Topics in nonlinear models for piezostructural systems

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Topics in nonlinear models for piezostructural systems
Zhang, Xiaoyan ( Dissertant )
Kurdila, Andrew ( Thesis advisor )
Niezrecki, Christopher ( Thesis advisor )
Cattafesta, Louis ( Reviewer )
Hager, William ( Reviewer )
Lind, Rick ( Reviewer )
Place of Publication:
Gainesville, Fla.
University of Florida
Publication Date:
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Subjects / Keywords:
Damping ( jstor )
Domain walls ( jstor )
Electric fields ( jstor )
Electric potential ( jstor )
Hysteresis ( jstor )
Mathematical variables ( jstor )
Modeling ( jstor )
Piezoelectric actuators ( jstor )
Stiffness ( jstor )
Vibration ( jstor )
Dissertations, Academic -- UF -- Mechanical and Aerospace Engineering
Mechanical and Aerospace Engineering thesis, Ph.D
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )


This dissertation includes two distinct topics: the frequency dependent hysteresis modeling of a stack actuator and averaging analysis of the state-switched piezoelectric structural systems. The PZT materials have widely been used for micro-positioning and precision control due to their special property of being able to deform after being applied voltage across it. However, the response of the PZT materials with respect to the driving force exhibits a hysteretic behavior, which considerably affects the resolution of positioning. This research aims to develop frequency-dependent nonlinear hysteresis models to predict the displacement response of a piezo stack in response to a driving force over a wide range of driving frequencies. Two hysteresis models are developed here: one includes only a nonlinear subsystem, the other one includes both a nonlinear subsystem and a linear subsystem. model validation shows that both model can give an accurate prediction of the stack position. The cascaded nonlinear-linear model can give more accurate and smoother prediction. The other topic is related to using PZT materials to achieve vibration absorption using state switch control strategy. Specifically, an averaging analysis was developed for qualitative and quantitative study of switched piezostructural systems. It is shown that averaging analysis provides a means of determining time domain as well as frequency domain response characteristics of switched piezostructural systems that include switched capacitive shunt circuits. The proposed switching architecture provides an essentially continuous range of tunable notch frequencies, in contrast to a finite and fixed collection of discrete notch frequencies available in some implementations of capacitively shunted piezostructures. ( , )
averaging, capacitance, duty, frequency, hysteresis, KP, nonlinear, notch, piezostructural, shunt, vibration
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Thesis (Ph.D.)--University of Florida, 2005.
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Full Text






Copyright 2005


Xiaoyan Zhang


I would like to thank my advisor, Dr. Andrew J. K~urdila, and Dr. lo

Niezrecki for their patience in teaching me the subject of dynamics and controls,

their effort in introducing me to this professional field, and their help in guiding

my research. I have learned a great deal front them, and I look forward to applying

this knowledge in my career.

I would also like to thank Dr. Rick Lind, Dr. Louis Cattafesta and Dr.

William Hager for taking the time and effort to serve on my coninittee. Addi-

tionally, I would like to thank my friends for their guidance and advice as I have

pursued my graduate studies. Finally, I would like to thank my husband and my

parents for their inineasurable advice, encouragement, and support.


ACK(NOWLEDGMENTS ......... .. iii

LIST OF TABLES ......... .. .. vii

LIST OF FIGURES ......... . .. viii

ABSTRACT ...... ...... ........... xi


1 INTRODUJCTION . ...... ... .. 1

1.1 Background ...... .. .... ........ 1
1.2 Constitutive Equation for Linear Piezoelectricity .. .. .. .. 4
1.3 Problem Statement and Motivation for Research .. .. .. .. 5
1.3.1 Application to Micropositioningf .. .. .. .. 5
1.3.2 Application in Vibration Suppression .. .. .. 6
1.4 Proposed Work and Contributions .. ... .. 6
1.4.1 Fr-equency Dependent Hysteresis Modeling .. .. .. .. 7
1.4.2 Averaging Analysis on the Senli-actively Tuned Vibration
System ........ ....... 9
1.5 Dissertation Outline ........ .. .. 9


2.1 Hysteresis Modeling and Literature Review .. .. .. 11
2.1.1 Early Theories about Hysteresis ... ... .. 1:3
2.1.2 Models to Simulate Hysteresis ... .. 14
2.1.3 Static Modeling of Hysteresis ... . .. 15
2.1.4 Rate-Dependent Hysteresis Models ... .. .. 22
2.2 Vibration Absorbers and Literature Review .. .. .. 24
2.2.1 Vibration Absorbers ..... .. 24
2.2.2 Vibration Absorber Literature Review .. .. .. 28
2.2.3 Switched Shunted Designs .... .. :32
2.2.4 Averaging Analysis ...... ... :37


:3.1 Linear Constitutive Equations of PZT Materials .. .. .. 4:3
:3.2 Nonlinear Presentation of Constitutive Equations .. .. .. .. 44
:3.3 K~rasnolselskii-Pokrovskii K~ernel Model ... ... .. 45

3.3.1 Fundamentals of the K(P kernel Method .. .. .. .. 45
3.3.2 Discretization of the Preisach Plane ... .. .. 46
3.3.3 Recursive Definition of Hysteresis .. .. .. 48
3.4 Developing the Fr-equency Dependent Model .. .. .. 50


4. 1 Structure of the Stack Actuator .... .. .. 52
4.2 1-D Constitutive Equation . ..... .. 53
4.3 Governing Equation of the Stack Actuator ... .. .. 55
4.4 Strong Form of Equation of Motion ... .. .. 62
4.5 Weak Form of Equation of Motion ... . .. 63
4.6 Model Well-Posedness . ... .. 64
4.7 Finite Element Approximation of Model .. .. .. 66
4.7.1 Construct [M~], [K] and [B(Q)] (t) .. .. . .. 67
4.7.2 Reduced Order To Approximate The Equation of Motion 70
4.7.3 Trapezoidal Rule to Solve the Equation .. .. .. 71
4.8 The Linear Model and Natural Frequency ... .. .. .. 72


5.1 Experiment Objectives Description ... .. .. 75
5.2 Experimental Setup .. .. .. 75
5.2.1 Equipment Used in the Experiment ... .. .. 75
5.2.2 Experiment Setup . .... .. 77
5.2.3 Sampling Fr-equency ...... .... 79
5.3 Experimental Results . ...... .. 79

TERESIS ......... ... 85

6.1 Quantifying the Hysteresis Function ... .. .. .. 86
6.1.1 Quantization of the S- Plane .. .. .. .. 86
6.2 The Nonlinear Frequeai~ s--dependent Hysteresis Model .. .. .. 88
6.2.1 Problem Formulation ...... .. 88
6.2.2 Convergence(l CI. II .:terization ... .. 89
6.2.3 Weight Identification Results ... . .. 90
6.2.4 Model Validation ... .. .. .. . . 91
6.3 Nonlinear-Linear Frequency Dependent Hysteresis Model .. .. 93
6.3.1 Problem Formulation and Convergence Characterization .93
6.3.2 Identification of the Weights .... .. 95
6.3.3 Summation of the Weighted K(P K~ernels .. .. .. .. 96
6.3.4 Ripples On The Optimization Curve .. .. .. 98
6.3.5 Model Validation . . .. 99
6.4 Trust Region Method to Adjust the Weights .. .. .. 99
6.5 Comparison of the two models ... .. .. .. 102


7. 1 Modeling of the Piezoceramic Vibration Absorption System .. 104
7.2 Shunted Capacitor Load Model .... .... .. 107
7.2.1 State-space Model for Zero-IC .. .. .. .. 108
7.2.2 State-Space Model for Non-Zero IC .. .. .. .. 112
7.3 Control Strategy of State-Switch ... . .. 112
7.4 Averaging All ll--;-; ...... ... .. 114
7.5 Averaging the Switched System ... .. .. 118
7.5.1 Ideal Switch .. . .. .. 118
7.5.2 Averaging the Vibration System with Shunted Capacitive
Loads ...... ... .. 118
7.5.3 Fr-equency Domain Results .... .. .. 123

8 SUMMARY ............ .......... 128

8.1 Free noi is li-dependent Hysteresis Model .. .. .. .. 128
8.2 Averaging Al ll--is on the State-Switched Piezostructural System 129

APP END IX ......... ... .. 130


A. 1 Construction of Atom and Axiomatic Definition of Polarization .130
A.2 Magnetization ......... .. .. 130


STIFFNESS SPRINGS ........ ... .. 134

D GRONWALL LEMMA ....... ... .. 135

REFERENCES ........ . .. 136

BIOGRAPHICAL SK(ETCH ......... .. .. 142

Table paget

1-1 Relation of subscripts ......... ... 5

1-2 Symbols and descriptions ........ .. .. 5

3-1 Symbols and descriptions ........ .. .. 44

4-1 Natural frequencies front analytical model and FE1\ model .. .. 74

5-1 Parameter list of the stack actuator ..... .. 75

7-1 Parameter values used in the simulation .. . .. 127


1-1 Sensor effect and actuator effect ......

1-2 Polarization process ......

2-1 Atomic force microscopy .....

2-2 A typical piezoelectric hysteresis loop ......

2-3 Orientation of dipole moment and electrical field .....

2-4 Singfle DOF vibration ......

























Classical vibration absorber .....

T.F. of displacement-force of mass al ....

T.F. of displacement-force of mass al ....

Active vibration absorber configuration .....

Dual mass configuration .....

Senli-active VA configuration ......

Passive shunted circuit (used by Hagood and vor

Passive shunted circuit (used by Wu) .....

Passive shunted circuit .....

State switch (a) Open-short circuit, (b) Open-res

Schematic of 1-D vibration system with PZT ..

A typical kernel in Preisach model ......

Discretization of the Preisach plane .....

Preisach kernels ......

Ridgfe function ......

?I i5 r loop and minor loop ......

The stack actuator .....

n Flotow)


circuit ...

4-2 Structure of the stacked actuator ...... .. 54

4-3 Axial piezoelectric rod ......... .. 56

4-4 FEM shape functions ......... .. 67

4-5 Shape function in one element . ..... .. 68

5-1 Voltage divider ......... . 77

5-2 Experimental setup ......... .. 78

5-3 Flow chart of experimental setup ...... .. 78

5-4 Driving voltage vs. time ........ .. .. 79

5-5 Velocity response of the stack actuator ... .. .. 80

5-6 Displacement response of the stack actuator .. .. .. 82

5-7 Relative displacement of the stack actuator-I ... .. .. 83

5-8 Relative displacement of the stack actuator-II ... .. .. 83

5-9 Relative displacement of the stack actuator-III .. .. .. 84

6-1 Freqe I-i i l--dependent hysteresis model .. .. .. 85

6-2 S- plane ......... . 87

6-3 K(P kernels .. ... . .. 88

6-4 Flow chart for weight identification .... .. .. 90

6-5 Weight distribution for selected frequencies .. .. .. 91

6-6 Reconstruct the displacement hysteresis .... .. .. 91

6-7 Weight identification for selected frequencies .. .. .. 92

6-8 Model prediction for training frequencies .... .. .. 93

6-9 Model prediction for desired frequencies .... .. .. 94

6-10 Weight identification for cascaded model .... .. .. 95

6-11 Weight distribution over the S- plane .... ... .. 96

6-12 Calculation of the reconstructed curve .... .. .. .. 97

6-13 Reconstructed data vs. experimental data ... ... .. 97

6-14 Ripples in the reconstructed curves ..... ... .. 98

6-15 Fr-equency components of the ripples .... .. .. 98

6-16 Model prediction for training frequencies ... ... .. 100

6-17 Model prediction for desired frequencies ... . .. 101

6-18 Comparison of the two models ...... .... .. 10:3

7-1 Idealized piezoceramic vibration absorber ... .. .. .. 104

7-2 FBD of absorber mass and structure mass ... .. .. 105

7-3 Vibration suppression with capacitor load ... .. .. .. 107

7-4 Structure mass response -changing actuator mass .. .. .. .. .. 110

7-5 Structure mass response -changing shunted capacitance .. .. .. 111

7-6 Comparison of the transfer function .... .... .. 11:3

7-7 Switched signal ......... .. .. 114

7-8Ideal switch ........ . .. 118

7-9 Switched signal ........ .. .. 119

7-10 Transfer function of the averaged system .. .. .. 125

7-11 Comparison of the two desigfns:discrete and continuous notch frequen-
cies. ............ ........... 126

A-1 Dipole moment ......... .. .. 1:30

B-1 Response of the 1-D vibration system .... .. .. 13:3

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy



Xiaoyan Zhang

August 2005

C'I I!1-: Andrew J. K~urdila
Cochair: Ch1~!I s-1.l. 1- Niezrecki
Il ini- Department: Mechanical and Aerospace Engineering

This dissertation includes two distinct topics: the frequency dependent

hysteresis modeling of a stack actuator and averaging analysis of the state-switched

piezoelectric structural systems.

The PZT materials have widely been used for micro-positioning and precision

control due to their special property of being able to deform after being applied

voltage across it. However, the response of the PZT materials with respect to

the driving force exhibits a hysteretic behavior, which considerably affects the

resolution of positioning. This research aims to develop free.ii~ Iv s -dependent

nonlinear hysteresis models to predict the displacement response of a piezo stack in

response to a driving force over a wide range of driving frequencies. Two hysteresis

models are developed here: one includes only a nonlinear subsystem, the other

one includes both a nonlinear subsystem and a linear subsystem. model validation

shows that both model can give an accurate prediction of the stack position. The

cascaded nonlinear-linear model can give more accurate and smoother prediction.

The other topic is related to using PZT materials to achieve vibration ab-

sorption using state switch control strategy. Specifically, an averaging analysis

was developed for qualitative and quantitative study of switched piezostructural

systems. It is shown that averaging an~ ll-k- provides a means of determining time

domain as well as frequency domain response characteristics of switched piezostruc-

tural systems that include switched capacitive shunt circuits. The proposed

switching architecture provides an essentially continuous range of tunable notch

frequencies, in contrast to a finite and fixed collection of discrete notch frequencies

available in some implementations of capacitively shunted piezostructures.


Within this chapter, piezoelectricity is briefly reviewed along with some cur-

rent technical applications of piezoelectric actuation. These examples provide the

motivation for this research. The proposed work and its contribution to engineer-

ing are also described. Lastly, an overview of the remainder of the dissertation is


1.1 Background

Since the discovery of piezoelectricity by Jacques and Pierre Curie over

100 years ago and the theoretical quantification of the relation between the

piezoelectricity and ( i-- II structure hv Woldermar Voigt in 1894, piezoceramic

materials have been widely used in lah instruments for micropositioning, sonar

devices, communication devices, etc. Some of their advantages include small size,

high actuation energy density, relatively low power consumption, high resolution,

and the capability of being used both as an actuator and sensor.

A piezoelectric ceramic is a material that demonstrates the piezoelectric effect

and the inverse piezoelectric effect. When a mechanical force is applied across a

piezoceramic element, electrical charges are generated. If electrodes are placed on

both sides of the material, a voltage is generated. Therefore, mechanical energy is

transformed into electrical energy. This is called the piezoelectric effect, or sensor

effect. Conversely, if an electrical field is applied across the material, a mechanical

deformation results. Thus, electrical energy is transformed into mechanical

energy. This is called the inverse piezoelectric effect, or actuator effect. Therefore,

piezoceramic materials can act as either a sensing element, or an actuation element,

or both. These effects are described by Figure 1-1.

Sensor effect

SMechanical Electrical
Energy Energy

Actuator effect

Figure 1-1: Sensor effect and actuator effect

The piezoelectric effect occurs naturally in quartz crystals but with very

low energy transferring efficiency. It can also be induced in other piezoelectric

materials such as lead zirconate titanate(PZT). Such materials usually have higher

energy transferring efficiency, so they are used more often in practice. Because

they are ceramics, they can be manufactured into virtually any wanted shape. To

induce the piezoelectric effect of the material, the piezoceramic materials are first

manufactured into the desired shape. Then electrodes are put on opposite sides of

the materials and a strong DC electric field is applied to the materials so that the

molecular dipoles in the ceramics are aligned along the direction of the electrical

field. This process is called polarization. After the electrical field is removed, the

materials exhibit piezoelectric properties. The material has the interesting property

that the dipoles remain, to a large extent, aligned. The polarized material, whose

polarization direction, aligned with the electrical field when being polarized, is

called "polar." Please refer to Appendix A for information about dipole moment

and polarization. Figure 1-2 shows the polarization process in a microscopic

view [1, 2].

To identify the directions in a piezoceramic element, according to the IEEE

standard on piezoelectricity, three axes labeled as "1," "2" and "3" are used,

associated with the x, y, and z axes in a classical three dimensional orthogonal

coordinate system. The polar direction is labeled as "3" and is parallel to the

direction of polarization within the ceramic.


Piezoceramic material, not Material is put in a strong After the electrical field is
poled. The dipoles exist in electrical field, the dipoles removed, the material is a
any directions. But the net are aligned in the electrical polar. The dipoles direction
effect is zero, direction, can vary away from the aligned
E: input electrical field direction, but the net effect is
along the aligned direction.

Figure 1-2: Polarization process

The piezoelectric constants relating the mechanical strain and the applied

electric field are termed the electromechanical coupling constants. By convention,

they are represented by the "d" coefficients. This is an important coefficient in the

positioning mechanism. For the piezoelectric effect, dij is defined as the ratio of the

electrical charge collected on the electrodes and the applied mechanical stress.

Short circuit (1,,, .I density7
is-Aplied mechanical stress

where i = 1, 2 6i andl j = 1, 2 -. Units ar~e often explressed as cmam et~

Conversely, in the inverse piezoelectric effect, dij is defined as how much deforma-

tion the material will generate when an electrical field is applied.

Strain developed
is-Aplied electrical field

Ther units of, this coelffil-cint are often e:xprecssed as neze eT arge: day coinst~ants

indicate large mechanical displacements which are usually favorable in motion

transducer devices. "(d33") and "d31" are the two constants often used in a position-

ing mechanism in response to the input voltage. The "(d33) motion means when an

input voltage is applied along the polar (or 3) axis, motion is produced in the polar

direction. Thus we ;?i that the actuator works in the "(d33 mode" or longitudinall

mode." When an input voltage is applied in the polar direction and the motion

we take advantage of is in the "1" direction, we ;?i the actuator works in the "(d31

mode" or ii ll!-i..~1--- mode" or "in-plane mode."

1.2 Constitutive Equation for Linear Piezoelectricity

The constitutive equations for linear piezoelectricity are available from many

sources, and they can take via Ilny different forms. For example, two forms of such

constitutive equations, which will be often referred to in this dissertation, from

Richard Holland and E. P. EerNisse [3], are

Eij = SE, ~,"kl dmijEm
Dn = dukl~kl + mnEm


aij =(E krl emijEm
Dn = enklEkl + nmEm

The equation is written in the i I .1I ::/vector" summation form. For an isotropic

media, the tensors have the following relations:

St kl =~.3 Ski 'l=SE

The constitutive equation can be reexpressed as

D, = d, p + Em Em


D, = el2pEsL + k~mEm

where, p, v = 1, 2 6, m, a = 1, 2, 3. The relation between p, v, and m, a is

described in Table 1-1. The meanings of the symbols are described in Table 1-2.

Table 1-1: Relation of subscripts

11 1 12 =21 4
22 2 1:3 =31 5

Table 1-2: Symbols and descriptions

symbols meanings
E Strain
o- Stress
E Electrical field
D Electric charge displacement
SE Adiahatic compliance at constant electric field
d Piezoelectric coupling tensor
e" The permittivity at constant stress
~E The permittivity at constant strain
PE Elastic coefficient under constant electrical filed

1.3 Problem Statement and Motivation for Research

Since piezoceramic elements can generate a very small deformation under

applied voltage, they can he applied to problems involving precision control and

high-resolution positioning. For example, the actuators can provide a positioning

resolution of up to several nanometers. Also, it can generate voltage in response

to the applied mechanical force. This characteristic makes piezoelectric actuators

a good candidate for vibration suppression of structures, machines and vehicles.

The result of the research in this dissertation provides a theoretical foundation for

several applications that include micropositioningf, precision design and vibration


1.3.1 Application to Micropositioning

In the application of micropositioning and precision control, the piezoceramic

transducers are typically used to position a mechanical component. At lower drive

levels, the piezoceramic materials exhibit nearly linear dynamics and negligible

hysteresis. Thus, a linear model is sufficient to study the dynamic behavior.

However, the materials are usually driven at moderate to high levels. At these

drive levels, due to their ferroelectric nature, piezoceramic materials show a

non-linear hysteretic behavior. Most smart materials exhibit hysteresis in the

relationship between the input electrical field and the output response, such as

velocity and displacement. Also it has been shown that the hysteresis response

of the materials is frequeai~ s -i-dependent. To provide a framework amenable to

characterize these relationships, one topic of current research in this dissertation is

related to modeling the rate-dependent hysteresis of a piezoelectric(PZT) stacked

actuator. A literature review of hysteresis modeling is presented in Section 2.1 of

1.3.2 Application in Vibration Suppression

When a PZT actuator is attached to a vibrating structure, the PZT is strained

because of the structural vibration, thus a voltage is generated. The generated

electrical energy can be dissipated through a shunt circuit and thus the vibration

of the structure is suppressed. M1 I.ny methodologies have been proposed for the use

of piezoelectric devices in motion control. A recently introduced methodology relies

on switching strategies to achieve vibration control. This dissertation provides a

theoretical foundation for some of these techniques. The shunt circuits, control

strategies and literature review relating to vibration absorption will be presented in

detail in Section 2.2 of C'!s Ilter 2.

1.4 Proposed Work and Contributions

The research in this dissertation is related to the two distinct topics that were

introduced in Section 1.3. This dissertation presents the necessary theoretical

foundations and derivations, numerical simulations and experimental results. Thus,

this work provides a contribution to the field of engineering in two primary r- :--s,

1. A frequency dependent hysteresis model is created to predict the displace-

ment of a PZT actuator over a wider range of frequency than currently exists. The

model can be used to improve the control performance of these actuators.

2. An averaging method is applied to a discontinuous state switched system to

obtain the effective response of some state-switched piezostructural systems. The

averaged system can provide a continuously tunable notch frequency.

1.4.1 Frequency Dependent Hysteresis Modeling

Contribution of this portion of the research is related to the applications

of the PZT materials in micropositioningf and precision control. One example

is the atomic force Microscope(AFM) positioning system. The response of the

PZT materials with respect to the input exhibits a hysteretic behavior, which

considerably affects the resolution of the positioning. There are some linear

models or static hysteresis models to characterize the behavior of the piezoceramic

materials have been derived. However, my experiment shows that the hysteresis

loop of the piezoceramic materials in response of an input voltage is frequency

dependent. There are very few frequency dependent hysteresis models, but those

models are good at extremely low frequency range. The frequency dependency

in hysteresis in piezoelectric actuators has not been studied extensively within

the controls literature. The objective of this research is to develop a frequency

dependent hysteresis model based on the elementary hysteresis operators. The

creation of such a model will allow us to predict the hysteresis displacement

response for the PZT actuators over a wider range of frequencies than currently


Two frequeai~ s -i-dependent hysteresis models are presented here. One model

just includes a nonlinear system, the other model includes a cascaded nonlinear-

linear model. For the first model, the displacement x of the stack in response to

the driving force is simply expressed as

i= 1

where, k-, 4(u () i s t he K~rasnol sels ki i-Pokrovs ki i (KP) kernel defined on t he S -

plane, which is generated by the driving voltage, which has the hysteresis property.

The weighting function I, ( v) is freenono,li- -dependent. In the second model, the

following tasks are performed:

1. A nonlinear constitutive equation is developed utilizing the concept of the
effective electrical field. And thus the nonlinear hysteresis subsystem can be
developed based on this nonlinear constitutive equation. The K(P kernels are
used as basis to quantify the nonlinear hysteresis quantity.
2. The dynamical governing equation of motion for the stack is derived in
C'!s Ilter 3 and 4. The governing equation of the stack is as follows:

[M~] -(t) + [C] -(t) + [K] c(t) = [B(Q)] (t) (1.6)

This is the linear subsystem of the hysteresis model.
3. Two approximation methodologies are derived.
a. Creating a finite element model of the governing equation, including
conventional elastic effects and hysteresis effects.
b. Expressing the control influence operator [B(Q)] (t) as

i= 1

where, I, is the value of discrete a pair over the S- plane and is free a 0. T-l-
dep endent .
4. A freenono,-~ Ii-dependent hysteresis model is created. This includes two ill i.r~
a. System Identification. Identify the weights via the least square method
from experimental data. The superposition of weighted kernels approximates
the experimental data in a least square sense.
b. Model Creation. The displacement of the stack for a desired frequency is
predicted using the identified weights.
5. The performance of the model is validated by using the empirical data. The
model validation shows that this model can be used to improve the control
performance of the stack actuator. It can improve the current positioning
error by an order of magnitude.

1.4.2 Averaging Analysis on the Semi-actively Tuned Vibration System

Contribution in this portion of the research is directed to reducing vibration

within a structure through dissipation using the PZT materials. This design is

motivated by Davis&Lesieutre [4]. In their design, an actively tuned vibration

absorber was developed. It applied on a vibration absorber system with a "ladder"

capacitive shunt circuit. Each switch can provide one notch frequency. Thus the

notch frequency is finite and discrete within the bandwidth, which is determined

by the short and open circuit. Our design still uses the "ladder" capacitive shunt

circuit. By controlling the switch, the state space model of the system is governed

by an N linear time invariant subsystems. Averaging analysis is applied to the state

space model of the system to find a continuous approximation to it. The resulting

system can provide a continuously tunable notch frequency range in one switch.

It is an effective design for a tunable vibration absorber. The analysis provides a

design procedure for such systems.

1.5 Dissertation Outline

Within OsI Ilpter 2 a comprehensive literature review pertaining to hysteresis

modeling and tunable vibration absorption is presented. Specifically, the hysteresis

generating mechanisms and previous work performed on the hvsteresis modeling

within the controls literature are reviewed. Models for tunable vibration absorbers

and new techniques in vibration absorption utilizing piezoceramic materials are

investigated. General analysis about modeling and averaging on state space models

for periodic switching control signals is also discussed.

The theoretical foundation about the Preisach method is reviewed in (I Ilpter

3. This chapter establishes the nonlinear constitutive equation of PZT materials

and presents the theoretical foundation for the current research. The Preisach

method to quantify the nonlinearity of the piezoceramic materials via an effective

electrical field is described.

C'!s Ilter 4 provides the theoretical derivation of the motion of the PZT stacked

actuator. The strong form and weak form of the governing equation are derived.

Analytical and numerical solutions are discussed. A finite element approach and

reduced order approximation will be applied to solve the governing equation.

Chapter 5 explains the experimental objectives. It also shows the experimental

set up and results.

In C'!s Ilter 6, the frequency dependent hysteresis model is developed based

on the theoretical derivation presented in C'!s Ilters 2, 3 and 4 along with the

experimental results that are provided in ('! .pter 5. Within this chapter, the

model is validated.

The governing equation of a 2-DOF vibration system, which employs a PZT

patch to suppress the vibration with a shunted switched circuit is derived in

C'!s Ilter 7. The averaging method is applied on the system. The performance of

the averaged systems is compared to the Davis&Lesieutre design.

Finally, within ('! .pter 8 a summary of the current research is presented.


This chapter reviews the related literature pertaining to hysteresis modeling

(Section 2.1) and tunable vibration absorbers (Section 2.2).

Hysteresis modeling is related to the application of piezocerantic materials for

micropositioning and precision control. In Section 2.1, static models via domain

wall method and Preisach method are first reviewed. Then several rate-dependent

models to simulate hysteresis are presented.

Piezocerantic materials have been recently attached to structures for the

purpose of vibration absorption. The state switch control strategy, a method of

senli-active control, combines the advantages of the passive and active vibration

absorption. However, this control strategy introduces a discontinuity to the system.

This is a disadvantage in analysis of the system properties. An averaging method is

used to solve this disadvantage.

2.1 Hysteresis Modeling and Literature Review

When a voltage is applied to a piezocerantic material a deformation will be

produced. But when the voltage is removed, it often does not return to its original

positions. This phenomenon is called hysteresis. Ovelic electrical loading and un-

loading generates a hysteresis loop between the input voltage and the displacement

response. To better understand this phenomenon, we take measurement front an

atomic force microscopy (AFM) as an example.

On application in which hysteresis is important is in measurements taken

with an atomic force microscope (AFM). The AFM requires precision positioning.

Figure 2-1 shows an AFM. In this application, the sample is first moved along an

x-y grid using a lateral positioning mechanism. The AFM feels the surface of the


Photodiode Cantilever with tip


3-D Scanner
Stack Actuator

Figure 2-1: Atomic force microscopy

sample, displacement in the cantilever tip is monitored using the photodiode and

corresponding forces are determined via Hooke's law. The sample is then displaced

in the Z-direction using a vertical positioning mechanism to maintain constant

forces with the displacement determined by a feedback law. In this way, the AFM

makes a topographic map of the sample surface [5].

Hysteresis loop for 160 Hz


-20 -10 0 40 Voltage(V) 140

Figure 2-2: A typical piezoelectric hysteresis loop

The accuracy with which the PZT element can position the sample is a key

factor to the resolution of the AFM. But the hysteresis behavior can make it

difficult to accurately determine the position of the stack. Figure 2-2 shows the

displacement hysteresis loop of a typical stack actuator after voltages are applied

front OV to 140V, then the voltage decreases to -10V, and then back to OV.

We can see that at 40V, the position of the actuator can he either at point "A"

when voltage increases, or at point "B" when voltage decreases. We ;? w that the

displacement of the actuator "1 .,,- behind the voltage as it oscillates. Unlike the

hardening of other materials such as in steel and copper, this displacement "1 I,,

is repeatable many times as the voltage increases and decreases. The hysteretic

loop of the actuator repeats itself essentially for the effective life of the actuator.

Hysteresis considerably degrades the actuator performance. To utilize the full

capability of piezocerantic actuators, nonlinear hysteresis needs to be investigated

and modeled.

2.1.1 Early Theories about Hysteresis

Early models and theories about piezocerantic materials have been based on

linear constitutive relations. Still, the hysteretic behavior has been known for a

long time. The hysteresis loop in piezocerantic materials was first demonstrated

by Tehhle and Craik [6] in 1969, which is the response that occurs frequently in

practice. Since then, many scientists and researchers have come up with many

hypotheses and models to explain this phenomenon and describe it niathentatically.

Much of the earlier research was based on models front magnetic hysteresis.

The early theories about the origin of hysteresis fell into two primary cate-

gories. One hypothesis is that hysteresis occurs due to a frictional type force. The

other hypothesis ell---- -1h I1 that hysteresis occurs due to strong mutual interactions

between the individual magnetic moments.

The idea of a frictional force originated with Weber in 1852, and was termed

a "restoringf force." He -II_t---- -1.. that there exists a restoring force which inhibits

the change of the magfnetization state. The restoring force could explain the

shape of the initial magfnetization curve. However, it failed to explain the residual

magnetization after the external field was removed. Wiedemann, in 1986, further

postulated that there exists a frictional resistance due to the rotation of the

magnetic molecules.

Maxwell and Ewingf sell__ -1. .1 that "mutual magnetic interactions" could ac-

count for the hysteresis phenomena. Ewing also gave some convincing calculations

to support his idea. Due to Ewing's contribution, the frictionall force" hypothesis

was discarded. The idea of mutual interaction of the in! sI_.!tic !!. .in, I-' was

well accepted, and mathematical models were explored and created to explain this

phenomenon theoretically. However, some remnant of the fractional model persists

'.?4 .vi. Actually it is believed that the smooth change in the hysteresis loops in

magnetization (which appears in most hysteresis loops) is due to a frictional force

which opposes the changes in magnetization.

2.1.2 Models to Simulate Hysteresis

Early attempts in modeling hysteresis could be viewed as attempts to curve

fit an equation to the experimental hysteresis data. Brauer used power series,

while Fisher and Morer, Widger, and Hopkins et al. used a rational polynomials

to approximate the experimental hysteresis loop. Later on, Rivas et al. added two

generating functions to fit the experimental data. The two generating functions

resemble the two .I-i-mmetric ODE functions in the current domain wall method.

This method will be discussed in detail shortly. These methods are equivalent to

curve fitting, and they did not inspect the hysteresis generating mechanism inside

the material. Current popular models to represent hysteresis in piezoceramics

(as well as in magnetization) include the domain wall model and Preisach model.

Preisach method is a kind of kernel method. Research in this dissertation utilizes

another kernel to develop the frequeai~ s -i-dependent hysteresis model. In the next

section, a brief review about the two methods will be provided. The theoretical

foundation about the kernel method will be discussed in detail within C'!s Ilter

3. Early models to simulate hysteresis are based on the assumption of static or

quasi-static operating conditions. They are called static models in this dissertation.

2.1.3 Static Modeling of Hysteresis

2.1.3 1 Domain Wall M~odel

The domain wall method appeared early in the 1960s. In 1962, Globus

originated the domain wall method to explain the general shape of hysteresis

behavior in ferromagnetic materials. In their paper, it is stated that "the domain

walls were pinned on grain boundaries by a frictional force. Under the action of the

magnetic field, the domain walls first underwent a reversible motion due to domain

wall bulging, then an irreversible motion due to domain wall displacement." It was

further -II_0-r-- -1. by Jiles and Atherton [7] that "Domain walls are also pinned

by inhomogeneities within a grain, for example tangles of dislocations, regions of

inhomogeneous strain and any precipitates or nonmagnetic inclusions within a

grain." This is the foundation of the current domain wall theory.

Jiles and Atherton [7] developed the mathematical model of the hysteresis

mechanism in ferromagfnetic materials based on the domain wall motion including

both bending and translation. In their paper, an alshli--1. i. l~ ic model is first created

using a mean field approach, and then a hysteresis model is created by considering

the effects of pinning a magnetic domain wall at defect sites.

Within the context described above, assis~i--r. i. --; is observed in an idealized

material that is devoid of inclusions or imperfections. The inclusions or imper-

fections will lead to the pinning sites, which will be used in the hysteresis model.

There are three techniques to model the nonlinear relation between the input

voltage or electrical field and the resulting polarization. The common features of

the three models are as follows: (i) they utilize Boltzmann statistics to quantify

the probability of dipoles occupying certain energy states; (ii) they are applicable

if the material is devoid of imperfections, and thus there is no pinning site in the

material. The difference lies in the possible directions of the dipole moments when

deriving the mathematical models. In the absence of an electrical field and above

Figure 2-3: Orientation of dipole moment and electrical field

Curie temperature, the dipole moments change their directions randomly, resulting

in a net zero polarization. In the presence of an electrical field, the change of direc-

tion is no longer random and microscopic polarization develops. Figure 2-3 shows a

dipole moment under the presence of electrical field E. For a dipole moment Po in

an electrical field E, the potential energy is given by Equation 2.1.

I = -P~oE = -PoEcos0 (2.1)

where, Po = IIo|, E = |E|1. 8 is the angle between P~o and E~. According to

Boltzmann statistics, the thermal energy under temperature T is ksT, where ks is

Boltzmann constant. The probability that a dipole occupies the energy state E is

p-(E) = Ce-E/(kpBT) = C6-PoE/(kRT) (2.2)

where, C is an coefficient which ensures that the integration with respect to 8

from 0 to xr results in the total number of dipole moments NV per unit volume. By

considering all the possible directions the dipole moments can orientate themselves,

we apply Boltzmann statistics on Equation 2.2, the three fashli--r. i. --; models: the

Langfevin model, Isingf Spin model and preferred orientation model are derived. The

angle 8 between P~o and E~ in Equation 2.2 will change accordingly.
Langevin Model: [8]

The Langevin model assumes that the material is isotropic. There are no

mutual interactions among the dipoles and the dipole moments can lie in any

direction. Under these assumptions, the following equation has been derived

for the .Ilnby:steresis polarization

P, I T(PoE TksPE(23

where, P, is the saturation polarization.

Ising Spin Model: [9, 10]

This model assumes that the dipole moments can only lie in two directions:

either in the direction of the applied electrical field or opposite to that

direction. The resulting althli--1. i. --; polarization expression is

Pn =,;,, s tanh (2.4)

Preferred Orientation Model: [9, 10]

This model combines the above two models, assuming that the grains are

randomly distributed but that the dipoles within each cell have only two

directions: in the electrical field direction or opposite to it. The following

equation is obtained

P,,n = ~ [log (1 + e-2E~" o/Tks) ]o0 (I 62EoTksi'"i
2g),,,~,, ,,~ ~,-,, (2.5)

where Li(x) represents the dilogarithm function, defined as

/olog(1 t)
Li~) dt .

However, according to Calusius-1\ossotti equation [11], the polarization and stress

contribute to the electrical field. The total electrical field, termed as effective

electrical field E, is given by

E, = E + caP + 2Q:3:3oP (2.6)

where o- is the axial stress, a~ is an inter-domain coefficient, indicates the contribu-

tion of polarization to change the electrical field, and Q:33 denotes the longitudinal

electrostrictive coefficient. It embodies the contribution of stress to the change of

the electrical field. So if the mutual interaction of the dipole moments is taken into

account, the Langevin and Ising spin model are expressed as

P,, [r~~l( 3T,EE, 3TETEo 27

P,, ,,;,,, [T,.E,Eo 2

where, T,. is the Curie temperature, Eo is a scaling of the electrical field. The

readers are referred to Hom and Smith [12] and Hom and Shankar [1:3] for a

detailed descriptions to the derivation of the three models.

In the hysteresis model, polarization P shows hysteresis, so E, shows hys-

teresis. And the effective charge displacement D, = eE,, also shows hysteretic

behavior. The current research in this dissertation uses this "effective field" idea

and Equation 2.6 to write the effective electrical field E, as a hysteresis function of

the input voltage Q, E, = 7 (Q). Then the hysteresis function is quantified.

Domain wall theory assumes that the nonhomogeneous properties of the

ferroelectric materials produce pinning sites for the domain wall. With the absence

of external field, the domain wall is at its equilibrium position. As external field

increases, the domain wall begins to bend. At the very beginning, this deformation

of domain wall is reversible, so if the external field is removed, it will spring back

to its original position. There is no hysteresis. When the external field is strong

enough, then domain wall is translated to a new position. This part of the energy

is irreversible. The polarization energy for a given effective field level equals the

total polarization energy minus loss energy necessary to overcome local energy

barriers. Hence we have the following equation

1 rDe 1 De rDe dP,
Pir doe PndDe k r doe (2.9)
Eg JO Eo Jo J doe

where, the first term is the net polarization energy for a given electrical field; the

second term represents the total polarization energy in the ideal case; the third

term is the polarization energy needed to break pinning sites. As detailed in [14],

Equation 2.9 can be simplified as the following ordinary differential equation for the

irreversible polarization Pirr:

drr=6 Pa ir(2.10)
dE 6k a~(Pn Pirr)

where, Pn represents the anhysteresis polarization, calculated by Langevin model

or Isingf spin model. The 6 is defined as 6 = 1 when dE > 0 and P > Pn, or

dE < 0 and P < Pn; otherwise it is zero. The variable 6 is introduced to ensure

that the above ODE yields a physical solution. The reversible polarization is

assumed very small, it is defined as Prey = cl(Pn Pirr), here cl is estimated by

a least square fit to experimental data. Thus the domain wall theory results in the

followingf equation for the total polarization

P = Pn + Pirr (2.11)

Equation 2.10 is the hysteresis generating function for irreversible polarization, and

Equation 2.11 calculates the total polarization. These two equations represents the

mathematical expressions of the domain wall method.

The domain wall method has been applied in many applications for different

structure. Smith and Hom [12], in 1999, [14], in 2002, derived a domain wall

model for characterizing the hysteresis behavior in ferroelectric materials. Smith

and Ounaies, [15] in 1999, presented a hysteresis model based on domain wall

considerations to quantify the hysteresis and nonlinearity inherent to the PZT

materials at moderate to high drive levels. They shew how to determine the five

parameters that defines the governing ODE. Smith and Salapaka [5] in, 2001,

developed a model via domain wall consideration for the positioning mechanism

emploi- II in the atomic force microscope(AFM). Massad and Smith, [16] in 2002,

applied the domain wall model to characterize the hysteresis behavior in shape

memory alloys.

2.1.3 2 Classical Preisach M~odel

The Preisach model was first seen in 1935, [17] to model the relation between

the input field P(t), and the resulting magnetization M~(t). Please refer to ap-

pendix A for the definition of magfnetization. The Preisach model is expressed

M~(t)= wr,)[k_, ()(tdr
0 -oo
The kernel [ks,,s,(u)] is a classical Preisach relay operator with threshold at and

82, and w(r, s) is its weight. Applications of the Preisach model to simulate the

hysteresis of active materials have been reported in [18-22] and many others.

Hughes et al. [18] applied the Preisach model for the hysteresis of shape

memory alloy materials. Ge et al. [20, 21] adapted the Preisach model to describe

the nonlinear hysteretic behavior of piezoceramic actuators and presented a

modified generalized Preisach model that was used in a linearizing control scheme.

Smith in 1997 presented a phenomenological characterization of hysteresis in

magnetostrictive transducer via the Preisach model. Smith et al. [23] applied the

Preisach method to quantify the hysteresis and constitutive nonlinearities for the

AFM. The hysteresis operator is given by

i= 1

The hysteresis is expressed as the superposition of weighted kernels. The weights

ws are chosen through a least-squares fit to the experimental data [24]. Within

their paper the authors show that that the Preisach model can accurately quantify

the nonlinear constitutive properties of the transducer, but at the expense that the

model is phenomenologfical.

Zhou et al. [25] applied the classical Preisach model to simulate the static

hysteresis behavior of the Thunder Actuator System@. They then used an inverse

classical Preisach model to do the real time micro-position tracking control. The

real time tracking control was achieved by combining a lead-lag feedback controller

and the inverse model.

Banks et al, [26] in 2003, studied consistent and convergent approximation

methods for the identification of the measure characterizing the hysteresis. The

continuity and convergence properties of the Preisach K(P kernels guarantee the

well-posedness of the identification problem.

2.1.3 3 Conclusion

The domain wall model is a more physical-based model. It provides more

insights regarding the actuating or sensing mechanisms. So it is theoretically

sound. However, this method yields complex equations of states. Also as it

uses two generating functions, it can not guarantee the closure of the minor

loops. Currently, several physical mechanisms in piezoceramic materials are not

well understood, thus motivating the phenomenologfical or empirical model to

characterize the behavior of the materials. For this purpose, the Preisach method is

more favorable.

The Preisach model is good for general material characterization. It guaran-

tees the closure of minor loops. As it is non-physical based, it is applicable when

the underlying physics are difficult to quantify or poorly understood. However,

the Preisach model is not physically based. It uses a rate-independent elementary

operator, and is difficult to incorporate the frequency and temperature dependence

exhibited by essentially all smart materials.

1\odels to simulate the hysteresis behavior presented so far have been based on

the assumptions that the operating conditions are static or quasi-static. But it has

long been recognized that the hysteresis behavior is related to how fast the input is

cycled. So recently, some work has been carried out on rate dependent hysteresis

modeling via modified domain wall methods or Preisach methods.

2.1.4 Rate-Dependent Hysteresis Models

The term "rate dependency" characterizes how the response hysteresis loop of

the PZT materials change according to how fast the input is cycled. It includes two

cases. One is that the loop depends on the frequency of the input signal for a fixed

input level; the other one is that the loop depends on the input level for a fixed

input frequency. Smith et al. [27] developed a rate dependent model to predict

the hysteresis in piezoceramic materials operating at low frequency. They stated

in the model that, in the absence of an electrical field E(t), the dipole orientation

changes due to the thermal fuctuation. The change is random. However in the

presence of an electrical field, the dipole number in the direction of electrical field

(or opposite to the direction of the electrical field) changes with frequency. This

fre I-i in. s- -dependent model gives an accurate prediction at low frequency range.

However, this model is only applicable at very low frequencies. For example, in

their paper, the authors gave model validation for frequencies from 0.1 Hz to 1 Hz,

which usually is not the operating frequency range of interest. Piezoceramics are

often utilized in frequency regimes that are measured in kilohertz or megahertz.

Ang et al. [28] developed a rate dependent hysteresis model for a piezoelectric

actuator in 2003. This paper used a modified Prandtl-Ishlinskii (PI) operator to

account for the hysteresis of the Piezoelectric actuator at varying frequencies.

In the application of employing the Preisach method to quantify the rate-

dependent hysteresis, there are two important cases, one uses rate-independent

elementary operators, and the probability measure is rate-dependent, as in Equa-

tion 2.12.

[P,(, (0)](t)= [s~v,(s)] (~dp~s)(2.12)

where, p,(s) indicates that the probability measure p is rate-dependent.

This expansion incorporates frequency dependence, but makes the model

extremely difficult to implement for material characterization and control design.

Moreover, Torre [29], shows that at low drive levels, modifications must be made to

ensure accuracy of minor loops inside the 1!! I i r loops. These difficulties motivate

the energy formulation for the Preisach kernels, thus comes the second case of

hysteresis representation. This representation uses rate-dependent operators.

However, the probability measure is rate-independent, as in Equation 2.13.

[P,(, ( 8)](t)= [s~v,((s)] (~dp~s)(2.13)

where I-ots) is a vector valued measure.

Smith et al. [24] used free energy representations for the dipole switching

to develop the dynamic hysteresis model. By employing the energy method,

the frequeai~ s -i-dependent, temperature-dependent and load-dependent terms

are incorporated in the kernels rather than in the probability measure. This

fact is demonstrated in Equation 2.13. This approach alleviates the difficulties

encountered when incorporating the varying terms in the probability measure.

Galinaitis [30] developed a rate dependent hysteresis model for a piezoelectric

stack actuator in 2004. This paper presents a rate-dependent hysteresis model

hy employing rate dependent elementary hysteresis operators. For this work, a

resistor-capacitor (RC) circuit is used for the elementary operator and the time

dependent charge on the capacitor is the operator output.

Model validation employing rate-dependent elementary operators have been

given in these papers. However, only the responses under different input levels are

given. They do not study the response of the models for differences in frequencies.

One objective of the current research is to study the frel-lo. n. s --dependency in

hysteresis of the piezoceramic materials. A frequency dependent model is created

that can he used for precision positioning and control.

2.2 Vibration Absorbers and Literature Review

Vibration can cause structural or mechanical failure, frequent and costly main-

tenance of machines, and unacceptable levels of noise. Therefore it is necessary to

control or eliminate excessive and undesirable vibrations. There are many r-wsi~ to

control vibration. One way is by the insertion of an isolator between the vibrating

mass and the vibration source to obtain the vibration reduction. This section

focuses on the study of a vibration absorber to reduce the undesirable vibrations.

2.2.1 Vibration Absorbers

((t)- x,(t)

Figure 2-4: Single DOF vibration

A vibration absorber(VA) has been shown to be an effective way to reduce or

eliminate the excessive vibrations that a machine or a system may encounter. The

vibration absorption is usually achieved by supplementing the original system an

additional spring mass system, or possibly a damper. If possible, the parameters

are carefully chosen so that the natural frequencies of the new system are i. !-~ 1

separated" from the excitation frequency. A vibration system is usually modeled
as a mass-spring system. Figure 2-4 shows a model for a vibration system. Now

a spring-mass system (mass m2) 1S attached to the original system, as shown in
Figure 2-5. The governing equation of the new system is

k) ( t

x me x/t)

Figure 2-5: Classical vibration absorber

m Oi 0 xl C2 C1 -C2i 1 kl k2 -k2] I 1~

The Laplace transform gives the transfer function from the input force to the

displacement of mass mi,

X1 k2 ~ C2 m022
F (kl + k2 C1 (C2 + ca 1 82i) (k2 C2 ca 28ma2) (k2 C2 cs2

In the frequency domain, the frequency response function is

X1 k2 C -0
F (kl + k2 C1 j C2 c m192) (k2 jC2w m292) (k2 jC2 22

When we divide the numerator and denominator of the Equation 2.16 by klk2, We

obtain Equation 2.17.

X1 1 -I 2 ,
w4lW~ 2nzw~ 2in~ 2 m2
1 1 ~ 41 2 m2 2 (1 2(
n,12 n, i w a ai aq,

where, Lc,4 = k /ml, c /mi = 2(40s~~,, i = 1, 2. Equation 2.17 shows that the

resonance of mass mi could be cancelled if we set us,i = Lcn2. The corresponding

frequency response function is shown in Equation 2.18

X1 1 -1 2 ,

F [j, 21 +~ 2(2I 3 -2 4(112 2I + j [2(1 +: 2(2 W


Figure 2-6 shows the bode plot of the damped 2-DOF damped vibration sys-

Vibration A~bsorber

Frequen yrdstec)

represent the 1-Dvibratin system.W caseththreiarsonefrte
single DO system a w V~ ?= T hetosldcre r h epneo

mass mi when a VA is attached. We can see that when ""'] = the resonance is

Still, in the undamped case, cl = c2 = 0, the transfer function reduces even further

to the form in Equation 2.19. Figure 2-6 shows the Bode plot of the mass mi of

the undamped vibration system(UVS), with "" = 1.

X1 k2 mw22
F (kl + k2 mw192 (k2 m292') k,

By observation, the motion of the absorber (mass m2) is al 4i- finite, eVeH at itS


Frqec (rdsc

Figure -7: T.. of diplacem nt-frc of mss m

reoatfrqec. hsi ecuete ytmha hne fo -DFsse

Weca asococldefrmFigure 2 -6 and. 2-7 thtifnodapig sreentfrc theas

resons ofth 2-DOF system isuboneda the new sse a w resonance frequencies, nihro

andc zerofor the originalmastth resonante frequency of thes orgia SO

system. However, the additional mass will still have a motion at the original

resonant frequency. However if damping is present in either mass-spring element,

the response of the main mass will be finite at the two new resonant frequencies,

and no longer be zero at the resonant frequency of the original system. Mass mi

is vibrating under an external force, but the motion is balanced by the force due

to the motion of mass m2 So motion of mass mi is attenuated (damped case) or

completely absorbed(undamped case). Mass m2 Vibrates freely without any force

applied to it, so it vibrates at its undamped natural frequency.

The undamped VAs are also called de-tuners. The damped vibration ab-

sorbers are sometimes simply called tuned vibration absorbers, dynamic vibration

absorbers, or vibration neutralizers.

2.2.2 Vibration Absorber Literature Review

2.2.2-1. Passive Vibration Absorbers

The passive, tuned, undamped vibration absorber has proven to be an effective

way to reduce undesirable vibration. However, it has a limitation: the absorber

is effective provided the system is operated under constant excitation frequency

or the excitation frequency remains in a narrow range. With the addition of an

absorber, there are two new resonant frequencies. One is smaller than the original

resonant frequency, the other one is larger than the original resonant frequency.

If the excitation frequency shifts, the response of the new system becomes large

quickly, and becomes unbounded if the excitation frequency shifts to one of the new

resonant frequencies. So there is a narrow frequency band over which the excitation

frequency may change, while allowing vibration reduction.

In 1928, Ormondroyd and den Hartog introduced a damping factor in the system to

avoid an unbounded response at the resonance at the two new resonant frequencies.

This method makes the frequency range wider, but doesn't completely absorb the

vibration at the resonant frequency of the original system.

Soont and Lee examined optimal parameters settings, which include choosing

absorber mass, spring and damping factor to nminintize the nmaxiniun response of

the system (for example, displacement and velocity).

2.2.2-2. Active Vibration Absorbers

Tuned vibration absorbers (TVAs) are extremely effective for attenuation of a

single known resonant frequency, but are only effective in a very narrow bandwidth.

Active vibration absorbers have been developed to control vibrations over a wider

bandwidth and with less knowledge of the system parameters.

One difference between a passive VA and an active VA is in the requirement

of external energy. In the passive VA, no external energy is needed. In the active

VA, an external force, denoted as f,, is used to control the motion of the absorber.

Then the overall system response is controlled using various control algorithms.

The specific control strategies will be reviewed in later part of this section. Figfure

k, c

Figure 2-8: Active vibration absorber configuration

2-8 shows an active VA configuration, with a force generating niechanisni between

absorber mass and main mass, parallel with the spring and damper. This makes

the system more flexible and amenable to control theory.

Feed back control laws designed using various methods are addressed in [:36, 37]

and [:38]. Stephen et al. [:36], 1991, uses the linear combination of the velocity and

acceleration of the original system as the control signal as feedback to the system.

Seto et al. [:37], 1989, Seto and Savatari, [:38], 1992, uses the linear quadratic (LQ)

optimal control theory to design the feedback control signal. It is shown from

experiment that the vibration of the controlled system is dramatically attenuated

compared to the uncontrolled system.

Burdisso and Heilmann design a dual-mass active vibration absorber for wide

bandwidth control [:39], 1998. It is shown in the paper that the dual mass strategy

requires less control force than the single mass active TVA to obtain the same

level of reduction vibration. Figure 2-9 gives a schematic view of the dual mass



Figure 2-9: Dual mass configuration

2.2.2-:3. Semi-Active Strategies

Active vibration control systems can have many limitations. While sometimes

highly effective, they can require costly and highly sophisticated control algorithms.

In addition, the stability and robustness of the system must he studied carefully

in an active control formulation. When subjected to an unanticipated excitation,

the control force can drive the system into instability. At least from an energy

consumption viewpoint, the control force should be minimal. However, a passive

VA, by its nature, induces negative feedback. Thus stability and robustness

are ensured. Due to the disadvantage of the potential instability that can he

induced in an active VA, the semi-active strategy was developed to reduce the

amount of external energy needed to achieve the desired performance. "The

semi-active strategy is to place the active force in active VA with a continually

adjustable element which can vary the rate of energy dissipation in response

to an instantaneous condition of motion" [40]. It is claimed that this method

combines the advantages of both passive and active VAs. First, it is effective over

a larger bandwidth than passive VA, moreover, it requires less force to drive the

system. Anecdotal evidence -II__- -0 that the system is much more stable and

simpler than the active VA. Figure 2-10 shows a schematic of the semi-active TVA

configuration. The system can alter its inertia, stiffness, damping factor or initial

conditions .

k, c

t1ii m; x,()

Figure 2-10: Semi-active VA configuration

A time varying stiffness was used by Hubbard and 1\argolis, [41] in 1976, for

vibration isolation applications. Fujino et al., 1993 used a varying stiffness for

cable vibration attenuation. Hino et al., 1994 used a varying stiffness for structural

control applications.

Variable damping seems quite effective in reducing the transient and steady

state vibrations. Applications of variable damping have been reported in many

sources. For example, Hrovat et al., [42], 1983 used semi-active dampers for

vibration control of civil engineering structures such as buildings and bridges.

Tanaka and K~ikushima, [43], 1992 developed a self-optimizing damper for reducing

machine tool oscillations and impact vibrations.

There are many techniques for obtaining a physical realization of the ad-

justable damper. These methods include the used of hydraulic actuators, electro-

rheological (ER) fluids, magneto-rheological (ilRt) fluids and piezoelectric materi-

als. Application of a hydraulic actuator is given in Patten, 1996.

Among the two rheologfical fluids, MR is promising due to the fact that

it can operate at low voltage, such as the voltage applied by a vehicle battery.

Details about the mechanics of the two fluids can be found in [40]. This paper

also addresses commonly used control techniques, including the on-off semi-active

strategy and optimal semi-active control. Jalili et al., [44], 2001, have achieved

vibration attenuation via an adjustable inertial absorber. Gavin [45], 2000,

designed a semi-active damper using electro-rheologfical fluid within the actuator.

Tentor investigated the use of an electromagnetic element as an absorber, which

he used with both semi-active and active control in his Ph. D. dissertation [46].

The semi-active VA is realized by altering the gap -1. II ;so thus changing the

system stiffness. Active VA is realized by providing the system a DC current. It

is also pointed out that the variable gap and current change the system transfer

function to minimize the primary system response. [46]. Vavreck developed a

dynamic vibration absorber with a magneto-rheological fluid damper instead of a

dashpot [47]. Two control schemes were discussed: the skyhook control technique

and linear quadratic optimal control. Simulation results were given for comparison.

Another adjustable scheme is the hybrid-switched treatment. This design has two

modes: an active mode and a passive mode. Both are realized by controlling a

switch on and off. The objective of this strategy is to simplify the control effort.

2.2.3 Switched Shunted Designs

2.2.3-1. Passive Shunted Design

As mentioned in ChI Ilpter 1, the high internal impedance and high energy

transmission efficiency of the piezoceramic materials make them a good candidate

for vibration suppression. When a PZT is attached to a vibrating structure, the

PZT material is strained because of the structural vibration generating a voltage.

This electrical energy can he dissipated through a shunted circuit. Thus, vibration

within a structure can he suppressed. 1\oreover, it has been argued that passive

electronic damping using piezoelectric ceramics is more tunable alternative to

viscoelastic damping treatments.

Early applications of PZT actuators to suppress vibrations utilized the passive

shunted design. Early in the 1970s, researchers investigated the possibility of using

PZT materials for vibration control. Forward originated the idea of using PZT

elements shunted with a passive electrical circuit for vibration control. He used a

very small transducer to successfully reduce the vibration in an optical system [48].

Hagood and von Flotow [49], in 1991, were the first to study quantitatively the

dissipation of energy with piezoceramic materials shunted with passive electrical

circuits. Two types of circuits are studied in their paper, a resistor circuit (RC)

and a resistor-inductor circuit (RLC). It was pointed out that, "For resistive shunt-

ing, the material properties exhibit frequency dependence similar to viscoelastic

materials, but are much stiffer and more independent of temperature. Shunting

with a resistor and inductor introduces an electrical resonance, which can he op-

timally tuned to structural resonances in a manner analogous to a mechanical

vibration absorber ". Also a cantilever beam experiment was conducted to validate

the two models. Figure 2-11 shows the two passive shunted circuits.

Hagood and von Flotow's initial work created a foundation for current passive

and adaptive shunted design. Since then, much research has been conducted to

improve the shunting technique and vibration suppression effect. Edherg et al.,

used an electronic circuit instead of the heavy inductor used by Hagood and von

R >R


a b

Figure 2-11: Passive shunted circuit (used by Hagood and von Flotow)

Flotow in a shunted RL circuit. Edberg et al., [50], 1991 and Hollkamp [51], 1994,

studied the possibility of dissipating multiple vibration modes using one tuned,

shunting circuit.

Wu [52], in 1996, found that the shunted circuit with a resistor in parallel with

an inductor(circuit is shown in Figure 2-12) is easier to tune than the circuit with

a resistor in series with an inductor as was used by Hagood an von Flotow in [49].

The amplitude of the displacement response versus frequency of a structural mode

was studied under several circumstances. Under the optimum tuning condition, the

peak of the curve decreases with an increase of the shunt resistance. It becomes

a plateau at the optimum resistance. The study by Wu [52] also shows how the

amplitude changes with the change of the shunted resistance, inductor and the




Figure 2-12: Passive shunted circuit (used by Wu)

In summary, four basic passive shunted circuits have appeared for the suppres-

sion of vibration. In addition to the two circuits shown in Figure 2-11, the other

two are comprised of a shunted capacitor(CC) and shunted inductor(LC), as shown

in Figure 2-13. "Each of these kinds of shunts results in characteristically different

dynamics behavior: a resistive shunt dissipates energy through 1.. .Ir;! which

has the effect of structural damping. An inductive shunt results in a resonant

LC circuit, the behavior of which is analogous to that of a mechanical vibration

absorber (tuned mass damper). A capacitive shunt changes the effective stiffness

of the piezoelectric element and can be used as a tunable mechanical vibration

absorber. A switched shunt offers the possibilities of controlling the energy transfer

to reduce frequency dependent behavior, or perhaps the conversion of energy to a

usable form". [53]


c d

Figure 2-13: Passive shunted circuit

2.2.3-2. Switched Shunted Strategy

As discussed before, the adaptive passive or semi-active control is an alter-

native to active control. Active control requires external power for activating the

control, and stability is not ensured. Frequently, the system is not robust to param-

eter change. The idea of the adaptive passive, or switch shunted strategy, is that

the equivalent stiffness and resistance of the PZT is altered to obtain the objective

of dissipating energy from the vibrating system. Since the actuators are passive

devices whose characteristics can be changed in real time, their performance can

conceivably be better than their passive counterparts. There is a slight difference

in concept between III Iltive pI I--i.; and state-switch" strategies. Adaptive

passive methods employ continuously variable shunts, which mean that the shunt

circuit parameters can he altered continuously according to environment or operat-

ing conditions. The -I II.- switch" technique is a comparatively new concept. The

term -I II.- switch" means that two or more circuits are switched according to a

certain switching control law.

The term -I II.- switch" was proposed by 1\unk in 1980 for -I II-- switched

acoustic source". It was first eniploi-. I1 in [54] by Larson and Rogers during

1994. In their paper, authors used an acoustic source with an active spring made

front PZT to demonstrate the concept of a state switch. Larson and Rogers also

developed an underwater transducer for high-aniplitude, low frequency signals, [55]

in 1998. This paper further developed the concept of state switch" systems and

demonstrated the state switching in an acoustic transducer. The switch strategy

can choose between two distinct values of spring stiffness so that the system has

two resonance frequencies, one associated with each configuration.

i i.R

a b

Figure 2-14: State switch (a) Open-short circuit, (b) Open-resistive circuit

Clark, [56, 57], in 1999, used a PZT actuator shunted with an electrical circuit

for the purpose of dissipating energy front a simple vibration system. The model of

the system was created, the control law for state switching was established and two

kinds of switching circuits were discussed in the paper: switching open and short

circuit (OC-SC), and open and resistive circuits (OC-RC), as shown in Figure 2-14.

The impulse and harmonic responses of the two switching systems are compared

with a passive, shunted, resistive circuit. Although the passive, shunted resistive,

circuit shows better results at the optimal resistance value, the state switched

systems are robust to change in the resistance value. The OC-R S strategy performs

slightly better than the OC-SC. The energy dissipation is obtained through the
following procedure: el.--! i. the system moves, the actuator is held in its high

stiffness state such that the energy could be stored in the actuator. When the

system motion could cause it to receive energy back from the actuator, the actuator
is switched to a low stiffness state, dissipating energy. [56, 57]. According to this,

the switching control law is developed. "When the system is moving away from

equilibrium, Ill > 0, the circuit is switched to high stiffness, and when the system
moving towards the equilibrium, the circuit is switched to low stiffness to dissipate

ei! n,3 ~ The system in this paper is a 1-D vibration system, shown in Figure 2-15.

K~urdila et al., [58], in 2002, addressed the stability problem of the vibration system

studied by Clark in [56]. The often used state-switch strategies are listed below:

J-V(t) k Ic J- vt)
shunted shunted
P ~ circuit circuit

Figure 2-15: Schematic of 1-D vibration system with PZT

Open-circuit to short circuit (OC-SC)
Open-circuit to resistive circuit (OC-RC)
Open-circuit to resistor-inductor circuit (OC-RLC)
Open-circuit to capacitance circuit (OC-CC).

In the short circuit case, as the wire ahr-l- .- has some impedance so it is usually

modeled as a resistor with low impedance.

2.2.4 Averaging Analysis

The resulting model of employing the state switch control scheme is a

"discrete-continuous" system, which is characterized by a discrete (or quan-

tized) variable specifying the state of the multi-stable elements and by a continuous

vector, which continuously changes in each state. However, the well-established

modern control theories are mostly based on linear time invariant(LTI) model,

which takes either continuous time forms or discrete time forms, but not both.

Thus it is desirable for us to use a LTI model of the system instead of the original

time-varying switched model. Modeling methods are thus needed to develop for

deriving the LTI model, among them the averaging method is the most popular

choice. In this part of my research, the averaging method is used to obtain the

quantitative analysis of the state switched piezostructural system.

Analysis of periodic switching has been reported in [59-62], etc. Most of the

relevant literature mentioned above is based on an averaging technique to simplify

the analysis. State-switch averaging (SSA) is the most widely used averaging

approach for modeling and analysis of conventional switching dc-dc converters.

Next, we introduce two concepts: the hybrid dynamical system and the

switched system. M ow!: systems can be described by one or several differential

equations. A hybrid system is a system whose states include continuous variables

and discrete variables. The evolution of an autonomous hybrid system can be

described by the system

x(t) = f (x(t), m(t))

m(t) = g(x(t), m(t-)) (2.21)

where, x E R"n, m e M = {ml, m2, mil}. The variable x represents a continuous

state, and m represents a discrete state. The notation t- means that m(t) is

piecewise constant from the right.

If for each x, several discrete states are possible, then the system is called a

hybrid system. However, if for each x, only one mi E ML is possible, the system is

known as a switched system. Switched systems include multi-modal systems, or

systems with variable structure, and constitute a special class of hybrid dynamic

systems (Pettersson and Lennartson, [63] in 1996, Hou et al., [64] in 1996). Next,

the general averaging method and the state space averaging method for a periodic

switch control system will be discussed.

2.2.4-1. General Averaging Method

Averaging methods have been widely used as to obtain approximate solutions

of differential equations for over 100 years. Usually switched and hybrid systems

can be represented by a collection of ordinary differential equations. These equa-

tions can be divided into two categories: one time scale (OTS) and two time scale

(TTS) equations. The state space model of a switched hybrid system has a discon-

tinuous right hand side(RHS). However the existence and uniqueness of solutions to

the ODE are frequently based on the assumption of continuity (Lipschitz continu-

ity). Therefore, it is very important to establish continuous approximations of the

original discontinuous system.

A one time scale system, also called a slow system, contains state variables

that evolve slowly with time due to a small parameter e. The state space model of

such a system can be written in the form of Equation 2.21.

=ef (t, X) (2.22)

subject to the initial condition X(to) = Xo. In this equation, X E R"n, f:

RW+ x RW" R I" may be a discontinuous function. The variable e > 0 is a very small

number. When we average the OTS state space equation, the resulting averaged

model is given by
efo(t, X,,) (2.23)

subject to the initial condition X,,(to) = Xo. The function f,, is defined as

for = "~lim f 7, Xd

If f is piecewise continuous and satisfies several other assumptions outlined in

Bainov and Milusheva, [65], Sun et al., [66], and Sanders, [67], it has been proven


lim || X(t) Xev (t) || = 0, f or Vt E [to To/e] (2.24)

We ;?i that the averaged system solution X,, approaches the real system solution

X(t) on a time scale
Two time scale systems is also called mixed systems. The state space equation

of such systems can be written in the form of Equation 2.25. State variables

contained in X evolve slowly because of the small parameter e. However state

space variables contained in Y evolve more quickly.

=" ef (t, X, Y)
d~ (2.25)
=g(t, X, Y)

These equations are subject to the initial conditions X(to) = Xo and Y(to) = Yo

We have X ER", f :RW+xRI~nx m R"n, g :RW+xRI~nx Im RIm and E>0Ois a

very small number. [68]

To obtain the averaged system for the mixed system described in Equation

2.25, we assume that cp(t, X, -r, co) is solution of the equation i = g(t, X, Y), with
an initial condition Y(-r) = co, at fixed X. where, co is defined on the domain of Y.

S= e fo(X,,) (2.26)

For this equation, the initial condition is X,,(to) = Xo, and fo is defined in

Equation 2.27.

fo= lm f l(v, X, (v, X, 7, co))dv (2.27)

Also with the same assumptions are assumed to hold for the slow system, Sun et

al., proved that the averaged system Xev (t) approaches the original system on a

time scale 1. [69]

2.2.4-2. State-Space Averaging for Periodic Switched System

If the converter switches periodically between N LTI sub-systems, each of

which corresponds to a switched state of the converter in one switching period Ts,

its state space model is piecewise linear and has the following form:

=AX + bilU (2.28)

where X E Rnx iis the state, Ai E 7. bi E 7. and U E R~mx is the input

vector of the system.

Suppose di is the duty in the switched cycle. To obtain the averaged system,

first, we need to write the state space equation into the standard form as described

in the Theorem, shown in Equation 2.26. For this purpose, we introduce a new

scale variable -r and let t = e-r, where e is a very small number. We will see

later that the small parameter e actually is the switching period T, = 1 s

When switching frequency is very fast, T, is a very small number. Denote that

X(e-r) = Z(-r). We have Equation 2.29.

dX(t) dX(e-r) 1 dZ(-r)
dt de-r s d-r


f (-, Z) = e (AiZ + bilU) (2.30)

where -r = 1, 2 r. The system has two discontinuous points when the switch is

on and when switch is off. In each phase, it is a continuous system. When we carry

out averaging of Equation 2.27, we obtain the averaged model for the new function

Z(t), shown in Equation 2.31.

i= 1 i= 1


If we change the variable back to X(t), we will get

dX~in) (
i= 1 i= 1

The semi-active absorber in this dissertation is a periodically state-switching

absorber (SSA). The averaging method for the state switched ODE introduced in

this section will be used in later chapters for a vibration absorption an~ ll-;


This chapter establishes a nonlinear constitutive equation for PZT materials,

and presents the theoretical foundation of the current research. The equations used

to develop a g. n. s-dependent hysteresis model are presented.

3.1 Linear Constitutive Equations of PZT Materials

As pointed out in ChI Ilpter 1, at low drive levels, piezoceramic materials exhibit

nearly linear properties. A linear model is sufficient to characterize the dynamic

behavior in these operating regimes. However, at moderate to high drive levels, the

piezoceramic materials show nonlinearities and hysteresis. Applications at these

drive levels must consider the nonlinearities and hysteresis.

Without considering structural damping and any constitutive nonlinearities,

thermodynamic arguments based on Gihhs free energy and Maxwell's equation of

electromagnetism [70] yield the following constitutive equation for PZT

P = XfoE + ya

In this equation, a, E, E and P denote the stress, strain, electrical field and

polarization in the PZT, respectively. Symbols and descriptions of the coefficients

are listed in Table :31. We can see that the relationship between P, a, E and E

is linear. Also, we have the linear relation Q = Eh, where b is the thickness of

the PZT across which the input voltage Q is applied. This equation applies at low

input voltages, when hysteresis and nonlinearities are negligible.

Table 3-1: Symbols and descriptions

c" Youngf's modulus at constant polarization
P dielectric nonpermittivities under constant strain
X Electric susceptibility
eo Permittivity of free space
Y Piezoelectric and electric couplingf coefficient

3.2 Nonlinear Presentation of Constitutive Equations

At moderate and high voltages, hysteresis is not negligible. To establish a

nonlinear constitutive equation, it is assumed that the stress-strain relation is

linear in the presence of the electrical field. However, the input voltage Q(t), the

resulting strain E and polarization P are assumed to exhibit ]!i.( -1' i. ic behavior.

Thus we can write the polarization as a nonlinear function of the input electrical

field and the generated stress, P = W(E, E), where, 7 quantifies the nonlinearity

and hysteresis inherent to the materials. Further experiments [70, 71] show that the

relationship between the polarization P and the generated strain or stress is linear,

so the nonlinearity lies only in the relationship between the resulting polarization P

and the input electrical field E. Including a structural damping term, we generalize

Equation 3.1 as
o- = cPE CDE. C PP
P = W(E)

In this equation, cD is the K~elven-Voigt damping factor. The history dependent

functional 7(-) quantifies the intrinsic nonlinearity of the materials. There are

numerous choices for the hysteresis functional 3. Two common choices include the

Domain Wall Method and the Preisach method [5]. Preisach method is a kind of

kernel method. It uses the ideal relay operator as kernel. But there is limitation

on this kernel. In this dissertation, we are interested in developing a free. ner-l-

dependent hysteresis model based on another kernel the K~rasnolselskii-Pokrovskii

(K(P) kernel. In next section, the K(P kernel method is reviewed.

3.3 Krasnolselskii-Pokrovskii Kernel Model

3.3.1 Fundamentals of the KP kernel Method

The kernel method, by its nature, is a superposition of simple nonlinear

functionals. A periodic signal can be expressed as the sum of weighted basis

functions, for example, to express a signal as a Fourier series. A similar philosophy

can be observed in the K(P kernel method. The K(P kernel method expresses

Ihi--r i. l ic behavior as the superposition of basis functionals. Because the response

of the stack to an input voltage is a multi-valued mapping problem, we need to

define a special ..-!- This ..-!- is comprised of a set of kernels which have

the "multi-valued mappingt property. Let C [0, T] denotes the space of continuous

functions on [0, T]. For an input signal E C [0, T] and an initial configuration

( e -1,+ 1}, a kernel, denoted as k, is a mapping from C [0, T] to {-1,+1}

The Preisach method uses the ideal relay as kernel, as shown in Figure 3-1. Note

Sk, (u)
Route 2 1_+1

-1 Route 1

Figure 3-1: A typical kernel in Preisach model

that the kernel is characterized by two constants (P, a~), with P > a~. Since the

Ihi--1. i. l~ic response also depends on (P, a~), we denote the kernel as kp,,(u, (). The

total hysteresis mapping from input to output is expressed as

[h~u()] t) = [kya~u,)] (~dy(3.3)

where, p = p(P, a~) is a weight function, also referred to as a density function. This

function depends on the physical properties of the material. The kernel kpy,a(u, () is

a "multi-valued" mapping. The Preisach plane is defined as

Po { (#,, a) IFe R2 <

which is an unbounded half-plane contained in RW2

The density function p(P, a~) is usually assumed to have compact support. For

some value < Po and a~ > cao, the contribution of p is very small, so we let p 0

for p < Po and a~ > cao. We therefore define the effective Preisach domain

P {(n, a) E ro| < a, B> o, a< ao}

P is a bounded, triangular region in RW2

3.3.2 Discretization of the Preisach Plane

For the purpose of simulation and identification, the Preisach plane often needs

to be discretized. The hysteresis quantity is approximated by an expression having

the form

i= 1
where, ni is a node in the finite Preisach domain. The term It is a weight value,

and is defined in the summation

i= 1

where, 6,, is the Dirac delta concentrated at node us. For the discrete grid, we use

ax, /SI=S2

u2, 6


Figure 3-2: Discretization of the Preisach plane

S instead of P. S is defined as

S' {(s1, 82 I21 S1 Smin + 8-Ia, s82 smin + (k 1)as, j < k}

where, j and k are integers. The mesh parameter as is defined as as = smax--mi

The range [smi, sm=,] is quantized uniformly into N pieces. The two thresholds

p and a~ correspond to sl and 82 TOSpectively. The two limits po and t~o define

smin and smax, respectively. The input range [umin umaz] is a strict subset of

[smi, sm=,]. The kernel defined on the discrete grid will be denoted as k,(u(t),(),

where = (sl, 82), iS One grid over the triangular plane S. Figure 3-2 shows a

triangular S- plane.

The smaller the mesh parameter as is, the better we are able to match ex-

perimental results due to the increased resolution. The trade off is that refining

the mesh significantly increases the computation time. This method of approxima-

tion has nice convergence properties under mild assumptions. Banks and K~urdila

studied the convergence and well-posedness of the identification problem in [72].

3.3.3 Recursive Definition of Hysteresis

The Preisach model uses the ideal relay operator, as depicted in Figure 3-1,

where at and 82 are the two thresholds of the relay operator, with 82 > 1. The

symbol u(t) is the input of the system, and is assumed to be continuous in time.

/ route 2

-route 1

a b

Figure 3-3: Preisach kernels

The input-output relation of the ideal relay operator is as follows: Assume at

the initial time to, the system input u(0) < sl, the hysteresis kernel k,(u, () has

value of -1. If the input u(t) monotonically increases, ks remains at -1 until the

input reacheS 82, and then the response switches to +1. If the input starts from

u(0) > 82, and decreases monotonically, ks remains at +1 until the input reaches a ,

and then k, switches to -1.

But as can be seen from the figure directly, this mapping from time to the

value of the kernel is not continuous. This can be undesirable in practice. Thus

two other candidate kernels are used for the kernel method, which are shown in


s / s+a

Figure 3-4: Ridge function

Figure 3-3 a and 3-3 b. The kernel depicted in Figure 3-3 b is the well-known

K~rasnolselskii-Pokrovskii operator, or K(P kernel. K(P kernel is widely used in

mathematical models for multi-valued problem. To define the K(P kernel, we first

introduce a shifted ridge function r, = r(u s). The K(P kernel is comprised of

two ridge functions r,, = r(u si) and r,, = r(u 82), With 82 > 1. FiguTO

3-4 depicts a plot of a single ridge function. The slope of this particular ridge

function is determined by a. The relation of the ridge function between the input

and output is given by Equation 3.5.

1, a
rs~, t = (u s;) 1, a < u < s + a (3.5)

+1, a2s + a

u (t) nks (u)

+1 S

a b

Figure 3-5: Major loop and minor loop

The input-output relationship associated with the K(P kernel in Figure 3-3.b

is described as follows: when the input monotonically increases from a < sl, the

output follows path 1, when it decreases from a > (s2 + a), the output follows

path 2. But for an input signal as in Figure 3-5(a), if a monotonically increasing

input decreases before it reacheS 82 + a, k, takes the minimum of the current value

and the upper envelope route. If a monotonically decreasing input increases before

it reaches sl, k, takes the maximum of the current value and the lower envelope

curve, as depicted in Figure 3-5(b). The upper and lower envelope curves define

the hysteresis 1!! li1 .r loop. The path consisting of (2),(3),(4) and (5) defines a

hysteresis minor loop. The mathematical expression for the hysteresis operator is

given by Equation 3.6.

[R(u, Rk-1)1 () maX Ek-1, r~) s2) if u iS nOn d6CTetSing (3.6)
min{Rk-1, r7~) s1) if u iS nOn inrCTaSing

where, R is defined by Equation 3.7.

[Rk k- k~l~)}, k = 2, 3-- j(37
[R~](t)= Ro = ~, k = 1, ( E {--1, +1} 37

The initial condition is given by Ro = ~. Definitions 3.6 and 3.7 ensure that the

minor loops rest inside the ill lij ur loop.

3.4 Developing the Frequency Dependent Model

The research in this dissertation develops a fregneav~li- dependent hysteresis

model. As noted in the domain wall model, with an input voltage Q(t), the

generated electrical field is E(t) = Q(t)/t, where t is the thickness of the PZT.

The electrical field E generates a polarization P = W(E), which shows hysteresis.

The polarization, conversely, generates an electrical field. The total electrical field,

denoted as effective electrical field Ee is expressed as Ee = E + a(P and effective

charge displacement is De = eE + P. In these expressions, a( is an inter-domain

coupling factor and e is the dielectricity of the material. Polarization P is derived

according to domain wall method in [32]. So both Ee and De have ]!i.'( i' (ic

behavior. In Equation 3.2, the nonlinear constitutive equation can also be written

o- = cPE CDE. cPPDe(E)
De = W(E)

o- = CEE CD. + Ee(D)
Ee = T(D)

In equation 3.8 and 3.9, Ee and De denote the effective electrical field and effec-

tive charge displacement, respectively. They both exhibit hysteretic responses

behaviors. However, E = Q/t and D = eE do not exhibit hysteresis.

Research in this dissertation uses Equation 3.9 to develop a fregnener-l-

dependent hysteresis model. The first equation of 3.9 is used to derive the govern-

ing equations. The second equation of 3.9 is used to quantify the nonlinearity and

hysteresis of the PZT materials, in which Ee is expressed by Equation 3.4.


This purpose of this chapter is to derive the equations of motion of the piezo-

electric stack actuator by utilizing the constitutive equation shown in Equation :3.9.

The form of the equation of motion (EOM) of the stack actuator is derived from

Hamilton's Principle. Subsequently, the weak form of the equations of motion are

derived based on a variational method. The finite element method and a modal

truncation are applied to the weak form of the governing equation to provide the

final approximate governing equations. Numerical approximation and simulation

results are given to compare the model with the experimental data.

This derivation is generic in nature, but ultimately applied to the P-11:3-00

stack actuator from the company Pi/. ..--.r/.;0 Jena. This specific actuator utilizes

the d:33 mode of the PZT component in the stack. When a voltage is applied to it,

the actuator will generate a displacement, which exhibits hysteresis. Since the cross

sectional area of the stack actuator is much smaller compared to its length, the

actuator is modeled as a linearly elastic rod which has longitudinal displacement.

In the experiment, the bottom plate of the stack is fixed through a screw onto a

rigid mount. There is no excitation force applied, so the boundary condition is

taken as fixed at one end and free at the other end. The stack actuator is assumed

to have uniform effective density p, cross sectional area A, and Young's modulus Y.

4.1 Structure of the Stack Actuator

Figure 4-1 shows the P-11:3-00 stack actuator. The piezostack consists of a

large number of ceramic discs inside a stainless steel housing. The electrodes are

arranged on both sides of the ceramic discs and are connected in a parallel line as

shown in Figure 4-2. Inside the housing, there are 2:30 1wa;-is of PZT wafers. Each

Figure 4-1: The stack actuator

lIn-;-r is a square, with cross sectional area of 5mm x 5mm, as shown in Figure

4-2(d). The thickness of each lIn-;-r is 100pm. The neighboring 111-- rs use common

voltage sources and the voltage potential across each lI.>. r is as indicated in Figfure

4-2(b) and (c). According to the model developed herein, the gravity is negligible.

4.2 1-D Constitutive Equation

The stack actuator is a one dimensional motion device. It utilizes the "(d33)

coefficient to induce motion. Thus, we use the constitutive Equation 1.4 in C'!. Ilter

1. From the first constitutive equation, we have

E33 d33E3 1 33
33, SE" SE E3 S E
33 33 33

Because there is only one mode of deformation, for convenience, we drop the

subscript. Then we have
1 d
a E (4.1)

Define cP = and CK = J Equation 4.1 can be written as

a = cPE CKE



das das mmx5mm

C Vd

Figure 4-2: Structure of the stacked actuator

To incorporate internal damping, an adhoc viscous damping factor cD is introduced

into the Equation 4.2. Thus we have Equation 4.3

o- = c E CDE


In Equation 4.3, cP is the equivalent elastic modulus under constant polarization,

s" = 1/cP is the elastic compliance under constant polarization and CK is a

coefficient representing the coupling efficiency between the mechanical domain

and the electrical domain. The typical values for the material lead zirconate

titanate (PZT) are: S3%, = 15.87 x 10-12 2 1, d33 = 6.3 x 10-10m/V, thus

cP = 6.3 x 1010NV/m2, and CK = 22.68N/lmV.

The definition of the other symbols appearing in the above equation can

be found in OsI Ilpter 1. We know that the input electrical field E induces a

polarization P in the material. Conversely, the induced polarization generates an

additional electrical field, the total electrical field, termed as effective electrical field

Ee is then expressed as Ee = E + a~P. For the purpose of establishing the nonlinear

and hysteresis model, we use the effective electrical field Ee in Equation 4.3 instead

of the input electrical field E. The effective electrical field Ee exhibits hysteresis

behavior and is expressed by Equation 3.9.

4.3 Governing Equation of the Stack Actuator

The governing equations of the stack actuator can be derived from Hamilton's

Extended Principle for Linear Piezoelectricity (HEPLP). To motivate HEPLP, let

us first recall Hamilton's Extended Principle (HEP).

6 (L.)dt +i GWNodt = 0 (4.4)

where ,

L = 7 7 stands for the Lagrangian. 7 is the total kinetic energy of the
system, 7 is the total potential energy, including the strain energy and the
potential energy of conservative external forces.
6WNC iS the virtual work of the nonconservative external forces.
6(-) denotes the variation or virtual change of the quantity in the parenthesis.
1 and t2 are the time when the configuration of the system is known, or
motion at time 11 and t2 are prescribed, therefore 6w(tl) = 0, Gw(t2) = 0.

The HEP states that the virtual variation of the integral of the Lagfrangfian and

the virtual work of all nonconservative forces must be equal to zero for all virtual

variations consistent with the kinematic constraints. For the linear piezoelectric

continue, a similar result holds. For this purpose, let us define the electrical

enthalpy H,


H = U EiDe


Figure 4-3: Axial piezoelectric rod

where U is the internal energy density which includes the stored mechanical energy

and electrical energy. Then we define a new Lagrangian for the linear piezoelectric


*C =6 po O- H dR (4.6)

In this equation, 3S denotes the body. The extended Hamilton's Principle for Linear

Piezoelectricity is written as shown in Equation 4.7.

6 L~t+ W~ci (4.7)
t1 t1

For the purpose of analysis and computation, we need an explicit expression for the

electrical enthalpy density. We define H as

1, 1
H = cl i~l-Cj~~k-- 4(4.8
2 2

in accordance with [73]. In this equation, H is a homogeneous and quadratic

function of strain and the electrical field, which means that Cijkl = Ck~lij = Cjikl =

c .,; eijk = eikj and Eij = Ej. Then we have the following relation

1 1
U = C ., Sj~kl+ -". Ey(4.9)
2 2

To prove this, we use the constitutive Equation 1.4 from Chapter 1,

We substitutive this expression into Equation 4.5,

H = U EiDi

=U eiklEiEkl r".F F y (4. 10)

When we compare Equation 4.10 with Equation 4.8, we have

1 1
20 Eiyckl eijkEiEjk 2'.FT Ej = U eiklEiEkl C .8 FT1F

Rearrange terms in the above equation, we have Equation 4.9. Substitute Equation

4.6 into the extended Hamilton's Principle Equation 4.5, we have

= 7 -Uda + EiD dR (4. 11)
Term 1~
Term 2 Term 3

Next we will use Hamilton's Extended Principle for Linear Piezoelectricity to derive

the governing equation of motion for the PZT stack actuator. There is only one

deformation which is associated with the "33" direction. Thus we have the velocity
vector v = 2ik. The internal energy density reduces to U = c3EE 7333

and the charge displacement becomes D3 = 633E3 + ~E3E3. The Term 1 of Equation

4.11 is the kinetic energy of the piezoelectric element in Figure 4.3. It is expressed


7 = /22pr2dV (4.12)

where, V/ indicates volume, dV

Adz. The variable w denotes the deformation in

the material. The kinetic energy can be expressed as

Term 2 in Equation 4.11, is expressed to the form




3WP~ 3 2rC Ii1"E3E3d


We have the relation

8z (1


Equation 4.14 can be re-expressed as



The quantity f EiDidR reduces to f E3D3da2. Substitute Equation 4.15 into this
a2 n
equation. We have

aQ(t) aw
e33A x

Term,, 3


Next, we calculate the variation of each term.


0 ti



Here, bwdt

d(6w), integrate Equation 4.17 by parts, we obtain Equation 4.18

L 2

/1 W 2
2 z

aQ (t>Adz



0 P

jjL 2m
pO~ dd t

p r'> t| dz


For Term 2, we obtain

6J Udadt

=6/t --c33/1 8 W 2d j

t1 L

aQ (t>Adzdt


2 Lw 2 E LQt d
= l 3 2Azt 6 e3A 2
1 0 ti 0
2.1 2.2
In the above expression, it is straigfhtforwad to calculate Term 2.1.

6 C33

// 2
1 0

,( 2 d ,,

i1 0


Integrate Equation 4.19 by parts. We obtain the following equation


= l C33A i'ia| 'dtr c33A 1 Gwdzdt

The variation of the quantity labeled "Term 2.2" is simply zero.

ti'.>< |] -, pwswdt d
0 1


Bw 86w
8z 8z

C33A~1 8z\) cwvo O

L~'"U \ h:'J
[ 8 (8w1l

As in the previous case, we subdivide variations into two quantities, Term (3.1) and
Term (3.2).

Term 3.1 can be calculated easily:

/ / AQ t) w e3Adzdt
dJ/cr, & 8z~ a2hi~
1 L

i;/,,, aQ (t)h
1 L

/l aQ(t ) 63A 8, l|,?


The variation of Term 3.2 is, again, just zero.

Let us put Equations 4.18, 4.20, 4.21 together and reorganize terms. We have

0 =

p r'> t| dz

0 ti

/JIt / A Q (t) aw aQ (t) 3A+ejAdd

/ A~t B A~te3Adzdt +6S ej3 A at dzdt
& 8z h
1 L 1

8I [iA Q (t) wd


6J ~E33A 2Q(~ dxd

2 2
C33 A /t,-| di 3 A 8.,,|,
tl tl

pt wu tdz+c3Awzd
0~ ll)ild SS drd 0 rl(71orcj~

n2] hudd

c""\33A St-|

pA@i~ -i c33 +i~ e33A awid (4.22

The first three terms are boundary terms. According to our definition in Equation
4.4, the first term vanishes since variation 6w vanish at time 11 and t2. That is

6wltz = 6w1,2 = 0. The second and third term are

p i,,2 Q~~

p iir/Q~~
e3A t
& A,

c33A(~) iw| ,dt

c33A bw|L dt
8zi a~ a jI"'

8 w ,l n'
-~i c33 a w~od _. U'W


In this problem, 6wlo

O thus the above expression is left with

ic:2 Q~~
e3A/ (t)

8 w
c33A bw|zdt

We will discuss this term later. Standard arguments imply that the equation of
motion is
-I pA +x c33A- e33A> = (4.24)

Let us assume constant structure properties of the stack actuator. In other words,
C33, A, and p, e33 arT COnStantS. Finally, We Want tO Show that the Equation of
motion 4.24 can be written as

pAic = c33A


To prove this interpretation, rewrite the equation of motion as
.. 8 Bwl 8 Qt
pI = c33 [e33lAQ
8zx 8z 8z

The right hand side can be further written as

8ix [( w\ aQ(t) j

[ C33A3 e33A3)

By observation, we see that c33E3 e333E3 = 3. Thus, the right hand side
is~~ (AU3 = N. Equation 4.25 is verified. Equation 4.25 i theclassical

equilibrium equation for the axial bar which can be found in many dynamics books.

4.4 Strong Form of Equation of Motion

According to Appendix C, from Hamilton's principle, we derived the governing

equation of an axial bar with fixed free boundary conditions as in Equation 4.26

82W i3
pA (4.26)

with boundary conditions

N\J~z=L = 0 (4.27)

By' employing constitutive Equation 4.3, and noting that E = =UA, the

equation of motion is

82W d2W d2
pA =cPA + CDA -CKA_ (E(t)) (4.28)
8t2 d2 dtd2

Equation 4.28 is the strong form of equation of motion. This is a second

order partial differential equation. The displacement w(z, t) must be at least twice

differentiable in the spacial variable, and two boundary conditions are needed for a

classical solution to exist. The weak form of the equations of motion is introduced

to reduce the number of derivatives required in the governing equation. The next

section derives the weak form of the equation of motion.

4.5 Weak Form of Equation of Motion

In principle, the derivation of the weak form of the governing equations is
straightforward. We multiply Equation 4.28 by a sufficiently smooth" function

cp, and integrate from 0 to L. As a result, we obtain Equation 4.29. Now the goal
is to shift some of the spatial derivatives from the solution to the test function

via integration by parts. Thus, the test function cp is restricted to be at least once

differentiable and must satisfy certain specific boundary conditions that will be

discussed shortly.

SpA pdz = cPA adz d+/ CDAX 60dz CKA_ (E (t)) cpdz

Term (1) Term (2) Term (3) Term (4)

When we integrate Terms (2) (3) and (4) by parts, we have:

SCDA cpdz =CDA drr |ft CDA dwdz

cKA (Gt)) dd
-CKAE.(t),- cK i A Edtpdz







Recall the boundary conditions on the solution

w |z=o = 0

N\J~z=L = 0

The first boundary condition is the kinetic boundary condition, while the second

is a force boundary condition. The functions cp selected in the weak formulation

must satisfy the kinematic boundary conditions just as the 6w must in a variational

formulation. We require

Equation 4.29 can be written as

pA dz +c' dzp +L~ CDA dzp
o t2 0 r 0rdr

= cKAE(t) dzp + cPA cp |z L + CDA cp |z L CKAE(t)(p |z=f4.33)
Jo 8z 8z 8z

The boundary terms can be organized and grouped as follows:

cPA P |Z=L

cPAecp |z=L +

A (cre + CDC

Ao- |z=L |P z=L

3 z=L ( z=L

+CDA (r I

CDA cp |z=L
- CKE (tj)) |=

- CKAE(t)cp |

- CKAE(t)cp |


According to boundary conditions in Equation 4.27, N~ z=L = 0, the above

boundary term is zero. Equation 4.33 is written in final form as

pA dz+ c' dz+ CD dz = KA t) dz (4.34)
o t2 0x 8x 8x 0 8 80

Equation 4.34 is the weak form governing equation.
4.6 Model Well-Posedness

For completeness, we want to establish that the weak form of the governing

equations is well-posed. This can be carried out using the general framework

outlined in [2, 28, 74]. Define the embedding


| o = 0

X = L2 (0, L)

V = H 1(0, L) = E HI (0, L) : ~(0) = 0

Each of these sets of function are, in fact, Hilbert spaces for the choice of inner


To provide a framework in which to establish model well-posedness, we
consider an abstract formulation of the models based upon stiffness and dampingf

sesquilinear forms. To this end, we define two bilinear forms

nl(o, #) = J cAi'9/dx

where, ai : V xV R I, i = 1, 2. The control influence operator [B(Q)] (t) is
defined as

([B(Q)] (t), 9) = [(Q)](t) cK"Ai'dz

Then Equation 4.34 can be written as

(ii)pv +((c' ii (, 9) +[ ) (CD, cpV*,V (4.35)
Term 1 Term 2 Tem3Term 4

To apply the formulation in [74], [28], and [2], we must show that the forms al and

a2 arT
1. Bounded
al( ) < c | |y |9| v, for cy ER

aa2 C2 V Va Il Pv, fOT C2 E
2. V-Elliptic
Rev ( )j > C3ll 1 fOT C3 > 0

Re62 (i, dj C4 l~i2, fOT C4 > 0

3. Symmetric

Alternatively, we can define operators

(As, ),,y e( ) i = 2

We can write Equation 4.35 equivalently as

ic(t) + A28(t) + A w(t) = [B(Q)] (t) (4.36)

where, Q(t) is the input voltage, and Q(t) E C[0, T], and B(Q) E L2 ((0, T); V*).

According to Theorem 4.1 of [75] or Theorem 2.1 and remark 2.1 of [76], there

exists a unique solution w to Equation 4.36 or equivalently, Equation 4.35 which

wEC ((0, T) ; V)

iEC ((0, T) ; X)

4.7 Finite Element Approximation of Model

To achieve a numerical approximation of Equation 4.34, express the displace-

ment as a linear combination of conventional finite element shape functions.

wi(z~t)= wyl(t)#(z) (4.37)

In this summation, NV is the total number of the shape functions. C!. .. --- cp(z) =

', (z). Upon substituting this series in the integrals in Equation 4.35, we obtain

(Term 1) = pAi(t)4; (z)."-, (z)dz CN (t Wy().- zd
j=1 j=1
(TLN 2 A / xN Lt 6C P X ,X
(Term~ 2) =" c'w (t ()4(zd w () U ()4(zd
j=1 j=1

(Term 3) = CDA Y ("t)4'.()1(zdz".(t CD '.(z) ((z\~IUdz
JOj=1 j=1 0

(Term 4) = KE-). J: = cKAE/Wt)01(zIUdz

Define the following matrices


[K~jk] = c 'Ad'.(z)41(z)dz

[B(k] = CKAE t)4 (z\dz

The equation of motion becomes

{IM, i'- C t-, + ,~~)f, tr- for j = 1 --

It is convenient to write this summation in terms of the matrices

[Mw] -t+ [C -t+ [ w t= IB t 4.38)

with w(t) = [ w (t) w2 t) wrN(t) ] The term [B] (t) is the control influence

operator. To emphasize that it is a function of the input voltage Q, we denote it as

[B(Q)] (t).
4.7. 1 Construct [M~], [K] and [B(Q)] (t)

ft f2 3 4 f5

11 2 34 5

Figure 4-4: FEM shape fumetions

Each lI e. r of PZT is divided into four elements, thus there are five nodal

points. The mass matrix [M~] and stiffness matrix [K] for each I -;r are 5 x 5

matrices. They are

[M~] =

c" A
and [K] =

1 0 0 0

1 2 -1 0

)0 -1 1

L 5x5 L 5x5

le here is the length of each element. Next we will calculate the control influence

operator, the [B(Q)] (t) matrix. We know that E(z, t) = -VQ, where Q is the

input voltage difference. We take one element for convenience, as shown in Figure

4-5. We introduce a local coordinate z', withl 0 < z' < le. The~ deri'vati've: #'tk\' Z '

-1 or +1, a constant over the element. So the element matrix B, is defined as

Be,i = 1 CKAE(t: l."-'.ez

=CKCAQ(z') toe

=CKA(QIz/,Ie Q z/=o)


Figure 4-5: Shape function in one element

21 0 0

14 1 0

0 1 41

0 01 4

0 0 01


B =i cKEt'.dz'
-e,2 '

=CKAl -Q(+1)dz'

-CKA(QIz= e -0 &z/=o)

The element matrix Be is then expressed as

Be = CKA (Qlz/=le) Q z/=o) +-1

and in one 1e. ;r of PZT, the global matrix B is

B = cKA (Qlz/=,- Qz/=o)[+1 0 0 0-1

The whole stack is comprised of 230 1 i- ris, the global matrices are given below:

2 1 0 0 1 -1 0 0

1 4 0 0 -1 2 0 0

0 0 4 1 0 0 2 -1

0 0 1 2 0 0 -1 1
-921x921 921x921

BG = cKA (Q(le) Q(0)) [-1, 0, 0, + 1] 921x1

As the piezo stack is fixed at the bottom, after applying the boundary condition,

first row and column of matrices [MoG] and [Ko] are cancelled. The first element of

B, is deleted. The global equation of motion is:

[MoG] W(t) + [CG] W(t) + [Ko] ty(t) = Be


with w(t) = [ wi(t) w2 t) N(t) ] [MoG] and [Ko] are 920 x 920, BG is
920 x 1.

4.7.2 Reduced Order To Approximate The Equation of Motion

The global matrices [Mo~] E RW920 x920 [G] 920 x920 and [KoG] E RW920 x920 rTO

large matrices. So in this section, modal truncation and mode-superposition are

Equation 4.38 can be taken as the coupled equations of motion of a 920 degree
of freedom system. There are 920 modes in total. Now only NV = 3 modes are

used to approximate the system. Introducing the coordinate transformation,

u(t) =~ y rl (t) T~he corresponding modal matirix 0 of thle syistemn is then the

size of 920 x 3.i = 1 2 3~ Eac 9ri risl a column vctlor of Ilength 920,
corresponding to one mode. We left multiply the equation of motion by the modal

matrix 0 and define Mr = @" [MoG] 4, Kr = @" [Ko] 4, Br = @" [BG]. We should

point out that this modal transformation diagfonalizes the mass matrix and stiffness
matrix. So Mr and Kr are diagonal matrices, which are called modal mass matrix

and modal stiffness matrix respectively. Both have the size of 3 x 3. The matrix Br
is 3x1i.

Now the system equation has been reduced to three degrees of freedom. The

reduced order equation of motion is

[Mr] i + [Cr] rl + [Kr] rl = Br (4.40)

State space form of the governing equation is given by Equation 4.41.

let y(t) = I q(t)

y~t) = = -

s-t 1 1 1
r(t) -~ x [Mr] [K]-[r][r (t) [Mr] F

yl(t) = [A] y(t) + [B] (t) (4.41)

4.7.3 Trapezoidal Rule to Solve the Equation

In Equation 4.41, [A] is a constant matrix, but [B] is a prescribed time varying
column vector. To simulate the above equation, a modified trapezoidal rule is

introduced. We suppose the time step is very small

=[A] ?l~ i+ [B] tj

yj+1 yj yj 7 +1 yj + 1j tj)Bj (4.42)
2 2

Reorganizing terms, we have

II -t~ j A yje = II (t+ l y, +j A (tj~ tj)Bj (4.43)

multiplying~ by rI (ty-t[A -1, the eqluatio:n becomes

y g = I I t~ Ij A t+l y A]

+I (tjax tj>)By

This equation can be written in the compact form

yj I = [A] yj + [B] (t,) (4.44)


II (t+1 tj) [A] (tj I tj) [A]


[B](ti) =! It~ 2tj) [A]i~~ (rt, t1)B

Equation 4.44 can be easily programmed and solved using programs like Matlab.

4.8 The Linear Model and Natural Frequency

Let us consider an ideal material without damping, so that CD = 0. If not

accounting for the hysteresis, the electrical field E is simply E(t) = Q(t)/1, where

the 1 is the thickness of the PZT. Then E(t) has nothing to do with the variable z.

The constant term doesn't affect the eigenvalues of the system, so we will drop this

term. Then the strong form of the governing equation for the stack actuator has

the form of the wave equation, as shown in Equation 4.45.

82W d2
pA = cPA (4.45)
8t2 dX2

c" A 8 2
() = L (4.46)
pA 8z2
L~ is a linear operator. Then Equation 4.45 becomes

= Lw

Let us find the eigenvalues and eigenvectors of the operator L~.

L~ + A =0o

Dein = K. The eigfenvalue problem requires that we solve

K z2+ A# = 0 (4.47)

Solution of Equation 4.47 has the form of Equation 4.48.

~(z) = A sin z+B cos z(4.48)

Finally, cp must satisfy the boundary conditions. So, the first boundary condition is

=o = 0

Also there is no external force at z = L, N

EA ', so #'|z=L = 0.

~'(z) = A cos z = 0

SinceV A a o ezro n ,tusw uthv

cos L = 0

So, L

(2p + 1) ~, p

0, 1, 2 So the solution for Equation 4.48 is clear,

m~)=A sin 2 ,,,

We have the spacial frequency

m +

Next, we calculate the natural frequency in time domain. Represent w(z, t) as

w(z, t) = 9,(z)q,(t)


Substitute Equation 4.49 into the wave Equation 4.45,




In the above equa~tion Ky"(z)

Lyc,(z). However, Lyc, + Ap,

0, so Equation

4.50 becomes

To solve the above equation, multiply by 4,(z) and integrate from 0 to L. We

obtain the following equation

0 =1

Table 4-1: Natural frequencies from analytical model and FEM model

mode # Analytical solution(Hz) FEM model(Hz) variation
1 31502.992921; !"IE 31502.996751;1:1 II 0.000012141
2 94508.9787794963 94509.0820951708 0.00010' I :2'
3 157514.964632494 157515.442948384 0.01 11 I :I I :1. .' .
4 220520.950485491 220r7. 1~' 's. .11 0.00059515' .
5 283526.'1:1 : : ;189 2 :". .7 I 001' u *.
6; 346532.922191487 346538.015319954 0011.
7 409538.908044484 409547.314973836 00 ''.

Since cp are eigenvectors, cp,(z) and cp,(z)

following equation

are orthogonal. Then we have the

i,(t) + A,rl,(t) = 0 q = 1, 2, 3


A, =q 1 2 K

q =1, 2, 3

The natural frequency is given by

2xr 21

Table 4-1 lists the natural frequencies calculated via the analytical model

and finite element method and compares the variation of the FEM model from

the analytical solution. We can see from the table that the intrinsic frequencies

of the FEM model are very near to those of the analytical solution. Values of

those parameters are cP = 6.3 x 1010NV/m2, p = 7500Kg/m3, K = c'/p and

L = 23 x 10-3m.


This chapter describes the objective of the experiment, gives the details of the

experimental setup, explains the experimental procedures, and provides a summary

and analysis of experimental results.

5.1 Experiment Ob jectives Description

The objective of the experiment described in this thesis is to measure and

characterize hysteretic response of a piezoelectric stack actuator P-113-00 for

several frequencies. The identification of a model that is frequency dependent

is subsequently developed. The experiment studies a P-113-00 stack actuator

that utilizes the "(d33 motion" of the PZT. The stack operates within a voltage

range from [-10 + 150] V. Table 5.1 gives a list of the dimensions and physical

parameters of the stack actuator.

5.2 Experimental Setup

5.2.1 Equipment Used in the Experiment

We briefly summarize the experimental apparatus and setup of the experiment

in this section.

Table 5-1: Parameter list of the stack actuator

Area per 117. ir 5 x 5mm2
Thickness per 1 e. ;r 100/mum
Total number of liives 230
Length of the PZT 230mm
Two stacks 5 x 5 x 7mm3 + 5 x 5 x 16mm3
Top plate and bottom plate 1mm each, mm in total
Total length of stack 27mm
Internal capacitance 2.5p-F
Input range [- 10 V 150 V]

1. Laser Vibrometer

The most complex measurement device in this experiment is the laser vi-

brometer, which is used to observe the velocity of the stack actuator. A laser

vibrometer utilizes the Doppler effect, and is a non-contact vibration measurement

device. Laser vibrometers are typically two-beam, interferometric devices which

detect the phase difference between an internal reference and the measurement

beam. The measurement beam is focused on the target and scattered back to the

interferometer. The velocity of the beam's target may be inferred from the phase


2. Function Generator

An HP33120A Function Generator is used as a sinusoidal voltage source for

the experiment. The output range is [-10 + 10] V.

3. Amplifier

There are two reasons we use an amplifier in the experiment. First, the operat-

ing voltage range of the function generator is [-10V + 150] V. Thus, the function

generator can provide in only a very limited range. An amplifier is required to

generate the high voltage and current for the PZT actuator. The amplifier LV-1200

from DSM is a true linear follower amplifier and has a maximum peak current of

1200mA. This is the amplifier that has been used in this experiment.

4. Potentiometer

A potentiometer is used in series with the stack actuator to protect the

amplifier from current spikes.

5. Voltage Divider

A voltage divider is used to scale down the input voltage so that the data

acquisition system is not overloaded. Figure 5-1 shows how the voltage divider is

created. With the oscilloscope or Laser Vibrometer we can only measure voltage

up to 10V. This upper limit is far below the stack's voltage range, which is

Stack Actuator

FG nrtorr .R~ 2 R R

Voltage Divider

Voltage relation between node 13 and node 23
is expressed as

Q 23= 13

Figure 5-1: Voltage divider

[-10V + 150] V. Also it is not feasible to measure the input voltage directly from

the function generator, and then multiply the gain of the amplifier to get the input

voltage signal. The reason is that the amplifier induces a phase shift. The voltage

applied to the actuator has a different phase from the one that is output by the

function generator. The voltage divider consists of two resistors, R1, and R2 With

R1 < R2. The resistors are in series with each other. The voltage output from R1

is measured. To make the voltage output by R1 measurable(< 10 volts), we should

have "'Fli < T0 To satisfy this condition, R1 is taken as 26ika anld R2 = 470~kR.

Thus the voltage range U23 arCOSS R1 iS [-0.5242 7.8629] V. The actual actuator

input voltage range Q is calculated by Q13 RQ 23. Sin~e the resistors are

linear devices, there is no phase shift introduced.

5.2.2 Experiment Setup

Figure 5-2 depicts the experimental set up and Figure 5-3 illustrates an

associated flow chart. In the experiment, the function generator sends a signal to

the stack actuator via an amplifier, which causes the motion of the stack actuator.

The laser vibrometer detects the motion of the actuator. The velocity signals are

sent to the data acquisition board (DAB). The DAB converts the velocity signal to

a voltage signal. Together with the input voltage, the velocity is sent to the data

Laser Vibrometer



Function Generator

Stack Actuator


Voltage Divider
as x2

Voltage divider makes
input voltage measurable

~Input Channel

Feed to Data Processor

Figure 5-2: Experimental setup


Voltage Divider
R1=22K. R2=474K

Figure 5-3: Flow chart of experimental setup

prcesso~r. For the purpose of creating the frequenol- -dependent model, the velocity

data at several input frequencies are measured.


5.2.3 Sampling Frequency

There are several reasons that the sampling frequency f, has to be high. If

the sampling frequency is not high, the higher order harmonic frequencies will be

aliased to the frequency range from zero to half of the sampling frequency with

the same magnitude. Thus new low frequency components are introduced due

to all; I-;li but can not he filtered out. This will degrade the accuracy of the

model prediction. The sampling frequency used in this experiment is 400 times the

driving signal.

5.3 Experimental Results

In this research, a huge amount of experimental data was collected for the

purpose of weight identification. The driving signals were pure sinusoidal waves,

as shown in Figure 5-4. Figure 5-5 shows the velocity hysteresis loops at some

of the selected frequencies :3Hx, :$OHx, 50Hx, 100Hx, 160Hx, 200Hx, 250Hx,

:300Hx respectively. The purpose of this research is to obtain the weights and


predict the disp~lacmn ytrsso eie rqec.Tu ene h

diplceen esone f h sac. h epeimntmasre te eoctydaa


so the displacement response is obtained from integration of the velocity. In this

case, there is no means to know the absolute displacement of the stack, because

the moment the velocity is taken, the initial position is not known. This is a

classical problem when velocity measurements are conversions to displacements.

So only the relative position of the actuator can be obtained via this method.

Before integration of the velocity, the DC component is subtracted. Figure 5-

Velocity response of the stack

x 102. x 10'

2 3 Hz 230 Hz

0 0

-2 -
11.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.03 1.04 1.05 1.06 1.07 1.06 1.09 1.1 1.1'1 1.12
x10' x10'
2 50 Hz 2~ 100 Hz

1.04 1.05 1.06 1.07 1.08 1.09 0.52 0.525 0.53 0.535 0.54 0.545
x10' x10'
2 140 Hz 2~ 200 Hz

0.262 0.264 0.266 0.268 0.27 0.272 0.274 0.276 0.278 0.28 0.282 0.26 0.262 0.264 0.266 0.268 0.27 0.272
x10 x1

1. O Hz 50 Hz

0.106 D 108 0.1'1 0.1'12 0.114 0.1'16 0.'104 0.105 0.106 0.107 0.108 0.109 0.11 0.1'11 0 1'12


Figure 5-5: Velocity response of the stack actuator

6 shows the relative displacement of the stack actuator for the corresponding

frequencies in Figure 5-5. We can see from the displacement versus time plots

that at low frequency, when the driving force increases, the displacement increases.

However, as frequency increases, there is a lag in the displacement response, this

is obvious at any frequency higher than 120Hz. Figure 5-7, 5-8 and 5-9 show

the displacement hysteresis loops from 3Hz to 350Hz. The x- axis is driving

voltage, the y- axis is the displacement response of the stack. We can see that

the displacement versus driving force curves are strongly free.ii~ no s -dependent.

For example, the width of the hysteresis loops is very small at low frequency,

nearly zero at 3Hz. However, the width increases as the driving voltage increases.

Another observation is that the shape of the loops is nearly an "e II'l1-- If we

draw a line along the main axis of the ellipse, the orientation of the line will rotate

clockwise as the driving frequency increases.

Displacement response of the stack



30 H~z

1510 10 10 0 0 0 1 9 1 1 1

11 12 13 14 15 16 17 18 19


. 1, 8 H z

1 04

1 05 1 06 1 07 1 08 1 09

14 H/ li

0 52 0 525 0 53 0 535

0 54 0 545

200 Hz

0 262 0 264 0 266 0 268 0 27 0 272 0 274 0 276 0 278 0 28 0 282


0 26 0 262 0 264 0 266 0 268 0 27 0 272

350 Hz

0 106 0 108 0 11 0 112 0 114 0 116 0 104 0 105 0 106 0 107 0 108 0 109 0 11 0 111 0 112


Figure 5-6: Displacement response of the stack actuator

Whigapd hpgrMIhai

Figure 5-7 Reatve islacmet o th sackacuatr-


5cs........ .

0 20 AD 50 90 100 120 140-
onnerg voltagewi)

Figure 5-8: Relative displacement of the stack actuator-II









- LL1*

---- 1MHI

Hystrears lops forfrequences



.. I

i/ i.

- 10Hz
-- 120Hz
-- 160Hz
-- 1B0Hz
--- 250Hz



i ;~

'TO 0 2) 40

Drkng oltagne)vls

00 100 120

Figure 5-9: Relative displacement of the stack actuator-III


This chapter presents the freenono,-~ Ii-dependent hysteresis model. Two methods

are used to develop the model. Those two methods can be understood in terms

of Figure 6-1, which depicts the structure of a system that has a "nonlinear to

lain )cascade structure. The first method does not include the linear dynamical

subsysteCm, that is. G 0. The second one includes both the nonlinear and linear




Figure 6-1: Freeno0.~ li-dependent hysteresis model

In the first method, the displacement x is simply expressed as

i= 1

where, ks,i(v(t),() is the K~rasnolselskii-Pokrovskii (K(P) kernel defined on a

generated by the driving voltage, which has the hysteresis property. The weighting

function I, ( v) is free0.i~ no, i-dependent. The variable v is frequency.

In the second method, the freenono,li- -dependent hysteresis model is based on

the analysis of ChI Ilpter 3, 4. In this method, the displacement x is the solution of

the dynamic governing equation

[M~]x(t) + [C]x(t) + [K]x(t) = [B (Q)](t) (6.2)

The hysteresis term arises in the driving term [B (Q)](t), which is expressed as

[B ll(Q)(t =/?I Com kr,i (vl(t), E: o (A ) (6.3)
i= 1

This chapter presents the models created by the two methods and compare the

accuracy of model predictions. The model validation shows that the models

developed using the two methods can accurately predict the position of the stack.

The positioning error can be improved by an order of magnitude. However, the

dynamical model provides more accurate position estimate of the stack.

6.1 Quantifying the Hysteresis Function

6.1.1 Quantization of the S- Plane

According to ChI Ilpter 4, we express any hysteresis quantity as & = '7(Q),

which quantifies the hysteresis phenomenon of the PZT. Write h as

where, ko,4(v(t), () is a relay operator, with threshold a~ and P are two threshold,

and p-(t) is a measure. For a discrete case, the measure p is approximated by

i= 1

where, p is defined on the half plane P, P = {(a~, )|lumin < a~ < P < umax}. 6i

is the Dirac delta function having the value of 1 at node i and 0 otherwise. is

discrete point position on the Preisach plane. In the discrete Preisach plane, the

hysteresis equation can be written as

i= 1

Fr-om this equation we can see that the hysteresis quantity is expressed as the

superposition of weighted kernels. As noted in OsI Ilpter 3, the input range

S- plane

1 4 0 * * * * * * 0 1

120-Fiur meas mee oommeme

The maximum 100 e of e stc is 15] eh *cua inu e

besticlywihi h* **** ** ***** *** *10] T e hrshl

slan ~ ae icrt vle***** ******hs ar s, ~ dtrmns n
E(Pkene dfie ********vngvotaee h oekresinti olcin

the ~ 40 betrreut e ae.Btmoekre* *giianl nres h

comutngtie.Th***osiee igacrc ftemde n h optt

~or achinpt vltae ut).Figure 6-2:sostedsrt S- plane n iu 6

Khw evrlEP kernels defined on the drivsoning voltage. The moekresihs colletionac


One example of histeron One example of histeron

126 4
I I 46.4

I I II I 66.4

sl=120, s2=130 I III Is=0 6

120 130 40 60
Histeron -- Input data Histeron -- Input data
One example of histeron One example of histeron



Is =0,s2=6 sl=120, S2=130
40 100 120 130
Histeron -- Input data Histeron -- Input data

Figure 6-3: KP kernels

ridge function is determined by as. The width of the kernels is an integer number

times as. This can be found in Figure 3-3.

6.2 The Nonlinear Frequency-dependent Hysteresis Model

6.2.1 Problem Formulation

Without the dynamics of the stack included, the displacement hysteresis is

simply expressed by Equation 6.1. We can write this equation into matrix form

S= [K-(v(t), E)] w( v) (6.5)

where [Ks(v(t), E)] = [kwm,(v(t,), E)] t RSxM, With NV the number of time samples

used, M~ the number of the S- plane points and 1 < m < M~, 1 < n < NV. w(s, v)

is a column vector of length M~. The weight identification procedure is equivalent

to findings the optimized weight vector w, which renders the superposition of the

weighted kernels best fit to the true data. Thus our problem becomes minimization

Full Text


Iwouldliketothankmyadvisor,Dr.AndrewJ.Kurdila,andDr.ChrisNiezreckifortheirpatienceinteachingmethesubjectofdynamicsandcontrols,theireortinintroducingmetothisprofessionaleld,andtheirhelpinguidingmyresearch.Ihavelearnedagreatdealfromthem,andIlookforwardtoapplyingthisknowledgeinmycareer.IwouldalsoliketothankDr.RickLind,Dr.LouisCattafestaandDr.WilliamHagerfortakingthetimeandeorttoserveonmycommittee.Addi-tionally,IwouldliketothankmyfriendsfortheirguidanceandadviceasIhavepursuedmygraduatestudies.Finally,Iwouldliketothankmyhusbandandmyparentsfortheirimmeasurableadvice,encouragement,andsupport. iii


TABLEOFCONTENTS page ACKNOWLEDGMENTS ............................. iii LISTOFTABLES ................................. vii LISTOFFIGURES ................................ viii ABSTRACT .................................... xi CHAPTER 1INTRODUCTION .............................. 1 1.1Background .............................. 1 1.2ConstitutiveEquationforLinearPiezoelectricity .......... 4 1.3ProblemStatementandMotivationforResearch .......... 5 1.3.1ApplicationtoMicropositioning ............... 5 1.3.2ApplicationinVibrationSuppression ............ 6 1.4ProposedWorkandContributions .................. 6 1.4.1FrequencyDependentHysteresisModeling ......... 7 1.4.2AveragingAnalysisontheSemi-activelyTunedVibration System ............................ 9 1.5DissertationOutline .......................... 9 2REVIEWOFRELATEDLITERATURE ................. 11 2.1HysteresisModelingandLiteratureReview ............. 11 2.1.1EarlyTheoriesaboutHysteresis ............... 13 2.1.2ModelstoSimulateHysteresis ................ 14 2.1.3StaticModelingofHysteresis ................. 15 2.1.4Rate-DependentHysteresisModels .............. 22 2.2VibrationAbsorbersandLiteratureReview ............. 24 2.2.1VibrationAbsorbers ...................... 24 2.2.2VibrationAbsorberLiteratureReview ............ 28 2.2.3SwitchedShuntedDesigns .................. 32 2.2.4AveragingAnalysis ...................... 37 3KRASNOLSELSKII-POKROVSKIIKERNELMETHOD ........ 43 3.1LinearConstitutiveEquationsofPZTMaterials .......... 43 3.2NonlinearPresentationofConstitutiveEquations ......... 44 3.3Krasnolselskii-PokrovskiiKernelModel ............... 45 iv


........... 45 3.3.2DiscretizationofthePreisachPlane ............. 46 3.3.3RecursiveDenitionofHysteresis .............. 48 3.4DevelopingtheFrequencyDependentModel ............ 50 4STACKACTUATORMOTION ....................... 52 4.1StructureoftheStackActuator ................... 52 4.21-DConstitutiveEquation ...................... 53 4.3GoverningEquationoftheStackActuator ............. 55 4.4StrongFormofEquationofMotion ................. 62 4.5WeakFormofEquationofMotion .................. 63 4.6ModelWell-Posedness ........................ 64 4.7FiniteElementApproximationofModel .............. 66 4.7.1Construct[M],[K]and[B(Q)](t) .............. 67 4.7.2ReducedOrderToApproximateTheEquationofMotion 70 4.7.3TrapezoidalRuletoSolvetheEquation ........... 71 4.8TheLinearModelandNaturalFrequency .............. 72 5EXPERIMENTALDESCRIPTIONANDRESULTS ........... 75 5.1ExperimentObjectivesDescription ................. 75 5.2ExperimentalSetup ......................... 75 5.2.1EquipmentUsedintheExperiment ............. 75 5.2.2ExperimentSetup ....................... 77 5.2.3SamplingFrequency ...................... 79 5.3ExperimentalResults ........................ 79 6MODELDEVELOPMENTFORFREQUENCYDEPENDENTHYS-TERESIS .................................. 85 6.1QuantifyingtheHysteresisFunction ................. 86 6.1.1QuantizationoftheSPlane ................ 86 6.2TheNonlinearFrequency-dependentHysteresisModel ....... 88 6.2.1ProblemFormulation ..................... 88 6.2.2ConvergenceCharacterization ................ 89 6.2.3WeightIdenticationResults ................. 90 6.2.4ModelValidation ....................... 91 6.3Nonlinear-LinearFrequencyDependentHysteresisModel ..... 93 6.3.1ProblemFormulationandConvergenceCharacterization .. 93 6.3.2IdenticationoftheWeights ................. 95 6.3.3SummationoftheWeightedKPKernels ........... 96 6.3.4RipplesOnTheOptimizationCurve ............. 98 6.3.5ModelValidation ....................... 99 6.4TrustRegionMethodtoAdjusttheWeights ............ 99 6.5Comparisonofthetwomodels .................... 102 v


................... 104 7.1ModelingofthePiezoceramicVibrationAbsorptionSystem .... 104 7.2ShuntedCapacitorLoadModel ................... 107 7.2.1State-spaceModelforZero-IC ................ 108 7.2.2State-SpaceModelforNon-ZeroIC ............ 112 7.3ControlStrategyofState-Switch ................... 112 7.4AveragingAnalysis .......................... 114 7.5AveragingtheSwitchedSystem ................... 118 7.5.1IdealSwitch .......................... 118 7.5.2AveragingtheVibrationSystemwithShuntedCapacitiveLoads ............................ 118 7.5.3FrequencyDomainResults .................. 123 8SUMMARY .................................. 128 8.1Frequency-dependentHysteresisModel ............... 128 8.2AveragingAnalysisontheState-SwitchedPiezostructuralSystem 129 APPENDIX ................................... 130 APOLARIZATIONANDMAGNETIZATION ................ 130 A.1ConstructionofAtomandAxiomaticDenitionofPolarization .. 130 A.2Magnetization ............................. 130 BCLASSICALVIBRATIONOSCILLATION ................ 132 CMAGNETO-RHEOLOGICALELASTOMERSUSEDASVARIABLESTIFFNESSSPRINGS .......................... 134 DGRONWALLLEMMA ............................ 135 REFERENCES ................................... 136 BIOGRAPHICALSKETCH ............................ 142 vi


Table page 1{1Relationofsubscripts ........................... 5 1{2Symbolsanddescriptions ......................... 5 3{1Symbolsanddescriptions ......................... 44 4{1NaturalfrequenciesfromanalyticalmodelandFEMmodel ...... 74 5{1Parameterlistofthestackactuator ................... 75 7{1Parametervaluesusedinthesimulation ................. 127 vii


Figure page 1{1Sensoreectandactuatoreect ..................... 2 1{2Polarizationprocess ............................ 3 2{1Atomicforcemicroscopy ......................... 12 2{2Atypicalpiezoelectrichysteresisloop .................. 12 2{3Orientationofdipolemomentandelectricaleld ............ 16 2{4SingleDOFvibration ........................... 24 2{5Classicalvibrationabsorber ....................... 25 2{6T.F.ofdisplacement-forceofmassm1 26 2{7T.F.ofdisplacement-forceofmassm1 27 2{8Activevibrationabsorberconguration ................. 29 2{9Dualmassconguration ......................... 30 2{10Semi-activeVAconguration ....................... 31 2{11Passiveshuntedcircuit(usedbyHagoodandvonFlotow) ....... 34 2{12Passiveshuntedcircuit(usedbyWu) .................. 34 2{13Passiveshuntedcircuit .......................... 35 2{14Stateswitch(a)Open-shortcircuit,(b)Open-resistivecircuit ..... 36 2{15Schematicof1-DvibrationsystemwithPZT .............. 37 3{1AtypicalkernelinPreisachmodel .................... 45 3{2DiscretizationofthePreisachplane ................... 47 3{3Preisachkernels .............................. 48 3{4Ridgefunction ............................... 48 3{5Majorloopandminorloop ........................ 49 4{1Thestackactuator ............................ 53 viii


.................... 54 4{3Axialpiezoelectricrod .......................... 56 4{4FEMshapefunctions .......................... 67 4{5Shapefunctioninoneelement ...................... 68 5{1Voltagedivider .............................. 77 5{2Experimentalsetup ............................ 78 5{3Flowchartofexperimentalsetup .................... 78 5{4Drivingvoltagevs.time ......................... 79 5{5Velocityresponseofthestackactuator ................. 80 5{6Displacementresponseofthestackactuator .............. 82 5{7Relativedisplacementofthestackactuator-I .............. 83 5{8Relativedisplacementofthestackactuator-II ............. 83 5{9Relativedisplacementofthestackactuator-III ............. 84 6{1Frequency-dependenthysteresismodel ................. 85 6{2Splane .................................. 87 6{3KPkernels ................................. 88 6{4Flowchartforweightidentication ................... 90 6{5Weightdistributionforselectedfrequencies .............. 91 6{6Reconstructthedisplacementhysteresis ................. 91 6{7Weightidenticationforselectedfrequencies .............. 92 6{8Modelpredictionfortrainingfrequencies ................ 93 6{9Modelpredictionfordesiredfrequencies ................. 94 6{10Weightidenticationforcascadedmodel ................ 95 6{11WeightdistributionovertheSplane ................. 96 6{12Calculationofthereconstructedcurve .................. 97 6{13Reconstructeddatavs.experimentaldata ............... 97 6{14Ripplesinthereconstructedcurves ................... 98 ix


.................. 98 6{16Modelpredictionfortrainingfrequencies ................ 100 6{17Modelpredictionfordesiredfrequencies ................. 101 6{18Comparisonofthetwomodels ...................... 103 7{1Idealizedpiezoceramicvibrationabsorber ................ 104 7{2FBDofabsorbermassandstructuremass ............... 105 7{3Vibrationsuppressionwithcapacitorload ................ 107 7{4Structuremassresponse{changingactuatormass ........... 110 7{5Structuremassresponse{changingshuntedcapacitance ....... 111 7{6Comparisonofthetransferfunction ................... 113 7{7Switchedsignal .............................. 114 7{8Idealswitch ................................ 118 7{9Switchedsignal .............................. 119 7{10Transferfunctionoftheaveragedsystem ................ 125 7{11Comparisonofthetwodesigns:discreteandcontinuousnotchfrequen-cies .................................... 126 A{1Dipolemoment .............................. 130 B{1Responseofthe1-Dvibrationsystem ................. 133 x


AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy TOPICS IN NONLINEARMODELSFORPIEZOSTRUCTURALSYSTEMS By XiaoyanZhang August2005 Chair:AndrewJ.Kurdila Cochair:ChristopherNiezrecki MajorDepartment:MechanicalandAerospaceEngineering Thisdissertationincludestwodistincttopics:thefrequencydependent hysteresismodelingofastackactuatorandaveraginganalysisofthestate-switched piezoelectricstructuralsystems. ThePZTmaterialshavewidelybeenusedformicro-positioningandprecision controlduetotheirspecialpropertyofbeingabletodeformafterbeingapplied voltageacrossit.However,theresponseofthePZTmaterialswithrespectto thedrivingforceexhibitsahystereticbehavior,whichconsiderablyaectsthe resolutionofpositioning.Thisresearchaimstodevelopfrequency-dependent nonlinearhysteresismodelstopredictthedisplacementresponseofapiezostackin responsetoadrivingforceoverawiderangeofdrivingfrequencies.Twohysteresis modelsaredevelopedhere:oneincludesonlyanonlinearsubsystem,theother oneincludesbothanonlinearsubsystemandalinearsubsystem.modelvalidation showsthatbothmodelcangiveanaccuratepredictionofthestackposition.The cascadednonlinear-linearmodelcangivemoreaccurateandsmootherprediction. xi


TheothertopicisrelatedtousingPZTmaterialstoachievevibrationabsorptionusingstateswitchcontrolstrategy.Specically,anaveraginganalysis wasdevelopedforqualitativeandquantitativestudyofswitchedpiezostructural systems.Itisshownthataveraginganalysisprovidesameansofdeterminingtime domainaswellasfrequencydomainresponsecharacteristicsofswitchedpiezostructuralsystemsthatincludeswitchedcapacitiveshuntcircuits.Theproposed switchingarchitectureprovidesanessentiallycontinuousrangeoftunablenotch frequencies,incontrasttoaniteandxedcollectionofdiscretenotchfrequencies availableinsomeimplementationsofcapacitivelyshuntedpiezostructures. xii


CHAPTER1 INTRODUCTION Withinthischapter,piezoelectricityisbrieyreviewedalongwithsomecurrenttechnicalapplicationsofpiezoelectricactuation.Theseexamplesprovidethe motivationforthisresearch.Theproposedworkanditscontributiontoengineeringarealsodescribed.Lastly,anoverviewoftheremainderofthedissertationis presented. 1.1Background SincethediscoveryofpiezoelectricitybyJacquesandPierreCurieover 100yearsagoandthetheoreticalquanticationoftherelationbetweenthe piezoelectricityandcrystalstructurebyWoldermarVoigtin1894,piezoceramic materialshavebeenwidelyusedinlabinstrumentsformicropositioning,sonar devices,communicationdevices,etc.Someoftheiradvantagesincludesmallsize, highactuationenergydensity,relativelylowpowerconsumption,highresolution, andthecapabilityofbeingusedbothasanactuatorandsensor. Apiezoelectricceramicisamaterialthatdemonstratesthepiezoelectriceect andtheinversepiezoelectriceect.Whenamechanicalforceisappliedacrossa piezoceramicelement,electricalchargesaregenerated.Ifelectrodesareplacedon bothsidesofthematerial,avoltageisgenerated.Therefore,mechanicalenergyis transformedintoelectricalenergy.Thisiscalledthepiezoelectriceect,orsensor eect.Conversely,ifanelectricaleldisappliedacrossthematerial,amechanical deformationresults.Thus,electricalenergyistransformedintomechanical energy.Thisiscalledtheinversepiezoelectriceect,oractuatoreect.Therefore, piezoceramicmaterialscanactaseitherasensingelement,oranactuationelement, orboth.TheseeectsaredescribedbyFigure1{1. 1


2 Figure1{1:Sensoreectandactuatoreect Thepiezoelectriceectoccursnaturallyinquartzcrystalsbutwithvery lowenergytransferringeciency.Itcanalsobeinducedinotherpiezoelectric materialssuchasleadzirconatetitanate(PZT).Suchmaterialsusuallyhavehigher energytransferringeciency,sotheyareusedmoreofteninpractice.Because theyareceramics,theycanbemanufacturedintovirtuallyanywantedshape.To inducethepiezoelectriceectofthematerial,thepiezoceramicmaterialsarerst manufacturedintothedesiredshape.Thenelectrodesareputonoppositesidesof thematerialsandastrongDCelectriceldisappliedtothematerialssothatthe moleculardipolesintheceramicsarealignedalongthedirectionoftheelectrical eld.Thisprocessiscalledpolarization.Aftertheelectricaleldisremoved,the materialsexhibitpiezoelectricproperties.Thematerialhastheinterestingproperty thatthedipolesremain,toalargeextent,aligned.Thepolarizedmaterial,whose polarizationdirection,alignedwiththeelectricaleldwhenbeingpolarized,is called\polar."PleaserefertoAppendixAforinformationaboutdipolemoment andpolarization.Figure1{2showsthepolarizationprocessinamicroscopic view[1,2]. Toidentifythedirectionsinapiezoceramicelement,accordingtotheIEEE standardonpiezoelectricity,threeaxeslabeledas\1,"\2"and\3"areused, associatedwiththe x y ,and z axesinaclassicalthreedimensionalorthogonal coordinatesystem.Thepolardirectionislabeledas\3"andisparalleltothe directionofpolarizationwithintheceramic.


Figure1{2: Polarizationprocess Thepiezoelectricconstantsrelatingthemechanicalstrainandtheappliedelectriceldaretermedtheelectromechanicalcouplingconstants.Byconvention,theyarerepresentedbythe\d"coecients.Thisisanimportantcoecientinthepositioningmechanism.Forthepiezoelectriceect,dijisdenedastheratiooftheelectricalchargecollectedontheelectrodesandtheappliedmechanicalstress.dij=Shortcircuitchargedensity Appliedmechanicalstress Appliedelectricalfield Volts=Meter.Largedijconstantsindicatelargemechanicaldisplacementswhichareusuallyfavorableinmotiontransducerdevices.\d33"and\d31"arethetwoconstantsoftenusedinaposition-ingmechanisminresponsetotheinputvoltage.The\d33"motionmeanswhenaninputvoltageisappliedalongthepolar(or3)axis,motionisproducedinthepolardirection.Thuswesaythattheactuatorworksinthe\d33mode"or\longitudinal


mode."Whenaninputvoltageisappliedinthepolardirectionandthemotionwetakeadvantageofisinthe\1"direction,wesaytheactuatorworksinthe\d31mode"or\transversemode"or\in-planemode." or Theequationiswritteninthe\matrix/vector"summationform.Foranisotropicmedia,thetensorshavethefollowingrelations:"ij="ji;ij=ji;dmij=dmjiSEijkl=SEklij=SEjikl=SElkij or where,;=1;26,m;n=1;2;3.Therelationbetween;,andm;nisdescribedinTable 1{1 .ThemeaningsofthesymbolsaredescribedinTable 1{2


Table1{1: Relationofsubscripts Table1{2: Symbolsanddescriptions symbolsmeanings


hysteresis.Thus,alinearmodelissucienttostudythedynamicbehavior.However,thematerialsareusuallydrivenatmoderatetohighlevels.Atthesedrivelevels,duetotheirferroelectricnature,piezoceramicmaterialsshowanon-linearhystereticbehavior.Mostsmartmaterialsexhibithysteresisintherelationshipbetweentheinputelectricaleldandtheoutputresponse,suchasvelocityanddisplacement.Alsoithasbeenshownthatthehysteresisresponseofthematerialsisfrequency-dependent.Toprovideaframeworkamenabletocharacterizetheserelationships,onetopicofcurrentresearchinthisdissertationisrelatedtomodelingtherate-dependenthysteresisofapiezoelectric(PZT)stackedactuator.AliteraturereviewofhysteresismodelingispresentedinSection 2.1 ofChapter 2 2.2 ofChapter 2 1.3 .Thisdissertationpresentsthenecessarytheoreticalfoundationsandderivations,numericalsimulationsandexperimentalresults.Thus,thisworkprovidesacontributiontotheeldofengineeringintwoprimaryways,


1.Afrequencydependenthysteresismodeliscreatedtopredictthedisplace-mentofaPZTactuatoroverawiderrangeoffrequencythancurrentlyexists.Themodelcanbeusedtoimprovethecontrolperformanceoftheseactuators. 2.Anaveragingmethodisappliedtoadiscontinuousstateswitchedsystemtoobtaintheeectiveresponseofsomestate-switchedpiezostructuralsystems.Theaveragedsystemcanprovideacontinuouslytunablenotchfrequency. Twofrequency-dependenthysteresismodelsarepresentedhere.Onemodeljustincludesanonlinearsystem,theothermodelincludesacascadednonlinear-linearmodel.Fortherstmodel,thedisplacementxofthestackinresponseto


thedrivingforceissimplyexpressedas where,k 1. Anonlinearconstitutiveequationisdevelopedutilizingtheconceptoftheeectiveelectricaleld.Andthusthenonlinearhysteresissubsystemcanbedevelopedbasedonthisnonlinearconstitutiveequation.TheKPkernelsareusedasbasistoquantifythenonlinearhysteresisquantity. 2. ThedynamicalgoverningequationofmotionforthestackisderivedinChapter 3 and 4 .Thegoverningequationofthestackisasfollows: [M]x Thisisthelinearsubsystemofthehysteresismodel. 3. Twoapproximationmethodologiesarederived. a.Creatinganiteelementmodelofthegoverningequation,includingconventionalelasticeectsandhysteresiseects. b.Expressingthecontrolinuenceoperator[B(Q)](t)as[B(Q)](t)=ConstNXi=1k where,wiisthevalueofdiscretespairovertheSplaneandisfrequency-dependent. 4. Afrequency-dependenthysteresismodeliscreated.Thisincludestwomajorsteps: a.SystemIdentication.Identifytheweightsviatheleastsquaremethodfromexperimentaldata.Thesuperpositionofweightedkernelsapproximatestheexperimentaldatainaleastsquaresense. b.ModelCreation.Thedisplacementofthestackforadesiredfrequencyispredictedusingtheidentiedweights. 5. Theperformanceofthemodelisvalidatedbyusingtheempiricaldata.Themodelvalidationshowsthatthismodelcanbeusedtoimprovethecontrolperformanceofthestackactuator.Itcanimprovethecurrentpositioningerrorbyanorderofmagnitude.


2 acomprehensiveliteraturereviewpertainingtohysteresismodelingandtunablevibrationabsorptionispresented.Specically,thehysteresisgeneratingmechanismsandpreviousworkperformedonthehysteresismodelingwithinthecontrolsliteraturearereviewed.Modelsfortunablevibrationabsorbersandnewtechniquesinvibrationabsorptionutilizingpiezoceramicmaterialsareinvestigated.Generalanalysisaboutmodelingandaveragingonstatespacemodelsforperiodicswitchingcontrolsignalsisalsodiscussed. ThetheoreticalfoundationaboutthePreisachmethodisreviewedinChapter 3 .ThischapterestablishesthenonlinearconstitutiveequationofPZTmaterialsandpresentsthetheoreticalfoundationforthecurrentresearch.ThePreisachmethodtoquantifythenonlinearityofthepiezoceramicmaterialsviaaneectiveelectricaleldisdescribed.


Chapter 4 providesthetheoreticalderivationofthemotionofthePZTstackedactuator.Thestrongformandweakformofthegoverningequationarederived.Analyticalandnumericalsolutionsarediscussed.Aniteelementapproachandreducedorderapproximationwillbeappliedtosolvethegoverningequation. Chapter 5 explainstheexperimentalobjectives.Italsoshowstheexperimentalsetupandresults. InChapter 6 ,thefrequencydependenthysteresismodelisdevelopedbasedonthetheoreticalderivationpresentedinChapters 2 3 and 4 alongwiththeexperimentalresultsthatareprovidedinChapter 5 .Withinthischapter,themodelisvalidated. Thegoverningequationofa2-DOFvibrationsystem,whichemploysaPZTpatchtosuppressthevibrationwithashuntedswitchedcircuitisderivedinChapter 7 .Theaveragingmethodisappliedonthesystem.TheperformanceoftheaveragedsystemsiscomparedtotheDavis&Lesieutredesign. Finally,withinChapter 8 asummaryofthecurrentresearchispresented.


Thischapterreviewstherelatedliteraturepertainingtohysteresismodeling(Section 2.1 )andtunablevibrationabsorbers(Section 2.2 ). Hysteresismodelingisrelatedtotheapplicationofpiezoceramicmaterialsformicropositioningandprecisioncontrol.InSection 2.1 ,staticmodelsviadomainwallmethodandPreisachmethodarerstreviewed.Thenseveralrate-dependentmodelstosimulatehysteresisarepresented. Piezoceramicmaterialshavebeenrecentlyattachedtostructuresforthepurposeofvibrationabsorption.Thestateswitchcontrolstrategy,amethodofsemi-activecontrol,combinestheadvantagesofthepassiveandactivevibrationabsorption.However,thiscontrolstrategyintroducesadiscontinuitytothesystem.Thisisadisadvantageinanalysisofthesystemproperties.Anaveragingmethodisusedtosolvethisdisadvantage. Onapplicationinwhichhysteresisisimportantisinmeasurementstakenwithanatomicforcemicroscope(AFM).TheAFMrequiresprecisionpositioning.Figure 2{1 showsanAFM.Inthisapplication,thesampleisrstmovedalonganx-ygridusingalateralpositioningmechanism.TheAFMfeelsthesurfaceofthe 11


12 Figure2{1: Atomicforcemicroscopy sample,displacementinthecantilevertipismonitoredusingthephotodiodeand correspondingforcesaredeterminedviaHooke'slaw.Thesampleisthendisplaced intheZ-directionusingaverticalpositioningmechanismtomaintainconstant forceswiththedisplacementdeterminedbyafeedbacklaw.Inthisway,theAFM makesatopographicmapofthesamplesurface[5]. Figure2{2: Atypicalpiezoelectrichysteresisloop


13 TheaccuracywithwhichthePZTelementcanpositionthesampleisakey factortotheresolutionoftheAFM.Butthehysteresisbehaviorcanmakeit diculttoaccuratelydeterminethepositionofthestack.Figure2{2showsthe displacementhysteresisloopofatypicalstackactuatoraftervoltagesareapplied from0 V to140 V ,thenthevoltagedecreasesto 10 V ,andthenbackto0V. Wecanseethatat40V,thepositionoftheactuatorcanbeeitheratpoint\A" whenvoltageincreases,oratpoint\B"whenvoltagedecreases.Wesaythatthe displacementoftheactuator\lags"behindthevoltageasitoscillates.Unlikethe hardeningofothermaterialssuchasinsteelandcopper,thisdisplacement\lag" isrepeatablemanytimesasthevoltageincreasesanddecreases.Thehysteretic loopoftheactuatorrepeatsitselfessentiallyfortheeectivelifeoftheactuator. Hysteresisconsiderablydegradestheactuatorperformance.Toutilizethefull capabilityofpiezoceramicactuators,nonlinearhysteresisneedstobeinvestigated andmodeled. 2.1.1EarlyTheoriesaboutHysteresis Earlymodelsandtheoriesaboutpiezoceramicmaterialshavebeenbasedon linearconstitutiverelations.Still,thehystereticbehaviorhasbeenknownfora longtime.Thehysteresisloopinpiezoceramicmaterialswasrstdemonstrated byTebbleandCraik[6]in1969,whichistheresponsethatoccursfrequentlyin practice.Sincethen,manyscientistsandresearchershavecomeupwithmany hypothesesandmodelstoexplainthisphenomenonanddescribeitmathematically. Muchoftheearlierresearchwasbasedonmodelsfrommagnetichysteresis. Theearlytheoriesabouttheoriginofhysteresisfellintotwoprimarycategories.Onehypothesisisthathysteresisoccursduetoafrictionaltypeforce.The otherhypothesissuggestedthathysteresisoccursduetostrongmutualinteractions betweentheindividualmagneticmoments.


14 TheideaofafrictionalforceoriginatedwithWeberin1852,andwastermed a\restoringforce."Hesuggestedthatthereexistsarestoringforcewhichinhibits thechangeofthemagnetizationstate.Therestoringforcecouldexplainthe shapeoftheinitialmagnetizationcurve.However,itfailedtoexplaintheresidual magnetizationaftertheexternaleldwasremoved.Wiedemann,in1986,further postulatedthatthereexistsafrictionalresistanceduetotherotationofthe magneticmolecules. MaxwellandEwingsuggestedthat\mutualmagneticinteractions"couldaccountforthehysteresisphenomena.Ewingalsogavesomeconvincingcalculations tosupporthisidea.DuetoEwing'scontribution,the\frictionalforce"hypothesis wasdiscarded.Theideaofmutualinteractionofthe\magneticmoments"was wellaccepted,andmathematicalmodelswereexploredandcreatedtoexplainthis phenomenontheoretically.However,someremnantofthefractionalmodelpersists today.Actuallyitisbelievedthatthesmoothchangeinthehysteresisloopsin magnetization(whichappearsinmosthysteresisloops)isduetoafrictionalforce whichopposesthechangesinmagnetization. 2.1.2ModelstoSimulateHysteresis Earlyattemptsinmodelinghysteresiscouldbeviewedasattemptstocurve tanequationtotheexperimentalhysteresisdata.Brauerusedpowerseries, whileFisherandMorer,Widger,andHopkinsetal.usedarationalpolynomials toapproximatetheexperimentalhysteresisloop.Lateron,Rivasetal.addedtwo generatingfunctionstottheexperimentaldata.Thetwogeneratingfunctions resemblethetwoasymmetricODEfunctionsinthecurrentdomainwallmethod. Thismethodwillbediscussedindetailshortly.Thesemethodsareequivalentto curvetting,andtheydidnotinspectthehysteresisgeneratingmechanisminside thematerial.Currentpopularmodelstorepresenthysteresisinpiezoceramics (aswellasinmagnetization)includethedomainwallmodelandPreisachmodel.


15 Preisachmethodisakindofkernelmethod.Researchinthisdissertationutilizes anotherkerneltodevelopthefrequency-dependenthysteresismodel.Inthenext section,abriefreviewaboutthetwomethodswillbeprovided.Thetheoretical foundationaboutthekernelmethodwillbediscussedindetailwithinChapter 3.Earlymodelstosimulatehysteresisarebasedontheassumptionofstaticor quasi-staticoperatingconditions.Theyarecalledstaticmodelsinthisdissertation. 2.1.3StaticModelingofHysteresis 2 : 1 : 3 1 DomainWallModel Thedomainwallmethodappearedearlyinthe1960s.In1962,Globus originatedthedomainwallmethodtoexplainthegeneralshapeofhysteresis behaviorinferromagneticmaterials.Intheirpaper,itisstatedthat\thedomain wallswerepinnedongrainboundariesbyafrictionalforce.Undertheactionofthe magneticeld,thedomainwallsrstunderwentareversiblemotionduetodomain wallbulging,thenanirreversiblemotionduetodomainwalldisplacement."Itwas furthersuggestedbyJilesandAtherton[7]that\Domainwallsarealsopinned byinhomogeneitieswithinagrain,forexampletanglesofdislocations,regionsof inhomogeneousstrainandanyprecipitatesornonmagneticinclusionswithina grain."Thisisthefoundationofthecurrentdomainwalltheory. JilesandAtherton[7]developedthemathematicalmodelofthehysteresis mechanisminferromagneticmaterialsbasedonthedomainwallmotionincluding bothbendingandtranslation.Intheirpaper,ananhystereticmodelisrstcreated usingameaneldapproach,andthenahysteresismodeliscreatedbyconsidering theeectsofpinningamagneticdomainwallatdefectsites. Withinthecontextdescribedabove,anhysteresisisobservedinanidealized materialthatisdevoidofinclusionsorimperfections.Theinclusionsorimperfectionswillleadtothepinningsites,whichwillbeusedinthehysteresismodel. Therearethreetechniquestomodelthenonlinearrelationbetweentheinput


16 voltageorelectricaleldandtheresultingpolarization.Thecommonfeaturesof thethreemodelsareasfollows:( i )theyutilizeBoltzmannstatisticstoquantify theprobabilityofdipolesoccupyingcertainenergystates;( ii )theyareapplicable ifthematerialisdevoidofimperfections,andthusthereisnopinningsiteinthe material.Thedierenceliesinthepossibledirectionsofthedipolemomentswhen derivingthemathematicalmodels.Intheabsenceofanelectricaleldandabove Figure2{3:Orientationofdipolemomentandelectricaleld Curietemperature,thedipolemomentschangetheirdirectionsrandomly,resulting inanetzeropolarization.Inthepresenceofanelectricaleld,thechangeofdirectionisnolongerrandomandmicroscopicpolarizationdevelops.Figure2{3showsa dipolemomentunderthepresenceofelectricaleld E .ForadipolemomentP 0 in anelectricaleld E ,thepotentialenergyisgivenbyEquation2.1. E = P 0 E = P 0 Ecos (2.1) where,P 0 = j P 0 j E = j E j istheanglebetween P 0 and E .Accordingto Boltzmannstatistics,thethermalenergyundertemperature T is k B T ,where k B is Boltzmannconstant.Theprobabilitythatadipoleoccupiestheenergystate E is ( E )= Ce E = ( k B T ) = Ce P 0 E= ( k B T ) (2.2) where, C isancoecientwhichensuresthattheintegrationwithrespectto from0to resultsinthetotalnumberofdipolemoments N perunitvolume.By consideringallthepossibledirectionsthedipolemomentscanorientatethemselves,


17 weapplyBoltzmannstatisticsonEquation2.2,thethreeanhysteresismodels:the Langevinmodel,IsingSpinmodelandpreferredorientationmodelarederived.The angle between P 0 and E inEquation2.2willchangeaccordingly. LangevinModel:[8] TheLangevinmodelassumesthatthematerialisisotropic.Thereareno mutualinteractionsamongthedipolesandthedipolemomentscanlieinany direction.Undertheseassumptions,thefollowingequationhasbeenderived fortheanhysteresispolarization P an =P s coth P 0 E Tk B Tk B P 0 E (2.3) where,P s isthesaturationpolarization. IsingSpinModel:[9,10] Thismodelassumesthatthedipolemomentscanonlylieintwodirections: eitherinthedirectionoftheappliedelectricaleldoroppositetothat direction.Theresultinganhysteresispolarizationexpressionis P an =P s tanh P 0 E k B T (2.4) PreferredOrientationModel:[9,10] Thismodelcombinestheabovetwomodels,assumingthatthegrainsare randomlydistributedbutthatthedipoleswithineachcellhaveonlytwo directions:intheelectricalelddirectionoroppositetoit.Thefollowing equationisobtained P an = P s 2 n k B T P 0 E log 1+ e 2 E P 0 =Tk B +log 1+ e 2 E P 0 =Tk B + 1 2 k B T P 0 E 2 Li e 2 E P 0 =Tk B Li e 2 E P 0 =Tk B (2.5) where Li ( x )representsthedilogarithmfunction,denedas Li ( x )= Z 0 x log(1 t ) t dt:


18 However,accordingtoCalusius-Mossottiequation[11],thepolarizationandstress contributetotheelectricaleld.Thetotalelectricaleld,termedaseective electricaleld E e isgivenby E e = E + P+2 Q 33 P(2.6) where istheaxialstress, isaninter-domaincoecient,indicatesthecontributionofpolarizationtochangetheelectricaleld,and Q 33 denotesthelongitudinal electrostrictivecoecient.Itembodiesthecontributionofstresstothechangeof theelectricaleld.Soifthemutualinteractionofthedipolemomentsistakeninto account,theLangevinandIsingspinmodelareexpressedas P an =P s coth 3 T c E e TE 0 TE 0 3 T c E e (2.7) P an =P s tanh T c E e E 0 T (2.8) where, T c istheCurietemperature, E 0 isascalingoftheelectricaleld.The readersarereferredtoHomandSmith[12]andHomandShankar[13]fora detaileddescriptionstothederivationofthethreemodels. Inthehysteresismodel,polarizationPshowshysteresis,so E e showshysteresis.Andtheeectivechargedisplacement D e = E e ,alsoshowshysteretic behavior.Thecurrentresearchinthisdissertationusesthis\eectiveeld"idea andEquation2.6towritetheeectiveelectricaleld E e asahysteresisfunctionof theinputvoltage Q E e = F ( Q ).Thenthehysteresisfunctionisquantied. Domainwalltheoryassumesthatthenonhomogeneouspropertiesofthe ferroelectricmaterialsproducepinningsitesforthedomainwall.Withtheabsence ofexternaleld,thedomainwallisatitsequilibriumposition.Asexternaleld increases,thedomainwallbeginstobend.Attheverybeginning,thisdeformation ofdomainwallisreversible,soiftheexternaleldisremoved,itwillspringback toitsoriginalposition.Thereisnohysteresis.Whentheexternaleldisstrong


19 enough,thendomainwallistranslatedtoanewposition.Thispartoftheenergy isirreversible.Thepolarizationenergyforagiveneectiveeldlevelequalsthe totalpolarizationenergyminuslossenergynecessarytoovercomelocalenergy barriers.Hencewehavethefollowingequation 1 0 Z D e 0 P irr dD e = 1 0 Z D e 0 P an dD e Z D e 0 k d P irr dD e dD e (2.9) where,thersttermisthenetpolarizationenergyforagivenelectricaleld;the secondtermrepresentsthetotalpolarizationenergyintheidealcase;thethird termisthepolarizationenergyneededtobreakpinningsites.Asdetailedin[14], Equation2.9canbesimpliedasthefollowingordinarydierentialequationforthe irreversiblepolarizationP irr : d P irr dE = e P an P irr k (P an P irr ) (2.10) where,P an representstheanhysteresispolarization,calculatedbyLangevinmodel orIsingspinmodel.The e isdenedas e =1when dE> 0andP > P an ,or dE< 0andP < P an ;otherwiseitiszero.Thevariable e isintroducedtoensure thattheaboveODEyieldsaphysicalsolution.Thereversiblepolarizationis assumedverysmall,itisdenedasP rev = c 1 (P an P irr ),here c 1 isestimatedby aleastsquarettoexperimentaldata.Thusthedomainwalltheoryresultsinthe followingequationforthetotalpolarization P=P an +P irr (2.11) Equation2.10isthehysteresisgeneratingfunctionforirreversiblepolarization,and Equation2.11calculatesthetotalpolarization.Thesetwoequationsrepresentsthe mathematicalexpressionsofthedomainwallmethod. Thedomainwallmethodhasbeenappliedinmanyapplicationsfordierent structure.SmithandHom[12],in1999,[14],in2002,derivedadomainwall


20 modelforcharacterizingthehysteresisbehaviorinferroelectricmaterials.Smith andOunaies,[15]in1999,presentedahysteresismodelbasedondomainwall considerationstoquantifythehysteresisandnonlinearityinherenttothePZT materialsatmoderatetohighdrivelevels.Theyshewhowtodeterminetheve parametersthatdenesthegoverningODE.SmithandSalapaka[5]in,2001, developedamodelviadomainwallconsiderationforthepositioningmechanism employedintheatomicforcemicroscope(AFM).MassadandSmith,[16]in2002, appliedthedomainwallmodeltocharacterizethehysteresisbehaviorinshape memoryalloys. 2 : 1 : 3 2 ClassicalPreisachModel ThePreisachmodelwasrstseenin1935,[17]tomodeltherelationbetween theinputeld P ( t ),andtheresultingmagnetization M ( t ).PleaserefertoappendixAforthedenitionofmagnetization.ThePreisachmodelisexpressed as M ( t )= 1 Z 0 + 1 Z w ( r;s )[ k s r;s + r ( u )]( t ) dsdr Thekernel[ k s 1 ;s 2 ( u )]isaclassicalPreisachrelayoperatorwiththreshold s 1 and s 2 ,and w ( r;s )isitsweight.ApplicationsofthePreisachmodeltosimulatethe hysteresisofactivematerialshavebeenreportedin[18{22]andmanyothers. Hughesetal.[18]appliedthePreisachmodelforthehysteresisofshape memoryalloymaterials.Geetal.[20,21]adaptedthePreisachmodeltodescribe thenonlinearhystereticbehaviorofpiezoceramicactuatorsandpresenteda modiedgeneralizedPreisachmodelthatwasusedinalinearizingcontrolscheme. Smithin1997presentedaphenomenologicalcharacterizationofhysteresisin magnetostrictivetransducerviathePreisachmodel.Smithetal.[23]appliedthe Preisachmethodtoquantifythehysteresisandconstitutivenonlinearitiesforthe


21 AFM.Thehysteresisoperatorisgivenby [P m ( u; )]( t )= m X i =1 [ k n i ( u; )]( t ) n i : Thehysteresisisexpressedasthesuperpositionofweightedkernels.Theweights sarechosenthroughaleast-squaresttotheexperimentaldata[24].Within theirpapertheauthorsshowthatthatthePreisachmodelcanaccuratelyquantify thenonlinearconstitutivepropertiesofthetransducer,butattheexpensethatthe modelisphenomenological. Zhouetal.[25]appliedtheclassicalPreisachmodeltosimulatethestatic hysteresisbehavioroftheThunderActuatorSystem R .Theythenusedaninverse classicalPreisachmodeltodotherealtimemicro-positiontrackingcontrol.The realtimetrackingcontrolwasachievedbycombiningalead-lagfeedbackcontroller andtheinversemodel. Banksetal,[26]in2003,studiedconsistentandconvergentapproximation methodsfortheidenticationofthemeasurecharacterizingthehysteresis.The continuityandconvergencepropertiesofthePreisachKPkernelsguaranteethe well-posednessoftheidenticationproblem. 2 : 1 : 3 3 Conclusion Thedomainwallmodelisamorephysical-basedmodel.Itprovidesmore insightsregardingtheactuatingorsensingmechanisms.Soitistheoretically sound.However,thismethodyieldscomplexequationsofstates.Alsoasit usestwogeneratingfunctions,itcannotguaranteetheclosureoftheminor loops.Currently,severalphysicalmechanismsinpiezoceramicmaterialsarenot wellunderstood,thusmotivatingthephenomenologicalorempiricalmodelto characterizethebehaviorofthematerials.Forthispurpose,thePreisachmethodis morefavorable.


22 ThePreisachmodelisgoodforgeneralmaterialcharacterization.Itguaranteestheclosureofminorloops.Asitisnon-physicalbased,itisapplicablewhen theunderlyingphysicsarediculttoquantifyorpoorlyunderstood.However, thePreisachmodelisnotphysicallybased.Itusesarate-independentelementary operator,andisdiculttoincorporatethefrequencyandtemperaturedependence exhibitedbyessentiallyallsmartmaterials. Modelstosimulatethehysteresisbehaviorpresentedsofarhavebeenbasedon theassumptionsthattheoperatingconditionsarestaticorquasi-static.Butithas longbeenrecognizedthatthehysteresisbehaviorisrelatedtohowfasttheinputis cycled.Sorecently,someworkhasbeencarriedoutonratedependenthysteresis modelingviamodieddomainwallmethodsorPreisachmethods. 2.1.4Rate-DependentHysteresisModels Theterm\ratedependency"characterizeshowtheresponsehysteresisloopof thePZTmaterialschangeaccordingtohowfasttheinputiscycled.Itincludestwo cases.Oneisthattheloopdependsonthefrequencyoftheinputsignalforaxed inputlevel;theotheroneisthattheloopdependsontheinputlevelforaxed inputfrequency.Smithetal.[27]developedaratedependentmodeltopredict thehysteresisinpiezoceramicmaterialsoperatingatlowfrequency.Theystated inthemodelthat,intheabsenceofanelectricaleld E ( t ),thedipoleorientation changesduetothethermaluctuation.Thechangeisrandom.Howeverinthe presenceofanelectricaleld,thedipolenumberinthedirectionofelectricaleld (oroppositetothedirectionoftheelectricaleld)changeswithfrequency.This frequency-dependentmodelgivesanaccuratepredictionatlowfrequencyrange. However,thismodelisonlyapplicableatverylowfrequencies.Forexample,in theirpaper,theauthorsgavemodelvalidationforfrequenciesfrom0 : 1Hzto1Hz, whichusuallyisnottheoperatingfrequencyrangeofinterest.Piezoceramicsare oftenutilizedinfrequencyregimesthataremeasuredinkilohertzormegahertz.


23 Angetal.[28]developedaratedependenthysteresismodelforapiezoelectric actuatorin2003.ThispaperusedamodiedPrandtl-Ishlinskii(PI)operatorto accountforthehysteresisofthePiezoelectricactuatoratvaryingfrequencies. IntheapplicationofemployingthePreisachmethodtoquantifytheratedependenthysteresis,therearetwoimportantcases,oneusesrate-independent elementaryoperators,andtheprobabilitymeasureisrate-dependent,asinEquation2.12. [P ( u;; )]( t )= Z S [ k s ( v; ( s ))]( t ) d f ( s )(2.12) where, f ( s )indicatesthattheprobabilitymeasure israte-dependent. Thisexpansionincorporatesfrequencydependence,butmakesthemodel extremelydiculttoimplementformaterialcharacterizationandcontroldesign. Moreover,Torre[29],showsthatatlowdrivelevels,modicationsmustbemadeto ensureaccuracyofminorloopsinsidethemajorloops.Thesedicultiesmotivate theenergyformulationforthePreisachkernels,thuscomesthesecondcaseof hysteresisrepresentation.Thisrepresentationusesrate-dependentoperators. However,theprobabilitymeasureisrate-independent,asinEquation2.13. [P ( u;; )]( t )= Z S [ k s ( v; ( s ))]( t ) d 0 ( s )(2.13) where 0 ( s )isavectorvaluedmeasure. Smithetal.[24]usedfreeenergyrepresentationsforthedipoleswitching todevelopthedynamichysteresismodel.Byemployingtheenergymethod, thefrequency-dependent,temperature-dependentandload-dependentterms areincorporatedinthekernelsratherthanintheprobabilitymeasure.This factisdemonstratedinEquation2.13.Thisapproachalleviatesthediculties encounteredwhenincorporatingthevaryingtermsintheprobabilitymeasure. Galinaitis[30]developedaratedependenthysteresismodelforapiezoelectric stackactuatorin2004.Thispaperpresentsarate-dependenthysteresismodel


24 byemployingratedependentelementaryhysteresisoperators.Forthiswork,a resistor-capacitor(RC)circuitisusedfortheelementaryoperatorandthetime dependentchargeonthecapacitoristheoperatoroutput. Modelvalidationemployingrate-dependentelementaryoperatorshavebeen giveninthesepapers.However,onlytheresponsesunderdierentinputlevelsare given.Theydonotstudytheresponseofthemodelsfordierencesinfrequencies. Oneobjectiveofthecurrentresearchistostudythefrequency-dependencyin hysteresisofthepiezoceramicmaterials.Afrequencydependentmodeliscreated thatcanbeusedforprecisionpositioningandcontrol. 2.2VibrationAbsorbersandLiteratureReview Vibrationcancausestructuralormechanicalfailure,frequentandcostlymaintenanceofmachines,andunacceptablelevelsofnoise.Thereforeitisnecessaryto controloreliminateexcessiveandundesirablevibrations.Therearemanywaysto controlvibration.Onewayisbytheinsertionofanisolatorbetweenthevibrating massandthevibrationsourcetoobtainthevibrationreduction.Thissection focusesonthestudyofavibrationabsorbertoreducetheundesirablevibrations. 2.2.1VibrationAbsorbers Figure2{4:SingleDOFvibration Avibrationabsorber(VA)hasbeenshowntobeaneectivewaytoreduceor eliminatetheexcessivevibrationsthatamachineorasystemmayencounter.The vibrationabsorptionisusuallyachievedbysupplementingtheoriginalsysteman additionalspringmasssystem,orpossiblyadamper.Ifpossible,theparameters


25 arecarefullychosensothatthenaturalfrequenciesofthenewsystemare\wellseparated"fromtheexcitationfrequency.Avibrationsystemisusuallymodeled asamass-springsystem.Figure2{4showsamodelforavibrationsystem.Now aspring-masssystem(mass m 2 )isattachedtotheoriginalsystem,asshownin Figure2{5.Thegoverningequationofthenewsystemis Figure2{5:Classicalvibrationabsorber 2 6 4 m 1 0 0 m 2 3 7 5 2 6 4 x 1 x 2 3 7 5 + 2 6 4 c 2 + c 1 c 2 c 2 c 2 3 7 5 2 6 4 x 1 x 2 3 7 5 + 2 6 4 k 1 + k 2 k 2 k 2 k 2 3 7 5 2 6 4 x 1 x 2 3 7 5 = 2 6 4 f 0 3 7 5 (2.14) TheLaplacetransformgivesthetransferfunctionfromtheinputforcetothe displacementofmass m 1 X 1 F = k 2 + c 2 s + m 2 s 2 ( k 1 + k 2 +( c 1 + c 2 ) s + m 1 s 2 )( k 2 + c 2 s + m 2 s 2 ) ( k 2 + c 2 s ) 2 (2.15) Inthefrequencydomain,thefrequencyresponsefunctionis X 1 F = k 2 + jc 2 m 2 2 ( k 1 + k 2 + j ( c 1 + c 2 ) m 1 2 )( k 2 + jc 2 m 2 2 ) ( k 2 + jc 2 ) 2 (2.16)


WhenwedividethenumeratoranddenominatoroftheEquation 2.16 byk1k2,weobtainEquation 2.17 !n;22+j2!n;2! 0B@!4 where,!n;i=k1=m1,ci=mi=2i!n;i,i=1;2.Equation 2.17 showsthattheresonanceofmassm1couldbecancelledifweset!n;1=!n;2.ThecorrespondingfrequencyresponsefunctionisshowninEquation 2.18 !n;22+j2!n;2! !n;13h2+412+k2 !n;12+j[21+22]! !n;1+1(2.18)Figure 2{6 showsthebodeplotofthedamped2-DOFdampedvibrationsysFigure2{6: T.F.ofdisplacement-forceofmassm1


massm1whenaVAisattached.Wecanseethatwhen!n;1 Still,intheundampedcase,c1=c2=0,thetransferfunctionreducesevenfurthertotheforminEquation 2.19 .Figure 2{6 showstheBodeplotofthemassm1oftheundampedvibrationsystem(UVS),with!n;1 Figure2{7: T.F.ofdisplacement-forceofmassm1 WecanalsoconcludefromFigure 2{6 and 2{7 that,ifnodampingispresent,theresponseofthe2-DOFsystemisunboundedatthenewresonancefrequencies,andzerofortheoriginalmassattheresonantfrequencyoftheoriginalSDOF


system.However,theadditionalmasswillstillhaveamotionattheoriginalresonantfrequency.Howeverifdampingispresentineithermass-springelement,theresponseofthemainmasswillbeniteatthetwonewresonantfrequencies,andnolongerbezeroattheresonantfrequencyoftheoriginalsystem.Massm1isvibratingunderanexternalforce,butthemotionisbalancedbytheforceduetothemotionofmassm2.Somotionofmassm1isattenuated(dampedcase)orcompletelyabsorbed(undampedcase).Massm2vibratesfreelywithoutanyforceappliedtoit,soitvibratesatitsundampednaturalfrequency. TheundampedVAsarealsocalledde-tuners.Thedampedvibrationab-sorbersaresometimessimplycalledtunedvibrationabsorbers,dynamicvibrationabsorbers,orvibrationneutralizers. Thepassive,tuned,undampedvibrationabsorberhasproventobeaneectivewaytoreduceundesirablevibration.However,ithasalimitation:theabsorberiseectiveprovidedthesystemisoperatedunderconstantexcitationfrequencyortheexcitationfrequencyremainsinanarrowrange.Withtheadditionofanabsorber,therearetwonewresonantfrequencies.Oneissmallerthantheoriginalresonantfrequency,theotheroneislargerthantheoriginalresonantfrequency.Iftheexcitationfrequencyshifts,theresponseofthenewsystembecomeslargequickly,andbecomesunboundediftheexcitationfrequencyshiftstooneofthenewresonantfrequencies.Sothereisanarrowfrequencybandoverwhichtheexcitationfrequencymaychange,whileallowingvibrationreduction. In1928,OrmondroydanddenHartogintroducedadampingfactorinthesystemtoavoidanunboundedresponseattheresonanceatthetwonewresonantfrequencies.Thismethodmakesthefrequencyrangewider,butdoesn'tcompletelyabsorbthevibrationattheresonantfrequencyoftheoriginalsystem.


SoomandLeeexaminedoptimalparameterssettings,whichincludechoosingabsorbermass,springanddampingfactortominimizethemaximumresponseofthesystem(forexample,displacementandvelocity).2.2.2-2.ActiveVibrationAbsorbers Tunedvibrationabsorbers(TVAs)areextremelyeectiveforattenuationofasingleknownresonantfrequency,butareonlyeectiveinaverynarrowbandwidth.Activevibrationabsorbershavebeendevelopedtocontrolvibrationsoverawiderbandwidthandwithlessknowledgeofthesystemparameters. OnedierencebetweenapassiveVAandanactiveVAisintherequirementofexternalenergy.InthepassiveVA,noexternalenergyisneeded.IntheactiveVA,anexternalforce,denotedasfa,isusedtocontrolthemotionoftheabsorber.Thentheoverallsystemresponseiscontrolledusingvariouscontrolalgorithms.Thespeciccontrolstrategieswillbereviewedinlaterpartofthissection.Figure Figure2{8: Activevibrationabsorberconguration 2{8 showsanactiveVAconguration,withaforcegeneratingmechanismbetweenabsorbermassandmainmass,parallelwiththespringanddamper.Thismakesthesystemmoreexibleandamenabletocontroltheory. Feedbackcontrollawsdesignedusingvariousmethodsareaddressedin[36,37]and[38].Stephenetal.[36],1991,usesthelinearcombinationofthevelocityand


accelerationoftheoriginalsystemasthecontrolsignalasfeedbacktothesystem.Setoetal.[37],1989,SetoandSavatari,[38],1992,usesthelinearquadratic(LQ)optimalcontroltheorytodesignthefeedbackcontrolsignal.Itisshownfromexperimentthatthevibrationofthecontrolledsystemisdramaticallyattenuatedcomparedtotheuncontrolledsystem. BurdissoandHeilmanndesignadual-massactivevibrationabsorberforwidebandwidthcontrol[39],1998.ItisshowninthepaperthatthedualmassstrategyrequireslesscontrolforcethanthesinglemassactiveTVAtoobtainthesamelevelofreductionvibration.Figure 2{9 givesaschematicviewofthedualmassconguration. Figure2{9: Dualmassconguration 2.2.2-3.Semi-ActiveStrategies Activevibrationcontrolsystemscanhavemanylimitations.Whilesometimeshighlyeective,theycanrequirecostlyandhighlysophisticatedcontrolalgorithms.Inaddition,thestabilityandrobustnessofthesystemmustbestudiedcarefullyinanactivecontrolformulation.Whensubjectedtoanunanticipatedexcitation,thecontrolforcecandrivethesystemintoinstability.Atleastfromanenergyconsumptionviewpoint,thecontrolforceshouldbeminimal.However,apassiveVA,byitsnature,inducesnegativefeedback.Thusstabilityandrobustnessareensured.DuetothedisadvantageofthepotentialinstabilitythatcanbeinducedinanactiveVA,thesemi-activestrategywasdevelopedtoreducethe


amountofexternalenergyneededtoachievethedesiredperformance.\Thesemi-activestrategyistoplacetheactiveforceinactiveVAwithacontinuallyadjustableelementwhichcanvarytherateofenergydissipationinresponsetoaninstantaneousconditionofmotion"[40].ItisclaimedthatthismethodcombinestheadvantagesofbothpassiveandactiveVAs.First,itiseectiveoveralargerbandwidththanpassiveVA,moreover,itrequireslessforcetodrivethesystem.AnecdotalevidencesuggeststhatthesystemismuchmorestableandsimplerthantheactiveVA.Figure 2{10 showsaschematicofthesemi-activeTVAconguration.Thesystemcanalteritsinertia,stiness,dampingfactororinitialconditions. Figure2{10: Semi-activeVAconguration AtimevaryingstinesswasusedbyHubbardandMargolis,[41]in1976,forvibrationisolationapplications.Fujinoetal.,1993usedavaryingstinessforcablevibrationattenuation.Hinoetal.,1994usedavaryingstinessforstructuralcontrolapplications. Variabledampingseemsquiteeectiveinreducingthetransientandsteadystatevibrations.Applicationsofvariabledampinghavebeenreportedinmanysources.Forexample,Hrovatetal.,[42],1983usedsemi-activedampersforvibrationcontrolofcivilengineeringstructuressuchasbuildingsandbridges.


TanakaandKikushima,[43],1992developedaself-optimizingdamperforreducingmachinetooloscillationsandimpactvibrations. Therearemanytechniquesforobtainingaphysicalrealizationofthead-justabledamper.Thesemethodsincludetheusedofhydraulicactuators,electro-rheological(ER)uids,magneto-rheological(MR)uidsandpiezoelectricmateri-als.ApplicationofahydraulicactuatorisgiveninPatten,1996. Amongthetworheologicaluids,MRispromisingduetothefactthatitcanoperateatlowvoltage,suchasthevoltageappliedbyavehiclebattery.Detailsaboutthemechanicsofthetwouidscanbefoundin[40].Thispaperalsoaddressescommonlyusedcontroltechniques,includingtheon-osemi-activestrategyandoptimalsemi-activecontrol.Jalilietal.,[44],2001,haveachievedvibrationattenuationviaanadjustableinertialabsorber.Gavin[45],2000,designedasemi-activedamperusingelectro-rheologicaluidwithintheactuator.Tentorinvestigatedtheuseofanelectromagneticelementasanabsorber,whichheusedwithbothsemi-activeandactivecontrolinhisPh.D.dissertation[46].Thesemi-activeVAisrealizedbyalteringthegapspacing,thuschangingthesystemstiness.ActiveVAisrealizedbyprovidingthesystemaDCcurrent.Itisalsopointedoutthatthevariablegapandcurrentchangethesystemtransferfunctiontominimizetheprimarysystemresponse.[46].Vavreckdevelopedadynamicvibrationabsorberwithamagneto-rheologicaluiddamperinsteadofadashpot[47].Twocontrolschemeswerediscussed:theskyhookcontroltechniqueandlinearquadraticoptimalcontrol.Simulationresultsweregivenforcomparison.Anotheradjustableschemeisthehybrid-switchedtreatment.Thisdesignhastwomodes:anactivemodeandapassivemode.Botharerealizedbycontrollingaswitchonando.Theobjectiveofthisstrategyistosimplifythecontroleort.


AsmentionedinChapter 1 ,thehighinternalimpedanceandhighenergytransmissioneciencyofthepiezoceramicmaterialsmakethemagoodcandidateforvibrationsuppression.WhenaPZTisattachedtoavibratingstructure,thePZTmaterialisstrainedbecauseofthestructuralvibrationgeneratingavoltage.Thiselectricalenergycanbedissipatedthroughashuntedcircuit.Thus,vibrationwithinastructurecanbesuppressed.Moreover,ithasbeenarguedthatpassiveelectronicdampingusingpiezoelectricceramicsismoretunablealternativetoviscoelasticdampingtreatments. EarlyapplicationsofPZTactuatorstosuppressvibrationsutilizedthepassiveshunteddesign.Earlyinthe1970s,researchersinvestigatedthepossibilityofusingPZTmaterialsforvibrationcontrol.ForwardoriginatedtheideaofusingPZTelementsshuntedwithapassiveelectricalcircuitforvibrationcontrol.Heusedaverysmalltransducertosuccessfullyreducethevibrationinanopticalsystem[48].HagoodandvonFlotow[49],in1991,werethersttostudyquantitativelythedissipationofenergywithpiezoceramicmaterialsshuntedwithpassiveelectricalcircuits.Twotypesofcircuitsarestudiedintheirpaper,aresistorcircuit(RC)andaresistor-inductorcircuit(RLC).Itwaspointedoutthat,\Forresistiveshunt-ing,thematerialpropertiesexhibitfrequencydependencesimilartoviscoelasticmaterials,butaremuchstierandmoreindependentoftemperature.Shuntingwitharesistorandinductorintroducesanelectricalresonance,whichcanbeop-timallytunedtostructuralresonancesinamanneranalogoustoamechanicalvibrationabsorber".Alsoacantileverbeamexperimentwasconductedtovalidatethetwomodels.Figure 2{11 showsthetwopassiveshuntedcircuits. HagoodandvonFlotow'sinitialworkcreatedafoundationforcurrentpassiveandadaptiveshunteddesign.Sincethen,muchresearchhasbeenconductedtoimprovetheshuntingtechniqueandvibrationsuppressioneect.Edbergetal.,usedanelectroniccircuitinsteadoftheheavyinductorusedbyHagoodandvon


Figure2{11: Passiveshuntedcircuit(usedbyHagoodandvonFlotow) FlotowinashuntedRLcircuit.Edbergetal.,[50],1991andHollkamp[51],1994,studiedthepossibilityofdissipatingmultiplevibrationmodesusingonetuned,shuntingcircuit. Wu[52],in1996,foundthattheshuntedcircuitwitharesistorinparallelwithaninductor(circuitisshowninFigure 2{12 )iseasiertotunethanthecircuitwitharesistorinserieswithaninductoraswasusedbyHagoodanvonFlotowin[49].Theamplitudeofthedisplacementresponseversusfrequencyofastructuralmodewasstudiedunderseveralcircumstances.Undertheoptimumtuningcondition,thepeakofthecurvedecreaseswithanincreaseoftheshuntresistance.Itbecomesaplateauattheoptimumresistance.ThestudybyWu[52]alsoshowshowtheamplitudechangeswiththechangeoftheshuntedresistance,inductorandthemass. Figure2{12: Passiveshuntedcircuit(usedbyWu) Insummary,fourbasicpassiveshuntedcircuitshaveappearedforthesuppres-sionofvibration.InadditiontothetwocircuitsshowninFigure 2{11 ,theothertwoarecomprisedofashuntedcapacitor(CC)andshuntedinductor(LC),asshown


inFigure 2{13 .\Eachofthesekindsofshuntsresultsincharacteristicallydierentdynamicsbehavior:aresistiveshuntdissipatesenergythroughheating,whichhastheeectofstructuraldamping.AninductiveshuntresultsinaresonantLCcircuit,thebehaviorofwhichisanalogoustothatofamechanicalvibrationabsorber(tunedmassdamper).Acapacitiveshuntchangestheeectivestinessofthepiezoelectricelementandcanbeusedasatunablemechanicalvibrationabsorber.Aswitchedshuntoersthepossibilitiesofcontrollingtheenergytransfertoreducefrequencydependentbehavior,orperhapstheconversionofenergytoausableform".[53] Figure2{13: Passiveshuntedcircuit 2.2.3-2.SwitchedShuntedStrategy Asdiscussedbefore,theadaptivepassiveorsemi-activecontrolisanalter-nativetoactivecontrol.Activecontrolrequiresexternalpowerforactivatingthecontrol,andstabilityisnotensured.Frequently,thesystemisnotrobusttoparam-eterchange.Theideaoftheadaptivepassive,orswitchshuntedstrategy,isthattheequivalentstinessandresistanceofthePZTisalteredtoobtaintheobjectiveofdissipatingenergyfromthevibratingsystem.Sincetheactuatorsarepassivedeviceswhosecharacteristicscanbechangedinrealtime,theirperformancecanconceivablybebetterthantheirpassivecounterparts.Thereisaslightdierenceinconceptbetween\adaptivepassive"and\state-switch"strategies.Adaptivepassivemethodsemploycontinuouslyvariableshunts,whichmeanthattheshunt


circuitparameterscanbealteredcontinuouslyaccordingtoenvironmentoroperat-ingconditions.The\stateswitch"techniqueisacomparativelynewconcept.Theterm\stateswitch"meansthattwoormorecircuitsareswitchedaccordingtoacertainswitchingcontrollaw. Theterm\stateswitch"wasproposedbyMunkin1980for\stateswitchedacousticsource".Itwasrstemployedin[54]byLarsonandRogersduring1994.Intheirpaper,authorsusedanacousticsourcewithanactivespringmadefromPZTtodemonstratetheconceptofastateswitch.LarsonandRogersalsodevelopedanunderwatertransducerforhigh-amplitude,lowfrequencysignals,[55]in1998.Thispaperfurtherdevelopedtheconceptof\stateswitch"systemsanddemonstratedthestateswitchinginanacoustictransducer.Theswitchstrategycanchoosebetweentwodistinctvaluesofspringstinesssothatthesystemhastworesonancefrequencies,oneassociatedwitheachconguration. Figure2{14: Stateswitch(a)Open-shortcircuit,(b)Open-resistivecircuit Clark,[56,57],in1999,usedaPZTactuatorshuntedwithanelectricalcircuitforthepurposeofdissipatingenergyfromasimplevibrationsystem.Themodelofthesystemwascreated,thecontrollawforstateswitchingwasestablishedandtwokindsofswitchingcircuitswerediscussedinthepaper:switchingopenandshortcircuit(OC-SC),andopenandresistivecircuits(OC-RC),asshowninFigure 2{14 .Theimpulseandharmonicresponsesofthetwoswitchingsystemsarecomparedwithapassive,shunted,resistivecircuit.Althoughthepassive,shuntedresistive,


circuitshowsbetterresultsattheoptimalresistancevalue,thestateswitchedsystemsarerobusttochangeintheresistancevalue.TheOC-RSstrategyperformsslightlybetterthantheOC-SC.Theenergydissipationisobtainedthroughthefollowingprocedure:\whenthesystemmoves,theactuatorisheldinitshighstinessstatesuchthattheenergycouldbestoredintheactuator.Whenthesystemmotioncouldcauseittoreceiveenergybackfromtheactuator,theactuatorisswitchedtoalowstinessstate,dissipatingenergy."[56,57].Accordingtothis,theswitchingcontrollawisdeveloped.\Whenthesystemismovingawayfromequilibrium,y_y0,thecircuitisswitchedtohighstiness,andwhenthesystemmovingtowardstheequilibrium,thecircuitisswitchedtolowstinesstodissipateenergy".Thesysteminthispaperisa1-Dvibrationsystem,showninFigure 2{15 .Kurdilaetal.,[58],in2002,addressedthestabilityproblemofthevibrationsystemstudiedbyClarkin[56].Theoftenusedstate-switchstrategiesarelistedbelow: Figure2{15: Schematicof1-DvibrationsystemwithPZT Intheshortcircuitcase,asthewirealwayshassomeimpedancesoitisusuallymodeledasaresistorwithlowimpedance.


vector,whichcontinuouslychangesineachstate.However,thewell-establishedmoderncontroltheoriesaremostlybasedonlineartimeinvariant(LTI)model,whichtakeseithercontinuoustimeformsordiscretetimeforms,butnotboth.ThusitisdesirableforustouseaLTImodelofthesysteminsteadoftheoriginaltime-varyingswitchedmodel.ModelingmethodsarethusneededtodevelopforderivingtheLTImodel,amongthemtheaveragingmethodisthemostpopularchoice.Inthispartofmyresearch,theaveragingmethodisusedtoobtainthequantitativeanalysisofthestateswitchedpiezostructuralsystem. Analysisofperiodicswitchinghasbeenreportedin[59{62],etc.Mostoftherelevantliteraturementionedaboveisbasedonanaveragingtechniquetosimplifytheanalysis.State-switchaveraging(SSA)isthemostwidelyusedaveragingapproachformodelingandanalysisofconventionalswitchingdc-dcconverters. Next,weintroducetwoconcepts:thehybriddynamicalsystemandtheswitchedsystem.Manysystemscanbedescribedbyoneorseveraldierentialequations.Ahybridsystemisasystemwhosestatesincludecontinuousvariablesanddiscretevariables.Theevolutionofanautonomoushybridsystemcanbedescribedbythesystem _x(t)=f(x(t);m(t))m(t)=g(x(t);m(t)) (2.21) where,x2Rn;m2M=fm1;m2;mlg.Thevariablexrepresentsacontinuousstate,andmrepresentsadiscretestate.Thenotationtmeansthatm(t)ispiecewiseconstantfromtheright. Ifforeachx,severaldiscretestatesarepossible,thenthesystemiscalledahybridsystem.However,ifforeachx,onlyonemi2Mispossible,thesystemisknownasaswitchedsystem.Switchedsystemsincludemulti-modalsystems,orsystemswithvariablestructure,andconstituteaspecialclassofhybriddynamic


systems(PetterssonandLennartson,[63]in1996,Houetal.,[64]in1996).Next,thegeneralaveragingmethodandthestatespaceaveragingmethodforaperiodicswitchcontrolsystemwillbediscussed.2.2.4-1.GeneralAveragingMethod Averagingmethodshavebeenwidelyusedastoobtainapproximatesolutionsofdierentialequationsforover100years.Usuallyswitchedandhybridsystemscanberepresentedbyacollectionofordinarydierentialequations.Theseequa-tionscanbedividedintotwocategories:onetimescale(OTS)andtwotimescale(TTS)equations.Thestatespacemodelofaswitchedhybridsystemhasadiscon-tinuousrighthandside(RHS).HowevertheexistenceanduniquenessofsolutionstotheODEarefrequentlybasedontheassumptionofcontinuity(Lipschitzcontinu-ity).Therefore,itisveryimportanttoestablishcontinuousapproximationsoftheoriginaldiscontinuoussystem. Aonetimescalesystem,alsocalledaslowsystem,containsstatevariablesthatevolveslowlywithtimeduetoasmallparameter".ThestatespacemodelofsuchasystemcanbewrittenintheformofEquation 2.21 dt="f(t;X)(2.22) subjecttotheinitialconditionX(t0)=X0.Inthisequation,X2Rn;f:R+Rn!Rnmaybeadiscontinuousfunction.Thevariable">0isaverysmallnumber.WhenweaveragetheOTSstatespaceequation,theresultingaveragedmodelisgivenby subjecttotheinitialconditionXav(t0)=X0.Thefunctionfavisdenedasfav=limT!11


IffispiecewisecontinuousandsatisesseveralotherassumptionsoutlinedinBainovandMilusheva,[65],Sunetal.,[66],andSanders,[67],ithasbeenproventhat lim"!0kX(t)Xav(t)k=0;for8t2[t0T0="](2.24) WesaythattheaveragedsystemsolutionXavapproachestherealsystemsolutionX(t)onatimescale1 Twotimescalesystemsisalsocalledmixedsystems.ThestatespaceequationofsuchsystemscanbewrittenintheformofEquation 2.25 .StatevariablescontainedinXevolveslowlybecauseofthesmallparameter".HoweverstatespacevariablescontainedinYevolvemorequickly. dt="f(t;X;Y)dY dt=g(t;X;Y)(2.25) TheseequationsaresubjecttotheinitialconditionsX(t0)=X0andY(t0)=Y0.WehaveX2Rn,f:R+RnRm!Rn,g:R+RnRm!Rmand">0isaverysmallnumber.[68] ToobtaintheaveragedsystemforthemixedsystemdescribedinEquation 2.25 ,weassumethat'(t;X;;c0)issolutionoftheequationdY dt=g(t;X;Y),withaninitialconditionY()=c0,atxedX.where,c0isdenedonthedomainofY. Forthisequation,theinitialconditionisXav(t0)=X0,andf0isdenedinEquation 2.27 Alsowiththesameassumptionsareassumedtoholdfortheslowsystem,Sunetal.,provedthattheaveragedsystemXav(t)approachestheoriginalsystemonatimescale1


2.2.4-2.State-SpaceAveragingforPeriodicSwitchedSystem IftheconverterswitchesperiodicallybetweenNLTIsub-systems,eachofwhichcorrespondstoaswitchedstateoftheconverterinoneswitchingperiodTs,itsstatespacemodelispiecewiselinearandhasthefollowingform: dt=AiX+biU(2.28) whereX2Rn1isthestate,Ai2Rnn;bi2RnmandU2Rm1istheinputvectorofthesystem. Supposediisthedutyintheswitchedcycle.Toobtaintheaveragedsystem,rst,weneedtowritethestatespaceequationintothestandardformasdescribedintheTheorem,showninEquation 2.26 .Forthispurpose,weintroduceanewscalevariableandlett=",where"isaverysmallnumber.Wewillseelaterthatthesmallparameter"actuallyistheswitchingperiodTs=1=fs.Whenswitchingfrequencyisveryfast,Tsisaverysmallnumber.DenotethatX(")=Z().WehaveEquation 2.29 Dene where=1;2r.Thesystemhastwodiscontinuouspointswhentheswitchisonandwhenswitchiso.Ineachphase,itisacontinuoussystem.WhenwecarryoutaveragingofEquation 2.27 ,weobtaintheaveragedmodelforthenewfunctionZ(t),showninEquation 2.31 d=""rXi=1diAi!Z+rXi=1diBi!U#(2.31)


IfwechangethevariablebacktoX(t),wewillgetdX dt=rXi=1diAi!X+rXi=1diBi!U


ThischapterestablishesanonlinearconstitutiveequationforPZTmaterials,andpresentsthetheoreticalfoundationofthecurrentresearch.Theequationsusedtodevelopafrequency-dependenthysteresismodelarepresented. 1 ,atlowdrivelevels,piezoceramicmaterialsexhibitnearlylinearproperties.Alinearmodelissucienttocharacterizethedynamicbehaviorintheseoperatingregimes.However,atmoderatetohighdrivelevels,thepiezoceramicmaterialsshownonlinearitiesandhysteresis.Applicationsatthesedrivelevelsmustconsiderthenonlinearitiesandhysteresis. Withoutconsideringstructuraldampingandanyconstitutivenonlinearities,thermodynamicargumentsbasedonGibbsfreeenergyandMaxwell'sequationofelectromagnetism[70]yieldthefollowingconstitutiveequationforPZT Inthisequation,,",EandPdenotethestress,strain,electricaleldandpolarizationinthePZT,respectively.SymbolsanddescriptionsofthecoecientsarelistedinTable 3{1 .WecanseethattherelationshipbetweenP,,"andEislinear.Also,wehavethelinearrelationQ=Eh,wherehisthethicknessofthePZTacrosswhichtheinputvoltageQisapplied.Thisequationappliesatlowinputvoltages,whenhysteresisandnonlinearitiesarenegligible. 43


Table3{1: Symbolsanddescriptions SYMBOLS DESCRIPTION 3.1 as Inthisequation,cDistheKelven-Voigtdampingfactor.ThehistorydependentfunctionalF()quantiestheintrinsicnonlinearityofthematerials.TherearenumerouschoicesforthehysteresisfunctionalF.TwocommonchoicesincludetheDomainWallMethodandthePreisachmethod[5].Preisachmethodisakindofkernelmethod.Itusestheidealrelayoperatoraskernel.Butthereislimitation


onthiskernel.Inthisdissertation,weareinterestedindevelopingafrequency-dependenthysteresismodelbasedonanotherkernel-theKrasnolselskii-Pokrovskii(KP)kernel.Innextsection,theKPkernelmethodisreviewed. 3.3.1FundamentalsoftheKPkernelMethod 3{1 .Note Figure3{1: AtypicalkernelinPreisachmodel thatthekernelischaracterizedbytwoconstants(;),with.Sincethehystereticresponsealsodependson(;),wedenotethekernelask;(u;).The


totalhysteresismappingfrominputtooutputisexpressedas [h(u;)](t)=ZP[k;(u;)](t)d(3.3) where,=(;)isaweightfunction,alsoreferredtoasadensityfunction.Thisfunctiondependsonthephysicalpropertiesofthematerial.Thekernelk;;(u;)isa\multi-valued"mapping.ThePreisachplaneisdenedasP0=f(;)2R2:g Thedensityfunction(;)isusuallyassumedtohavecompactsupport.Forsomevalue<0and>0,thecontributionofisverysmall,sowelet0for<0and>0.WethereforedenetheeectivePreisachdomainP=f(;)2P0j;0;0gPisabounded,triangularregioninR2. [hm(u;)](t)=mXi=1[kni(u;ni)](t)wni(3.4) where,niisanodeinthenitePreisachdomain.Thetermwniisaweightvalue,andisdenedinthesummation=mXi=1wnini


Figure3{2: DiscretizationofthePreisachplane SinsteadofP.SisdenedasS=f(s1;s2)2R2js1=smin+(j1)s;s2=smin+(k1)s;jkg 3{2 showsatriangularSplane. Thesmallerthemeshparametersis,thebetterweareabletomatchex-perimentalresultsduetotheincreasedresolution.Thetradeoisthatreningthemeshsignicantlyincreasesthecomputationtime.Thismethodofapproxima-tionhasniceconvergencepropertiesundermildassumptions.BanksandKurdilastudiedtheconvergenceandwell-posednessoftheidenticationproblemin[72].


3{1 ,wheres1ands2arethetwothresholdsoftherelayoperator,withs2s1.Thesymbolu(t)istheinputofthesystem,andisassumedtobecontinuousintime. Figure3{3: Preisachkernels Theinput-outputrelationoftheidealrelayoperatorisasfollows:Assumeattheinitialtimet0,thesysteminputu(0)s1,thehysteresiskernelks(u;)hasvalueof1.Iftheinputu(t)monotonicallyincreases,ksremainsat1untiltheinputreachess2,andthentheresponseswitchesto+1.Iftheinputstartsfromu(0)s2,anddecreasesmonotonically,ksremainsat+1untiltheinputreachess1,andthenksswitchesto1. Butascanbeseenfromtheguredirectly,thismappingfromtimetothevalueofthekernelisnotcontinuous.Thiscanbeundesirableinpractice.Thustwoothercandidatekernelsareusedforthekernelmethod,whichareshownin Figure3{4: Ridgefunction


Figure 3{3 aand 3{3 b.ThekerneldepictedinFigure 3{3 bisthewell-knownKrasnolselskii{Pokrovskiioperator,orKPkernel.KPkerneliswidelyusedinmathematicalmodelsformulti-valuedproblem.TodenetheKPkernel,werstintroduceashiftedridgefunctionrs=r(us).TheKPkerneliscomprisedoftworidgefunctionsrs1=r(us1)andrs2=r(us2),withs2s1.Figure 3{4 depictsaplotofasingleridgefunction.Theslopeofthisparticularridgefunctionisdeterminedbya.TherelationoftheridgefunctionbetweentheinputandoutputisgivenbyEquation 3.5 Figure3{5: Majorloopandminorloop Theinput-outputrelationshipassociatedwiththeKPkernelinFigure 3{3 .bisdescribedasfollows:whentheinputmonotonicallyincreasesfromus1,theoutputfollowspath1,whenitdecreasesfromu(s2+a),theoutputfollowspath2.ButforaninputsignalasinFigure 3{5 (a),ifamonotonicallyincreasinginputdecreasesbeforeitreachess2+a,kstakestheminimumofthecurrentvalueandtheupperenveloperoute.Ifamonotonicallydecreasinginputincreasesbeforeitreachess1,kstakesthemaximumofthecurrentvalueandthelowerenvelopecurve,asdepictedinFigure 3{5 (b).Theupperandlowerenvelopecurvesdene


thehysteresismajorloop.Thepathconsistingof(2),(3),(4)and(5)denesahysteresisminorloop.ThemathematicalexpressionforthehysteresisoperatorisgivenbyEquation 3.6 [R(u;Rk1)](t)=8><>:maxfRk1;r(u(t)s2)g;ifuisnondecreasingminfRk1;r(u(t)s1)g;ifuisnonincreasing(3.6) where,RisdenedbyEquation 3.7 [Rk](t)=8><>:Rfu;Rk1(tk)g;k=2;3jR0=;k=1;2f1;+1g(3.7) TheinitialconditionisgivenbyR0=.Denitions 3.6 and 3.7 ensurethattheminorloopsrestinsidethemajorloop. 3.2 ,thenonlinearconstitutiveequationcanalsobewrittenas


or Inequation 3.8 and 3.9 ,EeandDedenotetheeectiveelectricaleldandeec-tivechargedisplacement,respectively.Theybothexhibithystereticresponsesbehaviors.However,E=Q=tandD=Edonotexhibithysteresis. ResearchinthisdissertationusesEquation 3.9 todevelopafrequency-dependenthysteresismodel.Therstequationof 3.9 isusedtoderivethegovern-ingequations.Thesecondequationof 3.9 isusedtoquantifythenonlinearityandhysteresisofthePZTmaterials,inwhichEeisexpressedbyEquation 3.4


Thispurposeofthischapteristoderivetheequationsofmotionofthepiezo-electricstackactuatorbyutilizingtheconstitutiveequationshowninEquation 3.9 .Theformoftheequationofmotion(EOM)ofthestackactuatorisderivedfromHamilton'sPrinciple.Subsequently,theweakformoftheequationsofmotionarederivedbasedonavariationalmethod.Theniteelementmethodandamodaltruncationareappliedtotheweakformofthegoverningequationtoprovidethenalapproximategoverningequations.Numericalapproximationandsimulationresultsaregiventocomparethemodelwiththeexperimentaldata. Thisderivationisgenericinnature,butultimatelyappliedtotheP-113-00stackactuatorfromthecompanyPiezosystemJena.Thisspecicactuatorutilizesthed33modeofthePZTcomponentinthestack.Whenavoltageisappliedtoit,theactuatorwillgenerateadisplacement,whichexhibitshysteresis.Sincethecrosssectionalareaofthestackactuatorismuchsmallercomparedtoitslength,theactuatorismodeledasalinearlyelasticrodwhichhaslongitudinaldisplacement.Intheexperiment,thebottomplateofthestackisxedthroughascrewontoarigidmount.Thereisnoexcitationforceapplied,sotheboundaryconditionistakenasxedatoneendandfreeattheotherend.Thestackactuatorisassumedtohaveuniformeectivedensity,crosssectionalareaA,andYoung'smodulusY. 4{1 showstheP-113-00stackactuator.Thepiezostackconsistsofalargenumberofceramicdiscsinsideastainlesssteelhousing.TheelectrodesarearrangedonbothsidesoftheceramicdiscsandareconnectedinaparallellineasshowninFigure 4{2 .Insidethehousing,thereare230layersofPZTwafers.Each 52


Figure4{1: Thestackactuator layerisasquare,withcrosssectionalareaof5mm5mm,asshowninFigure 4{2 (d).Thethicknessofeachlayeris100m.TheneighboringlayersusecommonvoltagesourcesandthevoltagepotentialacrosseachlayerisasindicatedinFigure 4{2 (b)and(c).Accordingtothemodeldevelopedherein,thegravityisnegligible. 1.4 inChapter 1 .Fromtherstconstitutiveequation,wehave33="33d33E3 SEE(4.1) DenecP=1 SE.Equation 4.1 canbewrittenas


Figure4{2: Structureofthestackedactuator Toincorporateinternaldamping,anadhocviscousdampingfactorcDisintroducedintotheEquation 4.2 .ThuswehaveEquation 4.3 InEquation 4.3 ,cPistheequivalentelasticmodulusunderconstantpolarization,sP=1=cPistheelasticcomplianceunderconstantpolarizationandcKisacoecientrepresentingthecouplingeciencybetweenthemechanicaldomainandtheelectricaldomain.Thetypicalvaluesforthematerialleadzirconatetitanate(PZT)are:SE33=15:871012m2=N,d33=6:31010m=V,thuscP=6:31010N=m2,andcK=22:68N=mV.


ThedenitionoftheothersymbolsappearingintheaboveequationcanbefoundinChapter 1 .WeknowthattheinputelectricaleldEinducesapolarizationPinthematerial.Conversely,theinducedpolarizationgeneratesanadditionalelectricaleld,thetotalelectricaleld,termedaseectiveelectricaleldEeisthenexpressedasEe=E+P.Forthepurposeofestablishingthenonlinearandhysteresismodel,weusetheeectiveelectricaleldEeinEquation 4.3 insteadoftheinputelectricaleldE.TheeectiveelectricaleldEeexhibitshysteresisbehaviorandisexpressedbyEquation 3.9 where, TheHEPstatesthatthevirtualvariationoftheintegraloftheLagrangianandthevirtualworkofallnonconservativeforcesmustbeequaltozeroforallvirtualvariationsconsistentwiththekinematicconstraints.Forthelinearpiezoelectriccontinua,asimilarresultholds.Forthispurpose,letusdenetheelectricalenthalpyH, H=UEiDi(4.5)


Figure4{3: Axialpiezoelectricrod whereUistheinternalenergydensitywhichincludesthestoredmechanicalenergyandelectricalenergy.ThenwedeneanewLagrangianforthelinearpiezoelectriccontinuum: 2vvHd(4.6) Inthisequation,Bdenotesthebody.TheextendedHamilton'sPrincipleforLinearPiezoelectricityiswrittenasshowninEquation 4.7 Forthepurposeofanalysisandcomputation,weneedanexplicitexpressionfortheelectricalenthalpydensity.WedeneHas H=1 2cEijkl"ij"kleijkEi"jk1 2"ijEiEj(4.8) inaccordancewith[73].Inthisequation,Hisahomogeneousandquadraticfunctionofstrainandtheelectricaleld,whichmeansthatcijkl=cklij=cjikl=cijlk,eijk=eikjand"ij="ji.Thenwehavethefollowingrelation U=1 2cijkl"ij"kl+1 2"ijEiEj(4.9)


Toprovethis,weusetheconstitutiveEquation 1.4 fromChapter 1 ,Di=eikl"kl+"ijEj 4.5 WhenwecompareEquation 4.10 withEquation 4.8 ,wehave1 2Cijkl"ij"kleijkEi"jk1 2"ijEiEj=UeiklEi"kl"ijEiEj 4.9 .SubstituteEquation 4.6 intotheextendedHamilton'sPrincipleEquation 4.5 ,wehave {z }Term2+ZBEiDid| {z }Term3(4.11) NextwewilluseHamilton'sExtendedPrincipleforLinearPiezoelectricitytoderivethegoverningequationofmotionforthePZTstackactuator.Thereisonlyonedeformationwhichisassociatedwiththe\33"direction.Thuswehavethevelocityvectorv=_w^k.TheinternalenergydensityreducestoU=1 2c33"3"3+1 233E3E3,andthechargedisplacementbecomesD3=e33"3+"33E3.TheTerm1ofEquation 4.11 isthekineticenergyofthepiezoelectricelementinFigure 4.3 .Itisexpressedas: 2_w2d{V(4.12)


where,{Vindicatesvolume,d{V=Adz.Thevariablewdenotesthedeformationinthematerial.Thekineticenergycanbeexpressedas 2A(_w)2dz(4.13)Term2inEquation 4.11 ,isexpressedtotheform 2c33"23d+Z1 2"33E3E3d(4.14) Wehavetherelation @zE3=Q(t) Equation 4.14 canbere-expressedasZUd=Z(1 2c33@w @z2+1 2"33Q(t) 4.15 intothisequation.Wehave @z+"33AQ(t) Next,wecalculatethevariationofeachterm. 2LZ0A_w2dz1Adt=LZ0t2Zt11 2A_w2dtdz=LZ0t2Zt1A_w_wdtdz Here,_wdt=d(w),integrateEquation 4.17 byparts,weobtainEquation 4.18 =LZ0t2Zt1A_wd(w)dz


=LZ00@A_wwjt2t1t2Zt1Awwdt1Adz=LZ0A_wwjt2t1dzLZ0t2Zt1Awwdtdz ForTerm2,weobtaint2Zt1ZUddt=t2Zt1ZL(1 2c33@w @z2+1 2"33Q(t) 2c33@w @z2Adzdt| {z }2:1+t2Zt1LZ01 2"33AQ(t) {z }2:2 2c33@w @z2Adzdt=t2Zt1LZ01 2c33A@w @z2dzdt=t2Zt1LZ01 2c33A2@w @z@w @zdzdt=t2Zt1LZ0c33A@w @z@(w)dzdt IntegrateEquation 4.19 byparts.Weobtainthefollowingequation @zwjL0LZ0@ @zc33A@w @zwdz9=;dt=t2Zt1c33A@w @zwjL0dtt2Zt1LZ0@ @zc33A@w @zwdzdt Thevariationofthequantitylabeled\Term2.2"issimplyzero.t2Zt1LZ01 2"33AQ(t)


Asinthepreviouscase,wesubdividevariationsintotwoquantities,Term(3.1)andTerm(3.2).t2Zt1ZLe33AQ(t) @z+"33AQ(t) @zdzdt| {z }3:1+t2Zt1ZL"33AQ(t) {z }3:2 @zdzdt=t2Zt1ZLe33AQ(t) @ze33AQ(t) ThevariationofTerm3.2is,again,justzero.t2Zt1ZL"33AQ(t) LetusputEquations 4.18 4.20 4.21 togetherandreorganizeterms.Wehave0=LZ0A_wwjt2t1dzt2Zt1c33A@w @zwjL0dt+t2Zt1e33AQ(t) @zc33A@w @zwdzdtt2Zt1ZL@ @ze33AQ(t) 0=LZ0A_wwjt2t1dz+t2Zt1e33AQ(t) @zwjL0dt


@zc33A@w @z+@ @ze33AQ(t) Therstthreetermsareboundaryterms.AccordingtoourdenitioninEquation 4.4 ,thersttermvanishessincevariationwvanishattimet1andt2.Thatiswjt1=wjt2=0.Thesecondandthirdtermare @zwjL0dt=t2Zt1e33AQ(t) @zwjLdtt2Zt1e33AQ(t) @zwj0dt Inthisproblem,wj0=0,thustheaboveexpressionisleftwitht2Zt1e33AQ(t) @zwjLdt @zc33A@w @z@ @ze33AQ(t) Letusassumeconstantstructurepropertiesofthestackactuator.Inotherwords,c33;A,and;e33areconstants.Finally,wewanttoshowthattheEquationofmotion 4.24 canbewrittenas Toprovethisinterpretation,rewritetheequationofmotionasAw=@ @zc33A@w @z@ @ze33AQ(t)


Therighthandsidecanbefurtherwrittenas@ @zc33A@w @ze33AQ(t) @zAc33@w @ze33Q(t) @z[A(c33"33e33E3)] Byobservation,weseethatc33"3e33E3=3.Thus,therighthandsideis@ @z(A3)=@ @zN.Equation 4.25 isveried.Equation 4.25 istheclassicalequilibriumequationfortheaxialbarwhichcanbefoundinmanydynamicsbooks. 4.26 @t2=@N withboundaryconditions (4.27) ByemployingconstitutiveEquation 4.3 ,andnotingthat"=@w @z,N=zA,theequationofmotionis @t2=cPA@2w @z2+cDA@ @t@2w @z2cKA@ @z(E(t))(4.28) Equation 4.28 isthestrongformofequationofmotion.Thisisasecondorderpartialdierentialequation.Thedisplacementw(z;t)mustbeatleasttwicedierentiableinthespacialvariable,andtwoboundaryconditionsareneededforaclassicalsolutiontoexist.Theweakformoftheequationsofmotionisintroduced


toreducethenumberofderivativesrequiredinthegoverningequation.Thenextsectionderivestheweakformoftheequationofmotion. 4.28 bya\sucientlysmooth"function',andintegratefrom0toL.Asaresult,weobtainEquation 4.29 .Nowthegoalistoshiftsomeofthespatialderivativesfromthesolutiontothetestfunctionviaintegrationbyparts.Thus,thetestfunction'isrestrictedtobeatleastoncedierentiableandmustsatisfycertainspecicboundaryconditionsthatwillbediscussedshortly. @t2'dz| {z }Term(1)=ZL0cPA@2w @z2'dz| {z }Term(2)+ZL0cDA@2_w @z2'dz| {z }Term(3)ZL0cKA@ @z(E(t))'dz| {z }Term(4)(4.29) WhenweintegrateTerms(2)(3)and(4)byparts,wehave: @z2'dz=cPA@w @z'L0ZL0cPA@w @z@' @zdz @z2'dz=cDA@_w @z'L0ZL0cDA@_w @z@' @zdz @z(E(t))'dz=cKAE(t)'L0+ZL0cKA@' @zE(t)dz Recalltheboundaryconditionsonthesolutionwjz=0=0Njz=L=0 Therstboundaryconditionisthekineticboundarycondition,whilethesecondisaforceboundarycondition.Thefunctions'selectedintheweakformulationmustsatisfythekinematicboundaryconditionsjustasthewmustinavariational


formulation.Werequirej0=0 Equation 4.29 canbewrittenas @t2'dz+ZL0cPA@w @z@' @zdz+ZL0cDA@_w @z@' @zdz=ZL0cKAE(t)@' @zdz+cPA@w @z'jz=L+cDA@_w @z'jz=LcKAE(t)'jz=L Theboundarytermscanbeorganizedandgroupedasfollows:cPA@w @z'jz=L+cDA@_w @z'L0cKAE(t)'jz=L=cPAe'jz=L+cDA@_w @z'jz=LcKAE(t)'jz=L=AcPe+cD_ecKE(t)'jz=L=Ajz=L'jz=L=Njz=L'jz=L 4.27 ,Njz=L=0,theaboveboundarytermiszero.Equation 4.33 iswritteninnalformas @t2'dz+ZL0cPA@w @z@' @zdz+ZL0cDA@_w @z@' @zdz=ZL0cKAE(t)@' @zdz(4.34) Equation 4.34 istheweakformgoverningequation.


whereX=L2(0;L)V=H10(0;L)=2H1(0;L):(0)=0 Eachofthesesetsoffunctionare,infact,Hilbertspacesforthechoiceofinnerproductsh;iV=LR000dx8;2Vh;iX=LR0dx8;2X Toprovideaframeworkinwhichtoestablishmodelwell-posedness,weconsideranabstractformulationofthemodelsbaseduponstinessanddampingsesquilinearforms.Tothisend,wedenetwobilinearforms1(;)=LR0cPA00dx2(;)=LR0cDA00dx 4.34 canbewrittenas {z }Term1+cP;'V| {z }Term2+D(cD)_;'EV| {z }Term3=h[B(Q)](t);'iV;V| {z }Term4(4.35) Toapplytheformulationin[74],[28],and[2],wemustshowthattheforms1and2are 1. Bounded1(;)c1jjVj'jV;forc12R2(;)c2jjVj'jV;forc22R V-EllipticRe1(;)c3jj2V;forc3>0Re2(;)c4jj2V;forc4>0


3. Symmetric1(;)= Alternatively,wecandeneoperatorshAi;iV;V=i(;)i=1;2 WecanwriteEquation 4.35 equivalentlyas w(t)+A2_w(t)+A1w(t)=[B(Q)](t)(4.36) where,Q(t)istheinputvoltage,andQ(t)2C[0;T],andB(Q)2L2((0;T);V).AccordingtoTheorem4:1of[75]orTheorem2:1andremark2:1of[76],thereexistsauniquesolutionwtoEquation 4.36 orequivalently,Equation 4.35 whichsatisesw2C((0;T);V)_w2C((0;T);X) 4.34 ,expressthedisplace-mentasalinearcombinationofconventionalniteelementshapefunctions. Inthissummation,Nisthetotalnumberoftheshapefunctions.Choose'(z)=k(z).UponsubstitutingthisseriesintheintegralsinEquation 4.35 ,weobtain(Term1)=ZL0NXj=1Awj(t)j(z)k(z)dz=NXj=1wj(t)ZL0Aj(z)k(z)dz(Term2)=ZL0NXj=1cPAwj(t)0j(z)0k(z)dz=NXj=1wj(t)ZL0cPA0j(z)0k(z)dz


(Term3)=ZL0NXj=1cDA_wj(t)0j(z)0k(z)dz=NXj=1_wj(t)ZL0cDA0j(z)0k(z)dz(Term4)=ZL0cKAE(t)@ @zkdz=ZL0cKAE(t)0k(z)dz [M]w withw Figure4{4: FEMshapefunctions


EachlayerofPZTisdividedintofourelements,thustherearevenodalpoints.Themassmatrix[M]andstinessmatrix[K]foreachlayerare55matrices.Theyare[M]=Ale le266666666664110001210001210001210001137777777777555lehereisthelengthofeachelement.Nextwewillcalculatethecontrolinuenceoperator,the[B(Q)](t)matrix.WeknowthatE(z;t)=rQ,whereQistheinputvoltagedierence.Wetakeoneelementforconvenience,asshowninFigure 4{5 .Weintroducealocalcoordinatez0,with0z0le.Thederivative0k(z0)is1or+1,aconstantovertheelement.SotheelementmatrixB Figure4{5: Shapefunctioninoneelement


andB TheelementmatrixB le26666666666411001200...00210011377777777775921921BG [MG]w


withw Equation 4.38 canbetakenasthecoupledequationsofmotionofa920degreeoffreedomsystem.Thereare920modesintotal.Nowonly^N=3modesareusedtoapproximatethesystem.Introducingthecoordinatetransformation,u(t)=3Pr=1'rr(t)Thecorrespondingmodalmatrixofthesystemisthenthesizeof9203.='1'2'3.Each'isacolumnvectoroflength920,correspondingtoonemode.WeleftmultiplytheequationofmotionbythemodalmatrixT,anddeneMr=T[MG],Kr=T[KG],Br=T[BG].Weshouldpointoutthatthismodaltransformationdiagonalizesthemassmatrixandstinessmatrix.SoMrandKrarediagonalmatrices,whicharecalledmodalmassmatrixandmodalstinessmatrixrespectively.Bothhavethesizeof33.ThematrixBris31. Nowthesystemequationhasbeenreducedtothreedegreesoffreedom.Thereducedorderequationofmotionis [Mr] StatespaceformofthegoverningequationisgivenbyEquation 4.41 lety


or _y 4.41 ,[A]isaconstantmatrix,but[B]isaprescribedtimevaryingcolumnvector.Tosimulatetheaboveequation,amodiedtrapezoidalruleisintroduced.Wesupposethetimestepisverysmall 2yj+1+(tj+1tj)[A] 2yj+(tj+1tj)Bj Reorganizingterms,wehave 2yj+1=I+(tj+1tj)[A] 2yj+(tj+1tj)Bj(4.43) multiplyingbyhI(tj+1tj)[A] 2i1,theequationbecomesyj+1=I(tj+1tj)[A] 21I+tj+1tj[A] 2yj+I(tj+1tj)[A] 21(tj+1tj)Bj with[A]=I(tj+1tj)[A] 21I+(tj+1tj)[A] 2 21(tj+1tj)Bj 4.44 canbeeasilyprogrammedandsolvedusingprogramslikeMatlab.


4.45 @t2=cPA@2w @z2(4.45) Dene A@2 4.45 becomes@2w @t2=Lw DenecPA A=K.Theeigenvalueproblemrequiresthatwesolve @z2+=0(4.47) SolutionofEquation 4.47 hastheformofEquation 4.48 Kz!+Bcosr Kz!(4.48) Finally,'mustsatisfytheboundaryconditions.So,therstboundaryconditionisjz=0=0


Alsothereisnoexternalforceatz=L,N=EA0,so0jz=L=0.0(z)=r KAcosr Kz!=0 SinceAcannotbezero,andq K6=0,thuswemusthavecosr KL!=0 So,q KL=(2p+1) 4.48 isclear,m(z)=Asinm+1 2 2 SubstituteEquation 4.49 intothewaveEquation 4.45 Intheaboveequation,K'00q(z)=L'q(z).However,L'q+'q=0,soEquation 4.50 becomes1Xq=1(q(t)+qq(t))q(z)=0 Tosolvetheaboveequation,multiplybyr(z)andintegratefrom0toL.WeobtainthefollowingequationLZ01Xq=1(q(t)+qq(t))q(z)r(z)dz=0


74 Table4{1:NaturalfrequenciesfromanalyticalmodelandFEMmodel mode# Analyticalsolution(Hz) FEMmodel(Hz) variation% 1 31502.9929264988 31502.9967513644 0.000012141% 2 94508.9787794963 94509.0820951708 0.00010932% 3 157514.964632494 157515.442948384 0.00030366% 4 220520.950485491 220522.262986054 0.00059518% 5 283526.936338489 283529.725884032 0.00098939% 6 346532.922191487 346538.015319954 0.0014697% 7 409538.908044484 409547.314973836 0.0020258% Since areeigenvectors, r ( z )and q ( z )areorthogonal.Thenwehavethe followingequation q ( t )+ q q ( t )=0 q =1 ; 2 ; 3 where q = q + 1 2 l 2 Kq =1 ; 2 ; 3 Thenaturalfrequencyisgivenby f q = p q 2 = q + 1 2 2 l p K Table4{1liststhenaturalfrequenciescalculatedviatheanalyticalmodel andniteelementmethodandcomparesthevariationoftheFEMmodelfrom theanalyticalsolution.Wecanseefromthetablethattheintrinsicfrequencies oftheFEMmodelareveryneartothoseoftheanalyticalsolution.Valuesof thoseparametersare c P =6 : 3 10 10 N=m 2 =7500 Kg=m 3 K = c P = and L =23 10 3 m .


CHAPTER5 EXPERIMENTALDESCRIPTIONANDRESULTS Thischapterdescribestheobjectiveoftheexperiment,givesthedetailsofthe experimentalsetup,explainstheexperimentalprocedures,andprovidesasummary andanalysisofexperimentalresults. 5.1ExperimentObjectivesDescription Theobjectiveoftheexperimentdescribedinthisthesisistomeasureand characterizehystereticresponseofapiezoelectricstackactuatorP-113-00for severalfrequencies.Theidenticationofamodelthatisfrequencydependent issubsequentlydeveloped.TheexperimentstudiesaP-113-00stackactuator thatutilizesthe\ d 33 motion"ofthePZT.Thestackoperateswithinavoltage rangefrom[ 10+150] V .Table5.1givesalistofthedimensionsandphysical parametersofthestackactuator. 5.2ExperimentalSetup 5.2.1EquipmentUsedintheExperiment Webrieysummarizetheexperimentalapparatusandsetupoftheexperiment inthissection. Table5{1:Parameterlistofthestackactuator Areaperlayer 5 5 mm 2 Thicknessperlayer 100 =mum Totalnumberoflayers 230 LengthofthePZT 230 mm Twostacks 5 5 7 mm 3 +5 5 16 mm 3 Topplateandbottomplate 1 mm each, mm intotal Totallengthofstack 27 mm Internalcapacitance 2 : 5 F Inputrange [ 10 V 150 V ] 75


1.LaserVibrometer Themostcomplexmeasurementdeviceinthisexperimentisthelaservi-brometer,whichisusedtoobservethevelocityofthestackactuator.AlaservibrometerutilizestheDopplereect,andisanon-contactvibrationmeasurementdevice.Laservibrometersaretypicallytwo-beam,interferometricdeviceswhichdetectthephasedierencebetweenaninternalreferenceandthemeasurementbeam.Themeasurementbeamisfocusedonthetargetandscatteredbacktotheinterferometer.Thevelocityofthebeam'stargetmaybeinferredfromthephasedierence. 2.FunctionGenerator AnHP33120AFunctionGeneratorisusedasasinusoidalvoltagesourcefortheexperiment.Theoutputrangeis[10+10]V. 3.Amplier Therearetworeasonsweuseanamplierintheexperiment.First,theoperat-ingvoltagerangeofthefunctiongeneratoris[10V+150]V.Thus,thefunctiongeneratorcanprovideinonlyaverylimitedrange.AnamplierisrequiredtogeneratethehighvoltageandcurrentforthePZTactuator.TheamplierLV-1200fromDSMisatruelinearfolloweramplierandhasamaximumpeakcurrentof1200mA.Thisistheamplierthathasbeenusedinthisexperiment. 4.Potentiometer Apotentiometerisusedinserieswiththestackactuatortoprotecttheamplierfromcurrentspikes. 5.VoltageDivider Avoltagedividerisusedtoscaledowntheinputvoltagesothatthedataacquisitionsystemisnotoverloaded.Figure 5{1 showshowthevoltagedivideriscreated.WiththeoscilloscopeorLaserVibrometerwecanonlymeasurevoltageupto10V.Thisupperlimitisfarbelowthestack'svoltagerange,whichis


Figure5{1: Voltagedivider [10V+150]V.Alsoitisnotfeasibletomeasuretheinputvoltagedirectlyfromthefunctiongenerator,andthenmultiplythegainoftheampliertogettheinputvoltagesignal.Thereasonisthattheamplierinducesaphaseshift.Thevoltageappliedtotheactuatorhasadierentphasefromtheonethatisoutputbythefunctiongenerator.Thevoltagedividerconsistsoftworesistors,R1,andR2withR1

Figure5{2: Experimentalsetup Figure5{3: Flowchartofexperimentalsetup processor.Forthepurposeofcreatingthefrequency-dependentmodel,thevelocitydataatseveralinputfrequenciesaremeasured.


5{4 .Figure 5{5 showsthevelocityhysteresisloopsatsomeoftheselectedfrequencies3Hz,30Hz,50Hz,100Hz,160Hz,200Hz,250Hz,300Hzrespectively.Thepurposeofthisresearchistoobtaintheweightsand Figure5{4: Drivingvoltagevs.time predictthedisplacementhysteresisofadesiredfrequency.Thusweneedthedisplacementresponseofthestack.Theexperimentmeasuresthevelocitydata,


sothedisplacementresponseisobtainedfromintegrationofthevelocity.Inthiscase,thereisnomeanstoknowtheabsolutedisplacementofthestack,becausethemomentthevelocityistaken,theinitialpositionisnotknown.Thisisaclassicalproblemwhenvelocitymeasurementsareconversionstodisplacements.Soonlytherelativepositionoftheactuatorcanbeobtainedviathismethod.Beforeintegrationofthevelocity,theDCcomponentissubtracted.Figure 5{ Figure5{5: Velocityresponseofthestackactuator 6 showstherelativedisplacementofthestackactuatorforthecorresponding


frequenciesinFigure 5{5 .Wecanseefromthedisplacementversustimeplotsthatatlowfrequency,whenthedrivingforceincreases,thedisplacementincreases.However,asfrequencyincreases,thereisalaginthedisplacementresponse,thisisobviousatanyfrequencyhigherthan120Hz.Figure 5{7 5{8 and 5{9 showthedisplacementhysteresisloopsfrom3Hzto350Hz.Thexaxisisdrivingvoltage,theyaxisisthedisplacementresponseofthestack.Wecanseethatthedisplacementversusdrivingforcecurvesarestronglyfrequency-dependent.Forexample,thewidthofthehysteresisloopsisverysmallatlowfrequency,nearlyzeroat3Hz.However,thewidthincreasesasthedrivingvoltageincreases.Anotherobservationisthattheshapeoftheloopsisnearlyan\ellipse".Ifwedrawalinealongthemainaxisoftheellipse,theorientationofthelinewillrotateclockwiseasthedrivingfrequencyincreases.


Figure5{6: Displacementresponseofthestackactuator


Figure5{7: Relativedisplacementofthestackactuator-I Figure5{8: Relativedisplacementofthestackactuator-II


Figure5{9: Relativedisplacementofthestackactuator-III


Thischapterpresentsthefrequency-dependenthysteresismodel.Twomethodsareusedtodevelopthemodel.ThosetwomethodscanbeunderstoodintermsofFigure 6{1 ,whichdepictsthestructureofasystemthathasa\nonlineartolinear"cascadestructure.Therstmethoddoesnotincludethelineardynamicalsubsystem,thatisG0.Thesecondoneincludesboththenonlinearandlinearsubsystems. Figure6{1: Frequency-dependenthysteresismodel Intherstmethod,thedisplacementxissimplyexpressedas where,k Inthesecondmethod,thefrequency-dependenthysteresismodelisbasedontheanalysisofChapter 3 4 .Inthismethod,thedisplacementxisthesolutionof 85


thedynamicgoverningequation [M]x(t)+[C]_x(t)+[K]x(t)=[B(Q)](t)(6.2) Thehysteresistermarisesinthedrivingterm[B(Q)](t),whichisexpressedas [B(Q)](t)=ConstNXi=1k Thischapterpresentsthemodelscreatedbythetwomethodsandcomparetheaccuracyofmodelpredictions.Themodelvalidationshowsthatthemodelsdevelopedusingthetwomethodscanaccuratelypredictthepositionofthestack.Thepositioningerrorcanbeimprovedbyanorderofmagnitude.However,thedynamicalmodelprovidesmoreaccuratepositionestimateofthestack. 6.1.1QuantizationoftheSPlane 4 ,weexpressanyhysteresisquantityash=F(Q),whichquantiesthehysteresisphenomenonofthePZT.Writehash(t)=Z


Fromthisequationwecanseethatthehysteresisquantityisexpressedasthesuperpositionofweightedkernels.AsnotedinChapter 3 ,theinputrange Figure6{2: [uminumax]shouldbeastrictsubsetofthenitehalfplane[Smin;Smax].Themaximuminputrangeofthestackis[10150].Theactualinputshouldbestrictlywithinthisrange,thuswechooseS2[10+150].Thethresholds1ands2takediscretevaluesinthisrange.Eachspair(s1;s2)determinesoneKPkerneldenedonthedrivingvoltage.Themorekernelsinthiscollection,thebetterresultswecanhave.Butmorekernelscansignicantlyincreasethecomputingtime.Thus,consideringaccuracyofthemodelandthecomputationtime,toensurewehaveenoughkernels,thisvoltagerangeisdividedinto25parts,withs=6:4voltsseparation.Thereareconsequently351pointsinthediscretePreisachplane((26+1)26=2=351),whichmeanswewillhave351KPkernelsforeachinputvoltageu(t).Figure 6{2 showsthediscreteSplaneandFigure 6{3 showsseveralKPkernelsdenedbythecorrespondingspairs.Theslopeforeach

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Figure6{3: KPkernels ridgefunctionisdeterminedbys.Thewidthofthekernelsisanintegernumbertimess.ThiscanbefoundinFigure 3{3 6.2.1ProblemFormulation 6.1 .Wecanwritethisequationintomatrixform where[K

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oftheerrorfunctione,whichisdenedas 2k[K]w Intheaboveequation,wehavesuppressedtheargumentlistsothatw 6.7 ,wedenotetheconvergenceorderasp,where,pisthelargestnon-negativenumberforwhichanitelimitexists. 0limk!1kxk+1xk kxkxkp1(6.7) InEquation 6.7 ,whenp=1,thesequenceconvergeslinearlywith0<<1,andsuper-linearlywith=0.Whenp=2,thesequenceissaidtoconvergequadratically.Theconstantistheassociatedconvergenceratio.Thequadraticsequenceconvergesmorerapidlythansuper-linearsequence,andthesuper-linearsequenceconvergefasterthanlinearsequence.Wecanprovethattheoptimizationprobleminthismethodconvergesquadratically. 2k~x[K]w 2[~x[K]w 2~xTw 2~xT~x2~xT[K]w Rearrangingtheexpression,weseethate(w 2w

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Flowchartforweightidentication Figure 6{4 showstheowchartfortheidenticationoftheweights.Basicallytherearefourstepstothisprocess. 1. Giveaninitialguessfortheweightvector; 2. Foreachrelaypair,thenonlinearmodelgeneratesoneKPkernel.IfthereareLrowofdiscreteSpairs,therewillbeatotalofL(L+1) 2pairsintotal.ThustherewillbeL(L+1) 2KPkernelsintotal; 3. ThesuperpositionoftheweightedKPkernelsgivestheestimateddisplace-mentdisplacementx; 4. Comparexwiththeexperimentaldata~x,deneanerrorfunctionease=NPj=1(xj~xj)2.Ife
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Figure6{5: Weightdistributionforselectedfrequencies Figure6{6: Reconstructthedisplacementhysteresis tstheexperimentaldata~x.ThetrustregionmethodwillbebrieyreviewedinSection 6.4

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Figure6{7: Weightidenticationforselectedfrequencies frequenciesaremeasured:3Hz,5Hz,10Hz,15Hz,20Hz,25Hz,30Hz,35Hz,40Hz,45Hz,50Hz,55Hz,60Hz,70Hz,80Hz,90Hz,100Hz,120Hz,140Hz,160Hz,180Hz,200Hz,250Hz,300Hzand350Hz.Themodelisdevelopedusing10Hz,20Hz,30Hz,40Hz,50Hz,60Hz,80Hz,100HZ,140Hz,180Hz,200Hz,250Hz,300Hzand350Hz.Theotherfrequenciesareusedtovalidatethemodel.Figure 6{8 and 6{9 givethemodelvalidationresults.Figure 6{8 showsthemodelestimatesfortheselectedidenticationfrequencies.Usingthemodeltopredicttheidenticationdata,thepredictedhysteresisloopsshouldbenearlymatchtheexperimentaldata,justasshowninthegure.Figure 6{9 givesthemodel

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Figure6{8: Modelpredictionfortrainingfrequencies predictionforotherfrequencies.Weseethatthemodelcanaccuratelypredictthehysteresisloopatthedesiredfrequencies.ThemodelvalidationanderroranalysiswillbefurtherdiscussedinSection 6.5 6.3.1ProblemFormulationandConvergenceCharacterization 4 .TheresultinggoverningequationisEquation 4.38 .Thedrivingterm,whichisalsocalledthecontrolinuenceoperator,is[B(Q)](t),anditembodiesthehysteresis. [Bk(Q)](t)=lZ0cKAEe(t)0kdz(6.9)

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Figure6{9: Modelpredictionfordesiredfrequencies where,theeectiveelectricaleldEe(t)isexpressedas Thuswecanwritethecontrolinuenceoperatorinmatrixform: [B(Q)](t)=Const[k where,the[k 4.38 ,whichalsoshowshysteresis.Theweightidenticationisequivalenttondingtheoptimizedweightvectorw

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experimentaldata.Thusourproblembecomesminimizationoftheerrorfunctione,whichisdenedas 2kx~xk(6.12) Thedynamicalfrequency-dependenthysteresismodelusesthesamegridonS-planeandthesameexperimentaldataforidenticationofthemodel.TheprocedureissimilartotheprocessstudiedinSection 6.2 Weightidenticationforcascadedmodel Figure 6{10 illustratestheweightidenticationprocedureforthecascadednonlinear-linearmodel.Therearefourstepsforweightidentication: 1. Provideaninitialguessoftheweightvector; 2. Foreachrelaypair,thenonlinearmodelgeneratesoneKPkernel.ThereareL(L+1) 2KPkernelsintotal,thusthereareatotalofL(L+1) 2KPkernels; 3. ThesuperpositionofweightedKPkernels,whichwedenotey,providestheinputtothelinear,secondordersystemofordinarydierentialequation; 4. TheoutputofthelinearODEmodelistheestimateddisplacementx.Wedeneanerrorfunctionease=NPj=1(xj~xj)2.Iftheerrorinequalitye
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Figure6{11: WeightdistributionovertheSplane

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Figure6{12: Calculationofthereconstructedcurve coecientsoftheKPkernels,theinputiscalculated.ThesolutionxofthelinearPDErepresentsthebestttotheexperimentaldata.ThisprocedureisexplainedinFigure 6{12 .Figure 6{13 showsthereconstructedhysteresiscurves Figure6{13: Reconstructeddatavs.experimentaldata

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usingtheidentiedweightsforthefrequenciesinFigure 6{11 .Wecanseethatthereconstructeddatacanaccuratelyidentifytheexperimentaldata. Ripplesinthereconstructedcurves Figure6{15: Frequencycomponentsoftheripples Byobservation,thereareripplesintheoptimizationcurves,asshowninFigure 6{14 .Theseripplesembodythehigherfrequencydynamicsofthestackac-tuator.Thefrequencycomponentsoftheripplesareevidentfromadecompositionofnaturalfrequenciesofthestack.InChapter 4 ,afterderivingtheequationofmotion,modalreductionwasappliedtoreducethesizeoftheequationofmotion.

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Ifonlyonemodeistaken,therippleshaveonlyonefrequencycomponent,iftwomodesaretaken,therippleshavetwofrequencycomponents.Figure 6{15 showsthisrelation.Weknowthatmodalreductionsignicantlysavesthecomputationtime,however,thisputslimitationtotheapplicablefrequencyrangeofthemodel.Forexample,iftwomodesaretaken,thesecondfrequencyis94KHz,whichmeansthatthemodelisnotapplicablehigherthan94KHz.SincetheoperatingfrequencyofthePZTiswellbelowthisfrequency,usingtwomodesisacceptable. 6{16 givesthemodelpredictionforthetrainingfrequencies.Usingthemodeltopredictdatausedfortheidentication,theestimatedhysteresisloopsshouldbenearlythesameastheexperimentaldata.Inthisgure,wecanseethatthepredictedhysteresisloopstverywell.Figure 6{17 givesthemodelpredictionforfrequencies.Wecanseethatthemodelcanpredictaccuratelythestack'sdisplacementatthedesiredfrequency. minw1 2kF(w)~xk22=minw1 2Xi(F(wi)~xi)2(6.13) where,wand~xarevectorvaluedfunctions. Letf(w)=(F(w)~x),probleminEquation 6.13 becomesminimizethefunctionf(w).Thisisaunconstrainedminimizationproblem.Asubspacetrustregionmethodischosentodotheoptimization.Thismethodisbasedonthe

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Figure6{16: Modelpredictionfortrainingfrequencies interiorreectiveNewtonmethod[77].Supposewestartfromapointw0andwewanttomovetoalowerfunctionvalue.Thebasicideaistoapproximatef(w)withasimplerfunctionqwhichreasonablyreectsthebehavioroffunctionfinaneighborhoodNincludingthepointw0.ThisneighborhoodNiscalledthetrustregion. Inthestandardtrust-regionmethod,thequadraticapproximationqisdenedbythersttwotermsoftheTaylorapproximationtof(w)atw0.Mathematically,

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Figure6{17: Modelpredictionfordesiredfrequencies theproblembecomes min1 2sTHs+sTgsuchthate
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TheaboveproceduresarerepeateduntilEquation 6.13 isobtained.Nowtheweightvectoristheonewhichprovidesoptimizedsolutiontotheexperimentaldata~x. Fortheconvenienceofcomparison,themodelpredictioncurvesfrombothmodelsareputinoneguretogetherwiththeexperimentaldata.Figure 6{18 showsthemodelpredictionfromthetwomodelsandthehysteresisloopfromexperimentaldataforselectedfrequencies.Wecanseethattherearezig-zagteethinthesimplenonlinearmodelprediction.Thisisbecausethenonlinearmodelisthesumoftheweightedpiecewiselinearkernels.Inthecascadedmodel,thelinearsubsystemaddsalowpassltertothenonlinearmodel.Thus,thecascadednonlinear-linearmodelgeneratesmoreaccurateandsmoothercurvesthanthesimplenonlinearmodel.Butthecontributionofthecascadedmodelislimited.

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Figure6{18: Comparisonofthetwomodels

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Thischapterdevelopsanaveraginganalysisforthestudyofswitchedpiezostructuralsystems.Thetechniquedevelopedinthisresearchismotivatedbythesuccessofaveragingmethodsfortheanalysisofswitchedpowersupplies.Weshowthatbyadjustingthedutycycleofthecontrolsignal,thenotchfrequen-ciescanbecontinuouslyshiftedoverarangeoffrequencies.ThisresultstandsinaPZTvibrationabsorberincontrasttoaniteandxedcollectionofdiscretenotchfrequenciesdesignedin[4].Thus,anovelconceptualdesignforatunablevibrationabsorberisintroduced.Theanalysisalsoprovidesadesignprocedureforsuchsystems. 7{1 showstheidealizedmodelforthesystemandFigure 7{2 showsthefreebodydiagramsofthestructuremassMsandtheactuatormassMa.Accordingto Figure7{1: Idealizedpiezoceramicvibrationabsorber 104

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Figure7{2: FBDofabsorbermassandstructuremass Newton'slaw,theequationsofmotionforthetwomassesare (7.1) (7.2) whereadampingfactorbaisincludedandthusadampingforceFD=ba(_ya_ys)isintroduced.Thevariablesya,ysandyiaretheabsorbermassdisplacement,thestructuremassdisplacementandthebasedisplacementrespectively. ThePZTismodeledasadampingandaspring.TheforceFisrelatedtotheforcecarriedintheactuator,andisassociatedwiththeelectromechanicalpropertiesofthepiezoceramics.Thesystemutilizesthe\d33"motionofthematerial.UsingtherstequationoftheconstitutiveEquation 1.3 or 1.4 inthe\3"direction,wehave3="3d33E3

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where,tisthethicknessofthePZT,andQ(t)isthevoltagedierenceacrossthePZT.Substituterelation 7.3 into3,wehave3=1 SE33d[(yays)d33Q(t)](7.4) where,AdenotesthecrosssectionalareaofthePZTpatch.SubstituteEquation 7.4 intoEquation 7.2 ,wehave SE33d[(yays)d33Q(t)]=maya(7.5) Dene SE33t(7.6) tobetheequivalentstinessunderconstantvoltage.RewriteEquation 7.5 as FromtherstequationoftheconstitutiveEquations 1.3 ,wecancalculatethechargeacrossthePZT,whichisq(t)=AD3=A(d333+3E3)=Ad33"3d33E3 NotethatthersttermofEquation 7.8 isKscd33(yays).LetCp=A3

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Figure7{3: Vibrationsuppressionwithcapacitorload whichistermedasthezero-straincapacitance.DierentiateEquation 7.8 ,weobtain where,i(t)isthecurrentacrossthePZT.Equation 7.7 istermedasactuatorequationandEquation 7.9 sensorequation. Forthepurposeofdissipatingvibratingenergy,anelectricalload,suchascapacitor,resistororinductororthecombinationofthemisplacedparallelwiththePZTmaterial.ThedeformationofthePZTmaterialcausesavoltagedierence,thusthemechanicalenergyistransformedtoelectricalenergy,andhencevibrationsuppressionisachieved.Theportionoftheelectricalenergycouldbeharvestedandstoredforlateruse,suchasremotepowersupplies. Innextsection,wewillfocusontheshuntedcapacitor.Wederivethestatespacemodelandapplytheaveragingmethodtotheswitchedsystemtoanalyzeitsdynamicproperties. 7{3 .TheswitchinthisFigurecanbeconstruedassimplyacompactrepresentationofa\multivalued,butdiscrete"switch.Inthisscenario,theswitchcanassumestatesindexedbyintegersk=1;2;m,inwhichtheshuntcircuithasdiscretecapacitancevaluesCk=C1;C2;Cm.Thechargeq(t)acrossthecapacitorCkis

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where,eQ(t)indicatesthevoltagedierenceinthecapacitorsandQ(t)indicatesthevoltagedierencegeneratedinthePZTpatch.AccordingtoKirho'slaw,wealsohaveEquation 7.11 Weknowthatthecurrenti(t)acrossthePZTmaterialisdenedasi(t)=d dtq(t),soifwetakethederivativeofEquation 7.10 withrespecttotime,wehave CombineEquation 7.12 withEquation 7.9 ,weobtainKscd33(_ya_ys)+Cp_Q(t)=Ck_Q(t) whichcanberearrangedtoobtaintheexplicitsensorEquation 7.13 _Q(t)=Kscd33 Insummary,thegoverningequationsofthevibrationabsorberwithashuntedcapacitancecircuitis (7.14) 7.14 ,ifweintegratebothsidesofthethirdequation,wehaveQ(t)=Kscd33

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zeros,i.e.Q(0)=0,ya=0,andys(0)=0,thenweareleftwith (7.15) Deneka=Ksc+K2scd233 7.15 ,kaisobviouslytheeectivestinessofthepiezoelectricvibrationabsorbersystemwithashuntedcapacitornetwork.Wecanseethattheshuntedcapacitancechangestheeectivestiness.Thispropertyenableustodevelopatunablemechanicalvibrationabsorber.Forconvenienceinanalysis,werequirethatthegoverningequationsbewritteninrstorderform.Tothisend,wedenex _x Inthecaseofzerodamping,ba=0,bs=0.Onlyevenorderderivativesappearonthelefthandside.Theforcedmotion,orthesteadystateresponse,willhavetheformQp=Qcos(!drt),where!dristhedrivingfrequency.Assumethesteadystatesolutionys=Xssin(!drt)andya=Xasin(!drt).LetthedrivingforcehavetheformKsyi=F0sin(!drt),thenthetransferfunctionofthesystemis (ka+Ksms!2dr)(kama!2dr)k2aF0 AscanbeseenfromEquation 7.17 ,themotionofthestructuralmassXscanbecompletelyabsorbedbychoosingthetuningconditionka

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dampingcase.However,fromEquation 7.18 ,wecanseethatthemotionoftheactuatormassXawillneverbezero.WeknowthatXsKs 7.17 ,wehave Next,wedividetherighthandsideofEquation 7.19 byKaKsanddenema 7.20 1+!a Fromthisequation,wecanseethatthenotchfrequencyoccurswhenthedrivingfrequencyequals!a.Thenotchfrequency!acanbechangedbychangingeithertheactuatormassortheequivalentstinessofthesystemka,whichisafunctionoftheshuntedcapacitanceCk.Figure 7{4 showsXs=Xiversusdrivingfrequency Figure7{4: Structuremassresponse{changingactuatormass

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7{5 showsXs=Xiversusdrivingfrequency!drforvariousrvalues,where,risdenedastheratiooftheequivalentshuntedcapacitanceoverthezerostraincapacitance,r=CK Structuremassresponse{changingshuntedcapacitance

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isapproximately7:98%.ThedenitionsandvaluesfortheotherparametersarelistinTable 7{1 _x Itshouldbepointedoutthatthetransferfunctionsforthefour-stateandve-statearethesamebecausethetransferfunctionisobtainedfromzeroinitialconditions.ThisisveriedinFigure 7{6 ,wherethesolidlineistheplotofthe4-statesystem,thedash-dotlineistheplotofthe5-statesystem.Theymatcheachotherperfectly.

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Figure7{6: Comparisonofthetransferfunction thecircuitissettohighstinessstateforcollectingenergy.When thecircuitissettolowstinessstate,thusdissipatingenergy.TheeciencyisquitegoodduetothelargeinternalimpedanceofthePZT. Theabovecontrolstrategycanberepresentedbythefollowingfunction: where,n=1;2NandNTisthetotallengthofoperatingtime.Thecapaci-tanceCkisthenexpressedasCk=C0h(t).Whenswitchison,h(t)=1,Ck=C0.Whenswitchiso,h(t)=0,Ck=0.WhereC0istheeectivecapacitanceofthe

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capacitors.TheparameterDisthedutycycle.Itrepresentshowmuchpercentagetheswitchacts.Figure 7{7 depictsthecontrolsignal. Figure7{7: Switchedsignal Afterapplyingthecontrolstrategyintroducedabove,thesystemEquation 7.21 isbutaspecialcaseofthemoregeneralclassofpiecewiseanesystemssubjecttodisturbance. _x However,mostmoderncontroltheoryisbasedonaLTImodel,thusitismorefavorabletondanLTIapproximationtothehybridsystem.Therearemanymethodsdevelopedforthispurpose,amongthem,theaveragingmethodissuitableforthisproblem.Innextsection,theaveraginganalysisofasemi-activecontrolledsystemwillbegiven. dt=f(t;x)+2g(t;x;)(7.26)

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withtheinitialconditionx(t0)=x0.TheaverageofsystemEquation 7.26 isgivenby dt=f0(y)(7.27) where,thefunctionf0(y)isdenedas Letusmakethefollowingassumptions: 1. 2. 3. Werstintroduceanewfunctionu1(t;y)whichisdenedu1(t;y)=tZt0f(s;y)f0(y)ds

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=t0+TZt0f(s;y)dsf0(y)T However,f0(y)=1 Assumethatt=t0+nT+t0,with0t0T.Toshowthatku1(t;y)k2MT,wejustneedshowu1(t;y)=t0+nT+t0Zt0+nTf(s;y)f0(y)ds2MT Notethatx(t)z(t)=tRt0dx dsdz dsds.dx dtdz dt=f(t;x(t))+2g(t;x(t))dy dt@u1 dt@u1 dt=f0(y).Wehavedx dtdz dt=f(t;x(t))+2g(t;x(t))f0(y)2ru1(t;u)f0(y)f(t;y(t))

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Forthisequation,ifweaddandsubtracttheontermf(t;z(t)),wehave dtdz dt=f(t;x(t))+2g(t;x(t))f0(y)2ru1(t;u)f0(y)+f(t;z(t))f(t;z(t))f(t;y(t))=f(t;x(t))f(t;z(t))+R whereR=2g(t;x(t))f0(y)2ru1(t;u)f0(y)+f(t;z(t))f(t;y(t)).Theaboveequation,kf(t;z(t))f(t;y(t))kLkz(t)y(t)k=Lu1(t;y(t))L2MT dsdz dsdstZt0dx dsdz dsdstZt0f(t;x(t)f(t;z(t)+2dstZt0kf(t;x(t)f(t;z(t)kds+k2(tt0) (7.32) usingLipschitzinequalityonkf(t;x(t)f(t;z(t)kLkx(t)z(t)kkx(t)z(t)kLtZt0kx(t)z(t)kds+k2(tt0) AccordingtoGronwalllemma(AppendixD),herewehave1=L,2=k2,and3=0.Thenwehavekx(t)z(t)kk LeL(tt0)k L.Consequently,kx(t)y(t)kk LeL(tt0)k L+2MT.NowwecanconcludethatifL(tt0)

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isboundedbyaconstantindependentof,theny(t)approachesx(t)onthetimescale1 7.5.1IdealSwitch Idealswitch Usually,convertersconsistsofidealswitchesandlinearcomponentslikeresistors,inductorsandcapacitors.TheswitchesusedinthismodelaresketchedasinFigure 7{8 .Whentheswitchisclosed,thevoltageQ(t)=0.Whentheswitchisopen,thecurrenti(t)=0.Ineithercase,thepowerlossP(t)=Q(t)i(t)=0.Sothereisnopowerlossintheidealswitch. 7.21 isthestatespaceequationoftheactivevibrationabsorbersystem.Theterm1 7.26 .Theproblemwehavehereisevensimpler,weonlyhavetherstordertermdx dt=f(x),whichisEquation 2.22 .Lett=",and=1 2.22 andthushasalargevalue.Thusweintroduceanewcontrolsignal^h()=h(t)=h(").Thefunction~h()isexpressedinthefollowingequation Figure 7{9 depictsthefunction^h().where,Nisdenedasbefore.Substitute

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Figure7{9: Switchedsignal 7.21 .Wehavedx DeneY

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SubstitutetheabovedenitionsintoEquation 7.34 .wehavethefollowingequationdY +"0BBBBBBBBBB@000bs Equation 7.35 isthestandardform(Equation 2.22 ),withinitialconditionx NowapplytheaveragingmethodtoEquation 7.35 .First,theaveragingoftheconstanttermsarethemselves.Byobservation,therearefourtimevaryingtermsinthetheequationofmotionwhichneedtobeaveraged.TheyarecomposedofthethreeexpressionsKscd33 1.ApplyaveragingtoKscd33 1

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UseEquation 7.33 ,weobtainedthefollowingafterapplyingtheaveragingmethod 1 Cp+C0+(1D) Cp+C0+1D Cp(7.37) 2.ApplyaveragingtotermCs 1 (7.38) Changevariablesbacktot,1 Herewedenote1 and1 3.ApplyaveragingtoKs 1 Changevariablesbacktot,1 CombineEquation 7.38 and 7.39 .ThetimevaryingterminvectorF 7.35 isaveragedasbs

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or Changevariablesbacktot,Y 7.41 Cp+C0+1D CpKscd33D Cp+C0+1D Cp01CCCCCCCCCCCAy Theforcingtermiszeroiftheoperatingtimeisanintegernumberofdrivingperiodt=nT,whichisthecasewearestudyinghere.Thusouraveragesystemissimplyexpressedas_y

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ofEquation 7.27 .Inthiscaseg(t;x(t))=0,theproofisevensimpler.x dsdy dsds=tRt0f(t;x D ,inthiscase,2=0,1=L,3=2MT,thusx 7.41 isexpressedinEquation 7.42 ^Ys ( where,theeectivestinessofthedevice 7.43 Cp+C0+1D Cp(7.43) NotethatthedutycycleDintheeectivestinessofthedevicedeterminesaconvexcombination,oraverage,betweenthetwocapacitancevalues1=(Cp+C)and1=Cp.Thesearethecapacitancevaluesoftheclosedandopenswitchpositions,respectively.Fromthetransferfunctionoftheaveragedequationsofmotion,itisimportanttonotethatthefrequencyresponsefunctions(FRF)havethesameapparentformasthoseassociatedwiththeclassicalvibrationabsorber.However,

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thefrequencyresponsefunctionsinthiscaseareparameterizedbythedutycycleD.Figure 7{10 illustratestheFRFforseveraldierentvaluesofthedutycycleD.ThisgureshowsthattheFRFcharacterizingtheperformanceofthetunablevibrationabsorberisacontinuousfunctionofthedutycycleoftheswitchthatmodulatesthecapacitiveshuntcircuit.Wenowcomparetothereference[4].In[4],notchfrequenciesareacollectionofxed,niteanddiscretefrequencies.However,thedesigninthisresearchcancontinuouslyadjustthenotchfrequencies,asshowninFigure 7{11 .Figure 7{11 .ashowsthattheresultingnotchfrequenciesin[4]arediscrete.Figure 7{11 .bshowthenotchfrequencieschangewithinthesamerangebutarecontinuous.Inpractice,thedrivingfrequencyisnotunique,itusuallychangeswithinanarrowbandwidth.In[4],ifthedrivingfrequencydeviatesfromthenotchfrequency,thevibrationlevelwillbeup.However,thedesigndevelopedinthisresearchcanalwaysndaproperdutycycleDsuchthatthevibrationiskeptminimal.Thus,itcanachievebettervibrationreduction.

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Figure7{10: Transferfunctionoftheaveragedsystem

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Figure7{11: Comparisonofthetwodesigns:discreteandcontinuousnotchfrequen-cies

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127 Table7{1:Parametervaluesusedinthesimulation TermnamesSymbolsValuesUnits Crosssectionarea A 2 : 58 10 7 m 2 Thicknessofeachlayer L 5 : 08 10 4 m Compliance S E 33 1 : 52 10 11 m 2 =N Compliance S E 11 1 : 5 10 11 m 2 =N Shortcircuitstiness K sc = A LS 33 6 : 25 10 7 N=m Piezoelectriccoecient d 33 190 10 12 m=N Piezoelectriccoecient d 31 275 10 12 m=N Dielectricpermitivitiesunderconstantstress 33 1 : 53 10 8 N=m Zerostraincapacitance C p 6 : 884 10 12 F Shuntedcapacitance C k 13 : 77 10 12 F Actuatormass M a 1 : 026 10 6 Kg Structuremass M s 4 : 104 10 6 Kg Energycouplingfactor k 0 : 65 none Dampingofpiezoceramic b a 0 : 005 Ns=m Dampingofstructuremass b s 0 : 005 Ns=m Eectivestiness k a 3 : 5346 10 7 N=m

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Twotopicsarepresentedinthisdissertation.First,afrequencydependenthysteresismodeliscreatedtopredictthedisplacementhysteresisofaPZTactuatorforawiderangeofoperatingfrequencies.Second,anaveraginganalysisisappliedtoadiscontinuousstateswitchedsystem.Thedesignmethodologydevelopedhereprovidesaessentiallycontinuousrangeoftunablenotchfrequencies,incontrasttoaniteandxedcollectionofdiscretenotchfrequenciesavailablein[4]. Inthesecondmethod,thegoverningequationsarederivedfromrstprinciplestobe [M]x+[C]_x+[K]x=[B(Q)](t)(8.2) Thedisplacementxisthesolutionofthegoverningequation.Thedynamicsisrepresentedbytheordinarydierentialequation,whichislinear.Thenonlinearhysteresisarisesinthedrivingterm[B(Q)](t),whichisalsocalledthecontrol 128

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inuenceoperatorandisexpressedas [B(Q)](t)=NXk=1k Inbothofthecasesk Thisresearchcollectshugeamountofdataforthepurposeofthemodelidenticationandmodelvalidation.Twohysteresismodelsarepresentedhere:oneincludesonlyanonlinearsubsystem,theotheroneincludesbothanonlinearsubsystemandalinearsubsystem.Simulationandmeasurementsdemonstratethatbothmodelscanprovideveryaccuratedisplacementprediction.Thecascadednonlinear-linearmodelgivesamoreaccurateandsmootherprediction.Butthecontributionofthelinearsubsystemislimited.Thismodelappliesforawiderfrequencyrangethancurrentlyexists.

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A{1 showsconstructionofatomandaxiomaticdenitionofpolarization. FigureA{1: Dipolemoment 130

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whereNisthenumberofmagneticdipolesperunitvolumeandisthemagneticdipolemomentperdipole(Griths1998,p.220).Themagnetizationcanalsobewrittenas whereisthemagnetizability.ThemagneticdipolemomentcausedbyacurrentloopcarryingcurrentIisgivenby 2cIZrdl=AI c(A.3)

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Figure 2{4 showsadampedmass-springsystemwithaforcef1actingonthesystem.Ifweonlyconsidertheundampedsystem,c1=0,whentheforcingfrequencyequalsthenaturalfrequencyofmassm1,theresponseofthemassapproachesinnityorbecomesunbounded.Thisphenomenoniscalledresonance,anditcancausesevereproblemsforvibratingsystems.Inthefollowinganalysisandderivation,wewillstillkeepthedampinginducedbyc1forgenerality.Whenconsideringtheundampedcase,c1isassignedtobe\0".Thegoverningequationformassm1isgivenbytheprototypicalequation TheLaplacetransformofEquation B.1 givesthetransferfunctionofthesingledegreeoffreedom(DOF)vibratingsystem Infrequencydomain,wehave !n;1! !n;12(B.3) where,!n;1isdenedas!n;1=q

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andthephaseofthetransferfunctionistan=c1! k1m1!2: theresponseofthemassm1becomesunbounded.Figure B{1 showsthebodeplot FigureB{1: Responseofthe1-Dvibrationsystem ofanundampedanddampedmass-springsystemhavingdierentdampingfactors.Forsimplicityintheillustration,thesystemparametersaretakentobem1=1andk1=1.WecanseearesonancewhentheinputfrequencysatisesEquation B.4 Inthedampedcase,resonancestilloccursatfrequency!=!n;1=p

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Ironparticlesareaddedtoatwo-partsiliconepriortocuring.Thiscompositeliquidisthenplacedinaspecially-designedmoldtocure.Thepurposeofthismoldistogenerateastrongmagneticuxpaththatgoesthroughtheelastomerduringthecureprocess.Thiscausestheironparticleswithinthecompositetoalignalongthelinesofuxandthenbesolidiedintothesechains.Thisenhancesthestinesschangeeectoncecured.Thecuredsiliconecompositeundertheinuenceofnomagneticuxdensityissoft,whereasvariableamountsofmagneticuxcanincreaseitsstiness.Thestateswitchactuator(SSA)mustbedesignedsuchthatamagneticuxpathcanbegeneratedtorunthroughtheelastomer.Forthisreason,twohalf-cylindersoflow-carbonsteelwereconstructedasbaseandabsorbermasses.Roughly300turnsofgage-25magnetwirearewrappedaroundtheabsorbermass.Withthissetup,directcurrentthroughthemagnetwirecangenerateamagneticuxpaththrougheachsiliconehalf. 134

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supposethatfort0tt0+T(t)2(tt0)=1tZt0(s)ds+3 135

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BIOGRAPHICALSKETCH XiaoyanZhangwasborninLaiyang,Shandong,P.R.China,in1973.She attendedtheHarbinEngineeringUniversity,locatedinHeilongjiang,China, whereshereceivedherdegreeofBachelorofScienceinmechanicalengineering sciencein1995.SheattendedtheChineseAcademyofSciences,locatedinBeijing, China,whereshereceivedherdegreeofMasterofSciencein1998.Currently SheisaPh.D.candidateintheDynamicsandControlLabintheDepartmentof MechanicalandAerospaceEngineering,UniversityofFlorida.Herpresentresearch interestsareintheareasofnonlinearcontrolmethodologyforpiezoelectronic systemsandvibrationsuppressionusingpiezoelectricmaterials. 142