ACTIVE VIBRATION CONTROL OF ROTATING COMPOSITE

ACTIVE VIBRATION CONTROL OF ROTATING COMPOSITE
ACTIVE VIBRATION CONTROL OF ROTATING COMPOSITE
SHAFT SYSTEM
A THESIS SUBMITTED IN THE PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
Master of Technology
In
MECHANICAL ENGINEERING
[Specialization: Machine Design and Analysis]
By
Sikandar Kumar
211ME1158
Under the Supervisions of
Prof. T. Roy
Department Of Mechanical Engineering
National Institute of Technology Rourkela
Rourkela, Orissa, India – 769008
December, 2013
Abstract
Fiber reinforced polymer (FRP) composites shafts find many application in the modern
industries due to its flexibility and light weight. The present work deals with the study of
finite element analysis and active vibration control of rotating composite shaft system
under unbalance forces using three nodded beam element. The composite shafts are
modeled as a Timoshenko beam by mounting discrete isotropic rigid disks on it and
supported by flexible bearings that are modeled with viscous dampers and springs. Based
on first order shear deformation (FOSD) beam theory with transverse shear deformation,
rotary inertia, gyroscopic effect, strain and kinetic energy of shafts are derived by adopting
three-dimensional constitutive relations of material. The derivation of governing equation
of motion is obtained using Hamilton’s principle and solutions are obtained by three-node
finite element (FE) with four degrees of freedom (DOF) per node. Active vibration control
of the rotating composite shaft has also been implemented using electromagnetic actuator
and PD control technique. Various results have also been obtained such as Campbell
diagram, transverse displacements, transverse control responses, control currents and
control forces in the both directions. The effect of ply orientation on the Campbell
diagrams and the transverse responses has also been studied. The effect of number of
actuators on the control responses and the control forces has also been presented.
Keywords: FRP composite shaft, Finite element modeling, Vibration analysis,
Electromagnetic actuators, PD control scheme and Active vibration control.
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Contents
Chapter 1:
Chapter 2:
Chapter 3:
Chapter 4:
Page
List of figure
List of table
Nomenclature
Preface
Introduction
1.1 Composite Materials
1.1.1 Particulate composite
1.1.2 Fiber reinforced composite
1.1.3 Types of Laminate
1.1.3.1 Symmetrical laminates
1.1.3.2 Balanced Laminate
1.1.3.3 Anti-symmetrical laminates
1.1.3.4 Quasi-Isotropic laminate
1.1.4 Application areas of composites
1.2 Active Material
1.2.1 Piezoelectric material
1.2.2 Magneto-strictive materials
1.2.3 Electromagnetic actuator
1.3 Control technique
1.3.1 PD Controller
1.3.2 PID Controller
1.3.3 Linear Quadratic Regulation (LQR)
Advantages
of electromagnetic actuator
1.4
1.5 Finite element method
1.6 Outline of the Present Work
Literature review
2.1 General Introduction
Finite element modelling and vibration analysis of isotropic shaft
2.2
system
Finite element modelling and vibration analysis of FRP composite
2.3
shaft system
2.4 Active vibration control of rotor shaft system
2.5 Motivation and Objectives of the present work
Finite element modelling and analysis of FRP composite shaft system
3.1 Introduction
3.2 Mathematical Modelling of composite Rotor shaft system
3.3 Finite element model analysis of rotor shaft
Modelling of electromagnetic actuator
4.1 Introduction
4.2 The Electromagnetic Force
Linearization of the Electromagnetic Force about operating
4.2.1
Point
4.3 Governing equation of motion
4.4 The Proportional-Derivative Control Strategy
4.5 Linear Quadratic Regulator (LQR)
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Chapter 5: Results and discussions
Chapter 6:
5.1 Summary of various analyses
5.2 Code Validation
Uncontrolled and controlled responses of the various composite
5.3
shafts
5.3.1 Symmetric angle ply laminated shaft
5.3.2 Symmetric cross ply laminated shaft
5.3.3 Anti-symmetric cross ply laminated shaft
5.3.4 Quasi-isotropic laminated shaft
Conclusions and scope of further work
6.1 Conclusions
6.2 Scope of future work
References
List of publication from the present work
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LIST OF FIGURES
Fig. 1.1
Fig. 1.2
Fig. 1.3
Fig. 4.1
Fig.5.1
Fig. 5.2
Fig. 5.3
Fig. 5.4
Fig.5.5
Fig.5.6
Fig.5.7
Fig.5.8
Fig.5.9
Fig.5.10
Fig.5.11
Fig.5.12
Fig.5.13
Fig.5.14
Fig.5.15
Fig.5.16
Fig. 5.17
Fig. 5.18
Figure description
Page
Electro Magnetic levitation principle
Block diagram of PD controller with feedback
Block diagram of PID controller
(a) Stator configuration of actuator, (b) Arrangement of coils and poles
of the electromagnetic actuator
Campbell diagram of steel for first two pairs of modes
Campbell diagram of symmetrical angle ply laminated shaft
Uncontrolled displacement history in the direction of ‘V’ for the
symmetrical angle ply laminated shaft.
Uncontrolled displacement history in the direction of ‘w’ for the
symmetrical angle ply laminated shaft.
Controlled displacement history in the direction of ‘V’ for the
symmetrical angle ply laminated shaft using one actuator.
Controlled displacement history in the direction of ‘W’ for the
symmetrical angle ply laminated shaft using one actuator.
Controlled displacement history in the direction of ‘V’ for the
symmetrical angle ply laminated shaft using two actuators.
Controlled displacement history in the direction of ‘W’ for the
Symmetrical angle ply laminated shaft using two actuators
Controlled current in the direction of ‘Y’ for the symmetrical angle ply
laminated shaft using one actuator
Controlled current in the direction of ‘Z’ for the symmetrical angle ply
laminated shaft using one actuator.
Controlled current in the direction of ‘Y’ for the symmetrical angle ply
laminated shaft using two actuators.
Controlled current in the direction of ‘Z’ for the symmetrical angle ply
laminated shaft using two actuators.
Controlled Force in the direction of ‘Y’ for the symmetrical angle ply
laminated shaft using one actuator.
Controlled Force in the direction of ‘Z’ for the symmetrical angle ply
laminated shaft using one actuator.
Controlled Force in the direction of ‘Y’ for the symmetrical angle ply
laminated shaft using two actuators.
Controlled Force in the direction of ‘Z’ for the symmetrical angle ply
laminated shaft using two actuators.
Campbell diagram of symmetrical cross ply laminated shaft.
Uncontrolled displacement history in the direction of ‘V’ for the
symmetrical cross ply laminated shaft.
2
4
4
26
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33
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33
34
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35
35
35
36
36
36
37
38
38
Fig. 5.19
Fig. 5.20
Fig. 5.21
Fig. 5.22
Fig. 5.23
Fig.5.24
Fig.5.25
Fig.5.26
Fig.5.27
Fig.5.28
Fig.5.29
Fig.5.30
Fig.5.31
Fig. 5.32
Fig. 5.33
Fig. 5.34
Fig. 5.35
Fig. 5.36
Fig. 5.37
Fig. 5.38
Fig.5.39
Fig.5.40
Uncontrolled displacement history in the direction of ‘W’ for
symmetrical cross ply laminated shaft.
Controlled displacement history in the direction of ‘V’ for
symmetrical cross ply laminated shaft using one actuator.
Controlled displacement history in the direction of ‘W’ for
symmetrical cross ply laminated shaft using one actuator.
Controlled displacement history in the direction of ‘V’ for
symmetrical cross ply laminated shaft using two actuators.
Controlled displacement history in the direction of ‘W’ for
symmetrical cross ply laminated shaft using two actuators.
Controlled current in the direction of ‘Y’ for the symmetrical cross
laminated shaft using one actuator.
Controlled current in the direction of ‘Z’ for the symmetrical cross
laminated shaft using one actuator.
Controlled current in the direction of ‘Y’ for the symmetrical cross
laminated shaft using two actuators.
Controlled current in the direction of ‘Z’ for the symmetrical cross
laminated shaft using two actuators.
Controlled Force in the direction of ‘Y’ for the symmetrical cross
laminated shaft using one actuator.
Controlled Force in the direction of ‘Z’ for the symmetrical cross
laminated shaft using one actuator.
Controlled Force in the direction of ‘Y’ for the symmetrical cross
laminated shaft using two actuators.
Controlled Force in the direction of ‘Z’ for the symmetrical cross
laminated shaft using two actuators.
Campbell diagram of anti-symmetrical cross ply laminated shaft.
the
39
the
39
the
39
the
40
the
40
ply
40
ply
41
ply
41
ply
41
ply
42
ply
42
ply
42
ply
43
44
Uncontrolled displacement history in the direction of ‘V’ for the antisymmetrical angle ply laminated shaft.
Uncontrolled displacement history in the direction of ‘W’ for the antisymmetrical angle ply laminated shaft.
Controlled displacement history in the direction of ‘V’ for the antisymmetrical angle ply laminated shaft using one actuator.
Controlled displacement history in the direction of ‘W’ for the antisymmetrical angle ply laminated shaft using one actuator.
Controlled displacement history in the direction of ‘V’ for the antisymmetrical angle ply laminated shaft using two actuators.
Controlled displacement history in the direction of ‘W’ for the antisymmetrical angle ply laminated shaft using two actuators.
Controlled current in the direction of ‘Y’ for the anti-symmetrical cross
ply laminated shaft using one actuator.
Controlled current in the direction of ‘W’ for the anti-symmetrical cross
ply laminated shaft using one actuator.
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46
46
46
47
Fig.5.41
Fig.5.42
Fig.5.43
Fig.5.44
Fig.5.45
Fig.5.46
Fig. 5.47
Fig. 5.48
Fig. 5.49
Fig. 5.50
Fig. 5.51
Fig. 5.52
Fig. 5.53
Fig.5.54
Fig.5.55
Fig.5.56
Fig.5.57
Fig.5.58
Fig.5.59
Fig.5.60
Fig.5.61
Controlled current in the direction of ‘Y’ for the anti-symmetrical cross
ply laminated shaft using two actuators.
Controlled current in the direction of ‘W’ for the anti-symmetrical cross
ply laminated shaft using two actuators.
Controlled Force in the direction of ‘Y’ for the anti-symmetrical cross ply
laminated shaft using one actuator.
Controlled Force in the direction of ‘Z’ for the anti-symmetrical cross ply
laminated shaft using one actuator.
Controlled Force in the direction of ‘Y’ for the anti-symmetrical cross ply
laminated shaft using two actuators.
Controlled Force in the direction of ‘Z’ for the anti-symmetrical cross ply
laminated shaft using two actuators.
Campbell diagram of quasi-isotropic ply laminated shaft
Uncontrolled displacement history in the direction of ‘V’ for the quasiisotropic ply laminated shaft.
Uncontrolled displacement history in the direction of ‘W’ for the quasiisotropic ply laminated shaft.
Controlled displacement history in the direction of ‘V’ for the quasiisotropic ply laminated shaft using one actuator.
Controlled displacement history in the direction of ‘W’ for the quasiisotropic ply laminated shaft using one actuator.
Controlled displacement history in the direction of ‘V’ for the quasiisotropic ply laminated shaft using two actuators.
Controlled displacement history in the direction of ‘W’ for the quasiisotropic ply laminated shaft using two actuators.
Controlled current in the direction of ‘Y’ for the quasi-isotropic ply
laminated shaft using one actuator.
Controlled current in the direction of ‘Z’ for the quasi-isotropic ply
laminated shaft using one actuator.
Controlled current in the direction of ‘Y’ for the quasi-isotropic ply
laminated shaft using two actuators.
Controlled current in the direction of ‘Z’ for the quasi-isotropic ply
laminated shaft using two actuators.
Controlled Force in the direction of ‘Y’ for the quasi-isotropic ply
laminated shaft using one actuator.
Controlled Force in the direction of ‘Z’ for the quasi-isotropic ply
laminated shaft using one actuator.
Controlled Force in the direction of ‘Y’ for the quasi-isotropic ply
laminated shaft using two actuators.
Controlled Force in the direction of ‘Z’ for the quasi-isotropic ply
laminated shaft using two actuators.
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LIST OF TABLES
Table description
Page
29
Table 5.2
Mechanical Properties and Geometric dimension of Steel Rotor-shaft
system
Mechanical Properties and Geometric dimension of Composite shaft
Table.5.3
Parameters used for electromagnetic actuator and PD control techniques
31
Table 5.1
30
NOMENCLATURES
ux,
uy
uz
[M]
Flexural displacements of any point on the cross-section of the shaft in the x
direction.
Flexural displacements of any point on the cross-section of the shaft in the y
direction.
Flexural displacements of any point on the cross-section of the shaft in z
direction.
Inertia matrix of rotor shaft bearing system
{q}
Combined damping including gyroscopic effect, direct damping of internal
material damping, shaft bearing damping
Combined shaft stiffness, bearing stiffness, circulatory effect of the internal
material damping of shaft
Nodal displacement vector for the entire system
{f}
Global force vector due to unbalance
{fEMA}
ߤ௢
Force due to electromagnetic actuator
Magnetic permeability of free air
[D]
[K]
‫ܣ‬௣
Pole face area, m2
N
No. of coil turns
i
io
Current, A
Bias current, A
go
lg
Nominal radial air gap between stator poles and rotor, m
Length of magnetic flux lines through air gap, m
ߙ
Half of the angle between two adjacent poles
T
FY, FZ
FYmag, FZmag
KM
Ky, kz, ks
kg
ki
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Time, s
Linearized magnetic force along Y and Z direction, N
Magnetic force along Y and Z direction, N
Characteristic constant for a pair of poles in the exciter
Force displacement factor for magnetic exciters, N/m
Power amplifier gain, A/V
Force current factor, N/A
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kp
kv
V, w
J
Kopt
Icy
Icz
[Q]
[R]
FYC, FZC
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Gain of displacement feedback sensor, V/m
Gain of velocity feedback sensor, V.s/m
Translational displacement of a point of a point on rotor along Y and Z axes,
respectively, m
Cost function
Optimal gain of LQR
Control current in Y direction
Control current in Z direction
Semi-positive-definite matrix
Positive-definite matrix
Control force along Y and Z directions, N
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CHAPTER 1
INTRODUCTION
Rotating composite shaft play an important role in many different industries. Such as
in power production, turbines, and aircraft engines, process machines in heavy industry, fans,
pumps and ship engines. Now a day Composite rotor shaft materials are being increasingly
used. This is due to their good characteristics such as the high strength to mass ratio, the high
stiffness to mass ratio, good damping capacity and better resistance to fatigue. Composite
materials made of the stiff continuous or long fibers imbedded in soft matrix are used in the
structural components which give higher strength and stiffness but weight is lesser. In
engineering applications, however, the use of composite materials containing discrete
reinforcements may be warranted in order to reduce the cost, make parts of complicate shapes,
etc. Composites are formed by combining two or more materials to produce a new material
that retains important properties from the original components. These unique combinations
deliver significant advantages over traditional materials in structural applications. FRP
Composites consist of a polymer matrix material that is reinforced with fibers. The reinforcing
fibers provide the primary structural performance of the material, with the polymer
transferring the load from fiber-to-fiber and protecting the fibers from the operating
environment. Reinforcement may be continuous fibers, discontinuous fibers. Fiber Reinforced
Polymer (FRP) composite overturn our society and impinging our daily lives to use that,
which are lighter, more durable and have infinite design flexibility.
Vibration in the rotor–shaft–bearing systems is very important in industries, as it creates
operational difficulties, inaccuracies, power loss, fatigue and even failure of the system.
Reduction of rotor vibration is very important for safe and efficient functioning of all rotor
shaft system machines. Active vibration control is the active application of force in which an
equal and opposite forces imposed by external vibration. Active vibration control system uses
several components. A PID controller can be used to get better performance than a simple
inverting amplifier.
There are various types of active material such as: (1) thermo elastic material (2) piezo
electric material (3) Magneto-strictive materials (4) electromagnetic actuator (5) micro fiber
carbon.
In our present work electromagnetic actuator is used as an active material to control
the vibration of composite rotor shaft system in which exciters are mounted on the stator at a
plane around the rotor shaft for applying suitable force of actuation over an air gap to control
transverse vibration as shown in Fig. 1.1. Perini E. A. et al. [1] developed electromagnets
which is used for vibration control do not levitate the rotor and facilitate the bearing action,
which is provided by conventional bearings. Suitable force of actuation is achieved by varying
the control current in the exciters depending upon p-d control law applied to the displacement
of the rotor section with respect to the non-rotating position of the section taken as the
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reference. Thus this technique provides control force over an air gap and hence this technique
is free from the difficulties of maintenance, wear and tear and power loss. Electromagnetic
force is non-linear in nature which is very complex. Hence linearized expression of
electromagnetic force is used for finite element analysis. This application shows good
reduction in transverse response amplitude, postponement of instability caused by viscous
form of rotor internal damping as well as great reduction of support forces.
Fig. 1.1: Electro Magnetic levitation principle [1]
Thus this technique provides uncontact type control force over an air gap and hence it is free
from the difficulties of maintenance, wear and tear and power losses. There are many control
strategy such as PD control, PID control and LQR is used to control the unbalance response of
the composite rotor shaft system.
1.1 Composite Materials
Composite Materials can be defined as a combination of two or more dissimilar
materials having a distinct interface between them such that the properties of the resulting
materials are superior to the individual constituting components. That is insoluble in each
other and differs in form or chemical composition.
In other words composite material consists of two or more phase. Two phase composite
materials are classified as: (1) Particulate composite and (2) Fiber reinforced composite.
1.1.1
Particulate composite
Particulate composites are having various shape and size which are dispersed within
a matrix in a random fashion. Due to this, these composites are treated as quasi-homogeneous
and quasi-isotropic. Particulate composites are made of tungsten and molybdenum particles
dispersed in silver and copper matrices. It is used for electrical application. Example of
particulate composite is mica flakes reinforced with glass, aluminium particles in
polyurethane rubber, lead particles in copper alloys etc.
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1.1.2
Fiber reinforced composite
Fiber reinforced composite material consist of fibers of significant strength and stiffness
embedded in a matrix. Both fibers and matrix maintain their physical and chemical properties.
Fiber reinforced composite having continuous fibers are more efficient. FRP can be classified
into four categories according to matrix used such as: polymer matrix composites, metal
matrix composites, ceramic matrix composite and carbon/carbon composites.
Fiber reinforced composites possess high strength and stiffness. Some material
performs equally or better than many traditional metallic materials. Fatigue strength to weight
ratio as well as fatigue damage tolerance of many composite laminates is excellent.
Coefficient of thermal expansion of FRP composites is much lower than metals.FRP
composites possess high internal damping. This leads to better vibration energy absorption
within the material, and results in reduced transmission of noise, vibration.
A FRP laminate consists of a series of laminae or plies that are bonded together to act as
integral structural element. Each plies are oriented at different angle to produce different
strength and stiffness in the required direction of laminate. A lamina or a ply is formed by
combining more no. Of fibers in a thin layer of matrix and a laminate is formed by stacking
several laminas. It is the most common form of fiber reinforced composites.
1.1.3 Types of Laminates
There are different types of Laminates such as: Symmetrical laminates, balanced
laminate, Anti-Symmetrical laminates, Quasi-Isotropic laminate.
1.1.3.1 Symmetrical laminate
A laminates is termed as symmetric when the plies on one side of reference plane are
identical to another side in terms of thickness, orientation, properties and position. Also both
geometry and material properties should be same about the reference plane. It may contain
odd or even number of plies. The plies can all be of single composite or made of hybrid.
An example: [902|452|-454|452|902] or [902|452|-452]s
The laminate above is symmetric.
Symmetric laminates are of four types as: symmetric laminates with isotropic plies,
symmetric laminates with specially orthotropic plies (symmetric cross-ply laminates),
symmetric laminates with generally orthotropic layers (symmetric angle-ply laminates),
symmetric laminates with anisotropic layers.
1.1.3.2 Balanced Laminate
A laminates is termed as balanced if it contains pair of layers with identical thickness
and elastic properties, but the fibers are placed with orientation of + and – with respect to
the reference axes. The need not be placed at the same distance from the reference plane.
A balanced laminate may be of any of the three types: symmetric, anti-symmetric or
asymmetric.
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1.1.3.3 Anti-symmetrical laminate
An anti-symmetric laminate is made up of a number of plies in such a way that for each
ply below the middle surface, there is a ply at the same distance above it having identical
thickness and material properties, but opposite ply angle. Anti-symmetric laminates are
special case of balanced laminates.
There are two types of anti-symmetric laminates as: Anti-symmetric cross-ply laminates,
Anti-symmetric angle-ply laminates.
1.1.3.4 Quasi-Isotropic laminate
Quasi-isotropic laminates are combination of specially and generally orthotropic plies.
The result is that the inplane stiffnesses and compliances and engineering elastic constant
identical in all direction.With the help of orthotropic laminate, it can manufacture which
exhibits some elements of isotropic behaviour. Some examples of Quasi-isotropic laminates
are:
[60|-60|0]s, [-45|0|45|90]s, [0/-45/45/90]s
1.1.4 Application areas of composites
Fiber reinforced composites have many areas in which it is used such as commercial and
industrial. The main application areas may be broadly classified as follows:
• Aircraft and space
• Autpmotive
• Sporting goods
• Marine field
• Civil engineering structures
1.2 Active Material
There are various types of active material such as: (1) Piezo electric material (2)
Magneto-strictive materials (3) Electromagnetic actuator (4) Mircro carbon fiber.
1.2.1 Piezoelectric material
Piezoelectric materials develop charge if deformed by mechanical stress. The current
generation of piezoelectric materials is generally synthesized with polymeric fibrous
composite laminates which readily accommodate piezoelectric actuators and sensors. The
inverse effect in piezoelectricity deformation will occur due to the application of an electrical
field. External force applied on the structure will set vibrations and cause deformations in the
structure. These deformations will cause stresses and strains in the structure. If the structure is
embedded with smart crystals, the effects of the vibration can be controlled using a feedback
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mechanism. Since piezoelectric undergoes surface elongation, when an electric field is applied
and produces charge when surface strain is applied. They can be used as both actuators and
sensors. Some commonly used actuation materials are lead zirconate titanate and
polyvinylidenefluoride.
1.2.2 Magneto-strictive materials
Magnetostriction material has property of ferromagnetic material. Due to this, its shape
or dimension changes during the magnetization. This variation of magnetization due to
applied magnetic field will change the magnetostrictive strain until reaching the saturation
state. Magnetostriction is mostly found in the materials like iron, cobalt and nickel. Also it is
found in the rare earth materials like lanthanum and terbium. The grains of above materials
consist of numerous small randomly oriented magnetic constituent, which can be rotated and
aligned under the influence of magnetic field.
1.2.3 Electromagnetic actuator
Electromagnetic actuator is specially design as electromagnet that consists of a coil and
a movable iron core called the armature. When current flows through a coil, a magnetic field
is generated around the coil. When the coil of the solenoid is energized with current, the core
moves to increase the flux linkage by closing the air gap between the cores. The movable core
is usually spring-loaded to allow the core to retract when the current is switched off. The force
generated is approximately proportional to the square of the current and inversely proportional
to the square of the length of the air gap.
1.3 Control technique
There are various types of control technique such as:
(1) PD Control,
(2) PID Control and
(3) LQR
1.3.1 PD Controller
Fig.1.2 represents the block diagram of PD control with feedback. The stability and
overshoot problems that arise when a proportional controller is used at high gain can be
achieve by adding a term proportional to the time-derivative of the error signal. The value of
the damping can be adjusted to achieve a critically damped response. PD controllers are
slower than P, but less oscillation, smaller overshoot/ripple, Bennett et al. [20]
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Fig. 1.2: Block diagram of PD controller with feedback [2]
1.3.2 PID Controller
A PID controller calculates an "error" value as the difference between a
measured value and a desired set point. The PID controller involves three separate constant
parameters, the proportional, the integral and derivative denoted by P, I, and D. The
controller is used to minimize the error by adjusting the process control inputs. P depends on
the present error, I on the past errors, and D is on future errors, based on current rate of
change. In the absence of knowledge of the underlying process, a PID controller has
historically been considered to be the best controller. By tuning the three parameters in the
PID controller algorithm, the controller can provide control action designed for specific
process requirements K.H. Ang et al. [21].
The response of the controller can be described in terms of the responsiveness of
the controller to an error, the degree to which the controller overshoots the set point, and the
degree of system oscillation. Note that the use of the PID algorithm for control does not
guarantee optimal control of the system or system stability.
Fig. 1.3: Block diagram of PID controller [3].
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Where,
Kp= Proportional gain
gain, Ki= Integral gain, Kd= Derivative gain,
gain
integration, : Error
Error, : Time
: Variable of
1.3.3 Linear Quadratic Regulation (LQR)
LQR optimal control theory is used to determine the control
ol gains. In this, the
feedback control system has been designed to minimize a cost function or a performance
index, which is proportional to the required measure of the system's response.
Consider a state-space
space representation of the equations of motions of an N-DOF
N
linear
system. State-space
space model can be written asas
ɺ
X = A * X + B *U (t )
The cost function used in the present case is given by
1 tf
T
T
J = ∫ { X } [Q ]{ X } + {U } [ R ]{U } dt
t
2 0
Where [Q] and[R]] are the semi-positive-definite
semi
and positive-definite
definite weighting matrices on
the outputs and control inputs, respectively.
(
)
1.4 Advantages
dvantages of electromagnetic actuator
There are following advantages of electromagnetic actuator:
• No mechanical wear and friction due to non-contact
non
operation.
• No lubrication, and therefore non-polluting.
non
• Low energy
gy consumption.
• Operation in severe environments.
• Active vibration control and easy passing of critical speeds.
1.5 Finite element method
Finite element method (FEM) is a numerical method for solving a complex differential or
integral equation. It has been applied to a various number of physical problems, where the
governing differential equations (it may be linear or non-linear) are available. The method
essentially consists of assuming the elementary (i.e. no. of pieces) continuous function for the
solution and obtaining the functions so that it reduces the error in the solution.
solution In our present
work finite element method has been used for composite rotor shaft system.
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1.6 Outline of the Present Work
The outline of this thesis is divided into six chapters.
Chapter 1 represents an introduction of composite material, active material, and control
technique also brief introduction to rotor shaft system. The outline of the present work is also
given in the Chapter 1.
In Chapter 2 presents literature review on vibration control of isotropic rotor shaft
system, vibration control of composite rotor shaft system and, active vibration control of rotor
shaft system systems.
In Chapter 3 represents the finite element modelling and analysis of FRP composite
shaft system
Chapter 4 presents a modeling and analysis of electromagnetic actuator, governing
equations of motion, control strategy.
Chapter 5 discusses the results for many stacking sequence of laminated composite and
detailed report of results and discussion has been given.
Chapter 6 summarizes
further work are suggested.
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conclusions
of
this
project
work
and
scopes
for
Page 8
CHAPTER 2
LITERATURE REVIEW
2.1 General Introduction
Based on above research field many researchers have been presented about finite
element modelling and vibration analysis of isotropic as well as composite shafts system and
active vibration control of rotating shaft considering isotropic materials are also presented in
some important literature. These have been in the following sub section
2.2 Finite element modelling and vibration analysis of isotropic shaft system
Vibration caused by mass unbalance is a common problem in rotating shaft.
Rotor unbalance occurs when the principal inertia axis of the rotor does not coincide with its
geometrical axis and leads to synchronous vibrations and significant undesirable forces
transmitted to the mechanical elements and supports. Many researchers have been worked to
reduce the vibration caused by unbalance response.
Foeppl [4] developed an analytical method of the dynamic behaviour of the isotropic
de Laval rotor. Rotor dynamics as a subject first appeared in the last quarter of the 19th
Century due to the problems associated with the high speed turbine of Gustaf de Laval who
invented the elastically supported rotor, called de Laval Rotor, and observed its supercritical
operation. Foeppl explained analytically the dynamic behaviour of the de Laval rotor.
Serious research on rotor dynamics started in 1869 when Rankine published his paper on
whirling motions of a rotor. However, he did not realize the importance of the rotor
unbalances and therefore concluded that a rotating machine never would be able to operate
above the first critical speed. De Laval showed around 1900 that it is possible to operate
above critical speed, with his one-stage steam turbine.
Stodola [5] Presents the influence of gyroscopic effects on a rotating system was
presented in 1924 by Stodola. The model that was presented consists of a rigid disk with a
polar moment of inertia, transverse moment of inertia and mass. The disk is connected to a
flexible mass-less overhung rotor. The gyroscopic coupling terms in Stodola’s rotor model
resulted in the natural frequencies being dependent upon the rotational speed. Jeffcott [6]
developed the first paper where the theory of unbalanced rotors is described. Jeffcott derived
a theory which shows that it is possible for rotating machines to exceed the critical speeds.
However, in the Jeffcott model the mass is basically represented as a particle or a pointmass, and the model can’t correctly explain the characteristics of a rigid-body on a flexible
rotating shaft R.G. Kirk, et al. [7] analyzed he dynamic unbalance response and transient
motion of the single mass Jeffcott rotor in elastic bearings mounted on damped, flexible
supports are discussed. A steady state analysis of the shaft and the bearing housing motion
was made by assuming synchronous precession of the system. The conditions under which
the support system would act as a dynamic vibration absorber at the rotor critical speed were
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studied. Lateral vibrations in rotor systems have been analysed extensively by Tondl, Fritz,
and Lee, They considered different types of rotor systems, and in all those systems, lateral
vibrations are induced by the mass unbalance in a rotor. In all the systems considered it is
noticed that increase of mass unbalance can have destabilising effects. For example, Tondl,
and Lee, considered a simple disk with a mass unbalance connected to a shaft which is
elastic in the lateral direction and found out that in such systems, under certain conditions,
instabilities can appear if the mass unbalance increases.
He formed an expression for the work done in a small relative displacement by all
forces and obtained the differential equations by way of the calculus of variations. Solutions
of the differential equations of motion for an elastic solid were treated by Poisson who
founded the general theory of vibrations. The development of vibration theory, as a
subdivision of mechanics, came as a natural result of the development of the basic sciences it
draws from, mathematics and mechanics. Pythagoras of Samos conducted several vibration
experiments with hammers, strings, pipes and shells. He established the first vibration
research laboratory. That for a (linear) system there are frequencies at which the system can
perform harmonic motion was known to musicians but it was stated as a law of nature for
vibration systems by Pythagoras. Moreover, he proved with his hammer experiments that
natural frequencies are system properties and do not depend on the magnitude of the
excitation.
Nelson et.al [8] presented the Rayleigh beam theory to model the rotor shaft taking
into consideration the presence of viscous form of internal material damping, which is caused
to instability above the first critical speed.it presented a procedure for dynamic modelling of
rotor bearing systems which consisted of rigid disks, distributed parameter finite rotor
elements, and discrete bearings. They presented their formulation in both a fixed and rotating
frame of reference They developed a finite element model including the effects of rotary
inertia, gyroscopic moments. Yamamoto [9] developed rotor-bearing systems in which there
are many sources of nonlinearities, such as play in bearings and fluid dynamics in journal
bearings. The dynamic stiffness of the bearing which supports the rotating shaft has a
significant effect on the vibration. In particular it affects the machine critical speeds and the
vibration in between critical speeds suggested that rolling bearings, which are frequently used
in industry, sometimes have nonlinear spring characteristics due to coulomb friction and the
angular clearance between roller and ring. Holmes et al. [10] Analyzed a paper dealing with a
periodic behaviour in journal bearings. In their work, the symmetrical, steady state motion of
a rigid shaft supported by two short journal bearings was studied. The behaviour of this test
rig was found to be of two distinct types. For small eccentricity, the motion was
asymptotically periodic and consisted of a small number of components, principally at
synchronous and half-synchronous frequencies. For high eccentricity, the motion observed
was complex and did not settle to a limit cycle, remaining in a state of
Panda et.al [11] analyzed the viscoelastic support and their ability to provide good
support elements to rotor shaft systems by virtue of their efficiency in dissipating vibratory
energy. They studied that the in-phase stiffness and loss factor for such materials also change
with the frequency of excitation they are subjected to. They found the frequency dependent
characteristics of the polymeric supports by simultaneously minimizing the unbalanced
response and maximizing the stability limit speed (SLS). Optimum characteristics have been
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found for the rotor shaft system mounted on (a) rolling element bearings and (b) plain
cylindrical journal bearings at the ends having polymeric supports. The effects of viscous
internal damping in the shaft, support mass and gyroscopic effect due to non-symmetrical
location of the disc have been considered in the analysis. A procedure of controlling a slope of
the support characteristics versus frequency of excitation has been used and found to be very
suitable for obtaining feasible support characteristics.
Cunningham [12] developed a Rotor systems have been traditionally supported on oilfilm bearings due to their robustness. The oil-film bearings introduce some damping to the
rotor system, but can also lead to oil whip instability. In order to control the resonance and to
delay the onset of instability, passive devices such as squeeze-film bearings have been used to
augment the system damping. Stanway et al. [13] presented a paper in which supercritical
systems has several lateral bending modes of vibration are liable to be excited, and given a
single passive device it is not possible to select the stiffness and damping parameters so as to
exert a significant influence over all these modes and on the other hand, their success depends
on accurate knowledge of the dynamic behaviour of the machine. Additionally, passive
control techniques have low versatility, i.e., any change in the machine configuration or in the
loading condition may require a new damping device.
Viderman et al. and Subbiah et al. and [14-15] analyzed that a rotor has certain speed
ranges in which large and unacceptable amplitude of vibration could be developed. These
speed ranges are known as critical speeds (or critical frequencies) which could cause a bearing
failure or result in excessive rotor deflection. Under these circumstances, the problem of
ensuring that a rotor bearing system performs with stable and low-level amplitude of vibration
becomes increasingly important. Dutt J.K. et al. [16] presented a viscoelastic support
characteristics so that the frequency of excitation never coincides with any of the un damped
natural frequencies of the system, thus giving low unbalance response over a wide frequency
range and the support material can be chosen accordingly. Chen and Wang [17] presented an
analysis and design of rotor-bearing systems with gyroscopic effects. They transformed the
original problem of a damped system into state space form so that the transformed problem is
similar to the eigenvalue problem of an undamped system. The problem is then solved by
available eigenvalue solvers and the sensitivities needed for optimization were obtained. They
used sequential linear programming to solve the unique design optimization problem.
Childs D.W. et al. [18-19] presented that eigen values are in general complex, where
the sign of the real part decides stability. Negative real part confirms asymptotic stability
whereas a non-negative value indicates instability. So for a particular speed of rotation, the
maximum real part of the system eigenvalues should be negative for the stable operation of
the rotor at that speed. The lowest speed, at which the maximum real part of the eigenvalues
becomes nonnegative, is known as the stability limit speed (SLS) of the rotor-shaft system.
2.3 Finite element modelling and vibration analysis of FRP composite shaft system
There are many authors have been developed Finite element modelling and vibration
analysis of FRP composite shaft system as follows:-
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M. Y. Chang et al. [20] developed the vibration behaviors of rotating laminated
composite shaft model where transverse shear deformation, rotary inertia, gyroscopic effects
and coupling effect are incorporated. Bert developed governing equation of composite shaft,
including effect of gyroscopic, bending and torsion coupling and determines critical speeds of
composite shaft. Zinberg and Symmonds [21] analyzed a boron/epoxy composite tail rotor
driveshaft for a helicopter. Equivalent modulus beam theory (EMBT) is used for evaluation
of the critical speeds. Shaft is assumed thin walled circular tube simply supported at the ends.
Shear deformation was not taken into consideration. Dos Reis et al. [22] developed an
analytical analysis on thin-walled layered composite cylindrical hollow shaft. The beam
element was extended to formulate the problem of a rotor supported on general eight
coefficient bearings. Results were obtained for shaft configuration of Zinberg and Symmonds.
The authors have shown that bending-stretching coupling and shear-normal coupling effects
change with stacking sequence, and alter the frequency values. Gupta and Singh [23] studied
the effect of shear-normal coupling on rotor natural frequencies and modal damping. Kim and
Bert [24] have formulated the problem of determination of critical speeds of a composite shaft
including the effects of bending-twisting coupling. The shaft was modeled as a BresseTimoshenko beam. The shaft gyroscopic has also been included. In another study, Bert and
Kim [25] have analyzed the dynamic instability of a composite drive shaft subjected to
fluctuating torque and/or rotational speed by using various thin shell theories. The rotational
effects include centrifugal and Coriolis forces. Dynamic instability regions for a long span
simply supported shaft are presented.
C. Y. Chang et al. [26] published the vibration analysis of rotating composite shafts
containing randomly oriented reinforcements. The Mori-Tanaka mean-field theory is adopted
here to account for the interaction at the finite concentrations of reinforcements in the
composite material.
Pilkey et al [27] analyzed that Existence of support properties corresponding to
optimum dynamic performance was also reported. Out of many works on predicting the
optimum support properties an interesting one is by where the authors used the ‘‘linear
programming technique’’ to predict the optimum support characteristics. Damping is a
mechanism to dissipate mechanical energy in the form of heat. Muszynska [28] developed the
paper which states that Unbalance is a most common malfunction in rotating machines.
Unbalance in the rotating machine is a condition of unequal mass distribution at each section
of the rotor. In an unbalanced condition, the rotor mass center line does not coincide with the
axis of rotation. During rotation, rotor unbalance generates an inertia centrifugal force which
rotates at the rotor rotational frequency. Unbalance represents then the first, fundamental
mechanism to transfer the rotational energy into vibrations. Wettergren H. L. et al. [29]
analyzed for a rotor system with viscous form of internal damping, stability limit is not
expected until the first critical speed is reached. Instability can be postponed by introducing
external damping and using right combination of non-isotropic bearing stiffness and damping
coefficients along with some other factors.
Lananne and Ferraris [30] developed a finite element model of a multi-disc
viscoelastic rotor with bearings. The model was based on Euler-Bernoulli beam theory. The
study was based on a rotor with three discs with viscoelastic supports (Voigt Model) at both
ends. The internal damping of the material of the rotor was taken into consideration. Finite
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element model lead to Campbell Diagrams and Decay rate plot. These plots were used to
show the stability characteristics of the system. DOI [31] developed the dynamic response of
the considered mechanical system can be modelled by using variational mechanics equations
as based on the Hamilton`s principle. For this aim, the strain energy of the shaft and the
kinetic energies of the shaft and discs are calculated. An extension of Hamilton`s principle
makes possible to include the effect of energy dissipation. The parameters of the bearings are
considered in the model by using the principle of the virtual work.
Gustavsson [32] DeLaval and Jeffcott’s names are still in use as the name of the
simplified rotor model with the disc in the mid-span of the shaft. Jeffcott’s rotor is described
by Vance (1988), for example as one that consists of a flexible shaft, with zero mass,
supported at its ends. The supports are rigid and allow rotation around the centre axis of the
shaft. The mass is concentrated in a disk, fixed at the midpoint of the shaft. The system is
geometrically symmetrical with respect to its rotational axis, except for a mass imbalance
attached to the disk. When rotating the mass imbalance provides excitation to the system.
Nighil, et al. [33] analyzed that the magnetic force developed on the rotor by one pair of poles
of the electromagnet is dealt with in details; However, salient portions are given here in brief
for completeness, continuity and ease of understanding. Chang-Jian et al. [34] presented in a
rotor-bearing system, the hydrodynamic pressure in journal bearings is generated entirely by
the motion of the journal and depends on the viscosity of the lubricating fluid. However, the
hydrodynamic pressure around the bearing is nonlinear and hence the fluid film rotor-bearing
system has a strong nonlinearity which can cause substantial vibrations of the rotor and its
bearings. Bueno [35] developed an increase of research works in engineering dedicated to the
development of active vibration control techniques (AVC) is observed. This effort is
stimulated by the necessity of lighter structures associated to higher operational performance
and smaller operating costs. Das A.S. et al. [36] developed an arrangement of actuator poles is
intended to apply non-contact type control force on the rotor-shaft over an air-gap that can be
resolved into two orthogonal components along Y- and Z-directions, which can be computed
independent of one another. Details of derivation of the expressions of linearized force
components generated by the actuator with necessary assumptions are discussed. The
components of the linearized control force obtained from the actuator in terms of control
current and the displacement of the rotor-shaft at the location of the actuator of the controller.
Das A.S. et al. [37] analyzed that Rotor vibrations caused by large time-varying base
motion are of considerable importance as there are a good number of rotors, e.g., the ship and
aircraft turbine rotors, which are often subject to excitations, as the rotor base, i.e. the vehicle,
undergoes large time varying linear and angular displacements as a result of different
maneuvers. Due to such motions of the base, the equations of vibratory motion of a flexible
rotor–shaft relative to the base (which forms a non-inertial reference frame) contains terms
due to Coriolis effect as well as inertial excitations (generally asynchronous to rotor spin)
generated by different system parameters. The actuator does not levitate the rotor or facilitate
any bearing action, which is provided by the conventional suspension system. The equations
of motion of the rotor–shaft continuum are first written with respect to the non-inertial
reference frame (the moving base in this case) including the effect of rotor internal damping.
A conventional model for the electromagnetic exciter is used.
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Numerical simulations performed on the flexible rotor–shaft modelled using
beam finite elements shows that the control action is successful in avoiding the parametric
instability, postponing the instability due to internal material damping and reducing the rotor
response relative to the rigid base significantly, with sufficiently low demand of control
current in comparison with the bias current in the actuator coils. In the rotor dynamics
applications elastic behaviour of a rotating shaft is an important concern. There are several
works in the literature on the modelling and vibration control of a flexible shaft since the
analysis of flexible structure with lumped parameters does not reflects the real behaviour,
continuous time models with partial differential equations are used for more realistic analysis.
The well known classical beam theory also called as Euler-Bernoulli beam theory has been
used by many authors for the vibration analysis of the shafts. However, this beam theory is
not suitable for thick beams since the shear deformation and rotary inertia effects are not
included in the model.
2.4 Active vibration control of rotor shaft system
There are various methods have been adopted by many researchers to control active
vibration.
Schweitzer et al. et al. [38-39] developed an active magnetic bearings (AMB) provide an
active way of bearing action and vibration control over an air gap which is more elegant. That
measures the unbalance force with the help of sensor and applies the control force between the
outer race and the bearing housing. Schittenhelm et al [40] developed a linear quadratic
regulator is designed for a rotor system on the basis of a finite element model. The rotor is
subject to gyroscopic effect and is actively supported by means of piezoelectric actuators
installed at one of its two bearings. As a result of the first aspect, its dynamic behavior varies
with rotational frequency of the rotor. This aspect is challenging for linear time invariant
control techniques since it results in a demand for high robustness. Furthermore, if controller
and observer are calculated using a model at a specific design frequency, the separation
principle does not hold. In this article a proposal for combined Linear Quadratic Regulator
and Kalman Filter design on the basis of physical considerations is given. Maslen [41]
presented an Active Magnetic Bearings (AMB) not only as a main support bearing in a
machine but also as force actuators has become one of the useful devices in a control scheme
for active vibration control in rotating machinery such as suspension systems for shafts or
rotors. Several components of an AMB are characterized by nonlinear behavior and therefore
the entire system is inherently nonlinear.
Many methods have been proposed to reduce the unbalance-induced vibration, where
different devices such as electromagnetic bearings, active squeeze film dampers, lateral force
actuators, active balancers and pressurized bearings have been developed by Blanco et al.
[42], Guozhi et al. [43], Jinhao & Kwon, [44], Palazzolo et al. [45], Sheu et al. [46], Zhou &
Shi [47]. Passive and active balancing techniques are based on the unbalance estimation to
attenuate the unbalance response in the rotating machinery. The Influence Coefficient Method
has been used to estimate the unbalance while the rotating speed of the rotor is constant
presented by Lee et al. [48], Yu [49]. This method has been used to estimate the unknown
dynamics and rotor- bearing system unbalance during the speed-varying period Zhou et al.
[50]. On the other hand, there is a vast literature on identification methods developed by
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Ljung [51], Sagara & Zhao [52] , Sagara & Zhao [53] which are essentially asymptotic,
recursive or complex, which generally suffer of poor speed performance.
Lund J. [54] analyzed vibrations are of two types, active and passive vibration and
both passive and active means of vibration minimization techniques are reported in literature;
these techniques attempt to find suitable stiffness-damping combinations of the support to
avoid or minimize resonant response. The passive category includes the use of flexible–
damped supports, Squeeze-film dampers and viscoelastic bearing supports. The measures
require contact with the support element for its mechanical deflection to get the control force
and are applied for convenience between the bearing outer-race and the bearing housing.
Allaire et al. [55] developed and presented magnetic bearings in a multimass flexible rotor
both as support bearings and as vibration controller and demonstrated the beneficial effect of
reducing vibration amplitudes by using an electromagnet applied to a transmission shaft
respectively. They used two approaches to actively control flexible rotors. In the first
approach magnetic bearings or electromagnetic actuators are used to apply control forces
directly to the rotating rotor without contacting it. In the second approach, the control forces
of the electro-magnetic actuators are applied to the bearing housings.
Koroishi E. H. et al. [56] presented a paper proposes a simple model of an
electromagnetic actuator (EMA) for active vibration control (AVC) of rotor systems. For this
purpose, the actuator was linearized by adopting a behavior that is similar to the one used for
active magnetic bearings (AMB). The results show the validity of the proposed model and the
effectiveness of the control. Koroishi [57] developed In recent years, a number of new
methods dedicated to acoustic and vibration attenuation have been developed and proposed
aiming at handling several types of engineering problems related to the dynamic behavior of
the system. This is mainly due to the demand for better performance and safer operation of
mechanical systems. There are various types of actuators available. The present contribution
is dedicated to the electromagnetic actuator (EMA). EMA uses electromagnetic forces to
support the rotor without mechanical contact.
Keith et al. [58] analyzed that the use of electromagnetic bearings in lowering the
amplitude level has increased and) showed that they generate no mechanical loss and need no
lubricants such as oil or air as they support the rotor without physical contact. Cheung et al.
[59] presented the electromagnets are open loop unstable and all designs require external
electronic control to regulate the forces acting on the bearing. Abduljabbar et al. [60]
developed an optimal controller based on characteristics peculiar to rotor bearing systems
which take into account the requirements for the free vibration and the persistent unbalance
excitations. The controller uses as feedback signals, the states and the unbalance forces. A
methodology of selecting the gains on the feedback signals has been presented based on
separation of the signal effects: the plant states are the primary stimuli for stabilizing the rotor
motion and augmenting system damping, while the augmented states representing the
unbalance forces are the primary stimuli for counteracting the periodically excited vibration.
The results demonstrate that the proposed controller can significantly improve the dynamical
behaviour of the rotor-bearing systems with regard to resonance and instabilities. A passive
vibration control devices are of limited use. This limitation together with the desire to exercise
greater control over rotor vibration, with greatly enhanced performance, has led to a growing
interest in the development of active control of rotor vibrations.
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ROY T. et al [61] analyzed that the linear quadratic regulator (LQR) control approach
has been found to be effective in vibration control with appropriate weighting matrices, which
gives optimal control gain by minimizing the performance index. The weighting matrices [Q]
and [R] are the most important components in LQR optimization .The combinations of [Q]
and [R] matrices greatly affect the output performance and input cost of the system and hence
an optimal selection of these weighting matrices is of significant importance from the control
point of view. Serdar Cole M.O.T. et al [62] developed that the use of an Active Magnetic
Bearing (AMB) to attenuate the lateral vibration of a rotor under simultaneous excitations
from mass unbalance and an initial base impact, which is quasi-stationary in nature. For the
implementation and testing of the devised multi frequency vibration control strategy, a
flexible rotor facility was used. Multi frequency feedback control was implemented in order to
control rotor vibration at the synchronous frequency and the first two harmonics.
Jingjun Zhang et al. [63] analyzed that the transfer function is transformed to a state
space vector dynamic equation for state feedback control system design. To minimize the
displacement of the rotor shaft system, the Linear Quadratic Regulator (LQR) based on
independent mode space control techniques is designed. The control voltage for the actuators
is determined by the optimal control solution of the Linear Quadratic Regulator (LQR), which
is an effective and widely used linear control technique. Provided the full state vector is
observable, this method can be employed to meet specific design and performance criteria. A
quadratic cost function is used to minimize the performance index. The recent years have seen
the appearance of innovative systems for acoustic and vibration attenuation, most of actuators
them integrating new actuator technologies. In this sense, the study of algorithms for active
vibrations control in rotating machinery became an area of enormous interest, mainly due to
the countless demands of an optimal performance of mechanical systems in aircraft
automotive structures. Also, many critical machines such as compressors, pumps and gas
turbines continue to be used beyond ft, aerospace and automotive structures. Also, many
critical machines such as compressors, pumps and gas turbines continue to be used beyond
their expected service life despite the associated potential for failure due to damage
accumulation.
Koroishi, et al. [64] presented that the AMB is a feedback mechanism that supports a
spinning shaft by levitating it in a magnetic field. For its operation, the sensor measures the
relative position of the shaft and the measured signal is sent to the controller where it is
processed. Then, the signal is amplified and fed as electric current into the coils of the magnet,
generating an electromagnetic field that keeps the shaft in a desired position. The strength of
the magnetic field depends both on the air gap between the shaft and the magnet and the
dynamics of the system including the design
Janik et al. [65] analyzed that In order to invoke control force for reducing the
amplitude of rotor vibration an electromagnetic actuator is used which is capable of applying
non-contact type force of actuation over an air-gap on the rotor-shaft. The actuator consists of
four exciters (each having a pair of electromagnetic poles) symmetrically arranged within a
stator casing around the periphery of the rotor section. This actuator system can be placed at
any suitable location along the span of the rotor-shaft avoiding the bearing as well as disc
locations. Dutt and Toi [66] presented a Polymeric material in the form of sectors has been
considered in this work as bearing supports. Polymeric material has been considered in this
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work as both stiffness and loss factor of such materials varies with the frequency of excitation.
Stiffness and loss factor have been found out for the proposed support system comprising of
polymeric sectors.
Meirovitch, L. [67] presents mathematical modeling and numerically verified using
the feedback pole allocation control. The goal of linear feedback control is to place the closed
loop poles on the left half of the complex plane of the eigen values, so as to ensure asymptotic
stability of the closed loop system. One approach consists of prescribing first the closed-loop
poles associated with the modes to be controlled and then computing the control gains
required to produce these poles. Because this amounts to controlling a system by controlling
its modes, this approach is known as modal control.
In this way, this paper presents a developed approach for active vibration control in a
rotor using Active Magnetic Bearings (AMB) that is numerically verified. The feedback
technique is used for this controlling device and the controller gain is obtained by the pole
allocation. The AMB uses electromagnetic forces to support a rotor without mechanical
contact. It offers many advantages compared to fluid film and rolling element bearings, such
as no wear, the ability to operate in high temperature environments, and no contamination of
the working fluid due to the absence of lubricant in the system.
2.5 Motivation and Objectives of the present work
Based on the literature review it has been observed that many works have been carried
out for the modelling analysis of rotor shaft system using finite element method. In the
direction of composite rotor shaft system few works are also available. The use of composite
material is increasing day by day in the various industries because of FRP composites are
lighter, more durable and have infinite design flexibility. So details study of laminated
composite shaft is required. Due to its light weight and flexibility this types of shafts do not
sustains large vibration in the operational condition. Active vibration control of rotating
composite shaft is also very important in the area of research. Therefore the objectives of the
present works are laid down as follows:
• Finite element modelling for vibration analysis for rotating composite shaft, with disk
and bearing.
• Effect of ply angle on the responses of the composite shaft.
• Modelling of electromagnetic actuator and
• Study of active vibration control of rotating composite shaft using electromagnetic
actuators and PD control scheme.
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CHAPTER 3
FINITE ELEMENT MODELLING
COMPOSITE SHAFT SYSTEM
AND
ANALYSIS
OF
FRP
3.1 Introduction
The composite-material shafts have been sought as new potential subject for
replacement of the conventional metallic shafts in many application areas. This may be
attributed to the improved performance of the shaft system resulting from the use of the
composite materials. It was in fact demonstrated by Faust et al.via testing the composite
transmission shafts of twin-propeller helicopters that the composite shaft not only is lighter in
weight and has a lower vibration level, but also has greater strength and a longer service life
compared with its metallic counterparts. Accompanied by the development of many new
advanced composite materials, various mathematical models of spinning composite shafts
were also developed by researchers. Zinberg and Symonds who used an equivalent modulus
beam theory (EMBT) to model the composite shaft and compared the critical speeds with
those of the tests they had performed. dos Reis et al. incorporated the Timoshenko beam
theory with the Donnell thin shell theory to derive the stiffness matrix of rotating composite
shafts. They then adopted the approximate finite element approach of Ruhl and Booker to
derive the equations of motion of systems. The model was used to analyze the critical speeds
of thin-walled composite shafts developed [20].
The shaft is modelled as a Timoshenko beam with rotary inertia and gyroscopic effect is
used. The shaft rotates at constant speed about its longitudinal axis the shaft has a uniform,
circular cross-section.
3.2 Mathematical Modelling of composite Rotor shaft system
To derive the strain energy expression of the composite shaft, the following form of the
displacement fields of the shaft are assumed.
u x ( x , y , z , t ) = u ( x , t ) + z β x ( x , t ) − y β y ( x, t )
(3.1)
u y ( x, y, z, t ) = v ( x, t ) − zφ ( x, t )
u z ( x, y, z, t ) = w ( x, t ) + yφ ( x, t )
The stress-strain relations of the ith layer expressed in the cylindrical coordinate system can be
expressed as:
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σ xx = Q11iε xx + ksQ16iγ xθ
(3.2)
τ xθ = ks Q16iε xx + ksQ66iγ xθ
τ xr = ksQ55iγ xr
Where, ks is the shear correction factor, Qijn is the constitutive matrix which is related to
lamination angle η and elastic constants of principle axes.
Where,
n
2
2
A11 = π ∑ Q (ri +1 − ri )
11i
i =1
A
55
=
π
n
∑
2 Q
i =1
A66 =
B16 =
D11 =
D66 =
π
55i
n
2
∑Q
i =1
2π
3
∑Q
π
n
i =1
i =1
2
66 i
n
∑
4 Q
π
(ri 2+1 − ri 2 )
16 i
11i
i =1
( ri 3+1 − ri 3 )
( ri 4+1 − ri 4 )
n
∑Q
(ri 2+1 − ri 2 )
66 i
( ri 4+1 − ri 4 )
3.2 Finite element model analysis of rotor shaft
The equation of motion of the composite rotor-shaft-bearing system has been
derived using the finite element method and the rotor-shaft has also been modeled and
analyzed Timoshenko beam theory by three nodded element beams, each node having 4
degree of freedom. The governing equations of the motion are expressed below as ordinary
differential equations:
[M ]
{}
{}
q + ([C ] + Ω [G ]) q + [ K ]{q} = {F }
••
•
Where [ M ] denotes the mass matrix, [G ] denotes the gyroscopic matrix,
(3.3)
[C ]
denotes the
total damping matrix which include the material internal damping, bearing damping and
NIT ROURKELA
Page 19
actuator damping, [ K ] denotes the stiffness matrix , {q} denotes the displacement vector,
{F} represents the external force vector.
Mass matrices
To find out mass matrices assumes that field variable velocity is represents as,
•
•
n
v0 = ∑ψ n ( x ) v0 n
i =1
{}
2
2
•
•
1
1
n

T = ∫ ρ A v0 dx = ρ A∫ ∑ψ n ( x ) v0n  dx
20
2 0  i=1

L
Now, kinetic energy expression
L
So, mass matrices for an element
L
L
•
∂ ∂ 1
 n n

M ij =
ρ A∫ ∑ ∑ψ n ( x ) v0 n  dx = ∫ ρ Aψ ni ( x )ψ nj ( x ) dx
∂v0 i ∂v0 j 2

0  j =1 i =1
0
2
[ M 22 ]3×3

0
[ M e ] =  0

0

M
ij
22
=
ij
33
=
0
0
[ M 33 ]3×3
[ M 44 ]3×3
0
0
0
L
L
ND
0
0
i =1
l
ND




0

[ M 55 ]3×3 12×12
0
0
D
∫ I mψ i ( x )ψ j ( x ) dx + ∫ ∑ I m ψ i ( x )ψ j ( x )∆ ( x − x Di ) dx
l
M
0
∫
I mψ i ( x )ψ j ( x ) d x +
0
∫∑ I
i =1
0
l
l
ND
0
0
i =1
l
l
ND
0
0
i =1
D
m
ψ i ( x )ψ j ( x )∆ ( x − x D i ) d x
M 44ij = ∫ I dψ i ( x )ψ j ( x ) dx + ∫ ∑ I dD ψ i ( x )ψ j ( x )∆ ( x − x Di ) dx
M
ij
55
= ∫ I dψ i ( x )ψ j ( x ) dx + ∫ ∑ I dD ψ i ( x )ψ j ( x )∆ ( x − x Di ) dx
Stiffness Matrix
Now stiffness matrix expression K = ∫ [ B ] [ D ][ B ] dV
T
V
−1
 ∂  ∂η  ∂x   ∂ 
∂
∂
Now from the following relation   =     =     = J −1  
 ∂x   ∂η   ∂x   ∂η   ∂η 
 ∂η 
NIT ROURKELA
Page 20
 ∂x 
J −1 = 

 ∂η 
Where
−1
∂ v0
∂
=
(ψ 1v01 + ψ 2 v0 2 + ψ 3 v03 ) = (ψ '1 v01 + ψ '2 v0 2 + ψ '3 v03 )
∂η ∂η
∂ v0 i
= [ψ 'i ] {v0 i }
∂η
Now following

r sin θ J −1ψ ' −r cos θ J −1ψ '
0
0


B = − sin θ J −1ψ ' cos θ J −1ψ '
cos θ
sin θ

 cos θ J −1ψ ' sin θ J −1ψ '

sin
−
cos
θ
θ


Now following
[ B ] [ D ][ B ] =
T

−ks C16 sin θ J −1ψ '

ks C16 cosθ J −1ψ '


−1
 C11r sin θ J ψ '+ ks C16 cosθ
−C11r cosθ J −1ψ '+ k C16 sin θ
s

−ks C 66 sin θ J −1ψ '
ks C 66 cosθ J −1ψ '
ks C16 r sin θ J −1ψ '+ ks C 66 cosθ
−ks C16 r cosθ J −1ψ '+ ks C 66 sin θ
ks C 55 cosθ J −1ψ '

ks C 55 sin θ J −1ψ ' 
×
ks C 55 sin θ 
−ks C 55 cosθ 

0
0
r sin θ J −1ψ ' −r cosθ J −1ψ '


−1
−1
cosθ
sin θ
− sin θ J ψ ' cosθ J ψ '

−1
−1

sin θ
− cosθ 
 cosθ J ψ ' sin θ J ψ '
After matrix operation and simplifications, obtains
[K

[K
[ K ] =  K
[

 [ K
]3 × 3
3 2 ]3 × 3
4 2 ]3 × 3
5 2 ]3 × 3
22
[ K 2 3 ]3 × 3
[ K 3 3 ]3 × 3
[ K 4 3 ]3 × 3
[ K 5 3 ]3 × 3
[ K 2 4 ]3 × 3
[ K 3 4 ]3 × 3
[ K 4 4 ]3 × 3
[ K 5 4 ]3 × 3
[ K 2 5 ]3 × 3 
[ K 3 5 ]3 × 3 
[ K 4 5 ]3 × 3 

[ K 5 5 ]3 × 3  1 2 × 1 2
r 2π l
K
ij
22
=∫
∫
0 0
NB


−1
2
2
2
K yyBi ∆ ( x − xBi ) r .dθ dxdr
∫0 ks ( J ψ ') (C 66 sin θ + C 55 cos θ ) + ∑
i =1



∂ ψ i ∂ ψ j N B Bi
= ∫  k s ( A55 + A66 )
+ ∑ K yyψ iψ j ∆ ( x − x Bi ) dx
∂x ∂x
i =1

0 
l
r 2π l
∫∫
ij
K 23
=
0 0
l
=
NB
∫∑ K
0 i =1
ij
K 24
=
NB


−1
2
k
(
J
ψ
')
sin
θ
cos
θ
(
−
C
+
C
)
+
K yzBi ∆ ( x − x Bi ) r .d θ .dx .dr
66
55
∑
∫0  s
i =1

ψ iψ j ∆ ( x − x B i )d x
Bi
yz
r 2π l
∫ ∫ ∫ {k
s
}
( J −1ψ ')( − C 16 r sin 2 θ J −1ψ '− C 66 sin θ cos θ + C 55 sin θ cos θ ) r .d θ .dx.dr
0 0 0
l
=
1
∫ − 2K
0
s
A16
∂ψ i ∂ψ j
dx
∂x ∂x
NIT ROURKELA
Page 21
K
ij
25
r 2π l
∫ ∫ ∫ {k
=
s
}
( J −1ψ ')( C 16 r sin θ cos θ J − 1ψ '− C 66 sin 2 θ − C 55 cos 2 θ ) r .d θ .dx.dr
0 0 0
l
= ∫ − K s ( A55 + A66 )
0
∂ψ i
ψ j dx
∂x
r 2π l
K
ij
32
=∫
NB


−1
2
k
(
J
ψ
')
sin
θ
cos
θ
(
−
C
+
C
)
+
K yzBi ∆ ( x − x Bi ) r .dθ .dx.dr
66
55

∑
∫0  s
i =1

∫
0 0
l
NB
∫∑ K
=
0 i =1
ψ iψ j ∆ ( x − x B i )dx
Bi
yz
r 2π l
NB


−1
2
2
2
K = ∫ ∫ ∫ ks ( J ψ ') (C55 sin θ + C66 cos θ ) + ∑ KzzBi ∆( x − xBi )r.dθ dxdr
i =1

0 0 0
ij
33


∂ψ i ∂ψ j N B Bi
= ∫  k s ( A55 + A66 )
+ ∑ K zz ψ iψ j ∆ ( x − xBi ) dx
∂x ∂x
i =1

0 
l
r 2π l
{
}
l
∂ψi
ψ j dx
∂x
∂ψ i ∂ψ j
K34ij = ∫ ∫ ∫ ks (J −1ψ ')(C16r sinθ cosθ J −1ψ '+ C55 sin2 θ + C66 cos2 θ) r.dθdxdr = ∫ ks ( A55 + A66 )
0 0 0
r 2π l
0
{
}
l
1
K = ∫ ∫ ∫ ks ( J ψ ')(−C16 r cos θ J ψ '+ sin θ cos θ (C 66 − C 55 )) r.dθ .dx.dr = ∫ − ks A16
dx
2
∂x ∂x
0 0 0
0
ij
35
−1
−1
2
∂ψ ∂ψ j
1
K = ∫ ∫ ∫ ks ( J ψ ')(−C16r sin θ J ψ '+ sin θ cosθ (−C 66 + C 55 )) r.dθ .dx.dr = ∫ − ks A16 i
dx
2
∂x ∂x
0 0 0
0
r 2π l
{
−1
{
−1
ij
42
r 2π l
}
−1
2
l
}
l
K = ∫ ∫ ∫ ks ( J ψ ')(C16r sin θ cosθ J ψ '+ C 55 sin θ + C 66 cos θ ) r.dθ dxdr = ∫ ks ( A55 + A66 )
ij
43
−1
2
2
0 0 0
ij
K 44
=
0
 ( C 11 r sin θ ( J ψ ') + 2 k s C 16 r sin θ cos θ ( J ψ ') 

 r .d θ .dx.dr
∫0  + k C 66 cos 2 θ + k C 55 sin 2 θ )

 s
s
r 2π l
∫∫
0 0
∂ψ i
ψ j dx
∂x
2
2
−1
2
−1
l


∂ψ i ∂ψ j
= ∫  B11
+ k s ( A55 + A66 )ψ iψ j dx
∂x ∂ x

0 
2
−1
2
−1
2
2
( −C11r sin θ cos θ ( J ψ ') − k s C 16 r ( J ψ ')(cos θ − sin θ ) 
r.dθ .dx.dr
∫0 + k sin θ cos θ (C 66 − C 55 )
 s

r 2π l
ij
K 45
=∫
∫
0 0
∂ψ j 
1
∂ψ i
1
ψ j − ks A16
ψ i dx
= ∫  ks A16
∂x
∂x
2
2

0
l
r 2π l
K52ij = ∫ ∫ ∫ ks ( J −1ψ ')(C16 r sin θ cosθ J −1ψ '− C 66 sin 2 θ − C 55 cos2 θ )r.dθ .dx.dr
0 0 0
l
= ∫ −K s ( A55 + A66 )
0
NIT ROURKELA
∂ψ i
ψ j dx
∂x
Page 22
r 2π l
l
∂ψ ∂ψ j
1
K53ij = ∫ ∫ ∫ ks ( J −1ψ ')(−C16r cos2 θ J −1ψ '+ sin θ cosθ (C66 − C55 )) r.dθ .dx.dr = ∫ − ks A16 i
dx
2
∂x ∂x
0 0 0
0
{
}
( −C 11r 2 sin θ cos θ ( J −1ψ ') 2 + ks C 16 r ( J −1ψ ')(sin 2 θ − cos 2 θ ) 
K = ∫ ∫ ∫
r.dθ .dx.dr
θ
θ
k
sin
cos
(
C
C
)
+
−
66
55

0 0 0
 s

l
∂ψ j 
1
∂ψ
1
= ∫  ks A16 i ψ j − ks A16
ψ i dx
∂
x
∂
x
2
2


0
r 2π l
ij
54
K
ij
55
=
2
2
−1
2
−1
 (C 11 r cos θ ( J ψ ') − 2 k s C 16 rJ ψ ' sin θ cos θ 
r .d θ .dx.dr
∫0  + k C 66 sin 2 θ + k C 55 cos 2 θ
s
 s

r 2π l
∫∫
0 0
l
 ∂ψ i ∂ψ j

= ∫  B11
+ k s ( A55 + A66 )ψ iψ j dx
∂x ∂x

0
Damping Matrix of Bearing
[C 22 ]3×3

[C ]
B
 C e  =  32 3×3
 [0 ]
3× 3

0
[
]
3× 3

[C 23 ]3×3
[C33 ]3×3
[ 0 ]3×3
[ 0 ]3×3
xf
Now,
C
∫
=
ij
22
xi
C
=
∫
xi
xf
C 3ij2 =
∫
xi
xf
C
ij
33
=
[ 0 ]3×3 
[ 0 ]3×3 
[ 0 ]3×3 

[ 0 ]3×3 12×12
 N B Bi

 ∑ C yyψ iψ j ∆ ( x − x B i ) d x
 i =1

xf
ij
23
[ 0 ]3×3
[ 0 ]3×3
[ 0 ]3×3
[ 0 ]3×3
∫
xi
 N B Bi

 ∑ C y z ψ iψ j ∆ ( x − x B i ) d x
 i =1

 N B Bi

 ∑ C zy ψ iψ j ∆ ( x − x B i ) d x
 i =1

 N B Bi

 ∑ C zz ψ iψ j ∆ ( x − x B i ) d x
 i =1

Gyroscopic Matrix
[ 0]3×3

[ 0]
[Ge ] =  0 3×3
[ ]
 3×3
[ 0]3×3
NIT ROURKELA
[ 0]3×3 [ 0]3×3 [ 0]3×3 
[ 0]3×3 [ 0]3×3 [ 0]3×3 
[ 0]3×3 [ 0]3×3 [G45 ]3×3 

[ 0]3×3 [G54 ]3×3 [ 0]3×3 12×12
Page 23
In which
[G 4 5 ]3× 3
xf
∫
=
xf
I Pψ iψ j d x +
xi
ND
∫∑I
xi
i =1
xf
xf N
D
xi
xi
ψ iψ j ∆ ( x − x D i ) d x
D
P
[G54 ]3×3 = ∫ − I Pψ iψ j dx + ∫ ∑ − I PDψ iψ j ∆ ( x − x Di ) dx
i =1
Displacement Vector
{q } = {{v } {w } {β } {β } }
T
e
T
e
T
T
e
xe
ye
1×12
Force Vector
{F } = {{R } {R } {M } {M } }
e
e
T
y
NIT ROURKELA
e T
e
z
x
T
e
y
T
1×12
Page 24
CHAPTER 4
MODELLING OF ELECTROMAGNETIC ACTUATOR
4.1 Introduction
In principle, the actuator consists of a current driven coil placed between two
permanent magnets. Repellent forces are generated between the coil and the magnets,
centering the coil between the two magnets. The 2D finite element analyses are carried out to
predict the forces generated by this arrangement depending on coil current and coil position.
Force measurements are also made using the actual device.
Actuator forces as predicted by the finite element analyses are in excellent agreement
with the measured data, confirming the validity of the numerical model. Stiffness of the
actuator is defined as the increase of force per unit of coil displacement. Actuator stiffness
depends linearly on the coil current but in a nonlinear manner on the coil displacement. The
performance of the actuator is sufficient to demonstrate the effect of a so-called parametric
anti-resonance on a test stand.
4.2 The Electromagnetic Force
The magnetic force developed on the rotor by one pair of poles of the electromagnet.
The corresponding centres of the shaft at its nominal or reference position (when the shaft is
not vibrating) and the deflected position are OB and OC respectively. Assuming that (1) radial
air gap between rotor and stator poles is insignificant compared to the rotor radius R and there
is no fringing effect of magnetic flux lines near the pole-face, (2) flux leakage is negligible,
(3) all individual magnetic flux lines have equal length, (4) magnetic materials obey linear
relationship between flux density (B) and the magnetic-field-intensity (H) within the material,
i being the magnetic permeability [36].
Fmag ( t ) = -k M *
i (t )2
lg 2
(4.1)
Where,
kM =
µ0 Ap N 2
(4.2)
4
0, AP, and N are the absolute permeability of free air, pole face area, and number of coil turns
respectively. The negative sign indicates that electromagnetic force increases with decrease in
air gap. For nominal position of the rotor, uniform air gap go exists between the poles and the
rotor surface and a steady bias current of magnitude io passes through all the coils of the
exciters. Consequently the rotor is attracted equally by all the pairs of poles. At any given
instant during the vibration of the rotor, it comes closer to a particular pair of poles, and goes
NIT ROURKELA
Page 25
further from the opposite pair. The control force to oppose deflection of the shaft along any
direction (either Y or Z) is achieved by simultaneously stepping up the current in the coil
around the particular pair of poles, from which the rotor is farther, and stepping down the
current in the coil around the opposite pair, to which the rotor is nearer, by equal amount from
the steady bias current. The amount by which the current is stepped up or stepped down is
called the control current. The geometry has the advantage that the forces in the Y and Z
directions are (almost) uncoupled and can be calculated separately [36].
FYmag
 i + i 2  i − i 2 
= kmag  0 cY  −  0 cY  
 g 0 + y   g 0 − y  
(4.3)
FZmag
 i + i  2  i − i 2 
= kmag  0 cZ  −  0 cZ  
g + z   g0 − z  
 0

(4.4)
Due to an included angle of 2 between the poles, the magnetic exciter constant is:
kmag = kM cos(α )
4.2.1 Linearization of the Electromagnetic Force about operating Point
Above eg.(4) & eg.(5) are nonlinear relation of magnetic force. So we have to linearize
around an operating point. Let displacement Y=YOP and Z=ZOP and control current icY= iCyop
and icZ= icZOP along Y and Z directions [36]. Linearized expression of forces FY and FZ about
an operating point can be written as:
FY = kiicY + kY v
(4.5)
FZ = ki icZ + kZ w
Where, ki = 4kmag
(4.6)
i02
i0
k
=
k
=
k
=
−
4
k
;
v
Z
S
mag
g 03
g 02
Fig.4.1: (a) Stator configuration of actuator; (b) Arrangement of coils and poles of the
electromagnetic actuator [36]
NIT ROURKELA
Page 26
4.3 Governing equation of motion
Equations of motion of a general multiple-disc flexible rotor shaft- bearing system
is developed using finite element discretization technique. The flexible rotor shaft is modelled
with three-nodded Rayleigh beam finite with distributed inertia and stiffness properties. Rotor
discs are mounted on rotor-shaft and modelled as rigid, concentrated inertia elements. The
bearing is supposed to offer supporting forces to the rotor-shaft system which is considered to
be acting at a single node at the location of the bearing. The bearing forces are expressed in
terms of four stiffness and four damping coefficients associated with that particular bearing.
Equations of motion of such a rotor-shaft bearing system can be expressed as:
[ M ] {qɺɺ }
+ [ D ] {qɺ} + [ K ]{q} = { f } + { f EMA }
(4.7)
In the above equation,
[M]
[D]
[K]
{q}
{f}
{fEMA}
Inertia matrix of the rotor-shaft bearing system;
Matrix that takes into account of the gyroscopic effect, direct damping effect of
The internal damping of the shaft material bearing damping;
The shaft stiffness, bearing stiffness and circulatory effect of the internal material
damping of the shaft.
Nodal displacement vector for the entire system.
Global force vector due to mass-unbalance.
Force due to electromagnetic actuator.
4.4 The Proportional-Derivative Control Strategy
The proportional-derivative control action for each magnet by feeding back the rotor
displacement and its derivative with gains kp (V/m) and kv (V s/m) respectively using colocated displacement sensors [36].
Expression of the control currents are given by
I cY = −k g ( k P v + kv vɺ )
(4.8)
I cZ = −k g ( kP w + kv wɺ )
(4.9)
Where,
k g Is the power amplifier gain in (A/V). Putting these values in equations of currents in
equation (4.8) & (4.9). We get
FY = − ( k g ki kP − kS ) v − k g ki kv vɺ
FZ = − ( k g ki kP − kS ) w − k g ki kv wɺ
NIT ROURKELA
(4.10)
(4.11)
Page 27
(
)
So magnetic exciter may be modelled to have a combined stiffness of k g ki kP − kS and a
combined damping coefficient of k g ki kv .
4.5 Linear Quadratic Regulator (LQR)
The transfer function is transformed to a state space vector dynamic equation for state
feedback control system design. To minimize the displacement of the rotor shaft system, the
Linear Quadratic Regulator (LQR) based on independent mode space control techniques is
designed. The control voltage for the actuators is determined by the optimal control solution
of the Linear Quadratic Regulator (LQR), which is an effective and widely used linear control
technique. Provided the full state vector is observable, this method can be employed to meet
specific design and performance criteria. A quadratic cost function is used to minimize the
performance index.
State-space model can be written asXɺ = A * X + B *U (t )
(4.12)
Y = CX
(4.13)
LQR optimal control theory has been used to determine the control gains. In this, the
feedback control system has been designed to minimize a cost function or a performance
index, which is proportional to the required measure of the system’s response. The cost
function used in the present case is given by:
1 tf
T
T
(4.14)
J = ∫ { X } [Q ]{ X } + {U } [ R ]{U } dt
t
2 0
Where [Q] and[R] are the semi-positive-definite and positive-definite weighting matrices on
the outputs and control inputs, respectively.
The steady-state matrix Ricatti equation can be written as:
(
)
[ A] [ P ] + [ P][ A] − [ P][ B][ R] [ B][ P] + [Q] = 0
−1
T
(4.15)
After solving the recatti equation using potters method, optimal gain can be written as
Klqr = Kopt = [ R ]
−1
[ B] [ P]
T
(4.16)
The Matlab command in Equation (19) is used to compute the LQR gain matrix.
[ k , S , E ] = lqr ( A, B, Q, R)
Considering output feedback, actuation current can be written as:
v 
 w
 I CY 
 
 I  = [ − K ]lqr * vɺ 
 CZ 
 
 wɺ 
NIT ROURKELA
(4.17)
(4.18)
Page 28
CHAPTER 5
RESULTS AND DISCUSSIONS
Based on the above formulation, a complete MATLAB code has been developed and
validated with the available results in the literatures and Different analysis has been carried
out and presented in the following sub sections.
5.1 Summery of various Analyses
A composite shaft is supported on two bearings and a disk is rigidly mounted on the
centre of the shaft. The supporting bearings are modelled as springs and viscous dampers.
Vibration control of this rotor-shaft-bearing system due to unbalance forces has been studied
using one and two electromagnetic actuators with PD control strategy. The shaft is discretized
by 20 numbers of elements (it has been decided after convergence study). The total number of
nodes in this model is 41. The stiffness and damping coefficients of each bearing at the both
ends is considered as: KXX = 1.75 x107 N/m, KXY=0 N/m, KYY = 1.75 x107 N/m and CXX =500
Ns/m, CXY=500 Ns/m, CXY =500 Ns/m respectively.
5.2 Code Validation
In order to verify the FE developed code the following dimensions and mechanical properties
were considered for steel shaft [36] (details of which are given in Table 5.1). In order to
convergence study of the result, it has been observed that result from the present code has
been achieved an excellent agreement with the already published results [36] and thus
validates the correctness of the developed code. It is shown in Fig. 5.1.
Table 5.1 Mechanical Properties and Geometric dimension of Steel Rotor-shaft system
[36]
Parameter
Rotor shaft length (m)
Rotor shaft diameter (mm)
Young’s Modulus (Gpa)
Eccentricity (m)0.0002
Density (Kg/m3)
Outer diameter (m)
Thickness (mm)
Position from left end of rotor (m)
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Shaft
1.3
100
200
7800
0.24
5.0
0.2
Disk
7800
Page 29
Whirling Speed (rpm)
3
x 10
4
2.5
2
1.5
1
0.5
0
0
0.5
1
1.5
2
Rotating Speed (rpm)
2.5
3
4
x 10
Fig.5.1: Campbell diagram of steel shaft for first two pairs of modes
5.3 Uncontrolled and controlled responses of the various composite shafts
For validation of code different types of the stacking laminated structure has been
considered. Mechanical and geometrical dimension of all laminated composites are listed
[20], details of which are given in Table 5.2. Rotor shaft is assumed that rotating at 15000
RPM. A disk is placed at midpoint of the shaft i.e. at the node position no. 21. First of all one
actuator is placed at the node position no. 20 and response history is observed at the disk
position. After then two actuators is placed at the node position nos. 20 and 22 and response
history is observed at the disk position.
Table 5.2 Mechanical Properties and Geometric dimension of Composite shaft [20]
Parameter
Rotor shaft length, L (m)
shaft inner diameter, di (m)
shaft inner diameter, do (m)
Mean dia of shaft, D (m)
Longitudinal Young’s Modulus (Gpa)
Transverse Young’s Modulus (Gpa)
Density (Kg/m3)
Thickness of each composite layer (m)
Poisson’s ratio
Longitudinal shear modulus, (Gpa)
Transverse shear modulus, (Gpa)
Eccentricity (m)
Mass of disk in Kg
Diametrical inertia of disk in kg-m2
Polar inertia of disk in kg-m2
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0.72
0.028
0.048
0.1269
139x109
11x109
1578
0.001
0.313
6.05x109
3.78x109
Disk
5x10-5
2.4364
0.1901
0.3778
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After then, result for the controlled current and the controlled force have been developed and
compared with one and two actuator. The parameters for electromagnetic actuator [36] are
listed in the Table.5.3
Table.5.3 Parameters used for electromagnetic actuator and PD control techniques [36]
Parameters
N
Ap(cm2)
Kp (V/m)
Kv (V.s/m)
α (in degree)
Kg (A/V)
go (mm)
value
1000
5
5000
4000
22.5
2
2.5
In the following sub section result has been presented for the different stacking laminate.
5.3.1 Symmetric angle ply laminated shaft
Stacking sequence for symmetrical angle ply is considered as [45/-45/90/-90]2S.
Campbell diagram for the symmetrical angle ply rotor shaft system is shown in Fig. 5.2 and it
is found that first critical speed is 3500 r.p.m. The uncontrolled displacement history for the
symmetrical angle ply laminated shaft in the v-direction and in w-direction due to unbalance
forces is depicted in Fig. 5.3 and 5.4. The controlled displacement histories using one actuator
in v and w-directions are shows in Figs. 5.5 and 5.6 respectively. The controlled displacement
history using two actuators in v and w-directions are shows in Figs. 5.7 and 5.8 respectively.
Controlled currents for the present symmetrical angle ply laminated shaft using one actuator
in the direction of Y and Z shown in Fig 5.9 and 5.10 respectively. Controlled currents for the
present symmetrical angle ply laminated shaft using two actuators in the direction of Y and Z
shown in Fig 5.11 and 5.12 respectively. Controlled force for the present symmetrical angle
ply laminated shaft using one actuator in the direction of Y and Z shown in Fig 5.13 and 5.14
respectively. Controlled force for the present symmetrical angle ply laminated shaft using two
actuators in the direction of Y and Z shown in Fig 5.15 and 5.16 respectively.
It has been observed from the Fig.5.3, 5.5 and 5.7 that displacement in the direction of
V is reduced by 98.86 % using one actuator and by 99.58 % while using two actuators. Also it
has been found that from the Fig.5.4, 5.6 and 5.8 the displacement in the direction of W is
reduced by 99.36 % when using one actuator and by 99.74 % when using two actuators.
From Fig.5.9 and 5.11 it can be noticed that controlled current requirement in the
direction of Y for two actuators is reduced almost by 53.12 % of current requirement for one
actuator. From the Fig.5.10 and 5.12 the same phenomena is noticed while considering
controlled current requirement in the direction of Z for two actuator is reduced by 59 % of
current requirement for one actuator.
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From the Fig.5.13 and 5.15 it can be observed that controlled force requirement in the
Y direction is reduced by approx 66.66 % while using two actuators with respect to one
actuator. Also it can be observed that from the Fig.5.14 and 5.16 the controlled force
requirement in the direction of Z is reduced by 59.09 % while using two actuators with respect
to one actuator.
Fig. 5.2: Campbell diagram of symmetrical angle ply laminated shaft.
Fig. 5.3: Uncontrolled displacement history in the direction of ‘V’ for the symmetrical angle
ply laminated shaft.
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Fig. 5.4: Uncontrolled displacement history in the direction of ‘w’ for the symmetrical angle
ply laminated shaft.
Fig.5.5: Controlled displacement history in the direction of ‘V’ for the symmetrical angle ply
laminated shaft using one actuator.
Fig.5.6: Controlled displacement history in the direction of ‘W’ for the symmetrical angle ply
laminated shaft using one actuator.
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Fig.5.7: Controlled displacement history in the direction of ‘V’ for the symmetrical angle ply
laminated shaft using two actuators.
.
Fig.5.8: Controlled displacement history in the direction of ‘W’ for the Symmetrical angle ply
laminated shaft using two actuators.
Fig.5.9: Controlled current in the direction of ‘Y’ for the symmetrical angle ply laminated
shaft using one actuator.
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Fig.5.10: Controlled current in the direction of ‘Z’ for the symmetrical angle ply laminated
shaft using one actuator.
Fig.5.11: Controlled current in the direction of ‘Y’ for the symmetrical angle ply laminated
shaft using two actuators.
Fig.5.12: Controlled current in the direction of ‘Z’ for the symmetrical angle ply laminated
shaft using two actuators.
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Fig.5.13: Controlled Force in the direction of ‘Y’ for the symmetrical angle ply laminated
shaft using one actuator.
Fig.5.14: Controlled Force in the direction of ‘Z’ for the symmetrical angle ply laminated
shaft using one actuator.
Fig.5.15: Controlled Force in the direction of ‘Y’ for the symmetrical angle ply laminated
shaft using two actuators.
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Fig.5.16: Controlled Force in the direction of ‘Z’ for the symmetrical angle ply laminated
shaft using two actuators.
5.3.2 Symmetric cross ply laminated shaft
Stacking sequence for symmetrical angle ply is considered as [0/90/0/90]2S. In
symmetric cross ply laminates, plies are situated at 0o and 90°. Campbell diagram for the
symmetrical angle ply rotor shaft system is shown in fig. 5.17 and it is found that first critical
speed is 3000 r.p.m. The uncontrolled displacement history for the symmetrical angle ply
laminated shaft in the v-direction and in w-direction due to unbalance forces is depicted in
Fig. 5.18 and 5.19. The controlled displacement histories using one actuator in v and wdirections are shows in Figs. 5.20 and 5.21 respectively. The controlled displacement history
using two actuators in v and w-directions are shows in Figs. 5.22 and 5.23 respectively.
Controlled currents for the present symmetrical angle ply laminated shaft using one actuator
in the direction of Y and Z shown in Fig 5.24 and 5.25 respectively. Controlled currents for the
present symmetrical angle ply laminated shaft using two actuators in the direction of Y and Z
shown in Fig 5.26 and 5.27 respectively. Controlled force for the present symmetrical angle
ply laminated shaft using one actuator in the direction of Y and Z shown in Fig 5.28 and 5.29
respectively. Controlled force for the present symmetrical angle ply laminated shaft using two
actuators in the direction of Y and Z shown in Fig 5.30 and 5.31 respectively.
It can be found from the Fig5.18, 5.20 and 5.22 the displacement in the direction of V
is reduced by 97.5 % using one actuator and by 98.8 % while using two actuators. Also it has
been found that from the Fig.5.19, 5.21 and 5.23 the displacement in the direction of W is
reduced by 98.53 % when using one actuator and by 99.33 % when using two actuators.
From Fig.5.24 and 5.26 it is cleared that controlled current requirement in the
direction of Y reduced by 41.66 % while using two actuators with respect to one actuator.
From the Fig.5.25 and 5.27 also it can be observed that the controlled current requirement in
the direction of Z is reduced by 48.78 % while using two actuators with respect to one
actuator.
For the analysis of control force it can be observed from the Fig.5.28 and 5.30 the
controlled force requirement in the Y direction is reduced by approx 46.15 % while using two
actuators with respect to one actuator. Also it can be observed that from the Fig.5.29 and 5.31
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the controlled force requirement in the direction of Z is reduced by 40.90 % while using two
actuators with respect to one actuator.
Fig. 5.17: Campbell diagram of symmetrical cross ply laminated shaft.
Fig. 5.18: Uncontrolled displacement history in the direction of ‘V’ for the symmetrical cross
ply laminated shaft.
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Fig. 5.19: Uncontrolled displacement history in the direction of ‘W’ for the symmetrical cross
ply laminated shaft.
Fig. 5.20: Controlled displacement history in the direction of ‘V’ for the symmetrical cross ply
laminated shaft using one actuator.
Fig. 5.21: Controlled displacement history in the direction of ‘W’ for the symmetrical cross
ply laminated shaft using one actuator.
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Fig. 5.22: Controlled displacement history in the direction of ‘V’ for the symmetrical cross ply
laminated shaft using two actuators.
Fig. 5.23: Controlled displacement history in the direction of ‘W’ for the symmetrical cross
ply laminated shaft using two actuators.
Fig.5.24: Controlled current in the direction of ‘Y’ for the symmetrical cross ply laminated
shaft using one actuator.
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Fig.5.25: Controlled current in the direction of ‘Z’ for the symmetrical cross ply laminated
shaft using one actuator.
Fig.5.26: Controlled current in the direction of ‘Y’ for the symmetrical cross ply laminated
shaft using two actuators.
Fig.5.27: Controlled current in the direction of ‘Z’ for the symmetrical cross ply laminated
shaft using two actuators.
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Fig.5.28: Controlled Force in the direction of ‘Y’ for the symmetrical cross ply laminated shaft
using one actuator.
Fig.5.29: Controlled Force in the direction of ‘Z’ for the symmetrical cross ply laminated shaft
using one actuator.
Fig.5.30: Controlled Force in the direction of ‘Y’ for the symmetrical cross ply laminated shaft
using two actuators.
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Fig.5.31: Controlled Force in the direction of ‘Z’ for the symmetrical cross ply laminated shaft
using two actuators.
5.3.3 Anti-symmetric cross ply laminated shaft
Stacking sequence for anti-symmetrical cross ply is considered as [0/-90/0/90/0/90/0/90]2S. In anti-symmetric cross ply laminates, plies are situated at 0o and 90°. Let us
consider for 0o ply situated at a distance z above the reference plane, there is one ply 90o of
identical characteristics, made of the same material, same thickness below the reference plane.
Campbell diagram for the symmetrical angle ply rotor shaft system is shown in fig. 5.32 and it
is found that first critical speed is 300 r.p.m. The uncontrolled displacement history for the
symmetrical angle ply laminated shaft in the v-direction and in w-direction due to unbalance
forces is depicted in Fig. 5.33 and 5.34. The controlled displacement histories using one
actuator in v and w-directions are shows in Figs. 5.35 and 5.36 respectively. The controlled
displacement history using two actuators in v and w-directions are shows in Figs. 5.37 and
5.38 respectively. Controlled currents for the present symmetrical angle ply laminated shaft
using one actuator in the direction of Y and Z shown in Fig 5.39 and 5.40 respectively.
Controlled currents for the present symmetrical angle ply laminated shaft using two actuators
in the direction of Y and Z shown in Fig 5.41 and 5.42 respectively. Controlled force for the
present symmetrical angle ply laminated shaft using one actuator in the direction of Y and Z
shown in Fig 5.43 and 5.44 respectively. Controlled force for the present symmetrical angle
ply laminated shaft using two actuators in the direction of Y and Z shown in Fig 5.45 and 5.46
respectively.
It can be noticed from the Fig.5.33, 5.35 and 5.37 the displacement in the direction of
V is reduced by 97.4 % using one actuator and by 98.6 % while using two actuators. Also it
has been found that from the Fig.5.34, 5.36 and 5.38 the displacement in the direction of W is
reduced by 98.40 % when using one actuator and by 99.30 % when using two actuators.
From Fig.5.39 and 5.41 it is cleared that controlled current requirement in the
direction of Y reduced by 41.66 % while using two actuators with respect to one actuator.
From the Fig.5.40 and 5.42 also it can be observed that the controlled current requirement in
the direction of Z is reduced by 48.78 % while using two actuators with respect to one
actuator.
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For the analysis of control force it can be observed from the Fig.5.43 and 5.45 the
controlled force requirement in the Y direction is reduced by approx 46.15 % while using two
actuators with respect to one actuator. Also it can be observed that from the Fig.5.44 and 5.46
the controlled force requirement in the direction of Z is reduced by 40.90 % while using two
actuators with respect to one actuator.
Fig. 5.32: Campbell diagram of anti-symmetrical cross ply laminated shaft.
Fig. 5.33: Uncontrolled displacement history in the direction of ‘V’ for the anti-symmetrical
angle ply laminated shaft.
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Fig. 5.34: Uncontrolled displacement history in the direction of ‘W’ for the anti-symmetrical
angle ply laminated shaft.
Fig. 5.35: Controlled displacement history in the direction of ‘V’ for the anti-symmetrical
angle ply laminated shaft using one actuator.
Fig. 5.36: Controlled displacement history in the direction of ‘W’ for the anti-symmetrical
angle ply laminated shaft using one actuator.
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Fig. 5.37: Controlled displacement history in the direction of ‘V’ for the anti-symmetrical
angle ply laminated shaft using two actuators.
Fig. 5.38: Controlled displacement history in the direction of ‘W’ for the anti-symmetrical
angle ply laminated shaft using two actuators.
Fig.5.39: Controlled current in the direction of ‘Y’ for the anti-symmetrical cross ply
laminated shaft using one actuator.
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Fig.5.40: Controlled current in the direction of ‘W’ for the anti-symmetrical cross ply
laminated shaft using one actuator.
Fig.5.41: Controlled current in the direction of ‘Y’ for the anti-symmetrical cross ply
laminated shaft using two actuators.
Fig.5.42: Controlled current in the direction of ‘W’ for the anti-symmetrical cross ply
laminated shaft using two actuators.
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Fig.5.43: Controlled Force in the direction of ‘Y’ for the anti-symmetrical cross ply laminated
shaft using one actuator.
Fig.5.44: Controlled Force in the direction of ‘Z’ for the anti-symmetrical cross ply laminated
shaft using one actuator.
Fig.5.45: Controlled Force in the direction of ‘Y’ for the anti-symmetrical cross ply laminated
shaft using two actuators.
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Fig.5.46: Controlled Force in the direction of ‘Z’ for the anti-symmetrical cross ply laminated
shaft using two actuators.
5.3.4 Quasi-isotropic laminated shaft
Stacking sequence for quasi-isotropic ply is considered as [0/-45/45/90]2S. This is the
combination of specially and generally orthotropic plies.Campbell diagram for the
symmetrical angle ply rotor shaft system is shown in fig. 5.47 and it is found that first critical
speed is 3200 r.p.m. The uncontrolled displacement history for the symmetrical angle ply
laminated shaft in the v-direction and in w-direction due to unbalance forces is depicted in
Fig. 5.48 and 5.49. The controlled displacement histories using one actuator in v and wdirections are shows in Figs. 5.50 and 5.51 respectively. The controlled displacement history
using two actuators in v and w-directions are shows in Figs. 5.52 and 5.53 respectively.
Controlled currents for the present symmetrical angle ply laminated shaft using one actuator
in the direction of Y and Z shown in Fig 5.54 and 5.55 respectively. Controlled currents for the
present symmetrical angle ply laminated shaft using two actuators in the direction of Y and Z
shown in Fig 5.56 and 5.57 respectively. Controlled force for the present symmetrical angle
ply laminated shaft using one actuator in the direction of Y and Z shown in Fig 5.58 and 5.59
respectively. Controlled force for the present symmetrical angle ply laminated shaft using two
actuators in the direction of Y and Z shown in Fig 5.60 and 5.61respectively.
It can be noticed from the Fig.5.48, 5.50 and 5.52 the displacement in the direction of
V is reduced by 98.9 % using one actuator and by 99.55 % while using two actuators. Also it
has been found that from the Fig.5.49, 5.51 and 5.53 the displacement in the direction of W is
reduced by 99.38 % when using one actuator and by 99.76 % when using two actuators.
From Fig.5.54 and 5.56 it is cleared that controlled current requirement in the
direction of Y reduced by 30 % while using two actuators with respect to one actuator. From
the Fig.5.55 and 5.57 also it can be observed that the controlled current requirement in the
direction of Z is reduced by 50 % while using two actuators with respect to one actuator.
For the analysis of control force it can be observed from the Fig.5.58 and 5.60 the
controlled force requirement in the Y direction is reduced by approx 35.7 % while using two
actuators with respect to one actuator. Also it can be observed that from the Fig.5.59 and 5.61
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the controlled force requirement in the direction of Z is reduced by 50.58 % while using two
actuators with respect to one actuator.
Fig. 5.47: Campbell diagram of quasi-isotropic ply laminated shaft
Fig. 5.48: Uncontrolled displacement history in the direction of ‘V’ for the quasi-isotropic ply
laminated shaft.
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Fig. 5.49: Uncontrolled displacement history in the direction of ‘W’ for the quasi-isotropic ply
laminated shaft.
Fig. 5.50: Controlled displacement history in the direction of ‘V’ for the quasi-isotropic ply
laminated shaft using one actuator.
Fig. 5.51: Controlled displacement history in the direction of ‘W’ for the quasi-isotropic ply
laminated shaft using one actuator.
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Fig. 5.52: Controlled displacement history in the direction of ‘V’ for the quasi-isotropic ply
laminated shaft using two actuators.
Fig. 5.53: Controlled displacement history in the direction of ‘W’ for the quasi-isotropic ply
laminated shaft using two actuators.
Fig.5.54: Controlled current in the direction of ‘Y’ for the quasi-isotropic ply laminated shaft
using one actuator.
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Fig.5.55: Controlled current in the direction of ‘Z’ for the quasi-isotropic ply laminated shaft
using one actuator.
Fig.5.56: Controlled current in the direction of ‘Y’ for the quasi-isotropic ply laminated shaft
using two actuators.
Fig.5.57: Controlled current in the direction of ‘Z’ for the quasi-isotropic ply laminated shaft
using two actuators.
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Fig.5.58: Controlled Force in the direction of ‘Y’ for the quasi-isotropic ply laminated shaft
using one actuator.
Fig.5.59: Controlled Force in the direction of ‘Z’ for the quasi-isotropic ply laminated shaft
using one actuator.
Fig.5.60: Controlled Force in the direction of ‘Y’ for the quasi-isotropic ply laminated shaft
using two actuators.
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Fig.5.61: Controlled Force in the direction of ‘Z’ for the quasi-isotropic ply laminated shaft
using two actuators.
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CHAPTER 6
CONCLUSIONS AND SCOPE OF FURTHER WORK
This chapter presents few important observations based on the active vibration control
of composite rotor- shaft system using electromagnetic actuator, PD control scheme using
developed MATLAB code. Scope of further work in this direction has also been presented at
the end of this chapter
6.1 Conclusions
A three nodded beam element has been used for modeling and vibration analysis of rotating
composite shaft. First order shear deformation theory has been employed. The finite element
formulation is based on the Timoshenko beam theory. Various types of composite shafts have
been considered in order to study the effect of ply orientation on responses of the rotating
composite shaft. Active vibration control of the composite shaft has also been implemented
using PD control with electromagnetic actuators. The present work enables to arrive at the
following important conclusions:
• It has been found that from the Campbell diagram for the four laminated shaft the first
critical speed is maximum for the symmetric angle ply with respect to other three plies
and it is observed that 3600 r.p.m.
• It has been observed that from the four laminated shaft, the displacement (in both
directions) in case of symmetrical angle ply laminated shaft is more than other three.
• The maximum displacement in V direction is found that 1.2x10-4 m and in the W
direction is found that 1.9x10-4.
• It has also been observed that from the present analysis using two actuators the
reduction of response is more than that of single actuator.
• It is found that the reduction of response (in both directions) in case of symmetrical
angle ply is more than other thee using two actuators. And it is observed that
maximum response is 99.58%, 99.74% in V and W directions.
• Based on control current, maximum control occur using two actuators than that of
single actuator. And it has also been observed that maximum control occurs in case of
symmetrical angle ply laminated shaft than that of other three and it is found that
current requirement in Y and Z directions reduced by 53.12% and 59% respectively
using two actuators with respect to single actuator and
• In the same manner maximum control force occur using two actuators than that of
single actuator. And it is also observed that maximum control occur in case of
symmetrical angle ply laminated shaft than other three and it is found that force
requirement in Y and Z directions reduced by 66.66% and 59.09% respectively using
two actuators with respect to single actuator.
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6.2 Scope of future work
•
Optimization for optimal number of actuators and placement of actuators
•
Determination of optimal gain parameters
•
Determination of induced stress in the lamina/plies of the composite shaft and
•
Study of the delamination in the composite plies
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List of publication from the present work:
•
Sikandar Kumar, D. Koteswara Raoand, Tarapada Roy “Active Vibration Control
of Layered Composite Rotor Shaft System Using Electromagnetic Actuator” 1st
world conf. of WCFMAAE, IIT Delhi, 2nd February, 2013
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