Ride Quality and Drivability of a Typical Passenger Car

Ride Quality and Drivability of a Typical Passenger Car
Ride Quality and Drivability of a Typical Passenger Car
subject to Engine/Driveline and Road Non-uniformities
Excitations
Examensarbete utfört i Fordonssystem
vid Tekniska högskolan i Linköping
av
Neda Nickmehr
LiTH-ISY-EX--11/4477--SE
Linköping 2011
I
II
Ride Quality and Drivability of a Typical Passenger Car
subject to Engine/Driveline and Road Non-uniformities
Excitations
Examensarbete utfört i Fordonssystem
vid Tekniska högskolan i Linköping
av
Neda Nickmehr
LiTH-ISY-EX--11/4477--SE
Handledare:
Neda Nickmehr
ISY, Linköpings universitet
Examinator:
Jan Åslund
ISY, Linköpings universitet
Linköping, 7th June 2011
III
IV
Avdelning, Institution
Division, Department
Division of vehicular system
Department of Electrical Engineering
Linköpings universitet
SE-58183Linköping, Sweden
Rapporttyp
Report category
Språk
Language
Svenska/Swedish
Engelska/English
licentiatavhandling
Examensarbete
C-uppsats
D-uppsats
Övrigrapport
Datum
Date
2011-06-07
ISBN
ISRN
LiTH-ISY-EX--11/4477--SE
Serietitel och serienummer ISSN
Title of series, numbering
URL för elektronisk version
http:\\ urn:nbn:se:liu:diva-69499
Title
Ride Quality and Drivability of a Typical Passenger Car subject to Engine/Driveline and
Road Non-uniformities Excitations
Författare Neda Nickmehr
Author
Sammanfattning
Abstract
The aim of this work is to evaluate ride quality of a typical passenger car. This
requires both identifying the excitation resources, which result to undesired noise
inside the vehicle, and studying human reaction t applied vibration. Driveline linear
torsional vibration will be modelled by a 14-degress of freedom system while engine
cylinder pressure torques are considered as an input force for the structure. The
results show good agreement with the corresponding reference output responses
which proves the accuracy of the numerical approach fourth order Runge-kutta. An
eighteen-degree of freedom model is then used to investigate coupled motion of
driveline and the tire/suspension assembly in order to attain vehicle body
longitudinal acceleration subject to engine excitations. Road surface irregularities is
simulated as a stationary random process and further vertical acceleration of the
vehicle body will be obtained by considering the well-known quarter-car model
including suspension/tire mechanisms and road input force. Finally, ISO diagrams
are utilized to compare RMS vertical and lateral accelerations of the car body with
the fatigue-decreased proficiency boundaries and to determine harmful frequency
regions. According to the results, passive suspension system is not functional
enough since its behaviour depends on frequency content of the input and it provides
good isolation only when the car is subjected to a high frequency excitation.
Although longitudinal RMS acceleration of the vehicle body due to engine force is
not too significant, driveline torsional vibration itself has to be studied in order to
avoid any dangerous damages for each component by recognizing resonance
frequencies of the system. The report will come to an end by explaining different
issues which are not investigated in this thesis and may be considered as future
works.
Nyckelord
Keywords
Ride quality, Driveline, Engine excitations,
Road non-uniformities, Suspension
V
System, Torsional vibration, Random process
VI
Upphovsrätt
Detta dokument hålls tillgängligt på Internet – eller dess framtida ersättare – under 25 år från
publiceringsdatum under förutsättning att inga extraordinära omständigheter uppstår.
Tillgång till dokumentet innebär tillstånd för var och en att läsa, ladda ner, skriva ut
enstaka kopior för enskilt bruk och att använda det oförändrat för ickekommersiell forskning
och för undervisning. Överföring av upphovsrätten vid en senare tidpunkt kan inte upphäva
detta tillstånd. All annan användning av dokumentet kräver upphovsmannens medgivande.
För att garantera äktheten, säkerheten och tillgängligheten finns lösningar av teknisk och
administrativ art.
Upphovsmannens ideella rätt innefattar rätt att bli nämnd som upphovsman i den
omfattning som god sed kräver vid användning av dokumentet på ovan beskrivna sätt samt
skydd mot att dokumentet ändras eller presenteras i sådan form eller i sådant sammanhang
som är kränkande för upphovsmannens litterära eller konstnärliga anseende eller egenart.
För ytterligare information om Linköping University Electronic Press se förlagets
hemsida http://www.ep.liu.se/.
Copyright
The publishers will keep this document online on the Internet – or its possible replacement –
for a period of 25 years starting from the date of publication barring exceptional
circumstances.
The online availability of the document implies permanent permission for anyone to
read, to download, or to print out single copies for his/her own use and to use it unchanged
for non-commercial research and educational purpose. Subsequent transfers of copyright
cannot revoke this permission. All other uses of the document are conditional upon the
consent of the copyright owner. The publisher has taken technical and administrative
measures to assure authenticity, security and accessibility.
According to intellectual property law the author has the right to be mentioned when
his/her work is accessed as described above and to be protected against infringement.
For additional information about Linköping University Electronic Press and its
procedures for publication and for assurance of document integrity, please refer to its www
home page: http://www.ep.liu.se/.
© Neda Nickmehr
VII
VIII
Abstract
The aim of this work is to evaluate ride quality of a typical passenger car. This requires both
identifying the excitation resources, which result to undesired noise inside the vehicle, and
studying human reaction to applied vibration. Driveline linear torsional vibration will be
modeled by a 14-degress of freedom system while engine cylinder pressure torques are
considered as an input force for the structure. The results show good agreement with the
corresponding reference output responses which proves the accuracy of the numerical
approach fourth order Runge-kutta. An eighteen-degree of freedom model is then used to
investigate coupled motion of driveline and the tire/suspension assembly in order to attain
vehicle body longitudinal acceleration subject to engine excitations. Road surface
irregularities is simulated as a stationary random process and further vertical acceleration of
the vehicle body will be obtained by considering the well-known quarter-car model
including suspension/tire mechanisms and road input force. Finally, ISO diagrams are
utilized to compare RMS vertical and lateral accelerations of the car body with the fatiguedecreased proficiency boundaries and to determine harmful frequency regions.
According to the results, passive suspension system is not functional enough since its
behavior depends on frequency content of the input and it provides good isolation only when
the car is subjected to a high frequency excitation. Although longitudinal RMS acceleration
of the vehicle body due to engine force is not too significant, driveline torsional vibration
itself has to be studied in order to avoid any dangerous damages for each component by
recognizing resonance frequencies of the system. The report will come to an end by
explaining different issues which are not investigated in this thesis and may be considered as
future works.
IX
X
Acknowledgments
This work has been carried out at vehicular system division, ISY department, Linköping
University, Sweden. The thesis would not have been possible without the support of many
people and division laboratory facilities. I wish to express my gratitude to my examiner and
supervisor, Dr. Jan Åslund and PhD student Kristoffer Lundahl who were abundantly
helpful and offered invaluable assistance, support and guidance. Special thanks also to my
bachelor supervisor professor Farshidianfar for sharing the literature and invaluable
assistance. I would like to express my love and gratitude to my beloved parents Maryam
and Ahmad for their understanding and endless love, through the duration of my master
study.
Linköping, May 2011
Neda Nickmehr
XI
XII
Table of Contents
1
2
3
4
5
6
Chapter 1 ................................................................................................................................................. 1
1.1
Background ..................................................................................................................................... 1
1.2
Objective ......................................................................................................................................... 2
1.3
Assumptions and Limitations .................................................................................................... 2
1.4
Outline.............................................................................................................................................. 3
Chapter 2 .................................................................................................................................................. 5
2.1
Driveline and vehicle Modeling ............................................................................................... 5
2.2
Road Surface Irregularities ........................................................................................................ 5
2.3
Human Response to vibration ................................................................................................... 5
Chapter 3 .................................................................................................................................................. 7
3.1
Introduction .................................................................................................................................... 7
3.2
Driveline components .................................................................................................................. 7
3.2.1
Engine, flywheel and the main excitation torque........................................................ 7
3.2.2
Clutch Assembly ............................................................................................................... 18
3.2.3
Gearbox ................................................................................................................................ 19
3.2.4
Cardan (propeller) shaft and universal (Hooke’s) joints........................................ 19
3.2.5
Differential and final drive system ............................................................................... 20
3.2.6
Damping in the whole driveline system ..................................................................... 21
3.3
Overall driveline model ........................................................................................................... 21
3.4
Torsional vibration .................................................................................................................... 22
Chapter 4 ............................................................................................................................................... 23
4.1
Introduction ................................................................................................................................. 23
4.2
Mathematical model and system matrices .......................................................................... 23
4.3
Summary of Modal analysis ................................................................................................... 25
4.4
Natural frequencies.................................................................................................................... 27
Chapter 5 ............................................................................................................................................... 29
5.1
Introduction ................................................................................................................................. 29
5.2
Mathematical model for forced vibration of the driveline system............................... 29
5.3
Time responses of driveline at clutch and driving wheels ............................................. 30
5.4
Power spectral densities of time histories ........................................................................... 34
Chapter 6 ............................................................................................................................................... 37
6.1
Introduction ................................................................................................................................. 37
XIII
6.2
Coupled vibration of driveline and the vehicle body ...................................................... 37
6.3
Tire model and longitudinal force ......................................................................................... 37
6.4
18-degrees of freedom system for whole vehicle model and its equations of motion
38
6.5
Time response of the system .................................................................................................. 42
6.6
Studying the influence of stiffness and damping coefficients ...................................... 46
Chapter 7 ............................................................................................................................................... 49
7
7.1
Introduction ................................................................................................................................. 49
7.2
Quarter-car model and performance of suspension system ........................................... 49
7.3 Road roughness classification by ISO and the recommended single-sided vertical
amplitude power spectral density ....................................................................................................... 53
Typical passenger car driver RMS acceleration to an average road roughness ....... 54
7.4
Chapter 8 ............................................................................................................................................... 57
8
8.1
Introduction ................................................................................................................................. 57
8.2
International Standard ISO 2631-1:1985 ............................................................................ 57
8.3
Results and Discussion ............................................................................................................. 58
8.4
Thesis conclusion ....................................................................................................................... 60
8.5
Future works ................................................................................................................................ 60
References............................................................................................................................................. 63
9
10
Appendix........................................................................................................................................... 65
10.1 LTI object ..................................................................................................................................... 65
10.2 Driveline Modeling MATLAB code .................................................................................... 65
10.3
Power spectral density function .................................................................................... 68
10.4
Vehicle modeling MATLAB codeATLAB code ........................................................... 70
XIV
Figures
Figure 1-1, the ride dynamic system ......................................................................................................... 1
Figure 3-1, Front-engine rear-wheel-drive vehicle driveline [3] ...................................................... 7
Figure 3-2, Main engine parts [1] .............................................................................................................. 8
Figure 3-3, Original System of a crank [3] ............................................................................................. 8
Figure 3-4, Equivalent system of crankshaft and its compact model [3] ....................................... 9
Figure 3-5, output torque of a four-stroke single-cylinder engine [1] ............................................ 9
Figure 3-6, Line diagram of cylinders arrangement .......................................................................... 10
Figure 3-7, engine torque in the case of four cylinders .................................................................... 10
Figure 3-8, 10 seconds pressure recording from cylinder 1 ............................................................ 11
Figure 3-9, 10 seconds pressure recording from cylinder 2 ............................................................ 12
Figure 3-10, Crank mechanism ............................................................................................................... 13
Figure 3-11, Torque output of cylinder 1, total and fluctuating part, during 10 seconds and
one working cycle ....................................................................................................................................... 14
Figure 3-12, Torque output of cylinder 2, total and fluctuating part, during 10 seconds and
one working cycle ....................................................................................................................................... 15
Figure 3-13, Output torques from 4 cylinders in the same plot ..................................................... 16
Figure 3-14, Compact crankshaft model for a four-cylinder engine ............................................ 17
Figure 3-15, PSD for output torque from cylinder 1 ......................................................................... 17
Figure 3-16, PSD for output torque from cylinder 2 ......................................................................... 18
Figure 3-17, Clutch system [22] ............................................................................................................. 18
Figure 3-18, Gearbox model [3] ............................................................................................................. 19
Figure 3-19, Hooke's (cardan) joints [1] ............................................................................................... 20
Figure 3-20, propeller shafts and universal joints mathematical model [3] ............................... 20
Figure 3-21, Final drive system and its equivalent model [3] ........................................................ 21
Figure 3-22, Damped torsional vibration mathematical model of driveline system [3] ......... 22
Figure 5-1, Driveline model ..................................................................................................................... 29
Figure 5-2, Time response at the clutch, using different methods of solution: Black-> Modal
analysis, Blue-> ODE45, Green-> self-written Runge-Kutta code with nonzero initial
conditions and yellow-> with zero initial conditions ........................................................................ 31
Figure 5-3, zoomed version of figure 5.2 in order to see the instability of modal analysis and
ODE45 solutions ......................................................................................................................................... 31
Figure 5-4, Time response at the clutch with the aid of Runge-Kutta method.......................... 32
Figure 5-5, Time response at the driving wheels ............................................................................... 33
Figure 5-6, Power spectral density of the time response at the clutch ......................................... 34
Figure 5-7, Power spectral density of the time response at the driving wheels ........................ 35
Figure 6-1, tire model [3] .......................................................................................................................... 38
Figure 6-2, overall vehicle model [3] .................................................................................................... 39
Figure 6-3, Total engine excitation torques after applying filtration ........................................... 44
Figure 6-4, torsional velocity vibration of driving wheels due to engine excitation torques by
using table 6.1 data values ........................................................................................................................ 44
Figure 6-5, zoomed version of Figure 6-4 between seconds 8 to 9 .............................................. 45
XV
Figure 6-6, Longitudinal velocity vibration of the vehicle body and axle due to engine
excitation torques by using table 6-1 data values............................................................................... 45
Figure 6-7, zoomed version of Figure 6-6 ........................................................................................... 46
Figure 6-8, torsional velocity of driving wheels with low damping ............................................ 47
Figure 6-9, Longitudinal velocities of vehicle body and axle with low damping .................... 48
Figure 6-10, Longitudinal velocities of vehicle body and axle with low suspension system
stiffness........................................................................................................................................................... 48
Figure ‎7-1, Two-degrees of freedom model vehicle .................................................................... 49
Figure 7-2, Transmissibility as a function of frequency ratio for a single-degree of freedom
system ............................................................................................................................................................. 51
Figure 7-3, Modified quarter-car model including seat displacement ......................................... 51
Figure 7-4, Measured vertical acceleration of a passenger car seat traveling at 80 Km/hr over
an average road ............................................................................................................................................ 54
Figure 7-5, vehicle body vertical acceleration subject to an average road roughness with 80
Km/hr traveling speed ................................................................................................................................ 55
Figure 8-1, ISO 2631-1:1985 "fatigue-decreased proficiency boundary": vertical
acceleration limits as a function of frequency and exposure time [4].......................................... 57
Figure 8-2, ISO 2631-1:1985 "fatigue-decreased proficiency boundary": longitudinal
acceleration limits as a function of frequency and exposure time [4].......................................... 58
Figure 8-3, vehicle body vertical acceleration due to road excitation in comparison with ISO
ride comfort boundaries ............................................................................................................................. 58
Figure 8-4, Measured longitudinal acceleration of a passenger car body due to engine
excitation torques ........................................................................................................................................ 59
XVI
Tables
Table 3-1, Engine properties .................................................................................................................... 10
Table 4-1, Typical values for equivalent parameters of a vehicle driveline [3]........................ 24
Table 4-2, Undamped natural frequencies of whole driveline model using typical parameter
values for a passenger car ......................................................................................................................... 27
Table 4-3, first five natural modes of driveline system .................................................................... 28
Table 6-1, Overall vehicle properties [3] ............................................................................................. 43
Table 7-1, Tire/suspension properties [3]............................................................................................. 50
Table 7-2, Classification of road roughness proposed by ISO [4] ................................................ 53
XVII
XVIII
Chapter1
Introduction
1 Chapter 1
Introduction
1.1
Background
Ride quality is an important parameter for car manufacturers, which clarifies the
transmission level of unwanted noises and vibrations from vehicle body to the passengers.
The term unwanted is defined according to human response to vibration which is different
from one person to another and will be described more in the next chapters. Increasing
customer demands for more comfortable cars and better ride quality, not only requires “full
understanding of human response to excitation”, but also it is needed to study “different
sources which may result to vibration of vehicle body”, and “dynamic behavior of the
automobiles”.
In order to provide better realization of ride behavior [1], it is useful to show the ride
dynamic system as follows (Figure 1-1):
Figure ‎1-1, the ride dynamic system
According to Figure 1-1, there are four different excitation sources that may be divided
into two categories: 1) road surface irregularities and 2) on-board origins which result from
rotating parts (engine, driveline and non-uniformities (imbalances) of tire/wheel).
Since the days of first vehicles, the attempts have been made to isolate the car body from
road roughness1, and the car suspension system is responsible for this duty. Road profile is a
random function which acts as an input to suspension system, furthermore theory of
stochastic processes and power spectral densities have been utilized in the literatures to
model this random signal. The two degrees of freedom model (2-DOF) known as a quartercar model is used to simulate suspension system and vehicle body [2]. The goal is to
optimize the suspension system parameters to decrease the undesired effects on the vehicle
body (Chapter 7) according to ride comfort criterion that may be selected.
Driveline is one of the considerable sources of noise and vibration for any type of
automobiles, which is composed of everything from the engine to the driven wheels.
Driveline torsional oscillations fall into two broad categories: “gear rattle” and “driveline
vibration”. Idle gear rattle is a consequence of gear tooth impacts, and driveline vibrations
are noises which come from the driveline system parts such as engine, clutch and universal
1
roughness is described by the elevation profile along the wheel tracks over which the vehicle passes ‎[1]
1
Chapter1
Introduction
joints, while the vehicle is in motion at different running situations [3], we are interested in
the driveline system vibration in three aspects: 1) finding system natural frequencies in order
to avoid coincidence with forced frequencies and resonance occurrence, 2) to determine
forced response subject to engine oscillatory torque and universal joints and 3) transient
response. It should be noted that some components of driveline such as universal (Hooke’s)
joints result to nonlinear behavior of the system, and in addition the torsional vibration of
driveline can be coupled with the horizontal and vertical motions of vehicle body and rear
axle, these phenomenon may cause the complication of the system.
Modeling the system (quarter car model or driveline) and obtaining the response, it’s time
to evaluate ride quality of the system. Hence it is necessary to specify ride comfort limits.
Various methods have been developed over the years for assessing human tolerance to
vibration [4] which will be more explained in chapter 8.
1.2 Objective
The goal of this thesis is divided into two major parts:
1- To model the driveline and engine fluctuating torque in order to find free and forced
responses of the system and furthermore studying the sensitivity of driveline behavior by
changing design parameters. The attempt is made to simulate driveline as a 14-degrees-offreedom (DOF) system and the whole vehicle as an 18-DOF mechanism and at last
determining the horizontal acceleration of sprung mass (vehicle body) due to engine torque
using Runge-kutta numerical method. Acceleration time history is then converted to
frequency domain using power spectral density tool in order to compare with ride comfort
diagrams.
2- To use quarter car model and obtain driver response subject to road random irregularities
with the aid of random process theory. In this part of the report, the importance of the
suspension system to decrease the undesired motions will be illustrated.
1.3 Assumptions and Limitations
In this project a lumped-parameter model is used for studying the torsional vibration of
driveline system which assumed to be a set of inertia disks linked together by torsional,
linear and massless springs [3]. A normal four cylinder rear drive passenger car will be
considered and the system parameter values such as sprung and unsprung masses, all the
stiffness and damping coefficients have been chosen according to references [3] and [4] and
different vehicle companies database. It should also be noted that this work is based on the
PhD thesis by El-Adl Mohammed Aly Rabeih which is done in 1997.
The engine fluctuating torque (as will be described later) consists of two major parts: gas
pressure torque and inertia torque1 which are come from cylinder gas pressure and
reciprocating components of engine, respectively. however in the current report, we will
only study the effects of the pressure torque since there is no useful data for the mass of
reciprocating parts, furthermore the cylinders pressure are measured in the vehicular system
engine Laboratory for a four-stroke four-cylinder engine with the firing order of 1-3-4-2.
Moreover the nonlinear torque which is resulted by Hook’s joints has been introduced in this
thesis while defining the response of the system subject to this couple requires strong
nonlinear method which is beyond the aim of this work.
In order to investigate the road surface influences, three assumptions have been included:
1
Especially in high speed vehicles, the inertia torque is very important!
2
Chapter1



Introduction
The road profile is assumed to be a stationary ergodic random process, however in
reality the road’s profiles are non-stationary functions.
the amplitude distribution of the road roughness is assumed to be Gaussian
The car has a constant speed and travels on a straight line.
1.4 Outline
The thesis is composed of 8 chapters. This introductory chapter is followed by a short
literature review of what has been done so far associated to this work. In chapter 3, driveline
system and its different parts have been modeled as well as relations of converting cylinders
pressure to the torques which are delivered by crankshaft. Chapter 4 included of
mathematical simulation of driveline, and modal analysis to find natural frequencies of the
system. Chapter 5 consists of introducing different methods of obtaining forced response of
14-degrees of freedom system. In chapter 6 the whole vehicle model has been described and
horizontal vibration of the vehicle body and rear axle is obtained subject to driveline
torsional oscillations. Besides, the parameter values will be changed in order to study the
sensitivity of the system to stiffness and damping coefficients. Chapter 7 consists of using
quarter-car model to define the response of driver to road non-uniformities which is an input
to the system. The report will be ended by chapter 8 which has included international
standard ISO for evaluation of human exposure to whole-body vibration [4] and calculating
RMS vertical and horizontal accelerations of our model to compare with ride comfort
criteria. Moreover the conclusion section and future work suggestions are the last parts of
the final chapter.
3
Chapter1
Introduction
4
Chapter 2
previous works
2 Chapter 2
Previous work
2.1 Driveline and vehicle Modeling
During the last four decades and from the early days of automotive industry, the attempts
have been made to reach the desired ride comfort and quieter vehicles. Since driveline
torsional behavior is one of the major resources of unwanted vibrations in the cars,
significant researches have been done by different car companies and university scientists in
order to gain an acceptable model for driveline system and its components structure. Skyes
and Wyman in 1971, and Ergun in 1975 were among the first people who calculated natural
frequencies of a conventional automobile driveline system theoretically and experimentally,
respectively [3]. Reik in 1990 studied the main vibration sources of driveline system and he
considered the gas pressure torque and its effects [5]. Zhanqi et al in 1992, [6], constructed a
mathematical model including torsional, vertical and vehicle fore-aft vibrations to study the
coupling of those vibrations together, he also has investigated the influences of different
parameter values on the response of the vehicle chassis and axle. Significant experimental
measurements of system behavior and principal modes have been done by different
researchers in the years 1980-1990 [7]. El-Adl Mohammed Aly Rabeih has done a complete
research in 1997 concerning the driveline and compelete vehicle free and forced vibration
modelings, sensitivity analysis of driveline and suspension system parameter values. he used
runge-kutta numerical method in order to find displacement and velocity time histories at
different points of the model [3]. In the last decade, more investigations are focused on the
nonlinearity in driveline system such as gear rattle, backlash, and bahvior of the system
during clutch engagement.
2.2 Road Surface Irregularities
The health, protection, ride comfort and performance of both driver and passengers in
automobiles are influenced by the type of the surface, over which the vehicle moves. Earlier
researches in the automotive industry included of subjecting mathematical models to
deterministic inputs, however in real situation surface profiles are rarely simple forms. A
significant effort has been done between the years 1950-1970 in order to find the spectrum
of road roughness [10]. Furthermore attempts by various organizations have been done over
the years to divide the road surface irregularities into different classes, the international
organization for standardization (ISO) has presented this classification (A-H) based on the
power spectral density [4].
A huge amount of reports during the last four decades have included studies about
passive, semi-active and active suspension systems and vibration isolations in automobiles
as well as different optimizations methods to define optimum parameter values of
suspension system subject to random excitation from the road [13].
2.3 Human Response to vibration
Significant investigations have been conducted to acquire ride comfort limitations. There are
different standards to evaluate drivability and ride quality of a vehicle which are mentioned
5
Chapter 2
previous works
in references [4] and [17]. According to ISO 2631-1:1985, four parameters have trivial
influences on obtaining human response to vibration which are intensity, frequency,
direction and exposure time. In this work, these criterions will be used to study the ride
comfort of the desired model.
6
Chapter 3
Description of driveline and torsional phenomena
3 Chapter 3
Description of driveline and torsional phenomena
3.1 Introduction
In this chapter, torsional models of driveline system and its components are defined, since
driveline torsional vibration due to engine torque excitations is one of the main reasons of
undesired noise in the vehicles. Further, the necessary relations in order to obtain oscillatory
engine torques from cylinder pressures time histories are introduced.
3.2 Driveline components
The driveline function is to transmit mechanical energy of the engine to the wheels and that
will be occurred through different parts. A classical front-engine rear-wheel-drive vehicle
driveline is illustrated in Figure 3-1, Front-engine rear-wheel-drive vehicle driveline[3]. The
most common components of the system are engine, flywheel, clutch, gearbox, propeller (or
Cardan) shaft and universal joints, differential and rear axle assembly and tires. According
to the goal of the thesis and in order to simplify the analysis of driveline vibration, a lumped
parameter model is used for the whole driveline system. However from the vibration theory,
it is known that in the real world the systems are distributed, and therefore cannot be
modeled as point masses. Although, utilizing this simple simulation has some advantages
and the most important one is the ability of estimating natural frequencies and the forced
response of the system subject to different excitations without complicated mathematics. In
the following sections, each part is described in more detail and an appropriate model will be
suggested.
3.2.1 Engine, flywheel and the main excitation torque
Engine is the primary power source on a vehicle. Rotating behavior of the engine and
discrete strokes during its working cycle result to torsional vibration excitation of the
driveline.
Figure ‎3-1, Front-engine rear-wheel-drive vehicle driveline ‎[3]
7
Chapter 3
Description of driveline and torsional phenomena
The main parts of the engines are: cylinder, piston, connecting rod and crankshaft 1 which
are shown in Figure 3-2, Main engine parts. The crankshaft rotates by pushing the piston up
and down in the cylinder area, there are two dead points (at extreme down and up positions)
where the pressure on the piston will have no effects to force the crankshaft to turn, a stroke
is called to the movement of piston from one dead center to another dead center. Four-stroke
engine consists of induction, compression, power and exhaust steps, more description of
engine parts and strokes function can be found in the books of motor vehicle technology and
it is not the aim of this thesis.
Figure ‎3-2, Main engine parts ‎[1]
Since we are interested in torsional vibration of the driveline, the rotational dynamics of
the engine will be simulated by taking into account the crankshaft system which is shown in
Figure 3-3:
Figure ‎3-3, Original System of a crank ‎[3]
The compact crankshaft system can be modeled as follows (Figure 3-4), where the
rotational Jw (journal+ crank pin+ webs) and reciprocating parts Jr (piston+ connecting rod+
piston pin) are composed together as one final inertia disk2 J1.
The crankshaft (crank) is the part of an engine which translates reciprocating linear piston motion into
rotation, crankshaft connects to flywheel.
2 It should be noted that the engine mounts, which are important tools to decrease the unwanted effects
of engine vibration on the other pieces and to isolate the engine from the external excitations, will not be
considered here and their suitable influence can be perused in the next studies.
1
8
Chapter 3
Description of driveline and torsional phenomena
Figure ‎3-4, Equivalent system of crankshaft and its compact model ‎[3]
In the above figure, T(t) is the engine excitation torque, and k is the equivalent torsional
stiffness which is calculated in reference [3].
Because of the cyclic operation of a piston engine, the torque which is delivered at the
crankshaft is oscillatory and consists of a steady-state component (mean torque) plus
superimposed torque fluctuations T(t)1 (Figure 3-5):
Figure ‎3-5, output torque of a four-stroke single-cylinder engine ‎[1]
Concerning to practical issues, there are always more than one cylinder which are
arranged to have their power strokes in succession [20], the most common case is to have
four cylinders. the firing order of the engine illustrates the order in which the cylinders act,
in this thesis we will consider four-stroke four-cylinder engine with firing order 1-3-4-2,
consequently we will have 720 degrees per cycle of operation for this kind of engine and
each stroke takes 180 degrees. Figure 3-6 shows a simplified line-diagram of the cylinders
and cranks,
Moreover, the torsional vibration of the crankshaft due to longitudinal torque on the moving part of the
engine, is of particular importance because many crankshafts have failed subject to this torque.
1
9
Chapter 3
Description of driveline and torsional phenomena
(1)
(2)
(3)
(4)
Figure ‎3-6, Line diagram of cylinders arrangement
As it is seen in the above figure, pistons move in pairs: 1&4 and 2&3. The measured
pressures in engine laboratory are associated to cylinders 1 and 2 and the assumption of this
work is that: cylinders 1 & 4, and 2 & 3 have the same pressure distributions, respectively.
The following graph presents the expected torque for a four cylinder engine.
Mean
torque
Figure ‎3-7, engine torque in the case of four cylinders
Each cylinder excitation torque1 formed from two main parts2: gas-pressure torque
{Tg(t)} and inertia torque, however as it is also mentioned in section 1.3, in this report we
will study only the influence of the gas-pressure. Gas-pressure torque itself composed of
harmonics3 and steady-value, while the steady-value (mean value) will not excite torsional
vibration; it is omitted in the calculations.
Figure 3-8 and Figure 3-9 show the cylinders 1 and 2 pressures time history {pg1(t) &
pg2(t)} respectively, in addition the characteristics of the LAB engine are given in Table 3-1,
Table ‎3-1, Engine properties
Num. of
strokes
Num. of
cylinders
Piston
Diameter,
m
Crank
radius,
m
Connecting
rod length, m
Mean
Torque4,
N.m
4
4
0.086
0.043
0.043*4
100
Engine speed,
RPM, rad/sec
Expected fluctuating
torque fundamental
frequency5
2000
104.72 rad/s
ω=209.4395
or 16.66 Hz
which causes torsional vibration
the friction torque is assumed to be small compared to these two main components
3 which repeats themselves every complete working cycle, the interval of repetition is two revolutions of
the crankshaft (4π) and the period is 4π/ω ‎[3]
4 The useful engine mean torque is the steady part of cylinders net torque which is measured by a sensor
at flywheel point, this value is normally provided by the engine manufacturer
5 Refer to page 12
1
2
10
Chapter 3
Description of driveline and torsional phenomena
In this stage, the relations of converting cylinder pressure to delivered torque by the
crankshaft are presented. In addition the mean value of calculated torque will be subtracted
from the total torque in order to find the fluctuating part,
 Applying force on the piston (Fp) in Figure 3-10, Crank mechanism = Gas pressure
{p(t)} * piston area ( Ap).
Original figure
Zoomed version
Figure ‎3-8, 10 seconds pressure recording from cylinder 1



Gas torque {Tg(t)}= Fp * dxp/dφ where crank angle φ =ωt and ω is the constant
crankshaft speed, furthermore xp denotes the piston displacement
According to Figure 3-10, Crank mechanism, it is possible to derive the expression
for dxp/dθ [21],
In the above figure, r is the crank radius and l is the connecting rod length (these
values have been provided in Table 3-1).
11
Chapter 3

Description of driveline and torsional phenomena
Piston displacement in terms of crank angle can be estimated1 in the following form:
3.1
Therefore, the differentiation of
with respect to
is2:
3.2
Original figure
Zoomed version
Figure ‎3-9, 10 seconds pressure recording from cylinder 2
1
The exact expression is available in reference ‎[23]‎[21]
2
It should be noted that
12
Chapter 3
Description of driveline and torsional phenomena
Figure ‎3-10, Crank mechanism

Finally, the associated torque due to combustion,
is
3.3
Figure 3-11 and Figure 3-12 show the total torque outputs (in 10 seconds and in one
working cycle as well) and their fluctuating parts from cylinders 1 and 2 respectively.
As it was expected, the computed torque in one working cycle is similar to what was
demonstrated previously (Figure 3-5) for a four-cylinder four-stroke engine. In Figure 3-11
and Figure 3-12, different strokes are clearly distinguishable. Cylinders 3 and 4 output
torques are equal to cylinders 1 and 2 outputs, according to the assumption which was made
before. The noises which are seen in the plots are removable using filter commands, the
necessity of using filtration and the associated MATLAB commands will be described more
in detail in the next chapters. Finally to check the correctness of presented torque
calculations from the measured output pressures of cylinders 1&2, it is functional to find the
mean value for summation of torque 1,
torque 2, torque 3 and torque 4 {Tg1(t), Tg2(t),
Tg3(t), and Tg4(t)}. We expect that this mean value have to be around the value which was set
during experiment, 100N.m: with the aid of MATLAB command mean
(torque1+torque2+torque3 +torque 4) = 109.5114, which seems acceptable according to 100
N.m.
13
Chapter 3
Description of driveline and torsional phenomena
Figure ‎3-11, Torque output of cylinder 1, total and fluctuating part, during 10 seconds and one working cycle
14
Chapter 3
Description of driveline and torsional phenomena
Figure ‎3-12, Torque output of cylinder 2, total and fluctuating part, during 10 seconds and one working cycle
It would be useful to plot cylinders output torques in one graph (Figure 3-13): Tg1(t),
Tg2(t), Tg3(t), and Tg4(t).
15
Chapter 3
Description of driveline and torsional phenomena
Figure ‎3-13, Output torques from 4 cylinders in the same plot 1
The driveline system is subjected to these input excitation torques which are shown in
Figure 3-14. This figure represents compact crankshaft model of a four-cylinder engine [3].
It should be noted that Jd is the torsional damper and Jf is the flywheel mass moments of
inertia, respectively. Function of the flywheel is to decrease the magnitude of angular
accelerations produced by input excitation torques Tg1(t), Tg2(t), Tg3(t), and Tg4(t).
1
This diagram is similar to Figure ‎3-7
16
Chapter 3
Description of driveline and torsional phenomena
Figure ‎3-14, Compact crankshaft model for a four-cylinder engine
Furthermore, to see the excitation (forced) frequencies1 which are associated to the above
torques, the power spectral densities of the cylinder 1 and cylinder 2 time histories, in Figure
3-13, are obtained using MATLAB commands and are demonstrated in the following figure.
Figure ‎3-15, PSD for output torque from cylinder 1
1
fundamental frequency (refer to Table 3-1)and its multiplications
17
Chapter 3
Description of driveline and torsional phenomena
Figure ‎3-16, PSD for output torque from cylinder 2
According to Figure 3-15 and Figure 3-16, as it was expected from Table 3-1, Engine
properties, the first (fundamental) frequency is almost equal to 16.6 Hz (half engine speed1)
and the next excitation frequencies are 33.2, 50, 66.9, 83 … Hz.
3.2.2 Clutch Assembly
We have the clutch system (Figure 3-17) after flywheel which is made of two different
components, clutch disk and clutch mechanism[22]:
Figure ‎3-17, Clutch system ‎[22]
Therefore as it will be seen in chapter 5, resonance happens when half engine speed or half multiple of
engine speed is equal to one of the natural frequency of the system
1
18
Chapter 3
Description of driveline and torsional phenomena
The major duties of the clutch assembly are to join and disjoin the gearbox with the
engine, to transmit engine power to the input shaft, and to supply isolation from the
oscillatory engine torque oscillations. This function is achieved by two mechanisms
rotationally connected by an elastic and dissipative system which can rotate together (Figure
3-17, Clutch system), The first system is the clutch disc and rings connected to the flywheel,
and the second is the clutch hub connected to the input shaft via spline backlash [22]. Two
different working conditions can be considered for clutch: 1-clutch behavior during the
steady state running (linear action) and 2- clutch treatment during engagement (nonlinear
phenomenon1), however in this work we simply model the clutch system as an inertia disc
together with the flywheel which is connected to the gearbox via a spring and a damper [3]
and it is shown in the driveline overall model in section 3.3.
3.2.3 Gearbox
The third component in the driveline system is the gearbox which consists of various helical
gears in order to provide the ability of changing the speed ratio between the engine and
driving wheels for driver of the car. We have two major groups of the gearboxes: manual
and automatic. Since the dynamic model of gearbox mechanism is related to the purpose of
the study and concerning the aim of this thesis, which is analysis of the driveline torsional
behavior, therefore modeling of gears and carrying shafts as a simple torsional vibratory
system is the primary interest of this step of the report. The following mathematical model
is suggested for the torsional vibration of driveline, where the model included an equivalent
inertia disc for each of the gear that transmits torque. it should be noted that the inertia of
each disc contains also the inertia of the idling gears which results to the reduction of the
driving gear speed.
Figure ‎3-18, Gearbox model ‎[3]
3.2.4 Cardan (propeller) shaft‎and‎universal‎(Hooke’s)‎joints
Cardan shaft transmits the engine torque from the differential to the wheels. Since the engine
gearbox shaft, cardan shaft and back axle are not in line, a universal joint, which is shown in
Figure 3-19, has to be used in order to attach them. The Hooke’s joint suffer from one
important problem: even when the input shaft has a constant speed, the output shaft rotates
at a variable speed. We know that velocity change means acceleration and concerning
Newton’s law, acceleration results to force. Therefore a secondary couple will be created
and it is nonlinear. The magnitude of this produced torque is proportional to the torque
One of the most important purposes of torsional vibration of the driveline is during clutch engagement
in manual gear box mechanisms. The study of this topic is too complex and beyond the goal of this
report.
1
19
Chapter 3
Description of driveline and torsional phenomena
which is applied on the driveline and the Hooke’s joint angle, thus the variation of the
driveline excitation torque will cause new torque1 fluctuation.
Figure ‎3-19, Hooke's (cardan) joints ‎[1]
In order to find a simple model for propeller shaft and universal joints in the whole
driveline system, we assume that the mass moment of inertia of the joints is much larger
than the propeller shaft, therefore the system is regarded as an elastic massless shaft (like a
spring) between two inertias [3]. Moreover, the generated torque by the joints will be
applied on the two ends of the cardan shaft as it is shown in Figure 3-20, propeller shafts and
universal joints mathematical model,
Figure ‎3-20, propeller shafts and universal joints mathematical model ‎[3]
3.2.5 Differential and final drive system
A differential is a mechanism in automobiles, usually but not necessarily including gears,
which has the ability of transmitting torque and rotation through three shafts, normally it
receives one input and provides two outputs. the differential also allows each of the driving
roadwheels to rotate at different speeds. Final drive system consists of differential and two
similar shafts which are connected to the wheels, and a simple model for that, is
demonstarted in the following figure:
The derivation of the vibratory torque which is generated by universal joints is completely described
in ‎[3].
11
20
Chapter 3
Description of driveline and torsional phenomena
Figure ‎3-21, Final drive system and its equivalent model ‎[3]
3.2.6 Damping in the whole driveline system
Damping is a resisting force which acts on the vibrating body and may arise from different
sources such as friction between dry sliding surfaces, friction between lubricated surfaces,
air or fluid resistance, electric damping, and internal friction due to imperfect elasticity.
Regarding the damping type, the mathematical model is different and may depend on the
velocity of the motion, material, viscosity of the lubricant and etc... The cases, in which the
friction forces are proportional to velocity, are named as viscous damping. In the current
driveline system, we will only consider the effects of viscous damping in different
components1 and other kinds of damping are neglected. The equivalent viscous torsional
damping coefficients are given in reference [3].
3.3 Overall driveline model
As it was described before, the overall torsional model for driveline system is based on
discretisation and lumped masses are used. The suggested 14-degrees of freedom linear
model for a four-cylinder rear-drive passenger car is shown in Figure 3-22 which composed
of inertia discs and massless torsional springs and viscous dampers.
1
Crankshaft, engine, clutch disc, gearbox, propeller shaft and differential units, and tires.
21
Chapter 3
Description of driveline and torsional phenomena
Figure ‎3-22, Damped torsional vibration mathematical model of driveline system ‎[3]
This model is used throughout this thesis in order to study the free and forced vibration of
driveline and whole vehicle.
3.4 Torsional vibration
There are different excitation sources (linear and nonlinear) for the torsional vibration of the
driveline model which is shown in Figure 3-22, Damped torsional vibration mathematical
model of driveline system. However as it was mentioned in section 3.2.1, engine torque
oscillations1 (linear behavior), is the main reason of torsional vibration. Studying the
nonlinear purposes such as Hooke’s joints is beyond the scope of this thesis. It should be
noted that torsional vibration is in primary interest since firstly, it may cause harmful effects
on the different parts of the system and secondly, it will be coupled with the whole body
motions of the vehicle and results to longitudinal vibration which is investigated in the next
chapters.
1
due to different strokes
22
Chapter 4
Undamped natural frequencies of the overall driveline system
4 Chapter 4
Undamped natural frequencies of the overall
driveline system
4.1 Introduction
This chapter includes solving driveline differential equations of motion in order to find
natural frequencies of the system. To avoid resonance, which is a harmful phenomenon for
mechanical systems, it is necessary to obtain natural frequencies of the structure. Modal
analysis is used to study undamped model of the driveline system.
4.2 Mathematical model and system matrices
The governing differential equation for torsional vibration of the overall driveline system
(14-degrees of freedom) which is shown in Figure 3-22, is
4.1
where , ,
and
are the symmetric mass moment of inertia1, torsional damping,
stiffness, and applying force (engine fluctuating torque) matrices, respectively and finally
is the 14-dimensional column vector of generalized coordinates. In Table 4-1,
parameter values are given for mass moment of inertia, stiffness and damping coefficients of
different components in driveline. In order to obtain the system matrices for this multidegrees of freedom system, two methods have been described in vibration theory books
[21]: Newton’s procedure and energy method, the closer one applies Newton’s laws on the
free body diagram of each component and it is straightforward but time consuming, while
the energy method is based on Lagrange’s equations and more practical for large systems. In
this thesis Newton’s method is utilized and the inertia, stiffness and damping matrices have
been determined as follows:
1
Since we have torsional vibration, M matrix is briefly called inertia matrix
23
Chapter 4
Undamped natural frequencies of the overall driveline system
Table ‎4-1, Typical values for equivalent parameters of a vehicle driveline ‎[3]
Equivalent stiffness
coefficient
Equivalent system damping
coefficient
Equivalent moment of inertia
(kg.m2)
(N/m or N.m/rad)
(N.s/m or N.m.s/rad)
Parameter
Value
Parameter
Value
Parameter
Value
k1
0.2e6
J1
0.3
c1
3
k2
1e6
J2
0.03
c2
2
k3
1e6
J3
0.03
c3
2
k4
1e6
J4
0.03
c4
2
k5
1e6
J5
0.03
c5
2
k6
0.05e6
J6
1.0
c6
4.42
k7
2e6
J7
0.05
c7
1
k8
1e6
J8
0.03
c8
1
k9
0.1e6
J9
0.05
c9
1
k10
0.1e6
J10
0.02
c10
1.8
k11
0.2e6
J11
0.02
c11
1.8
k12
0.5e6
J12
0.3
c12
2
k13
0.5e6
J13
2
c13
10
k14
0.2e6
J14
2
c14
10
k15
0.2e6
K
 k1
 k
 1
 0

 0
 0

 0
 0

 0

 0
 0

 0
 0

 0
 0

 k1
0
k1  k 2
 k2
 k2
k 2  k3
 k3
0
 k3
k3  k4
 k4
0
 k4
k 4  k5
 k5
0
0
0
0
0
0
0
0
0
0
 k5
k5  k6
 k6
0








0


0

0


0


0

0

 k12 

0

k12  k14 
0
0
0
0
0
 k6
k6  k7
 k7
 k7
k 7  k8
 k8
 k8
k8  k9
 k9
 k9
k 9  k10
 k10
0
0
0
0
24
0
0
 k10
k10  k11
 k11
 k11
k11  k12  k13
 k13
0
 k12
 k13
k13  k15
0
Chapter 4
Undamped natural frequencies of the overall driveline system
C
 c1
 c
 1
 0

 0
 0

 0
 0

 0

 0
 0

 0
 0

 0
 0

 c1
0
0
0
0
0
c6
 c6
 c6
c6  c7
0
0
0
0
0
0
c1  c 2
c3
c4
c5
c8
c9
c10
c11
c12
0
0
0
0
0
0
0
0
0
0
0
0
0
0
c13
0
0 
0 
0 

0 
0 

0 
0 

0 

0 
0 

0 
0 

0 
c14 
M
J1
0

0

0
0

0
0

0

0
0

0
0

0
0

0
0
0
0
0
0
0
0
0
J2
J3
J4
J5
J6
J7
J8
J9
J 10
J 11
J 12
0
0
0
0
0
0
0
0
0
0
0
J 13
0
0 
0 
0 

0 
0 

0 
0 

0 

0 
0 

0 
0 

0 
J 14 
Now to find undamped natural frequencies of the system, the right hand side of equation
4.1 and damping matrix are set to be zero. it should be mentioned that, since the damping
matrix is not a linear combination of the inertia and stiffness matrices (
), it is
1
not possible to use modal analysis to decouple the equations, therefore undamped natural
frequencies are to be determined, however they are almost the same with damped
frequencies which are provided in reference [3].
4.3 Summary of Modal analysis
Modal analysis is a procedure to find the natural frequencies of the system by decoupling the
system differential equations of the motion which is given in equation 4.1. The problem is
that when the equations are coupled, it is not possible to solve them separately at the same
1
which is described in section ‎4.3
25
Chapter 4
Undamped natural frequencies of the overall driveline system
time. There are different types of coupling: static coupling1 and dynamic coupling2,
according to the mass and stiffness matrices which are already given, we have static
coupling in driveline system of equations. It is also useful to mention that the selection of
coordinate system influences on the existence or nonexistence of the coupling.
3
As it is known from vibration theory [21], we can substitute
in the equations of
motion (equation 4.1), and further we reach to the following expression:
4.2
By removing the scalar value
,
4.3
In order to solve equation 4.3, it is converted to the form
which is the familiar
form for Eigen-value problems. To do so, both sides of relation 4.3 are multiplied by the
term
from the left as follows:
4.4
or
4.5
and finally we obtain:
4.6
in this case
and
. Therefore the natural frequencies of the multidegrees of freedom system are the inverse of the positive square roots of the Eigen-values of
matrix . It is possible to find these values by using eig command in MATLAB. In order to
determine the corresponding mode shape4 for each frequency
, the following equation
has to be solved:
4.7
using these mode shapes, we can form the modal vectors matrix that is the base of modal
equations5 for modal analysis6. however (as it was noted in previous section) in the current
system, this method of solution is not usable to attain the forced response of driveline
mechanism since the damping matrix is un-proportional and the above procedure do not
decouple the equations which are coupled by damping coefficients. Although, in the next
When the stiffness matrix is not diagonal
When the mass matrix is not diagonal
3 Since we know that the response of the undamped vibratory system, x will be sinusoidal, therefore it
can be shown by exponential form
4 finding the mode shapes of one mechanical system provides the information about the positions at
which large displacement will occur and therefore, it would be possible to represent a solution to
attenuate the harmful vibration
5 Modal equations are n independent relations for an n-degree of freedom system which are solvable
separately, the new coordinate are called principal coordinates.
6 More description is available in vibrations book ‎[21]
1
2
26
Chapter 4
Undamped natural frequencies of the overall driveline system
chapter, forced response of the system is determined using both modal analysis and
numerical method, and the precision of the solutions is studied according to reference [3], in
order to see the inaccuracy of modal method and to evaluate the precision of numerical
method.
4.4 Natural frequencies
Resonance is a harmful phenomenon which happens in mechanical systems, this results to
failure of the mechanism and is very dangerous in the case of passenger cars. As it is known,
resonance occurs when natural frequencies of the system coincident with the forced
frequencies, therefore in order to avoid this unwanted situation, it is necessary to know
natural frequencies of the system (using the method of previous section for a multi-degree of
freedom system). Moreover we will find the mode shapes of driveline to be aware about the
positions which have considerable vibration amplitude.
Among the undamped natural frequencies for torsional vibration of the whole driveline
model Table 4-2, the first six frequencies are important while they are in operating range of
the driveline system.
7
8
9
10
11
12
13
14
679.986
838.878
955.638
1421.383
1730.087
1845.537
109.682
6
432.407
5
427.421
4
139.041
3
50.856
(Hz)
2
12.585
Natural
Frequency
1
9.416
Mode number
4.03
Table ‎4-2, Undamped natural frequencies of whole driveline model using typical parameter values for a passenger
car
These values have a very good agreement with the damped natural frequencies for the
system, which are given in [3]- table 5.2. However there are fewer calculations in the case of
finding undamped frequencies for the vibratory systems. Further, the first five mode shapes
of the system are obtained as in Table 4-3.
27
Chapter 4
Undamped natural frequencies of the overall driveline system
Table ‎4-3, first five natural modes of driveline system
1st Mode
2nd Mode
3rd Mode
4th Mode
5th Mode
1.0000
0
-1.9806
1.0000
-2.8087
0.9990
0
-1.9621
0.8468
-0.8078
0.9982
0
-1.9579
0.8136
-0.3961
0.9986
0
-1.9535
0.7779
0.0212
0.9983
0
-1.9487
0.7398
0.4383
0.9981
0
-1.9435
0.6994
0.8491
0.9799
0
-1.5969
-1.5362
1.0000
0.9794
0
-1.5879
-1.5882
0.9919
0.9785
0
-1.5698
-1.6873
0.9616
0.9686
0
-1.3836
-2.5919
0.4299
0.9586
0
-1.1956
-3.4436
-0.1426
0.9535
0
-1.1009
-3.8342
-0.4221
0.8337
-1
1
0.0972
0.0022
0.8337
1
1
0.0972
0.0022
In addition, the second mode only includes the driving wheels vibration (with a common
frequency) and other components of the system are in rest, therefore it can be concluded that
the 2nd mode has been never excited by any excitation torques.
28
Chapter 5 Steady-state Response of linear driveline model due to engine
fluctuating torque
5 Chapter 5
Steady-state Response of linear driveline model due
to engine fluctuating torque
5.1 Introduction
In this chapter, forced response of driveline model subject to the engine cylinder pressure
input torques is determined, using the same system of differential equations as chapter4,
however now excitation forces exist in the right hand side of the equations. According to the
final solutions, the results of Runge-kutta numerical method provide desired accuracy.
5.2 Mathematical model for forced vibration of the driveline system
In order to obtain forced response of the linear damped system, again we consider the main
differential matrix equation for torsional vibration of the driveline mechanism (Figure 5-1):
5.1
here the force matrix is a column vector as follows:
0

Tg1 (t ) Tg 2 (t ) Tg 3 (t ) Tg 4 (t ) 0 0 0 0 0 0 0 0 0
1
4.2
where Tg1 (t ) , Tg 2 (t ) , Tg 3 (t ) and Tg 4 (t ) are engine fluctuation torques from different
cylinders which are given in Figure 3-13. the applying forces are not sinusoidal but periodic.
Figure ‎5-1, Driveline model
29
Chapter 5 Steady-state Response of linear driveline model due to engine
fluctuating torque
5.3 Time responses of driveline at clutch and driving wheels
Steady-state response of the equation 5.1 is found using 3 methods in this section: Modal
analysis as an analytical method, and 2 numerical procedures1: ode45 function in MATLAB
and self-written fourth order Runge-Kutta method. Although the excitation forces are not
exactly similar to reference [3]2, there is a good agreement between the responses from
numerical methods that have been utilized here and the results which are given in [3],
however the Modal analysis answer is not satisfactory. In the following paragraphs, a brief
description is provided for each of the three solution methods:

Modal analysis
Decoupling the system of differential equations 5.1, by using Modal procedure in
section 4.3. In addition, the modal forces have to be obtained.
II.
Finding the forced response of each independent equation using lsim3 function
in MATLAB, the associated command is given in Appendix 10.1.
m(1,j),c(1,j)and k(1,j)are read from modal inertia, damping and stiffness matrices
respectively. The big assumption is that damping matrix will be decoupled
through modal process.
III.
The obtained solution is represented in Figure 5-2 for clutch torsional
displacement together with the solutions from the numerical methods. The output
result from modal analysis is not correct in the amplitude value, and also it is
unstable (which is more clear in Figure 5-3). All the corresponding matlab codes
are attached in appendia ‎10.2‎10.2.
 ode 45 function
I.
MATLAB software includes a series of functions which are called solvers in
order to solve ordinary differential equations of each order, they use Runge-Kutta
numerical method with variable time step to find the solution of the equations.
ode 23, ode 45 and ode 113 are the most famous functions of these solvers. In
this thesis ode 45 is used to find the forced response of the 14-degrees of freedom
driveline system. To do so, first it is necessary to convert the equations of the
motion to a set of first order differential equations (this form is named statevariable form or Cauchy form), this new form4 is saved in a separate function.
II.
The second step is to manipulate the force data in order to use them as an input
for the system.
III.
Finally the command
[T,Y]=ode45(@(t,z)sde(t,z,t1,F),[0,10],[zeros(14,1).zeros(14,1)])is written to
find the solution for the system of equations in the desired time interval [0,10]
with zero intial postions and velocities, there are two problems that this method
suffers from: ”a little unstability which appears in Figure 5-2” and ”it is vey time
consuming5”.
I.
However the base of both numerical methods is Runge-kutta method.
He has used sinusoidal terms to simulate the engine torques and the solution is obtained using Modal
analysis method.
3 According to MATLAB help, lsim is an useful function in this software which is able to determine the
vibratory system response to any arbitrary point data input as
4 28 first order differential equations
5 For 10 seconds, it took around 2 days to find the answer
1
2
30
Chapter 5 Steady-state Response of linear driveline model due to engine
fluctuating torque
Figure ‎5-2, Time response at the clutch, using different methods of solution: Black-> Modal analysis, Blue-> ODE45,
Green-> self-written Runge-Kutta code with nonzero initial conditions and yellow-> with zero initial conditions
Figure ‎5-3, zoomed version of figure 5.2 in order to see the instability of modal analysis and ODE45 solutions

Self-written Runge-Kutta code
I.
In order to improve the calculations time, a MATLAB m-file is written to find
the solutions for a system of fourteen second order differential equations using
numerical method which is called Runge-Kutta1, in this work the fourth order
Runge-Kutta will be utilized which means that the error per step is on the order
of h5, while the total accumulated error has order h4.
From Wikipedia -> In numerical analysis, the Runge–Kutta methods are an important family of
implicit and explicit iterative methods for the approximation of solutions of ordinary differential
equations. These techniques were developed around 1900 by the German mathematicians C. Runge and
M.W. Kutta.
1
31
Chapter 5 Steady-state Response of linear driveline model due to engine
fluctuating torque
II.
III.
For this method we have to do the same manipulations like the previous
procedure to convert all the 14 second order equations to 28 first order equations
(State-variable form).
The final result is shown in Figure 5-2, which has not only less instability1 in
comparison with ode 45 function solution, but also it takes a few minutes to
obtain the answer which is very valuable regarding the computer program
efficiency. Another point which is important is that, according to Figure 5-4,
there is no difference in the response of the system, either we consider zero intial
velocities or when a non-zero matrix2 is set to be as initial velocities.
Figure ‎5-4, Time response at the clutch with the aid of Runge-Kutta method
displacement vibrates around zero position which is preferable
We are studying steady-state running and the force data are not from a period of starting and stopping
the engine, we have only picked 10 seconds of running engine.
1
2
32
Chapter 5 Steady-state Response of linear driveline model due to engine
fluctuating torque
In comparison with the clutch response, which is given in reference [3], smaller amplitude is
achieved here because the engine excitation force do not include the generated torque due to
reciprocating components, however the FFT plot contains the same frequencies.
Angular vibration of the driving wheels has also significant importance since it may
produce longitudinal vibration of the vehicle body1, therefore the displacement plots are
presented in Figure 5-5 by using the last method.
Figure ‎5-5, Time response at the driving wheels
1
this will be described more in chapter ‎6
33
Chapter 5 Steady-state Response of linear driveline model due to engine
fluctuating torque
As it was expected the amplitude is smaller compare to clutch response, since between
clutch and driving wheels there are number of components that reduce and damp the effects
of engine excitation torque, moreover the response plot is almost stable.
5.4 Power spectral densities of time histories
We are always interested in the responses of the mechanical systems in frequency domain in
order to find harmful frequencies which are within the operating range of the machine.
Power spectral density is a tool to obtain Fourier transform of random signal time histories
[4]. After determining time responses of clutch and driving wheels in the previous section, it
would be useful to plot PSD diagrams. we expect to see frequencies of the applying force
which are different for different engine speeds (in this report for 2000 RPM, according to
Figure 3-16, 16.6 Hz is the fundamental frequency of the engine excitation torque and
further we have 33.2, 50, 66.9, 83 … Hz) since the natural frequencies of the driveline are
known, it is obvious that high amplitudes will occur if applying force frequencies coincident
with the natural frequencies of the system. There are different built-in functions in
MATLAB to attain power spectral densities of random signals. PSD function is used in this
thesis which is attached in appendix 10.310.2. Figure 5-6 and Figure 5-7 show the PSDs of
clutch response and driving wheels1, respectively.
Figure ‎5-6, Power spectral density of the time response at the clutch
As it was expected all the forcing frequencies have created sharp peaks in the frequency
domain of the system response.
In this section only the PSD diagrams have been obtained for the time domain results from the RungeKutta numerical method.
1
34
Chapter 5 Steady-state Response of linear driveline model due to engine
fluctuating torque
Figure ‎5-7, Power spectral density of the time response at the driving wheels
There are two important points that have to be mentioned at the end of this chapter:
I.
Since the engine speed is not constant in a car, therefore the forcing frequencies are
not fixed and in addition, even at the constant engine speed, there exist more than
one excitation frequency. as a result, the problem of controlling driveline system
becomes complicated and it is not possible to use passive control method.
II.
On the other hand, while the response of the system is predictable, it is feasible to
find an optimization technique to decrease the unwanted effects as well as achieving
active algorithm to control the output vibrations.
35
Chapter 5 Steady-state Response of linear driveline model due to engine
fluctuating torque
36
Chapter 6
driveline
Vibration of the whole vehicle system due to torsional vibration of the
6 Chapter 6
Vibration of the whole vehicle system due to
torsional vibration of the driveline
6.1 Introduction
The main goal of this thesis, as it was mentioned in section 1.1, is to reduce the effects of
unwanted vibrations on the automobiles passengers. So far, we have studied the engine
fluctuation torques from different cylinders, as the main excitation source of driveline, while
failure of each component of driveline system can result to a dangerous occurrence.
furthermore it was noted that driveline vibration beside road surface irregularities is the
main reason of undesired noises inside the passenger cars, accordingly it is necessary to
obtain the vehicle body response due to driveline vibration in order to use the RMS
acceleration of the vehicle body (sprung mass of the car) through ride comfort diagrams and
study the drivability of a specific car.
6.2 Coupled vibration of driveline and the vehicle body
Torque fluctuations at the driving axle will result to variation of the drive forces at the
ground and therefore may generate longitudinal vibrations in the vehicle. Hence torsional
vibration of the driveline is coupled with the vibrations of the vehicle body. In order to
achieve the response of sprung mass subject to engine excitations, first the whole vehicle
should be modeled which is the aim of the following sections.
6.3 Tire model and longitudinal force
The whole vehicle model is divided into two main parts: driveline part which consists of all
the components between engine and differential and tire-suspension-body system part. This
section is devoted to tire modeling.
One powertrain component which is, most of the time, simplified in both old and new
torsional models of the automobiles, is the influence of the tires. However tires are the most
important element in the quest to get a car to handle well, since they are the only link
between the vehicle and the ground. By the efforts of Pacejka [23], Delft University of
Technology has significantly contributed to tire research. In this thesis, Pacejka principles
are used to gain a model for tire which consists of longitudinal and vertical stiffness and
damping coefficients, two different inertias are taken into account in order to have more
accurate model: one for the wheel and the other for the tire tread bands [3], all the necessary
properties for the calculations are given in Table 6-1. It should be noted that, an ideal
tire/wheel assembly is considered here, although practically the imperfections in the
manufacturing of tires, wheels, hubs, brakes and other parts of the rotating assembly, may
cause to three main groups of irregularities: mass imbalance, dimensional variations and
stiffness variations which are more described in reference [1].
Using the 2-degrees of freedom model for tire in Figure 6-1, according to Pacejka
formula, the total longitudinal force relation is found to be:
37
Chapter 6
driveline
Vibration of the whole vehicle system due to torsional vibration of the
6.1
where
is the coefficient of tire rolling resistance which is available in the database, and
all other parameters are demonstrated in Figure 6-1, the corresponding values are provided
in Table 6-1. The above expression shows that if the tire has an oscillatory rotational/vertical
movement, then the longitudinal tire/road contact force will be various which result to
longitudinal vibration of the vehicle body and this is not desired in a large scale according to
ride comfort criteria.
Figure ‎6-1, tire model ‎[3]
6.4 18-degrees of freedom system for whole vehicle model and its
equations of motion
In order to achieve an overall model including torsional vibration of driveline coupling with
tire, suspension and vehicle body motion, first we consider the substructure which includes
sprung (vehicle body) and unsprung masses,
and
respectively (Figure 6-2),
moreover it is assumed that vehicle body motion is limited in longitudinal and vertical
directions by suspension system characteristics. The second substructure as it was already
explained is driveline from torsional damper to differential (12-degrees of freedom) and it is
shown in Figure 6-2 as well.
38
Chapter 6
driveline
Vibration of the whole vehicle system due to torsional vibration of the
Driveline substructure (12-degrees of freedom)
Tire, suspension and vehicle body substructure (6-degrees of freedom)
Figure ‎6-2, overall vehicle model ‎[3]
According to the above figure, two substructures are related to each other by a torque
named
which is equal to:
6.2
furthermore, while it is supposed that two wheels have the same movements in Figure 6-2, it
would be reasonable to regard them as single wheel.
Now we follow the same Newton procedure as section 4.2 in order to attain differential
equations of motion for the overall 18-degrees of freedom system. Mass, stiffness and
damping matrices in equation 6.3 are given in equations 6.4, 6.5 and 6.6 respectively.
6.3
where
39
Chapter 6
driveline
 k1
 k
 1
 0

 0
 0

 0
 0

 0

 0
 0

 0
 0

 k1
k1  k 2
 k2
0
Vibration of the whole vehicle system due to torsional vibration of the
0
 k2
k 2  k3
 k3
0
0
 k3
k3  k 4
 k4
0
0
0
 k4
k 4  k5
 k5
0
 k5
k5  k6
 k6
0
 k6
k6  k7
 k7
0
 0
 0

 0

 0
 0

 0
 0

 0

 0
 0

 0
 k
 12
0
 k7
k 7  k8
 k8
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
 k8
k8  k9
 k9
0
0
0
 k9
k 9  k10
 k10
0
 k10
k10  k11
 k11
0


0


0

0


0

0


0

0


0


0

 k11

k11  k12  k13 
6.4
0
0
0

0
0

0
0

0

0
0

0
0
=
k12

 0
 0

 0
 0

 0
 R 2 k13
0
Rk13
0
0
0
0
0
k15
0
 k15
0
 Ar Rk14
 Ar Rk14
Ar k14
k14  k16
0
 k16
0
0
 k15
0
k15
0
0 

0 
0 

 k16 
0 
k16 
,
where
=
40
6.5
Chapter 6
driveline
Vibration of the whole vehicle system due to torsional vibration of the
 c1
 c
 1
 0

 0
 0

 0
 0

 0

 0
 0

 0
 0

 c1
c1  c 2
0
0
0
0
0
0
c6
 c6
 c6
c6  c7
0
0
0
c3
c4
c5
c8
c9
c10
0
0
0
0
0
0
0
0
0
c11
0
0 
0 
0 

0 
0 

0 
0 

0 

0 
0 

0 
c12 
=
=
0  R 2 c13

0
0
0
Rc13

0
0
0
0

0
0
0
0
c15
0
 c15
0
 Ar Rc14
 Ar Rc14
Ar c14
c14  c16
0
 c16
0
0
 c15
0
c15
0
0 

0 
0 

 c16 
0 
c16 
and finally the stiffness matrix is defined as follows:
where
6.6
=
M
J1
0

0

0
0

0
0

0

0
0

0
0

0
J2
0
0
0
0
0
0
0
0
0
J3
J4
J5
J6
J7
J8
J9
J 10
0
0
0
0
0
0
41
0
0
0
J 11
0
0 
0 
0 

0 
0 

0 
0 

0 

0 
0 

0 
J 12 
Chapter 6
driveline
Vibration of the whole vehicle system due to torsional vibration of the
=
=
J t








 J 13
Jt
0
0
0
0
Jt
Jt
0
0
0
0
0
0
mu
0
0
0
0
0
0
mu
0
0
0
0
0
0
mb
0
0 
0 
0 

0 
0 

mb 
It is useful to denote that and matrices in the above expression contain terms associated
to tire/road contact force which has been obtained already.
6.5 Time response of the system
This section is devoted to solve the differential equations of 18-degrees of freedom system
subject to engine fluctuation torques. In order to avoid nonlinearity we ignore the effects of
Hook’s joints and we only consider the excitation torques from the engine. Also, steadystate running is regarded, and studying the system behavior during clutch engagement
(transient running) is beyond the aim of this report.
In contrast with the differential equations for the 14-degrees of freedom driveline
mechanism which was solved in section 5.3, the current stiffness and damping matrices are
not symmetric, as a result finding the time response of the system is more difficult and time
consumable. Considering the excitation torques data from Figure ‎3-13, Output torques from
4 cylinders in the same plot, again the solution of the system has been obtained via fourth
order Runge-kutta method and by substituting the given data in Table 6-1 [3] in equation 6.1
and in its associated matrices.
42
Chapter 6
driveline
Vibration of the whole vehicle system due to torsional vibration of the
Table ‎6-1, Overall vehicle properties ‎[3]
Equivalent stiffness
coefficient
Equivalent moment of inertia
(kg or kg.m2)
(N/m or N.m/rad)
Equivalent system
damping coefficient
(N.s/m or N.m.s/rad)
Parameter
Value
Parameter
Value
Parameter
Value
k1
0.2e6
J1
0.3
c1
3
k2
1e6
J2
0.03
c2
2
k3
1e6
J3
0.03
c3
2
k4
1e6
J4
0.03
c4
2
k5
1e6
J5
0.03
c5
2
k6
0.05e6
J6
1.0
c6
4.42
k7
2e6
J7
0.05
c7
1
k8
1e6
J8
0.03
c8
1
k9
0.1e6
J9
0.05
c9
1
k10
0.1e6
J10
0.02
c10
1.8
k11
0.2e6
J11
0.02
c11
1.8
k12
1e4
J12
0.3
c12
2
k13
0.5e6
J13
4
c13
4000
k14
0.4e6
Jt
0.12
c14
600
k15
1e8
mu
c15
100000
k16
9e4
c16
4000
180
(for both wheels)
1800
mb
rolling friction coefficient
Typical tire radius in a normal
passenger car
30 cm
The force resisting the motion when a
body rolls on a surface is called the
rolling resistance or rolling friction,
some typical rolling coefficient are
provided in
www.engineeringtoolbox.com/rollingfriction
0.03
In order to reduce the computer efforts and upcoming difficulties due to instability for
solving the second order differential equations of 18-degrees of freedom vehicle system, the
cylinders pressures embedded noises are filtered1 before converting to torques and
consequently applying as an input to the system, the associated MATLAB commands are
attached in appendix 10.210.1, cut-off frequency is chosen depending on the degree of
filtration and 1000 Hz is suitable for our study, the filtered version of the torques time
histories, which were shown in Figure 3-13, are demonstrated in Figure 6-3,
We used an order n low-pass digital Butterworth filter with normalized cutoff frequency ωn.
Butterworth filters sacrifice roll-off steepness for monotonicity in the pass- and stop-bands.
1
43
Chapter 6
driveline
Vibration of the whole vehicle system due to torsional vibration of the
Figure ‎6-3, Total engine excitation torques after applying filtration
It is clear that the mean value of the torques summation should not change after filtration
and remain around the experiment set value (100 N.m). Here we have 93.12 N.m which is
meaningful. In the following plots, torsional vibration of driving wheels and longitudinal
oscillations of the vehicle body and axle are represented for the nominal values which are
given in Table 6-1. Zero initial displacements and velocities are used to determine the
solution of equations.
Figure ‎6-4, torsional velocity vibration of driving wheels due to engine excitation torques by using table 6.1 data
values
44
Chapter 6
driveline
Vibration of the whole vehicle system due to torsional vibration of the
Figure ‎6-5, zoomed version of Figure 6-4 between seconds 8 to 9
The result became stable in the second five seconds. To have a better look one second has
picked out and zoomed in Figure 6-5.
According to Figure 6-5, there is a good agreement between the amplitude of torsional
velocity in our result and reference [3] (Fig 8.5), however due to different input force, the
frequency content is dissimilar.
As it was earlier denoted number of times, vehicle body vibration in every direction
(sprung-mass) is very important in the level of ride comfort and we need vehicle body
accleration to go through drivability diagrams, hence firstly it is necessary to obtain the plots
in Figure 6-6:
Figure ‎6-6, Longitudinal velocity vibration of the vehicle body and axle due to engine excitation torques by using
table 6-1 data values
45
Chapter 6
driveline
Vibration of the whole vehicle system due to torsional vibration of the
From the above figure, it takes considerable seconds for achieving stable results. Huge
amount of input data to the system and thirty six differential equations that have to be
processed at the same time is one reason, the last three seconds is almost stable which is
zoomed in Figure 6-7. Accordingly fourth order Runge-kutta approach is not that powerful
to solve this system and as a future work another predictor corrector numerical method has
to be used to study this kind of systems. Again the results agree with reference [3].
Figure ‎6-7, zoomed version of Figure 6-6
6.6 Studying the influence of stiffness and damping coefficients
From manufacturing view, sensitivity analysis which studies the effects of different
parameter values has a significant importance. Moreover there are a variety of structural
optimization techniques to find the optimum value of each parameter in any kind of system,
this can be regarded as a future work for this thesis, here we will only change stiffness and
damping coefficients and resolve the equations of the vehicle system to see the influence on
the output results.
In Figure 6-8 and Figure 6-9, damping coefficients of the system are decreased notably
which result to great higher amplitude in torsional and longitudinal velocities of driving
wheels and vehicle body, respectively. Low damping may also generate instability in system
output response since the dissipated energy is not enough in comparison to the added
nonlinear longitudinal force,
46
Chapter 6
driveline
Vibration of the whole vehicle system due to torsional vibration of the
Figure ‎6-8, torsional velocity of driving wheels with low damping
47
Chapter 6
driveline
Vibration of the whole vehicle system due to torsional vibration of the
Figure ‎6-9, Longitudinal velocities of vehicle body and axle with low damping
Figure ‎6-10, Longitudinal velocities of vehicle body and axle with low suspension system stiffness
It should be noted that the solution method may not be enough powerful to show accurate
sensitivity to the data value change.
In Figure 6-10, Longitudinal velocities of vehicle body and axle with low suspension
system stiffness the suspension system stiffness value has been reduced to see the influence
on the longitudinal vibration of the vehicle body and axle. According to Figure 6-10,
decreasing suspension system longitudinal stiffness may cause to vibration increase of the
vehicle body (sprung mass) subject to engine excitation torques, which is not desired. On the
other hand, appropriate suspension system vertical stiffness coefficient, as it will be seen in
the next chapter, when the input excitation is from the ground1, depends on the range of
force frequency as well, and consequently high or low vertical stiffness may be required.
1
road surface non-uniformities
48
Chapter 7
car model
Vehicle suspension system response due to ground input using quarter-
7 Chapter 7
Vehicle suspension system response due to ground
input using quarter-car model
7.1 Introduction
The results of previous chapter and this one will be used in ride comfort diagrams to study
the level of induced discomfort due to engine oscillatory torques and road irregularities.
7.2 Quarter-car model and performance of suspension system
In order to evaluate ride quality of normal passenger cars, it is necessary to consider the
possibility for different kinds of vibration which may occur in vehicle body (sprung mass)1
system as well as front and rear wheels (unsprung mass)2 mechanisms. it is useful to note
that aerodynamic, driveline and engine forces are applied to the sprung mass, however
ground non-uniformities input excitation is applied to the tire and consequently suspension
system. For the 18-degrees of freedom overall vehicle model in section 6.4, longitudinal and
vertical vibrations of the sprung an un-sprung masses were studied due to engine excitations
torques since the goal was investigating the coupled behavior of the driveline torsional
vibration and tire/suspension system. The aim of this chapter is looking more specifically
into suspension system response subject to the ground input and finding the effects of
raising/reducing stiffness and damping coefficients to have better performance of suspension
mechanisms in automobiles. a simple two-degrees of freedom quarter–car model which is
suitable for our study is represented in . In this model, sprung and unsprung masses are
denoted respectively by m2 and m1 while all the corresponding parameter values are given in
Table 7-1, using Table 6-1, Overall vehicle properties ,
Figure ‎7-1, Two-degrees of freedom model vehicle
1
2
pitch, bounce and roll
bounce and roll
49
Chapter 7
car model
Vehicle suspension system response due to ground input using quarter-
Table ‎7-1, Tire/suspension properties ‎[3]
Vehicle body mass (m2) : sprung mass = 1800/4 kg (quarter of the whole body mass)
Wheel (axle) mass : Unsprung mass = 90 kg
Suspension
Tire
Stiffness :
(N/m)
9e6
Stiffness :
(N/m)
4e5
Damping :
(N.s/m)
4000
Damping :
(N.s/m)
600
the equations of motion for the above system in vertical direction are found according to
Newton’s law and free body diagram of the separate masses (Figure ‎7-1):
7.1
7.2
The undamped natural frequencies of the quarter car model for sprung (vehicle body) and
un-sprung masses (wheels) are as follows (Table 7-1):
7.3
7.4
as it is seen, there is a wide difference between natural frequencies of the vehicle body and
tire/wheels assembly, therefore in the case of high excitation frequency (such as the input
impulse by a bumpy road), according to Figure 7-2, since the frequencies ratio
(fexcitation/fnatural )sprung mass is high, transmissibility will be very low and consequently we will
achieve desired vibration isolation for vehicle body with the aid of suspension mechanism.
On the other hand, when the excitation frequency is low and near to vehicle body natural
frequency (in the situation of traveling over an undulating surface), the transmitted force is
equal to the input or even could be amplified, and hence a conventional suspension system
with fixed properties has not good performance in this case. This is the reason for a huge
amount of research activities in the field of active suspension system1 (in contrast to passive
system) which has variable stiffness and damping2 regarding to the input frequency.
Hydraulic systems which is now available in the new vehicles to have active suspension system
In order to attain desired high or low natural frequency, stiff or soft properties are required for the
suspension mechanism
1
2
50
Chapter 7
car model
Vehicle suspension system response due to ground input using quarter-
Figure ‎7-2, Transmissibility as a function of frequency ratio for a single-degree of freedom system
As it was already known, vertical acceleration of vehicle body is important in ride
comfort diagrams and it is related to the level of isolation by the suspension system which
was described precisely in previous paragraph, however the passenger will not feel vehicle
body motion but rather the displacement of his seat, thus we use the model suggested in
reference [2] in this thesis and it is shown in Figure 7-3, the necessary steps to find
corresponding transfer functions for obtaining output responses are explained as follows:
Figure ‎7-3, Modified quarter-car model including seat displacement

first, it is necessary to convert the equations 7.1 and 7.2 to Laplace domain:
7.5
7.6

now the only unknowns are
and
, by applying Cramer’s rule and
substituting s=i in equations 7.5 and 7.6, we have:
51
Chapter 7
car model
Vehicle suspension system response due to ground input using quarter-
7.7
7.8

there are different versions of transfer functions :Receptance, Mobility and
Accelerance, expressions 7.7 and 7.8 are receptances, now to determine the
mobilities and accelerances we have the following converting relations:
obility
7.9
ccelerance

7.10
now according to random process theory [24], for a stationary1 random input (such
as Gaussian function which will be later considered as ground excitation for the
suspension mechanism in ) in the case of linear system, there is a well-known
formula in order to attain power spectral density of output response by using power
spectral density (PSD) of input and the appropriate transfer function among the
above three:
7.11
where
is the PSD of desired system point displacement, velocity or
acceleration,
is the corresponding transfer function, and
is the PSD
of input force,
 Using the same approach, the complex transfer function between
and is:
where from reference [2] for a normal passenger car, we have
and
rad sec and consequently we arrive into:
7.12
7.13
One important point is that: we cannot change the suspension system stiffness and
damping coefficients, to reach desired vibration isolation, without taking into consideration
two other aspects of suspension mechanism which are important in its performance: road
holding2 and suspension working space3, these parameters are to be considered as
Stationary random process: for this type of random data, mean value and variance are constant and
independent of time
2 it is important for a safe ride that the contact forces between the wheels and the road are so large that
horizontal forces on the vehicle can be balanced by frictional forces at the wheels ‎[2]
3 practically, the working space of vehicle suspension system is limited ‎[2]
1
52
Chapter 7
car model
Vehicle suspension system response due to ground input using quarter-
constraints in the problem of finding optimum values of suspension damping and stiffness
which can be studied in a future work.
7.3 Road roughness classification by ISO and the recommended singlesided vertical amplitude power spectral density
As it was already explained, in section 1.3, by assuming the availability of three conditions,
we can use the road surface profile which is recommended by ISO in a form of single-sided
power spectral density as follows [4]:
7.14
where
is a fixed datum spatial frequency equal to 1 (2π) cycles m and
is attained
according to road roughness classification proposed by ISO which is also represented in
Table 7-2 using reference [4]. In the analysis of vehicle vibration, it is more practical to
work with temporal frequency in Hz rather than spatial frequency in cycles/m. if the car
travels with a constant speed v (m/s), then
where
, accordingly we
reach to:
7.15
Table ‎7-2, Classification of road roughness proposed by ISO ‎[4]
Degree of roughness
, 10-6m2/cycles/m
Road Class
Range
Geometric mean
A (very good)
<8
4
B (good)
8-32
16
C (average)
32-128
64
D (poor)
128-512
256
E (very poor)
512-2048
1024
F
2048-8192
4096
G
8192-32768
16384
H
>32768
53
Chapter 7
car model
Vehicle suspension system response due to ground input using quarter-
7.4 Typical passenger car driver RMS acceleration to an average road
roughness
We are interested in RMS vertical acceleration of the car seat which is obtainable by
knowing power spectral density of the seat acceleration and using the following formula1:
7.1
where
is found from equation 7.13 by substituting
, we assume that the
travelling velocity of the vehicle is 80 Km/hr (22.22 m/s) and thus the value of the
f0=vn0=3.5368 Hz. Figure 7-4 demonstaretes RMS acceleration (m/s2) graph of a typical
passenger car seat in a certain frequency range, an average road roughness class is picked,
therefore
*10-6m2/cycles/m.
Figure ‎7-4, Measured vertical acceleration of a passenger car seat traveling at 80 Km/hr over an average road
As it is seen in Figure 7-4, the peak value of the car seat RMS acceleration plot happens
around frequency 2 Hz which is the natural frequency of the sprung/vehicle body mass (first
mode frequency), hence it can be concluded that one of the road surface input excitation
frequencies is in the region of 2 Hz. further in order to control the system response
amplitude for this class of road roughness and avoid high acceleration, the natural frequency
of the suspension system have to be regularized2 with respect to input force frequency by
taking into account road holding and suspension working space limitations at the same time.
Now, for checking the solution method3 accuracy, the effects of passenger seat is ignored
in equation 7.13 and the represented model in is applied to obtain the vehicle body vertical
acceleration power spectral density subject to surface irregularities. In this case, it is possible
to compare the result with the plot which is given in reference [4]. Figure 7-5 illustrates the
fc is the center frequency, and the RMS value is calculated in one-third octave band, it is necessary to
accumulate the spectrum between the lower and upper bands ‎[24].
2 by changing stiffness or damping coefficient in an active suspension system
3 the procedure of finding power spectral density of the output
1
54
Chapter 7
car model
Vehicle suspension system response due to ground input using quarter-
sprung/vehicle body mass RMS vertical acceleration in similar conditions with Figure 7-4.
The plot shape has a good agreement with Fig. 7.32 in reference [4], however the
acceleration amplitude is somehow greater due to different road surface conditions and
suspension system characteristics. In contrast to Figure 7-4, in Figure 7-5 two peaks exist:
one “at the sprung mass natural frequency around 2 Hz (1st modal frequency)” and the
second one “at unsprung mass natural frequency around 10 Hz (2nd modal frequency)”. The
reason for this phenomenon is that if we include passenger seat mass in the model, it will
perform as a absorber which is a very important subject in vibration theory, in other words
passenger seat mass has absorbed the movement of unsprung mass in the region of
resonance frequency and consequently no peak will occur in the 2nd modal frequency
around 10 Hz1.
Figure ‎7-5, vehicle body vertical acceleration subject to an average road roughness with 80 Km/hr traveling speed
.
Although this function for the absorber is strongly related to the mass ratio and if it was not in the
appropriate region, then absorber has negative influence which happened in the current system for the
1st modal frequency (2Hz) and the amplitude of acceleration is greater in Figure 7-4 compare to Figure
7-5
1
55
Chapter 7
car model
Vehicle suspension system response due to ground input using quarter-
56
Chapter 8
criteria
Evaluation of typical passenger car comfort with respect to ride quality
8 Chapter 8
Evaluation of typical passenger car comfort with
respect to ride quality criteria
8.1 Introduction
This chapter is devoted to use the proposed results in Figure 7-4 and Figure 6-6 in ride
quality diagrams for evaluating drivability of the typical passenger car with the given overall
vehicle properties in Table 6-1, Overall vehicle properties.
8.2 International Standard ISO 2631-1:1985
According to ISO 2631-1:1985 [19], four physical factors have significant effects on human
response to applied vibration: the strength (power), frequency, direction and interval of
exposure, there are also three different issues which are important to evaluate the human
reaction to the vibratory displacements [3]:
 preservation of working efficiency
 preservation of health or safety
 preservation of comfort
In Figure 8-1 and Figure 8-2, the fatigue-decreased proficiency boundaries for various
exposure times are represented in vertical (along the z2- axis in ) and longitudinal (along the
x2- axis in ) directions, respectively. As it is clear in the proposed diagrams, the comfort
boundary will decrease by rising the vibration duration. It should be mentioned that also the
human body is more sensitive to vibration in some frequency ranges than in others, for
example according to the following figures, for vertical vibration the critical frequency
region is 4 to 8 Hz while for longitudinal vibration this frequency area is less than 2 Hz.
Center frequency of one-third octave band
Figure ‎8-1, ISO 2631-1:1985 "fatigue-decreased proficiency boundary": vertical acceleration limits as a function of
frequency and exposure time ‎[4]
57
Chapter 8
criteria
Evaluation of typical passenger car comfort with respect to ride quality
Center frequency of one-third octave band
Figure ‎8-2, ISO 2631-1:1985 "fatigue-decreased proficiency boundary": longitudinal acceleration limits as a
function of frequency and exposure time ‎[4]
8.3 Results and Discussion
This section of the report contains appropriate figures in order to evaluate ride comfort of a
typical passenger car1 at a certain frequency interval by using the sprung mass responses2
and the ISO criteria which are shown in the previous section.
Measured vertical RMS acceleration of the vehicle body, with 80 Km/hr traveling speed
on an average road roughness, is shown again in Figure 8-3 but together with ISO fatiguedecreased boundaries to investigate the level of comfort for this specific passenger car,
Figure ‎8-3, vehicle body vertical acceleration due to road excitation in comparison with ISO ride comfort
boundaries
with the given properties in Table 6-1
which have been obtained already for different directions in chapters 6 and 7 (Figure 6-6, Figure 7-4
and Figure 7-5) subject to engine and road irregularities excitations.
1
2
58
Chapter 8
criteria
Evaluation of typical passenger car comfort with respect to ride quality
Regarding Figure 8-3, it can be concluded that for this type of passenger car and passive
suspension system properties, in the situation of average road roughness, it is desired to have
exposure time less than 2.5 hours, however if the vibration duration is more than 2.5 hours,
the vertical RMS acceleration is beyond the limitations, specifically around the modal
frequencies or in the other words low frequency region, consequently the ride quality level is
low. one good suggestion is moving the second modal frequency to the outside of the critical
region for human sensitivity (4-8 Hz), thus it will exist wider band to the allowed
boundaries.
Now, the longitudinal RMS acceleration of the vehicle body due to engine excitations is
compared to ISO proposed boundaries to evaluate the ride comfort in lateral direction. to
achieve this goal first, “it is necessary to differentiate the velocity data in Figure 6-6 and
obtain time history for longitudinal acceleration using
TL B command gradient”,
secondly, “we have to determine acceleration power spectral density in order to calculate
RMS value in equation 7.161, one-third octave band rule is again utilized for each frequency
in the interval 0f 1-80 Hz”. Figure 8-4 illustrates the corresponding longitudinal RMS
acceleration of the vehicle body.
Figure ‎8-4, Measured longitudinal acceleration of a passenger car body due to engine excitation torques
As it was already explained, for high frequency force input, suspension system has good
vibration isolation, and since the engine excitation frequencies are high in comparison with
natural frequency of the sprung mass (in longitudinal direction), the acceleration amplitude
is too low in Figure 8-4, which is strongly desired, and completely below the allowed ISO
fatigue-decreased boundaries which is shown in Figure 8-2.
In order to obtain RMS value from discrete-time power spectral density, we have to sum up the PSD
values in the desired interval ‎[24].
1
59
Chapter 8
criteria
Evaluation of typical passenger car comfort with respect to ride quality
8.4 Thesis conclusion
In this report, two main excitation resources of a passenger car were studied in order to
evaluate drivability of the specified vehicle according to ISO ride quality criteria:
1-“torsional vibration of the driveline system” due to engine oscillatory torques which was
modeled by using 14-degrees of freedom system, and further the influence of driveline
excitation on the longitudinal displacement of the vehicle body was investigated with the aid
of coupled driveline and tire/suspension 18-degress of freedom mechanism.
2-“road surface non-uniformities vertical input force” effects on the vehicle body were
attained by using two-degrees of freedom quarter-car model including suspension and tire
assembly vertical properties.
Regarding the achieved results in Figure 8-3 and Figure 8-4, and by considering the
specific assumptions1 which have been described in the introductory chapter, the produced
vertical RMS acceleration of the vehicle body subject to road surface irregularities (low and
high frequency content) is much greater than the longitudinal acceleration of sprung mass
due to engine excitation (high frequency content), and furthermore in some frequency region
vertical response of the system is beyond the ISO limitations. Therefore, since the desired
performance of suspension system is dependent to various input excitations, and noting that
for a vehicle, there exist different input forces with different frequency content at the same
time, it can be concluded that a passive suspension system with fixed stiffness and damping
coefficients is not proper and an active mechanism is required. Although, it should be
mentioned that the importance of driveline torsional oscillation modeling and knowing its
natural frequencies is not only for the corresponding influence on the vehicle body vibration
and induced noise, but also for dangerous damages which may occur for different driveline
components due to high vibration amplitude and resonance phenomenon. Consequently it is
necessary to obtain the response sensitivity of each driveline part to high or low parameter
values as it was done in section 6.6, Studying the influence of stiffness and damping
coefficients .
The fourth order Runge-kutta numerical approach which was used for solving 14-degrees
and 18-degrees of freedom mechanical systems in Chapter 4 and Chapter 6, is a powerful
iterative method for solving ordinary differential equations and it was functional here since
the non-proportional damping matrix prevents from using Modal analysis method to find the
system response. The problem is that, this method is not sensitive enough for showing the
effects of changing different parameter values.
8.5 Future works
There exist different subjects that may be considered as a future work for this master thesis
while a few numbers of them are explained in the following items:


The effects of engine mounts, which have been ignored here. Nowadays, in new car
generations, engine mounts are very helpful in order to reduce the unwanted
transmitted vibration of the engine and it would be practical to model and optimize
their structure.
As it was already noted in chapter 7, it is required to introduce an appropriate
optimization technique for determining suspension system optimum parameter
values in order to have the best vibration isolation with taking into account the
We are only studying the engine vibratory pressure torques not the effects of reciprocating parts of the
engine and, moreover we ignore the nonlinear generated torques by Hooke’s joints and the other
possible components.
1
60
Chapter 8
criteria


Evaluation of typical passenger car comfort with respect to ride quality
suspension working space and road holding constraints. Furthermore, investigating
about an active suspension system is needed.
It is necessary to find more powerful numerical method to solve the system and
decrease the level of instability, however in some cases the lack of reference data to
evaluate the accuracy of numerical solution results to requirement of analytical
method, thus recently a lot of research activities have been focused on analytical
approaches for liner and nonlinear systems.
Modeling nonlinear effects of different parts in driveline system has a primary
interest, since they may cause to harmful phenomenon due to corresponding chaotic
behaviors.
61
Chapter 8
criteria
Evaluation of typical passenger car comfort with respect to ride quality
62
References
9 References
[1]. D.Gillespie, Thomas. Fundamentals of vehicle dynamics. Warrendale,PA:
Society of Automotive Engineers, Inc.
[2]. Dahlberg, T. "Ride Comfort and Road Holding of a 2-DOF Vehicle Travelling
on a Randomly Profiled Raod." Journal of Sound and vibration, (1978) 58 (2): 179187.
[3]. El-Adl Mohammed Aly Rabeih, BSc. & M. Sc. Torsional Vibration Analysis
of Automotive Driveline. PhD thesis, Leeds, UK: Mechanical Engineering
Department, the university of Leeds, 1997.
[4]. Wong, J. Y. Theory of Ground Vehicles. New Jersey: John Wiley & Sons, Inc.,
2008.
[5]. Reik, W. "Torsional Vibrations in the Drive train of Motor Vehicles, principle
considerations." 4th International Symposium. Baden-Baden, 1990. 5-28.
[6]. Zhanoi, W., et al. "Research on the Coupling Power Train Torsional Vibration
With Body Fore-Aft and Vertical Vibration of a 4x2 Vehicle." IMechE,C3 89/450,
Vol. 2, Paper No. 925067. 1992. 219-223.
[7]. Parkins, D. W. "Automobile Drive-Line Vibration and Internal Noise." Annual
Conf. of the stress Analysis Group of the Inst. of physics, "Stress, Vibration and
Noise Analysis in Vehicles". 165-179.
[8]. Healy, S. P., et al. "An Experimental Study of Vehicle Driveline Vibrations."
IMechE, Paper No. C 13 2/79. 1979. 69-81.
[9]. Exner, W. and Ambom, P. "Noise and Vibration in the Driveline of Passenger
Cars." IMechE Automotive Congress, Seminar 37, Paper No. C 427/37/025. 1991. 8
pages.
[10]. J.H. Walls, J. C. Houbolt and H. Press. Some measurements and power spectra
of runway roughness. NACA Technical Note 3305, 1954.
[11]. Houbolt, J. C. "Runway roughness studies in the Aeronautical field." Journal
of the Air Transport Division Proceedings, American Society of Civil Engineers 87,
AT-1, 1961.
[12]. Colledge, A. Craggs and R. B. Random vibration analysis applied to road
surface measurement and vehicle suspension performance. Motor Industries
Research Association Report MIRA , 1964/69.
[13]. Bender, E. K. Optimization of the random vibration characteristics of vehicle
suspensions using random process theory. ScD Thesis, MIT, Cambridge, MA, 1967.
[14]. Hrovat, D. "Application of optimal control to advanced automotive suspension
design." Journal of dynamical systems, measurements and control, (1993) 115: 328342.
[15]. Rill, G. "The influence of correlated random excitations processes on
dynamics of vehicles." proceedings of the eighth IAVSD symposium on the dynamics
of vehicles on road and railway tracks. 1983. 449-459.
[16]. Turkey, S. and Akcay, H. "A study of random vibration characteristics of the
quarter car model." Journal of sound and vibration, (2005) 282: 111-124.
63
References
[17]. Van Deusen, B.D. "Human response to vehicle vibration." SAE transactions ,
1969 vol.77: paper 680090.
[18]. Society of automotive engineers, "Ride and vibration ata manual." SAE J6a,
1965.
[19]. International Standard ISO, 2631/1-1985. "Evaluation of human exposure to
whole-body vibration, part1: General requirements." International Organization for
standardization, 1985.
[20]. Hillier, V.A.W. Fundamentals of motor vehicle technology. Cheltenham, UK:
Nelson Thornes Ltd, fourth edition, 2001.
[21]. Den Hartog, J. P. Mechanical Vibrations. Newyork & London: McGRAW-Hill
Boo Company, 1947.
[22]. Ph. Couderc, J. Callenaere, J. Der hagopian and G. Ferraris. "Vehicle Driveline
Dynamic Behaviour: Experimentation and Simulation." Journal of Sound and
Vibration, (1998) 218(1): 133-157.
[23]. Pacejka, H. B. "Tire factors and front wheel vibrations." International journal
of vehicle design, 1980: 97-119.
[24]. Brandt, A. Introductory noise and vibration analysis. Saven Edu Tech Ab,
Täby, Sweden.
64
Appendix
10 Appendix
10.1 LTI object
[x,t]=lsim(sys,f)
where
sys
is
an
LTI
sys=tf(1,[m(1,j),c(1,j),k(1,j)]) and f is the input data.
object1,
10.2 Driveline Modeling MATLAB code
clc
clear all
% close all
global i A F M H
K=[7000 0 -5000 0 0 0 0 0 0 0 0 0 0 0. 0 7000 -5000 0 0 0 0 0 0 0 0 0 0
0.-5000 -5000 210000 -200000 0 0 0 0 0 0 0 0 0 0.0 0 -200000 300000 100000 0 0 0 0 0 0 0 0 0.0 0 0 -100000 200000 -100000 0 0 0 0 0 0 0 0.0 0
0 0 -100000 1100000 -1000000 0 0 0 0 0 0 0.0 0 0 0 0 -1000000 3000000 2000000 0 0 0 0 0 0.0 0 0 0 0 0 -2000000 2050000 -50000 0 0 0 0 0.0 0 0 0
0 0 0 -50000 1050000 -1000000 0 0 0 0.0 0 0 0 0 0 0 0 -1000000 2000000 1000000 0 0 0.0 0 0 0 0 0 0 0 0 -1000000 2000000 -1000000 0 0.0 0 0 0 0 0
0 0 0 0 -1000000 2000000 -1000000 0.0 0 0 0 0 0 0 0 0 0 0 -1000000 1200000
-200000.0 0 0 0 0 0 0 0 0 0 0 0 -200000 200000].
K1=rot90(K,2).
M=[0.3 0 0 0 0 0 0 0 0 0 0 0 0 0.0 0.03 0 0 0 0 0 0 0 0 0 0 0 0.0 0 0.03 0
0 0 0 0 0 0 0 0 0 0.0 0 0 0.03 0 0 0 0 0 0 0 0 0 0.
0 0 0 0 0.03 0 0 0 0 0 0 0 0 0.0 0 0 0 0 1 0 0 0 0 0 0 0 0.0 0 0 0 0 0
0.05 0 0 0 0 0 0 0.0 0 0 0 0 0 0 0.03 0 0 0 0 0 0.
0 0 0 0 0 0 0 0 0.05 0 0 0 0 0.0 0 0 0 0 0 0 0 0 0.02 0 0 0 0. 0 0 0 0 0 0
0 0 0 0 0.02 0 0 0. 0 0 0 0 0 0 0 0 0 0 0 0.3 0 0.
0 0 0 0 0 0 0 0 0 0 0 0 2 0. 0 0 0 0 0 0 0 0 0 0 0 0 0 2].
C=[3 -3 0 0 0 0 0 0 0 0 0 0 0 0.-3 5 0 0 0 0 0 0 0 0 0 0 0 0.0 0 2 0 0 0 0
0 0 0 0 0 0 0.0 0 0 2 0 0 0 0 0 0 0 0 0 0.
0 0 0 0 2 0 0 0 0 0 0 0 0 0.0 0 0 0 0 4.42 -4.42 0 0 0 0 0 0 0.0 0 0 0 0 4.42 5.42 0 0 0 0 0 0 0.
0 0 0 0 0 0 0 1 0 0 0 0 0 0.0 0 0 0 0 0 0 0 1 0 0 0 0 0.0 0 0 0 0 0 0 0 0
1.8 0 0 0 0. 0 0 0 0 0 0 0 0 0 0 1.8 0 0 0.
0 0 0 0 0 0 0 0 0 0 0 2 0 0.0 0 0 0 0 0 0 0 0 0 0 0 10 0.0 0 0 0 0 0 0 0 0
0 0 0 0 10].
According to MATLAB help, LTI object is a tool to define transfer function for differential equation of a
Linear Time Invariant systems, for example You can create transfer function (TF) models by specifying
numerator and denominator coefficients.
num = [1 0].
den = [1 2 1].
sys = tf (num,den)
1
Transfer function:
s
------------s^2 + 2 s + 1
65
Appendix
A=zeros(14,14).
A(1:14,15:28)=eye(14,14).
A(15:28,1:14)=-M^(-1)*K1.
A(15:28,15:28)=-M^(-1)*C.
[U,w2]=eig(K1,M).
omega=sqrt(w2).
f=omega/(2*pi).
for j=1:14
U(:,j)=U(:,j)/max(U(:,j)).
end
for j=1:14
m(1,j)=U(:,j)'*M*U(:,j).
k(1,j)=U(:,j)'*K1*U(:,j).
c(1,j)=U(:,j)'*C*U(:,j).
end
for j=1:14
gamma(1,j)=1/sqrt(m(1,j)).
U1(:,j)=gamma(1,j)*U(:,j).
end
load neda2000rpm100nm_110210
t1=linspace(0,(1279999/128000),1280000).
w=2000*2*pi/60.
degree=w*t1*180/pi.
torque1=hp1433_convdata.p_cyl_1*0.043.*(sin(w*t1)+1/4*sin(w*t1).*cos(w*t1)
)'*pi*0.086^2/4*10^5.
%
%
%
%
%
%
t_sample = t1(2). % time sample step length
n = 1. % order
fn = 500. % [Hz] cutoff frequency
Wn = fn/(2*pi)*t_sample.
[b,a] = butter(n,Wn).
torque1= filtfilt(b, a, torque1).
mean1=mean(torque1).
torque1_fluc=torque1-mean1.
torque2=hp1433_convdata.p_cyl_2*0.043.*(sin(w*t1)+1/4*sin(w*t1).*cos(w*t1))'*pi*0.
086^2/4*10^5.
%
%
%
%
%
%
t_sample = t1(2). % time sample step length
n = 1. % order
fn = 500. % [Hz] cutoff frequency
Wn = fn/(2*pi)*t_sample.
[b,a] = butter(n,Wn).
torque2= filtfilt(b, a, torque2).
mean2=mean(torque2).
torque2_fluc=torque2-mean2.
66
Appendix
torque3(1:3841)=torque2(3841:7681).
torque3(3842:7681)=torque2(1:3840).
torque3=torque3'.
for j=1:165
torque3((7681*j)+1:(j+1)*7681)=torque3(1:7681).
end
torque3(1275047:1280000)=0.
mean3=mean(torque3).
torque3_fluc=torque3-mean3.
torque4(1:3841)=torque1(3841:7681).
torque4(3842:7681)=torque1(1:3840).
torque4=torque4'.
for j=1:165
torque4((7681*j)+1:(j+1)*7681)=torque4(1:7681).
end
torque4(1275047:1280000)=0.
mean4=mean(torque4).
torque4_fluc=torque4-mean4.
Force(1,1:1280000)=0.
Force(2,1:1280000)=torque1_fluc.
Force(3,1:1280000)=torque2_fluc.
Force(4,1:1280000)=torque3_fluc.
Force(5,1:1280000)=torque4_fluc.
for j=6:14
Force(j,1:1280000)=0.
end
F=zeros(28,1280000).
F(15:28,1:1280000)=M^(-1)*Force.
N=U'*Force.
for j=1:14
sys=tf(1,[m(1,j),c(1,j),k(1,j)]).
sys_ss = ss(sys).
[x(:,j),t1]=lsim(sys_ss,N(j,:),t1,[0.0]).
end
x=x'.
response=U1*x.
fs=128000.
Runge-kutta integration
tsol=zeros(1,1280000).
zsol=zeros(28,1280000).
t=0. z(1:14)=0.z(15:28)=0.
tstop=10.
h=1/fs*2.
[tsol,zsol]=runkut4_driveline(t,z,tstop,h).
function [tsol,zsol]=runkut4_driveline(t,z,tstop,h)
global F A i M
if size(z,1)>1. z=z'. end
67
Appendix
tsol=zeros(1280002,1).zsol=zeros(1280002,length(z)).
tsol(1)=t.zsol(1,:)=z.
F1=zeros(1,28).
i=1.
while t<tstop
F1(1:28)=A*z'+F(1:28,i).
h=min(h,tstop-t).
K1=h*F1.
z1=z+K1/2.
F1(1:28)=A*z1'+F(1:28,i).
K2=h*F1.
z2=z+K2/2.
F1(1:28)=A*z2'+F(1:28,i).
K3=h*F1.
z3=z+K3.
F1(1:28)=A*z3'+F(1:28,i).
K4=h*F1.
z=z+(K1+2*K2+2*K3+K4)/6.
t=t+h.
i=i+1
tsol(i)=t.zsol(i,:)=z.
end
10.3 Power spectral density function
function [Pxx,f_v] = psd(x,Fs,nfft,noverlap)
%PSD Power Spectral Density estimate.
%
PSD has been replaced by SPECTRUM objects. PSD still works but may be
%
removed in the future. Use SPECTRUM (or its functional form PWELCH)
%
instead. Type help SPECTRUM for details.
%
%
See also SPECTRUM.
%
%
%
Author(s): T. Krauss, 3-26-93
Copyright 1988-2005 The MathWorks, Inc.
$Revision: 1.12.4.5 $ $Date: 2007/12/14 15:05:42 $
%
%
%
%
%
%
%
%
%
NOTE 1: To express the result of PSD, Pxx, in units of
Power per Hertz multiply Pxx by 1/Fs [1].
%
%
%
%
%
%
References:
[1] Petre Stoica and Randolph Moses, Introduction To Spectral
Analysis, Prentice hall, 1997, pg, 15
[2] A.V. Oppenheim and R.W. Schafer, Discrete-Time Signal
Processing, Prentice-Hall, 1989, pg. 731
[3] A.V. Oppenheim and R.W. Schafer, Digital Signal
NOTE 2: The Power Spectral Density of a continuous-time signal,
Pss (watts/Hz), is proportional to the Power Spectral
Density of the sampled discrete-time signal, Pxx, by Ts
(sampling period). [2]
Pss(w/Ts) = Pxx(w)*Ts,
68
|w| < pi. where w = 2*pi*f*Ts
Appendix
%
Processing, Prentice-Hall, 1975, pg. 556
% error(nargchk(1,7,nargin,'struct'))
% x = varargin{1}.
% [msg,nfft,Fs,window,noverlap,p,dflag]=psdchk(varargin(2:end),x).
% if ~isempty(msg), error(generatemsgid('SigErr'),msg). end
dflag='none'.
% compute PSD
window = flattopwin(nfft).
n = length(x).
% Number of data points
nwind = length(window). % length of window
if n < nwind
% zero-pad x if it has length less than the window
length
x(nwind)=0. n=nwind.
end
% Make sure x is a column vector. do this AFTER the zero-padding
% in case x is a scalar.
x = x(:).
k = fix((n-noverlap)/(nwind-noverlap)). % Number of windows
%k = fix(n/nwind). %for
noverlap=0
%
if 0
%
disp(sprintf('
x
= (length %g)',length(x)))
%
disp(sprintf('
y
= (length %g)',length(y)))
%
disp(sprintf('
nfft
= %g',nfft))
%
disp(sprintf('
Fs
= %g',Fs))
%
disp(sprintf('
window
= (length %g)',length(window)))
%
disp(sprintf('
noverlap = %g',noverlap))
%
if ~isempty(p)
%
disp(sprintf('
p
= %g',p))
%
else
%
disp('
p
= undefined')
%
end
%
disp(sprintf('
dflag
= ''%s''',dflag))
%
disp('
--------')
%
disp(sprintf('
k
= %g',k))
%
end
index = 1:nwind.
KMU = k*norm(window)^2. % Normalizing scale factor ==> asymptotically
unbiased
% KMU = k*(sum(window))^2.% alt. Nrmlzng scale factor ==> peaks are about
right
Spec = zeros(nfft,1). % Spec2 = zeros(nfft,1).
for i=1:k
if strcmp(dflag,'none')
xw = window.*(x(index)).
elseif strcmp(dflag,'linear')
xw = window.*detrend(x(index)).
else
xw = window.*detrend(x(index),'constant').
end
index = index + (nwind - noverlap).
Xx = abs(fft(xw,nfft)).^2.
Spec = Spec + Xx.
%
Spec2 = Spec2 + abs(Xx).^2.
end
69
Appendix
% Select first half
if ~any(any(imag(x)~=0)),
% if x is not complex
if rem(nfft,2),
% nfft odd
select = (1:(nfft+1)/2)'.
else
select = (1:nfft/2+1)'.
end
Spec = Spec(select).
%
Spec2 = Spec2(select).
%
Spec = 4*Spec(select).
% double the signal content - essentially
% folding over the negative frequencies onto the positive and adding.
%
Spec2 = 16*Spec2(select).
else
select = (1:nfft)'.
end
f_v = (select - 1)*Fs/nfft.
% find confidence interval if needed
% if (nargout == 3) || ((nargout == 0) && ~isempty(p)),
%
if isempty(p),
%
p = .95.
% default
%
end
%
% Confidence interval from Kay, p. 76, eqn 4.16:
%
% (first column is lower edge of conf int., 2nd col is upper edge)
%
confid = Spec*chi2conf(p,k)/KMU.
%
%
if noverlap > 0
%
disp('Warning: confidence intervals inaccurate for NOVERLAP >
0.')
%
end
% end
Pxx = Spec*(2/KMU).
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
% normalize
set up output parameters
if (nargout == 3),
Pxx = Spec.
Pxxc = confid.
f = freq_vector.
elseif (nargout == 2),
Pxx = Spec.
Pxxc = freq_vector.
elseif (nargout == 1),
Pxx = Spec.
elseif (nargout == 0),
if ~isempty(p),
P = [Spec confid].
else
P = Spec.
end
newplot.
plot(freq_vector,10*log10(abs(P))), grid on
xlabel('Frequency'), ylabel('Power Spectrum Magnitude (dB)').
end
10.4 Vehicle modeling MATLAB codeATLAB code
clc
clear all
% close all
global A F i
70
Appendix
k=[7000 0 -5000 0 0 0 0 0 0 0 0 0 0 0. 0 7000 -5000 0 0 0 0 0 0 0 0 0 0
0.-5000 -5000 210000 -200000 0 0 0 0 0 0 0 0 0 0.0 0 -200000 300000 100000 0 0 0 0 0 0 0 0 0.0 0 0 -100000 200000 -100000 0 0 0 0 0 0 0 0.0 0
0 0 -100000 1100000 -1000000 0 0 0 0 0 0 0.0 0 0 0 0 -1000000 3000000 2000000 0 0 0 0 0 0.0 0 0 0 0 0 -2000000 2050000 -50000 0 0 0 0 0.0 0 0 0
0 0 0 -50000 1050000 -1000000 0 0 0 0.0 0 0 0 0 0 0 0 -1000000 2000000 1000000 0 0 0.0 0 0 0 0 0 0 0 0 -1000000 2000000 -1000000 0 0.0 0 0 0 0 0
0 0 0 0 -1000000 2000000 -1000000 0.0 0 0 0 0 0 0 0 0 0 0 -1000000 1200000
-200000.0 0 0 0 0 0 0 0 0 0 0 0 -200000 200000].
K3=rot90(k,2).
K=K3(1:12,1:12).
K(13:18,13:18)=zeros(6,6).
K(12,13)=-10000.
K(13,12)=-10000.
K(13,13)=10000.
K(13,14)=-.3^2*5e5.
K(13,16)=-0.03*.3*4e5.
K(14,16)=-0.03*.3*4e5.
K(15,14)=.3*5e5.
K(15,15)=1e8.
K(15,16)=0.03*4e5.
K(15,17)=-1e8.
K(16,16)=49e4.
K(16,18)=-9e4.
K(17,15)=-1e8.
K(17,17)=1e8.
K(18,16)=-9e4.
K(18,18)=9e4.
M=[0.3 0 0 0 0 0 0 0 0 0 0 0 0 0.0 0.03 0 0 0 0 0 0 0 0 0 0 0 0.0 0 0.03 0
0 0 0 0 0 0 0 0 0 0.0 0 0 0.03 0 0 0 0 0 0 0 0 0 0.
0 0 0 0 0.03 0 0 0 0 0 0 0 0 0.0 0 0 0 0 1 0 0 0 0 0 0 0 0.0 0 0 0 0 0
0.05 0 0 0 0 0 0 0.0 0 0 0 0 0 0 0.03 0 0 0 0 0 0.
0 0 0 0 0 0 0 0 0.05 0 0 0 0 0.0 0 0 0 0 0 0 0 0 0.02 0 0 0 0. 0 0 0 0 0 0
0 0 0 0 0.02 0 0 0. 0 0 0 0 0 0 0 0 0 0 0 0.3 0 0.
0 0 0 0 0 0 0 0 0 0 0 0 2 0. 0 0 0 0 0 0 0 0 0 0 0 0 0 2].
M=M(1:12,1:12).
M(13:18,13:18)=zeros(6,6).
M(13,13)=4.12.
M(13,14)=0.12.
M(14,13)=0.12.
M(14,14)=0.12.
M(15,15)=180.
M(16,16)=180.
M(17,17)=1800.
M(18,18)=1800.
C=[3 -3 0
0 0 0 0 0
0 0 0 0 2
4.42 5.42
0
0
0
0
0 0
0.0
0 0
0 0
0
0
0
0
0
0
0
0
0
2
0
0
0 0 0 0 0.-3 5 0 0 0 0 0 0 0 0 0 0 0 0.0 0 2 0 0 0 0
0 0 0 0 0 0 0 0 0 0.
0 0 0.0 0 0 0 0 4.42 -4.42 0 0 0 0 0 0 0.0 0 0 0 0 0.
71
Appendix
0 0
1.8
0 0
0 0
0
0
0
0
0
0
0
0
0 0 0 1 0 0 0 0 0 0.0 0 0 0 0 0 0 0 1 0 0 0 0 0.0 0 0 0 0 0 0 0 0
0 0. 0 0 0 0 0 0 0 0 0 0 1.8 0 0 0.
0 0 0 0 0 0 0 2 0 0.0 0 0 0 0 0 0 0 0 0 0 0 10 0.0 0 0 0 0 0 0 0 0
10].
C=C(1:12,1:12).
C(13:18,13:18)=zeros(6,6).
C(13,14)=-0.3^2*4000.
C(13,16)=-0.03*0.3*600.
C(14,16)=-0.03*0.3*600.
C(15,14)=0.3*4000.
C(15,15)=100000.
C(15,16)=0.03*600.
C(15,17)=-100000.
C(16,16)=4600.%
C(16,18)=-4000.%
C(17,15)=-100000.
C(17,17)=100000.
C(18,16)=-4000.%4->2
C(18,18)=4000.%
A=zeros(18,18).
A(1:18,19:36)=eye(18,18).
A(19:36,1:18)=-M^(-1)*K.
A(19:36,19:36)=-M^(-1)*C.
load neda2000rpm100nm_110210.mat
t1=linspace(0,(1279999/128000),1280000).
w=2000*2*pi/60.
degree=w*t1*180/pi.
t_sample = t1(2). % time sample step length
n = 1. % order
fn = 1000. % [Hz] cutoff frequency
Wn = fn/(2*pi)*t_sample.
[b,a] = butter(n,Wn).
pres1= filtfilt(b, a, hp1433_convdata.p_cyl_1).
torque1=pres1*0.043.*(sin(w*t1)+1/4*sin(w*t1).*cos(w*t1))'*pi*0.086^2/4*10
^5.
mean1=mean(torque1).
torque1_fluc=torque1-mean1.
t_sample = t1(2). % time sample step length
n = 1. % order
fn = 1000. % [Hz] cutoff frequency
Wn = fn/(2*pi)*t_sample.
[b,a] = butter(n,Wn).
pres2= filtfilt(b, a, hp1433_convdata.p_cyl_2).
torque2=pres2*0.043.*(sin(w*t1)+1/4*sin(w*t1).*cos(w*t1))'*pi*0.086^2/4*10^5.
mean2=mean(torque2).
torque2_fluc=torque2-mean2.
72
Appendix
torque3(1:3841)=torque2(3841:7681).
torque3(3842:7681)=torque2(1:3840).
torque3=torque3'.
for j=1:165
torque3((7681*j)+1:(j+1)*7681)=torque3(1:7681).
end
torque3(1275047:1280000)=0.
mean3=mean(torque3).
torque3_fluc=torque3-mean3.
torque4(1:3841)=torque1(3841:7681).
torque4(3842:7681)=torque1(1:3840).
torque4=torque4'.
for j=1:165
torque4((7681*j)+1:(j+1)*7681)=torque4(1:7681).
end
torque4(1275047:1280000)=0.
mean4=mean(torque4).
torque4_fluc=torque4-mean4.
Force(1,1:1280000)=0.
Force(2,1:1280000)=torque1_fluc.
Force(3,1:1280000)=torque2_fluc.
Force(4,1:1280000)=torque3_fluc.
Force(5,1:1280000)=torque4_fluc.
for j=6:18
Force(j,1:1280000)=0.
end
F=zeros(36,1280000).
F(19:36,1:1280000)=M^(-1)*Force.
fs=128000.
% tsol=zeros(1,1280000).
% zsol=zeros(36,1280000).
%runge-kutta integration
t=0. z(1:18)=0.z(19:36)=0.
tstop=10.
h=1/fs*2.
[tsol,zsol]=runkut4(t,z,tstop,h).
acc_axle_long=gradient(zsol(1:640002,33)).
acc_vehicle_long=gradient(zsol(1:640002,35)).
x1=acc_vehicle_long.
[Pxx_ovp,f_ovp]=psd_1(x1,2^18,fs,hann(2^18),2^17).
Pxx_ovp=Pxx_ovp/fs.
[n,m]=min(abs(f_ovp-0.89*(33))).
[n1,m1]=min(abs(f_ovp-1.12*(33))).
Pxx_sum=sum(Pxx_ovp(m:m1)).
RMS_psd=sqrt(Pxx_sum*fs/(2^18))
73
Was this manual useful for you? yes no
Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertisement