⎥
⎢
⎥
⎢⎣φ 2 roll ⎥⎦
⎡U v ⎤
U = ⎢⎢U H ⎥⎥
⎢⎣U Z ⎥⎦
These equations can be then incorporated with those for the seat and body.
64
Semi-active Suspension Control
3.8.5 Head Accelerations During Avoidance Manoeuvre
The acceleration of the driver’s head during an avoidance manoeuvre (modelled as
a rapid lane change) are shown in Figure 3.13 as a function of vehicle speed. The
performance of three different algorithms for the semi-active control (discussed
subsequently in Chapters 4 and 7) of the suspension, namely skyhook, balance
control cancelling (BCC) and balance control additive (BCA).
Skyhook was designed to achieve improved ride, and the benefit of skyhook
control compared with the passive case is evident in all three directions —
longitudinal, vertical and lateral. The BCC algorithm was designed to limit
dynamic tyre loads and hence reduce road damage in the case of heavy vehicles. It
is of no help to the driver in terms of comfort, but is very helpful in improving
handling and thus achieving the intended rapid lane change. The BCA algorithm is
employed to reduce vehicle roll. This is important for a laden freight vehicle which
will have a higher centre of gravity, but is of secondary importance for a passenger
vehicle. The switching between different algorithms depending on the driving
conditions could be implemented automatically on the basis of some feedback
indicators. This could be made in a variety of ways. A possible approach could be
using a rapid steering input at high speeds to select the BCC algorithm (to achieve
improved safety). Conversely at lower speeds, skyhook could be selected for driver
comfort.
0.15
0.1
0.05
0
2
(a)
R M S a c c e ln (m / s 2 )
R M S a c c e ln (m /s 2 )
0.2
0
10
20
30
Vehicle velocity (m/s)
R M S a c c e ln (m / s 2 )
1.5
40
(b)
1.5
1
0.5
0
0
Passive
BCC
Skyhook
BCA
1
10
20
30
Vehicle velocity (m/s)
40
(c)
0.5
0
0
10
20
30
Vehicle velocity (m/s)
40
Fig. 3.13. RMS acceleration of the driver head-and-neck complex: (a) longitudinal direction,
(b) vertical direction, (c) lateral direction [copyright IMechE (2008), reproduced from
Tsampardoukas G, Stammers CW and Guglielmino E, Semi-active control of a passenger
vehicle for improved ride and handling, accepted for publication in Proceedings of the
Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering, Publisher:
Professional Engineering Publishing, ISSN 0954/4070, Vol 222, D3/2008, used by
permission]
4
Semi-active Control Algorithms
4.1 Introduction
Feedback control radically alters the dynamics of a system: it affects its natural
frequencies, its transient response as well as its stability. These aspects must be
carefully studied whenever a closed-loop control system is designed. This is the
aim of this chapter which focusses on the “brain” of the semi-active suspension,
the control algorithm.
A brief and succinct overview of control fundamentals (PID, adaptive and
robust control) is firstly provided, then robust control algorithms will be analysed,
with a particular focus on variable structure control. Subsequently the classical
semi-active damper control algorithms will be reviewed with emphasis on balance
logic (which is a spring force cancellation strategy).
A sound mathematical analysis of the balance logic algorithm will be
presented. The theory is discussed based on a friction damper (FD) but the results
can be applied straightforwardly to a magnetorheological damper (MRD) as there
are deep analogies between the FD and MRD mathematical models.
In Chapter 2 it was stated that the simplest suspension model has one degree of
freedom. Such a system can be regarded as a non-linear oscillator constituted by a
spring–mass–damper system whose governing equation is second order
m1 x1 + c( x1 − y ) + k ( x1 − y ) + Fd ( x1 − y , x1 − y , x1 ) = 0 .
(4.1)
Figure 4.1 depicts the system described by Equation 4.1: k and c are,
respectively, the elastic and viscous coefficients, m1 is the sprung mass, y and y
represent the external inputs, x1 the mass displacement, x1 its velocity and x1 its
acceleration. The function Fd(·) is the controlled damping force generated by the
variable damper in the figure. The coefficient c can be regarded as a residual
passive viscous damping).
66
Semi-active Suspension Control
Fig. 4.1. 1DOF spring–mass–damper system model
From a sheer mathematical viewpoint the suspension control problem can be
regarded as finding a piecewise continuous function Fd ( x1 − y , x1 − y , x1 ) that
forces the solutions of the equation (chassis position) and its derivatives (i.e.,
velocity and acceleration) to behave in a predefined manner, according to a set of
design specifications. Ideally these quantities should all be minimised.
The function Fd(·) is physically a controlled damping force that adds up to the
spring, inertial and residual passive viscous damping forces of the system. The
specification of this function (in terms of ride and road holding) is the target of the
control. Equation 4.1 is in general nonlinear and its coefficients may not be known
precisely (parameter uncertainties), hence the system parameters cannot be
identified precisely and may deviate significantly from their actual values, thus
hampering the controller performance.
Control theory offers a number of tools which can be employed to cope with
this class of systems. They can be grouped into three families:
•
•
•
PID controllers
adaptive controllers
robust controllers
The following sections will provide some qualitative insights into each.
Semi-active Control Algorithms
67
4.2 PID Controllers
Proportional–integral–derivative (PID) controllers are used in a wide range of
applications. These controllers are frequently designed in the frequency domain,
and on the hypothesis of linearity. Therefore they work well if this hypothesis is
close to the actual behaviour of the system. If this is not the case, performance is
likely to be poor. A linear system is defined in the frequency domain by its transfer
function G(iω), which is often the result of a linearisation of a mathematical model.
The actual response of a PID-controlled system depends on how close the linear
model represents the actual behaviour of the real system. The major sources of
uncertainty are due to parameter changes, unmodelled dynamics, time delays,
changes in the operating point, sensor noise and unpredicted disturbance inputs.
Although a PID controller is not supposed to perform well in conditions different
from those for which it has been designed, its performance can be optimised using
a robust design approach or by including an additional adaptive loop (see Section
4.3). In this way, it can be made to exhibit the desired performance over a larger
range.
A controller is said to be robust (Dorf and Bishop, 1995) if it has low
sensitivities, is stable and continues to meet its nominal specification over a typical
range of parameter variations. A robust control design is one that satisfactorily
meets its control specification, even in the presence of parameter uncertainties and
other modelling errors. A measure of the robustness to small parameter variations
is the sensitivity, defined as:
Sα =
∂G / G
∂α / α
(4.2)
with α being a parameter of the system. The design of a PID controller entails the
choice of the proportional, integral and derivative gains. More generally the design
of the controller transfer function entails defining its static gain, and its poles and
zeroes. Classical linear control system theory offers a number of tools for
controller design and stability analysis of the closed-loop response, such as the root
locus, Bode and Nyquist diagrams etc., as well as several design criteria.
The desirable features of a robust design in the frequency domain are the
largest possible bandwidth and the largest possible loop gain, attained primarily in
the controller and in the forward path transfer functions (in this way the
disturbance rejection is increased). However these specifications are in conflict and
trade-offs must be pursued in order to guarantee both a swift response and system
stability. Several design and tuning methods have been developed over the years
for PID controllers. The most classical tuning method involves the so-called
Ziegler and Nichols rules (Ziegler and Nichols, 1942); they permit the tuning of the
PID parameters, based on the partial knowledge of the transfer function, which can
be obtained by simple tests on the system; another classical method is based on the
Cohen and Coon rules (Cohen and Coon, 1953); in this method the PID parameters
are tuned by fixing a damping ratio of 0.25 in response to a disturbance input. In
the subsequent years several methods have been developed, such as the Smith
68
Semi-active Suspension Control
predictor (Smith, 1957), the Haalman method (Haalman, 1965) and the Dahlin
regulator (Dahlin, 1968). Other methods exist that minimise a performance index
of the error (ISE, IAE etc.). A review of all these and other methods can be found
in Åström et al. (1993). Research is still active on PID controller tuning and new
methods are being proposed due to the widespread industrial use of these
controllers worldwide.
As opposed to linear frequency-domain-based methods, typically employed in
PID controller design, in the past 40 years non-linear control methods, namely
adaptive and robust algorithms (Slotine and Li, 1992), have been developed as a
consequence of the derivation (circa 1960–1980) of various mathematical methods
(state space and phase plane methods, Lyapunov stability theory etc.). Other
control methods have been developed in those years including optimal control
(Berkovitz, 1974; Hocking, 1991; Kirk, 2004) and neuro-fuzzy control (Passino
and Yurkovich, 1998; Abdi et al., 1999).
4.3 Adaptive Control
A possible approach to deal with nonlinear systems is adaptive control. This
control philosophy is opposite to robust control. Although this work is focussed on
the use of the latter type of control, it is worthwhile describing the basic principles
of adaptive control in the context of this chapter. Adaptive controllers are used if
the parameters of the process are time varying and a fixed-parameter controller
would not yield acceptable results.
Adaptive control can be thought as control with on line estimation of uncertain
parameters (Slotine, 1984). The parameters, estimated on the basis of the measured
plant signals, are then used in the control action. Systems with constant or slowly
varying uncertain parameters are the most likely to have an improvement in their
responses with this type of control. In such systems the on line estimation requires
a small amount of computational power and therefore the controller learns more
quickly (the more the adaptation goes on, the more an adaptive controller improves
its performance). The target of the adaptive control is to try to keep consistent
performance in systems with unknown variations in its parameters. A controller
with fixed parameters (such as a PID) can become inaccurate or even unstable for
large parameter variations; however an adaptive loop can be added to a PID
controller, enhancing its performance.
Typical applications where adaptive algorithms could be successfully employed
can be found in robotics (where the payload can vary), electrical power systems
(where the power demand varies slowly over the day) and process control (e.g.,
chemical or metallurgical processes). Two main approaches exist to design
adaptive controllers. The first one is referred to as model-reference adaptive
control; a reference model is used to set the ideal response of the system, with the
error between the ideal and the actual response being the input to the adaptation
law. The second approach is self-tuning control where the estimation of the
parameters is obtained by measuring the inputs and the outputs to the plant. These
are then used for the on line tuning of the controller, which is continually updated
as parameters change with operating conditions. There is an enormous amount of
Semi-active Control Algorithms
69
literature on the topic. For a comprehensive overview on adaptive and other nonlinear control methods the reader can refer to the work by Slotine and Li (1992).
4.4 Robust Control
A robust controller is meant to provide a reasonable level of performance in
systems with uncertain parameters, no matter how fast they vary, but it usually
requires the knowledge (or an estimate) of the bounds of the uncertainty. The
typical structure of a robust control law is composed of a nominal part (such as a
state feedback or an inverse model control law), plus a term which deals with
model uncertainty.
A small degree of robustness can be achieved with PID controllers, as
discussed before. However robust control is often associated with H∞ control and
variable structure control (VSC). In this book only the latter control strategy is
considered. Details on H∞ control can be found in Chen (2000).
Variable structure controllers are a very large class of robust controllers (Gao
and Hung, 1993; Hung et al., 1993; De Carlo et al., 1998). The distinctive feature
of VSC is that the structure of the system is intentionally changed according to an
assigned law. This can be obtained by switching on or cutting off feedback loops,
scheduling gains and so forth (Itkis, 1976). By using VSC, it is possible to take the
best out of several different systems (more precisely structures), by switching from
one to the other. The control law defines various regions in the phase space and the
controller switches between a structure and another at the boundary between two
different regions according to the control law. Therefore the designer is no longer
forced to trade off between static and dynamic requirements, as is the case in linear
control systems design. It is possible to synthesize a wide range of trajectories by
switching between two or more systems according to a predefined law, improving
therefore the transient and steady-state responses of the new system with respect to
the responses of the original systems.
In principle the laws governing the change in the structure can be very
different; however the law that is given greatest attention is that producing the socalled sliding regime. The reason why it has been so widely studied lies in the fact
that, when the sliding motion occurs, the system becomes insensitive (ideally
invariant) to parametric and external disturbances of the plant. This particular type
of VSC is known as sliding mode control (Utkin, 1992). It must be stressed that a
VSC system can be devised without a sliding mode, but in this general case the
robustness properties are not guaranteed.
The principles of adaptive and robust control have been now outlined. There is
no general rule to make a decision on which control is more suitable to a particular
type of system to be controlled; however some rules of thumb can be given. In very
broad terms adaptive control performs better with constant or slowly varying
parameters. An adaptive controller improves its performance as adaptation
proceeds, whereas a robust controller only attempts to keep good performance.
Robust control however can cope better with quickly varying parameters and
unmodelled dynamics. Some advanced controllers exist which combine adaptive
and robust features; they are known as robust adaptive controllers. Furthermore,
70
Semi-active Suspension Control
the implementation of an adaptive controller requires more computational effort to
perform the on line identification and this require more computational power,
whilst robust control algorithms are usually simpler from a computational
viewpoint.
4.5 Balance, Skyhook and Groundhook
After having overviewed control fundamentals, attention is now centred on semiactive suspension control algorithms. In particular the focus is on balance,
skyhook, groundhook logic and their numerous variants. They can all be
categorised as VSC algorithms.
4.5.1 Balance Logic
This logic first was introduced by Rakheja and Sankar (1985) and developed by the
authors (Stammers and Sireteanu, 1997). With reference to Figure 1.3 (the case of
a friction damper) balance logic aims at reducing chassis acceleration. If the
relative displacement x and relative velocity x across the suspension are
measured then, according to the balance logic, the damping force Fd (x , x ) must be
modulated sequentially such that:
⎧ k x sgn x
Fd = Fbalance = ⎨ s
⎩0
xx ≤ 0
xx > 0
(4.3)
4.5.2 Skyhook Logic
This logic, originally devised by Karnopp et al. (1974) is called skyhook, as the
damper is regarded as being hooked to a fixed point in the sky. As this is obviously
not possible, the approximation to an ideal skyhook is given by the following
control logic:
⎧csky x1
Fd = Fskyhook = ⎨
⎩0
xx1 > 0
xx1 ≤ 0
(4.4)
where x is the relative velocity and x1 the absolute chassis velocity.
4.5.3 Groundhook Logic
This logic (Novak and Valasek, 1996) aims at reducing dynamic tyre force, thus
improving handling and at the same time reducing road damage (particularly useful
in the case of heavy vehicles). Like skyhook, the groundhook damper is supposed
to be hooked to a fixed point, in this case the ground. The algorithm is:
Semi-active Control Algorithms
− xx 2 > 0
⎧cgnd x 2
Fd = Fgroundhook = ⎨
⎩0
− xx 2 ≤ 0
71
(4.5)
where x is the wheel absolute displacement and x 2 the relative displacement.
4.5.4 Displacement-based On–Off Groundhook Logic
This is an interesting algorithm based on an on–off switched damping logic: it is a
form of groundhook blended with balance. The switching condition is given by the
product of a displacement and a velocity rather than two velocities (as in the pure
groundhook). The algorithm was originally proposed by Koo et al. (2004). The
algorithm is the following:
Fd = ccontrollable x ,
(4.6)
where
⎧con
ccontrollable = ⎨
⎩coff
x2 x > 0
x1 x ≤ 0
(4.7)
4.5.5 Hybrid Skyhook–Groundhook Logic
This logic is aimed at reducing both body acceleration and dynamic tyre force and
is obtained by combining skyhook and groundhook:
[
]
Fhybrid = G μFskyhook + ( 1 − μ )Fgroundhook ,
(4.8)
with the following four-state switching condition:
⎧ Fskyhook = x1
⎪
⎪ Fskyhook = 0
⎨
⎪ Fgroundhook = x 2
⎪F
⎩ groundhook = 0
xx1 > 0
xx1 ≤ 0
− xx 2 > 0
(4.9)
− xx 2 ≤ 0
If μ is set to 1, the hybrid control policy is switched to pure skyhook. On the
other hand, if μ is set to 0, the hybrid control is switched to pure groundhook.
An interesting problem is to choose μ so as to minimise a performance index
which takes into account, with different weight terms proportional to dynamic tyre
force, suspension working space and chassis acceleration (coefficients α, β and γ in
Equation 4.10) such as
72
Semi-active Suspension Control
T
J ( μ ) = ∫ {[α ( x2 - z 0 )]2 + [ β ( x1 - x2 )]2 + (γ x1 ) 2 }dt ,
(4.10)
0
If the integral is solved numerically over an integer number of periods or a
sufficiently long time period, the functional J(μ) is a function of μ only, and hence
it can be plotted, and the minimum can be easily found. The main drawback of this
approach is that the optimal set depends heavily on the type of road (and clearly on
the type of performance index too). Hence, in order to implement it, it would be
necessary to add a further adaptive loop which chooses in real time the most
appropriate set of values (previously calculated for various road conditions and
stored in a look-up table in the microcontroller memory) by the knowledge of the
type of road, inferred via some type of observer. This is complex and of
questionable benefit.
4.6 Balance Logic Analysis
The previous section has outlined three classical damper semi-active algorithms
and two variants of these logics. The focus is now placed on the balance logic. The
subsequent theory will be developed considering a friction damper (using a
Coulomb friction model and not a Bouc–Wen model for the mathematical
developments to simplify the notation) but the results can readily be extended also
to an MRD or any damper which can be modelled via a Bouc–Wen model.
The basic idea of this semi-active control strategy is to balance the elastic force
by means of the damping force (as long as these forces act in opposite directions)
and to set the damping force to a minimum value (possibly zero) otherwise.
Therefore, the force transmitted through the isolation system is significantly
reduced or even cancelled during the motion sequences in which the damper is
active and is only slightly higher (ideally equal) to the elastic force otherwise. As
firstly introduced by Federspiel (1976), this type of damping modulation is also
known as sequential damping. Analytical models for sequential damping have been
proposed and analysed (Stammers and Sireteanu, 1997; Guglielmino, 2001),
showing better vibration isolation properties than the optimal passive damping, for
both deterministic and random excitations.
The effectiveness of using balance logic in the semi-active control of vibration
can be intuitively illustrated for the 1DOF models of a vehicle suspension and a
machine foundation shown in Figure 4.2.
Semi-active Control Algorithms
a
73
b
Fig. 4.2. a. 1DOF vehicle suspension model; b. 1DOF machine foundation model (copyright
Publishing House of the Romanian Academy (2003), reproduced from Topics in Applied
Mechanics, Vol I, Ch 12, edited by Sireteanu T and Vladareanu L, used by permission)
The motion of the sprung mass M is described by:
Mx1 + Fd (x , x ) + Fe (x ) = P(t ) ,
(4.11)
where x1 and x are the absolute and relative displacements of the sprung mass,
Fe (x ) is the passive elastic force, Fd (x , x ) is the semi-active damping force and
P(t ) is the exciting force.
For a vehicle suspension model:
x1 (t ) = x(t ) + x0 (t ) , P(t ) ≡ 0 ,
(4.12)
where x0 (t ) is the imposed displacement of the wheel centre, assumed to follow
the road surface (without losing contact with it).
In the case of a machine foundation model
x1 (t ) = x(t ) , P(t ) = me eω 2 sinω t ,
(4.13)
where the perturbation of the sprung mass (machine and foundation block) is
produced by a rotor unbalance, modelled via an eccentric mass me spinning with
angular frequency ω and placed at a distance e from the machine rotation axis.
The absolute acceleration of the vehicle body x1 and the transmitted force FT
to the foundation base are given by:
x1 = −
1
[Fd (x , x ) + Fe (x )] ,
M
(4.14)
74
Semi-active Suspension Control
FT = Fd (x , x ) + Fe (x ) .
(4.15)
In both cases the vibration isolation problem entails a reduction of the elastic
and damping forces across the suspension. In the balance semi-active control
scheme, this can be achieved by controlling the damping force so as to balance the
elastic force when these forces act in opposite direction i.e., when xx < 0 , and by
setting the damping force to a minimum value when xx ≥ 0 .
If the relative displacement and relative velocity across the vibration isolation
system are measured then, according to the control logic, the damping force
Fd (x , x ) must be modulated sequentially such that
⎧⎪ Fe (x ) sgnx
Fd (x , x ) = ⎨
⎪⎩ Fd min (x , x )
if xx < 0
if xx ≥ 0
,
(4.16)
where Fe (x ) is the elastic force and Fd min is the minimum possible setting of the
damping force. From an intuitive point of view, Equation 4.16 implies that the
damping force is zero or very small as long as the sprung mass is moving away
from its static equilibrium position. This force suddenly increases when the stroke
reverses and then gradually decreases as the system returns to its static equilibrium
position.
The semi-active closed-loop control law defined by (4.16) can be practically
implemented by using a controllable friction damper or an MR damper.
For a friction coefficient μ (assuming a Coulomb friction model), the normal
force Fn (x , x ) applied to the friction plates of the device must be controlled by an
actuator such that:
⎧⎛ 2α ⎞
⎟ Fe (x )
⎪⎜
Fn (x , x ) = ⎨⎜⎝ μ ⎟⎠
⎪F
⎩ d min
if xx < 0
,
(4.17)
if xx ≥ 0
where α is a dimensionless gain factor. Theoretically, complete (ideal) balance is
achieved for α = 0.5 and Fd min = 0 . However, in order to avoid damper lock-up
(no motion across the damper), which could occur when the relative displacement
shifts from bound to rebound stroke (or conversely), it is sufficient to ensure that
the friction force produced is always less than the spring force, i.e., α < 0.5 .
Assuming Fe (0) = 0 , the switching logic (4.17) can be rewritten in the form:
⎛ 2α ⎞
⎟⎟ Fe (x ) ,
Fn (x , x ) = sw (x , x )⎜⎜
⎝ μ ⎠
(4.18)
Semi-active Control Algorithms
75
where sw(·) is the two-valued condition function which takes on the value 1 when
the friction damper is active and 0 otherwise
sw (x , x ) =
1
[sgn (xx ) − 1]sgn (xx ) .
2
(4.19)
Equation 4.17 implies an instantaneous switch between the on and off settings
of the force. A more realistic model must take the switching time into account. The
gain factor α has to be replaced by a demanded value α dem since the force
achieved is not that demanded by the logic. The variation of the damping force is
assumed to be described by the following first-order differential equations:
Tc Fd + Fd = − αdem Fe ( x)
Tc Fd + Fd = Fdmin
if xx < 0
if xx ≥ 0
,
(4.20)
where Tc is the switching time constant.
The control law (4.17) is discontinuous and consequently the controller prone
to chatter. Chattering occurs in the neighbourhood of a switch point (e.g., when
either x or x is zero). The smaller the switching time constant Tc the more likely
chatter is to occur. On the other hand, increasing the switching time constant Tc
too much will inevitably lessen the effectiveness of the semi-active controller
because the elastic force will be only partially balanced by the damping force in the
“on” sequences. The drawbacks of the switchable semi-active system are all related
with the rate of change between the two values of the condition function.
Unfortunately, a condition function as that involved in the balance logic, i.e.,
sw (xx ) has a large rate of change between 0 and 1, mainly due to the derivative x ,
especially when the input x0 is a filtered white noise.
4.7 Chattering Reduction Strategies
The control law outlined above is of the switched type and consequently prone to
induce chatter. Furthermore fast switching produces acceleration and jerk peaks
which overall negatively affect ride quality. Chattering occurs in the
neighbourhood of a switch point (i.e., when either x or x is zero). The faster the
drive dynamics the more likely chatter is to occur. On the other hand, sluggish
drive dynamics will inevitably lessen damper performance. Therefore appropriate
anti-chatter strategies need to be devised. Amongst the practical anti-chatter
methods applied to smooth the variation of the condition function, the following
can be mentioned (Sireteanu et al., 1997; Guglielmino et al., 2005):
•
•
•
filtering of feedback signals
introduction of a dead band around the switching points
introduction of a fuzzy semi-active controller
76
Semi-active Suspension Control
The first method consists of either low-pass filtering both the feedback signals
x(t ) and x (t ) or only the relative velocity which, in most cases, has a broader
frequency spectrum than the displacement. It should be pointed out that any signal
filtering would result in a lag between the filtered input and output, and therefore
in an increase of the switching time constant Tc . This method, although the most
straightforward, penalises the performance of the dampers, as the effective working
bandwidth is reduced.
The second method allows for relative displacement and velocity dead bands
− xε ≤ x ≤ xε , − xε ≤ x ≤ xε , within which the condition function is given only the
value 0, irrespective of sgn( xx ) . Then the new condition function is:
⎧⎪0
sw (x , x , xε ) = ⎨
⎪⎩ sw (x , x )
if x ≤ xε , x ≤ xε
if x > xε , x > xε
.
(4.21)
In most cases it is sufficient to introduce only a displacement dead band since
the elastic force is not very high for small relative displacements and therefore
system performance is not significantly affected by the spring force not being
balanced. This approach is somewhat similar to the introduction of a boundary
layer in sliding mode controllers (which are themselves particular variable
structure algorithms; Utkin, 1992).
The third method is the synthesis of a hybrid fuzzy-variable structure controller
(Guglielmino et al., 2005). The variable structure algorithm can be fuzzified by
choosing as fuzzy variables the relative displacement and velocity. This makes it
possible to soften the fast switching action of the VSC “crisp” controller, without
low-pass filtering which would result in a system bandwidth reduction. An
application of this method is presented in Chapter 7.
In order to illustrate qualitatively how the first two anti-chatter policies work,
let us consider the time histories of the relative displacement and of the unfiltered
and filtered relative velocity across a vehicle suspension, depicted in Figures 4.3,
4.4 and 4.5. Since in this example the semi-active control is implemented to
mitigate body absolute acceleration within the low-frequency range, the controller
dynamic response can be assumed to be instantaneous.
The effects of the anti-chatter policies are readily observed in Figures 4.3, 4.4
and 4.5, showing the time trends of displacement and velocity (both filtered and
unfiltered) and in Figures 4.6, 4.7 and 4.8 plotting the associated condition
functions. As can be evinced from Figure 4.8 the number of on–off switches could
be reduced by a factor of four (compared with Figure 4.6) in the same time interval
if a combination of the first two anti-chattering strategies is applied.
Semi-active Control Algorithms
77
Displ. (cm)
2
0
-2
0
1
Time (s)
2
Fig. 4.3. Relative displacement versus time (copyright Publishing House of the Romanian
Academy (2003), reproduced from Topics in Applied Mechanics Vol I, Ch 12, edited by
Sireteanu T and Vladareanu L, used by permission)
Vel. (cm/s)
20
0
0
1
-20
2
Time (s)
Vel. (cm/s)
Fig. 4.4. Unfiltered relative velocity versus time (copyright Publishing House of the
Romanian Academy (2003), reproduced from Topics in Applied Mechanics Vol I, Ch 12,
edited by Sireteanu T and Vladareanu L, used by permission)
20
0
0
1
Time (s)
-20
2
sw
Fig. 4.5. Filtered relative velocity versus time (copyright Publishing House of the Romanian
Academy (2003), reproduced from Topics in Applied Mechanics Vol I, Ch 12, edited by
Sireteanu T and Vladareanu L, used by permission)
0
0.5
1
Time (s)
1.5
2
Fig. 4.6. Condition function versus time without anti-chatter logic (copyright Publishing
House of the Romanian Academy (2003), reproduced from Topics in Applied Mechanics
Vol I, Ch 12, edited by Sireteanu T and Vladareanu L, used by permission)
Semi-active Suspension Control
sw
78
0
0.5
1
Time (s)
1.5
2
sw
Fig. 4.7. Condition function versus time with dead band anti-chatter logic (copyright
Publishing House of the Romanian Academy (2003), reproduced from Topics in Applied
Mechanics Vol I, Ch 12, edited by Sireteanu T and Vladareanu L, used by permission)
0
0.5
1
Time (s)
1.5
2
Fig. 4.8. Condition function versus time with combined filtering and dead band anti-chatter
logic (copyright Publishing House of the Romanian Academy (2003), reproduced from
Topics in Applied Mechanics Vol I, Ch 12, edited by Sireteanu T and Vladareanu L, used by
permission)
Figure 4.9 shows on the same graph a linear restoring force having a spring rate
k = 120 kN/m and the force generated by the variable friction damper. The balance
logic control is applied applying a low pass filter with a cut-off frequency of 8 Hz
and a displacement dead band xe = 0.1 x max .
Force (kN)
2
0
-2
0
0.5
1
Time (s)
1.5
2
Fig. 4.9. Elastic force (thin line) and damping force (thick line) in balance logic (copyright
Publishing House of the Romanian Academy (2003), reproduced from Topics in Applied
Mechanics Vol I, Ch 12, edited by Sireteanu T and Vladareanu L, used by permission)
Semi-active Control Algorithms
79
4.8 SA Vibration Control of a 1DOF System with Sequential Dry
Friction
The sequential semi-active (SA) damping force Fd (x , x ) of the 1DOF systems
pictured in Figure 4.2 will be considered as a function of the relative
displacement x(t ) and the relative velocity x (t ) , having the general form:
Fd (x , x ) = cx +
1
[1 − sgn (xx )] f1 (x ) f 2 (x ) , c ≥ 0 ,
2
(4.22)
where f1 (x ) ≥ 0 , xf 2 (x ) ≥ 0 and f1 (x ) = f 2 (x ) = 0 if and only if x = x = 0.
Moreover, f1 (x ) and f 2 (x ) are continuous functions on ℜ (with possible
exception of x = 0 ) and monotonic on each x and x semi-axis.
The motion of the sequentially damped 1DOF oscillator having linear spring
force is described by
Mx + Fd (x , x ) + kx = Pext (t ) ,
(4.23)
where the exciting force Pext (t ) is the force P(t ) acting directly on the sprung
mass of the machine foundation model (Figure 4.2b). In the case of the vehicle
suspension model shown in Figure 4.2a the exciting force is given by
Pext (t ) = − Mx0 (t ) .
(4.24)
From Equations 4.22 and 4.23 it is obvious that perfect semi-active balance of
the spring force by the damping force is achieved if:
Fd (x , x ) = cx + f1 (x ) f 2 (x ) = −kx
for
xx < 0 .
(4.25)
The simplest physically implementable solution of the functional equation
(4.25) is obtained for:
c = 0 , f 2 (x ) = sgn x , f1 (x ) = k x
for
xx < 0 ,
(4.26)
i.e., when no viscous damping is present, the energy dissipation takes place only
when the relative displacement and velocity are of opposite sign, and this can be
achieved by a controllable friction damper. If a low level of viscous damping cmin
is added, Equation 4.22 can be written in the form:
Mx + cmin x + α dem k x sgn x + (1 − α dem )kx = Pext (t )
Mx + cmin x + kx = Pext (t )
if xx < 0
if xx ≥ 0
,
(4.27)
80
Semi-active Suspension Control
where α dem is the demanded gain factor for the normal force applied to the friction
plates. It should be chosen so as to obtain as much cancellation as possible of the
damping and elastic forces in the “on” sequences.
It should be noticed that viscous damping can be also emulated by a suitable
control of the normal force applied to the friction plates:
Fn (x ) =
cmin
μ
x .
(4.28)
Such damping should be more appropriately called pseudo-viscous damping, as
it is obtained by velocity-controlling a friction damper.
Equation 4.27 can be regarded as describing the motion of an oscillator having
in the “on” sequences (i.e., when xx < 0 ) the dissipative characteristic
cmin x + αdem k x sgn x and the elastic characteristic (1 − α dem ) kx . In the “off”
sequences (i.e., when xx ≥ 0 ) these characteristics are cmin x and kx, respectively.
The above form of the equation of motion shows a dynamic weakening of the
spring stiffness in the “on” sequences caused by partial balance. This results in a
reduction of the resonant frequency of the system, without any modification of the
sprung mass static deflection. This is a very beneficial effect if a low tuned
vibration isolation system is envisaged. Equation 4.27 shows that α dem < 1 is
required in order to have a positive elastic coefficient. In fact α dem should be
chosen to be close to 0.5 if the minimisation of sprung mass acceleration is the
main control objective.
For sake of generality the subsequent analysis will be carried out using
dimensionless equations, using the following notation:
ω = k M , ς min = cmin 2 kM , τ = ωt ,
y (τ ) =
P (τ ω )
dy
x(τ ω )
,
, y ′(τ ) =
, z (τ ) = ext
dτ
aMω
a
(4.29)
where a is the unit of length. The equation of motion can be written in the
following dimensionless form
y ′′ + δ ( y , y ′) + y = z (τ ) -
(4.30)
The dimensionless sequential damping characteristic is:
⎧⎪ 2ς min y ′ + 2α dem y sgn y ′
⎪⎩ 2ς min y ′
δ ( y , y ′) = ⎨
if yy ′ < 0
if yy ′ ≥ 0
(4.31)
For ease of notation in the following developments the dimensionless
parameters will be referred to by the corresponding physical parameters (e.g., τ
Semi-active Control Algorithms
81
time, y relative displacement or elastic force, δ ( y , y ′) damping force, y1′′ = y ′′ − z
absolute acceleration or transmitted force etc.).
4.8.1 Sequential Damping Characteristics
The behaviour of the damping force δ ( y , y ′) as the relative displacement y varies
sinusoidally is important not only from a theoretical but also from a practical point
of view since most testing machines can generate such a relative motion between
the mounting ends of the shock absorber.
For an imposed cyclic sinusoidal motion:
y (τ ) = Y0 sinντ
(4.32)
the damping force δ ( y , y ′) varies as shown in Figures 4.10a and 4.10b in terms of
the relative displacement y and the relative velocity y ′ respectively, for
ς min = 0.25 and α dem = 0.45
The energy loss per cycle is:
2π ν
ΔE =
∫ δ ( y , y′)y′dτ = 2Y (πνς
0
min
+ α dem ) .
(4.33)
0
From Equation 4.33 it can be deduced that the dissipative effect of the
additional semi-active dry friction could be significant at low frequency but
becomes less important when the frequency of the imposed motion increases.
a
b
Fig. 4.10. Sequential damping characteristics (a) damping force versus displacement; (b)
damping force versus velocity [copyright Publishing House of the Romanian Academy
(2003), reproduced from Sireteanu T, Stoia N, Damping optimization of passive and semiactive vehicle suspension by numerical simulation, Proc Romanian Academy, Series A, Vol
4, N 2, used by permission]
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Semi-active Suspension Control
The damping coefficient ς eq of the equivalent linear damper, which provides the
same energy loss per cycle as the sequential damper, is given by
ς eq = ς min +
α dem
πν
(4.34)
for a pure on–off damping characteristic ( ς min = 0) ς eq < 1 πν , since α dem < 1 .
The peak value of the sequential damping force is reached at the time interval
Δτ =
1 ⎡π
-1 α dem ⎤
)⎥
⎢ − tan (
ν ⎣2
νς min ⎦
(4.35)
from the instantaneous switch of the friction force from zero to its demanded value,
which is given by:
2
δ max = 2Y0ν ς min
+
2
α dem
ν2
(4.36)
The peak value of the equivalent linear damping force is:
⎛
δ eq max = 2Y0νς eq = 2Y0ν ⎜ ς min +
⎝
α dem ⎞
⎟.
πν ⎠
(4.37)
α dem
,
π
(4.38)
If ς min = 0 , then
δ max = 2Y0α dem , δ eq max = 2Y0
and therefore the peak value of the sequential friction force is less than 2Y0 since
α dem < 1 . It is π times larger then the peak value of the linear equivalent damping
force and independent of the excitation frequency.
4.8.2 Free Vibration: Phase Plane Trajectories
The phase plane trajectories of Equation 4.30 can be determined analytically for
the pure on–off damper. In this case, the free response of the system is described
by the Cauchy problem:
y 1′ = y 2
y 2′ = − δ ( y 1 , y 2 ) − y 1
y 1 (0 ) = y 10
y 2 (0 ) = y 20 .
(4.39)
Semi-active Control Algorithms
83
The analytical expressions of the phase trajectories are:
( ) + (y )
y12 + y 22 = y10
2
0 2
2
(1 − α dem )y12 + y22 = (1 − α dem )(y10 )2 + (y20 )2
if y10 y 20 ≥ 0 , y 20 ≠ 0
if y10 y 20 ≤ 0, y10 ≠ 0.
(4.40)
These trajectories are piecewise circular and elliptic curves as shown in Figure
4.11 for different initial conditions. The ratio between two successive maxima or
minima of the relative displacement y (τ ) is constant and given by:
ρ=
1
.
1 − α dem
(4.41)
Therefore the free motion amplitude decreases linearly. This behaviour is also
encountered in the free response of a passive damping system with dry friction.
The main difference is that the semi-active system with sequential dry friction will
always return to its equilibrium position since there is no friction force when y = 0
(unlike a pure dry friction oscillator).
Fig. 4.11. Phase plane trajectories for several different initial conditions [copyright
Publishing House of the Romanian Academy (2003), reproduced from Sireteanu T, Stoia N,
Damping optimization of passive and semi-active vehicle suspension by numerical
simulation, Proc Romanian Academy, Series A, Vol 4, N 2, used by permission]
4.8.3 Free Vibration: Shock Absorbing Properties
In this section the shock absorbing properties of the 1DOF suspension model with
sequential dry friction will be compared to those of an optimally damped linear
system. This aspect is very important as it is desirable to reduce not only the peak
84
Semi-active Suspension Control
value of a transmitted impulsive force but also the number of after-shock free
oscillations. It can be shown that the optimum value of the relative damping
coefficient ς , which minimises the peak value y ′′ of the transmitted force
y1′′(τ ) = y ′′(τ ) in the case of a passive linear vibration isolation system excited by a
Dirac impulse force δ (τ ) applied to the sprung mass, is ς 0 = 0.25 (Sireteanu and
Balas, 1991). Since the first sequence of the motion described by Equation 4.30 for
z (τ ) = δ (τ ) and initial conditions y (0) = y′(0) = 0 is always governed by a linear
′′ can be obtained by taking ς min = ς 0 in the
equation, the same optimal value y opt
sequential damping characteristic (4.31). The effect of the additional sequential dry
friction is then observed by comparing the evolution of the motion after the first
peak value of the transmitted force is reached.
a
b
Fig. 4.12. Free vibration versus time for (a) passive case (linear); (b) semi-active case (nonlinear) [copyright Publishing House of the Romanian Academy (2003), reproduced from
Sireteanu T, Stoia N, Damping optimization of passive and semi-active vehicle suspension
by numerical simulation, Proc of the Romanian Academy, Series A, Vol 4, N 2, used by
permission]
Figure 4.12 depicts the optimal passive system free vibration time histories with
ς = 0.25 and those of the semi-active system with same viscous linear damping, to
which a sequential dry friction with α dem = 0.45 is added. As shown in Figure
4.12b, the free motion of the semi-active system with sequential dry friction
Semi-active Control Algorithms
85
resembles that of an almost critically damped passive system, but the peak value of
the transmitted force in the semi-active isolation system is significantly lower. For
example, in the case of the passive system with ς = 0.7 , the peak value of the
transmitted force is yˆ ′′ = 1.4 , i.e., 1.7 times larger than the optimal value
′′ = 0.82 .
ŷopt
4.8.4 Harmonically Excited Vibration
4.8.4.1 Time Histories
The steady-state solution of Equation 4.30 with harmonic excitation
z (τ ) = Z sinντ
(4.42)
has been determined by numerical integration using Newmark’s method. Figure
4.13 depicts the time histories of the transmitted force y1′′(τ ) and of the damping
force δ ( y, y ′) in the case of the linear system with ς = 0.25, α dem = 0.25, for
Z = 0.2 and ν = 1 (i.e., when the excitation frequency is equal to the undamped
natural frequency of the system). It can be noticed that the additional sequential dry
friction leads to a reduction of the transmitted force (46% for the peak value and
54% for the RMS value) for only a 3% increase in the damping force peak value.
4.8.4.2 Amplitude–Frequency Characteristics
Since the interest lies in the reduction of the loads transmitted through the vibration
isolation system from the system base (road) to the sprung mass (in the case of a
suspension) or conversely (in the case of a rotating machinery), the force
transmissibility factor is the key indicator. For a linear system the force amplitude
ratio T (ν ) is given by:
T (ν ) = ŷ1′′ Z = 2 ~y1′′ Z ,
where ŷ1′′ and ~
y1′′ are the peak and RMS values of y1′′(τ ) , respectively.
(4.43)
86
Semi-active Suspension Control
a
b
Fig. 4.13. (a) Response to harmonic excitation of passive system; (b) response to harmonic
excitation of semi-active system (copyright Publishing House of the Romanian Academy
(2003), reproduced from Sireteanu T, Stoia N, Damping optimization of passive and semiactive vehicle suspension by numerical simulation, Proc Romanian Academy, Series A, Vol
4, N 2, used by permission)
In the case of a semi-active system similar amplitude–frequency characteristics can
be defined which, in general, depend on both input amplitude and frequency.
However since the equation of motion (4.30) is piecewise linear, these functions
are independent of the input amplitude Z , but they are not equal as in the case of a
linear system:
T̂ (ν ) = ŷ1′′ Z ,
~
T (ν ) = 2 ~y1′′ Z .
(4.44)
In Figure 4.14 the force transmissibility factors T (ν ) are plotted for ς = 0.25
~
and T (ν ) for ς min = 0.25, α dem = 0.45.
Semi-active Control Algorithms
87
Fig. 4.14. Force transmissibility factors for passive (dashed line) and semi-active (solid line)
systems (copyright copyright Publishing House of the Romanian Academy (2003),
reproduced from Topics in Applied Mechanics Vol I, Ch 12, edited by Sireteanu T and
Vladareanu L, used by permission)
Figure 4.14 indicates a noteworthy result: when the sequential dry friction damping
is added to the passive damping a resonance shifting towards the lower-frequency
range occurs. This effect is a consequence of the dynamic weakening shown by
Equation 4.30 and is a very advantageous effect for both vehicle suspensions and
machine foundations. In this manner, the system resonant frequency can be
reduced without an increase in the static deflection for a given value of the sprung
mass. Another important feature of semi-active vibration isolation system is the
remarkable reduction of the RMS force amplification factor in the resonance range
of the initial passive system. For ν > 2 and same viscous damping the
transmissibility factor of both passive and semi-active systems are virtually equal.
This aspect is important, since for low values of ς, passive suspensions are very
effective vibration isolators in the higher frequency range.
4.8.5 Random Vibration
In this section the mean square response to a stationary Gaussian white noise
excitation for the 1DOF passive and semi-active suspension models is evaluted
(Sireteanu and Stoia, 2003), using Newmark’s method and Monte Carlo
simulation.
The aim of this numerical analysis is to optimise the suspension damping with
respect to the classical ride comfort criterion (minimum RMS body acceleration). It
will be shown that the semi-active suspension with sequential dry friction can
achieve a significant comfort enhancement in comparison with the optimum
settings of linear or non-linear passive suspensions.
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Semi-active Suspension Control
The equation of motion (4.11) of a 1DOF vehicle suspension model with a
linear spring can be written as:
Mx + Fd (x , x ) + kx = − My(t ) .
(4.45)
If y (t ) is modelled as a stationary Gaussian random process with a power
spectral density approximated by the analytical expression
S y (ω ) = S 0 ω −4 , ω ≠ 0 ,
(4.46)
then the spectral density of the system excitation y(t ) can be approximated by a
stationary Gaussian white noise with auto-correlation function and spectral density
given by:
R (τ ) = 2π S 0 δ (τ ) ;
S y (ω ) = S 0 ,
(4.47)
where S 0 is a constant dependent upon the road roughness and the vehicle speed.
The damping characteristics of the passive benchmark suspensions considered
are linear and quadratic:
Fd (x ) = cx
Fd (x ) = qx x .
(4.48)
(4.49)
Defining:
f (x , x ) =
ω=
1
Fd (x , x ) , z (t ) = − y(t ),
M
k
c
q
a,
, ς=
, β=
2Mω
M
Mω 2
(4.50)
where a is the acceleration unit (1 m/s2), Equation 4.45 can be rewritten as:
x + 2ω ς x + ω 2 x = z (t )
(4.51)
for linear damping,
x + β a -1 ω 2 x x + ω 2 x = z (t )
for quadratic damping, and
(4.52)
Semi-active Control Algorithms
x + α ω 2 x sgn x + (1 − α )ω 2 x = z (t ) if
xx < 0
if
xx ≥ 0
x + x = z (t )
89
(4.53)
for sequential damping.
The damping optimisation procedure consists in determining the parameters
ς , β and α such that the RMS absolute acceleration of the sprung mass is
minimum. This quantity is given by:
[
] [
]
σ x = E ( x − z ) 2 = E ( f (x , x ) + ω 2 x) 2 ,
2
1
(4.54)
where E [ ] is the mathematical expectation operator.
In the analysis of real systems, random processes describing physical
phenomena represent families of individual realisations. Each realisation (sample
function) of the system input leads to a unique solution trajectory if the problem is
well posed. The collection of these solution trajectories is an output random
process. If the output random process is a second-order process than this is the
solution of the stochastic equation of motion in the mean square sense (Soong,
1973).
The numerical simulation methods used to evaluate the statistical properties of
the output random process usually imply a numerical integration of the
deterministic differential equation of motion for numerically simulated trajectories
of the system random excitation. The statistical properties of the solution process
are then determined by standard estimation procedures (Bendat and Piersol, 1980).
In this section, the Newmark’s method and a pseudo-random number
generation algorithm are used for the numerical integration of Equations 4.51, 4.52
and 4.53 and the simulation of the discrete-time trajectories of the white noise
excitation z(t).
In order to verify the accuracy of the mean square response evaluation, the
approximate solution is compared with the exact solution (Dinca and Teodosiu,
1973), known for the system (4.51):
2
σx =
(
)
π S0
π S0
π S 0 ω 1 + 4ς 2
2
2
.
, σ x =
, σ x =
3
2ω ς
2ς
2ω ς
1
(4.55)
The minimum value of mean square absolute acceleration is obtained for
ς = 0.5 and the corresponding mean square response is:
σ x (0.5) =
2
π S0
π S0
2
2
, σ x (0.5) =
, σ x1 (0.5) = 2πω S 0 .
ω
ω3
(4.56)
4.8.5.1 Simulation of White Noise Sample Functions
A discrete-time history (sample function) of the stationary Gaussian white noise
excitation
90
Semi-active Suspension Control
z n +1 = z (nΔt ) , n = 1, ..., N
(4.57)
can be derived approximately from a sequence of pseudo-random numbers
U n , n = 1, ... ,N , uniformly distributed on the unit interval [0,1] (Abramowitz and
Stegun, 1970). The sequence can be obtained using a linear congruential pseudorandom number generator (Monte Carlo simulation). This has the recursive form
X n +1 = aX n + b (mod c ) ,
(4.58)
where a and c are positive integers and b a non-negative integer. For an integer
initial value or seed X 0 , the algorithm (4.58) generates a sequence taking integer
values from 0 to c − 1 (the remainders when the aX n + b are divided by c ). When
the coefficients a, b and c are chosen appropriately, the numbers
Un = X n c
(4.59)
seem to be uniformly distributed on the unit interval [0,1]. Since the number
sequences are finite, the modulus c should be chosen as large as possible. To
prevent cycling with a period smaller than c , the multiplier a should be also taken
to be relatively prime to c .
According to the Box–Muller method, if U 1 and U 2 are two independent
uniformly-distributed random variables on [0,1], then N1 and N 2 defined by:
N1 = − 2lnU 1 cos 2 π U 2
N 2 = − 2lnU 1 sin 2 π U 2
(4.60)
are two independent standard Gaussian random variables. Therefore:
z 2 k −1 = 2π S 0 Δt
z2k
− 2lnU 2 k −1 cos2π U 2 k
= 2π S 0 Δt − 2lnU 2 k −1 sin 2 π U 2 k
(4.61)
are independent Gaussian random variables with:
E [z n ] = 0 , E [z m z n ] = 2π S 0 Δt δmn .
(4.62)
As can be seen from (4.58) and (4.59), the discrete random process defined by
z Δ (t ) = z n for (n − 1)Δt ≤ t ≤ nΔt
is mean square convergent to the white noise process z (t ) when Δt → 0 .
(4.63)
Semi-active Control Algorithms
91
4.8.5.2 Numerical Solution of the Equation of Motion
Newmark’s discrete-time method in five steps is now applied in order to obtain the
approximate solution of the equation
x + f ( x , x ) + kx = z (t )
(4.64)
with the initial conditions
x(0) = x0 ,
x (0) = x 0 .
(4.65)
At the initial time t0 = 0 , the initial value of the acceleration x(0) = x0 is
evaluated from:
x0 = z (t 0 ) − f ( x0 , x 0 ) − kx0 .
(4.66)
The principle of the method consists in the approximation of the discrete values
x n +1 , x n +1 , xn +1 by using the values obtained at the time step tn. The steps of the
method are:
•
Initialisation of xn+1 with an arbitrary value xn+1,i
•
Evaluation of x n+1 from
x n +1 = x n + ( xn + xn +1,i )
•
Δt
2
Approximation of xn+1 by
xn +1 = x n + x n Δt + ( xn + xn +1,i )
•
(4.67)
(Δt )2
4
With xn+1 and x n+1 from (4.67) and (4.68) the following value of
acceleration is found:
xn +1,c = z (t n +1 ) − f ( xn +1 , x n +1 ) − kxn +1
•
(4.68)
(4.69)
The values xn+1,c and xn+1,i are then compared. If the difference is not
sufficiently small, xn+1,i is replaced by xn+1,c and the algorithm is repeated
from (4.67), otherwise, a new iteration is initiated. Usually, for the
initialisation of the unknown acceleration value, the value at the previous
step is used.
92
Semi-active Suspension Control
4.8.5.3 Numerical Results
The aim of the numerical analysis is to compare the mean square response of the
suspension analytical models (4.51), (4.52) and (4.53) for optimal linear, quadratic
and sequential damping characteristics. The optimal values of the parameters ς , β
and α are determined so as to minimise the RMS absolute acceleration for
different values of the excitation intensity S 0 . In order to determine a realistic
variation range of the excitation intensity S 0 , appropriate RMS values of the
sprung mass absolute acceleration x1 must be considered. The measured values of
σ x for a medium-sized saloon car were obtained within the range 0.9–2.15 m/s2.
1
Assuming that the undamped natural frequency of the car suspension is 1 Hz, then
ω = 2π . Therefore, considering Equation 4.57, values of S 0 between 0.01 m2/s3
and 0.1 m2/s3 seem reasonable.
The optimal values of the damping coefficients ς , β and α determined by
using the numerical solutions of Equations 4.51–4.53, are almost constant for all
values of S 0 within the range ς 0 = 0.5 , β 0 = 0.9 , α 0 = 0.8.
The mean square response values for the three optimal damping characteristics
(L – linear, Q – quadratic and S – sequential) are given in Table 4.1.
Table 4.1. RMS response of passive and semi-active systems (copyright Publishing House
of the Romanian Academy (2003), reproduced from Sireteanu T, Stoia N, Damping
optimization of passive and semi-active vehicle suspension by numerical simulation, Proc
Romanian Academy, Series A, Vol 4, N 2, used by permission)
L
Q
S
S 0 [m/s2]
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
σ x [mm]
11
16
19
22
24
27
29
31
33
36
σ x [m/s]
0.07
0.11
0.12
0.14
0.16
0.17
0.19
0.20
0.21
0.23
σ x1 [m/s2]
0.61
0.89
1.06
1.24
1.38
1.52
1.66
1.77
1.87
2.05
σ x [mm]
19
24
26
28
30
31
33
35
38
39
σ x [m/s]
0.12
0.15
0.16
0.17
0.18
0.19
0.20
0.22
0.23
0.24
σ x1 [m/s2]
0.83
1.06
1.21
1.36
1.49
1.63
1.76
1.87
1.98
2.17
σ x [mm]
44
58
67
74
80
86
92
97
101
108
σ x [m/s]
0.13
0.17
0.19
0.21
0.23
0.25
0.26
0.27
0.29
0.30
σ x1 [m/s2]
0.45
0.65
0.78
0.92
1.03
1.14
1.26
1.34
1.43
1.58
Figure 4.15 shows a comparison of the minimum RMS absolute acceleration
versus excitation intensity S 0 for the three methods.
Semi-active Control Algorithms
93
Fig. 4.15. RMS body acceleration for optimum passive (L), quadratic (Q) and semi-active
(S) damping (copyright Publishing House of the Romanian Academy (2003), reproduced
from Sireteanu T, Stoia N, Damping optimization of passive and semi-active vehicle
suspension by numerical simulation, Proc Romanian Academy, Series A, Vol 4, N 2, used
by permission)
As can be observed from Figure 4.15, the semi-active suspension can provide a
reduction of approximately 25–30% of the RMS sprung mass acceleration over the
whole excitation intensity range, when compared with both linear and non-linear
passive suspensions. The improvement in terms of comfort is clear. It should be
mentioned that this improvement is obtained at the expense of a significant
increase in the suspension relative displacement. Therefore, a certain compromise
between comfort and working space requirements should be considered in the
optimisation cost function.
The results obtained by the numerical simulation presented in this section show
that the acceleration experienced by a system controlled by a semi-active control
strategy can be significantly lower than that of a passive system (either linear or
non-linear).
4.9 Stability of SA Control with Sequential Dry Friction
In this paragraph the stability properties obtained for the semi-active control
strategy with sequential dry friction is discussed. The stability is studied employing
the direct Lyapunov method and it is shown that the closed-loop system is
asymptotically stable, even for a more general class of damping forces, which
includes sequential dry friction as a particular case. Consider the system described
by
Mx + cx + kx + Fd = 0 .
(4.70)
94
Semi-active Suspension Control
where Fd is assumed to be a generic nonlinear controlled damping force of the type
Fd = f ( x) g ( x ) with f ( x) ≥ 0, xg ( x ) ≥ 0, g (0) = 0 , subject to the switching law
⎧ f ( x) g ( x )
Fd = ⎨
⎩ 0
if
if
xx < 0
xx ≤ 0 .
(4.71)
The previous relation can be written as:
1
Fd = f ( x ) g ( x ) [1 − sgn( xx )] .
2
(4.72)
The vehicle suspension with semi-active dry friction is a particular case
obtained for
f ( x) = 2αk x ,
g ( x ) = sgn x .
(4.73)
However the sgn (signum) function is a non-smooth discontinuous function and
this makes it very difficult to use classical Lyapunov theory as the equation is
piecewise linear. Hence for the purpose of the proof the sgn function is replaced by
a saturation-shaped function which is smooth and continuous over its entire
domain. Such an approximation is often used also in numerical simulation in order
to make system equations less numerically stiff.1
If the saturation function is defined as:
sat(⋅) =
2tan−1 (⋅)
(4.74)
π
the discontinuous sgn functions can be replaced by the correspondent sat functions.
Using the expression (4.72), Equation 4.70 can be rewritten as follows:
Mx + cx + kx + f ( x) g ( x )
1 − sat( xx )
=0.
2
(4.75)
1. A system of ordinary differential equations x = f ( x, t ) is said to be stiff (according to Lambert,
1973) if the eigenvalues λi of the Jacobian matrix J =
∂f
max[− Re( λi )]
satisfy Re(λi)<1 and
<< 1 . In
∂x
min[− Re( λi )]
more operative terms (Richards et al., 1990) the stiffness can be measured via the dimensionless
quantity
Integration range
.
Smallest time constant
Semi-active Control Algorithms
95
By introducing the state variables x1 = x and x 2 = x , Equation 4.75 becomes:
x1 = x2 ,
x 2 = −
(4.76)
f ( x1 ) g ( x2 )
k
c
x2 −
[1 − sat( x1 x2 )] .
x1 −
2M
M
M
Consider the Lyapunov function associated with Equation 4.76:
V ( x1 , x2 ) =
1 k 2
( x1 + x22 ) .
2 M
(4.77)
From the definition it follows directly that V is a continuous and positivedefinite function. The proof of stability is based on the following theorem
(Barbashin–Krasowsky).
If V ≤ 0 (negative semi-definite) and if the set of the points of the state space
for which V = 0 does not include any complete trajectory (except the origin), then
the system is asymptotically stable.
The first derivative of V is:
f ( x1 ) g ( x2 )
∂V
∂V
c 2
V =
x1 +
x 2 = −
x2 − x2
[1 − sat( x1 x2 )] .
∂x1
∂x2
M
2M
(4.78)
Using the conditions f ( x) ≥ 0, xg ( x ) ≥ 0 , g (0) = 0 it is obvious from (4.78)
that V ≤ 0 and the set of points for which V = 0 is A = {( x1 , x 2 ) x2 = 0}. Next,
according to the previous theorem, in order to prove the asymptotic stability, it is
sufficient to show that the trajectories of system defined by (4.76) are not
contained within the abscissa axis. This can be readily verified considering the
k
x1 ) , and
tangent vector to the trajectory for x 2 = 0 . Its components are (0 , −
M
hence it is orthogonal to the abscissa axis. Therefore the hypotheses of the above
stability theorem are fulfilled and hence the equilibrium O (0, 0) is asymptotically
stable.
4.10 Quarter Car Response with Sequential Dry Friction
The theory developed for a 1DOF system can be readily extended to the classical
quarter car model with viscous damping and sequential dry friction damping. As
shown in Figure 4.16, the equations of motion are:
Mx + Fd + k (x − x1 ) = 0 ,
M 1 x1 − Fd − k (x − x1 ) + k1 (x1 − r ) = 0 ,
(4.79)
96
Semi-active Suspension Control
where
⎧⎪cmin ( x − x1 ) + 2α dem x − x1 sgn (x − x1 ) if (x − x1 )(x − x1 ) < 0
Fd = ⎨
⎪⎩cmin ( x − x1 )
if (x − x1 )(x − x1 ) ≥ 0
(4.80)
and r (t ) is the excitation induced by the road profile for a constant vehicle speed
V.
Fig. 4.16. Quarter car model with controlled friction damper (copyright Publishing House of
the Romanian Academy (2003), reproduced from Topics in Applied Mechanics Vol I, Ch
12, edited by Sireteanu T and Vladareanu L, used by permission)
By using the notation
ω = k M , ς min = cmin 2 kM ,
τ = ωt , γ =
k
M
,χ= 1,
M1
k
x (τ ω )
x(τ ω )
, y1 (τ ) = 1
,
a
a
r (τ ω )
z (τ ) =
,
a
y (τ ) =
(4.81)
Equation 4.79 can be written in the dimensionless form:
y ′′ + δ + y − y1 = 0 ,
y1′ − γ δ − γ ( y − y1 ) + γ χ ( y1 − z ) = 0 ,
(4.82)
Semi-active Control Algorithms
97
where
⎧⎪2ς min ( y ′ − y1′ ) + 2α dem y − y1 sgn ( y ′ − y1′ ) if ( y − y1 )( y ′ − y1′ ) < 0
⎪⎩2ς min ( y ′ − y1′ )
if ( y − y1 )( y ′ − y1′ ) ≥ 0.
δ =⎨
(4.83)
Equation 4.83 assumes instantaneous switching of the friction damper force
from zero to its demanded value − 2α dem ( y − y1 ) and conversely. If switching
dynamics is modelled (via a first-order lag) the equations become:
τ cδ ′ + δ = −2α dem ( y − y1 )
τ cδ ′ + δ = 0
if
if
( y − y1 )( y − y ′) < 0
( y − y1 )( y − y ′) ≥ 0 ,
where τ c = ωTc is the dimensionless switching time constant.
(4.84a)
(4.84b)
5
Friction Dampers
5.1 Introduction
This chapter deals with the design of a controlled automotive friction damper. A
damper of this type is indeed non-conventional in the world of semi-active
suspensions (as opposed to the magnetorheological damper, the use of which is
more widespread). Controlling friction involves revisiting an early-day damping
technology and enhancing it in the light of the latest mechatronic advances.
An overview of the phenomenology and modelling of the friction force is
firstly given. Subsequently its modelling and design are illustrated. The focus is on
the methodological approach required when designing a novel damping system:
from its modelling and experimental concept proofing to building a working
prototype, including designing an appropriate electrohydraulic driving system and
eventually enhancing its design and optimising damper performance.
5.2 Friction Force Modelling
Friction arises as a consequence of the dissipative microscopic phenomena taking
place between the microscopical asperities of two nominally smooth surfaces. The
contact between two surfaces can be of two types: conformal or punctual. The
former occurs, for instance, in a slider; in this case the macroscopic contact area is
proportional to the dimensions of the object. The second type of contact is defined
as punctual, e.g., the (ideal) contact between tyre and road or between two gear
teeth, where the actual contact area is proportional to load and material strength.
The latter is usually referred to as a Hertzian contact (Hertz, 1881).
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Semi-active Suspension Control
5.2.1 Static Friction Models
A large number of friction models have been developed over the years to describe
the frictional interaction between two materials. In broad terms essentially two
main friction regimes exist: the pre-sliding regime and the sliding regime. In the
pre-sliding regime there is no true sliding; this motion arises with surface
deformation: the asperity junctions deform elastoplastically behaving as springs
and friction force appears to be a function of displacement rather than velocity. As
the displacement increases, more and more junctions will break resulting
eventually in sliding. In the sliding regime all the asperity junctions are broken and
the friction force becomes also a function of the velocity (Lampaert et al., 2004).
In this brief review the approach and categorisation proposed by ArmstrongHelouvry et al. (1994) is followed. In a static model the common approach is to
describe friction level in terms of a velocity-dependent friction coefficient. Friction
is predominantly a function of velocity (although it does depend on other factors,
such as pressure, temperature, wear, type of lubrication and so forth) because the
physical process of shear at the junction changes with velocity. This can be
mathematically expressed through an expression of the type
Fd ( x ) = − μ ( x ) Fn sgn x
(5.1)
Fd being the friction force, Fn the normal force and x the relative sliding velocity
between the two surfaces. The minus sign indicates that friction always opposes
the motion.
Dynamic effects occur at the breakaway, in response to change in velocity and
normal force.
The classical Coulomb friction model was originally envisaged by Leonardo da
Vinci (1519), revisited by Amontons in 1699 and subsequently rediscovered and
formalised by Coulomb (1785). In this model the function μ (x ) is taken to be
constant. The Coulomb model, although very approximate, is still largely used
because it is easy to handle and provides a sufficient level of approximation in
many engineering problems. In numerical simulations the discontinuity expressed
by the sgn(·) function may create problems, therefore it is sometimes replaced by
an equivalent continuous function around the origin (for instance a saturationshaped function as mentioned in Chapter 4).
The stiction, or zero-velocity friction, was introduced by Morin (1833).
Subsequently Reynolds (1886) took viscous effects into account and Stribeck
(1902) further improved the model. His model describes the friction regimes
between two lubricated surfaces with grease or oil and is usable in a variety of
applications. The Stribeck model accounts for a negative slope of the frictional
characteristics and can be described by:
Fd ( x ) = − μ ( x ) Fn sgn x − ( Fs − Fc )e
(−
x δ
)
vs
sgn x − k v x ,
(5.2)
Friction Dampers
101
where Fs and Fc are the stiction and the minimum level of friction, kv is a velocitydependent coefficient, vs and δ are model parameters; vs is known as the Stribeck
velocity, which corresponds to the minimum frictional force.
Following the approach of Armstrong-Helouvry, Dupont and Canudas De Wit
(1994) the frictional interaction between two lubricated surfaces as a function of
velocity can be described by four regimes: pre-sliding or static friction, boundary
lubrication, partial fluid lubrication and full fluid lubrication.
Pre-sliding or static friction: in this regime the asperities of the two materials
deform elastically producing pre-sliding micro-displacements and plastically
causing rising static friction (stiction). A junction in the static friction regime
behaves in a spring-like manner and hence friction force is proportional to microslip (order of microns):
Fd ( x ) = − k tan x ,
(5.3)
ktan being the tangential stiffness of the contact, which depends upon asperity,
geometry, material elasticity and normal load (Johnson, 1987). If tangential force
increases, there exists a breakaway force which causes the actual sliding to begin
(hence friction passes from static to sliding friction). In this situation the undesired
stick–slip effect may occur.
From a control viewpoint the effect of pre-sliding can be significant in position
and pointing servos where leverage systems can amplify this micro-slip to an
actual displacement of the order of millimetre (Canudas de Wit et al., 1993), which
can trigger the feedback loop. Dynamic effects occur as well in the pre-sliding to
actual sliding transition, which are described in Subsection 5.2.2.
Boundary lubrication: this regime arises at very low velocities, when velocity
is not large enough to maintain a fluid film between the two materials. A high
stiction force occurs when lubricants which provide small boundary lubrication are
employed. If lubricants having low-stiction additives (Wills, 1980) are chosen,
stick–slip can be reduced or even eliminated. Stick–slip can also be reduced by
increasing the stiffness of the system.
Partial fluid lubrication: this is the most complicated regime to model (Sadeghi
and Sui, 1989). In this regime a dynamic effect occurs in response to change of
velocity in the form of a delay between a change of velocity and a change of
friction force. This phenomenon is known as frictional memory (Rice and Ruina,
1983).
Full fluid lubrication: in this regime the fluid film between the two materials is
fully developed and dictates the trend of the friction characteristic, which is of
viscous type. The transition between the partial and full fluid lubrication regime
may produce a negative slope in the friction characteristic (the aforementioned
Stribeck effect), which can result in a destabilising effect because under such
conditions the system has a negative damping term. Hence if the working velocity
of the system is in that range, the system may become unstable and self-sustained
oscillations and limit cycles arise.
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Semi-active Suspension Control
5.2.2 Dynamic Friction Models
Dynamic phenomena occur in the transition from static to Coulomb friction and in
response to changes in velocity. These phenomena are known as relaxation
oscillations and were first investigated by Rabinowicz (1958). Dynamic
phenomena also exist in response to changes in the normal force.
The dynamics associated with the breakaway transition have been modelled by
a number of researchers using, for instance, exponential (Kato et al., 1972) or
linear (Armstrong-Helouvry, 1990) models. The rising time is known as the dwell
time.
The dynamic effect in response to changes in velocity is the so-called frictional
memory: it results in a lag between a change of velocity and the subsequent change
of the friction force to a new steady value; this delay can range from milliseconds
to seconds depending upon the materials and appears to be independent of the
input frequency (Hess and Soom, 1990).
From a control viewpoint frictional memory can help reduce the destabilising
effect of the negative slope of the Stribeck effect. Rabinowicz (1965) verified that,
if the time constants of the system are short compared to the frictional memory
(i.e., if the system is stiff), the resulting limit cycle is unstable.
Dynamics associated with changes in the normal force also exist. Anderson and
Ferri (1990) studied this effect. However Dupont and co-workers (1997) verified
that friction dynamics associated with variations in normal force are fast in
comparison with the dynamics associated with relaxation oscillations; therefore
this effect is negligible in many engineering applications, including vehicle
suspensions.
5.2.3 Seven-parameter Friction Model
Taking into account both static and dynamic effects a very comprehensive model
known as the seven-parameter friction model has been proposed (ArmstrongHelouvry et al., 1994). This model is summed up by the following set of equations.
Pre-sliding displacement:
Fd ( x ) = − k tan x
(5.4)
Sliding (Coulomb, viscous, Stribeck effects and frictional memory):
Fd ( x, t )= − [ Fd + k v x − Fs (γ, t 2 )
1
] sgn x
x (t- TL ) 2
1+ (
)
x s
(5.5)
Rising static friction (friction levels at breakaway):
Fs (γ ,t 2 ) = Fs, a + ( Fs, ∞ -Fs, a )
t2
t2 + γ
(5.6)
Friction Dampers
103
Parameters are defined in Table 5.1. In this model a polynomial model has
replaced the exponential model used to represent the Stribeck effect.
Table 5.1. Seven-parameter friction model coefficients
Fs
Friction force at the breakaway [N]
Fs,a
Breakaway friction force at the end of the previous sliding period [N]
Fs, ∞
Friction force at the breakaway after a long time at rest [N]
kv
Viscous coefficient in Stribeck friction model [Ns/m]
TL
Frictional memory [ms]
t2
Dwell time [s]
γ
Friction model coefficient [–]
In recent decades several other friction models (both static and dynamic models)
have been developed to meet different application needs, here briefly summarised.
The interested reader can refer to the specialised literature cited in the references.
Several other friction models exist; amongst the static ones, Karnopp (1985)
developed a model to deal with the zero-speed issue in computer simulations. If
hysteresis is present in the static friction characteristic the Bouc–Wen model (Wen,
1976) described in Chapter 2 can be usefully used. Dahl (Dahl, 1968 and 1976)
developed a dynamic model based on the stress–strain curve of materials. Another
dynamic friction model is the LuGre model (so called because it was developed at
the universities of Lund and Grenoble; Canudas de Wit, Olsson et al., 1995) which
captures internal frictional dynamics as well as friction velocity hysteresis, springlike response at stiction and varying breakaway force. Another dynamic model is
the Leuven model (Swevers et al., 2000) based on the experimental findings that
the friction force in the pre-sliding regime is a hysteresis function of the position,
with non-local memory. The Leuven model attempts to fit this specific behaviour
into the LuGre model. Other comprehensive models are the generalised Maxwell slip
friction model (Lampaert et al., 2003) and the elasto-plastic model (2000) which
renders both pre-sliding displacement and stiction.
Several other friction models exist, such as the Bliman–Sorine model (1991 and
1995). It is also finally worth mentioning other models recently developed for
haptic interface applications. Haessig and Friedland (1991) proposed two models
known as the bristle model and reset-integrator model; another model was
prooposed in 1998 by Hollerbach and two years later Hayward (2000) developed a
computationally efficient model for the same type of applications.
This brief survey of friction models has outlined the main phenomena occuring
in the sliding friction between two surfaces and their modelling. Depending upon
the application the designer can adopt a more or less advanced friction model.
However, whatever the model adopted, a major problem is the identification of the
correct numerical values for the coefficients of the model. An effective method is
to carry out experimental tests on the materials under the working conditions of the
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Semi-active Suspension Control
system. This approach has been undertaken in order to identify the most suitable
friction model for an automotive friction damper.
5.3 The Damper Electrohydraulic Drive
In order to implement a proportional-type semi-active control logic (balance,
skyhook etc.) friction force must be controlled in a proportional fashion, which can
be achieved with an electrohydraulic drive. Hence the force control problem
translates into a pressure control problem, i.e., the modulation of pressure in a
constant-volume chamber. In a pressure control system the flow and hence the
required power is negligible. This results in smaller components (pump, pipes,
valves). In principle a force-controlled electrical drive could be an alternative
solution but this would need a large solenoid to generate the forces required for the
application. A pneumatic actuation would not be effective because of the slow
dynamic response of pneumatic systems due to the compressibility of air.
Various hydraulic circuits can be envisaged to perform the task of controlling
pressure. A typical solution is based on a pressure control valve mounted in
parallel with a pump (Figure 5.1). The main drawback of this configuration is that,
if several independent pressures are to be controlled, an equivalent number of
pumps and valves would be necessary. This solution is therefore convenient only
when a single FD is to be controlled.
Fig. 5.1. Pressure control valve system
An alternative configuration employs two two-way valves (Figure 5.2): one for the
loading phase (to control the pressure rise) and the other for the unloading phase
(to control the pressure decrease). In a simplified version the unloading valve can
be replaced by a fixed orifice. Such a circuit configuration is common in braking
systems.
Friction Dampers
105
Fig. 5.2. Two two-way valves system
A third type of drive can be devised which minimises the number of components if
several pressures (i.e., several FDs) are to be controlled independently. This
solution is based on a three-way proportional flow control underlapped valve used
in pressure control mode driving a single-chamber actuator (Figure 5.3). With such
a configuration it is possible to control several independent pressures using only
one pump and a number of valves equivalent to the number of pressures to be
controlled, as Figure 5.4 depicts.
The control valve behaves in a manner analogous to a resistive potential
divider, i.e., the actuator chamber pressure is modulated by metering both the inflow and the out-flow. Pressure versus spool current demand characteristics
(pressure gain) have a saturation-shaped trend with saturation limits at supply and
return pressures and a gradient dependent upon the leakage flows (an ideal leakfree valve would have an on–off characteristic). Such a characteristic allows
pressure to be controlled in an almost proportional fashion for small demand
signals, whereas for large signals pressure can be switched in a relay-like fashion.
This allows the implementation of both proportional and bang–bang type control
algorithms (Guglielmino and Edge, 2000 and 2001). Detailed modelling of this
hydraulic drive is presented in the following section.
106
Semi-active Suspension Control
Fig. 5.3. Three-way underlapped valve system (copyright International Federation of
Automatic Control (2005), reproduced with minor modifications from Guglielmino E,
Stammers CW, Edge KA, Sireteanu T, Stancioiu D, Damp-by-wire: magnetorheological vs.
friction dampers, used by permission)
Fig. 5.4. Hydraulic circuit for the independent control of four friction dampers
From the viewpoint of controlling the pressure the three configurations are
virtually equivalent, provided valves with sufficient bandwidth are employed.
When designing a hydraulic circuit a major limiting factor is the valve dynamic
response which could be negatively affected by external factors independent of the
component chosen such as the presence of air in the hydraulic oil. However costly
and sophisticated the system, the presence of free air in the circuit results in a
reduction of the hydraulic oil bulk modulus and consequently adversely affects the
system response. All these issues will be dealt with in detail in Section 5.9.
Friction Dampers
107
5.4 Friction Damper Hydraulic Drive Modelling
The mathematical model of the three-way valve-based hydraulic drive is now
presented. The model here described has been implemented in the Simulink®
software.
In this design an electrohydraulic proportional underlapped valve, supplied by a
pump in parallel with a relief valve, drives a single-chamber actuator. The
hydraulic circuit behaves as a resistive potential divider (Figure 5.5).
From consideration of Figure 5.5, the continuity equation can be written as:
Qp = Q1 + Qrv + Qc
for
Ps > Pc ,
(5.7)
Pc being the relief valve cracking pressure and Qc the compressibility flow in the
line connecting the outlet of the pump with the inlet of the relief valve and with the
supply port of the control valve (also Qp = Q1 + Qc for Ps<Pc, however in the
model the pump is supposed to supply enough flow so as to keep the relief valve
always open).
Fig. 5.5. Equivalent hydraulic circuit [copyright Elsevier (2003), reproduced from
Guglielmino E, Edge KA, Controlled friction damper for vehicle applications, Control
Engineering Practice, Vol 12, N 4, pp 431–443, used by permission]
The governing equation of the relief valve is (refer to Table 5.2 for notation):
Ps = Pc + k rv (Qp − Q1 − Qc ) ,
(5.8)
where Qp is the rated flow of the pump and krv is the relief valve override
coefficient.
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Semi-active Suspension Control
Relief valve dynamics have been not included because they are typically very
fast (around 200 Hz). The compressibility flow in the connecting hose is:
Qc =
Vhose dPs
.
B dt
(5.9)
Applying the continuity equation at the second node of the circuit of Figure 5.5,
yields
Q1 = Q2 + Q3
(5.10)
Table 5.2. Key parameters used in the hydraulic drive simulation [copyright Elsevier
(2003), reproduced from Guglielmino E, Edge KA Controlled friction damper for vehicle
applications, Control Engineering Practice, Vol 12, N 4, pp 431–443, used by permission]
Parameter
Value
Underlap (u)
0.1 [m]
Actuator area (Ac)
6.28 × 10-4 [m2]
Pump flow (Qp)
9 × 10-5 [m3/s]
Cracking pressure (Pc)
64 [bar]
Relief valve override (krv)
104 [bar⋅s/m3]
Chamber volume (Vt)
10-4 [m3]
Connecting hose volume (Vhose)
10-3 [m3]
Bulk modulus (B)
1.6 × 109 [N/m2] (supply); 5 × 107 [N/m2] (actuator)
Discharge coefficient (Cq)
0.62 [–]
Leakage coefficient (k1s)
1.5 [–]
Valve spool damping ratio
0.6 [–]
Valve spool resonant frequency
105 [Hz]
Hydraulic oil density
870 [kg/m3]
A sufficiently accurate model of the flow past the valve is crucial for capturing the
behaviour of the valve. When it works in pressure control mode and the spool
moves around the central position with small displacements, leakage flows play an
important role. Leakage flow could be considered to a first approximation as being
Friction Dampers
109
laminar and therefore expressed by a linear relationship between flow and pressure
drop. However depending upon the length of the leakage flow path the regime can
be either laminar, turbulent or transitional. The model described here was proposed
by Eryilmaz and Wilson (2000) and uses a turbulent model and an empirical
correlation to model the valve opening. Based on this model, the governing
equations are (see Figure 5.6):
Q1 = Cqπ D( u + z )
Q1 = Cqπ D
Q2 =
2( PS - PA )
ρ
2( PS − PA )
ρ
u2
(u − k1s z )
if z ≥ 0
(5.11a)
if z < 0
(5.11b)
Vt dPA
B dt
Q3 = CqπD
(5.12)
2( PA - PT )
ρ
Q3 = Cqπ D(u − z )
u2
(u + k1s z )
2( PA − PT )
ρ
if z ≥ 0
if z < 0
(5.13a)
(5.13b)
with –u ≤ z ≤ u, where
k1s =
1
2
PS + PA (u ) − PT
−1 .
PS − PA (u ) − PT
(5.14)
110
Semi-active Suspension Control
Fig. 5.6. Three-way control valve internal geometry [copyright ASME (2001), reproduced
from Guglielmino E, Edge KA, Modelling of an electrohydraulically-activated friction
damper in a vehicle application, Proc ASME IMECE 2001, New York, used by permission]
Spool–solenoid electromechanical dynamics can be expressed through a secondorder linear model with transfer function
z
( s) = 2
i
s
ωn 2
kz
ξ
+ 2 v s +1
ωn
(5.15)
z being the spool displacement and i the solenoid current. The valve amplifier
voltage–current dynamics are fast enough to be neglected.
The transfer function relating pressure to current is obtained to show which
parameters (valve lap, volume, bulk modulus) affect the dynamic response of the
drive. A linearised model cannot be used for detailed performance assessment
because the pressure variations are too large to allow a linearisation procedure.
Neglecting leakage flows past the valve, linearising (see Equation 5.10) around
the equilibrium position yields (lower-case letters denote small variations):
q1 = q2 + q3 ,
(5.16)
where
q1 =
∂Q1
∂z
∂Q1
PA
z + ∂P
A
z
pA = k c1 z + k q1 pA
(5.17)
Friction Dampers
111
and
q3 =
∂Q3
∂z
∂Q3
PA
z + ∂P
A
z
pA = k c3 z + k q 3 pA
(5.18)
hence
pA
(s) =
z
k q1 − k q 3
k c 3 − k c1
Vt
s
1+
B (k c3 − k c1 )
.
(5.19)
Therefore, the overall transfer function between the solenoid current and
pressure is third order:
pA
( s) =
i
k q1 − k q 3
kz
,
2
Vt
ξ
s
v
1+
s(
+2
s + 1)
B (k c3 − k c1 ) ω n2
ωn
k c 3 − k c1
(5.20)
where
k q1 = Cq π D
k c1 = −
ρ
Cq π D(u + z )
2 ρ( PS - PA )
k q 3 = −Cqπ D
k c3 =
2( PS - PA )
,
,
2( PA - PT )
ρ
Cqπ D(u − z )
2 ρ ( PA − PT )
(5.21)
(5.22)
,
.
Linearising around ( z = 0 ; PA =
k q1 = Cqπ D
Ps
ρ
,
(5.23)
(5.24)
PS
) yields
2
(5.25)
112
Semi-active Suspension Control
k c1 = −
Cqπ Du
ρ Ps
Ps
k q3 = −Cqπ D
k c3 =
Cqπ Du
ρ Ps
,
ρ
.
(5.26)
,
(5.27)
(5.28)
The pressure-to-current transfer function (5.20) indicates that the dynamic
response is dependent on the lap size and on the volume of the actuator chamber
and connecting hose characteristics. A block diagram of the whole system, made
with the Simulink® software, is depicted in Figure 5.7.
Fig. 5.7. Simulink® model of the hydraulic drive
Friction Dampers
113
114
Semi-active Suspension Control
The expression of pressure versus demand signal (pressure gain) can be readily
obtained. Under static conditions Q1 = Q3 (there is no compressibility flow under
static conditions). If leakage flows are neglected, the following equality holds:
Cqπ D(u + z )
2( PS - PA )
ρ
= Cqπ D(u − z )
2( PA - PT )
ρ
,
(5.29)
which yields (supposing for simplicity PT=0)
(u + z ) 2 ( PS − PA ) = (u − z ) 2 − PA .
(5.30)
Hence the analytical expression of the pressure gain as a function of spool
displacement and underlap is
PA ( z ) = PS
z 2 + u 2 + 2uz
2(u 2 + z 2 )
with –u ≤ z ≤ u .
(5.31)
In the limit u → 0 this gives an ideal on–off characteristic. Figure 5.8 shows a
typical trend for the pressure gain.
Pressure gain
10
9
8
pressure [bar]
7
6
5
4
3
2
1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
spool position [mm]
0.6
0.8
1
Fig. 5.8. Three-way underlapped valve pressure gain [copyright ASME (2001), reproduced
from Guglielmino E, Edge KA, Modelling of an electrohydraulically-activated friction
damper in a vehicle application, Proc ASME IMECE 2001, New York, used by permission]
Friction Dampers
115
If leakage flows defined by (5.11b) and (5.13a) are included, it is still possible to
obtain an analytical expression for the pressure gain but it is far more involved
(Eryilmaz and Wilson, 2000).
It is interesting to note that the pressure versus opening characteristic of an
underlapped valve inherently constitutes a sliding-mode controller with boundary
layer proportional to valve lap; an ideal zero-lapped valve would behave as an on–
off controller. If piloted with small demand signal it behaves as an almost
proportional controller. Hysteresis, if present, only affects the motion inside the
boundary layer and can contribute to reduction of chattering (Gerdes and Hedrick,
1999). However from the viewpoint of minimising chattering (to improve
comfort), a strategy based on a switched proportional controller is of greater
interest. Balance logic, skyhook and almost all the other control strategies
described in Chapter 4 can be classified as proportional-type variable structure
controllers, or in more classical control terms as switched state feedback
controllers.
5.4.1 Power Consumption
A major benefit of the pressure-controlled semi-active suspension is the low power
required compared not only to an active suspension but also to other semi-active
suspensions whose control involves throttling flows past orifices. The hydraulic
circuit described controls only pressure. Flow is negligible in the force control
hydraulic drive described, and therefore the power required is smaller than that in a
more conventional flow system.
It is possible to roughly estimate the power consumption. The power required is
given by:
W=
PSQP
η
,
(5.32)
where PS is the supply pressure, QP is the pump flow and η its efficiency. The
power dissipated is an increasing function of the underlap. Using typical values
with an operating pressure of 15 bar the power required is about 150 W per
damper.
5.4.2 The Feedback Chain
The friction force control system works in closed-loop mode. In a semi-active
control scheme (e.g., balance logic) relative displacement and velocity signals are
fed back into the controller, which issues a demand signal to the valve solenoid.
Position can be measured either with an inductive transducer such as an LVDT or
with a resistive displacement transducer based on a potentiometric system.
Velocity can be either measured or obtained numerically (by differentiating
displacement). Alternatively accelerometers can be used and velocity and
displacement obtained via integration.
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Semi-active Suspension Control
Transducer dynamics should be included in a dynamic model. However such
transducers are extremely fast relative to vehicle dynamics, and their dynamics can
be either neglected or simply represented via a first-order lag in simulation studies.
5.5 Pilot Implementation of Friction Damper Control
In this section the development of a semi-active friction device for conceptproofing purposes is presented. This step is necessary before designing a more
realistic vehicle FD. The scope is to have an experimental verification in a simple
system that dry friction can be controlled to produce a damper force which can be
exploited to reduce vibration.
The basis of this type of control is the generation of a friction force by means of
a controlled normal force upon a pair of plates. As described in Chapter 4 the
appeal of controlling friction is twofold: in principle any desired force can be
generated, including zero force, and a control force can be produced even when the
relative velocity is very low.
While any type of control logic can be used, the main interest here is in the
balance logic where friction force is used to reduce or even cancel the spring force
on the sprung mass. The semi-active friction device has an inherent physical
limitation: it can only oppose the motion and not assist it and therefore it is not
possible to apply the control force continuously, but only when the following
condition is met:
xx ≤ 0
(5.33)
(otherwise the control force would have the same direction as the spring force).
In order to achieve tracking, the control action must be proportional to the
elastic force, hence
⎧⎪bk s x
Fn (x , x ) = ⎨
⎪⎩ 0
if xx < 0
(5.34)
if xx ≥ 0 .
The coefficient b is a gain defining the level of cancellation of the spring force,
inversely proportional to the friction coefficient. Assuming Coulomb friction and
that the friction coefficient is perfectly known, then in order to obtain perfect
spring force cancellation b must be chosen to be b = 1 . However there will
μ
always be a mismatch between the assumed friction coefficient and the actual one.
and the actual amount of spring
Hence in reality it will be b = 1
μ assumed
cancellation will be dictated by the ratio μ
μ assumed .
Figure 5.9 depicts a simple laboratory friction damper solely for conceptproofing purposes.
Friction Dampers
117
Fig. 5.9. Photo of the pilot experimental set-up
The sprung mass of a quarter car model is represented by a 33 kg mass suspended
by springs, which resembles in scale the main vehicle suspension. The system has
a natural frequency of 1.35 Hz, typical of vehicle suspensions. The frame from
which the mass is suspended is mounted on a hydraulically activated shake table,
which simulates wheel motion.
The mass runs on two vertical rails, which prevent horizontal motion, while
inducing only a low level of rolling friction in the vertical direction. The
controllable damping is produced by a friction pad pressed against a steel plate
rigidly connected to the mass. The friction pad is mounted on a linear actuator,
driven by a proportional pressure relief valve (the hydraulic drive depicted in
Figure 5.1). The valve is placed in parallel with a pump, which supplies a rated
constant flow. In the off position, the valve is open and the piston is in light contact
with the plate. The diameter of the piston is made as small as is practical in order to
increase the pressures required so that seal friction is not a significant fact.
118
Semi-active Suspension Control
The demanded normal force is set according to the balance logic strategy by
measuring the relative displacement and velocity. An eddy-current transducer is
used to measure displacement and an inductive device for relative velocity (both
LVDT-type transducers). The rated bandwidth of the valve is 100 Hz.
The schematic diagram of the experimental set-up is depicted in Figure 5.10.
The friction coefficient has to be assumed and the controller gain adjusted
consequently. A difficulty in carrying out measurements is due to the stiction force,
especially in the presence of a high control force. Such a phenomenon can produce
an undesired lockup of the mass from time to time; such behaviour occurs when
the stiction force is higher than the inertial force. Care has to be taken in choosing
the values of the control force to ensure that they do not lie extremely close to the
maximum theoretical value that produces full cancellation. The control of the
device under the conditions of heat, moisture and wear makes this a challenging
control problem in on-road operation.
Fig. 5.10. Schematic diagram of the pilot experimental set-up [copyright IFToMM (1999),
reproduced from Stammers CW, Guglielmino E, Sireteanu T, A semi-active friction system
to reduce machine vibration Tenth World Congress on The Theory of Machines and
Mechanisms, Oulu, Finland, used by permission]
Figure 5.11 displays position and velocity controlled transient responses following
a step input. By analysing Figure 5.11 it can be noticed that firstly the control is off
because position and velocity have the same sign; subsequently it is turned on.
When the product of position and velocity becomes positive again, the control is
turned off for the second time. Therefore the structure of the system changes three
times during this transient. When the values of position and velocity become small
enough to enter the chosen dead zone, the control remains switched off and the
Friction Dampers
119
structure does not change any longer. Obviously the transition between on and off
is not instantaneous, but depends upon the pump–valve–actuator dynamics.
Fig. 5.11. Experimental free response [copyright IFToMM (1999), reproduced from
Stammers CW, Guglielmino E, Sireteanu T, A semi-active friction system to reduce
machine vibration Tenth World Congress on The Theory of Machines and Mechanisms,
Oulu, Finland, used by permission]
Sinusoidal frame inputs in the range 1–5 Hz are then applied. Figure 5.12 shows,
plotted on the same axis, both passive and semi-active acceleration at a frequency
of 1.5 Hz. The acceleration has been obtained by numerical differentiation. The
passive response is not exactly sinusoidal because of the actual non-sinusoidal
motion of the shaker, of the parasitic friction effects in the rig (e.g., the rolling
friction in the vertical rails) and because of the noise amplification as a
consequence of the numerical differentiation. The acceleration at this frequency in
the semi-active case is much reduced even if it has a larger harmonic content due
to the presence of the control action. It is worth remarking that qualitatively the
switching control has not provoked a very harsh response. This is favourable from
the point of view of comfort.
120
Semi-active Suspension Control
Fig. 5.12. Sprung mass acceleration versus time; input frequency: 1.5 Hz [copyright
IFToMM (1999), reproduced from Stammers CW, Guglielmino E, Sireteanu T, A semiactive friction system to reduce machine vibration Tenth World Congress on The Theory of
Machines and Mechanisms, Oulu, Finland, used by permission]
The amplification factor (transmissibility) of the sprung mass acceleration in the
semi-active case is compared in Figure 5.13 with that in the passive case for
sinusoidal inputs.
Fig. 5.13. Amplification factor for the sprung mass acceleration [copyright IFToMM (1999),
reproduced from Stammers CW, Guglielmino E, Sireteanu T, A semi-active friction system
to reduce machine vibration Tenth World Congress on The Theory of Machines and
Mechanisms, Oulu, Finland, used by permission]
Friction Dampers
121
The control strategy is particularly effective in the neighbourhood of the system
resonant frequency, where a reduction of the transmitted forces of about 35% with
respect to the passive case is achieved. Far from the resonant frequency the
behaviour is the same as in the case without control. This is a remarkable result,
compared with the passive case, in which the achievement of such a big reduction
at resonance (by increasing the damping) would produce the drawback of larger
amplitudes at higher frequencies.
The issue of algorithm robustness can be experimentally checked for instance
by adding random offsets to the feedback signals, thus emulating a situation in
which feedback signals are corrupted (e.g., because of a faulty transducer). The
uncompensated offsets give rise to a switching condition of the form
(5.35)
( x + o1 )( x + o2 ) ≤ 0 ,
where o1 and o2 are two arbitrary offsets. Therefore the condition (5.33) is not
exactly respected. However the results, depicted in Figure 5.14, are very similar to
those shown in Figure 5.13, except for a small increase in the frequency at which
the peak transmissibility occurs. Therefore this VSC scheme has proven to be
robust to corrupted feedback signals.
9
Acc. ratio
8
7
6
5
passive
semi-active
4
3
2
1
0
0
1
2
3
4
Frequency [Hz]
Fig. 5.14. Experimental frequency response with corrupted feedback signals
By varying the coefficient b in Equation 5.34, which defines the rate of control
force applied (as mentioned before it can be thought of as related to the reciprocal
of friction coefficient μ), it is possible to control the rate of cancellation and
consequently vary the acceleration experienced by the mass. In order to check this,
measurements have been carried out changing this parameter; the results are shown
in Figure 5.15. If b is increased the acceleration ratio at that frequency decreases,
tending to an asymptotic value.
122
Semi-active Suspension Control
4
3.5
Acc. ratio
3
2.5
2
1.5
1
0.5
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
b
Fig. 5.15. Pilot rig RMS acceleration versus balance gain
5.6 Automotive Friction Damper Design
After having illustrated the initial experimentation on a pilot rig to test the concept,
the design of a vehicle friction damper will be described. The original vehicle
suspension unit (MacPherson strut) for a saloon car will be replaced with a semiactive device. An automotive friction damper is to be constructed in such a manner
as to be able to replace a conventional viscous damper in a vehicle. The main
objective is to retain the maximum dimensions of the original damper in order to
avoid difficulty in fitting the new device into the car as the upper and lower points
cannot be modified and the room available in the car is limited.
The basic idea is to create a relative sliding movement between a part fixed to
the chassis and one fixed to the wheel; the major problem is that in a car the
motion of the sprung mass with respect to the unsprung mass is not a pure vertical
translation: the composite motion of the car prevents a good contact between the
two conjugated surfaces (the pad and the plate).
A possible solution involves trying to place the friction device directly inside
the existing shock absorber, installing the cylinder, the friction pad and the supply
pipe inside the original cylinder of the viscous damper and leaving the external
geometry unmodified. Unfortunately the damper cross-section is too small to
permit this, and hence the damper diameter must be enlarged, while retaining the
spring in its original position. Placing the friction damper within the spring coils
eliminates the risk of contact between the car body and the modified damper, as the
overall external dimensions of the original suspension unit are not modified.
The embodiment of the concept is a piston in a cylindrical housing which
contains two diametrically opposed pistons to which the friction pads are bonded
(Figure 5.16). The pistons are supplied with hydraulic oil through the centre of the
piston rod, with the control valve mounted remotely.
Friction Dampers
123
The choice of the friction material is crucial. A material that is the least
susceptible to wear with similar properties to those used in car brake pads is
required. The material chosen has a nominal friction coefficient of 0.40.
Fig. 5.16. Drawing of the friction damper prototype [copyright ASME (2004), reproduced
with modifications from Ngwompo RF, Guglielmino E, Edge KA, Performance
enhancement of a friction damper system using bond graphs, Proc ASME ESDA 2004,
Manchester, UK, used by permission]
In order to identify a model to support controller design, a bench test assessment of
the static characteristic of the device needs to be undertaken on a hydraulically
powered shaker and a series of tests conducted to measure the damper response.
The device must be tested on a bench rig. The damper test rig with the damper
fitted is depicted in Figure 5.17. It consists of a frame into which the damper under
test is fitted. A position-controlled actuator provides the vertical excitation. The
constant pressure necessary to produce the normal force is provided by an external
pressure source. A load cell measures the total force and an inductive-type
transducer (LVDT) measures the vertical displacement. Velocity is obtained by
numerical differentiation within the data acquisition package.
The force versus velocity characteristics of the friction damper for a constant
supply pressure are shown in Figure 5.18.
124
Semi-active Suspension Control
The overall damping characteristic is not solely due to the friction damping;
some hysteresis is also present because of the elasticity of the rubber bushes
(viscoelastic material) at the mounting ends and the frictional memory effect (i.e.,
the dynamic effect following a change in the sign of velocity).
Fig. 5.17. Friction damper on the bench rig
The force and relative velocity time histories for a 1.2-Hz sinusoidal input,
depicted in Figure 5.19, reveal that sticking between the friction pads and the
cylinder surface is negligible. The asymmetry in the friction characteristic has been
found also in pneumatic actuators (which have similar geometry and dry frictional
interfaces), as reported in a work by Brun et al. (1999). The low sticking is due to
the material employed, manufactured using low-sticking material (Wills, 1980). In
response to changes in the sign of velocity, the friction force lags the demand with
a delay of around 40 ms. This delay is found to be frequency independent.
0.3
Force [kN]
0.2
0.1
-60
-40
0
-20 -0.1 0
20
40
60
-0.2
-0.3
Velocity [mm/s]
Fig. 5.18. Damping force versus velocity characteristics [copyright International Federation
of Automatic Control (2005), reproduced from Guglielmino E, Stammers CW, Edge KA,
Sireteanu T, Stancioiu D, Damp-by-wire: magnetorheological vs. friction dampers, IFAC
16th World Congress, Prague, Czech Republic, used by permission]
Friction Dampers
125
0.6
0.4
0.2
0
-0.2 0
0.5
1
-0.4
-0.6
Time [s]
Fig. 5.19. Force (solid) and relative velocity (dashed) time histories at 1.2 Hz
It is not readily possible to test the friction force dynamic response to a dynamic
change in the normal force. However as previously stated, Dupont et al. (1997)
verified that these friction dynamics are usually fast. Therefore this dynamic
behaviour is negligible for this automotive application. It is instead interesting to
measure the static dependence of friction coefficient μ with pressure. In the
pressure range of interest to the control (up to 20–25 bar), the variations were not
very large (Figure 5.20).
% friction coefficient variation
120
MU/MUnom %
100
80
60
40
20
0
0
5
10
15
20
25
30
Pressure [bar]
Fig. 5.20. Friction coefficient pressure dependency
An 8-hour endurance test was also carried out using a sinusoidal input and 5-bar
constant pressure. The repeatability was good and wear negligible. Increase in
temperature was minimal because of the good heat transfer properties of the steel
wall. Besides, in a car suspension the air convection would also increase the heat
dissipation when the vehicle is riding.
126
Semi-active Suspension Control
The purpose of these tests was to identify the static and dynamic friction
characteristics of the friction damper designed for the vehicle suspension.
The overall damping characteristic of the damper is not solely dependent on the
friction damping effect; some non-linear viscous damping is also present, because
of the rubber bushes.
Therefore the friction force in the damper has the required characteristics for
the application: negligible stiction and some dynamics in the desired range (not so
fast to be negligible, but not too slow to compromise the overall performance of
the system). The pure delay from a dynamic viewpoint behaves approximately as a
first-order lag (Padè approximations; Baker and Graves-Morris, 1996) which can
contribute together with the hydraulic dynamics to smooth the sharp transitions of
the valve switching control.
The friction model employed can be identified based on the experimental
measurement of the friction force on the friction damper itself and expressed by:
Fd (t ) = − μFn (t − TL )sgn x ,
(5.36)
where the friction coefficient μ is velocity independent and TL is a delay due to the
frictional memory effect. Stiction is not included in the model because it was found
to be negligible (temperature effects have been not considered either in this
analysis).
This model can be seen as a particular case of the more general sevenparameter model of Section 5.2.4 (a Bouc–Wen model could be used as well).
It is worth remarking that, in a vehicle, friction force is a function of relative
and not of the absolute velocity. This is because of the actual mounting of the
friction device. In a real-life situation of a car, in fact, the hydraulic actuator with
the pad is connected to the sprung mass, while the plate is fixed to the unsprung
mass (or vice versa). Hence, the actuator is not stationary but follows the motion of
the sprung mass while the plate is fixed to the unsprung mass. Consequently the
sign of the friction force depends on the relative velocity. Conversely, if the
cylinder were ideally fixed to a stationary reference, the sign of the friction force
would depend only upon the sign of the sprung mass velocity. This could be the
case, for instance, of a friction damper used for vibration isolation of rotating
machinery. To take into account the vertical motion of the cylinder in the
simulation, a mathematical model of a cylinder placed on a moving frame should
be developed; however the problem has been more readily solved by considering
the frictional force as a function of the relative velocity between the sprung and the
unsprung mass, rather than of the absolute velocity of the body of the car.
5.7 Switched State Feedback Control
It has previously been shown that, because of the inherent physical limitation of the
semi-active friction device, which can only oppose to the motion and not assist it, it
is not possible to apply the control force continuously in order to obtain spring
force cancellation, but only when the condition (5.33) is fulfilled.
Friction Dampers
127
The second step in the controller design, after the definition of the allowed
regions of the phase plane via Equation 5.33, is the definition of the analytical
expression of the control logic in these two regions. Recall Equation 5.34:
⎧⎪bk x
Fn (x , x ) = ⎨ s
⎪⎩ 0
if xx < 0
if xx ≥ 0 .
(5.37)
Equation 5.37 can be written as:
Fn =
bk s
bk
x − s x sgn ( xx ) ,
2
2
(5.38)
Friction force is expressed by (for ease of notation the time delay TL in the
friction force model is omitted in the subsequent equations)
Fd = − μ
bk s
bk
x sgn x + μ s x sgn( xx )sgn x .
2
2
(5.39)
Noting that:
x sgn( xx )sgn x = x
(5.40)
hence
Fd = − μ
bk s
bk
xsgn( xx ) + μ s x .
2
2
(5.41)
Therefore Equation 5.41 is composed of two terms: a switching term and a state
feedback controller. The product (x x ) can be interpreted as a particular non-linear
sliding surface. With this strategy the valve mainly works in the (nearly) linear
zone of its pressure gain characteristic. Additional viscous damping may be added
either in the first and third quadrants of the phase plane diagram or in the second
and fourth quadrants (Guglielmino and Edge, 2001). Hence the overall logic can be
regarded as non-linear switched state feedback; it is actually a partial state
feedback because only one state is used in the control action; the other state, the
velocity, only dictates the switching condition. It is worth noting that this logic
does not require pressure feedback (nor acceleration feedback).
The balance controller is designed to track the spring force. The target of the
controller is to reduce chassis acceleration by performing spring force tracking,
thus improving comfort.
Strictly, the tracking law ought to take into account the hydraulic system
dynamic behaviour. However this would result in a very complicated control law
with questionable benefit. Provided that hydraulic dynamics are fast enough, it
128
Semi-active Suspension Control
should be possible to reduce the RMS values of the overall response, rather than
perform a perfect instantaneous tracking.
Comfort is not easy to quantify and, although standards exist, it is an inherently
subjective matter. Several criteria have been proposed (see Chapter 1 for a survey
on the topic) based on minimising different combinations of position, velocity,
acceleration and jerk. In this work chassis acceleration is used as the comfort
improvement criterion.
In principle it is possible to modify the logic, taking into account also the
friction force in the control loop, and possibly increasing the robustness to friction
coefficient variations. Considering that spring and friction forces depend upon
displacement and velocity (with opposite sign), a switching condition based on the
product of spring and friction force can be defined, yielding a control force defined
by
bk s
bk
x − s x sgn( Fs Fd ) ,
2
2
(5.42)
Fs Fd = −k s x( M 1 x1 + 2ξω m1 x + k s x ) .
(5.43)
Fn =
where
However in this case acceleration feedback is also required in order to deduce
the friction force.
The controller described by Equation 5.37 does not introduce any velocity
feedback (velocity needs to be measured for implementing the switching condition,
but the control signal is only a function of displacement). However a small amount
of velocity feedback could be advantageous to help smooth further the transition
between the on and the off state. For this reason, a more general controller can be
devised whose general form is:
⎧⎪bk s x + 2 z1ω1m1 x
Fn (x , x ) = ⎨
⎪⎩ 2 z 2ω1m1 x
if xx < 0
if xx ≥ 0
(5.44)
with b, z1, z2 ≥ 0, or alternatively:
Fn = (bk s x + 2 z1ω1m1 x )
1 − sgn( xx )
1 − sgn( xx )
.
+ 2 z 2ω1m1 x
2
2
(5.45)
This control law combines position damping in the second and fourth quadrants
of the phase plane (which results in a spring force reduction effect) blended with
some pseudo-viscous damping; additional viscous damping is added in the first and
third quadrants (there is no position damping in these quadrants because it would
be in phase with the spring force). The latter not only introduces a viscous-like
effect, but also contributes to smooth the sudden transitions between the two
structures defined by (5.44). By tuning the coefficients b, z1 and z2 it is possible to
Friction Dampers
129
either emphasise the position-dependent damping or the pseudo-viscous damping
effect.
The tuning of the coefficients b, z1 and z2 is not a trivial problem. As already
introduced in Subsection 4.5 by using an approach which recalls the theory of
optimal control the set {b, z1 , z2} could be chosen to minimise a performance index
such as:
T
∫
J (b , z1 , z 2 ) = − {[α ( x2 − z 0 )]2 + [ β ( x1 − x2 )]2 + (γx1 ) 2 }dt .
(5.46)
0
Such an index achieves a trade-off between the reduction of chassis
acceleration and dynamic tyre force within the constraint of a set working space.
Formally speaking this is not an optimal control problem, because the
supposedly optimal function is assumed to be defined by (5.44). Therefore it is not
necessary here to employ the methods of the variational calculus but only finding
the optimal value of b, z1 and z2 which minimises the integral (5.46).
The main problem of this optimisation method is that the optimal set depends
heavily on the type of road (and also on the type of performance index).
5.8 Preliminary Simulation Results
This section is concerned with the analysis of the performance of the VSC balance
control law outlined above, using a quarter car model. The performance is analysed
both in the frequency and time domains. The general form of the controller (in
terms of normal force) is given by Equation 5.44.
Subsequent figures show transmissibility curves for body acceleration, working
space and dynamic tyre force. These curves give an evaluation of the performance
of the controller in terms of RMS signals. They are plotted in the frequency range
1–5 Hz, the main range of interest in a car suspension, for the purpose of
controlling chassis resonance.
Figures 5.21, 5.22 and 5.23 have been plotted for variations in the friction
coefficient μ of ±20% around its nominal value (0.40). The scope of this sensitivity
analysis is to assess how an uncertainty in the knowledge of the friction coefficient
affects the results. Figure 5.21 shows that semi-active RMS acceleration is
generally smaller than in the passive case. It is worth drawing an analogy between
the behaviour of the semi-active response, when the friction coefficient varies, and
the response of a linear system when the viscous damping ratio is varied: close to
the resonance a highly damped system (μ = 0.5) performs better; conversely above
the resonance frequency a low damped system (μ = 0.3) behaves better and,
analogously to a linear system, there exists a frequency where the three trends tend
to cross one another and the behaviour is almost independent of the value of the
friction coefficient.
130
Semi-active Suspension Control
RMS Body Acceleration
2.6
RMS Body acceleration [m/s 2]
2.4
passive
2.2
2
1.8
mu=0.5
1.6
mu=0.4
mu=0.3
1.4
1.2
1
1
1.5
2
2.5
3
3.5
Frequency [Hz]
4
4.5
5
Fig. 5.21. RMS chassis acceleration transmissibility curves varying friction coefficient (0.3,
0.4, 0.5). Controller with b = 2.5, z1 = 0, z2 = 0 [copyright Kluwer (2004), reproduced from
Meccanica, Vol 39, N 5, pp 395–406, Guglielmino E, Edge KA and Ghigliazza R, On the
control of the friction force and used with kind permission of Springer Science and Business
Media]
RMS Working Space
0.035
0.03
RMS Working space [m]
passive
0.025
mu=0.3
0.02
mu=0.4
0.015
mu=0.5
0.01
0.005
1
1.5
2
2.5
3
3.5
Frequency [Hz]
4
4.5
5
Fig. 5.22. RMS working space transmissibility curves varying friction coefficient (0.3, 0.4,
0.5). Controller with b = 2.5, z1 = 0, z2 = 0
Friction Dampers
131
RMS Dynamic Tyre Force
450
RMS Dynamic tyre force [N]
400
passive
350
mu=0.5
300
mu=0.4
250
mu=0.3
200
150
100
1
1.5
2
2.5
3
3.5
Frequency [Hz]
4
4.5
5
Fig. 5.23. RMS dynamic tyre force transmissibility curves varying friction coefficient (0.3,
0.4, 0.5). Controller with b = 2.5, z1 = 0, z2 = 0
The RMS semi-active working space (Figure 5.22) is smaller than the
corresponding passive one in the vicinity of the resonance; after the resonance, the
controlled response tends asymptotically to the passive response. The reduction in
RMS working space is somewhat dependent on the value of the friction coefficient.
The controlled RMS dynamic tyre force (Figure 5.23) is reduced with respect to
the passive case only in the neighbourhood of the resonance, but at higher
frequencies the semi-active system, depending upon the value of the friction
coefficient, can be better or worse than the passive system.
In Figures 5.24, 5.25 and 5.26 the performance of various types of controllers is
compared in terms of RMS frequency response.
RMS Body Acceleration
2.6
RMS Body acceleration [m/s 2]
2.4
passive
2.2
2
b=2.5, z1=0.2, z2=0
b=2.5, z1=0, z2=0
1.8
b=2.5, z1=0.2, z2=0.2
1.6
b=2.5, z1=0, z2=0.2
1.4
1.2
1
1
1.5
2
2.5
3
3.5
Frequency [Hz]
4
4.5
5
Fig. 5.24. RMS chassis acceleration transmissibility curves for different controllers
132
Semi-active Suspension Control
RMS Working Space
0.035
RMS W orking space [m]
0.03
passive
0.025
b=2.5, z1=0.2, z2=0.2
0.02
0.015
b=2.5, z1=0, z2=0
b=2.5, z1=0.2, z2=0
b=2.5, z1=0, z2=0.2
0.01
0.005
1
1.5
2
2.5
3
3.5
Frequency [Hz]
4
4.5
5
Fig. 5.25. RMS working space transmissibility curves for different controllers
RMS Dynamic Tyre Force
500
RMS Dynamic tyre force [N]
450
passive
400
b=2.5, z1=0.2, z2=0
350
b=2.5, z1=0.2, z2=0.2
300
b=2.5, z1=0, z2=0
250
b=2.5, z1=0, z2=0.2
200
150
100
1
1.5
2
2.5
3
3.5
Frequency [Hz]
4
4.5
5
Fig. 5.26. RMS dynamic tyre force transmissibility curves for different controllers
The benchmark controllers used in the following assessment are a pure position
feedback controller (b = 2.5, z1 = 0, z2 = 0) and other controllers where a certain
amount of velocity feedback (equivalent to a damping ratio of 0.2) has been added
(either in the second and fourth quadrants or in the first and third quadrants or in
both). From consideration of Figure 5.24 it is clear that the controller with b = 2.5,
z1 = 0, z2 = 0.2 achieves the best results in terms of acceleration reduction,
particularly close to the resonance; however its performance is not as outstanding
in terms of working space and tyre force (see Figures 5.25 and 5.26). At the other
extreme the controller with b = 2.5, z1 = 0.2, z2 = 0 produces the least acceleration
reduction among the four types of controllers investigated, but it performs better
Friction Dampers
133
with respect to working space and tyre force. The other two controllers, the pure
position feedback controller (b = 2.5, z1 = 0, z2 = 0) and that with b = 2.5, z1 = 0.2,
z2 = 0.2 provide a reasonable trade-off among the requirements of minimising
acceleration and reducing working space and dynamic tyre force.
Figures 5.27–5.30 depict the time trends for spring and normal forces, in order
to assess qualitatively how abrupt the transition is between the two structures for
the four benchmark inputs above. Sharp transitions in the normal force are an
indirect and qualitative measure of the ride quality achievable with the different
controllers. Abrupt transitions may result in spiky acceleration time histories.
Spring and normal forces
800
600
normal force
force [N]
400
200
0
spring force
-200
-400
0
0.2
0.4
0.6
0.8
time [s]
1
1.2
1.4
Fig. 5.27. Spring and normal force time trends with a controller with b = 2.5, z1 = 0, z2 = 0
Spring and normal forces
800
normal force
600
force [N]
400
200
0
spring force
-200
-400
0
0.2
0.4
0.6
0.8
time [s]
1
1.2
1.4
Fig. 5.28. Spring and normal force time trends with a controller with b = 2.5, z1 = 0, z2 = 0.2
134
Semi-active Suspension Control
The control force Fn in Figure 5.27 results from the pure spring force cancellation
controller (b = 2.5, z1 = 0, z2 = 0); when the control is set to the off state the
transition is smoothed only by the hydraulic and frictional dynamics. Some viscous
damping is helpful as Figure 5.28 shows (case with b = 2.5, z1 = 0, z2 = 0.2). The
additional viscous action in the first and third quadrants smoothes the control force
trend, although it cannot compensate for the sudden rise when the controller is set
to the on state.
Spring and normal forces
800
600
normal force
force [N]
400
200
0
spring force
-200
-400
0
0.2
0.4
0.6
0.8
time [s]
1
1.2
1.4
Fig. 5.29. Spring and normal force time trends with a controller with b = 2.5, z1 = 0.2, z2 = 0
Spring and normal forces
800
600
normal force
force [N]
400
200
0
spring force
-200
-400
0
0.2
0.4
0.6
0.8
time [s]
1
1.2
1.4
Fig. 5.30. Spring and normal force time trends with a controller with b = 2.5, z1 = 0.2, z2=0.2
The controller with b = 2.5, z1 = 0.2, z2 = 0 (Figure 5.29) is probably the worst from
a ride quality point of view. The additional viscous damping in the second and
Friction Dampers
135
fourth quadrants sharpens both the on and off transitions. The controller of Figure
5.30 (b = 2.5, z1 = 0.2, z2 = 0.2) performs fairly well, as the control force is never
set to zero and the trend is relatively smooth.
A pseudo-random road input is now considered. The road model is expressed
by the following relationship (Horrocks et al., 1997):
z 0 (t ) = 2
45
∑
i =1
π ΔfV 1.5
(Δfi ) 2.5
10 −3 sin[(2π Δf i )t + p (i )] ,
(5.47)
the velocity V being in m/s.
This is a multiharmonic input where the amplitude is a decreasing function of
the frequency (Figures 5.31 and 5.32); the increment Δf is 0.25 Hz and the phase
p(i) is a random number between –π and π.
The spectrum defined by Equation 5.47 constitutes a discrete approximation to
the continuous spectral density defined by Equation 2.34. Lower harmonics are the
most important; higher harmonics, although present, are mainly filtered by the tyre.
Amplitude spectrum
20
18
16
Amplitude [mm]
14
12
10
8
6
4
2
0
0
2
4
6
frequency [Hz]
8
10
12
Fig. 5.31. Amplitude spectrum of the road vertical profile
It must be stressed that, with the controlled system being non-linear, the property
of the response to this input cannot be inferred in any way from the properties of
the response to each sinusoidal input separately, i.e., the RMS and peak value
percentage reductions (or increases) with respect to the passive system will be
different, because the superposition principle does not hold.
136
Semi-active Suspension Control
Vertical road profile
60
road vertical elevation [mm]
40
20
0
-20
-40
-60
0
1
2
3
time [s]
4
5
6
Fig. 5.32. Road vertical profile time trace; vehicle speed: 50 km/h
Figures 5.33–5.36 show the working space and acceleration time responses to this
input for the passive and the controlled systems with b = 2.5, z1 = 0, z2 = 0. These
results correspond to a vehicle forward constant speed of 50 km/h.
Working space
0.04
0.03
0.02
x [m]
0.01
0
-0.01
-0.02
-0.03
-0.04
-0.05
0
1
2
3
time [s]
4
5
6
Fig. 5.33. Working space time trace for the passive system; vehicle speed: 50 km/h
Friction Dampers
137
Body Acceleration
5
4
3
2
a [m/s 2]
1
0
-1
-2
-3
-4
-5
0
1
2
3
time [s]
4
5
6
Fig. 5.34. Chassis acceleration time trace for the passive system; vehicle speed: 50 km/h
Working space
0.04
0.03
0.02
displacement [m]
0.01
0
-0.01
-0.02
-0.03
-0.04
-0.05
0
1
2
3
time [s]
4
5
6
Fig. 5.35. Working space time trace for the controlled system with b = 2.5, z1 = 0, z2 = 0;
vehicle speed: 50 km/h
138
Semi-active Suspension Control
Body Acceleration
6
5
4
acceleration [m/s2]
3
2
1
0
-1
-2
-3
-4
0
1
2
3
time [s]
4
5
6
Fig. 5.36. Chassis acceleration time trace for the controlled system with b = 2.5, z1 = 0, z2=0;
vehicle speed: 50 km/h
The responses of the different state feedback controllers to this input can be more
appropriately compared in terms of RMS and peak values rather than from their
time histories. Tables 5.3 and 5.4 list RMS and peak values for acceleration and
working space for the different controllers.
Table 5.3. RMS and peak acceleration for a pseudo-random input [copyright Kluwer
(2004), data from the first two lines reproduced from Meccanica, Vol 39, N 5, pp 395-406,
Guglielmino E, Edge KA and Ghigliazza R, On the control of the friction force, used with
kind permission of Springer Science and Business Media]
Controller
RMS acceleration [m/s2]
Max. and min. peak
acceleration [m/s2]
Passive
1.59
4.79/–4.24
b = 1, z1 = 0, z2 = 0
1.66
4.51/–3.53
B = 1, z 1 = 0.2, z 2 = 0
1.71
4.80/–3.35
b = 1, z 1 = 0, z 2= 0.2
1.41
4.46/–3.59
b = 1, z 1 = 0.2, z 2 = 0.2
1.45
4.65/–3.44
The pseudo-random test is a much more severe test than the sinusoidal one. The
performance of the controlled system is somewhat less outstanding compared to
that in response to the sinusoidal input and in some cases even worse: the RMS
acceleration is only slightly better in two out of the four benchmark controllers
considered. The RMS working space response is however always slightly smaller
than the corresponding passive one in all but one case.
Friction Dampers
139
Table 5.4. RMS and peak working space for a pseudo-random input
Controller
RMS working space
[mm]
Max. and min. peak
working space [mm]
Passive
16.7
35.7/–43.7
b = 1, z1 = 0, z2 = 0
16.6
35.4/–45.4
B = 1, z 1 = 0.2, z 2 = 0
17.7
34.5/–48.3
b = 1, z 1 = 0, z 2= 0.2
14.7
32.4/–40.9
b = 1, z 1 = 0.2, z 2 = 0.2
15.3
32.0/–42.4
Last but not least, the response to a bump must be considered. This input, which
represents a discrete event in a road profile, must be analysed seperately. The
relevant quantities to minimise are the peak value of the acceleration and the
number of oscillations after the bump. Figures 5.37 and 5.38 show the chassis
acceleration response in the two cases. The input is a 50-mm-high sinusoidalshaped bump. The bump horizontal length is 0.5 m and the car is assumed to pass
over it at a constant speed of 20 km/h. This is equivalent to saying that the car is
excited with half a sinusoid having a frequency of 11.11 Hz (as the frequency is
equal to the car forward velocity divided by the bump wavelength).
Body Acceleration
15
10
acceleration [m/s 2]
5
0
-5
-10
-15
-20
0
0.1
0.2
0.3
0.4
0.5
time [s]
0.6
0.7
0.8
0.9
Fig. 5.37. Bump response acceleration time trace for the passive system. Bump amplitude:
50 mm; speed: 20 km/h [copyright Kluwer (2004), reproduced from Meccanica, Vol. 39,
N 5, pp 395–406, Guglielmino E, Edge KA and Ghigliazza R, On the control of the friction
force, used with kind permission of Springer Science and Business Media]
140
Semi-active Suspension Control
Body Acceleration
10
8
6
acceleration [m/s 2]
4
2
0
-2
-4
-6
-8
-10
0
0.1
0.2
0.3
0.4
0.5
time [s]
0.6
0.7
0.8
0.9
Fig. 5.38. Bump response acceleration time trace for the controlled system. Bump
amplitude: 50 mm; speed: 20 km/h. Controller with b = 2.5, z1 = 0, z2 = 0 [copyright Kluwer
(2004), reproduced from Meccanica, Vol. 39, N 5, pp 395–406, Guglielmino E, Edge KA
and Ghigliazza R, On the control of the friction force, used with kind permission of Springer
Science and Business Media]
The controlled system acceleration overshoot and undershoot are much smaller
although the number of oscillations is almost the same in both cases and in the
controlled system some spikes are present because of the switching logic.
The values of the overshoot and undershoot for the different controllers are
listed in Table 5.5.
Table 5.5. Peak acceleration value for a bump input
Controller
Max. and min. peak acceleration [m/s2]
Passive
13.77/–17.39
b = 1, z1 = 0, z2 = 0
8.39/–8.58
b = 1, z 1 = 0.2, z 2 = 0
8.39/–8.38
b = 1, z 1 = 0, z 2= 0.2
6.83/–8.18
b = 1, z 1 = 0.2, z 2 = 0.2
6.83/–8.19
The controllers with b = 2.5, z1 = 0, z2 = 0.2 and with b = 2.5, z1 = 0.2, z2 = 0.2
provide the largest reduction of the peak values among the four types of
controllers.
Friction Dampers
141
However a remark should be made on the validity of the results of this test. A
bump can be viewed as a sort of impulsive input which produces a high
acceleration peak. When a tyre undergoes high accelerations, it should be modelled
taking into account its dynamic characteristics. A tyre model which includes only
wheel mass and the tyre static stiffness characteristic is no longer adequate (Genta,
1993).
Furthermore during a bump the point of contact between the wheel and the road
is not aligned with the vertical axis passing through the centre wheel, hence
horizontal contact forces are generated that cannot be taken into account by the
quarter car model (nor by the 7DOF model). Therefore the results from this test
must be considered carefully and taken only as an indication of the actual
behaviour of the car.
5.9 Friction Damper Electrohydraulic Drive Assessment
In automotive suspension control, there is scope to explore simple and cheap
solutions, based on low-cost valves which can be appealing to the automotive
market. The purpose of this section is how to assess, tune and optimise hydraulic
drive performance.
It was previously concluded that the use of a proportional underlapped valve
supplied by a pump in parallel with a relief valve, driving a single-chamber
actuator is suitable for an effective friction damper hydraulic actuation system.
Such a hydraulic circuit behaves in an analogous manner to an electric potential
divider (Figure 5.5).
The valve employed in the hydraulic drive (depicted in Figure 5.3) has been
designed for directional flow control; hence its performance when used in the
pressure control mode is not known, (nor usually provided in manufacturers’
datasheets). A potentially major limiting factor in the system is the pressure
dynamic response of the valve as well as its pressure versus demand signal static
characteristics (the pressure gain).
These real-life engineering issues are crucial for designing a properly working
system and therefore a bench test assessment must be undertaken. This section
describes a methodological approach to verify if a hydraulic system can meet the
requirements of the application. Firstly the performance of a system based on a
cheap 2-way valve is investigated and subsequently the same analysis is presented
for a more costly proportional valve with incorporated electronics; it will also be
shown that a drive using a costly valve requires an accurate optimisation process to
actuate the system properly.
The solution based on a low-cost two-way valve requires an additional return
orifice, the reason being that, for the control of the friction damper, a three-way
valve is necessary. Hence the hydraulic circuit needs to be completed with the
connection of an additional orifice to the service port of the valve, which allows
the flow to tank to be damped. In this way the valve together with the orifice has a
geometry equivalent to that of a three-way valve. For testing purpose the restriction
can be created with an adjustable needle valve (i.e., a variable orifice). The valve
needs to be mounted on a flow bench test rig and both downstream pressure and
142
Semi-active Suspension Control
spool position measured. Figure 5.39 a schematic of the measurement system
shows.
Fig. 5.39. Schematic of the measurement circuit for a two-way valve
The valve is supplied at nominally constant pressure by a volumetric pump (the
pressure being around 60 bar in this initial test) and the service port connected to
tank through the orifice. Downstream pressure is recorded by a variable-reluctance
pressure transducer. A sinusoidal voltage source provides the input to a current
amplifier. Solenoid valves are typically current driven and, if not embedded in the
valve, an external current amplifier is required to provide a current driving signal.
Figure 5.40 shows an example of a circuit for a current driver. It is worth noting
that as the spool valve is spring-preloaded, the equilibrium position does not
correspond to the null signal but to a biased constant value; therefore the amplifier
needs to be supplied with two non-symmetric voltages.
Fig. 5.40. Electrical scheme of a typical valve current amplifier
Friction Dampers
143
Data can be captured and stored using commercial data acquisition software,
acquiring data from pressure transducers and a current probe. Good engineering
practice suggests installing additional pressure gauges and oscilloscopes to check
whether pressure and other signals are in the correct range.
The first characteristic to be established is the pressure versus demand curve,
which defines the non-linear gain and the tracking ability of the controller. The
local slope of the pressure gain around the valve null position depends upon the
valve underlap u and the leakage coefficient k1s. As this is a static characteristic, it
is crucial to carry out the measurement under quasi-static conditions, otherwise the
phase lag introduced at higher frequencies would result in a distortion of the
characteristic. This can be achieved by driving the valve with a low-frequency
sinusoidal signal. The pressure gain is expected to be saturation shaped with some
hysteresis depending upon the magnetic properties of the solenoid core and on the
friction within the spool. A typical measured characteristic is plotted in Figure
5.41.
Pressure gain
60
Pressure [bar]
50
40
30
20
10
0
-1,2
-1
-0,8
-0,6
-0,4
-0,2
0
Current [A]
Fig. 5.41. Pressure versus current static characteristic of a two-way valve
Hysteresis is large, as expected for a low-cost valve. Such a hysteresis would
compromise its usage in applications where good tracking capability is crucial.
However in the FD control application good tracking is not a vital issue. The
second problem is the slope of the characteristics, which is very high, meaning that
the valve is almost critically lapped; this makes it more difficult to use it in a
proportional fashion (it should be driven with low-amplitude signals).
Dynamic performance must be assessed as well. A very convenient way to do
this is the measurement of the frequency response. The system being non-linear, its
measurement requires the deduction of the first harmonic of the input and output
signals and calculation of the modulus and phase of the resulting sinusoidal signals
for each frequency.
144
Semi-active Suspension Control
Because of the strong non-linearity of the system, it is advisable to measure the
frequency response for at least three different amplitudes of the driving signal, as
in non-linear systems the response is not amplitude independent (as opposed to
linear systems). Results are depicted in Figure 5.42. By analysing the figure it can
be noticed that different input amplitudes produce frequency response traces which
are not superimposable. This occurs because for each input the valve works at a
different point of its non-linear pressure-flow characteristics.
The valve is very slow (the average bandwidth is less than 3 Hz). The reasons
for the low bandwidth can be attributed to the combined effects of both the
relatively cheap type of valve and also the likelihood of the presence of free air in
the circuit.
Press [bar]/Current [A]
Amplitude
40
35
30
25
20
15
10
5
0
0.4 V
0.5 V
0.55 V
1
10
100
Frequency [Hz]
Fig. 5.42. Modulus Bode diagram of the pressure-to-current transfer function of the twoway valve (driving signals: 0.4 V, 0.5 V, 0.55 V)
Slightly better performance would have been obtained by restricting the crosssectional area of the fixed orifice (the hydraulic time constant depends on the
inverse of this area), but this would have had as a drawback a larger pressure drop
and hence higher energy consumption at the pump.
The tests indicate that the overall characteristics of this valve are not good
enough for the application; the static characteristic is too sharp and the bandwidth
is insufficient. Such a valve is not suitable for the application.
It would be reasonable to think that a more sophisticated and costly valve,
namely a solenoid proportional three-way valve with incorporated conditioning
electronics (the current amplifier embedded in the valve) could meet the
specification. However it will be shown that this is not always the case.
It is anticipated that an unexpected behaviour at low supply pressure can arise:
dynamic performance could be extremely poor with the bandwidth severely limited
despite the valve being of a relatively sophisticated design.
Friction Dampers
145
The working supply pressure needs to be set at a value consistent with that
required for the suspension application, which is about 10 bar; will be investigated
the feasibility of using this system at low pressure.
Pressure gain must first be assessed. Hysteresis is expected to be negligible in
this valve. Hysteresis is due to both a poor magnetic material (not the case for a
fairly costly valve) and to friction in the spool and other dissipative phenomena.
However, inaccurately designed tests can result in measuring an amount of
hysteresis larger than the actual one. In such cases the measurement chain must be
thoroughly verified. First the inherent hysteresis of the pressure transducer must be
assessed and if necessary the transducer must be replaced with a more accurate one
(such as a semiconductive strain gauge transducer) with very low hysteresis. It is
also useful to repeat the test by superimposing a high-frequency dither signal
(square waves at 200 Hz, 300 Hz and 1 kHz) on the quasi-static signal, as dither
helps reduce the amount of hysteresis. If no appreciable improvement is recorded
then the input test frequency must be drastically reduced. The response plotted in
Figure 5.43 depicts the static characteristic at the supply pressure of 10 bar taken at
the extremely low frequency of 0.004 Hz. This test at such a low frequency was
necessary because of the presence of circuits in the electronics which introduced
high phase margin at very low frequency, thus resulting in an apparent hysteresis.
This stresses the importance of making a test as close as possible to an ideal static
test when measuring static characteristics.
Pressure Gain
Pressure [bar]
10
9
8
7
6
5
4
3
2
1
0
-5
-4
-3
-2
-1
0
1
2
3
4
5
Demand [V]
Fig. 5.43. Three-way valve pressure gain
Dynamic performance is the next step to gain an understanding of the hydraulic
drive behaviour. Before evaluating pressure dynamic response it is important to
measure the valve spool dynamic response (with oil, so as to account for flow
forces in the spool).
Spool frequency response can be readily measured using the LVDT embedded
into the valve (if an external measurement connection is present). Figure 5.44
shows a recorded spool dynamic response, which is second order, with a
146
Semi-active Suspension Control
bandwidth of about 100 Hz, a damping ratio of about 0.60 and virtually supply
pressure independent.
Spool Frequency Response
1,4
Spool pos. [V]/Demand [V]
1,2
1
X/V (0.5 V)
0,8
X/V (1V)
0,6
X/V (1.5 V)
0,4
0,2
0
0,1
1
10
100
1000
Freq. [Hz]
Fig. 5.44. Modulus Bode diagram of the spool position-to-voltage transfer function of the
three-way valve
Pressure dynamic response depends upon volume and bulk modulus: from the
linearised analysis presented before, the pressure-to-spool opening transfer
function is a first-order lag where the time constant is directly proportional to
volume and inversely to bulk modulus. The penalising effects of large volumes can
be reduced only via a compact circuit layout, if compatible with other system
constraints.
Bulk modulus dependency is a more delicate matter. A standard value for a
hydraulic oil is 1.6 × 109 N/m2. However experimental work shows that this value
is hardly attainable in the low-pressure range.
Preliminary tests carried out at a supply pressure of 10 bar (more properly with
the relief valve cracking pressure set at 10 bar) have revealed a bandwidth of only
4 Hz, virtually independent of demand amplitude (this also implies that the system
behaves fairly linearly in those working conditions): Figure 5.45 depicts the
frequency response for three different input voltages.
Such a poor bandwidth is unexpected considering that the valve is fairly
sophisticated and is expected to have a bandwidth up to 100 Hz. The reasons for
such a slow response need to be thoroughly investigated.
Friction Dampers
147
Load Pressure Frequency Response
Press. [V]/Demand [V]
9
8
7
6
5
4
3
2
1
0
0,1
Pa/V (10 bar;0.1V )
Pa/V (10 bar;0.2V )
Pa/V (10 bar;0.3V )
1
10
100
1000
Freq. [Hz]
.
Fig. 5.45. Modulus Bode diagram of the pressure-to-voltage transfer function of the threeway valve; demand signals 0.1 V, 0.2 V, 0.3 V [copyright Elsevier (2003), reproduced with
minor modifications from Guglielmino E, Edge KA, Controlled friction damper for vehicle
applications, Control Engineering Practice, Vol. 12, N 4, pp 431–443, used by permission]
Experimentally testable hypotheses need to be made on the nature of this
phenomenon and a set of tests designed to verify experimentally their consistency.
Sensible hypotheses based on the physics of the system are:
•
•
•
Presence of air bubbles trapped inside the circuit: a small free air
percentage can produce a large reduction of the bulk modulus (McCloy and
Martin, 1980)
Not fully developed turbulent flow because of the small pressure drop past
the valve
Ripple and dynamic effects on the supply pressure caused by relief valve
dynamics, which in turn could affect the downstream pressure
The previous test depicted in Figure 5.45 has also allowed the dependency upon
the demand signal amplitude to be established (the bandwidth usually decreases if
the system is driven hard). If this dependency is found to be negligible (as here) the
valve has fairly linear behaviour.
Considering that the spool response is fast and independent of the measurement
conditions then the problem of the limited pressure bandwidth cannot be caused by
the valve itself but is a fluid mechanics problem. In this case ad hoc tests must be
designed to test the hypotheses above and identify the possible causes.
The first hypothesis made was that of the presence of an air pocket at low
pressure. This can be verified by mounting the valve upright to prevent the air
stagnating at the outlet port and the mounting connection of the pressure
transducer.
Different types of pressure transducers also need to be tested to ascertain
whether transducer dynamics (or also a fault in the instrumentation) can be
148
Semi-active Suspension Control
responsible for the result. An initial flushing of the valve was made before starting
the measurement of the frequency response. The test resulted in no appreciable
changes (the bandwidth was still around 4 Hz). Figure 5.46 depicts the
experimental frequency response under these test conditions.
Load Pressure Frequency Response
Press. [V]/Demand [V]
5
4,5
4
3,5
3
2,5
2
1,5
1
0,5
0
0,01
0,1
P/V (0.2 V)
P/V (0.3V)
P/V (0.5 V)
1
10
Freq. [Hz]
Fig. 5.46. Modulus Bode diagram of the pressure-to-voltage transfer function of the threeway valve measured in the air bubble test
In order to verify the consistency of the second hypothesis the spool must be driven
with a biased signal so as to create a narrower flow path, to help promote
turbulence as the small pressure drop past the orifice may not be sufficient to fully
develop the turbulent flow regime. The low-Reynolds-number flow arising may be
related to the small bandwidth. In order to verify this hypothesis the valve has to be
driven with a biased signal so that the spool does not move symmetrically with
respect to the centre of the land. This creates a narrower flow path and hence a
larger pressure drop, helping to promote a turbulent flow regime. However, even in
this situation the dynamic response does not show any improvement at all.
A third test to be carried out is to quantify the amount of dynamic variation in
the supply pressure induced by the combined effects of the volumetric pump
pressure ripple and the relief valve characteristics. If this is not negligible and is in
phase opposition with the load pressure, it may slow the dynamic performance.
However the measured pressure variation is negligible, as depicted in Figure 5.47.
A relief valve dynamic effect (the second-order spring–mass–damper dynamics of
the relief valve spool) can be excluded to be a co-cause, because relief valve
dynamics are typically at much higher frequencies (around 200–300 Hz).
Friction Dampers
149
12
Pressure [bar]
10
8
Load press. [bar]
Supply press. [bar]
6
4
2
0
0
0,2
0,4
0,6
0,8
1
1,2
Time [s]
Fig. 5.47. Supply pressure dynamic variation
To double check the hypothesis a quick test consisting of replacing the relief valve
with a more sophisticated one, having two stages (hence smaller pressure override)
which produces a smaller ripple on the supply pressure showed that the bandwidth
did not change. Hence not even the relief valve was responsible of the poor
pressure dynamic performance. This test, however, also enables it to be established
that a single-stage relief valve is sufficient for the application, without recourse to a
more costly two-stage relief valve.
However in none of these tests an appreciable improvement has been recorded.
In this case a solution is to increase the supply pressure (by appropriately setting
the relief valve) up to 100 bar. This produced an increase of the bandwidth up to
around 40 Hz. Under these conditions the response is finally acceptably fast: the
bandwidth increases with supply pressure. Figure 5.48 shows the frequency
response measured with a relief valve cracking pressure of 60, 80 and 100 bar.
Load Pressure Frequency Response
12
Press. [V]/Demand [V]
10
8
Pa/V (60 bar; 0.1 V )
Pa/V (80 bar; 0.1 V )
Pa/V (100 bar; 0.1 V )
6
4
2
0
0,1
1
10
100
1000
Freq. [Hz]
Fig. 5.48. Modulus Bode diagram of the pressure-to-voltage transfer function of the threeway valve at 60, 80, 100 bar
150
Semi-active Suspension Control
The bandwidth improved with increasing supply pressure rising to around 40 Hz.
An increase in pressure helps the free air to pass into solution as stated by Henry’s
law (McCloy and Martin, 1980): the greater the pressure, the higher the mole
fraction of air in solution.
Hence in such cases the reason for the slow dynamic response can be postulated
to be due to the presence of air bubbles, probably arising from air release occurring
within the valve. This accounts also for flushing of the system with oil being
ineffective. Therefore although the system pressure for the application is around
10–20 bar, it may be necessary to increase it, thus creating a trade-off between the
higher power consumption (associated with the higher pressure) and the valve
system bandwidth. As a consequence all subsequent tests on the car described in
Chapter 7 were carried out with a supply pressure of 65 bar.
Figure 5.49 depicts the step response having a trend similar to a first-order
linear response, which is desired for the application.
Step Response
60
50
40
30
Press. [bar]
Demand *100 [V]
20
10
0
-10
0
0,2
0,4
0,6
0,8
1
1,2
-20
Time [s]
Fig. 5.49. Three-way valve step response at 60 bar
The volume of oil downstream of the valve has a significant effect on performance
as well. Tests previously described are performed on the valve alone, without the
presence of any additional volume. This is the most favourable condition. Any
further volume increases produce a bandwidth reduction.
Hence further tests need to be performed in order to establish the influence of
the volume on the dynamic response. The impact of additional volume is presented
in Figure 5.50, which depicts tests carried out in three different situations, namely
with the valve load port blanked off (the ideal situation), for an operating condition
with the valve mounted inside the car and connected to the damper via a 0.5-m
long 0.5-inch-diameter hose and finally connecting the valve to the damper via a 2m hose. An increase of the volume by an order of magnitude in the simulation
model reduces the bandwidth by about a decade. This is consistent with a
linearised analysis of the system, which shows that the hydraulic time constant is
directly proportional to the volume. Therefore in a final design the volume should
Friction Dampers
151
be minimised, with the valve integrated into the assembly such that it is as close as
possible to the friction pad piston(s). The minimum volume is the internal pipe
inside the damper plus the volume of the hose connecting the valve to the damper.
Frequency Response Volume Dependency
Press. [V]/Demand [V]
5
4,5
4
3,5
3
2,5
2
1,5
1
0,5
0
0,1
P/V (min. vol)
P/V (nom. vol)
P/V (max. vol)
1
10
100
Freq. [Hz]
Fig. 5.50. Modulus Bode diagram of the pressure-to-voltage transfer function of the 3-way
valve varying volume
Last but not least the problem of back-pressure must be mentioned. A 2-bar backpressure is present, as is evident from the measured pressure gain trend of Figure
5.43. This creates a constant-amplitude friction force present also when the
demand is set to zero. The larger the back-pressure, the bigger the residual friction
force. This can produce an undesired lockup of the damper. The back-pressure is
due to the loss of head in the return line and to the leakage flow. This problem can
however be addressed in the design phase by mounting a pre-loaded spring
between the friction pads. However from a dynamic viewpoint, this additional
spring would introduce two further complex conjugated poles, which would affect
the overall system response. If these poles are of the order of magnitude of the
valve bandwidth, they can affect the dynamic performance of the hydraulic drive.
This design enhancement will be described subsequently.
5.10 Electrohydraulic Drive Parameters Validation
In this section the validation of the experimental tests is described. A prime
objective of the experimental data collected is to validate the mathematical models
developed. This permits to verify that the dynamic order of the model, its
complexity and the precision in the knowledge of the system parameters and inputs
are sufficient to give a reasonably accurate representation of its physical behaviour
over the range of the operating conditions. Backed up by this validation process, if
simulation is sufficiently accurate over the working range, further optimisation
work can be solely made via simulation.
152
Semi-active Suspension Control
From an opposite viewpoint, experimental tests on single components or parts
of the suspension system also permit the identification of system parameters
otherwise difficult to identify with sufficient precision (for instance, the friction
characteristic). Within this context, such experimental results can be logically seen
as an a priori stage, providing reliable input data to the simulation. The friction and
tyre characteristics employed in the simulation were estimated experimentally.
The first characteristic to be assessed for the electrohydraulic drive is the
pressure gain. The local slope of the pressure gain depends upon the valve underlap
u and the leakage coefficient k1s. Since these two parameters cannot be readily
measured, they must be identified via a trial-and-error procedure, by performing a
sensitivity analysis. The initial guess values chosen were a 0.6 mm underlap and a
unit leakage coefficient (estimated with Formula 5.14).
The dependence of the pressure gain upon valve lap is shown in Figure 5.51.
Pressure gain (vs demand)
70
u=0.1
60
u=1.1
u=2.1
pressure [bar]
50
40
experimental
30
20
10
0
-10
-5
-4
-3
-2
-1
0
1
demand [V]
2
3
4
5
Fig. 5.51. Comparison of experimental and predicted control valve pressure gain, varying
the underlap from 0.1 to 2.1 mm [copyright Elsevier (2003), reproduced from Guglielmino
E, Edge KA, Controlled friction damper for vehicle applications, Control Engineering
Practice, Vol. 12, N 4, pp 431–443, used by permission]
Lap values of 0.1 mm, 1.1 mm and 2.1 mm are assumed. The upper-bound region
of the characteristics is affected more by a change in the underlap width than the
lower-bound region, which is the one of interest. The asymmetry in the behaviour
is caused by the leakage term, the relief valve pressure override, the spool
dynamics and the compressibility flow. A symmetrical behaviour would have
Friction Dampers
153
resulted if the leakage had not been considered, the supply pressure had been
perfectly constant and the spool dynamics had been neglected.
In such an ideal situation, pressure gain can be expressed as a function of the
demand in an analytical form and the behaviour in the upper- and lower-bound
regions would be symmetrical. From the analysis of Figure 5.51, the best
agreement is obtained when u is equal to 0.1 mm, although a larger value would
not have affected the lower-bound region.
Figure 5.52 depicts the dependence upon the leakage coefficient, varied
between 0.5 and 2.5 (taking an underlap of 0.1 mm). From consideration of the
figure, the best fit is obtained when k1s assumes the value of 1.5.
Pressure gain (vs demand)
70
K1s=2.5
60
K1s=1.5
pressure [bar]
50
experimental
40
30
20
10
K1s=0.5
0
-5
-4
-3
-2
-1
0
1
demand [V]
2
3
4
5
Fig. 5.52. Comparison of experimental and predicted control valve pressure gain, varying
the leakage coefficient from 0.5 to 2.5 [copyright Elsevier (2003), reproduced from
Guglielmino E, Edge KA, Controlled friction damper for vehicle applications, Control
Engineering Practice, Vol. 12, N 4, pp 431–443, used by permission]
The dynamic response of the valve is the other critical issue. The approximations
made should be recalled. While the valve experimental response is a real frequency
response, obtained by extracting the first harmonic of input and output signals, the
non-linear valve dynamics have been simulated and the “frequency response”
obtained by computing the actual RMS, including therefore the contribution of all
the harmonics. The approximation however is not so critical (and in any case
comparable with that produced by a linearisation of the model) because the static
characteristic of the valve is fairly linear in its central region and has negligible
154
Semi-active Suspension Control
hysteresis, hence it is inherently low-pass and therefore the overall contribution of
the higher harmonics to the RMS is small.
The trends of the frequency responses are portrayed in Figures 5.53 and 5.54.
Figure 5.53 shows the spool-to-voltage demand transfer function frequency
response. The measured response is close to second order: the simulation of a
second-order linear model with a natural frequency of 105 Hz and a damping ratio
of 0.60 matches the results well.
RMS spool pos
0.11
0.1
0.09
simulated
RMS spool pos [V]
0.08
0.07
experimental
0.06
0.05
0.04
0.03
0.02
0.01
-1
10
0
10
1
10
Frequency [Hz]
2
10
3
10
Fig. 5.53. Comparison of experimental and predicted spool position-to-voltage demand
frequency response [copyright Elsevier (2003), reproduced from Guglielmino E, Edge KA,
Controlled friction damper for vehicle applications, Control Engineering Practice, Vol. 12,
N 4, pp 431–443, used by permission]
The suitability of a linear model also confirms that friction of the spool within the
valve sleeve is negligible.
The pressure-to-valve demand frequency response presented in Figure 5.54
shows close correspondence to a first-order system with a dominant pole at around
20 Hz in cascade with the two complex conjugated poles corresponding to the
spool dynamics.
Friction Dampers
155
RMS press vs demand
90
RMS Pressure vs Demand [bar/V]
80
70
60
50
simulated
40
30
20
experimental
10
0
-1
10
0
10
1
10
Frequency [Hz]
2
10
3
10
Fig. 5.54. Comparison of experimental and predicted pressure-to-voltage demand frequency
response [copyright Elsevier (2003), reproduced from Guglielmino E, Edge KA, Controlled
friction damper for vehicle applications, Control Engineering Practice, Vol. 12, N 4, pp
431–443, used by permission]
In the experimentation the bandwidth was substantially penalised by the presence
of air bubbles formed in the oil because of the low system pressure. A simulation
using standard values for the valve parameters and considering the actual volume
formed by the valve with the load port blanked off results in an extremely large
bandwidth, which is absolutely unrealistic when compared to the experimental
results. Such a pressure response would have been even faster than the spool
dynamic response. In such a situation, the pressure response would have been close
to second order with its dynamics dictated by the two complex poles of the spool
mechanical response, which would become dominant. In order to match the
simulation with the experimental results, it is necessary to include in the dynamic
model the presence of air, whose effect is dynamically manifested through a drastic
reduction in the bulk modulus.
Its effect can be accounted for by a dramatic reduction in the effective bulk
modulus of the fluid. A very low bulk modulus of around 5 × 107 N/m2 (compared
to 1.6 × 109 N/m2 for the oil alone) leads to reasonable agreement between the
simulated and experimental responses. The actual value is probably higher as the
volume (with the port blanked off) is not precisely known (possibly
underestimated). An error in the estimation of the volume would result in a bulk
modulus being a few times higher. Such low values for the bulk modulus are
physically possible in the presence of air (McCloy and Martin, 1980). The
equivalent bulk modulus is expressed by the following equation:
156
Semi-active Suspension Control
V V
1
1
1
,
=
+ air tot +
Beq Bhose
Bair
Boil
(5.48)
where Vair and Vtot are, respectively, the volume of the free air and the total volume.
If Bhose= 8 × 103 bar and Boil= 16 × 103 bar, while the ratio Vair over Vtot is equal to
2 × 10-2, which yields an equivalent bulk modulus of around 5 × 102 bar.
In Figure 5.55 the impact of an increase in volume, the other critical parameter,
is investigated. The volume has been increased by an order of magnitude and this
produces a reduction of the bandwidth by about a decade. This is reasonable,
recognising that in a linear approximation the hydraulic time constant is directly
proportional to the volume.
RMS press vs demand
90
RMS Pressure vs Demand [bar/V]
80
70
V=1e-5
60
V=4e-4
50
40
V=9e-4
30
20
experimental
10
0
-1
10
0
10
1
10
Frequency [Hz]
2
10
3
10
Fig. 5.55. Comparison of experimental and predicted pressure-to-voltage demand frequency
response varying volume [copyright Elsevier (2003), reproduced from Guglielmino E, Edge
KA, Controlled friction damper for vehicle applications, Control Engineering Practice, Vol.
12, N 4, pp 431–443, used by permission]
5.11 Performance Enhancement of the Friction Damper System
Previous experimental work showed that the theoretically predicted system
performance does not entirely reflect actual system behaviour because of both the
residual back-pressure, which produces an undesired constant-amplitude damping
force, and reduced hydraulic system bandwidth due to presence of free air within
the hydraulic oil. This can motivate the research on the improvement of the friction
damper performance, outlined in the last part of this chapter.
Friction Dampers
157
The first issue can be addressed by spring-preloading the friction damper so as
to compensate for the constant force caused by the residual back-pressure. The
second issue can be tackled by closing a local pressure control loop around the
control valve (at the price of an additional pressure transducer in the loop). In this
section bond graphs (Karnopp et al., 1990; Gawthrop and Smith, 1996) are
introduced and it is shown how to employ them as a tool for modelling and
designing the modifications mentioned above.
A residual back-pressure is likely to be present in the system when the
controller requires flow to be returned to tank. This is due to the pressure losses in
the pipework. A certain amount of back-pressure is always present and its negative
impact is higher at low supply pressures. Back-pressure produces a noncontrollable constant friction force superimposed on the time-varying controlled
damping action. A possible solution is to insert a preloaded spring into the friction
damper actuation system. In the case of the design described above, a tension
spring can be conveniently placed between the friction actuators. The spring force
would ideally compensate the residual back-pressure, but on the other hand another
dynamic element is introduced into the system, which can affect controller
performance.
The second problem is the presence of free air in the hydraulic oil, which
produces a significant reduction in the value of the bulk modulus. This causes a
bandwidth reduction (bandwidth is directly proportional to bulk modulus). It was
previously shown that the problem occurred when using a relatively sophisticated
valve.
The bulk modulus of a hydraulic oil is generally accepted to be 16,000 bar.
However experimental work showed that this value is not attainable in the system
under normal working conditions (with a relatively low supply pressure). The
improved bandwidth attained by increasing the supply pressure is just acceptable
for the application. It was shown that, in order to match simulation with
experimental results, the bulk modulus has to be reduced to a value as low as 500
bar.
This experimental investigation showed that the main cause of the very low
bandwidth was the presence of air bubbles at low supply pressure. Air release can
occur due to the throttling action of the valve and its presence in small quantities
can spoil the performance of the most sophisticated valves. If the problem cannot
be solved by an appropriate re-design of the hydraulic system, then a solution can
be sought at control level, by closing a pressure control loop to counteract this
effect.
5.11.1 Damper Design Modification
Figure 5.56 shows a possible damper design modification to insert a spring
between the two friction actuators.
158
Semi-active Suspension Control
Fig. 5.56. Schematic of the original and modified friction damper [copyright ASME (2004),
reproduced from Ngwompo RF, Guglielmino E, Edge KA, Performance enhancement of a
friction damper system using bond graphs, Proc ASME ESDA 2004, Manchester, UK, used
by permission]
In this optimisation study, a two-degree-of-freedom mass–spring–damper system
modelling a quarter car is considered. If the relative displacement is defined as
x = x1 − x2 , the quarter car together with the friction damper is described by a
system of two second-order non-linear ordinary differential equations as follows
analogous to (2.7a) and (2.7b):
m1 x1 = − μFn sgn x − 2ξω1m1 x − k s x ,
m2 x2 = μFn sgn x + 2ξω1m1 x + k s x − k t (x2 − y ) ,
(5.49a)
(5.49b)
where
Fn = AC PA (x, x , x1 ) − k p d 0 ,
(5.50)
k p being the preloaded spring stiffness and d 0 the spring length (i.e., distance
between friction pads).
The preloaded spring to compensate residual back-pressure is sized such that:
k p d 0 = AC PT
(5.51)
The bond graph model of the system (including the modified design of the
friction damper as shown in Figure 5.56) is represented in Figure 5.57.
Friction Dampers
body_position
Algorithm
control
Ø
x
R
MR
MR
1
MR
Orif_Q3
1
0
C
Sf
0
1
0
Pump
Pa
TF
1
TF_hydrau_to_meca
R
v_pad
C
0
I
Force_body
mass_body
C
F_spr_damp
1
v_w
I
mass_wheel
I
0
mass_pad
F_w
C
spring_wheel
C
pad_friction preloaded_spring
Pipe_volume
v_body
xdot
suspension_spring
Q1
K
visquous_damper
friction_damper
Ps
1
xdot
R
relief_valve Orif_Q1
body_acceleration
Ø
x
159
Ø
1
v_road
road_p
Actuator_chamber
MSf
road_v
road_profile
Design modification
Fig. 5.57. Bond graph model of the modified friction damper [copyright ASME (2004),
reproduced from Ngwompo RF, Guglielmino E, Edge KA, Performance enhancement of a
friction damper system using bond graphs, Proc ASME ESDA 2004, Manchester, UK, used
by permission]
The dynamics of the preloaded spring and friction pad are taken into account and
represented in terms of bond graphs as a TF-element to achieve the conversion
from the hydraulic domain (pressure in the actuator chamber) to the mechanical
domain (force on the friction pad) connected via a 1-junction (common velocity) to
I, C and R elements representing, respectively, the mass of the pad, the preloaded
spring and the friction between the pad and the housing. The pump is modelled as a
source of flow; each volume in the hydraulic circuit is represented by a C-element
and the valve orifices as well as the friction damper are modelled as modulated
resistive MR-elements, respectively controlled by a feedback signal and the
actuator chamber pressure.
5.11.2 Hydraulic Drive Optimisation
The poor valve dynamics can be tackled by closing a local pressure control loop.
Working on the assumption that the system can be approximated by a first-order
lag, the simplest controller possible that allows to achieve an increment in
bandwidth is a proportional controller. The controller as represented by the block
diagram in Figure 5.58 is quite simple (a proportional controller).
160
Semi-active Suspension Control
kc1
k q1
iref
+
⊗
K
-
i
s2
ωn 2
+2
ξv
z
s +1
ωn
⊗
+
+
+
⊗
-
k q3
B
Vt S
pA
+
⊗
+
kc3
Ki
Fig. 5.58. Valve control loop block diagram [copyright ASME (2004), reproduced from
Ngwompo RF, Guglielmino E, Edge KA, Performance enhancement of a friction damper
system using bond graphs, Proc ASME ESDA 2004, Manchester, UK, used by permission]
The purpose of this inner control loop is only to increase valve bandwidth. An
external variable structure control loop (or some other suitable controller according
to the application) is then used for the damper control.
The effect of a reduction of bulk modulus on the system bandwidth is shown by
the frequency response Bode diagram in Figure 5.59 (obtained with a simplified
linear analysis) where a decrease of the bulk modulus from 16,000 bar to 160 bar
causes a reduction of the bandwidth from 120 to 30 Hz.
Fig. 5.59. Pressure-to-voltage frequency response of the valve actuation system for different
bulk moduli [copyright ASME (2004), reproduced from Ngwompo RF, Guglielmino E,
Edge KA, Performance enhancement of a friction damper system using bond graphs, Proc
ASME ESDA 2004, Manchester, UK, used by permission]
Friction Dampers
161
Figure 5.60 shows the frequency response Bode diagrams with and without the
controller for the case where the bulk modulus is reduced to 160 bar.
Fig. 5.60. Pressure-to-voltage frequency response of the valve actuation system in open and
closed loop for B = 160 bar [copyright ASME (2004), reproduced from Ngwompo RF,
Guglielmino E, Edge KA, Performance enhancement of a friction damper system using
bond graphs, ASME ESDA 2004, Manchester, UK, used by permission]
As expected, the bandwidth is increased at the price of a reduction in the gain. This
is a well-known trade-off of a linear proportional controller. This implies that, in
order to achieve a reasonable bandwidth, the closed-loop supply pressure has to be
increased with respect to the open-loop supply pressure, with a subsequent increase
in the power demand. On the other hand, problems arising from the presence of air
and poor bandwidth are more common in systems working at low supply pressure.
Therefore, the increase in the supply pressure should not necessarily produce a
large increase in the absolute value of power demand. Furthermore this particular
system is a pressure control system, not a flow control system, and hence flows can
be maintained at low levels.
5.11.3 Friction Damper Controller Enhancement
The following figures show the key simulation results with the enhanced friction
damper using balanc logic.
Figures 5.61 and 5.62 show the RMS chassis acceleration and relative
displacement versus frequency in the range 0.5–5 Hz. The response with a
conventional viscous damper is compared with the semi-active response with ideal
and reduced bulk modulus. The reduction in bulk modulus causes a worsening of
the performance, as expected.
Semi-active Suspension Control
RMS body acceleration
[m/s2]
162
7,00
passive
6,00
5,00
semi-active B =
16,000 bar
4,00
3,00
semi-active B =
160 bar
2,00
1,00
0,00
0
2
4
6
Frequency [Hz]
RMS suspension rel. displacement
[m]
Fig. 5.61. Comparison of passive and semi-active RMS body acceleration (B = 16,000 bar
and B = 160 bar) [copyright ASME (2004), reproduced from Ngwompo RF, Guglielmino E,
Edge KA, Performance enhancement of a friction damper system using bond graphs, Proc
ASME ESDA 2004, Manchester, UK, used by permission]
0,1
0,09
0,08
0,07
0,06
0,05
0,04
0,03
0,02
0,01
0
Rel. displ. B =
16,000 bar
passive
0
2
4
6
Frequency [Hz]
Fig. 5.62. Comparison of passive and semi-active RMS suspension relative displacement
[copyright ASME (2004), reproduced from Ngwompo RF, Guglielmino E, Edge KA,
Performance enhancement of a friction damper system using bond graphs, Proc ASME
ESDA 2004, Manchester, UK, used by permission]
Figure 5.63 shows the impact of an imprecise residual back-pressure on the
performance of the system.
RMS body acceleration [m/s2]
Friction Dampers
163
3,5
back pressure (PT
= 2 bar)
3
2,5
undercompensatio
n (PT = 1 bar)
2
overcompensation
(PT = 3 bar)
1,5
1
perfect
compensation
0,5
0
0
2
4
6
Frequency [Hz]
Fig. 5.63. Effect of back-pressure on the system performance [copyright ASME (2004),
reproduced from Ngwompo RF, Guglielmino E, Edge KA, Performance enhancement of a
friction damper system using bond graphs, Proc ASME ESDA 2004, Manchester, UK, used
by permission]
Assuming a residual back-pressure PT = 2 bar, various compensation scenarios are
considered: (a) perfect compensation when the spring is sized according to
Equation 5.51, (b) under-compensation when the stiffness of the spring is such that
k p d 0 < AC PT , and (c) the over-compensation case. In case (a) the system behaves
as if there is no back-pressure, while in case (b), there is still a residual backpressure k p d 0 − AC PT causing the pad to stick and apply a residual friction
(
)
damping force. The over-compensation situation corresponds to the case where the
friction pad is not in contact with the plate. This leads to an underdamped response,
because no controlled frictional damping is present and the only damping present
in the system is the residual viscous damping of the vehicle, which is very low.
6
Magnetorheological Dampers
6.1 Introduction
Magnetorheological dampers (MRD) are among the most promising semi-active
devices used nowadays in automotive engineering. As mentioned in Chapter 1 a
number of vehicles currently employs them. The key feauture of an MRD is the
magnetorheological oil, a particular type of oil whose rheological properties can be
altered by applying a magnetic field; by controlling the field (i.e., the current in a
solenoid) variable damping can be produced.
The aim of this chapter is to provide insights into MRD modeling,
characteristic identification and design. In the first section an overview of the basic
properties of the MR fluids and the fluid behaviour under different flow regimes is
presented. A design procedure for an MRD is then illustrated. The chapter closes
with a presentation of the phenomenological models for MR dampers and their
identification.
6.2 Magnetorheological Fluids
Magnetorheological fluids belong to a family of fluids whose properties depend on
the strength of an electrical or magnetic field. This family includes also ferrofluids,
electrorheological fluids (ER) as well as magnetorheological fluids (MR).
Ferrofluids (Melzner et al., 2001) are colloidal suspensions of magnetic
particles smaller than 10 nm (usually made of magnetite) floating in an appropriate
carrier liquid such as water, hydrocarbons, esters etc. In the presence of an external
magnetic field the viscosity of these fluids increases, the increase of the viscosity
being proportional to the local intensity of the magnetic field.
Electrorheological and magnetorheological fluids are non-colloidal suspensions
of particles having a size of the order of a few microns (5–10 μm) where Brownian
166
Semi-active Suspension Control
motion is not present (in contrast with ferrofluids). The discovery of these fluids
dates back to the 1940s (Winslow, 1947 and 1949; Rabinow, 1948).
ER and MR fluid respond to, respectively, an applied electric or magnetic fields
with a dramatic change in their rheological behaviour. The main characteristic of
these fluids is their ability to change reversibly from free-flowing, linear viscous
liquids, to semi-solids with the yield strength swiftly and continuously controllable
(milliseconds scale dynamics) when exposed to either an electric or magnetic field.
In the absence of an applied field, the ER and MR fluids exhibit Newtonian-like
behaviour. When an external electric or magnetic field is applied, this results in a
polarisation of the suspended particles which therefore move so as to reduce the
stored energy of the ensemble. A minimum-energy configuration consists of
particle chains aligned in the direction of the external field. These chain-like
structures modify the motion of the fluid, thereby changing its rheological
properties through an increase of the yield stress in the direction perpendicular to
the applied field. The mechanical energy needed to yield these chain-like structures
increases as the applied field increases.
Since yield stress can be varied reversibly by controlling the field strength, both
ER and MR fluids are potentially an excellent means to dynamically interface
mechanical and electrical systems.
The rheological behaviour of such fluids can be separated into two distinct preand post-yield regimes. In the pre-yield regime, both fluids behave like elastic
solids as a result of the chain stretching with some occasional ruptures, while in the
post-yield there is equilibrium between chain ruptures and chain reformations and
the fluids behave like a viscous Newtonian fluid.
Thus the behaviour of MR and ER fluids can be modelled as a Bingham plastic
having variable yield strength (Jolly et al., 1998):
τ = τ y (field )sgn (γ ) + ηγ , τ ≥ τ y ,
(6.1)
where field is the magnitude of either the electric or the magnetic field, γ is the
fluid shear rate, η is the plastic viscosity (at zero field) and τ y is the yield stress.
Below the yield stress (at strains of the order of 10 −3 ) the material behaves
viscoelastically:
τ = Gγ, τ < τ y ,
(6.2)
where G is the complex material modulus. Note that the complex modulus is also
field dependent (Weiss et al., 1994).
MR fluids were initially less investigated than ER fluids but in recent years MR
fluids have been extensively studied owing to a number of interesting properties,
essentially linked to their robustness for real-life engineering applications. Table
6.1 (Carlson et al., 1996) lists some properties of typical ER and MR fluids.
Magnetorheological Dampers
167
Table 6.1. Typical properties of ER and MR fluids [copyright Lord Corp. (www.lord.com),
used by permission]
Property
ER fluid
MR fluid
Response time
Milliseconds
Milliseconds
Plastic viscosity, η
(at 25ºC)
0.2–0.3 Pa·s
0.2–0.3 Pa·s
Operable temperature
range
+10 to +90ºC (ionic, DC)
–40 to +150ºC
–25 to +125ºC (non-ionic,
AC)
Max. yield stress, τ y
2–5 kPa
50–100 kPa
(at 3–5 kV/mm)
(at 150–250 kA/m)
Maximum field
~ 4 kV/mm
~ 250 kA/m
Power supply
(typical)
2–5 kV
2–25 V
1–10 mA
1–2 A
(2–50 W)
-7
-8
(2–50 W)
-10
–10-11 s/Pa
η / τ y2
10 –10 s/Pa
Density
1 × 103 – 2 × 103 kg/m3
3 × 103 – 4 × 103 kg/m3
Max. energy density
103 J/m3
105 J/m3
Stability
Cannot tolerate impurities
Unaffected by most impurities
10
6.3 MR Fluid Devices
6.3.1 Basic Operating Modes
Virtually all devices that use MR fluids can be classified as operating in three flow
regimes:
• Pressure-driven flow mode with either fixed poles or valve mode (Figure
6.1):
Fig. 6.1. Pressure-driven flow mode [copyright Lord Corp. (www.lord.com), used by
permission]
168
Semi-active Suspension Control
•
This mode corresponds to the Hagen–Poiseuille flow. Devices that operate
in this mode include dampers, shock absorbers, valves etc.
Direct shear mode, with relatively movable poles, translating or rotating
perpendicular to the field (Figure 6.2).
The flow is driven by virtue of attraction forces acting on the fluid between
the magnetisable particles. This mode corresponds to the Couette flow.
Devices operating in this mode include clutches, brakes, locking devices,
dampers for small displacements and medium-frequency applications.
Fig. 6.2. Direct shear mode [copyright Lord Corp. (www.lord.com), used by permission]
•
Squeeze-film mode with relatively movable poles top-down, in the field
direction.
Fig. 6.3. Squeeze-film mode [copyright Lord Corp. (www.lord.com), used by permission]
Devices operating in this mode include dampers used in high-force and
low-motion applications.
6.3.2 Flow Simulation
6.3.2.1 Pressure-driven Flow Mode with Either Pole Fixed
The pressure-driven flow mode can provide large-magnitude damping forces and
large displacements. Hence this mode is suitable for automotive suspension
applications.
Magnetorheological Dampers
a
169
b
Fig. 6.4. Mode of flow through an MR fluid damper working in pressure-driven flow mode;
(a) pre-yield flow; (b) post-yield flow [copyright Lord Corp. (www.lord.com), used by
permission]
In Figure 6.4 the mode of flow through an MR damper working in pressure-driven
flow mode is shown.
To capture the damper force–velocity behaviour for an MR damper working in
pressure-driven flow mode an axisymmetric model for the flow through an annular
duct can be developed (Constantinescu, 1995). This model is based on the Navier–
Stokes equations for the Hagen–Poiseuille flow.
For incompressible viscous fluids the governing equations are:
⎧∇ ⋅V = 0 continuity equation ,
⎪
⎨ DV
⎪ ρ Dt = ρf − ∇p + ∇τ ij Navier - Stokes equation ,
⎩
(6.3)
where V is the velocity vector, ρ MR fluid density, t is the time, f f is the
external force vector, p the pressure and τ ij is the viscous stress tensor. For
axisymmetric flow u x = u (r ), u r = 0, uω = 0 , hence Equation 6.3 becomes:
⎧ ∂u x
⎪ ∂x = 0 continuity equation ,
⎪
(6.4)
⎨
⎪ ρ ∂u x = ρf x − ∂p + ∂ ⎛⎜ 2η ∂u x + λ∇ ⋅V ⎞⎟ + 1 ∂ (rτ x ,r ) Navier - Stokes eq. ,
∂x ∂x ⎝
∂x
⎠ r ∂r
⎩⎪ ∂t
where ρ and η are constant, f x = 0 , r is the radial co-ordinate, x is the
∂p
longitudinal co-ordinate and
the pressure gradient.
∂x
To analyse the quasi-static motion of the flow inside the damper, the fluid
⎛ ∂u
⎞
inertia can be neglected ⎜ x = 0 ⎟ , and hence Equation 6.4 can be reduced to
⎝ ∂t
⎠
170
Semi-active Suspension Control
−
∂p ∂τ x ,r (r ) τ x ,r (r )
+
+
=0
∂x
∂r
r
(6.5)
i.e.,
∂τ x ,r (r ) τ x ,r (r ) ∂p
+
=
,
∂r
r
∂x
(6.6)
which has solution
τ x ,r =
C
1 dp(x )
r+ 1 ,
2 dx
r
(6.7)
where C1 is a constant which can be evaluated by the boundary conditions.
To describe the MR fluid field-dependent characteristics and shear effects the
Bingham viscoplastic model is employed whose governing equation is:
∂u (r )
⎛ ∂u x (r ) ⎞
⎟ +η x .
∂r
⎝ ∂r ⎠
τ x ,r (r ) = τ y (H , r )sgn⎜
(6.8)
Fig. 6.5. Typical velocity profile along with a typical shear stress diagram for viscoplastic
fluid flow through an annular gap for pressure-driven flow mode [copyright John Wiley &
Sons Limited (1998), reproduced with minor modifications from Spencer BF, Yang G,
Carlson JD and Sain MK, Smart dampers for seismic protection of structures: a full-scale
study, in Proceedings of the Second World Conference on Structural Control edited by
Kobori T, Inoue Y, Seto K, Iemura H and Nichitani A, reproduced with permission]
A typical velocity profile along with a typical shear stress diagram for a
viscoplastic fluid (MR fluid) flow through an annular gap is depicted in Figure 6.5
(Constantinescu, 1995; Yang et al., 2002).
In regions I and II, the shear stress exceeds the yield stress and is given by
Equation 6.8. The velocity profile results from (6.7) and (6.8).
• region I with boundary condition at r = R1 , u x (r ) = v0
Magnetorheological Dampers
u x (r ) =
(
171
r
)
C
1 dp 2
r 1
r − R12 + 1 ln −
τ y (r )dr − v0 ,
4η dx
η R1 η R
∫
1
(6.9)
for R1 ≤ r ≤ r1
•
region II with the boundary condition at r = R2 u x (r ) = 0
u x (r ) = −
(
)
R2
C
R
1 dp 2
1
R2 − r 2 − 1 ln 2 −
τ y (r )dr ,
4η dx
r η r
η
∫
(6.10)
for r2 ≤ r ≤ R2 .
In region C (the plug flow region) the shear stress is lower than the yield stress
so that no shear flow occurs. In this region the yield stress is assumed to be a
monotonic function of the radius r. In this region u x (r ) = const. , u x (r1 ) = u x (r2 ) ,
τ x ,r (r1 ) = τ y (r1 ) and τ x ,r (r2 ) = −τ y (r2 ) .
Since the yield stress τ y in axisymmetric flow is related to r, due to the radial
distribution of the magnetic field in the gap and, in general, the dimension of the
gap R2 − R1 is small compared with the magnetic pole radius R1 , the variation of
the yield stress with r in the gap can be neglected. Hence:
τ y (r1 ) = τ y (r2 ) = τ y (H ) .
(6.11)
From (6.7) and (6.11):
C1 =
r1r2
τ y (H )
r2 − r1
(6.12)
and, from Equation 6.6:
dp
(r2 − r1 ) = 2τ y (H ) .
dx
(6.13)
Therefore:
(r2 − r1 ) =
2τ y (H )
dp
dx
(6.14)
In (6.13) and (6.14) τ y = τ y (H ) is the yield stress, H is the value of the applied
magnetic field and (r2 − r1 ) is the plug thickness.
172
Semi-active Suspension Control
The plug thickness varies with the fluid yield stress τ y (and with the intensity
of the magnetic field). Flow can only be established when (r2 − r1 ) < ( R2 − R1 ) i.e.,
the plug flow needs to be within the gap.
With the conditions (6.11) and (6.12), the equations (6.9) and (6.10) yield
u x (r1 ) =
(
)
C
r
1 dp 2
1
r1 − R12 + 1 ln 1 − τ y (r1 − R1 ) − v0 ,
4η dx
η R1 η
u x (r2 ) = −
(
)
C
R
1 dp 2
1
R2 − r22 − 1 ln 2 − τ y (R2 − r2 ).
4η dx
η
r2 η
(6.15)
In the region C, from (6.15) with the condition (6.11), u x (r ) = const. ,
u x (r1 ) = u x (r2 ) yields
rr
rR
dp ⎡
1
⎤
hRm − (r2 − r1 )(r2 + r1 )⎥ + 2τ y 1 2 ln 1 2 +
dx ⎢⎣
2
r
−
r
R1r2
⎦
2
1
(6.16)
+ 2τ y (2 Rm − r1 − r2 ) − 2ηv0 = 0 ,
where
R2 − R1 = h (gap thickness )
R2 + R1 = 2 Rm (gap average radius )
r1 − R1 = h1 (region I thickness )
(6.17)
r2 − R1 = h2 (region I + region C thickness )
r2 − r1 = h2 − h1 = hp (plug flow thickness)
The volumetric flow rate Q is given by
R2
Q = 2π ∫ ru x (r )dr .
(6.18)
R1
Hence
R
r
⎡r
⎤
I
II
Q = Q + Q + Q = 2π ⎢ ru x (r )dr + ru x (r )dr + ru xC (r )dr ⎥ .
⎢⎣ R
⎥⎦
r
r
1
I
II
C
2
2
∫
∫
∫
1
2
1
(6.19)
Substituting (6.9) and (6.10) into (6.19) and considering the boundary
conditions as well as the condition (6.11) yields
Magnetorheological Dampers
R
⎡ r du I (r )
du II ⎤
Q = π R12 v0 − π ⎢ r 2 x dr + r 2 x dr ⎥
dr
dr
⎢⎣ R
⎥⎦
r
1
173
2
∫
∫
1
2
(6.20)
Hence
Q = π R12 v0 −
[
[
]
π ⎧d p
2hRm (h 2 + 2 R1 R2 ) − (r24 − r14 ) +
⎨
8η ⎩ d x
8τ y
2
2
2
(
4r r
+ 1 2 2 gRm − r − r
r2 − r1
2
1
)]
[ (
)]
⎫
2 Rm h + R1 R2 − r + r ⎬ .
+
3
⎭
) (
3
2
3
1
(6.21)
dp
and the plug thickness can be obtained numerically by using
dx
Equations (6.13) and (6.14) or (6.16) and (6.21).
If a new co-ordinate system O1 x1 y1 is introduced with the origin O1 in the
surface of the piston (fixed point), where O1 x1 is the axis oriented in direction of
u x , O1 y1 the axis oriented in direction of the Or axis, then:
The term
∂τ x ,y
=
∂y
∂p
.
∂x
(6.22)
From Equation 6.22:
τ x ,y =
dp
y + D1 .
dy
(6.23)
For region C, u xC = const. Hence
d u xI
dy
y = h1
d u xII
=
dy
y = h2
d u xC
=
dy
y = h2
=0.
(6.24)
y = h1
Therefore
h2 − h1 = −
2τ y
.
dp
dx
Equation 6.8 becomes
(6.25)
174
Semi-active Suspension Control
⎛ d ux
⎝ dy
τ x , y = τ y ( y , H )sgn ⎜⎜
⎞
du
⎟⎟ + η x .
dy
⎠
(6.26)
Hence from (6.23) and (6.26)
⎛ du
d ux 1 d p
1⎡
=
y + ⎢ D1 ± τ y sgn ⎜⎜ x
dy η dx
η ⎣⎢
⎝ dy
⎞⎤
⎟⎟⎥ .
⎠⎦⎥
(6.27)
The resulting velocity profiles in the three regions are:
• region I with the boundary condition at y = 0 , u x ( y ) = v0 and at
y = h1 , u x ( y ) = const. = u xC
u xI ( y ) =
v − u xC
1 dp
y ( y − h1 ) − 0
y + v0
2η dx
h1
(6.28)
for 0 ≤ y ≤ h1
•
region C:
u xC = u xI (h1 ) = u xII (h2 ) = −
•
1 dp
(h − h2 )2 = const.
2η dx
(6.29)
region II with the boundary condition at y = h2 , u x ( y ) = u xC and at
y = h , u x (h ) = 0
u xII ( y ) = −
1 dp
(h − y )( y − h2 ) + u xC h − y
2η dx
h − h2
(6.30)
for h2 ≤ y ≤ h
In (6.28), (6.29) and (6.30) the yield stress in the gap is taken to be constant.
d ux
= 0 ( u xC = const. ) and from (6.28) and (6.30)
In region C,
dy
v − uC
dp
= 2η 0 2 x
dx
h1
and
(6.31)
Magnetorheological Dampers
175
u xC
dp
,
= −2η
dx
(h − h2 )2
(6.32)
2η v0
dp
=−
dx
(h − h2 )2 − h12
(6.33)
i.e.,
or
η v0
h τy
+
+
2 dp h dp + 2τ
y
dx
dx
h1 =
(6.34)
and
h2 =
η v0
h τy
.
−
+
d
d
p
p
2
+ 2τ y
h
dx
dx
(6.35)
The fluid flux in the x-axis direction is:
h
h1
h2
0
0
h1
h
q = u x ( y )dy = u xI ( y )dy + u xC ( y )dy + u xII ( y )dy ,
∫
∫
∫
∫
(6.36)
h2
i.e.,
q=−
[
]
v
uC
1 dp 3
3
h1 + (h − h2 ) + 0 h1 + x (h + h2 − h1 ) .
12η d x
2
2
(6.37)
The volumetric flow rate Q is
Q = Ap vp = 2π Rm q ,
(6.38)
where Ap is the cross-sectional area of the piston head and vp = v0 is the piston
head velocity.
By considering Equation 6.38 and replacing u xC , h1 and h2 from (6.25), (6.29),
dp
.
dx
Introducing dimensionless variables (Yang et al., 2002):
(6.34) and (6.35) an expression can be obtained for
176
Semi-active Suspension Control
N1 = −
N2 = −
N3 =
π Rm hv0
Q
=−
π Rm h
Ap
,
π Rm h 3 d p
π Rm h 3 d p
=−
,
6ηQ d x
6ηAp vp d x
(6.39)
π Rm h 2τ y π Rm h 2τ y
=
,
6ηQ
6ηAp vp
The following equation can be obtained:
[
]
3(N 2 − 2 N 3 ) N 23 − (1 + 3 N 3 − N1 )N 2 + 4 N 3 + N12 N 22 N 3 = 0 ;
2
when
(6.40)
2
N1 > 3(N 2 − 2 N 3 ) / N 2 , the pressure gradient is independent of the
dimensionless yield stress N 3 . Therefore a controllable yield stress cannot affect
the resisting force of the MR damper.
Equation 6.40 cannot be solved analytically. However for 0 < N 3 < 1000 and
−0.5 < N1 < 0 (i.e., flow in the opposite direction to the piston velocity) an
approximate solution can be found for the pressure gradient:
N 2 (N1 , N 3 ) = 1 + 2.07 N 3 − N1 +
N3
.
1 + 0.4 N 3
(6.41)
This solution encompasses most practical design solutions with a maximum error of
4%.
The pressure drop in a device working in pressure-driven flow mode is:
Δp =
dp
L,
dx
(6.42)
where L is the length of magnetic pole.
From (6.41) the pressure drop is
Δp = Δpη + Δpτ =
(
6vp Ap + π Rm h
π Rm h
3
)ηL + c τ (H )L
h
y
(6.43)
i.e., the sum of a viscous component Δpη and a field-dependent induced yield
stress component Δpτ .
In Equation 6.43 the parameter c is a function of the flow velocity profile. This
parameter can be expressed as (Jolly et al., 1998)
Magnetorheological Dampers
c = 2.07 +
⎛
12ηAp vp
1
= ⎜ 2.07 +
1 + 0.4 N 3 ⎜⎝
12ηAp vp + 0.4 Ag hτ y
⎞
⎟.
⎟
⎠
177
(6.44)
Its value ranges from a minimum value of 2 (for Δpτ / Δpη < 1 ) to a maximum
value of 3 (for Δpτ / Δpη > 100 ).
Observing that 2π Rm h = Ag is the gap area, from (6.43) it follows that
Δpη =
12ηAR vp
m
Ag h 2
where AR = π Rm2 = Ap +
(6.45)
L
Ag
.
2
Consequently, the force developed by a damper working in pressure-driven
flow mode is:
m
F = ΔpAp = Δp η Ap + Δp τ Ap = Fη + Fτ .
(6.46)
Hence
Fη =
12ηARm vp
Ag h
2
Ap L and Fτ =
cτ y (H )Ap
h
(6.47)
L
Defining the control ratio λ (for pressure-driven flow mode) as λ =
from (6.43) and (6.45), with Wm = QΔpτ and k =
12 AR
m
c 2 Ap
Δpτ
and,
Δpη
= const. the equation
defining the minimum active fluid volume can be obtained as:
⎛η
V = k⎜ 2
⎜τ y
⎝
⎞
⎟ λ Wm .
⎟
⎠
(6.48)
where V = LAg is the active fluid volume necessary to achieve the desired control
ratio λ at a required controllable mechanical power level Wm .
6.3.2.2 Direct Shear Mode with Relatively Movable Poles
The direct shear mode can provide small displacements and is suitable for mediumfrequency applications. In Figure 6.6 the mode of flow through an MR damper
working in direct shear mode is depicted.
178
Semi-active Suspension Control
a
b
Fig. 6.6. Mode of flow through an MR damper working in direct shear mode; (a) pre-yield
flow; (b) post-yield flow [copyright Lord Corp. (www.lord.com), used by permission]
A typical velocity profile along with a typical shear stress diagram for a
viscoplastic fluid (MR fluid) flow through the annular gap is shown in Figure 6.7
(Constantinescu, 1995):
Fig. 6.7. Typical velocity profile along with a typical shear stress diagram for a viscoplastic
fluid (MR fluid) flow through the annular gap in direct shear mode
dp
= 0 , if no external forces ( f x = 0) are present and in
dx
d ux
= 0),
quasi-static flow motion (hence fluid inertia can be neglected, i.e.,
dt
Equation 6.4 becomes:
In the direct shear mode
⎧ ∂u x
⎪ ∂x = 0 ,
⎪
⎨
⎪ ∂ ⎛⎜ 2η ∂u x + λ∇ ⋅V ⎞⎟ + 1 ∂ (rτ x ,r ) = 0 .
⎪⎩ ∂x ⎝
∂x
⎠ r ∂r
(6.49)
Hence
∂
(rτ x ,r ) = 0 or ∂τ x ,r (r ) + τ x ,r (r ) = 0 .
∂r
∂r
r
(6.50)
Magnetorheological Dampers
179
Substituting (6.8) into (6.49) yields
−
∂τ y
+
∂r
∂ ⎛ d ux ⎞ τ y η d ux
= 0.
⎜η
⎟− +
∂r ⎝ d r ⎠ r r d r
Since η = const. and
∂τ y
∂r
(6.51)
=0
d2 ux 1 d ux τ y
+
−
=0
d r 2 r d r ηr
(6.52)
with boundary conditions r = R1 , u x (R1 ) = v0 and r = R2 , u x (R2 ) = 0 , leading to
ux =
τy
η
⎤ ln R2 − ln r
⎡ ln R2 − ln r
− R2 + r ⎥ +
v0
⎢h
⎦ ln R2 − ln R1
⎣ ln R2 − ln R1
(6.53)
and shear stress
τ x ,r =
τ yh
r (ln R2 − ln R1 )
−
ηv0
(6.54)
r (ln R2 − ln R1 )
Considering the small gap-damper piston diameter ratio, the axisymmetric flow
can be approximated as a flow through parallel plates, as shown in Figure 6.2.
Hence it follows that
∂τ x , y
∂y
= 0 or
d2 ux
=0
d y2
Equation 6.55, with boundary conditions y = 0 , u (0) = v0 ,
u x (h ) = 0 , yields
(6.55)
y=h
v
y⎞
⎛
u x ( y ) = ⎜1 − ⎟v0 and τ x , y = −τ y − η 0 .
h
h
⎝
⎠
and
(6.56)
The force developed by a damper working in direct shear flow mode is
F = Fη + Fτ (H ) =
2π Rmη vr L
+ 2π Rm Lτ y (H ) ,
h
(6.57)
180
Semi-active Suspension Control
where L is the length of the magnetic pole and vr is relative velocity of magnetic
pole (if the velocity u x (h ) = 0 then vr = v0 ).
The control ratio λ is defined, for direct shear mode, as λ =
ΔFτ
and, from
ΔFη
Equation 6.57, with Wm = Fτ vr and k = 1 the equation defining the minimum
active fluid volume is:
⎛η
V = k⎜ 2
⎜τ y
⎝
⎞
⎟λWm ,
⎟
⎠
(6.58)
where V = LAg is the active fluid volume necessary to achieve the desired control
ratio λ at a required controllable mechanical power level Wm and the term
η
(present in (6.48) and (6.58)) depends on the type of fluid.
τ y2
6.3.3.3 Squeeze-film Mode
This operational mode is not analysed in this book. It should be noted that in
applications where large forces are required (e.g., civil engineering applications)
the expected damping forces and displacement are considerably large, MR
dampers operating in squeeze mode might be impractical.
6.4 MR Damper Design
After having overviewed the basics of MR fluid dynamics, the focus is now on the
design of an automotive MR damper. The design of a prototype MR damper will
now be described (Figure 6.8). It is essentially composed of a hydraulic cylinder
through which an electromagnetic piston pumps MR fluid.
Magnetorheological Dampers
181
Fig. 6.8. MR damper prototype for experimental studies
The MR damper is a prototype for experimental studies. A design with two piston
rods is chosen; the piston is wrapped in copper wire and forms two electromagnetic
coils. When the current flows in the coils, a magnetic flux in the piston cylinder is
generated and the yield stress value of the MR fluid increases.
The design procedure of the MR damper involves the following steps:
•
•
•
•
determination of input data and choice the design solution
choice of the working MR fluid
determination of the optimal gap size and hydraulic design
magnetic circuit design.
6.4.1 Input Data and Choice of the Design Solution
The input data are defined by a specification and the design solutions are chosen
from operating and limit conditions. In the case described the basic design data are
the following:
•
•
controllable mechanical power level: Wm = 80 W
maximum damper controllable force: Fτ = 2700 N
•
•
maximum displacement of MR damper: c max = ±0.05 m
maximum working frequency: f max = 5 Hz
The design conditions are
•
in
internal diameter of damper cylinder: d cyl
= 0.04 m
•
piston rod diameter: d rod = 0.02 m
•
•
working temperature: T = −20 − 150 o C
compatibility with synthetic rubbers and elastomer
6.4.2 Selection of the Working MR Fluid
In order to choose the most appropriate MR fluid, a criterion to compare the
nominal behaviour of different MR fluids must first be established. Such an
182
Semi-active Suspension Control
indicator is the figure of merit (Jolly et al., 1998). The other parameters to be
considered in the fluid choice are stability, durability, temperature range and
compatibility with the other damper materials.
6.4.2.1 MR Fluid Figures of Merit
Firstly the fluid efficiency is defined as:
W
We
α = m ,
(6.59)
where W = W / V is the power density, Wm = τ y γ the mechanical power density
BH
the electrical power density, γ is the shear strain rate ( γ ≅ vr / h ), B
and We =
2t c
the magnetic field flux density in the fluid, H is the magnetic field intensity in the
fluid and t c is the characteristic time for the establishment of the magnetic field
within the fluid. Hence:
α = 2tcγ
τy
BH
,
(6.60)
where τ y and B are defined at a common H. The following figures of merit can be
defined:
• Figure of merit based on the active fluid volume, defined as
F1 =
•
(6.61)
This figure of merit is inversely proportional to the minimum active fluid
volume V. If F1 increases, the minimum active fluid volume V decreases.
The decrease of V results in a decrease of the damper size and the electrical
power consumption. For a given active fluid volume V, if F1 increases the
desired control ratio λ increases as well.
Figure of merit based on the mass of the active fluid volume, defined as:
F2 =
•
τ y2
.
η
τ y2
.
ηρ
(6.62)
This figure of merit is inversely proportional to the minimum active fluid
mass.
Figure of merit based on the power efficiency, defined as:
Magnetorheological Dampers
F3 =
τy
BH
.
183
(6.63)
The maximisation of this figure of merit results in a minimisation of the
electrical power consumption of the MR damper for a given delivered
mechanical power.
6.4.2.2 Choice of the MR Fluid
For the purpose of evaluating the different figures of merit the Lord Corporation
132DG MR fluid (http://www.lordfulfillment.com/upload/DS7015.pdf) is chosen.
The characteristics of this MR fluid are listed in Table 6.2 and in Figures 6.9, 6.10
and 6.11.
The MR fluid dampers are usually designed such that, in normal conditions, the
MR fluid is magnetically saturated. It is under this condition that the fluid will
generate its maximum yield stress τ y .
Table 6.2. Rheological properties of MR fluid 132DG [copyright Lord Corp.
(www.lord.com), used by permission]
184
Semi-active Suspension Control
Fig. 6.9. Magnetic flux versus magnetic field for MR fluid 132DG [copyright Lord Corp.
(www.lord.com), used by permission]
Fig. 6.10. Yield stress versus magnetic field for MR fluid 132DG [copyright Lord Corp.
(www.lord.com), used by permission]
Fig. 6.11. Viscosity versus shear strain rate for MR fluid 132DG [copyright Lord Corp.
(www.lord.com), used by permission]
Magnetorheological Dampers
185
Choosing the values γ = 140 s-1 (for the shear strain rate) and H = 250 kA/m (for
the magnetic field), the following parameter values can be obtained from Figures
6.9, 6.10 and 6.11:
•
•
the magnetic flux B ≅ 0.83 T (from Figure 6.9)
the yield stress τ y = 44.1 kPa (from Figure 6.10)
•
the off-state plastic viscosity η = 0.25 Pa⋅s (from Figure 6.11)
6.4.3 Determination of the Optimal Gap Size and Hydraulic Design
When selecting an MR damper the dynamic range and the maximum value of the
controllable force are two key parameters.
6.4.3.1 Controllable Force and Dynamic Range
The controllable force Fτ represents the force due to the field-induced yield stress
τy.
The dynamic range D is defined as the ratio between the total damper output
force F and the uncontrollable force Fuc where uncontrollable forces include the
fluid viscosity force Fη and the friction force Ff . Hence
D=
F + Fτ
Fτ
F
.
= uc
= 1+
Fuc
Fuc
Fη + Ff
(6.64)
From (6.64), considering (6.47) yields
D = 1+
cτ y LAp
12ηAR m vp Ap L + Ag h 2 Ff
m
(6.65)
[(d − 2h) − d ]
4
π
= [(d − h ) − d ] and A = π (d
4
where c is defined by (6.44), Ap =
friction force, AR
,
in
cyl
2
π
in
cyl
2
rod
2
g
2
rod
in
cyl
is the piston area, Ff is the
)
− h h . It is worth noticing
that the values of both the friction force Ff and of the magnetic pole length L do
not affect the optimal value of the gap.
To maximise the effectiveness of the MR damper, the controllable range of the
force should be as large as possible. A small gap size will increase the controllable
force range but, when the gap size h is very small the viscous force Fη increases
much faster than the controllable force Fτ and the dynamic range D decreases. If
the gap size h is large, both Fη and Fτ decrease. Hence there exists an optimal gap
size which maximises the dynamic range D.
186
Semi-active Suspension Control
In Figure 6.12 the trend of the dynamic range for the basic design data is
plotted as a function of the gap size.
From Figure 6.12 g = 0.4 × 10 −3 m for D max ≅ 65. With this choice the
dimensional and functional parameters of the hydraulic circuit have been
determined.
Fig. 6.12. Dynamic range D versus gap dimension g [copyright Publishing House of the
Romanian Academy (2004), reproduced from Topics in Applied Mechanics, Vol. II, Ch 5,
edited by Chiroiu L and Sireteanu T, used by permission]
6.4.3.2 Parameters of the Hydraulic Circuit
The damper has a fairly simple geometry in which the outer cylindrical housing is
part of the magnetic circuit. The effective fluid orifice is the entire annular space
between the piston outer diameter and the inside of the damper cylinder housing.
The motion of the piston causes the fluid to flow through the entire annular region.
The damper is double ended (Figure 6.8). This arrangement has the advantage
that a rod-volume compensator does not need to be incorporated into the damper.
The following design steps entail the choice of the remaining damper
dimensions (L and h) to achieve the requirements given above.
With the aforementioned data the key design parameters of the MR damper are
obtained. The parameters are given in Table 6.3. Note that seal and bearing friction
and other parasitical effects may worsen the dynamic range λ .
Magnetorheological Dampers
187
Table 6.3. MR damper parameters
Ap = 8.927 × 10 -4 m 2
Q = 2.678 × 10 −5 m3/s
AR m = 9.175 × 10 -4 m 2
η = 0.25 Pa ⋅ s
Ag = 4.976 × 10 -5 m 2
τ y = 44.1 × 10 3 Pa
h = 4 ×10 −4 m
Wm = 80 W
vp = 3 × 10 −2 m/s
Fτ = 2700 N
γ = 140 s -1
Δpτ = 29.87 × 10 5 Pa
L = 1.2 ×10 −2 m
Δpη = 1.244 × 10 5 Pa
Δp = Δpη + Δpτ = 31.114 × 10 5 Pa
F = Fτ + Fη = 2777.55 N
λ=
Δpτ
= 24
Δpη
c = 2.256
V = 5.82 ×10 −7 m 3
6.4.4 Magnetic Circuit Design
The MR damper magnetic circuit is shown in Figure 6.13. With this design two
magnetic flux paths can be defined. The coil design solution (two identical coils with
one joint pole) allows two functional characteristics for the MR damper.
The flux in the magnetic circuit flows axially through the piston steel core
(whose diameter is d c ) beneath the windings, radially through the piston poles of
length L1 and L2 , through a gap of thickness h where the MR fluid flows, and
axially through the cylinder wall.
The design of a magnetic circuit requires the determination of Lc (length of
coil), d c (inner diameter of coil) and the total number of turns N in the coil.
188
Semi-active Suspension Control
Fig. 6.13. Magnetic circuit geometry [copyright Publishing House of the Romanian
Academy (2004), reproduced from Topics in Applied Mechanics, Vol. II, Ch 5, edited by
Chiroiu L and Sireteanu T, used by permission]
Figure 6.14 shows the equivalent magnetic circuit.
Fig. 6.14. Equivalent magnetic circuit [copyright Publishing House of the Romanian
Academy (2004), reproduced from Topics in Applied Mechanics, Vol. II, Ch 5, edited by
Chiroiu L and Sireteanu T, used by permission]
The design process for sizing the magnetic circuit involves the following steps:
•
•
•
Selection of the operating point of the MR fluid: from consideration of
Figure 6.9, in correspondence with H = 250 kA/m, the magnetic flux
density B = 0.83 T is obtained.
Choice of the steel for the piston and cylinder (for the magnetic circuit). A
low-carbon steel (the carbon content of the steel should be less than 0.15%)
having a high magnetic permeability is desirable.
Choice of the magnetic pole length. For the MR damper prototype a variant
with two equal poles and the middle pole having a length double that of the
others was considered. The middle pole has a length L2 = 6× 10 -3 m while
both the others have a length of L1 = 3 ×10 −3 m (the total length of the pole
is L = 1.2 × 10 −2 m).
Magnetorheological Dampers
189
•
Calculation of the magnetic flux density for the steel core from the
continuity of magnetic flux φgap = φsteel = φ .
•
The effective area of the pole is found to be Af' = 2.41×10−4 m2 , and
Bsteel =
•
•
φ
A0
=
BAf'
= 0.96 T .
A0
From the B versus H magnetic curves for the steel it can be deduced that
H steel = 272.8 A/m .
By using Kirchoff’s law for magnetic circuits the required ampere-turns for
magnetic coil can be determined: NI = ∑ H i Li =124 A-turns.
•
•
Taking I = 2A, yields N= 62.
Determination of the inner coil diameter for a coil with two layers:
d c = 3.82 × 10 −2 m.
•
Determination of the coil length: Lc = 2.48 × 10 −2 m.
•
Calculation of the piston length: Lp = 2 Lc + L = 6.2 × 10 −2 m.
It should be noticed that both the hydraulic circuit design and the magnetic
circuit design involve an iterative calculation.
6.5 MRD Modelling and Characteristics Identification
In this section the modelling of MR dampers is studied. Several methods have been
proposed over the years for obtaining MR damper models and identifying their
parameters. Spencer and co-workers (1997) developed a phenomenological model
that accurately portrays the response of an MR damper in response to cyclic
excitations. This is a modified Bouc–Wen (BW) model governed by ordinary
differential equations. BW-based models in semi-active seismic vibration control
have proven to be easy to use and numerically amenable. Employing a different
approach to Spencer’s work, Kyle et al. (2000) modelled an MR damper in the
form of a Takagi–Sugeno–Kang fuzzy inference system. The training and checking
data of the model were generated using Spencer’s model of MR damper.
Other authors studied the behaviour of the MR damper, emphasising the
difference between the pre-yield viscoelastic region and the post-yield viscous
region as a key aspect of the damper.
Starting from Spencer’s studies, Butz and Stryk (1999) showed that the
difference between using the modified BW model and the Bingham plastic model
in a 2DOF quarter car dynamic system is very small. This justifies in many
applications the interest in a simple description of the model rather than a more
accurate but more complex one.
190
Semi-active Suspension Control
6.5.1 Experimental Data
The damper used for modelling and characteristic identification studies is the
magnetorheological damper model RD 1005-3 produced by Lord Corp.
(http://www.lord.com/Home/MagnetoRheologicalMRFluid/Products/MRDamper/
tabid/3361/Default.aspx). Tests were carried out under the following conditions:
•
•
•
the maximum current value considered was 1.75 A,
the force variation in response to the current variations from 0.02 to 1.75 A
was about 300 N,
the measured internal resistance of the device was 5.2 Ω at 29°C.
Figure 6.15 portrays the RMS force response to sinusoidal loads versus current
values for different frequencies and different excitation amplitudes. It can be
readily observed that this variation is non-linear within the test range.
The force response time history of the MR damper to an imposed cyclic motion
at the frequency of 1.5 Hz and 16 mm amplitude is shown in Figure 6.16 for 0.1 A,
0.2 A, 0.6 A and 1.75 A current values.
The force versus velocity loops reveal some interesting aspects (Figure 6.17).
The effect of the accumulator can be noted, producing an offset in the experimental
data. This effect is more evident for small current values.
Fig. 6.15. RMS force [N] variation versus current values [A] [copyright IMechE (2004),
reproduced from Giuclea M, Sireteanu T, Stancioiu D and Stammers CW, Model parameter
identification for vehicle vibration control with magnetorheological dampers using
computational intelligence methods, Proceedings of the Institution of Mechanical Engineers,
Part I: Journal of Systems and Control Engineering, Publisher: Professional Engineering
Publishing, ISSN 0959-6518, Vol. 218, N 7/2004, pp 569–581, used by permission]
Magnetorheological Dampers
191
Fig. 6.16. Experimentally measured force [N] versus time [s] [copyright IMechE (2004),
reproduced from Giuclea M, Sireteanu T, Stancioiu D and Stammers CW, Model parameter
identification for vehicle vibration control with magnetorheological dampers using
computational intelligence methods, Proceedings of the Institution of Mechanical Engineers,
Part I: Journal of Systems and Control Engineering, Publisher: Professional Engineering
Publishing, ISSN 0959-6518, Vol. 218, N 7/2004, pp 569–581, used by permission]
Fig. 6.17. Experimentally measured force [N] versus velocity [m/s] [copyright IMechE
(2004), reproduced from Giuclea M, Sireteanu T, Stancioiu D and Stammers CW, Model
parameter identification for vehicle vibration control with magnetorheological dampers
using computational intelligence methods, Proceedings of the Institution of Mechanical
Engineers, Part I: Journal of Systems and Control Engineering, Publisher: Professional
Engineering Publishing, ISSN 0959-6518, Vol. 218, N 7/2004, pp 569–581, used by
permission]
192
Semi-active Suspension Control
Considering the large force and large velocity values region of the diagram, it is
interesting to note that those forces vary almost linearly with velocity. In the smallvelocity range, the diagrams display a significant hysteresis. Werely et al. (1998)
emphasised that these are two distinct rheological regions over which dampers
operate: the pre-yield and the post-yield regions. The pre-yield region exhibits a
strong hysteresis typical of a viscoelastic material while the post-yield region
exhibits a rather viscous behaviour.
6.5.2 Parametric Model Simulation
The behaviour of MR fluids is often described as a Bingham plastic model having
a variable yield strength, which depends upon the magnetic field H . At fluid shear
stresses above the field-dependent yield stress τ y (H ) the fluid flow is governed by
the Bingham plastic equation. At fluid stresses below the yield stress the fluid acts
as a viscoelastic material. This behaviour is described by Equation 6.66:
⎧ τ y ( H ) + ηγ
⎩ Gγ
τ =⎨
τ >τ y
τ <τ y ,
(6.66)
where H is the magnetic field, γ is the fluid shear rate and η is the plastic
viscosity (i.e., viscosity at H = 0), G is the complex material modulus (which is
also field dependent).
Based on this model of the rheological behaviour of smart fluids, an idealised
model, known also as Bingham model, was proposed in 1985. This model consists
of a Coulomb friction element placed in parallel with a viscous dashpot. In order to
obtain a better approximation of the experimentally measured data, an elastic
element was added in parallel with these two elements (Figure 6.18). In the model,
the damping force is generated by:
F = f csgn x + c0 x + K0 x + f 0 ,
(6.67)
where c0 is the damping coefficient, f c is the frictional force directly related to
the yield stress, K 0 is the elastic coefficient, f 0 is the offset force, x is the
imposed relative displacement and x its time derivative.
Magnetorheological Dampers
193
Fig. 6.18. Bingham plastic model of an MR damper [copyright IMechE (2004), reproduced
with minor modifications from Giuclea M, Sireteanu T, Stancioiu D and Stammers CW,
Model parameter identification for vehicle vibration control with magnetorheological
dampers using computational intelligence methods, Proceedings of the Institution of
Mechanical Engineers, Part I: Journal of Systems and Control Engineering, Publisher:
Professional Engineering Publishing, ISSN 0959-6518, Vol. 218, N 7/2004, pp 569–581,
used by permission]
Figures 6.19 and 6.20 depict the force versus time and the force versus velocity
diagrams for a current value of 0.8 A (both numerically predicted and
experimentally measured). The agreement between predicted and experimental
data was used as a criterion for fitting the model parameters. The parameters
chosen are f c = 200 N, c0 = 0.7 Ns/mm, K 0 = 0.3 N/mm, f 0 = 0 N.
Fig. 6.19. Force [N] versus time [s], experimental data (dashed) and Bingham plastic model
(solid) [copyright IMechE (2004), reproduced from Giuclea M, Sireteanu T, Stancioiu D
and Stammers CW, Model parameter identification for vehicle vibration control with
magnetorheological dampers using computational intelligence methods, Proceedings of the
Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering,
Publisher: Professional Engineering Publishing, ISSN 0959-6518, Vol. 218, N 7/2004, pp
569–581, used by permission]
194
Semi-active Suspension Control
Fig. 6.20. Force [N] versus velocity [m/s], experimentally measured data (dashed) and
Bingham plastic model (solid) [copyright IMechE (2004), reproduced from Giuclea M,
Sireteanu T, Stancioiu D and Stammers CW, Model parameter identification for vehicle
vibration control with magnetorheological dampers using computational intelligence
methods, Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems
and Control Engineering, Publisher: Professional Engineering Publishing, ISSN 0959-6518,
Vol. 218, N 7/2004, pp 569–581, used by permission]
Although (6.67) is a simple formula, it is difficult to use it in numerical simulations
because of the presence of the discontinuous term sgn x . Such a term needs to be
replaced with a continuous one which approximates it. Thus, the formula (6.67)
becomes:
F = fc
2tan−1 (bx)
π
+ c0 x + K 0 x + f 0 ,
(6.68)
where f c , c0 , K 0 and f 0 have the same physical meaning as in (6.67), and b is a
form factor. Figure 6.21 shows that this substitution does not significantly change
the sinusoidal input response.
Magnetorheological Dampers
195
Fig. 6.21. Force [N] versus velocity [m/s], experimentally measured data (dashed) and
modified Bingham plastic model (solid) [copyright IMechE (2004), reproduced from
Giuclea M, Sireteanu T, Stancioiu D and Stammers CW, Model parameter identification for
vehicle vibration control with magnetorheological dampers using computational intelligence
methods, Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems
and Control Engineering, Publisher: Professional Engineering Publishing, ISSN 0959-6518,
Vol. 218, N 7/2004, pp 569–581, used by permission]
The parameters chosen are f c = 210 N, c0 = 0.65Ns/mm, K 0 = 0.3 N/mm, f 0 = 0 N
and b = 10. Clearly, the main disadvantage of this model is the poor description of
the pre-yield region. However, when the velocity is large, i.e., if the damper works
at rather large force values, this model can be successfully employed.
Another model based on Equation 6.67 is the Bingham viscoplastic model. This
model replaces the friction element by a block (Figure 6.22), which generates a
force governed by the equation
⎧⎪ cpr x
f =⎨
⎪⎩ cpo x + f c
x < v y
x ≥ v y ,
(6.69)
where cpo and cpr are the post-yield and pre-yield damping coefficients and v y is
the yielding velocity. It should be noticed that the parameters included in this
model are not independent.
196
Semi-active Suspension Control
Fig. 6.22. Bingham viscoplastic model of an MR damper [copyright IMechE (2004),
reproduced from Giuclea M, Sireteanu T, Stancioiu D and Stammers CW, Model parameter
identification for vehicle vibration control with magnetorheological dampers using
computational intelligence methods, Proceedings of the Institution of Mechanical Engineers,
Part I: Journal of Systems and Control Engineering, Publisher: Professional Engineering
Publishing, ISSN 0959-6518, Vol. 218, N 7/2004, pp 569–581, used by permission]
The yield force must satisfy the condition:
f y (1 −
cpo
cpr
)=f c.
(6.70)
Therefore, the damping force is defined by the equation
F = f + K0 x + f0 ,
(6.71)
where the terms K 0 and f 0 have the same meaning as in the Bingham equation
(6.67). There is no significant improvement in the model description, except that a
yield velocity appears. As with the Bingham plastic model, this model predicts
large forces accurately.
A model that leads to results similar to the Bingham viscoplastic model is the
delayed Bingham model. This model is similar to the modified Bingham model
except that the velocity is delayed so that hysteresis occurs.
Figure 6.23 shows the time response of the model with a delay of 0.01 s. The
force versus velocity loops (Figure 6.24) show that hysteresis is generated at low
velocities. However the main disadvantage of Bingham-equation-based models is
the poor accuracy in the pre-yield region.
The bi-viscous model (6.69) has also been considered for MR behaviour
description. Although this model can predict well the MR damper response to
sinusoidal excitations, it is a non-smooth model. This makes it difficult to use the
bi-viscous model in random vibration problems. Good results can be obtained
instead using the delayed model.
Magnetorheological Dampers
197
Fig. 6.23. Force [N] versus time [s], experimental data (dashed) and modified Bingham
plastic model (solid) [copyright IMechE (2004), reproduced from Giuclea M, Sireteanu T,
Stancioiu D and Stammers CW, Model parameter identification for vehicle vibration control
with magnetorheological dampers using computational intelligence methods, Proceedings of
the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering,
Publisher: Professional Engineering Publishing, ISSN 0959-6518, Vol. 218, N 7/2004, pp
569–581, used by permission]
Fig. 6.24. Force [N] versus velocity [m/s], experimental data (dashed) and delayed Bingham
plastic model (solid) [copyright IMechE (2004), reproduced from Giuclea M, Sireteanu T,
Stancioiu D and Stammers CW, Model parameter identification for vehicle vibration control
with magnetorheological dampers using computational intelligence methods, Proceedings of
the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering,
Publisher: Professional Engineering Publishing, ISSN 0959-6518, Vol. 218, N 7/2004, pp
569–581, used by permission]
198
Semi-active Suspension Control
A model that can capture a large variety of hysteretic behaviour is the Bouc–Wen
(BW) model, which has been introduced in Chapter 2. A schematic representation
of a BW-based model of an MR damper is shown in Figure 6.25.
Fig. 6.25. Schematic of Bouc–Wen model of an MR damper [copyright IMechE (2004),
reproduced from Giuclea M, Sireteanu T, Stancioiu D and Stammers CW, Model parameter
identification for vehicle vibration control with magnetorheological dampers using
computational intelligence methods, Proceedings of the Institution of Mechanical Engineers,
Part I: Journal of Systems and Control Engineering, Publisher: Professional Engineering
Publishing, ISSN 0959-6518, Vol. 218, N 7/2004, pp 569–581, used by permission]
The model equation is given by:
z = −γ x z z
n −1
n
−ν x z + Ax .
(6.72)
By changing the parameters γ , ν and A , the shape of the evolutionary
variable z can vary from a sinusoidal to a quasi-rectangular function of the time.
When the model is completed with viscous dashpots or springs the system response
can predict a wide range of hysteretic behaviour. The damping force is given by
F = cx + K 0 x + α z + f 0 ,
(6.73)
where the parameter α determines the influence of the model on the final force
value.
Magnetorheological Dampers
199
Fig. 6.26. Force [N] vs. time [s] experimental data (dashed) and Bouc–Wen (solid) model
[copyright IMechE (2004), reproduced from Giuclea M, Sireteanu T, Stancioiu D and
Stammers CW, Model parameter identification for vehicle vibration control with
magnetorheological dampers using computational intelligence methods, Proceedings of the
Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering,
Publisher: Professional Engineering Publishing, ISSN 0959-6518, Vol. 218, N 7/2004, pp
569–581, used by permission]
To determine the parameters that fit the MR damper model response to the
experimentally measured response at a frequency of 1.5 Hz and with a 16 mm
amplitude sinusoidal input, a least-mean-squares nonlinear algorithm was used.
The set of parameters chosen to fit the 0.8 A current value response was n = 2,
γ = 1.2 mm-2, ν = 1 mm-2, A = 15, α = 80 N/mm, c0 = 0.65 Ns/mm, K 0 = 0.3
N/mm, f 0 = 0 N.
A comparison between the experimentally determined data and the BW-based
model response is shown in Figure 6.26. The predicted response fits the measured
data well, even in the pre-yield region. The force versus velocity loops are
displayed in Figure 6.27. It is worth noting that BW model provides also a smooth
transition between the viscoelastic state and the post-yield state.
200
Semi-active Suspension Control
Fig. 6.27. Force [N] versus velocity [m/s] experimentally measured data (dashed) and
Bouc–Wen (solid) model [copyright IMechE (2004), reproduced from Giuclea M, Sireteanu
T, Stancioiu D and Stammers CW, Model parameter identification for vehicle vibration
control with magnetorheological dampers using computational intelligence methods,
Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and
Control Engineering, Publisher: Professional Engineering Publishing, ISSN 0959-6518, Vol.
218, N 7/2004, pp 569–581, used by permission]
6.5.3 Fuzzy-logic-based Model
The use of a fuzzy-logic-based modelling scheme offers a number of benefits, the
first and foremost being that is suitable to incorporate the qualitative aspects of
human experience within its mapping laws. A neuro-adaptive learning technique
can be used to train a family of membership functions (MF) to simulate a nonlinear mapping function. The neuro-adaptive learning technique works similarly to
neural networks. This technique provides a method for the fuzzy modelling
algorithm to learn information on a data set, in order to fit the membership function
parameters to the fuzzy inference system to track the input/output data. The
parameters associated with the membership functions will change through the
learning process. Data for training and checking are obtained from experimentally
measured data. The fuzzy-logic model is based on a Takagi–Sugeno–Kang
architecture. Displacement and velocities were used as input functions. The
number of MFs was set at two for displacements and five for velocities. The afterlearning input MFs that fit the experimentally measured data are plotted in
Figure 6.28.
Magnetorheological Dampers
201
Fig. 6.28. Membership functions after training [copyright IMechE (2004), reproduced from
Giuclea M, Sireteanu T, Stancioiu D and Stammers CW, Model parameter identification for
vehicle vibration control with magnetorheological dampers using computational intelligence
methods, Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems
and Control Engineering, Publisher: Professional Engineering Publishing, ISSN 0959-6518,
Vol. 218, N 7/2004, pp 569–581, used by permission]
Fig. 6.29. Force [N] versus time [s] experimental (dashed) and fuzzy-logic-based model
(solid) [copyright IMechE (2004), reproduced from Giuclea M, Sireteanu T, Stancioiu D
and Stammers CW, Model parameter identification for vehicle vibration control with
magnetorheological dampers using computational intelligence methods, Proceedings of the
Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering,
Publisher: Professional Engineering Publishing, ISSN 0959-6518, Vol. 218, N 7/2004, pp
569–581, used by permission]
The result of simulation in terms of the force-time dependence is shown in Figure
6.29. It is possible to see that an almost perfect fit is obtained. The good accuracy
202
Semi-active Suspension Control
of the representation can also be seen for the force versus velocity (Figure 6.30)
and force versus displacement loops (Figure 6.31).
Fig. 6.30. Force [N] versus velocity [m/s] experimental (1) and fuzzy-logic-based model (2)
[copyright IMechE (2004), reproduced from Giuclea M, Sireteanu T, Stancioiu D and
Stammers CW, Model parameter identification for vehicle vibration control with
magnetorheological dampers using computational intelligence methods, Proceedings of the
Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering,
Publisher: Professional Engineering Publishing, ISSN 0959-6518, Vol. 218, N 7/2004, pp
569–581, used by permission]
Fig. 6.31. Force [N] versus displacement [m] experimental (1) and fuzzy-logic-based model
(2) [copyright IMechE (2004), reproduced from Giuclea M, Sireteanu T, Stancioiu D and
Stammers CW, Model parameter identification for vehicle vibration control with
magnetorheological dampers using computational intelligence methods, Proceedings of the
Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering,
Publisher: Professional Engineering Publishing, ISSN 0959-6518, Vol. 218, N 7/2004, pp
569–581, used by permission]
Magnetorheological Dampers
203
Therefore, it can be stated that the fuzzy-logic-based model predicts well the
behaviour of the damper in all the regions, including the pre-yield one.
6.5.4 Modelling the Variable Field Strength
To take full advantage of the MR damper models in active or semi-active control
applications the effect of a continuously variable magnetic field must be
considered. Spencer et al. (1997) generalised the model for such cases by imposing
for the four parameters a linear dependence on the applied voltage. Seven
parameters were considered to fit the measured data. None of the BW equation
parameters was assumed to be voltage dependent. With this approach the model of
the hysteresis loops and of the transition between the pre-yield and post-yield states
are de facto unchanged when voltage varies and this is also in agreement with
measured data.
Kyle and Roschke (2000) used a fuzzy-logic-based model with three inputs:
displacement, voltage and velocity. The results obtained are very accurate, but the
modelling is based on data generated by simulation. It is also worth noting that an
increase in neuro-fuzzy modelling precision takes into consideration even the
measured noise.
In order to determine the parametric models for variable field, it must be
assumed that some of the parameters chosen are continuously variable functions of
the current. From the experimentally measured data, it is possible to see that the
hysteresis loop is not field dependent (Figure 6.17). Hence, the BW model
parameters γ , ν and A are considered to be independent of the current.
Fig. 6.32. α [N/m] parameter current [A] dependency [copyright IMechE (2004),
reproduced from Giuclea M, Sireteanu T, Stancioiu D and Stammers CW, Model parameter
identification for vehicle vibration control with magnetorheological dampers using
computational intelligence methods, Proceedings of the Institution of Mechanical Engineers,
Part I: Journal of Systems and Control Engineering, Publisher: Professional Engineering
Publishing, ISSN 0959-6518, Vol. 218, N 7/2004, pp 569–581, used by permission]
204
Semi-active Suspension Control
Assuming that these model parameters are constant, the α parameter, which
determines the yield force, is field dependent. Its current dependency can be
supposed to be polynomial, having almost a linear variation for current values
above 0.6 A (Figure 6.32).
The parameter c0 that describes the damping dashpot has also a nonlinear
variation. The current dependency of this parameter is similar to that of the α
parameter shown in Figure 6.32.
The spring elastic constant, denoted K 0 in the BW-based model (Figure 6.33),
exhibits an almost linear variation with the supplied current. Therefore the fielddependent parameters for the BW-based model were set to be α , c0 , K 0 and f 0 .
Polynomial variation functions having different degrees were used to fit the model.
Fig. 6.33. K 0 [N/m] parameter current [A] dependency [copyright IMechE (2004),
reproduced from Giuclea M, Sireteanu T, Stancioiu D and Stammers CW, Model parameter
identification for vehicle vibration control with magnetorheological dampers using
computational intelligence methods, Proceedings of the Institution of Mechanical Engineers,
Part I: Journal of Systems and Control Engineering, Publisher: Professional Engineering
Publishing, ISSN 0959-6518, Vol. 218, N 7/2004, pp 569–581, used by permission]
For variable control field, the BW model response to a random variable load is
shown in Figure 6.34. The current amplitude variation is rectangular.
Magnetorheological Dampers
205
Fig. 6.34. Bouc–Wen model for variable control field. Force [N] vs. time [s] [copyright
IMechE (2004), reproduced from Giuclea M, Sireteanu T, Stancioiu D and Stammers CW,
Model parameter identification for vehicle vibration control with magnetorheological
dampers using computational intelligence methods, Proceedings of the Institution of
Mechanical Engineers, Part I: Journal of Systems and Control Engineering, Publisher:
Professional Engineering Publishing, ISSN 0959-6518, Vol. 218, N 7/2004, pp 569–581,
used by permission]
Fig. 6.35. Modified Bingham model for variable control field. Force [N] versus time [s];
[copyright IMechE (2004), reproduced from Giuclea M, Sireteanu T, Stancioiu D and
Stammers CW, Model parameter identification for vehicle vibration control with
magnetorheological dampers using computational intelligence methods, Proceedings of the
Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering,
Publisher: Professional Engineering Publishing, ISSN 0959-6518, Vol. 218, N 7/2004, pp
569–581, used by permission]
206
Semi-active Suspension Control
In the same way, the modified Bingham model can also be generalised to a
variable-field-strength model. The results for the generalisation of the modified
Bingham model are similar, but less accurate (Figure 6.35) than those of the BW
model.
Fig. 6.36. The learning data [copyright IMechE (2004), reproduced from Giuclea M,
Sireteanu T, Stancioiu D and Stammers CW, Model parameter identification for vehicle
vibration control with magnetorheological dampers using computational intelligence
methods, Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems
and Control Engineering, Publisher: Professional Engineering Publishing, ISSN 0959-6518,
Vol. 218, N 7/2004, pp 569–581, used by permission]
In order to develop the fuzzy-logic model with fluctuating field strength, the neuroadaptive learning method is used. The established learning and checking data are,
respectively, plotted in Figure 6.36 and in Figure 6.37. Here the measurement units
of the variables are chosen such that they can be represented on the same graph.
A three-input fuzzy-logic-based model with two functions for voltage and
displacement and three functions for velocity is found to achieve the best results.
The after-learning input membership functions are displayed in Figure 6.38.
The force–time history of the learning data compared to the fuzzy-logic model
generated data is depicted in Figure 6.39. The checking data time history compared
to the model predicted data are portrayed in Figure 6.40.
Magnetorheological Dampers
207
Fig. 6.37. The checking data [copyright IMechE (2004), reproduced from Giuclea M,
Sireteanu T, Stancioiu D and Stammers CW, Model parameter identification for vehicle
vibration control with magnetorheological dampers using computational intelligence
methods, Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems
and Control Engineering, Publisher: Professional Engineering Publishing, ISSN 0959-6518,
Vol. 218, N 7/2004, pp 569–581, used by permission]
Fig. 6.38. The after-learning membership functions (voltage–displacement–velocity);
[copyright IMechE (2004), reproduced from Giuclea M, Sireteanu T, Stancioiu D and
Stammers CW, Model parameter identification for vehicle vibration control with
magnetorheological dampers using computational intelligence methods, Proceedings of the
Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering,
Publisher: Professional Engineering Publishing, ISSN 0959-6518, Vol. 218, N 7/2004, pp
569–581, used by permission]
208
Semi-active Suspension Control
Fig. 6.39. The learning data validation. Force [N] versus time [s] [copyright IMechE (2004),
reproduced from Giuclea M, Sireteanu T, Stancioiu D and Stammers CW, Model parameter
identification for vehicle vibration control with magnetorheological dampers using
computational intelligence methods, Proceedings of the Institution of Mechanical Engineers,
Part I: Journal of Systems and Control Engineering, Publisher: Professional Engineering
Publishing, ISSN 0959-6518, Vol. 218, N 7/2004, pp 569–581, used by permission]
Fig. 6.40. The checking data validation. Force [N] versus time [s] [copyright IMechE
(2004), reproduced from Giuclea M, Sireteanu T, Stancioiu D and Stammers CW, Model
parameter identification for vehicle vibration control with magnetorheological dampers
using computational intelligence methods, Proceedings of the Institution of Mechanical
Engineers, Part I: Journal of Systems and Control Engineering, Publisher: Professional
Engineering Publishing, ISSN 0959-6518, Vol. 218, N 7/2004, pp 569–581, used by
permission]
Magnetorheological Dampers
209
6.5.5 GA-based Method for MR Damper Model Parameters Identification
As was shown by Spencer et al. (1997), the best results portraying the hysteretic
behaviour of MR dampers are obtained with the BW-modified model, which is
depicted in Figure 6.41.
In this subsection, a method for finding the BW-modified model parameters is
proposed by using a genetic algorithm (GA) optimisation procedure. Initially, the
values of the parameters are found for a set of constant values of the applied
current and an imposed cyclic motion, such that the predicted response optimally
fit the experimental data. The second step consists in obtaining the variation law
for each parameter as a function of the current, considering the corresponding
values from the first step. The resulting model is validated by comparison of the
predicted and experimentally obtained responses for some cases with variable
control current.
Fig. 6.41. BW-modified mechanical model [copyright IMechE (2004), reproduced from
Giuclea M, Sireteanu T, Stancioiu D and Stammers CW, Model parameter identification for
vehicle vibration control with magnetorheological dampers using computational intelligence
methods, Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems
and Control Engineering, Publisher: Professional Engineering Publishing, ISSN 0959-6518,
Vol. 218, N 7/2004, pp 569–581, used by permission]
In this model the damping force generated by the device is given by
F = c1 y + k1 ( x − x0 ) ,
(6.74)
where x is the total relative displacement and x0 the initial deflection of the
accumulator gas spring with stiffness k1 . The partial relative displacement y and
the evolutionary variable z are governed by the coupled differential equations:
y =
1
[c0 x + k 0 ( x − y ) + kz ] ,
c0 + c1
z = −γ x − y z z
n −1
n
−ν ( x − y ) z + A ( x − y ) ,
(6.75)
(6.76)
where k is a stiffness coefficient associated with the displacement z . The
parameters n , γ , ν , A and k1 are considered fixed, and the parameters c0 , c1 ,
210
Semi-active Suspension Control
k and k0 are assumed to be functions of the delayed current i = i (t ) applied to the
MR damper: c0 = c0 (i ) , c1 = c1 (i ) , k = k (i ) , k 0 = k 0 (i ) . If i0 (t ) is the supplied
current, then the dynamics of the MR fluid to reach its rheological equilibrium are
modelled by a first-order lag ( T is the time constant):
di
1
= − (i − i0 ) .
dt
T
(6.77)
The BW-modified model includes the above-mentioned parameters ( c0 , c1 , k ,
k 0 ) that depend on the applied current. In order to determine the corresponding
functional dependencies, a GA-based inverse method is applied. For different
constant current values ( i = 0.02 A, 0.06 A, 0.1 A, 0.2 A, 0.4 A, 0.6 A, 0.8 A, 1.05
A, 1.45 A, 1.75 A) and cyclic imposed motion (frequency 1.5 Hz, amplitude 16
mm), the values of parameters c0 , c1 , k , k0 are determined by using a GA, such
that the predicted force Fp fits the measured force Fexp . Subsequently, by
analysing the dependence between current and the values obtained for the
parameters, the functions c0 = c0 (i ) , c1 = c1 (i ) , k = k (i ) , k 0 = k 0 (i ) can be
determined.
In the method proposed, the parameter identification is treated as a black-box
optimisation problem. It is well known that genetic algorithms are robust
probabilistic search techniques with very good results in black-box optimisation
problems (Goldberg, 1989). They are based on the mechanism of natural genetics
and natural selection, starting with an initial population of (encoded) problem
solutions and evolving towards better solutions. For the considered optimisation
procedure, a real-coded GA is employed with four real genes corresponding to the
four coefficients c0 , c1 , k , k0 . By using appropriate scaling factors, it can be
assumed that the parameters take on values within the interval [0, 1]. The other
characteristics of the GA applied are an averaged crossover with probability 0.8,
uniform mutation, Monte Carlo selection and an objective function of the RMS
error between the predicted and experimental response.
For numerical simulations the following values of the fixed coefficients were
chosen: n = 2, d = 500 N/m, g = 613000 m-2, α = 30.56, k1 = 540 N/m. By
applying the proposed GA, after about 500 generations, the values given in Table
6.4 were produced. The accuracy of GA optimisation can be evinced from Figures
6.42, 6.43, 6.44 and 6.45, showing the predicted and experimentally obtained
responses for 0.06 A and 1.75 A. Since the parameter k0 has a fairly irregular
variation with respect to the current the average value k0 =1050 N/m was
considered.
Magnetorheological Dampers
211
Table 6.4. Parameter values obtained by GA [copyright IMechE (2004), reproduced from
Giuclea M, Sireteanu T, Stancioiu D and Stammers CW, Model parameter identification for
vehicle vibration control with magnetorheological dampers using computational intelligence
methods, Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems
and Control Engineering, Publisher: Professional Engineering Publishing, ISSN 0959-6518,
Vol. 218, N 7/2004, pp 569–581, used by permission]
i [A ]
⎡ Ns ⎤
c0 ⎢ ⎥
⎣m⎦
⎡ Ns ⎤
c1 ⎢ ⎥
⎣m⎦
⎡N⎤
k⎢ ⎥
⎣m⎦
⎡N⎤
k0 ⎢ ⎥
⎣m⎦
0.02
121
10300
2950
527
0.06
340
8350
15300
306
0.10
465
13900
23900
22.3
0.20
966
33600
34000
468
0.40
1690
83900
58500
988
0.06
2880
93300
89800
1990
0.80
3220
101800
104900
1240
1.05
3500
107800
114500
1330
1.45
4730
111600
114400
1630
1.75
4050
122500
133900
2010
200
Force [N]
100
0
0.01
0.21
0.41
0.61
0.81
-100
-200
Time [s]
Fig. 6.42. Force vs. time for 0.06 A _____ experimental; .….. predicted [copyright IMechE
(2004), reproduced from Giuclea M, Sireteanu T, Stancioiu D and Stammers CW, Model
parameter identification for vehicle vibration control with magnetorheological dampers
using computational intelligence methods, Proceedings of the Institution of Mechanical
Engineers, Part I: Journal of Systems and Control Engineering, Publisher: Professional
Engineering Publishing, ISSN 0959-6518, Vol. 218, N 7/2004, pp 569–581, used by
permission]
212
Semi-active Suspension Control
200
Force [N]
100
0
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
-100
-200
Velocity [m/s]
Fig. 6.43. Force [N] versus velocity [m/s] for 0.06 A _____ experimental; …… predicted
[copyright IMechE (2004), reproduced from Giuclea M, Sireteanu T, Stancioiu D and
Stammers CW, Model parameter identification for vehicle vibration control with
magnetorheological dampers using computational intelligence methods, Proceedings of the
Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering,
Publisher: Professional Engineering Publishing, ISSN 0959-6518, Vol. 218, N 7/2004, pp
569–581, used by permission]
2000
Force [N]
1000
0
0.01
0.21
0.41
0.61
0.81
-1000
-2000
Time [s]
Fig. 6.44. Force [N] versus time [s] for 1.75 A _____ experimental; ……. predicted
[copyright IMechE (2004), reproduced from Giuclea M, Sireteanu T, Stancioiu D and
Stammers CW, Model parameter identification for vehicle vibration control with
magnetorheological dampers using computational intelligence methods, Proceedings of the
Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering,
Publisher: Professional Engineering Publishing, ISSN 0959-6518, Vol. 218, N 7/2004, pp
569–581, used by permission]
Magnetorheological Dampers
213
2000
Force [N]
1000
0
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
-1000
-2000
Velocity [m/s]
Fig. 6.45. Force [N] versus velocity [m/s] for 1.75 A _____ experimental; ……. predicted
[copyright IMechE (2004), reproduced from Giuclea M, Sireteanu T, Stancioiu D and
Stammers CW, Model parameter identification for vehicle vibration control with
magnetorheological dampers using computational intelligence methods, Proceedings of the
Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering,
Publisher: Professional Engineering Publishing, ISSN 0959-6518, Vol. 218, N 7/2004, pp
569–581, used by permission]
The functional approximations obtained for the remaining parameters c0 , c1 , k ,
are
c0 (i ) = 26.134 i + 5.164 ,
(6.78)
c1 (i ) = 1150 tanh(1.95 i ) ,
(6.79)
k (i ) = 1297.2 tanh(1.3 i ) .
(6.80)
Therefore, the BW-modified model is completely defined and can be used for
different loading conditions and variation patterns of the applied current.
In the remainder of this section, a comparison is made between the measured
and predicted forces computed for the identified model. Simulations were
performed using the experimentally determined displacement x and the calculated
velocity x of the piston rod in determining the force generated by the damper
model. The effectiveness of the model in describing the dynamic behaviour of the
MR damper has been investigated in a variety of deterministic tests. When
compared with experimental data, the model is shown to accurately predict the
response of the MR damper over a wide range of operating conditions.
214
Semi-active Suspension Control
18
V e lo c it y [ c m / s ]
9
V o lt a g e [ V ]
0
1
1 .2 5
1 .5
1 .7 5
2
2 .2 5
2 .5
2 .7 5
3
T im e [ s ]
-9
D is p la c e m e n t [ c m ]
-1 8
Fig. 6.46. Time histories of voltage [V], displacement [cm] and velocity [cm/s] in the first
test case [copyright IMechE (2004), reproduced from Giuclea M, Sireteanu T, Stancioiu D
and Stammers CW, Model parameter identification for vehicle vibration control with
magnetorheological dampers using computational intelligence methods, Proceedings of the
Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering,
Publisher: Professional Engineering Publishing, ISSN 0959-6518, Vol. 218, N 7/2004, pp
569–581, used by permission]
1800
1200
Force [N]
600
0
1
1.25
1.5
1.75
2
2.25
2.5
2.75
3
-600
-1200
-1800
Time [s]
Fig. 6.47. Force [N] versus time [s] in the first test case _____ experimental; …… predicted
[copyright IMechE (2004), reproduced from Giuclea M, Sireteanu T, Stancioiu D and
Stammers CW, Model parameter identification for vehicle vibration control with
magnetorheological dampers using computational intelligence methods, Proceedings of the
Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering,
Publisher: Professional Engineering Publishing, ISSN 0959-6518, Vol. 218, N 7/2004, pp
569–581, used by permission]
Magnetorheological Dampers
215
15
Voltage [V]
10
5
0
0.38
0.88
Time [s]
1.38
1.88
-5
Velocity [cm/s]
Displacement [cm]
-10
-15
Fig. 6.48. Time histories of voltage [V], displacement [cm] and velocity [cm/s] in the
second test case [copyright IMechE (2004), reproduced from Giuclea M, Sireteanu T,
Stancioiu D and Stammers CW, Model parameter identification for vehicle vibration control
with magnetorheological dampers using computational intelligence methods, Proceedings of
the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering,
Publisher: Professional Engineering Publishing, ISSN 0959-6518, Vol. 218, N 7/2004, pp
569–581, used by permission]
1500
1000
Force [N]
500
0
1
1 .2 5
1 .5
1 .7 5
2
-5 0 0
-1 0 0 0
-1 5 0 0
T im e [s ]
Fig. 6.49. Force [N] versus time [s] in the second test case _____ experimental;……
predicted [copyright IMechE (2004), reproduced from Giuclea M, Sireteanu T, Stancioiu D
and Stammers CW, Model parameter identification for vehicle vibration control with
magnetorheological dampers using computational intelligence methods, Proceedings of the
Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering,
Publisher: Professional Engineering Publishing, ISSN 0959-6518, Vol. 218, N 7/2004, pp
569–581, used by permission]
216
Semi-active Suspension Control
10
V o lta g e [V ]
V e lo c ity [c m /s ]
5
T im e [s ]
0
0 .0 1
0 .5 1
1 .0 1
1 .5 1
2 .0 1
2 .5 1
-5
D is p la c e m e n t [c m ]
-1 0
Fig. 6.50. Voltage [V], displacement [cm] and velocity [cm/s] for the third test case
[copyright IMechE (2004), reproduced from Giuclea M, Sireteanu T, Stancioiu D and
Stammers CW, Model parameter identification for vehicle vibration control with
magnetorheological dampers using computational intelligence methods, Proceedings of the
Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering,
Publisher: Professional Engineering Publishing, ISSN 0959-6518, Vol. 218, N 7/2004, pp
569–581, used by permission]
1500
1000
Force [N]
500
0
0.01
-500
0.51
1.01
1.51
2.01
2.51
-1000
-1500
Time [s]
Fig. 6.51. Force [N] versus time [s] in the third test case_____ experimental; …… predicted
[copyright IMechE (2004), reproduced from Giuclea M, Sireteanu T, Stancioiu D and
Stammers CW, Model parameter identification for vehicle vibration control with
magnetorheological dampers using computational intelligence methods, Proceedings of the
Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering,
Publisher: Professional Engineering Publishing, ISSN 0959-6518, Vol. 218, N 7/2004, pp
569–581, used by permission]
Magnetorheological Dampers
217
To illustrate this statement, the results obtained in three test cases are presented.
The time histories of the displacement, velocity and of the voltage applied to the
MR damper controller for each test case are plotted in Figures 6.46–6.51. The
current i0 supplied by the device controller is proportional to the applied voltage,
the output–input ratio being 0.4 A/V. The units of measurement of Figures 6.36,
6.37, 6.46, 6.48 and 6.50 (daN, cm, cm/s) have been chosen to allow all the
relevant quantities to be plotted on the same graph.
7
Case Studies
7.1 Introduction
A number of case studies are now presented to show applications of the concepts
illustrated in the previous chapters. After a succinct introduction of some aspects of
data acquisition and digital control, three case studies are presented, namely a
friction damper-based suspension unit for a saloon car, a magnetorheological
damper-based seat suspension for vehicles not equipped with primary suspensions
and a semi-active suspension system with magnetorheological dampers for heavy
vehicles where the emphasis is not only on ride comfort but also on road damage
reduction.
7.1.1 Some Aspects of Data Acquisition and Control
The reader is assumed to be familiar with the fundamental theoretical concepts of
computer-based data acquisition and digital control (Shannon theorem, Nyquist
theorem and digital-to-analogue and analogue-to-digital conversion). For
furthering the topic the reader can refer to the numerous textbooks on the subject
(e.g., Leigh, 1992). Data acquisition and real-time control can be implemented
using an embedded microprocessor architecture, a PC or a programmable logic
controller (PLC) equipped with one or more I/O cards. Commercial data
acquisition and control cards are made to acquire a variety of input signals and
issue a variety of output signals including not only digital and analogue (e.g., 0–10
V, ± 5 V, 4–20 mA) signals but also more specialised signals (e.g., those produced
by thermocouples, thermoresistances, solenoid valves, servovalves, vibration
sensors, frequency modulated signals etc.). Signals are properly conditioned
(filtered, amplified, modulated, demodulated etc.) by appropriate hardware within
the card or via software.
In the case studies described an off-the-shelf card was used. The card was
capable of receiving up to 16 single-ended channels with 12-bit resolution.
Depending upon the transducers output and actuators input range, the card I/O
220
Semi-active Suspension Control
range could be configured. For instance in the applications described, analogue
inputs were configured to have a unipolar range of 0–10 V and analogue outputs a
bipolar range of ±5 V. This implies that the mean value of the quantisation error is
0.025 V, which is negligible with respect to the range of the converter2 and in the
suspension application does not produce any appreciable deterioration of the
controller performance.
The card can be made to operate in clocked D/A mode; in this mode its clock is
set up to produce the required sampling frequency. In the light of the Shannon
theorem, considering the frequency content of the vehicle dynamic response, the
sampling frequency can be set to 100 Hz. The total delay introduced in the
sampling, holding and multiplexing phases is negligible.
For data acquisition, effective software is key. In the work described, a
commercial data acquisition package and an in-house frequency response analyser
software developed at the University of Bath (UK) were employed.
Digital closed-loop control must be implemented in real time, and therefore an
environment with a deterministic operating system is required. This can be
achieved with an embedded architecture or a PLC. Typically PLCs are used in
industrial plants where a vast number of loops need to be controlled or monitored
within the framework of a hierarchical (pyramidal type) distributed architecture. A
PLC is often interfaced with a PC (often referred to as a HMI, a human–machine
interface) for graphic display of the controlled plant parameters. I/O signals can be
either hardwired or connected via a network (sometimes referred to as remote I/O).
Signals, appropriately routed, can also be transferred to other PCs or PLCs over the
network, using a variety of transmission media (e.g., serial links, Ethernet or fibre
optics) and appropriate protocols. A typical automotive protocol is the controller
area network (CAN bus). Control software can be written in a graphical language
(Ladder-like) or in C/C++. Fast dynamics routines can also be written in Assembly
language to optimise performance.
Closed-loop control software in the applications described was written in C.
Good programming practice suggests to write software in modular form, and
develop a user interface that allows the entry of the key configurable parameters of
the control algorithm, enabling data to be saved to file for off-line post-processing.
In automotive applications embedded architectures are used and feedback
signals are either hardwired using shielded cables (and additional low-pass filtering
stages are present in the on-board electronics) or transmitted over the network.
Furthermore dedicated circuitries are present to increase the noise rejection.
2. The resolution V0 of a converter can be calculated with the following expression that links range of
the converter, resolution and number of bits:
[K] Log10 2=Log10{(Vmax – Vmin)/V0 +1},
where [K] is an integer that represents the number of bits and Vmax – Vmin is the range of the converter.
Case Studies
221
7.2 Car Dynamics Experimental Analysis
A car, thought of as a rigid body unilaterally constrained by the road, is a 6DOF
system having three translational (forward and backward motion, side slip and
bounce) and three rotational (roll, pitch and yaw) degrees of freedom.
From the viewpoint of assessing the ride performance of a suspension system,
bounce, roll and pitch motions are the most pertinent. Side slip, which occurs only
for a lack of adhesion, and yaw are of interest mainly to handling studies.
At the outset it is necessary to specify the inputs which excite car dynamic
responses of interest and which best represent real conditions on the road. Vehicle
driving tests in controlled conditions would appear to be the most natural solution.
However it is also possible to emulate the road input with a road simulator. This
allows measurements to be made in more controlled conditions; besides, it permits
tests which are not feasible on a road to be carried out (e.g., sine wave inputs or the
excitation of single suspension units separately to mimic a quarter-car-like
behaviour). Moreover on a road simulator the level of external disturbances is
reduced or even eliminated (e.g., wind gust) compared to a road test, and hence
tests are more repeatable.
Preliminary tests on the car equipped with its original viscous damper are
necessary in order to have a set of data to benchmark a semi-active system. In the
following developments it will be described how to practically carry out vehicle
measurements and how to process data effectively.
7.2.1 The Experimental Set-up
The heart of the vehicle testing equipment is the four-poster road simulator. This is
essentially composed of four hydraulic actuators on which the vehicle under testing
(a Ford Orion) is placed (Figure 7.1).
The main features of the road simulator available at the University of Bath are
now briefly described to identify its potential and its limits. Each hydraulic
actuator is position-controlled through a servovalve. The rig is operated via a
human–machine interface PC where software manages and supervises all the
operations. It is possible to select elementary inputs (sine waves, half-sine waves,
square waves) in the 0–25 Hz frequency range and ±125 mm amplitude range. The
relative phase shift among the actuators can be controlled as well. In this way pure
bounce, roll and pitch inputs, and other customised inputs, can be generated.
Furthermore, it is possible to create user-defined waveforms by composing
elementary functions via a graphical editor.
In order to gain an understanding of the vehicle dynamic phenomena of interest
and to provide a comprehensive set of data to assess suspension performance, tests
with the following inputs are essential:
•
•
•
•
sinusoidal input to a single wheel
bounce, roll and pitch tests with sinusoidal input
pseudo-random input
bump test
222
Semi-active Suspension Control
Sinusoidal waveforms do not represent any realistic road condition. However
they are simple and well-known inputs and they allow the use of frequency-domain
methods for manipulating data. Sinusoidal tests give a first clue to the effectiveness
of the suspensions: they permit an assessment of how controlled suspensions
behave with respect to passive suspensions.
Fig. 7.1. Test vehicle on the four-poster road simulator
It must be remarked that, because of the non-linear behaviour of the four-poster
shaker, it is not possible to obtain a truly sinusoidal motion in spite of a sinusoidal
driving input voltage applied to the shaker. In addition the ride dynamic response
of the passive car is linear only to a first approximation (the main non-linearities
being the tyre characteristics and the hysteretic viscous damper characteristics,
including mountings and rubber bushes). As a consequence of the whole system
being non-linear, it is not possible to define a proper frequency response (in the
control sense). For this reason the most significant plot is the RMS value (or the
ratio of two RMS values, for instance the ratio of chassis and wheel accelerations
versus the fundamental harmonic of the excitation frequency for a fixed input
amplitude). This experimental frequency response should more appropriately be
called the acceleration transmissibility ratio, because it includes the effect of all the
harmonic content of the signals. However for ease of terminology the term
frequency response will be used hereafter.
The passive system frequency responses are ideally expected to be worse (i.e.,
with larger values) than the semi-active frequency responses in the working
frequency range.
This is a first crucial assessment in the frequency domain before further
investigations in order to evaluate the properties of the responses in the time
domain (harmonic content, peak values, higher-derivative trends, peak values etc.).
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223
The vehicle is instrumented with four accelerometers and one displacement
transducer mounted on the axle where the semi-active damper is fitted. Two
accelerometers are mounted on each rear wheel, connected through small cylinders
bolted on the wheels. The connections to the vehicle must be made as compact and
stiff as possible in order to reduce unwanted vibration which could affect the
measurement. The other two accelerometers are fixed on the chassis of the car,
close to each rear suspension. Accelerometers have a principal axis of sensitivity,
therefore they need to be mounted vertically in order to measure the vertical
component of chassis and wheel acceleration. The calibration of these devices is
based on gravity (by interpolating readings at +1 g, 0 and –1 g). Because of their
sensitivity to acceleration due to gravity, it might be necessary to compensate (in
hardware or in software) the offset of 1 g that they may introduce.
Signals from the transducers are manipulated in conditioning cards. As
mentioned in Section 7.1.1, most conditioning circuitries are essentially composed
of an amplification stage followed by a filtering stage. A stabilised power supply
provides the required constant voltage to the controller. Pre-amplification is
necessary to make the signal more robust, increasing its immunity to noise and
making it less interference sensitive. The amplifier gain must be suitably chosen
considering the order of magnitude of car vertical accelerations (a good choice
could be 1g =10 V for instance). The grounding is done through an earth resistance
on the metallic trolley carrying the equipment; the ground is unique to avoid earth
loops.
A critical stage of the conditioning chain is filtering. Acceleration is typically a
noisy signal: stray effects due to the mounting, rubber bushes, friction etc. are
picked up by the transducer and therefore it is necessary to low-pass filter the
signal. The break frequency of the filter must be chosen trading off among
different conflicting requirements: if it is too low it introduces an undesired phase
lag and further it smoothes the waveforms by cutting off the high-frequency
harmonic content. This only slightly affects integral parameters like the RMS
(higher-order harmonics typically have small amplitudes). However it can diminish
the actual value of some time-domain information such as peak acceleration values
or jerk content which are comfort related. These need to be quantified properly
(particularly in a controlled suspension with a switching control logic). On the
other hand if the break frequency is too high, a significant level of noise can be
superimposed on the signal, affecting not only its readability but also the accurate
evaluation of its RMS. In addition, a high level of noise could introduce chattering
if acceleration signals were used in a switching logic.
The choice of an appropriate break frequency is a trial-and-error process and
can be initially performed with an analogue, manually-adjustable filter. In the
applications investigated a second-order Butterworth filter with a break frequency
of 40 Hz appeared to be appropriate. With this choice the phase lag introduced is
only 10˚ at 5 Hz. Digital filtering is another option too.
The relative displacement is measured with a position transducer mounted
between body and wheel. An externally mounted LVDT is not appropriate as,
when the car undergoes testing, tyre torsional modes may turn the wheels, with a
risk of damaging the device. Mounting connections through plastic bolts and rose
joints are not always sufficient to provide protection. A pull-wire displacement
224
Semi-active Suspension Control
transducer is instead a better choice. This potentiometric device is safe, since the
position is measured with a flexible wire, and not a rigid rod. Potentiometers
(depending on the materials they are made of, e.g., hybrid conductive plastic) have
an extremely high resolution, resulting in a very clean measured signal; no filtering
is in fact necessary. In order to calibrate it accurately, a stepper motor can be
employed. Finally measurements need to be carried out with tyres inflated at their
rated pressures and with wheel brakes on to minimise tyre torsional vibration,
which tends to rotate them.
7.2.2 Post-processing and Measurement Results
As far as sinusoidal tests are concerned the most meaningful data are the time
responses and the ratio of chassis to wheel RMS acceleration. The ride model of
the car is supposed to be linear to a first approximation; this implies that the
acceleration ratio is equal to the velocity and displacement ratios:
a1
jω v1 ω 2 x1
,
=
=
a2
jω v 2 ω 2 x 2
(7.1)
ai , vi , xi being, respectively, chassis and wheel (subscripts 1 and 2) acceleration,
velocity and displacement. Hence transmissibility curves can be readily obtained.
Acc ratio
Acc. ratio
1.5
1
0.5
0
0
5
10
15
20
25
Freq. [Hz]
Fig. 7.2. Body-to-wheel acceleration ratio of the rear right corner of the car. Sinusoidal input
to one wheel, amplitude: 3 mm
A simple test to verify the amount of linear behaviour of the passive car can be
carried out by applying inputs with three different amplitudes. The closer the three
responses are to one another the more the system approximates linear behaviour.
Figure 7.2 shows the response of the car when only one wheel is excited; the ratio
of body over wheel RMS acceleration is plotted.
The four-poster shaker actuator generates a sinusoidal waveform with a peak
value of 3 mm in the range 1–22 Hz. In the car response three resonances are
evident; the first one is the chassis resonance at about 3 Hz; this is fairly high:
Case Studies
225
typically this resonance occurs at about 1.5 Hz. This shift is essentially due to the
small portion of sprung mass loading the suspension strut, since only the rear right
corner of the car is shaken. The second resonance is at about 8.5 Hz; this is due to
cross-coupling effects: around that frequency range a compound motion of bounce,
roll and pitch is present; an engine mounting resonance is also possible. Eventually
the wheel-hop resonance occurs at 18 Hz, although it is not readily recognisable on
the graph of Figure 7.2; however if the wheel acceleration response is plotted
(Figure 7.3) the resonance becomes clear.
Wheel Acc. [m/s^2]
W heel acceleration response
3.5
3
2.5
2
1.5
1
0.5
0
0
5
10
15
20
25
F req . [H z ]
Fig. 7.3. Rear right wheel acceleration response. Sinusoidal input to one wheel, amplitude:
3 mm
Figure 7.4 shows the same response as above when three different input amplitudes
(3 mm, 5 mm and 7 mm) are applied (for the purpose of testing linearity). The
differences among the three graphs are reasonably small; hence it is possible to
state that the non-linear effects, although present, are not so significant and the
hypothesis of linearity is fairly realistic within the amplitude range of the signals.
Non-linear effects occur typically with large signals (although small signals can
also excite non-linear phenomena).
Acceleration ratio
1.6
1.4
Acc. ratio
1.2
1
input =3 mm
0.8
input =5 mm
0.6
input =7 mm
0.4
0.2
0
0
5
10
15
Freq. [Hz]
Fig. 7.4. Body-to-wheel acceleration ratio of the rear right corner of the car. Sinusoidal input
to one wheel, amplitudes: 3 mm, 5 mm, 7 mm
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Semi-active Suspension Control
Figures 7.5 and 7.6 depict two responses in the time domain: relative body-towheel displacement and body acceleration (units have been scaled), in response to
a 3 mm amplitude sinusoidal signal at a frequency of 2.5 Hz. The traces are nearly
sinusoidal. This further confirms the hypothesis of linearity.
Relative displacement
Amplitude
10
5
0
-5 0
0.5
1
1.5
2
2.5
3
-10
Tim e [s]
Fig. 7.5. Relative displacement time trend. Sinusoidal input to one wheel, amplitude: 3 mm,
frequency: 2.5 Hz
C h a s s is a c c e le r a t io n
0 .0 6
0 .0 4
Acc.
0 .0 2
0
-0 .0 2 0
1
2
3
-0 .0 4
-0 .0 6
T im e [s ]
Fig. 7.6. Chassis acceleration time trend. Sinusoidal input to one wheel, amplitude: 3 mm,
frequency: 2.5 Hz
For subsequent studies, the frequency range is limited to 6 Hz. Such a range is
sufficient to show all the main dynamic phenomena associated with the chassis, in
response to bounce, roll and pitch inputs. Figure 7.7 shows the rear right and rear
left corner responses of the car to a pure bounce input, applied to all four wheels.
Case Studies
227
Acc. ratio
Rear right and left corner bounce response
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
body R/wheel R(0.3V)
body L/wheel L(0.3V)
0
2
4
6
8
Freq. [Hz]
Fig. 7.7. Body-to-wheel acceleration ratio of the rear of the car. Sinusoidal bounce input to
all wheels, amplitude: 3 mm
The right and left corner trends exhibit some differences; this is mainly because the
centre of gravity of the car does not exactly pass through the vertical plane of
symmetry. Two resonances are evident from the figure: the first (around 2.3 Hz) is
the bounce resonance while the second resonance (around 3.8 Hz) is due to the
pitch–bounce coupled motion. In order to identify which is the dominant parasitic
motion (with respect to pure bounce) responsible for that spurious resonance, two
more accelerometers can be mounted on the front of the car above each suspension
and the relative phase shift recorded. A phase shift of about 180˚ is associated with
roll motion, while a null phase shift corresponds to pitch motion. At the spurious
resonance of 3.8 Hz the measured phase shift was about 40˚, and hence this
resonance is mainly due to the induced pitch motion.
Pitch response is presented in Figure 7.8. Two resonances again are present as
before: the pitch resonance at around 2.3 Hz, plus another one at around 3.8 Hz due
to cross-coupling effects.
Acc. ratio
Rear right corner pitch response
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
1
2
3
4
5
6
7
Freq. [Hz]
Fig. 7.8. Body-to-wheel acceleration ratio of the rear left corner of the car. Sinusoidal pitch
input to all wheels, amplitude: 3 mm
Figure 7.9 shows the roll response for three different inputs applied from an
external voltage signal generator (0.2 V, 0.3 V and 0.4 V corresponding to
2.74 mm, 4.11 mm, and 5.48 mm, the calibration factor being 1 V = 13.7 mm).
Two resonances are clear from the graph: the roll resonance at around 2.3 Hz and
another spurious cross-coupling resonance at about 6 Hz. The closeness of the
three responses confirms once again the hypothesis of linearity.
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Semi-active Suspension Control
Roll response
2.5
Acc. ratio
2
body L/wheel L(0.2V)
1.5
body L/wheel L(0.3V)
1
body L/wheel L(0.4V)
0.5
0
0
2
4
6
8
Freq. [Hz]
Fig. 7.9. Body-to-wheel acceleration ratio of the rear left corner of the car. Sinusoidal roll
input to all wheels, amplitude: 0.2 V, 0.3 V, 0.4 V (1 V = 13.7 mm)
Figure 7.10 depicts the rear right relative displacement (body-to-wheel) response to
a roll input. This is proportional to the rolling angle (relative to the wheel). The
rolling angle is usually referred to the ground. However in the frequency range of
interest the wheel can be considered almost steady so that the rolling angle with
respect to the ground is not expected to differ very much from the one measured
with respect to the wheel. As expected the resonance measured in terms of
acceleration occurs at the same frequency as that of Figure 7.9.
Rear right corner rel. displacement
Displacement [V]
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
Freq. [Hz]
Fig. 7.10. Relative displacement of the rear-right corner of the car. Sinusoidal roll input to
all wheels, amplitude: 0.2 V, 0.3 V, 0.4 V (1 V = 13.7 mm)
For the pseudo-random input test, 25-Hz filtered white noise, produced with an
external noise generator, is applied to the rear-right wheel. The body acceleration
time trend is plotted in Figure 7.11; its RMS value is 0.75 m/s2.
Case Studies
229
Acc. [m/s^2]
Chassis acceleration passive response to random
input
1.5
1
0.5
0
-0.5 0
-1
-1.5
-2
2
4
6
8
10
12
Time [s]
Fig. 7.11. Chassis acceleration time trend. Random input, 25-Hz filtered white noise
The last test is the bump input. A sinusoidal bump with a slight offset has been
generated; the demanded input is filtered by actuator dynamics, resulting in a
smoother signal. Figure 7.12 shows the response of the car to a bump applied at the
rear right wheel.
Chassis acceleration bump response
Acc. [m/s^2]
1
0.5
0
-0.5
0
2
4
6
8
10
12
-1
Time [s]
Fig. 7.12. Chassis acceleration time trend. Sinusoidal bump input, amplitude: 30 mm
The most meaningful parameters in a bump test are peak acceleration and number
of oscillations. In this case the max overshoot and undershoot are symmetrical with
a value of ±0.85 m/s2 and the decay of oscillations indicate an equivalent damping
ratio of approximately 0.3.
7.2.3 Suspension Spring and Tyre Tests
If accurate figures for spring and tyre stiffness are required, an experimental bench
test assessment must be undertaken. This subsection describes a simple method to
measure the suspension spring stiffness and the tyre stiffness.
Spring stiffness tests can be carried out by cyclically loading the spring in a
material testing machine with a periodic force applied quasi-statically. The
compression of the spring under the load can be measured with an extensometer or
an LVDT and the force applied with a strain gauge. The spring characteristic is
expected to be linear in the measured range. Experiments showed linearity up to a
deflection of 200 mm. Such a deflection is far larger than the one experimented in
a saloon car under normal conditions. This validates the hypothesis of linearity and
allows a linear model to be used for the spring. The spring static stiffness recorded
230
Semi-active Suspension Control
for the vehicle under test was 19 kN/m. No hysteresis is present, the spring being a
steel coil.
The same type of test can be carried out on the tyre. It is possible to define a
tyre radial stiffness, which is the one of interest in ride analysis, as well as a tyre
lateral stiffness, which is more of interest in handling behaviour.
The experimental set-up is roughly the same; however, in this case, it is
important to specify how the load is applied (concentrated or distributed) as the
stress distribution is different. A load distributed over a small area, however, better
represents the forces transmitted from the road.
The typical trend is expected to be slightly non-linear with some hysteresis
because rubber is a viscoelastic material. The equivalent linear value of stiffness,
measured as the slope of the line interpolating the points corresponding to null
force and maximum force, was 74 kN/m. The effective stiffness is twice this value
because the tyre as loaded for the test can be thought of as two springs in parallel.
If the tyre is loaded from the top the stiffness measured will be half that
experienced by a load at the axle, which is what is needed for vehicle dynamics.
7.3 Passively Damped Car Validation
Experimental data can be used for the validation of the 7DOF mathematical model
described in Chapter 2. The validation process ought to be carried out over the
whole range of the expected operating conditions. However in practice a selection
of relevant inputs must be made, extrapolating the model behaviour for all other
possible inputs.
The accuracy achieved by the vehicle model has to be assessed both in the time
and frequency domains. In the time domain, the simulated trends ought to
reproduce the behaviour of the measured quantities, following as closely as
possible the trends of the measured variables. In the frequency domain a good
match of the frequency response is expected in the range of interest.
Figure 7.13 shows a comparison of measured and predicted rear right body
acceleration time history at a frequency of 2.5 Hz. The experimental trend is
almost sinusoidal; this confirms again the hypothesis that the behaviour of the car
is reasonably linear for typical road profile amplitudes. Under such conditions the
simulated behaviour is quite close to the experimental response. The model ought
to work in a range of input amplitudes of the order of 10 mm; if the inputs are too
large or too small they can excite unmodelled dynamics.
Case Studies
231
Rear right body acceleration
3
experimental
acceleration [m/s 2]
2
simulated
1
0
-1
-2
-3
1.3
1.4
1.5
1.6
time [s]
1.7
1.8
1.9
Fig. 7.13. Rear right passive acceleration; sinusoidal input to one wheel, amplitude: 7 mm;
frequency: 2.5 Hz
Turning to the frequency-domain analysis, the measured RMS acceleration of the
right rear body acceleration for an input excitation of 7 mm is presented in Figure
7.14 (stars on the graph). Predicted behaviour, for four different levels of viscous
damping, is also shown. Up to 4 Hz the match is fairly good. At higher
frequencies, the simulation overestimates the acceleration. This is presumably due
to unmodelled non-linearities, principally associated with the viscous damper
characteristic (the viscous damper characteristic employed in the model is obtained
by linearising its actual characteristic around the origin, while the real
characteristic is non-linear with different trends for bound and rebound strokes). At
higher frequencies the local slope of the actual viscous characteristic is smaller
than that modelled, resulting in lower damping in practice. Furthermore in a real
viscous damper some hysteresis is present, attributable partly to the rubber bushes
and parasitic friction as well as to the dissipative internal forces within the
hydraulic oil. This explains the mismatch between the experimental data and
simulation at frequencies higher than the chassis resonance frequency.
At 8.5 Hz a further resonance is present, due to either the engine mounting or to
the induced yaw; such a resonance is not predicted by the model, which would
need further degrees of freedom to include these dynamic phenomena. The tyre
model is not sophisticated since ride is the main issue here. However highly
complicated tyre models are essential in handling studies (as well as in
brake/traction analysis) where lateral forces play a major role and not in ride
studies where the dynamics of interest are vertical. A small amount of hysteresis is
present in the tyre characteristics, but this mainly affects the behaviour around the
wheel-hop resonance. The results of a sensitivity analysis, by changing the viscous
coefficient in the range 350–425 Ns/m, are also depicted in the figure. Best
232
Semi-active Suspension Control
agreement between predicted and measured behaviour is obtained with a viscous
damping coefficient of 400 Ns/m.
Therefore, the 7DOF car model captures the main features of the response both
in the time and the frequency domain, up to the chassis resonance frequency. For
more accurate modelling of the higher-frequency range a more complicated model
would be required.
RMS rear right acceleration
2.5
RMS rear right acceleration [m/s 2]
c=350
c=375
2
c=400
c=425
1.5
1
0.5
0
1
2
3
4
5
6
Frequency [Hz]
7
8
9
10
Fig. 7.14. Rear-right passive acceleration frequency response; sinusoidal input to one wheel,
amplitude: 7 mm; c is the viscous damping coefficient in Ns/m [copyright Elsevier (2003),
reproduced from Guglielmino E, Edge KA, Controlled friction damper for vehicle
applications, Control Engineering Practice, Vol. 12, N 4, pp 431–443, used by permission]
7.4 Case Study 1: SA Suspension Unit with FD
This section reports the performance of a semi-active suspension unit equipped
with a friction damper. After the experimental work on the damper and on its
hydraulic drive described in Chapter 5, the friction damper has been installed on a
vehicle under test (Figure 7.15).
Case Studies
233
Fig. 7.15. Friction damper installed on the experimental vehicle
The system supply pressure defined by the relief valve is set to 60 bar for the
reasons outlined in Chapter 5. The control valve is connected to the FD, using a
hose as short as possible to reduce the volume. The instrumentation is the same
employed in the tests on the passively damped car. The results are presented in the
following subsections. Firstly the frequency-domain performance of the semiactive FD is discussed. In this domain it is possible to make an initial comparative
assessment with the benchmark viscous damper response in terms of RMS values.
Subsequently an analysis in the time domain is carried out. Particular care has been
given to the issue of ride comfort assessment. This issue is critical in a suspension
whose control is based on a switching logic, as time trends can potentially be nonsmooth and spiky, causing an uncomfortable ride.
7.4.1 Frequency-domain Analysis
The analysis in the frequency domain is based on RMS values. A comparative
analysis of acceleration and working space transmissibility curves in the semiactive and passive case is presented. Figure 7.16 shows the experimentally
determined acceleration transmissibility ratio for an input amplitude of 7 mm in the
range 1–5 Hz compared with the original (passive) system. The controlled system
out-performs the passive system over most of frequency range considered although
the passive system response is marginally better up to 1.8 Hz. The controlled
response exhibits three peaks: the first, at 1.7 Hz, is the semi-active system chassis
resonance; this frequency is lower than the corresponding passive resonance and
the peak amplitude is smaller. The inferior behaviour of the semi-active system at
low frequency is due to the hydraulic circuit back-pressure, which causes a residual
234
Semi-active Suspension Control
constant-amplitude friction force. The small amplitudes at the lowest frequencies
do not produce very significant pressure variations: in this range the constantamplitude friction force is not negligible. When the frequency increases, the
feedback signal is larger and the FD works properly. Hence, the residual friction
force deteriorates the performance of the damper if the disturbance is not large, i.e.,
on smooth roads and at low speed. This confirms that the compensation of the
residual friction force via a pre-loaded spring inside the damper would be an
effective remedy. Two more peaks are evident in the semi-active curve. This is a
non-linear effect of the semi-active system, but in fact the resonances do not create
any problem, because they are far lower in size than the corresponding passive
values.
Acceleration ratio
1.6
1.4
Acc. ratio
1.2
1
Acc ratio ACTIVE (7mm)
0.8
Acc ratio PASS (7mm)
0.6
0.4
0.2
0
0
1
2
3
4
5
6
Freq. [Hz]
Fig. 7.16. Acceleration ratio transmissibility for passive and semi-active systems. Sinusoidal
input to one wheel, amplitude: 7 mm [copyright Elsevier (2003), reproduced with minor
modifications from Guglielmino E, Edge KA, Controlled friction damper for vehicle
applications, Control Engineering Practice, Vol. 12, N 4, pp 431–443, used by permission]
Figure 7.17 portrays the rear-right suspension working space responses (scaled
units) in both cases. Over the frequency range considered the wheel motion can be
assumed to be almost steady, hence working space is a good approximation to
absolute chassis displacement. The results presented show that the semi-active
response is much the same as the passive response, except at the resonant
frequency. This is because of the force tracking performed by the controller which
inherently promotes higher displacements.
Case Studies
235
Working space
1.2
Displacement
1
0.8
Work space, semi-active
0.6
Work space, passive
0.4
0.2
0
0
2
4
6
Freq. [Hz]
Fig. 7.17. Working space for passive and semi-active systems. Sinusoidal input to one
wheel, amplitude: 7 mm [copyright Elsevier (2003), reproduced with minor modifications
from Guglielmino E, Edge KA, Controlled friction damper for vehicle applications, Control
Engineering Practice, Vol. 12, N 4, pp 431–443, used by permission]
Figure 7.18 depicts the acceleration response for three different input amplitudes.
Within reasonable approximations they are fairly similar. This means that
nonlinear effects are not very strong. Actually balance logic governing differential
equation is piecewise linear and for this particular class of equations the frequency
response does not depend on the input amplitude as is the case in nonlinear
systems3.
Acceleration ratio
1.4
1.2
Acc. ratio
1
Acc. ratio (9 mm)
0.8
Acc. ratio (7 mm)
0.6
Acc. ratio (5 mm)
0.4
0.2
0
0
1
2
3
4
5
6
Freq. [Hz]
Fig. 7.18. Acceleration ratio transmissibility for semi-active system. Sinusoidal input to one
wheel; amplitudes, 5 mm, 7 mm, 9 mm
3. In a linear system the RMS and peak transmissibility ratios (e.g., body over wheel acceleration) are
equal and independent of the input signal amplitude. In a piecewise-linear system RMS and peak
transmissibility ratios are not equal but still input amplitude independent. In a nonlinear system RMS
and peak transmissibility ratios are not equal and both input amplitude dependent.
236
Semi-active Suspension Control
Figures 7.19 and 7.20 show the effect of a change in the control law and in the
friction properties. Figure 7.19 examines the effect of changing the closed-loop
coefficient b from 2 to 3.33; b is the reciprocal of the friction coefficient μ, and
hence its increase from 2 to 3.33 can be treated as equivalent to a decrease of the
friction coefficient from 0.5 to 0.3. The performance with the reduced friction
shows some deterioration over part of the frequency range: this is to be expected
since, with a lower assumed friction coefficient, the force cancellation is smaller,
resulting in higher accelerations.
It is of interest to evaluate how a change in the frictional characteristic would
affect the performance of the control. The test of Figure 7.20 is carried out in a
situation of lubricated friction. The response in dry friction regime is here
compared to that in lubricated friction regime. This test is important, because
lubricated friction is a realistic alternative to pure dry friction: it helps reduce
stiction and is potentially advantageous in terms of heat dissipation between the
friction surfaces.
At low frequencies the dry friction system response is better than the lubricated
friction system response although both are worse than the passive response. In the
central frequency band the lubricated friction response is better, but at higher
frequencies the opposite holds.
Overall, the performance of the controlled suspension is superior to the passive
case and is not worsened by lubrication.
Acceleration ratio
1.6
1.4
Acc. ratio
1.2
1
Acc. ratio b = 3.3
0.8
Acc. ratio b = 2
0.6
Acc. ratio, passive
0.4
0.2
0
0
1
2
3
4
5
6
Freq. [Hz]
Fig. 7.19. Acceleration transmissibility ratio for passive and semi-active systems varying
controller gain; sinusoidal input to one wheel, amplitude: 7 mm [copyright Elsevier (2003),
reproduced with minor modifications from Guglielmino E, Edge KA, Controlled friction
damper for vehicle applications, Control Engineering Practice, Vol. 12, N 4, pp 431–443,
used by permission]
From the foregoing investigations involving changes in the feedback coefficient b
(equivalent to a change in dry friction coefficient) and the nature of the lubrication
regime, the robustness of the scheme, in a practical sense, has been experimentally
verified. Moreover, the semi-active system remains generally superior to the
passive system, even in the presence of these changes.
Case Studies
237
Accelerationratio
1.6
1.4
Acc. ratio
1.2
1
Acc. ratio (dry)
Acc. ratio (lubr)
Acc. ratio, passive
0.8
0.6
0.4
0.2
0
0
1
2
3
4
5
6
Freq. [Hz]
Fig. 7.20. Acceleration ratio transmissibility for passive and semi-active systems; sinusoidal
input to one wheel, amplitude: 7 mm [copyright Elsevier (2003), reproduced with minor
modifications from Guglielmino E, Edge KA, Controlled friction damper for vehicle
applications, Control Engineering Practice, Vol. 12, N 4, pp 431–443, used by permission]
Figure 7.21 shows the effect on RMS chassis acceleration of changing the control
law.
Body Acc. [m/s^2]
Chassis acceleration
2
1.8
1.6
1.4
1.2
1
0.8
0.6
body X
body XV
body XVV
body PASSIVE
0.4
0.2
0
0
1
2
3
4
5
6
Freq. [Hz]
Fig. 7.21. Chassis acceleration transmissibility for passive and semi-active systems;
sinusoidal input to one wheel, amplitude 7 mm
The performance of three types of controllers are compared. These are:
•
•
•
a pure position feedback (b = 2.5, z1 = 0, z2 = 0) controller (X in the
legend)
a controller with position feedback in two quadrants and velocity feedback
in the other two (b = 2.5, z1 = 0, z2 = 0.1; XV in the legend)
a controller with position plus velocity feedback in two quadrants and
velocity feedback in the other two (b = 2.5, z1 = 0.1, z2 = 0.1; XVV in the
legend)
238
Semi-active Suspension Control
The pure position feedback control logic achieves the best results in terms of
chassis acceleration reduction because it aims for pure spring force cancellation.
The other controller (b = 2.5, z1 = 0, z2 = 0.1) introduces a small viscous effect,
and thus produces a slightly higher chassis acceleration. It is almost comparable
with the previous controller in terms of wheel acceleration: over a part of the
frequency range it is worse, but in other parts slightly better.
The final controller (b = 2.5, z1 = 0.1, z2 = 0.1) produces the least chassis
acceleration reduction among the three types of controllers investigated and also
the worst wheel acceleration. Therefore the additional viscous damping in the first
and third quadrants is not advantageous.
7.4.2 Time-domain Analysis
Time domain analysis typically permits an assessment of the transient behaviour.
In this context it is mainly used to assess ride quality. Figures 7.22 and 7.23 show
jerk trends in the passive and semi-active cases. The latter trend presents slightly
higher peak values. Jerk is obtained by numerical differentiation of the measured
acceleration signal.
Jerk
20
15
Jerk [m/s^3]
10
5
0
-5
0
0.5
1
1.5
2
2.5
3
3.5
Jerk [m/s^3]
-10
-15
-20
-25
time [s]
Fig. 7.22. Experimental jerk time trace for the passive system; sinusoidal input, amplitude:
3 mm, frequency: 2.2 Hz
Case Studies
239
Jerk
20
15
Jerk [m/s^3]
10
5
0
-5 0
1
2
3
Jerk [m/s^3]
4
-10
-15
-20
-25
Time [s]
Fig. 7.23. Experimental jerk time trace for the controlled system; sinusoidal input,
amplitude: 3 mm, frequency: 2.2 Hz
Next a Fourier analysis of the acceleration waveforms is presented. Although the
spectrum is more appropriate to assess the degree of non-linearity rather than
comfort, it is possible to establish a qualitative correlation between a high
harmonic content and discomfort.
Figures 7.24–7.27 depict experimental time traces and spectra for the passive
and semi-active suspension. Figures 7.24 and 7.25 show measured acceleration
time histories and spectra for the passive case at frequencies of 2.2 and 4.5 Hz. At
2.2 Hz (Figures 7.24) the behaviour is fairly linear (only a negligible third
harmonic is present in the spectrum).
Body acceleration
FFT body acceleration (rectangular window)
0.8
0.7
0.6
0.6
0.5
Acc. amplitude [m/s2]
Acceleration [m/s 2]
0.4
0.2
0
-0.2
0.3
0.2
-0.4
0.1
-0.6
-0.8
0.4
0
0.5
1
1.5
Time [s]
2
a
2.5
3
0
0
5
10
15
20
25
30
Frequency [Hz]
35
40
45
50
b
Fig. 7.24. (a) Experimental acceleration time trace for the passive system; (b) experimental
acceleration FFT for the passive system; sinusoidal input, amplitude: 3 mm, frequency: 2.2
Hz
240
Semi-active Suspension Control
The same is true at 4.5 Hz (Figure 7.26). This further validates the hypothesis of
linearity of the passive system: the Fourier transform of the passive system
acceleration has a small harmonic content (and to a greater extent the
displacement, since acceleration is the noisiest signal).
Body acceleration
FFT body acceleration (rectangular window)
0.6
0.7
0.4
0.6
0.5
Acc. amplitude [m/s 2]
Acceleration [m/s 2]
0.2
0
-0.2
-0.4
0.3
0.2
-0.6
-0.8
0.4
0.1
0
0.1
0.2
0.3
0.4
0.5
Time [s]
0.6
0.7
0.8
0
0.9
0
5
10
15
a
20
25
30
Frequency [Hz]
35
40
45
50
b
Fig. 7.25. (a) Experimental acceleration time trace for the passive system; (b) experimental
acceleration FFT for the passive system; sinusoidal input, amplitude: 3 mm, frequency:
4.5 Hz
Figures 7.26 and 7.27 depict the same graphs for the controlled system. The
harmonic content is not very large at 2.2 Hz, but is certainly larger at 4.5 Hz and
actually the acceleration time trend is fairly spiky. This spectral analysis of the
acceleration has hence confirmed the qualitative correlation between jerk time
trends and richer higher-harmonic content.
Body acceleration
FFT body acceleration (rectangular window)
0.8
0.7
0.6
0.6
0.4
Acc. amplitude [m/s 2]
Acceleration [m/s 2]
0.5
0.2
0
-0.2
0.3
0.2
-0.4
0.1
-0.6
-0.8
0.4
0
0.5
1
1.5
Time [s]
a
2
2.5
3
0
0
5
10
15
20
25
30
Frequency [Hz]
35
40
45
50
b
Fig. 7.26. (a) Experimental acceleration time trace for the controlled system; (b)
experimental acceleration FFT for the controlled system; sinusoidal input, amplitude: 3 mm,
frequency: 2.2 Hz
Case Studies
Body acceleration
241
FFT body acceleration (rectangular window)
1.5
0.9
0.8
1
Acc. amplitude [m/s 2]
Acceleration [m/s 2]
0.7
0.5
0
-0.5
0.6
0.5
0.4
0.3
0.2
-1
0.1
-1.5
0
0.1
0.2
0.3
0.4
0.5
Time [s]
0.6
0.7
0.8
0
0.9
0
5
10
a
15
20
25
30
Frequency [Hz]
35
40
45
50
b
Fig. 7.27. (a) Experimental acceleration time trace for the controlled system; (b)
experimental acceleration FFT for the controlled system; sinusoidal input, amplitude: 3 mm,
frequency: 4.5 Hz
Figure 7.28 shows the semi-active response to a pseudo-random input. The RMS of
the acceleration is 0.58 m/s2, which is smaller than in the passive case (0.75 m/s2).
Figure 7.29 shows the semi-active system response to a sinusoidal bump. The
bump input is the same as in the passive case. The acceleration overshoot and
undershoot for the controlled system (in the case b = 2.5, z1 = 0, z2 = 0) are 1.1
m/s2 and –0.8 m/s2 respectively, whereas for the passive system the values are
±0.85 m/s2. The number of oscillations is the same for both cases. Thus the
controlled system is slightly worse in response to a bump. This is because of the
relatively slow response of the pressure control circuit. A bump can be thought as a
high-frequency half-wave input (the higher the vehicle velocity, the higher the
frequency) and above a certain frequency the servo-system response is not swift
enough.
Chassis response to random input
Acc. [m/s^2]
3
2
1
0
-1 0
2
4
6
8
10
12
-2
-3
Time [s]
Fig. 7.28. Semi-active chassis acceleration time trend; random input: 25-Hz filtered white
noise [copyright Elsevier (2003), reproduced with minor modifications from Guglielmino E,
Edge KA, Controlled friction damper for vehicle applications, Control Engineering Practice,
Vol. 12, N 4, pp 431–443, used by permission]
242
Semi-active Suspension Control
Acceleration bump response
Acc. [m/s^2]
1.5
1
0.5
0
-0.5 0
2
4
6
8
10
12
-1
-1.5
Time [s]
Fig. 7.29. Bump response acceleration time trace for the controlled system [copyright
Elsevier (2003), reproduced with minor modifications from Guglielmino E, Edge KA,
Controlled friction damper for vehicle applications, Control Engineering Practice, Vol. 12,
N 4, pp 431–443, used by permission]
7.4.3 Semi-active System Validation
Having validated separately the electrohydraulic drive and the passive vehicle, the
final step is the validation of the whole system. Figure 7.30 shows the predicted
and measured sinusoidal time responses after the start-up transient has decayed.
The simulation follows the overall trend of the measured acceleration well. The
spikes in the experimental results are not captured by the model, but most of them
arise from noise present in the measurements. The overall agreement can be
considered acceptable.
acceleration
simulated
2.5
experimental
2
acceleration [m/s 2]
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
1.6
1.8
2
2.2
time [s]
2.4
2.6
2.8
Fig. 7.30. Rear right semi-active acceleration; sinusoidal input to one wheel, amplitude:
7
mm, frequency: 2.5 Hz [copyright Elsevier (2003), reproduced with minor modifications
from Guglielmino E, Edge KA, Controlled friction damper for vehicle applications, Control
Engineering Practice, Vol. 12, N 4, pp 431–443, used by permission]
Case Studies
243
Considering now the frequency domain, a sensitivity analysis needs to be carried
out to identify the most suitable values for the critical parameters. Figure 7.31
represents the frequency response for different friction coefficients μ between 0.1
and 0.2. At low frequency the trend is virtually independent of the friction
coefficient but after the resonance the dependency gets stronger. A smaller friction
coefficient produces a larger resonance peak but at higher frequency it tracks the
experimental response better, and vice versa for a larger coefficient. The mismatch
occurring for frequencies higher than 3.8 Hz is due to the limitations in the car
model and the over-simplified model for the other three (passive) dampers, rather
than to the hydraulic model of the friction damper drive. The actual value of the
nominal friction coefficient of the material was 0.4. The mismatch between
simulation and experiments is related to the
μ
ratio (see Section 5.5).
μ assumed
Figure 7.32 portrays the frequency response for different levels of delay
between velocity and friction force created by the frictional memory effect. A
change of ±50% does not produce any significant change except at the lowest
frequencies.
In Figure 7.33 the effect of a change in the actuator and connecting pipe
volume is considered. An increase of an order of magnitude in the volume
produces a noticeable effect for frequencies above 2.5 Hz. The impact of the
corresponding reduction of the valve-actuator bandwidth, following an increase in
volume, is a higher acceleration. This physically occurs because the valve-actuator
system cannot catch up with the higher-frequency input; therefore the effect of the
residual constant friction force plays the dominant role.
RMS rear right acceleration
2.2
2
RMS rear right acceleration [m/s 2]
1.8
1.6
mu=0.2
1.4
mu=0.15
1.2
1
mu=0.1
0.8
0.6
0.4
0.2
1
1.5
2
2.5
3
3.5
Frequency [Hz]
4
4.5
5
Fig. 7.31. Rear right semi-active acceleration frequency response varying friction
coefficient; sinusoidal input to one wheel, amplitude: 7 mm [copyright Elsevier (2003),
reproduced with minor modifications from Guglielmino E, Edge KA, Controlled friction
damper for vehicle applications, Control Engineering Practice, Vol. 12, N 4, pp 431–443,
used by permission]
244
Semi-active Suspension Control
RMS rear right acceleration
2
RMS rear right acceleration [m/s 2]
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
1
1.5
2
2.5
3
3.5
Frequency [Hz]
4
4.5
5
Fig. 7.32. Rear right semi-active acceleration frequency response varying frictional
memory; sinusoidal input to one wheel, amplitude: 7 mm [copyright Elsevier (2003),
reproduced with minor modifications from Guglielmino E, Edge KA, Controlled friction
damper for vehicle applications, Control Engineering Practice, Vol. 12, N 4, pp 431–443,
used by permission]
Therefore it can be concluded that simulation provides accurate results in the time
domain and in the frequency domain up to almost 4 Hz. Beyond that frequency the
predicted acceleration is too great.
RMS rear right acceleration
2
RMS rear right acceleration [m/s 2]
1.8
1.6
V=9*10-4
1.4
1.2
V=10-4
1
0.8
0.6
0.4
0.2
1
1.5
2
2.5
3
3.5
Frequency [Hz]
4
4.5
5
Fig. 7.33. Rear right semi-active acceleration frequency response varying volume;
sinusoidal input to one wheel, amplitude: 7 mm [copyright Elsevier (2003), reproduced with
minor modifications from Guglielmino E, Edge KA, Controlled friction damper for vehicle
applications, Control Engineering Practice, Vol. 12, N 4, pp 431–443, used by permission]
Case Studies
245
7.5 Case Study 2: MR-based SA Seat Suspension
This case study concerns an investigation of the use of a controlled MR damper for
a semi-active seat suspension in vehicles not equipped with primary suspensions
(e.g., some types of agricultural, forestry and roadwork vehicles). The control
strategy is targeted to improve driver comfort based on a hybrid variable structure
fuzzy logic controller. Variable structure control is inherently a switching logic,
hence more prone to cause chattering. Fuzzy logic helps reduce chatter without
penalising damper dynamic response.
Fig. 7.34. Suspension, seat and driver model [copyright Inderscience (2005), reproduced
from Guglielmino E, Stammers CW, Stancioiu D, Sireteanu T and Ghigliazza R, Hybrid
variable structure-fuzzy control of a magnetorheological damper for a seat suspension,
International Journal of Vehicle Autonomous Systems, Vol. 3, N 1, used by permission]
The whole system, depicted in Figure 7.34, is composed of the suspension, the seat
and the driver (i.e., seated body model) and can be modelled as a 3DOF system.
The seat is mounted on the semi-active suspension constituted by a linear spring
and the MR damper. The seat and driver model (Wei and Griffin, 1998a) consists
of a rigid frame m1 to which two masses, m2 and m3 are suspended. The two masses
cannot be associated with particular organs of human body (refer to Chapter 3 for
further details). For convenience they will be denoted as upper mass and lower
mass. A driver seat suspension usually includes a linkage mechanism between the
suspension unit and the seat. Its presence can be taken into account in the model by
employing an equivalent value for the suspension stiffness and by appropriate
scaling factors for the damping force which take into account the geometry of the
linkage. Setting x = x1 − x0 , the equations of motion can be written as
246
Semi-active Suspension Control
m3 x3 = B3 ( x1 − x3 ) + K 3 ( x1 − x3 ) = f 3 ,
m2 x2 = B2 ( x1 − x 2 ) + K 2 ( x1 − x2 ) = f 2 ,
(7.2)
m1 x1 = F1 ( x , x , z ,u ) + K1 x − f 3 − f 2 ,
where F1 ( x , x , z ,u ) is the seat suspension damping force (a Bouc–Wen model has
been employed), K1 the suspension stiffness, K2 and K3 the lower and upper body
stiffness and B2 and B3 the lower and upper body damping. The value of the
suspension stiffness is chosen so its natural frequency is around 1.5 Hz. Such a
value corresponds to a good passive design, because it filters out the frequencies in
the range 3–6 Hz, which are the worst for human comfort. The reason for this
choice is to show that the controlled system can produce an improvement of the
response also in presence of a good passive seat suspension. The simulation
parameters are listed in Table 7.1.
Table 7.1. Key parameters employed in simulation [copyright Inderscience (2005),
reproduced with minor modifications from Guglielmino E, Stammers CW, Stancioiu D,
Sireteanu T and Ghigliazza R, Hybrid variable structure-fuzzy control of a
magnetorheological damper for a seat suspension, International Journal of Vehicle
Autonomous Systems, Vol. 3, N 1, used by permission]
Passive linear damper
B1 = 180 Ns/m
Body upper damping
B2 = 761 Ns/m
Body lower damping
B 3 = 458 Ns/m
Suspension stiffness
K1 = 4500 N/m
Body upper stiffness
K 2 = 35776 N/m
Body lower stiffness
K 3 = 38374 N/m
Seat mass
m1 = 6 kg
Body upper mass
m 2 = 33.4 kg
Body lower mass
m 3 = 10.7 kg
Bouc–Wen model coefficients
A = 2000; β = γ = 2000 m-2
Bouc–Wen exponent and offset force
n = 2; f0 = 15 N
With reference to Figure 7.34, the variable the value of which must be minimised
is the arithmetic mean of the RMS of lower and upper mass accelerations, x2 and
x3 , as this quantity is related to driver vertical acceleration. The frequency range
of interest for comfort is up to around 6 Hz (far within the MRD bandwidth).
The aim of the control is to reduce the forces transmitted to the seat by
generation of a spring-like (position-dependent damping) control force of sign
opposite to that of the spring force.
The logic is implemented by controlling the solenoid current; the control logic
can be expressed by the following functional equation:
Case Studies
F1 ( x, x , z , u ) = F ( x, x , z , u ( x, x )) ,
247
(7.3)
where the current u ( x, x ) is expressed by:
⎧ a x if xx ≤ 0
u ( x , x ) = ⎨
if xx > 0 ,
⎩0
(7.4)
a being a gain proportional to suspension spring stiffness. This balance controller
is switching type and this may cause chattering problems when the controller
switches from one structure to the other.
MRD dynamic response is extremely swift since it is mainly dependent upon
electromagnetic dynamics and the time necessary for the oil to reach rheological
equilibrium. The fast switching produces periodical acceleration and jerk peaks
which degrade ride quality. Chattering problems in fast-dynamics dampers
controlled via switched-type algorithms have been reported in work by Choi et al.
(2000), who pointed out the problem in a study on sliding-mode control of
electrorheological dampers. The problem can be tackled at the control level by
smoothing the control action by using fuzzy logic. In this way it is possible to
soften the fast switching action of the crisp balance controller, without the need for
low-pass filters which would reduce system bandwidth, which is one the major
benefits of using an MRD.
The fuzzy-controlled damping force is expressed as:
F1 ( x , x , z ,u ) = F ( x , x , z ,η x u ( x , x )) ,
(7.5)
where u ( x , x ) is the fuzzy controller current input and η is a gain.
The variable structure algorithm has been fuzzified by choosing as fuzzy
variables the relative displacement and velocity; the linguistic variables are:
negative (neg) and positive (pos). The membership functions are depicted in
Figure 7.35.
248
Semi-active Suspension Control
Fig. 7.35. Fuzzy logic membership functions [copyright Inderscience (2005), reproduced
from Guglielmino E, Stammers CW, Stancioiu D, Sireteanu T and Ghigliazza R, Hybrid
variable structure-fuzzy control of a magnetorheological damper for a seat suspension,
International Journal of Vehicle Autonomous Systems, Vol. 3, N 1, used by permission]
The fuzzy controller function is a two-value function u ( x(t ) , x (t )) ∈ [0 ,u max ] where
Small = 0 and Big = umax. The fuzzy set rules (Table 7.2) are obtained by fuzzifying
the control logic defined in Equation 7.4. In this way, the transitions between the
two structures are not abrupt.
Table 7.2. Fuzzy logic rules [copyright Inderscience (2005), reproduced with minor
modifications from Guglielmino E, Stammers CW, Stancioiu D, Sireteanu T and Ghigliazza
R, Hybrid variable structure-fuzzy control of a magnetorheological damper for a seat
suspension, International Journal of Vehicle Autonomous Systems, Vol. 3, N 1, used by
permission]
Velocity
Negative
Positive
Negative
Small
Big
Positive
Big
Small
Displacement
7.5.1 Numerical Results
The system is tested with a random input x0(t) obtained from a measured power
spectral density of the seat base acceleration in operating conditions. The excitation
Case Studies
249
RMS value is σ x = 0.02 m. Table 7.3 reports the power spectral density RMS of
0
the controlled variables for the different controllers.
Table 7.3. Performance assessment of passive and semi-active systems [copyright
Inderscience (2005), reproduced with minor modifications from Guglielmino E, Stammers
CW, Stancioiu D, Sireteanu T and Ghigliazza R, Hybrid variable structure-fuzzy control of a
magnetorheological damper for a seat suspension, International Journal of Vehicle
Autonomous Systems, Vol. 3, N 1, used by permission]
σ x [m/s2]
2
σ x [m/s2]
3
σ x + σ x
2
3
2
[m/s2]
Passive linear viscous damper
3.06
3.23
3.14
Passive MRD
3.68
3.94
3.81
VSC (crisp) with MRD
2.03
2.52
2.27
Fuzzy control with MRD
2.20
2.75
2.48
A reduction of the RMS of the controlled variable of about 21% is achieved with
fuzzy control, with respect to a passive system with a traditional viscous damper.
The crisp controller performs slightly better (27% reduction). However the merits
of the fuzzy controller are in terms of chattering reduction as is evident from
Figure 7.36 where the demands for the VSC and fuzzy algorithms are presented.
Fig. 7.36. Current demand for VSC (
) and fuzzy (
) controllers [copyright
Inderscience (2005), reproduced from Guglielmino E, Stammers CW, Stancioiu D,
Sireteanu T and Ghigliazza R, Hybrid variable structure-fuzzy control of a
magnetorheological damper for a seat suspension, International Journal of Vehicle
Autonomous Systems, Vol. 3, N 1, used by permission]
The fast switching action of the crisp controller is softened by the fuzzy algorithm
and the current transitions between the on and the off state are smoother; this in
250
Semi-active Suspension Control
2.5
2.5
2
2
RMS acc. [m/s^2]
RMS acc. [m/s^2]
turn reduces peaks of acceleration arising in the instants of transition, thus causing
improvement in the ride quality, although the RMS is slightly higher.
It is also worthwhile noting that an MRD without control performs worse than a
linear viscous damper. This is due to the large forces at low velocity as well as to
the large hysteresis in its characteristics.
Finally Figure 7.37 shows the transmissibility trends (RMS acceleration versus
frequency) for upper and lower mass accelerations, superimposed on the response
of the passive system. A reduction in the magnitude of the resonance peak is
noticeable, which will produce an increase in comfort. The peak shift to about 1 Hz
is beneficial as the body is a little less sensitive to vertical vibration at 1 Hz than at
1.5 Hz.
1.5
1
0.5
1.5
1
0.5
0
0
0
1
2
3
Frequency [Hz]
a
4
5
6
0
1
2
3
4
5
6
Frequency [Hz]
b
Fig. 7.37. (a) Upper mass acceleration transmissibility; passive (
), fuzzy (
);
(b) lower mass acceleration transmissibility; passive (
), fuzzy (
), [copyright
Inderscience (2005), reproduced from Guglielmino E, Stammers CW, Stancioiu D,
Sireteanu T and Ghigliazza R, Hybrid variable structure-fuzzy control of a
magnetorheological damper for a seat suspension, International Journal of Vehicle
Autonomous Systems, Vol. 3, N 1, used by permission]
7.5.2 Conclusions
The semi-active seat suspension with a controllable MRD can satisfactorily reduce
the vertical acceleration experienced by the driver. By using the hybrid variable
structure fuzzy control strategy a RMS reduction of about 21% is achieved. Using
crisp control slightly better results can be obtained in terms of RMS but the spikes
induced by the variable structure controller switching action worsen ride quality. In
order to eliminate these spikes filtering would be necessary, but this would reduce
the MRD bandwidth; fuzzy logic instead allows full use of the MRD bandwidth.
Case Studies
251
7.6 Case Study 3: Road Damage Reduction with MRD Truck
Suspension
7.6.1 Introduction
Heavy vehicles travel over a variety of road surfaces and experience a wide range
of vibration. This affects not only ride quality, but also causes road damage.
Weather conditions as well as vehicle motion are two key causes of road
pavement damage. Road damage requires significant investment every year to
repair the road surface and causes delays to traffic while it is being done.
Heavy vehicle suspensions ought to be able to isolate the sprung mass from
road-induced disturbances as well as improving handling and minimising road
damage by reducing dynamic tyre force within the constraint of a set working
space.
The reduction of dynamic tyre forces is a challenging field. Cole and Cebon
(1996a and 1996b) did extensive work on it, both theoretical and experimental. A
thorough analysis of the damage due to dynamic tyre forces and other co-factors is
presented by Cebon (1999): an instrumented vehicle was employed to measure the
dynamic tyre forces at both low and high excitation frequencies. The study
concluded that the wheel dynamic load increases with both vehicle speed and road
roughness.
Algorithms are aimed at the reduction of tyre load oscillations, which improves
handling on one hand and on the other reduces road damage caused by vehicle
wheels. This latter application is particularly important in the case of heavy freight
vehicles. Extended groundhook control logic was investigated by Valasek et al.
(1998) to reduce dynamic tyre forces.
This case study presents a hybrid balance algorithm to reduce road damage and
investigate the performance of a heavy articulated vehicle equipped with both MR
dampers and passive viscous dampers. A half truck model is employed and system
performance investigated via numerical simulation. A variation of the balance
logic strategy based on dynamic tyre force tracking has been devised. Algorithm
robustness to parametric variations as well as to real-life implementation issues
such as feedback signals noise are discussed.
Given the modest cost of MR dampers compared to the overall cost of the
vehicle, and the fact that road maintenance requires significant investment
worldwide (an estimated of several hundred millions pounds of road damage in the
UK was judged to be due to heavy vehicles; Potter et al., 1995) these benefits are
cost effective.
A model is developed for the semi-active control of the suspension of a threeaxle tractor–trailer combination. Control is applied via an MR damper at either the
tractor rear axle or the trailer axle. The load on the tractor front axle is smaller than
that on the other two axles and is much less significant from the point of view of
road damage.
Control reduces dynamic tyre forces and hence road damage, while improving
handling. Trailer sprung mass acceleration (heave and pitch) is also reduced.
Robustness of control is established by adding noise to the computed sensor inputs.
252
Semi-active Suspension Control
7.6.2 Half Truck and MR Damper Model
The half truck model is based on that described in Chapter 2 (Equations 2.31 and
2.32) and pictured in Figure 2.18. Lateral and yaw motions are neglected at this
stage, reducing the model complexity. The roll motion is also neglected. Half truck
vehicle parameters employed in the simulation are listed in Table 7.4.
The vehicle travels in a straight course with constant speed and is modelled as a
three-axle vehicle, the steer axle, the drive tractor axle and the trailer axle,
assuming two MR dampers, one fitted on the tractor drive axle and one on the
trailer axle of the half truck; the steer axle is equipped with a passive viscous
damper.
The objective is to investigate the vehicle performance in terms of ride and
road damage using two semi-active dampers controlled by a hybrid logic for
various road profiles. The MR damper model employed is based on the work by
Lau and Liao (2005), who designed and modelled a prototype damper for a train
suspension. Such a damper develops forces of the same order of magnitude as
those required in a truck application and in this respect it could be potentially
suitable for heavy vehicle applications as well. The schematic diagram of the
damper model is shown in Figure 7.38.
Fig.7.38. MR damper schematic model [copyright IMechE (2008), reproduced from
Tsampardoukas G, Stammers CW and Guglielmino E, Semi-active control of a passenger
vehicle for improved ride and handling, accepted for publication in Proceedings of the
Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering, Publisher:
Professional Engineering Publishing, ISSN 0954/4070, Vol. 222, D3/2008, pp 325–352,
used by permission]
It is a Bouc–Wen model coupled with a non-linear viscous damper with
exponential characteristics and a linear spring term. The governing equations are as
follows:
z = − γ x z
n −1
z − β x z
n
C ( x ) = − a 1 exp( − a 2 x )
+ A x ,
(7.6)
p
(7.7)
,
F − f 0 = az + kx + C ( x ) x + m x .
(7.8)
Case Studies
253
Table 7.4. Key parameters employed in simulation (copyright Elsevier, reproduced from
Tsampardoukas G, Stammers CW and Guglielmino E, Hybrid balance control of a
magnetorheological truck suspension, accepted for publication in Journal of Sound and
Vibration, used by permission)
Parameter
Value
Tractor chassis mass
mc = 4400 kg
Trailer chassis mass (laden)
mt = 12500 kg
Tractor pitch inertia
Jc = 18311 kgm2
Trailer pitch inertia
Jt = 251900 kgm2
Steer tractor unsprung mass
mu1 = 270 kg
Drive tractor unsprung mass
mu2 = 520 kg
Trailer unsprung mass
mu3 = 340 kg
Distance from steer tractor axle to tractor C.G.
l1 = 1.2 m
Distance from tractor C.G. to drive tractor axle
l2 = 4.8 m
Distance from tractor C.G. to articulation point
l4 = 4.134 m
Distance from articulation point to trailer C.G.
l5 = 6.973 m
Distance from trailer C.G. to trailer axle
l6 = 4 m
Tyre stiffness of steer tractor wheel
ktf = 847 kN/m
Tyre stiffness of drive tractor wheel
ktr = 2 MN/m
Tyre stiffness of trailer wheel
ktt = 2 MN/m
Suspension spring stiffness of steer tractor axle
kf = 300 kN/m
Suspension spring stiffness of drive tractor axle
kr = 967430 N/m
Suspension spring stiffness of trailer axle
kt = 155800 N/m
Suspension damping rate of steer tractor axle
cf = 10 kNs/m
Suspension damping rate of drive tractor axle
cr =27627 Ns/m
Suspension damping rate of trailer axle
ct = 44506 Ns/m
Stiffness of articulated connection
k5 = 20 MN/m
Damping of articulated connection
c5 = 200 kNs/m
As explained in Chapter 2, the variable z is an evolutionary variable while the
parameters β, γ, Α and n define the shape of the hysteresis loop. Equation 7.7
models the post-yield plastic damping coefficient, which depends on the relative
velocity. This equation is used to describe the MR fluid shear thinning effect,
which results in the roll-off of the resisting force of the damper in the low-velocity
region. The total exerted force is described by Equation 7.8, which takes into
account the evolutionary variable z and the post-yield plastic model, expressed by
254
Semi-active Suspension Control
(7.7). Table 7.5 lists the numerical value of the constant parameters of the MRD
(while the parameters a , a1 , a2 , n and fo are current dependent).
Table 7.5. Constant parameters for MR damper (copyright Elsevier, reproduced from
Tsampardoukas G, Stammers CW and Guglielmino E, Hybrid balance control of a
magnetorheological truck suspension, accepted for publication in Journal of Sound and
Vibration, used by permission)
Parameter
Value
-2
Parameter
Value
γ
32000 m
m
100 kg
β
22 m-2
k
2500 N/m
Α
220
p
0.54
15
15
10
10
Damper force (kN)
Damper force (kN)
The simulated characteristics are depicted in Figure 7.39.
5
0
-5
0
-5
-10
-10
-15
-30
5
-20
-10
0
10
Displacement (mm)
a
20
30
-15
-0.2
-0.1
0
Velocity (m/s)
0.1
0.2
b
Fig. 7.39. (a) MR damper force velocity and (b) force displacement characteristics for a
22-mm 1-Hz sinusoidal displacement excitation (copyright Elsevier, reproduced from
Tsampardoukas G, Stammers CW and Guglielmino E, Hybrid balance control of a
magnetorheological truck suspension, accepted for publication in Journal of Sound and
Vibration, used by permission)
A benchmark viscous damper having different damping coefficients for the bound
and rebound strokes was used (the equivalent damping ratios are, respectively, 0.15
and 0.35).
Appropriate road profile models are required to assess truck performance under
realistic operative conditions. Two types of road are considered in this case study:
a smooth highway and a highway with gravel. The spectral densities of both road
profiles are expressed by Sg(Ω)=CspΩ-n, the parameters Csp and n determining the
road quality. The smooth highway is described by n = 2.1 and Csp=4.8· × 10-7
whereas the gravel road is determined by setting Csp=4.4· × 10-6 and n = 2.1 (Wong,
1993). Heavy goods vehicles are mainly designed to operate on smooth rather than
on poor roads. The vehicle operation on highway with gravel is a scenario to
Case Studies
255
examine the performance of the semi-active suspension on damaged roads or in
off-road operation.
7.6.3 Road Damage Assessment
It is of paramount importance to establish a quantitative criterion to assess road
damage (Cebon, 1989). The most widely employed is the fourth power law. This is
a result of the experimental work undertaken by the American Association of State
Highway and Transportation Officials (AASHO) (Gillespie, 1985). This law shows
that the pavement serviceability decreases every time a heavy vehicle axle passes
on the road. This reduction is assumed to be related to the fourth power of its static
load (Cebon, 1999). Another criterion is known as the aggregate fourth power
force (Cole et al., 1994) while Potter et al. (1995) give a simplified approach to
road damage. It is expressed by the following formula:
A kn =
Na
∑P
n
jk
,
(7.9)
j =1
where k =1, 2, 3,…, Na is the location along the road. The exponent n is chosen
depending upon the type of pavement and ranges from n = 4 (suitable for fatigue
damage) to n = 1 (permanent deformation caused by static load). In this work in
order to describe the fatigue damage, the aggregate fourth power law with n = 4 is
used, normalised with respect to the static force i.e.,
A k4 =
4
⎛ Static force + Dynamic force ⎞
⎟
⎜
∑
Static force
⎠ jk
j =1 ⎝
Na
(7.10)
7.6.4 Road Damage Reduction Algorithm
The algorithm outlined here is a variant of the balance logic described in
Chapter 4. This hybrid version aims at cancelling the drive tractor and trailer axle
tyre forces. The essence of the proposed control algorithm is to cancel the tyre
force fluctuations on each axle by ensuring that the wheel follows the road profile
closely. The dynamic tyre forces are balanced by applying a controlled damping
force in the opposite direction. This is only possible when the control force and the
relative velocity have opposite signs and hence energy dissipation takes place.
⎧br (− Fsr − mu 2 xwr ) + b2 Fdr if Fc rear ⋅ rear_rel_vel < 0
Fc rear = ⎨
if Fcrrear ⋅ rear_rel_vel > 0
⎩b3 Fdr
(7.11)
⎧bt (− Fst − mu 3 xwt ) + b2 Fdt if Fc trailer ⋅ trailer_rel_vel < 0
(7.12)
Fc trailer = ⎨
if Fc trailer ⋅ trailer_rel_vel > 0
⎩b3 Fsd
256
Semi-active Suspension Control
A pseudo-viscous damping term is added to the control forces to reduce
transients, particularly when inputs are near to the wheel-hop frequency. Studies
(not reported here) have indicated that the optimal values of b2 and b3 (to minimise
road damage) should be 20% of critical passive damping when the vehicle travels
on smooth or gravel roads. Smaller or higher values of b2 and b3 result in larger
dynamic tyre forces and higher vibration levels. However, the optimum values of
those parameters (in terms of road damage) alter when the vehicle wheels come
into contact with bumps or potholes.
7.6.5 Time Response
In this section, and in the following ones, the response of the semi-active
controlled MRD system is benchmarked against the passive system. The
performance of the controlled system is assessed numerically both in the time
domain and in the frequency domain.
The time response to a sinusoidal road surface is firstly investigated.
Figures 7.40–7.42 depict the time histories for a 5-mm 2-Hz sinusoidal road input.
The hybrid control logic significantly reduces the dynamic tyre forces of the tractor
drive and trailer axles. The same trend is also observed for the trailer chassis
acceleration. In contrast, the controlled suspension increases the chassis
acceleration of the tractor unit. As expected a higher harmonic content is present in
the semi-active time trends.
4
(a)
Dynamic tyre force (kN)
Dynamic tyre force (kN)
2
1
0
-1
-2
-3
-4
(b)
10
3
5
0
-5
-10
9.8
10
10.2
10.4 10.6
Time (s)
a
10.8
11
10
10.2
10.4 10.6
Time (s)
10.8
11
b
Fig. 7.40. Dynamic tyre forces on tractor drive and trailer axles: (a) dynamic tyre force of
tractor drive axle; (b) dynamic tyre forces of trailer axle; (
) semi-active
) passive suspension (copyright Elsevier, reproduced from
suspension, (
Tsampardoukas G, Stammers CW and Guglielmino E, Hybrid balance control of a
magnetorheological truck suspension, accepted for publication in Journal of Sound and
Vibration, used by permission)
Case Studies
a
257
b
Fig. 7.41. Tractor and trailer chassis heave acceleration: (a) tractor chassis heave
acceleration; (b) tractor chassis pitch acceleration; (
) semi-active suspension,
) passive suspension (copyright Elsevier, reproduced from Tsampardoukas G,
(
Stammers CW and Guglielmino E, Hybrid balance control of a magnetorheological truck
suspension, accepted for publication in Journal of Sound and Vibration, used by permission)
a
b
Fig. 7.42. Trailer chassis heave and pitch accelerations (a) trailer chassis heave acceleration;
) semi-active suspension, (
) passive
(b) trailer chassis pitch acceleration; (
suspension (copyright Elsevier, reproduced from Tsampardoukas G, Stammers CW and
Guglielmino E, Hybrid balance control of a magnetorheological truck suspension, accepted
for publication in Journal of Sound and Vibration, used by permission)
258
Semi-active Suspension Control
7.6.6 Truck Response on Different Road Profiles
RMS pitch acceln (rad/s2)
0.7
(a)
0.6
0.5
0.4
0.3
0.2
5
10
15
20
Vehicle velocity (m/s)
2.5
25
(c)
2
1.5
1
0.5
0
5
10
15
20
Vehicle velocity (m/s)
25
RMS pitch acceln (rad/s2)
RMS heave acceln (m/s2)
RMS heave acceln (m/s2)
The percentage reduction achieved by the semi-active case relative to both passive
cases (MRD with current I = 0 A and passive viscous damping) is given by
Figures 7.43 and 7.44 for vehicle accelerations and road damage. The benefits of
semi-active control are evident, particularly at moderate speed, but limited in
regard to road damage because the predominant load (and hence damage) is the
static one.
0.5
(b)
0.4
0.3
0.2
0.1
5
10
15
20
Vehicle velocity (m/s)
0.4
25
(d)
0.3
0.2
0.1
0
5
10
15
20
Vehicle velocity (m/s)
25
Fig. 7.43. RMS heave and pitch accelerations: (a) tractor chassis; (b) tractor chassis; (c)
) conventional passive viscous damper, (
)
trailer chassis; (d) trailer chassis; (
passive MR damper (current = 0 A), (
) semi-active MR damper (copyright Elsevier,
reproduced from Tsampardoukas G, Stammers CW and Guglielmino E, Hybrid balance
control of a magnetorheological truck suspension, accepted for publication in Journal of
Sound and Vibration, used by permission)
The semi-active MR damper significantly reduces the trailer unit heave and pitch
accelerations. This is due to the improved isolation achieved by the two MR
dampers fitted on the drive tractor and trailer axles. However, at high vehicle speed
the semi-active suspension degrades the vehicle performance in terms of tractor
chassis accelerations. This occurs because the controllable damper is installed only
on the drive tractor axle. It has been verified that the tractor performance would
improve if a third MR damper was fitted on the steer axle as well. However truck
driver seat are often equipped with a seat damper (ideally semi-active; see case
study 2), which in practice reduces the vibration level experienced by the driver.
Case Studies
259
Normalised road damage
2
(a)
1.8
1.6
1.4
Normalised road damage
5
10
15
20
Vehicle velocity (m/s)
6
25
Normalised road damage
Normalised road damage
The application of the road damage criterion given in Figure 7.44 reveals that
the damage caused by the dynamic tyre forces is significantly reduced when the
balance control cancels them by 100%, while the coefficients b2 and b3 are equal to
0.2 of the critical damping force. Figure 7.44 shows that the semi-active
suspension reduces significantly the road damage caused by each individual axle as
well as the total vehicle damage. It is important to note that the vehicle suspension
employing passive MR dampers (i.e., I = 0 A) degrades the vehicle response due to
its low damping.
The amount of dynamic tyre force cancellation is a critical parameter which
affects the system response as Figure 7.45 shows. At low and medium vehicle
velocities 100% cancellation is the best option because both RMS and max
dynamic tyre forces at each axle are significantly reduced relative to the passive
system, resulting in lower road damage with respect to the damage criterion used.
On the other hand, it is beneficial to reduce the amount of cancellation for the
damper fitted on the drive tractor axle in order to improve tractor unit comfort,
particularly at high vehicle velocities, but such a reduction adversely affects road
damage at high velocities. Consequently, the optimal choice for the amount of
spring force cancellation depends on the control objective.
(c)
4
2
0
5
10
15
20
Vehicle velocity (m/s)
25
4
(b)
3
2
1
5
10
15
20
Vehicle velocity (m/s)
10
25
(d)
8
6
4
2
5
10
15
20
Vehicle velocity (m/s)
25
Fig. 7.44. Maximum normalised road damage: (a) steer axle; (b) drive axle; (c) trailer axle;
) passive viscous damper, (
) passive MR damper (current =
(d) total vehicle: (
0 A), (
) semi-active MR damper (copyright Elsevier, reproduced from
Tsampardoukas G, Stammers CW and Guglielmino E, Hybrid balance control of a
magnetorheological truck suspension, accepted for publication in Journal of Sound and
Vibration, used by permission)
A design solution which achieves a compromise between these two requirements
entails the use of a suspended driver cab and seat to help reduce the vibration
260
Semi-active Suspension Control
3
2
1
5
10
15
20
Vehicle velocity (m/s)
25
(c)
10
8
6
4
5
10
15
20
Vehicle velocity (m/s)
25
RMS dynamic tyre force (kN)
4 (a)
MAX dynamic tyre force (kN)
MAX dynamic tyre force (kN)
RMS dynamic tyre force (kN)
levels transmitted to the human body. In that case, 100% cancellation is the best
solution in terms of lower maximum dynamic tyre forces for the semi-active device
located at the tractor drive axle, while 50% cancellation is the best solution overall
for the same device at the trailer axle. The variation between the amount of
cancellation between the units is mainly affected by the coupled vehicle motion
and the model (half truck) used. Consequently, a full truck model should be
developed in order to evaluate the performance of the hybrid control logic not only
for ride but also for handling manoeuvres.
8
(b)
6
4
2
5
10
15
20
Vehicle velocity (m/s)
25
10
15
20
Vehicle velocity (m/s)
25
(d)
15
10
5
5
Fig. 7.45. RMS and max dynamic tyre forces due to partial cancellation: (a) tractor drive
axle; (b) trailer axle; (c) tractor drive axle; (d) trailer axle. (
) passive viscous damper,
) semi-active 25% cancellation, (
) semi-active 50% cancellation, (
)
(
semi-active 75% cancellation, (
) semi-active 100% cancellation (copyright
Elsevier, reproduced from Tsampardoukas G, Stammers CW and Guglielmino E, Hybrid
balance control of a magnetorheological truck suspension, accepted for publication in
Journal of Sound and Vibration, used by permission)
The vehicle performance is now investigated when the vehicle operates on a gravel
road. A smooth road surface may have sections where the surface is rough due to
maintenance or resurfacing work. It is essential to assess the vehicle and the
control logic behaviour under these conditions. Heavy good vehicles are mainly
designed to operate on smooth highways rather than on poor roads. Consequently,
the vehicle operation on highway with gravel is a scenario to assess the
performance of the semi-active suspension either off-road or under large
amplitude-road inputs.
Simulation results show that the heave and pitch accelerations of the tractor and
trailer units are reduced by control. Figure 7.46 presents the dynamic tyre forces of
the vehicle in three different cases. The semi-active suspension performs better
Case Studies
261
60
40
20
0
5
10
15
20
Vehicle velocity (m/s)
25
(c)
100
50
0
5
10
15
20
Vehicle velocity (m/s)
25
RMS dynamic tyre forces (kN)
150
(a)
40
RMS dynamic tyre forces (kN)
MAX dynamic tyre force (kN)
80
MAX dynamic tyre forces (kN)
than the passive system over the velocity range investigated; however, the control
logic becomes ineffective when the vehicle velocity is higher than 25 m/s. The
vehicle performance is similar to that on smooth road in terms of normalised road
damage.
80
(b)
30
20
10
0
5
10
15
20
Vehicle velocity (m/s)
25
10
15
20
Vehicle velocity (m/s)
25
(d)
60
40
20
0
5
Fig. 7.46. Dynamic tyre forces: (a) tractor drive axle; (b) tractor drive axle; (c) trailer axle;
) passive viscous damper, (
) passive MR damper (I = 0 A),
(d) trailer axle; (
(
) semi-active MR damper (copyright Elsevier, reproduced from Tsampardoukas G,
Stammers CW and Guglielmino E, Hybrid balance control of a magnetorheological truck
suspension, accepted for publication in Journal of Sound and Vibration, used by permission)
The results presented so far can be summarised as follows:
•
•
•
•
Semi-active hybrid balance control is beneficial to heavy vehicle
performance: chassis accelerations are significantly reduced and road
damage moderately so,
The semi-active MR damper significantly reduces axle loads at all speeds
on both harsh and smooth roads,
By employing balance logic on the drive tractor and trailer controlled
dampers, trailer chassis acceleration is substantially reduced. The tractor
chassis acceleration slightly increases because a passive viscous damper is
used on the steer tractor axle,
The partial cancellation of the dynamic tyre forces is mainly affected by the
vehicle speed; 100% cancellation is the optimal solution in terms of lower
road damage in the moderate vehicle speed range (from 12.5 m/s to 20 m/s)
while 75% cancellation is the best solution in the low (from 5m/s to 10m/s)
and high vehicle speed ranges (from 20 m/s to 25 m/s),
262
Semi-active Suspension Control
•
The passive MR damper is not able to produce high forces, causing
excessive load on axles. However an MR damper operates in this mode
only because of an electrical fault (e.g., if the control system power supply
fails). The system performs poorly but this is a provisional fail-safe
condition. The on-board electronics will spot the failure immediately,
warning the driver.
7.6.7 Truck Response to Bump and Pothole
The vehicle response to bump and to pothole inputs is another essential test to
assess suspension performance. The bump and pothole are modelled as follows
(Pesterev et al., 2002):
2π s
⎧ a
)
⎪ − (1 − cos
r(s) = ⎨
⎪
⎩
2
0
b
0≤ s≤b
(7.13)
s < 0, s > b
The parameters a and b determine the depth and the width of the pothole, while
the negative value of a corresponds to a bump. Table 7.6 lists the parameters used
in the results presented subsequently.
Table 7.6. Parameters for bump and pothole
Pothole
Bump
a = 0.008 m
a = –0.008 m
b=1m
b=1m
Vehicle velocity = 10 m/s
100% cancellation of dynamic tyre forces
br = 1 and bt = 1
b2 = 0.5, amount of additional critical damper force when damper is on
b3 = 0.2, amount of critical damper force when damper is off
Figures 7.47 and 7.48 show that the semi-active control scheme reduces the peak
values of the dynamic tyre forces on both drive tractor and trailer axles. The
controlled suspension reduces the number of oscillations with respect to the
passive suspension. Similar trends can be observed for vehicle speeds ranging from
5 to 25 m/s (results not shown here). Control produces high damping during the
settling time but more or less the same level of damping as the passive case during
the impact stage.
Case Studies
263
Drive tractor dynamic tyre forces (kN)
8
6
4
2
0
-2
-4
-6
-8
10
10.2
10.4
10.6
10.8
Time (s)
11
11.2
11.4
11.6
Fig. 7.47. Tractor drive dynamic tyre forces due to bump; (
) passive suspension,
(
) semi-active suspension (copyright Elsevier, reproduced from Tsampardoukas G,
Stammers CW and Guglielmino E, Hybrid balance control of a magnetorheological truck
suspension, accepted for publication in Journal of Sound and Vibration, used by permission)
10
Trailer dynamic tyre forces (kN)
8
6
4
2
0
-2
-4
-6
-8
-10
9.5
10
10.5
11
11.5
Time (s)
12
12.5
13
Fig. 7.48. Trailer drive dynamic tyre forces due to bump; (
) passive suspension,
(
) semi-active suspension (copyright Elsevier, reproduced from Tsampardoukas G,
Stammers CW and Guglielmino E, Hybrid balance control of a magnetorheological truck
suspension, accepted for publication in Journal of Sound and Vibration, used by permission)
Simulation has also shown that an improvement is present on the trailer body
heave and pitch accelerations too (graphs not depicted here): the free oscillations
264
Semi-active Suspension Control
are significantly reduced with the semi-active control as this results in an additional
damping. However, the controlled tractor body heave and the pitch accelerations
(graphs not shown here) are not any better than those with the passive suspension.
In fact, the semi-active suspension increases the peak values of the heave and pitch
accelerations. This is not unexpected because the control logic also increases the
tractor body pitch acceleration. Analogous reults have been obtained in response to
a pothole input.
7.6.8 Robustness Analysis
Instrumentation noise (due to electromagnetic interference, electrical component
damage or any other reason) is a real-life issue. The algorithm robustness to
injected white noise into the control loop is examined. White noise is added to
feedback signals, i.e., measured relative velocity and the axle heave acceleration.
The level of noise is indicated on the velocity signal in Figure 7.49 for a vehicle
velocity of 15 m/s.
A measure of how noise can degrade the overall performance is the increase in
the number of switches of the controller (i.e., increased chattering)
0.25
0.2
Relative velocity (m/s)
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
4
4.1
4.2
4.3
4.4
4.5
Time (s)
4.6
4.7
4.8
4.9
5
Fig. 7.49. Relative velocity across the tractor drive semi-active damper; (
) signal
) signal with noise (copyright Elsevier, reproduced from
without added noise, (
Tsampardoukas G, Stammers CW and Guglielmino E, Hybrid balance control of a
magnetorheological truck suspension, accepted for publication in Journal of Sound and
Vibration, used by permission)
A moderate penalty due to noise is observed in the heave tractor chassis
acceleration.
Case Studies
0.25
0.2
0.15
5
0.6
2
0.15 (b)
2
0.3
0.1
RMS heave acceln (m/s )
RMS pitch acceln (rad/s )
(a)
0.35
10
15
20
Vehicle velocity (m/s)
0.1
0.05
0
25
(c)
RMS heave acceln (m/s2)
2
RMS pitch acceln (rad/s )
0.4
265
0.55
0.5
0.45
0.4
0.35
Passive
BCC without noise
BCC with noise
5
10
15
20
Vehicle velocity (m/s)
25
10
15
20
Vehicle velocity (m/s)
25
(d)
0.7
0.6
0.5
0.4
0.3
0.2
5
10
15
20
Vehicle velocity (m/s)
5
25
RMS dynamic tyre force (kN)
3
2
Passive
BCC without noise
BCC with noise
1
0
5
10
15
20
Vehicle velocity (m/s)
(c)
8
6
4
2
6
5
4
3
2
1
25
(b)
7
5
18
10
Max dynamic tyre force (kN)
8
(a)
4
Max dynamic tyre force (kN)
RMS dynamic tyre force (kN)
Fig. 7.50. Chassis RMS heave and pitch acceleration on both vehicle units: (a) tractor
chassis; (b) trailer chassis; (c) tractor chassis; (d) trailer chassis; (
) passive, (
)
) semi-active (BCC) with noise
semi-active (BCC) without imposed noise, (
(copyright Elsevier, reproduced from Tsampardoukas G, Stammers CW and Guglielmino E,
Hybrid balance control of a magnetorheological truck suspension, accepted for publication
in Journal of Sound and Vibration, used by permission)
10
15
20
Vehicle velocity (m/s)
25
25
10
15
20
Vehicle velocity (m/s)
25
(d)
16
14
12
10
8
6
4
5
10
15
20
Vehicle velocity (m/s)
5
Fig. 7.51. RMS and max dynamic tyre forces on tractor drive and trailer axles: (a) tractor
drive axle; (b) trailer axle; (c) tractor drive axle; (d) trailer axle; (
) passive, (
)
) semi-active (BCC) with noise
semi-active (BCC) without imposed noise, (
(copyright Elsevier, reproduced from Tsampardoukas G, Stammers CW and Guglielmino E,
Hybrid balance control of a magnetorheological truck suspension, accepted for publication
in Journal of Sound and Vibration, used by pemission)
266
Semi-active Suspension Control
The system response in respect of chassis acceleration, depicted in Figure 7.50, is
somehow affected by the presence of noise: the heave and pitch acceleration of the
trailer drive in the semi-active case are slightly increased relative to the semi-active
case without noise.
Finally the RMS and maximum dynamic tyre forces on each axle are examined
from the algorithm robustness angle. Figure 7.51 shows that the imposed noise
only moderately affects the peak tyre forces, and has almost no effect on the RMS
values. Consequently, the system performance in terms of dynamic tyre forces,
comparing both semi-active cases, is not significantly influenced by the imposed
noise to the feedback signals.
A reduction in the number of switches is essential to avoid over-heating and
wear, and reduce component life. The switches between the on and off state in the
two cases are measured and compared. For graphic clarity only, the on state of the
semi-active damper is equal to unity while the off state is designated zero.
Simulation results show that the semi-active case with the added noise increase the
chattering as observed in Figure 7.52 in the time interval 3.7–4.2 s.
Semi-active damper state
1
0
3.7
3.8
3.9
Time (s)
4
4.1
4.2
Fig. 7.52. On/off states of the semi-active damper; (
) semi-active (BCC without noise),
(
) semi-active (BCC with noise) (copyright Elsevier, reproduced from Tsampardoukas
G, Stammers CW and Guglielmino E, Hybrid balance control of a magnetorheological truck
suspension, accepted for publication in Journal of Sound and Vibration, used by permission)
7.6.8.1 Trailer Mass Variation
The payload in a truck can vary significantly (either fully loaded, partially loaded
or unladen). The effects of the trailer mass variations are vital in terms of road
damage. For the passive half vehicle model, the mass of the unladen trailer was
taken as 5000 kg, the fully loaded trailer 12500 kg and a partly-loaded trailer
8500 kg.
The variations of the trailer mass are considered in the simulation process to
examine the response of the system in the three cases. Figure 7.53 shows the
reduction of the dynamic forces at a specified vehicle speed when the three
Case Studies
267
15
(a)
10
5
0
5
20
10
15
20
Vehicle velocity (m/s)
10
5
5
(b)
4
2
5
8
(c)
15
0
6
0
25
Trailer axle (kN)
Trailer axle (kN)
Tractor drive axle (kN)
Tractor drive axle (kN)
different trailer masses A convergence in the trends is observed at high vehicle
speeds.
10
15
20
Vehicle velocity (m/s)
25
10
15
20
Vehicle velocity (m/s)
25
10
15
20
Vehicle velocity (m/s)
25
(d)
6
4
2
0
5
Fig. 7.53. Dynamic tyre forces for different values of the trailer mass: (a) maximum values;
(b) RMS values; (c) maximum values; (d) RMS values; (
) mt = 5000 kg, (
) mt
) mt = 12500 kg (copyright Elsevier, reproduced from Tsampardoukas G,
= 8500 kg, (
Stammers CW and Guglielmino E, Hybrid balance control of a magnetorheological truck
suspension, accepted for publication in Journal of Sound and Vibration, used by permission)
7.6.8.2 Tyre Stiffness Variation
In order to simulate the vertical tyre motion due to road irregularities the tyre is
modelled as a spring with high stiffness. In real operating conditions the tyre
pressure is not constant and slowly changes over time, resulting in variations of
the tyre stiffness. A tyre stiffness of 2 MN/m has normally been used in the
simulation, while lower or higher values result in different tyre pressures.
The variation of the tyre stiffness of the drive tractor wheel shows that the
dynamic tyre forces are slightly affected at low and moderate vehicle speeds
(Figure 7.54). The system response alters at high vehicle velocities, producing
larger tyre forces, resulting in higher road damage (as expressed by Equation 7.9).
The simulation results plotted in Figure 7.54 show that the dynamic tyre forces at
both axles are little affected by this variation. Consequently, the road damage (see
road damage criterion) is also unaffected at low vehicle speeds with a moderate
increase at high vehicle velocities.
(a)
10
8
6
4
5
16
10
15
20
Vehicle velocity (m/s)
3
2
5
8
(c)
14
12
10
8
6
5
4 (b)
1
25
Trailer axle (kN)
Trailer axle (kN)
Tractor drive axle (kN)
Semi-active Suspension Control
Tractor drive axle (kN)
268
10
15
20
Vehicle velocity (m/s)
25
10
15
20
Vehicle velocity (m/s)
25
10
15
20
Vehicle velocity (m/s)
25
(d)
6
4
2
5
Fig. 7.54. Dynamic tyre forces for different values of the tyre stiffness: (a) maximum
values; (b) RMS values; (c) maximum values: (d) RMS values; (
) ktr = 1.6 MN/m,
) ktr = 2 MN/m, (
) ktr = 2.4 MN/m (copyright Elsevier, reproduced from
(
Tsampardoukas G, Stammers CW and Guglielmino E, Hybrid balance control of a
magnetorheological truck suspension, accepted for publication in Journal of Sound and
Vibration, used by permission)
7.6.8.3 MRD Response Time
The response time of MR dampers for vehicle applications is a critical factor
because it determines the effectiveness of the MR damper used. The aim is now to
assess the effect of the damper response to the vehicle performance.
The response time is defined as the time required for the MR damper to reach
64% or 95% (Goncalves et al., 2003) of the final exerted force, starting from the
initial state. The time response depends both on the fluid transformation from a
mineral-oil-like consistency (but not majorly as this time is less than 1 ms) and on
the inductance of the electromagnetic circuit as well as the output impedance of the
driving electronics.
Extensive experimental work by Koo et al. (2004) showed that the time
response of the MR damper is affected by several parameters such as the response
of the driving electronics, the applied current, the piston velocity and the system
compliance. The time response of MR damper for commercial vehicle application
is less than 25 ms (effective time response to reach 95% of its final value).
In the simulation work a first-order lag is employed in the semi-active control
schemes in order to model the MR damper dynamics. The time constant of the
first-order lag is chosen so that the system reaches 64% of the final value in 11 ms
(one time constant), which corresponds to 25 ms to reach 95% of its final value
Case Studies
269
8
6
4
2
5
10
15
20
Vehicle velocity (m/s)
25
16 (c)
14
12
10
8
6
4
5
10
15
20
Vehicle velocity (m/s)
25
RMS dynamic tyre forces (kN)
10 (a)
5
RMS dynamic tyre forces (kN)
Max dynamic tyre forces (kN)
Max dynamic tyre forces (kN)
(effective time constant). In the current study the time constant quoted is that to
reach 64% of the final state.
Simulation results indicate that a fast-response MR damper (5 ms time
constant) is the preferable device to improve the vehicle response in terms of lower
dynamic tyre forces. Figure 7.55 shows that the maximum values rather than the
RMS values of the dynamic tyre forces are more sensitive to the time response of
the semi-active damper.
A damper with time constant equal to 20 ms produces larger maximum tyre
forces because it cannot respond fast enough to exert the required control force in
order to cancel the dynamic tyre forces.
However, the latter value of the time constant is high and is the typical of a
bigger damper, more suitable for controlling structural vibration. A time constant
of 15 ms, which corresponds to a 37.5 ms effective time constant, might be a more
realistic value for semi-active dampers used in heavy-vehicle applications.
8
(b)
4
3
2
1
0
5
10
15
20
Vehicle velocity (m/s)
25
10
15
20
Vehicle velocity (m/s)
25
(d)
6
4
2
0
5
Fig. 7.55. Max dynamic tyre forces for different time constants: (a) tractor drive axle; (b)
) passive damper, (
)
tractor drive axle; (c) trailer axle; (d) trailer axle; (
Tc = 5 ms (
), Tc = 10 ms (
), Tc = 15 ms (
), Tc = 20 ms
(copyright Elsevier, reproduced from Tsampardoukas G, Stammers CW and Guglielmino E,
Hybrid balance control of a magnetorheological truck suspension, accepted for publication
in Journal of Sound and Vibration, used by permission)
270
Semi-active Suspension Control
7.7 Conclusions
This case study centred on road damage reduction has shown that the semi-active
truck suspension response is superior overall to the passive response on both road
profiles (smooth and gravel). The maximum and RMS values of the dynamic tyre
forces are substantially reduced by the control logic and the road damage follows
the same pattern as the dynamic tyre forces. Similarly, a reduction of the trailer
chassis acceleration is obtained because the trailer unit is well isolated from the
ground irregularities. Conversely, the tractor chassis acceleration slightly increases
because the steer axle is assumed to be equipped with conventional viscous
dampers.
Additionally, the simulation results show that the MRD has poor performance
when the applied current is zero (passive operation) due to the very low damping
provided by the semi-active device. However, this scenario occurs only in the
event of a failure of the MRD control system hardware.
The partial cancellation of the dynamic tyre forces is also examined to establish
the optimal amount of cancellation on smooth and gravel road profiles while the
other control parameters (b2 and b3) are kept constant. The results indicate that
100% cancellation of tyre force fluctuation is always preferable in terms of lower
road damage and dynamic tyre force fluctuations.
The vehicle was also tested with pothole and bump inputs. The results show
that the semi-active suspension with the hybrid control algorithm reduces the
amplitude of the free oscillations while the peak values of dynamic tyre forces are
slightly reduced. Also, the vehicle response is mainly affected by the parameter b2
rather than the amount of cancellation.
From a robustness viewpoint, chassis accelerations and the RMS dynamic tyre
forces are slightly affected by imposed noise while the maximum values of the
dynamic tyre forces are slightly affected at moderate vehicle velocities. Robustness
to sprung and unsprung mass variation was also assessed and the algorithm was
found to be not very sensitive to noise.
In terms of dynamic response a fast-response semi-active damper (one time
constant equal to 5 ms) is extremely beneficial in reducing dynamic tyre forces.
However, this value of time constant is optimistic not only for a large force damper
needed for heavy vehicles but also for dampers suitable for passenger vehicles.
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Authors’ Biographies
Dr Emanuele Guglielmino received his PhD in Mechanical Engineering from the
University of Bath (UK) in 2001, and a master’s degree in Electrical Engineering
from the University of Genoa (Italy) in 1998. His doctoral research regarded the
robust control of hydraulically actuated friction damper systems for vehicle
applications. He subsequently joined Westinghouse Brakes (UK), where he worked
as an R&D engineer on controlled braking systems. In 2004 he joined General
Electric (Florence, Italy) where he held positions as a control engineer and an
application engineer. In 2008 he joined the Italian Institute of Technology (IIT) in
Genoa as a team leader.
He has authored over 25 publications in the fields of semi-active suspensions,
fluid power systems, mechatronics and robust control, and co-authored a chapter in
a book of applied mechanics and control. For his work he won the ASME Best
Paper Award (Fluid Power Systems and Technology Division) in 2001. He was
invited Guest Editor in a special issue on semi-active suspensions of the
International Journal of Vehicle Design. He is also recipient of an award from
IMechE, and an entrepreneurship award from the Italian Industrial Association for
an outstanding business plan as a spin-off of a research project.
Dr Tudor Sireteanu graduated in Mathematics from the University of Bucharest
(Romania) in 1966. He subsequently joined the Institute of Solid Mechanics of the
Romanian Academy in Bucharest. In 1971 he was a Fulbright fellow at the
California Institute of Technology, Pasadena.
In 1981 he was awarded a PhD in non-linear random vibration from the
University of Bucharest. Since 1992 he has been a PhD advisor in applied
mathematics. In 2000 he was awarded the Aurel Vlaicu Romanian Academy prize
for a series of publications in the field of vibration control. At present he is
Director of the Institute of Solid Mechanics and honorary member of the Academy
of Technical Sciences of Romania.
His research interests include random vibration and semi-active damping
systems, in particular friction dampers and magnetorheological dampers. He is coauthor of two books on automotive random vibration and magnetorheological
290
Authors’ Biographies
fluids and dampers, editor of three books of applied mechanics and author of over
100 publications in scientific journals and international conference proceedings.
Dr Charles W. Stammers worked at the Institute of Sound and Vibration,
University of Southampton (UK) from 1963 to 1969, in 1968 being awarded a PhD
for a thesis on the stability of rotor systems. He then joined Westland Helicopters
Ltd. studying machine and rotor vibration problems. In 1973 he joined the
Department of Mechanical Engineering at the University of Bath. Projects
undertaken have included the manufacture of an ambulance stretcher suspension
and a robot for disabled users. Recent work has centred on vibration control in
machines and vehicles utilising smart semi-active control systems.
Since 1996 he has headed a collaborative programme with the Institute of Solid
Mechanics in Bucharest, Romania, supported by the Royal Society of London.
This collaboration has resulted in two books dealing with research topics in applied
mechanics. Current work concerns experimental systems to protect historic
buildings from seismic inputs. He has 100 publications (journals and international
conferences).
Dr Gheorghe Ghita graduated in Aeronautical Engineering from Politehnica
University of Bucharest (Romania) in 1975. After graduating he was employed by
the aircraft company Aerostar, and since 1982 he has been a researcher in mechanical
engineering. At present he is employed by the Institute of Solid Mechanics of the
Romanian Academy in Bucharest. In 2003 he received his PhD in mechanical
engineering from Politehnica University with a thesis in the field of semi-active
vibration control.
His research interests focus on experimental methods, signal processing and
applications of computational intelligence to semi-active vibration control. He is a
co-author of a book on magnetorheological fluids and dampers and of over 50
journal and conference papers.
Dr Marius Giuclea graduated in Mathematics from the University of Bucharest,
Romania in 1994 and obtained his MSc degree in Mathematics in 1995. Between
1994 and 2001 he worked as a researcher at the Institute of Microtechnology and
from 2001 as a lecturer at the Academy of Economic Studies. In 2004 he was
awarded a PhD in applications of intelligent techniques in dynamic systems control
by the Institute of Mathematics, Bucharest. His research interests include
intelligent techniques and their applications in modelling and control of dynamic
systems. He is author of over 30 journal and conference publications.
Index
accelerometer, 113, 219, 223
anti-roll bars, 4
Appel function, 59, 60, 62
axisymmetric flow, 166, 168, 176
back-pressure, 149, 154, 155, 156,
160, 161, 230
balance logic, 6, 64, 65, 71, 71, 72,
73, 74, 76, 78, 79, 80, 103,
113, 116, 125, 127, 231, 243,
247, 251, 255, 257
Barbashin–Krasowsky theorem, 95
Bingham model, 167, 186, 189, 192,
193, 203
Bliman–Sorine friction model, 102
bond graphs, 121, 155, 156, 157,
158, 159, 160
Bouc–Wen model, 21, 22, 24, 27,
58, 72, 102, 124, 186, 196,
197, 198, 202, 206, 242, 248
Box–Muller method, 95
breakaway friction force, 100
bulk modulus, 105, 109, 144, 145,
153, 154, 155, 158, 159
bump input, 137, 139, 217, 225, 237,
252, 258, 266
chattering, 75, 219, 241, 243, 245,
260, 262
Chua–Stromsoe model, 22
Cohen and Coon rules, 67
comfort, 1, 2, 3, 5, 6, 7, 8, 9, 10, 28,
29, 31, 43, 51, 64, 87, 93,
113, 117, 125, 126, 215, 219,
229, 235, 241, 242, 246, 255
absorbed power method, 9
Janeway's criterion (also SAE
criterion), 8
control,
adaptive, 5, 6, 65, 66
proportional–integral–derivative,
65, 66, 67, 68, 69
robust, 5, 67, 68, 69, 70, 119
sliding mode, 6, 69, 76, 243
variable structure, 69, 76, 113,
158, 241, 243, 246
cushion, 43, 44, 48, 49, 50, 54, 55,
56, 57, 58
Dahlin regulator, 68
damper
electrorheological, 2, 11, 243
friction, 12, 14, 15, 65, 70, 72,
74, 78, 79, 80, 97, 98 103,
104, 114, 115, 120, 121,
124, 139, 141, 154, 156,
157, 159, 195, 196, 200,
210, 215, 229, 230 239
magnetorheological, 5, 11, 12, 14,
15, 16, 27, 51, 65, 72, 74, 98,
162, 166, 174, 177, 178, 180,
182, 183, 184, 186, 187, 193,
195, 196, 206, 210, 214, 215,
241, 242, 243, 246, 247, 248,
292
Index
250, 252, 254, 255, 257, 258,
264, 265, 266
damping,
position, 11, 127, 242
pseudo-viscous, 80, 126, 127,
252
sequential, 72, 79, 80, 82, 84, 87,
89, 95
dead band, 75, 76, 78
displacement-based on/off
groundhook, 71
direct shear mode, 174, 175
driver body, 57, 59
driver seat, 59, 241, 254
Duffing equation, 45
dynamic tyre force, 2, 3, 6, 28, 29,
70, 71, 74, 127,129, 131, 247,
255, 263, 264, 265
FD, see damper, friction,
fifth wheel, 36, 38
free vibration, 83, 84
frequency response, 45, 46, 50, 129,
141, 142, 143, 147, 151, 152,
158 216, 217, 218, 226, 228,
231, 239
friction,
boundary lubrication, 100
coefficient, 12, 74, 99, 114, 116,
121, 123, 124, 126, 127,
128, 129, 232, 239
Coulomb friction, 18, 72, 99,
101, 114
Dahl model, 102
dynamic friction, 102
frictional memory, 101, 102, 122,
124
full fluid lubrication, 100
LuGre model, 102
partial fluid lubrication, 100
pre-sliding or static, 101
relaxation oscillations, 101
seven-parameter friction model,
101, 102
static, 100, 101
Stribeck, 100, 101, 102
full car see model, full car
fuzzy logic, 7, 239, 241, 243
Gaussian white noise, 87, 88, 89
genetic algorithm, 27, 206
groundhook, 70, 71
Hagen–Poiseuille flow, 165
half car, see model, half car
half truck, see model, half truck
Haalman method, 68
handling, 1, 2, 3, 7, 14, 28, 29, 39,
64, 70, 217, 226, 227, 247,
256
head-and-neck complex, 57, 59, 60,
63
heave, 6 28, 36, 38, 59, 247, 254,
257, 260, 261, 262
Hertzian contact, 98
HNC, see head-and-neck complex
Hooke’s law, 11
hybrid skyhook–groundhook, 71
hydraulic drive, 104, 106, 113, 115,
139, 143, 149, 150, 228
hysteresis, 17, 18, 19, 20, 22, 24,
102, 113, 122, 141, 143, 152,
189, 193, 200, 226, 227, 246,
249
ISO 2631 (standard), 8
isolation, 1, 43, 72, 74, 80, 84, 85,
124, 254
jerk, 8, 10, 75, 126, 219, 234, 236,
243
Lagrange, Lagrangian formalism,
33, 60
linear-quadratic-gaussian scheme, 6
loss factor, 44, 46, 49, 54
Lotus, 4
Luenberger state observer, 51
LVDT, 113, 116, 121, 143, 219, 225
Lyapunov, 68, 93, 94, 95
MacPherson, Earle S, 4
MacPherson strut, 4, 120
mechatronics, 13
Index
mesentery, 44
model,
Bouc–Wen, 22, 24, 27, 58, 72,
102, 124, 186, 242, 248
full car, 5, 29, 32
half car, 31
half truck, 36, 39, 247, 248, 256
quarter car, 5, 6, 28, 29, 30, 31,
95, 115, 127, 139, 156,
186, 217
seated human, 56, 57
Monte Carlo simulation, 87
MRD, see damper,
magnetorheological
MR damper, see damper,
magnetorheological
MR fluid figures of merit, 179, 180
Navier–Stokes equations, 166
Newmark’s method, 85, 87, 89. 91
Newtonian fluid, 163
Nyquist theorem, 215
phase plane, 68, 82
PID, see proportional–integral–
derivative
pitch, 7, 28, 29, 31, 32, 34, 36, 59,
60, 217, 221, 222, 223, 247,
254, 256, 259, 260, 262
pothole, 252, 258, 260, 266
Preisach model, 22
pressure-driven flow mode, 164,
165, 166, 171, 173
pressure gain, 104, 112, 113, 125,
139, 141, 143, 149, 150, 151
pressure-to-current transfer function,
111
protocol, 216
pseudo-random input, 40, 89, 90.
133, 136, 139, 217, 224, 237
quarter car, see model, quarter car
random vibration, 87
reduced-order state observer, 52
Reynolds number, 146
rheological behaviour, 163, 189
293
ride, 1, 3, 7, 14, 18, 28, 29, 32, 39,
64, 66, 75, 87, 131, 132, 215,
217, 218, 220, 226, 227, 229,
234, 243, 246, 247, 248, 256
road damage, 29, 40, 64, 70, 244,
215, 247, 248, 251, 252, 254,
255, 257, 258, 264, 266
road holding, 1, 2, 3, 5, 7, 28, 66
robustness, 67, 69, 232, 247, 260,
262, 266
roll, 4, 6, 7, 29, 32, 34, 60, 217, 221,
222, 223, 224, 248
rubber bush, 4, 122, 124
SA, see suspension, semi-active
SAE criterion, see Janeway’s
comfort criterion
safety, 1, 64
seat, 43, 48, 52, 53, 56, 57, 58, 59,
63
sequential damping, 84
Shannon theorem, 215, 216
shock absorber, 1, 81, 120, 165
skeleton, 44, 48, 52
skyhook, 6, 64, 70, 71, 103, 113
Smith predictor, 67
squeeze-film mode, 165
state observer, 51, 52
sprung mass, 28, 32, 34, 36, 49, 50,
53, 63, 74, 80, 80, 84, 87, 89,
93, 95, 114, 115, 118, 119,
221, 247, 266
stability, 1, 65, 68, 93, 95, 164
suspension,
active, 4, 5, 6, 113
dependent, 4
hydragas, 4
hydropneumatic, 4
independent, 4
passive, 2, 28, 30, 87, 93, 218,
258, 260
semi-active, 1, 2, 4, 6, 10, 16, 17,
65, 70, 87, 93, 98, 113,
215, 228, 235, 241, 251,
254, 255, 256, 260, 266
smart, 2
switched state feedback, 113, 125
294
Index
switching condition, 119, 125, 126
tractor, 29, 36, 247, 248, 251, 252,
254, 255, 256, 257, 258, 260,
263, 266
trailer, 29, 36, 247, 248, 251, 252,
254, 256, 257, 258, 259, 262,
263, 266
transmissibility, 85, 87, 118, 127,
218, 229, 231
tyre, 2, 3, 7, 19, 28, 29, 30, 32, 35,
39, 40, 42, 64, 98, 133, 139,
150, 218, 219, 220 263
underlap, 112, 113, 141, 150, 151
unsprung mass, 28, 29, 32, 34, 36,
49, 120, 124, 266
validation, 149, 226, 238
valve
two-way, 103, 139
three-way, 104, 106, 139, 142
pressure control, 103, 115
relief, 106, 107, 115, 139, 144,
145, 146, 147, 150, 229
servovalve, 215, 217
static characteristic, 139, 141,
142, 143, 151
valve lap, 109, 113, 150
valve spool, 107
viscera, 43, 44, 45, 48, 50, 51, 52,
53, 54
viscoelastic material, 123, 189, 226
VSC, see control, variable structure
wheel-hop, 7, 31, 48, 221, 227, 252
working space, 2, 28, 30, 71, 93,
129, 130, 131, 136, 137, 138,
139, 228, 229, 230, 247
wrist, 44
yaw, 28, 29, 217, 227, 248
yield strength, 163, 189
yield stress, 15, 162, 163, 167, 168,
169, 171, 173, 178, 181, 182,
189
Ziegler and Nichols rules, 67
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