Design, modeling and analysis of piezoelectric forceps actuator

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Content DESIGN, MODELING AND ANALYSIS OF
PIEZOELECTRIC FORCEPS ACTUATOR

by

Ken Suwarno Susanto

A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(MECHANICAL ENGINEERING)

May 2007

Copyright 2007 Ken Suwarno Susanto

DEDICATION

To Endless Love and Support of

My Parents and Family

ii
ACKNOWLEDGEMENTS

I would like to thank my Ph.D. advisor, Prof. Bingen Yang, for his guidance,
many suggestions, patience, encouragement, support and friendship over years
during my Ph.D. study. Without his expert knowledge in applied mechanics,
vibration and control, this dissertation would lack what completeness, insight, and
continuity it has.
I am very much grateful to both Prof. Henryk Flashner and Prof. Petros Ioannou,
for teaching me “What is all about the control system?” while I took their courses
few years ago. Their teachings, time, support, valuable comments and discussions to
improve part of the dissertation throughout the years are greatly appreciated.
In addition, I would like to thank to both Prof. Yan Jin and Prof. Geoffrey
Shiflett, for their insight, time, support, valuable comments and discussions about the
design point of view and experimental testing discussions to improve part of the
dissertation throughout the years are greatly appreciated.
Finally, I would like to thank to Prof. George Bekey for his valuable comments,
kindness, willingness, support and time to serve as one of the committee members
for my Ph.D. qualification exam.
Special appreciation and acknowledgment to the Charles Lee Powell
Foundation, Design News and ANSYS, inc., to sponsor my work partially through a
college design engineering award.
Special thanks to Ms. Elsie Reyes, Ms. Marrietta Penoliar and Ms. Silvana
Martinez of Aerospace and Mechanical Engineering Department and my colleagues,
iii
Yaubin Yang, Zhengxiang Chen, Zheng Liu, Hongli Ding and Yuhong in Dynamic
System Lab for their friendship throughout the years.

iv
TABLE OF CONTENTS
Dedication ii
Acknowledgements iii
List of Tables vii
List of Figures viii
Abstract xi
Chapter 1: Introduction 1
1.1 Background and Motivation 1
1.2 Current State of the Art 6
1.3 Previous Research on Curved Beams 11
1.4 Objectives and Scope of the Study 15

Chapter 2: Review of Shell Theory 17
2.1 Derivation of the Basic Shell Equation 17
2.2 Strain-Displacement Relations 24
Chapter 3: Piezoelectric Laminated Beam Models 32
3.1 Review of Piezoelectric Materials 32
3.2 Piezoelectric Laminated Slightly Curved Beam Model 45
3.3 Piezoelectric Laminated Straight Beam Model 55
Chapter 4: Transfer Function Formulation of Piezoelectric Laminated Beam 66
4.1 Transfer Function Formulation of Piezoelectric Forceps Actuator 66
4.2 Transfer Function of Piezoelectric Straight Beam: Static 72
4.3 Transfer Function of Piezoelectric Straight Beam: Dynamic 78
Chapter 5: Models of Piezoelectric Forceps Actuator 83
5.1 Radius of Curvature Displacement Model of the PFA 83
5.2 Grasping Force Model of the PFA 91

Chapter 6: Characteristic Performance Analysis of 93
Piezoelectric Laminated Straight Beam
6.1 Performance Analysis Piezoelectric Straight Beam: Static 96
6.2 Performance Analysis Piezoelectric Straight Beam: 109
Natural Frequencies
6.3 Conclusion 109

v
Chapter 7: Characteristic Performance Analysis of 111
Piezoelectric Forceps Actuator
7.1 Exact Natural Frequencies by DTFM 111
7.2 Approximation Natural Frequencies by Rayleigh-Ritz Method 113
7.3 Conclusion 115
Chapter 8: Piezoelectric Forceps Actuator Experimental Testing 116
8.1 Radial Deflection Measurements 116
8.2 Grasping Force Measurements 120
8.3 Natural Frequencies Measurements 123
8.4 Conclusion 125
Chapter 9: Control System Formulation of Piezoelectric Forceps Actuator 126
9.1 Open-Loop Transfer Function: DTFM Model 127
9.2 Open-Loop Transfer Function: Rayleigh-Ritz Model 128
9.3 Feedback Control Formulation of the PFA: DTFM Model 132
9.4 Feedback Control Formulation of the PFA: Rayleigh-Ritz Model 135
9.5 Numerical Simulations 136
9.6 Conclusion 137
Chapter 10: Concluding Remarks 139
Chapter 11: Future Research 141
Bibliography 143
Appendices
Appendix A 150
Appendix B 154

vi
LIST OF TABLES
Table 1.1: Existing miniature/ micro gripper or forceps 5
Table 1.2: Literature review of curved beams 34
Table 3: IEEE standard on piezoelectricity symbolic 36
Table 6.1: Material properties of multimorph 95
Table 7.1: Material properties of Piezoelectric Forceps Actuator 112
Table 7.2: Theoretical natural frequencies of the PFA 115

vii
LIST OF FIGURES
Figure 1.1 Novel concept idea of surgeon doing minimal invasive surgery 3
Figure 1.2 Schematic drawing of Piezoelectric Forceps Actuator 9
Figure 1.3 Piezoelectric Forceps Actuator 10
(a) grasped a thin wire through long narrow beaker;
(b) integrated with an endoscope;
(c) controlled with data glove;
(d) as a miniature robotic gripper attached to the mobile robot
Figure 2.1 Coordinate system of reference shell element 21
Figure 2.2 Distance between two points with height zn

from the reference surface element 21
Figure 3.1 Dipole alignment in piezoelectric material 34
Figure 3.2 Piezoceramic Material 37
Figure 3.3 A piezoceramic expands as a result of 39
an applied voltage differential
Figure 3.4 A short circuited (E = 0) piezoelectric sample 40
Figure 3.5 An open circuit (D = 0) piezoelectric sample 40
Figure 3.6 Electrodes attached to the 2-faces and a field 42
of intensity applied
2
E

Figure 3.7 Piezoelectric laminated slightly curved beam 47
Figure 3.8 Piezoelectric laminated straight beam 57
Figure 4.1 Free body diagram of the Piezoelectric Forceps Actuator 68
Figure 4.2 Schematic of the multimorph 75
(a) clamped-free ends; (b) simply-supported ends;
(c) clamped-elastically constrained; (d) clamped-clamped ends
Figure 4.3 Cantilevered multimorph 79
viii
Figure 5.1 The Piezoelectric Forceps Actuator 84
(a) closed its jaws; (b) opened its jaws
Figure 5.2 Free body diagram of the cantilevered PFA 85
for radial deflection and grasping force analysis
Figure 5.3 Lower part portion of the PFA (A-B segment) 87
under mechanical and piezoelectric moments
Figure 6.1 Schematic diagram of the cantilevered multimorph 94
case I: EI-varies for arbitrary number of layers;
case II: EI-constant for arbitrary number of layers
Figure 6.2 Case I: The spatial distribution of cantilevered multimorph 97
Figure 6.3 Case II: The spatial distribution of cantilevered multimorph 98
Figure 6.4 Cases I and II: Bending stiffness of cantilevered multimorph 100
Figure 6.5 Case I: Maximum tip displacement and number of layers 101
Figure 6.6 Case II: Maximum tip displacement and number of layers 102
Figure 6.7 Cases I and II: Bending moment of cantilevered multimorph 103
Figure 6.8 The effect of each piezoelectric layer thickness 104
(a) bending moment; (b) tip displacement

Figure 6.9 The effects of ON-OFF activation of piezoelectric 105
induced moment of simply supported bimorph
Figure 6.10 The spatial distribution of clamped-elastic 106
constrained multimorph
Figure 6.11 Case I: Force and number of layers 107
Figure 6.12 Case I: Force and input voltage 108
Figure 6.13 Case I and II: Natural frequencies and number of layers 110
Figure 8.1 Schematic diagram of the experimental set up 117
with a curvature fiber optic sensor mounted on the PFA

ix
Figure 8.2 Theoretical radial deflection of the PFA verified with 118
experimental measurements
(a) 0V to 100V; (b)-300V to 300V
Figure 8.3 Experimental grasping force of the PFA measured by 121
Milinewton  force sensor
(a) schematic diagram of the PFA grasps the force sensor
(b) top view, tip jaw of the PFA ready to grasp the force sensor
Figure 8.4 Theoretical grasping force model of the PFA 121
verified with experimental result
Figure 8.5 Measured frequency response of the PFA 123
Figure 8.6 Theoretical predicted frequency response of the PFA 124
Figure 9.1 Schematic of the PFA under position control system 126
Figure 9.2 Frequency responses of the PFA 137
Figure 9.3 Step responses of the PFA with and without feedback control 138
Figure 9.4 Error position tracking of the PFA under PD feedback control 138
Figure 11.1 Data glove fuzzy controlled the PFA 142
Figure 11.2 Future PFA attached to the snake-type robot and its tip jaws 142
embedded with micro-machined force sensor

x
ABSTRACT
Meso/micro grasping of tiny soft objects such as biological tissues or biopsy,
which ranges from hundreds to thousands micro-millimeters in dimension, plays a
significant role in the fields of tele-surgery, minimally invasive surgery (MIS), and
biomedical instrumentation. In this dissertation, a proposed novel piezoelectric
forceps actuator (PFA) is fabricated to improve one of the existing surgery tools used
in MIS. One of the advantages of the PFA over conventional MIS forceps lies in that
it can be remotely controlled to achieve precision deflection and grasping force.
Furthermore, it does not have any moving parts such as gears and hinges, and hence
avoids problems in operation like friction, backlash, lubrication, leakage and
sterilization. A new general piezoelectric laminated slightly curved beam model is
derived to predict the natural frequencies of the PFA, and verified with experimental
results. By setting the curvature of the proposed model to zero (i.e., the radius of
curvature goes to infinity) and neglecting stretch-bend couple, a piezoelectric
laminated straight beam model can be derived. With a distributed transfer function
formulation for the proposed models, the effects of the layer number, the layer
thickness and the bending stiffness on natural frequencies, deflection, bending
moment and output force of the system can be investigated. Based on solid
mechanics and Castiliagno’s theorems, radius of curvature displacement and
grasping force models of the PFA is derived and verified with experimental results.
Finally, the control system formulations of the PFA based on the distributed transfer
function method (DTFM) and Rayleigh-Ritz method are provided to show the
xi
usefulness of the proposed models for further used in control system design and
analysis.
xii
CHAPTER 1
INTRODUCTION

1.1. Background and Motivation
Minimally invasive surgery (MIS) has recently been advanced with the use of
endoscopes, improvements in camera equipment and advances in medical
robotics. The advantages to the patient are less pain, smaller incisions rather than
through large cavities as in traditional open surgery, fewer complications, shorter
hospital stays, less cost and a more rapid return to normal activity when compared
to conventional surgery. If necessary, a bit of tissue can be taken for a biopsy or
relatively minor surgery performed, without the need for fully invasive surgery.
Robotics and computers are now playing an expanding role in assisting the
surgeon in these minimally invasive procedures. The idea of robotics in surgery
got its start in the military. The idea was to develop technology where a surgeon
could perform an operation from a remote location on an injured soldier in the
battlefield. This concept has evolved into robotics to enhance surgical
performance. In this instance, a robotic arm performs the procedure with the
surgeon guiding the robotic arm from a location in or adjacent to the operating
room. The surgeon sits at a station peering at a monitor that shows a magnified
view of the surgical field. A computer mimics and enhances his hand movements
1
through joystick interface. The computer in this instance makes the movements
more precise by dampening even a tiny tremor in the surgeon's hands, which
might increase the difficulty in performing procedures under high power
microscopic magnification.
Since 1987, the MIS has rapidly been applied to many procedures previously
performed only with traditional “open” surgical techniques. While providing
important improvements, currently MIS tools generally consist of small parts,
actuation handle via long tendon wires or push rods, rigid components connected
by mechanical hinge joints and gaps which make proper sterilization difficult and
costly, (Frecker and Aguilera, 2004). By incorporating compliant mechanisms
into the tool structure, the capability of MIS instruments could be enhanced. In
contrast to traditional rigid-link mechanisms that use hinge joints to allow for
relative motion between rigid link members, compliant mechanisms are single
piece flexible structures that exploit elastic deformation to achieve motion
transmission. The compliant mechanisms could be easier to construct and sterilize
because they are single piece mechanisms. Compliant mechanisms also lend
themselves to actuation by systems other than the standard push-rod, such as
piezoelectric actuators. The focus of this dissertation is on the design, modeling
and performance analysis of the prototype piezoelectric forceps actuator (Susanto,
2002; Titus, 2003; Susanto and Yang, 2003; Susanto and Yang, 2004a, b; Susanto
and Yang, 2007) that can be controlled by surgeon remotely via data glove
interface in minimally invasive surgery procedure as shown in Figure 1.1.
2

Figure 1.1 Novel concept idea of surgeon doing minimal invasive surgery or tele-
surgery by simply wearing the data glove to manipulate the Piezoelectric Forceps.

3
Several researchers have explored innovative designs for MIS tools to address
the limitations of current instruments. Table 1 lists some existing miniature/micro
grippers that are designed for MIS procedures. Compared to these grippers and
conventional minimum invasive surgery forceps, the proposed piezoelectric
forceps actuator or known as the PFA is simple in design, inexpensive in
manufacture, small size, light weight, being able to achieve precision deflection
and grasping force with computer controlled or data glove (Susanto and Yang,
2004b).
Furthermore, the PFA does not have moving parts such as mechanical gears,
hinges, bearings, racks and pinions, and thus avoids problems in operation like
friction, backlash, lubrication, leakage and sterilization.

4
Authors Device Actuation Method Advantages Disadvantages
Henein
(2003)
Micro gripper
Inch-worm
piezoelectric
Integrated force
sensor
A lot of hinges
Ku
(1995)
Micro gripper Solenoid, hydraulic
High grasping
force
Mini solenoid,
costly, leakage
Miyata
(2005)
Micro forceps
Push and pull
wire drive
Translation and
rotation
Complex design,
Manual drive
Ikuta
(2002)
Hyper-finger of
remote minimal
invasive surgery
Small motor and wire
drive
Multiple degrees
of freedom
Hinged, friction,
sterilization
Kobayashi
(2002)
Endoscopic
robotics
DC-servo motors,
linear guides
High stiffness
Hinged and ball
screw design
Dario
(2003)
Da Vinci surgical
robot
Motor and
wire drive
Redundancy
Hinged, friction,
sterilization
Carrozza
(2000)
Micro gripper
Piezoelectric stack
actuator
Integrated with
strain gage sensor
LIGA
fabrication,
highly cost

Edinger
(2000)
Piezoelectric
actuator for
minimally invasive
surgery
Inchworm
piezoceramic stack
actuators and servo
motor
Stack actuator
increases force
Heavier weight
because of the
stacks actuators
and less
deflection

Frecker
(2005)

Multifunctional
compliant
instrument for
minimally invasive
surgery
Push and pull
No hinge joints,
easier for
sterilization
Manually
actuated

Tanikawa
(1999)
Two-fingered
micro hand
Inchworm
piezoceramic stack
actuators
Grasp and rotate
Complex and a
lot of
piezoelectric
stacks actuators

Table 1.1 Existing Miniature/Micro Gripper or Forceps

5
1.2. Current State of the Art
Smart structures are the field that have the potential to make a significant
impact on our world, a field that heralds the “next engineering revolution”
(Rogers, 1995). In this field, the structures are engineered to sense and react to the
world around them. To perform these functions, a smart structure requires
embedded sensors, actuators, and control logic.
In the last decade, applications have appeared in the field of smart structures,
includes applications in vibration control, noise control, shape control, flow
control and biomedical device applications:
• Vibration control is an important application where smart actuators are actively
used to damp vibration. Vibration control, where actuators are used to dampen the
movement of structures, is the most popular application in the field of smart
structures (Tani, 1998). This technology has been applied to applications ranging
from simple beam and plates (Herold, 1996) to active automotive suspensions
(Fukami, 1994) to helicopter blade smart flaps (Hall, 1996) to vibration
dampening skis (Ashley, 1995).
• Noise control refers to the use of smart structures to attenuate sound (Liang,
1992). Smart materials have been used to reduce noise from items as simple as
individual beams (Wang, 1994) and as complex as the inside of jet aircraft
(Fuller, 1996).
6
• Shape control refers to the use of actuators to flex a structure, thus altering its
overall shape. Shape control is used to reduce wing drag in supersonic aircraft by
altering wing shape (Kudva, 1996), to optically correct deformable mirrors
(Burke, 1991) and to steer antennas (Washington, 1996).
• Flow control refers to the use of smart structures to alter the flow of fluid. Flow
control applications include minimization of drag in aircraft (Austin, 1995).
• Rapid developments in intelligent biomedical devices using smart materials
have been proposed, which include a piezoelectric cell stretching device (Clark,
2000), a stimulator for cultured bone cells (Tanaka, 1999), a device for straining
flexible cell culture membranes (Schaffer, 1994) and a novel piezoelectric forceps
actuator for minimum invasive surgery (Susanto, 2002; Titus, 2003; Susanto and
Yang, 2003; Susanto and Yang, 2004a, b; Susanto and Yang, 2007).
Many of the new breakthroughs have been possible due to the development of
actuators constructed from high energy density smart materials, such as
piezoceramics, shape memory alloys, electrostrictives, and magnetostrictive
materials. Piezoelectric actuators are the actuators of choice because they are
small, have low power requirements, respond quickly, and generate relatively
high forces (Baz, 1988; Damjanovic, 1992).
A number of new actuator architectures, such as leveraged stacks,
flextensional actuators and tapered benders have been created. Unfortunately,
these type of actuators either have low efficiency, have a limited performance
7
range, or are poorly characterized. To find an actuator architecture that can
perform as a MIS grasping forceps, a prototype design of the PFA as shown in
Figure 1.2 is proposed and designed, where the actuator is composed of two
symmetric curved beams laminated with piezoceramic (PZT or Plumbum
Zirconate Titanate) layers.
The purpose of using a curved beam structure in the PFA design is to produce
higher transverse displacement and force outputs, as demonstrated in the studies
on an actuator called THUNDER (Thin Layer Composite Unimorph Ferroelectric
Drive and SensoR) (Shakeri, 1999; Mossi, 1999; Ounaies, 2001). The
piezoelectric laminated curved beam is stiffer than an equivalent straight bender
of the same cross-section, and less stiff than an equivalent stack due to its nature
of the curved shape. Thus, the piezoelectric laminated curved beam is very ideal
to be fabricated as the MIS forceps due to its capability to produce forces and
deflections between those of the straight bender and the stack.
The PFA is especially capable of grasping a very thin wire (such as a blood
vessel or vein) through the narrow small diameter beaker (Figure 1.3a), which
makes the actuator a promising device for meso/ micro grasping tiny object,
tissues or cells in many biomedical applications. Furthermore, the PFA also can
be integrated with endoscope (Figure 1.3b) and fuzzy controlled via human-
computer interface data glove (Figure 1.3c). Besides of biomedical applications,
the PFA also has potential applications in Mars exploration rover mission in
8
collecting and return of minute rock and soil samples back to Earth for further
laboratory analysis (Figure 1.3d).

Figure 1.2 Schematic Drawing of Piezoelectric Forceps Actuator (PFA)

9

(a) (b)

(c) (d)

Figure 1.3 Piezoelectric Forceps Actuator (PFA) (a) grasped a thin wire through
long narrow beaker. (b) integrated with an endoscope. (c) controlled with data
glove. (d) acted as a miniature robotic gripper attached to the mobile robot.

10
1.3. Previous Research on Curved Beams
The dynamic behavior of curved beam is a more complicated subject than the
static behavior. The vibration of curved beam is a topic that has been investigated
by a number of researchers, as illustrated in Table 1.2. This table illustrates the
breakdown of research on curved beams into different categories, with the
primary division being between individual curved beams. The research on
individual curved beams can be further broken down into composite curved
beams, composed of a number of layers; and homogeneous curved beams,
fabricated from one solid material. Finally, the research can be categorized on the
basis of the assumptions made to determine the equations of motion, which range
from neglecting the effects of rotatory inertia and transverse shear and assuming
an inextensible neutral axis to including the effects of rotatory inertia and
transverse shear and assuming an extensible neutral axis.
The body of research on individual curved beam vibration has been reviewed
quite comprehensively. The latest, most complete review was by Chidamparam
and Leissa (1993), including over 400 references. Other reviews of curved beam
literature were done by Aucielo and De Rosa (1993), Markus and Nanasi (1981),
and Laura and Maurizi (1987). These reviews present a wealth of information on
homogeneous curved beams, but illustrate the comparative scarcity of literature
on composite curved beams.

11
Homogeneous Curved Beams Composite Curved Beams
Non-Piezoelectric Non-Piezoelectric Piezoelectric
Inexten.
Neutral
Axis
Exten.
Neutral
Axis
With
Rotatory
Inertia
With
Trans.
Shear
Semi
Circular
Slight
Curved
Semi
Circular
Slight
Curved
Love
(1944)
Waltking
(1934)
Philipson
(1956)
Buckens
(1950)
Dym
(1980)
Qatu
(1992)
Larson
(1993)
NONE
Lamb
(1888)
Nelson
(1962)
Hammoud
(1963)
Seidel
(1964)
Qatu
(1993)

Wash-
ington
(1996)

Den
Hartog
(1928)
Veletsos
(1972)

Austin
(1973)

Mitchell
(1997)

Archer
(1960)
Chidam-
param
(1995)

Rao
(1969)

Moska-
lik
(1999)

Laura
(1987)

Irie
(1983)

Aucielo
(1993)

Table 1.2 Literature Review of Curved Beams

12
The research conducted on the composite slightly curved beams model is the
most immediately useful in modeling the PFA because the PFA consists of a
symmetric pair of composite slightly curved beams laminated with piezoelectric
layers. However, many researchers examining the dynamic behavior of composite
curved beams, like Dym (1980), merely derived equations of motion for the
curved beam, but without a piezoelectric term. As mentioned previously, these
equations of motion for composite curved beams are identical in form to those for
homogeneous beams, but incorporate laminate stiffness in place of the
homogeneous extensional and bending stiffness.
Qatu (1993) used these equations of motion to find the natural frequencies of
specific geometries of pinned composite beams, and extended the theory with the
inclusion of rotatory inertia and transverse shear to find the natural frequencies of
thicker pinned composite beams. Vinson and his colleagues included piezoelectric
terms in the equations of motion of composite shells (Larson and Vinson, 1993a),
composite curved beams and rings (Larson and Vinson, 1993b). Although Larson
and Vinson (1993b) use these equations of motion to develop solutions for the
static behavior of curved composite beams, they did not investigate the dynamic
behavior at all.
Washington (1996) added the piezoelectric term to the composite circular
beams by Qatu (1993) to study the quasi-static behavior of satellite antenna.
Mitchell and Chattopadhyay (1997) improved the composite piezoelectric circular
beams model by including first order shear deformation. But none of them
13
determined any dynamic response or natural frequencies of the composite
piezoelectric-laminated curved beams.
Moskalik and Brei (1999) added the piezoelectric term to the composite
circular beams by Qatu (1993) to study their C-block actuator in terms of the
static and dynamic behavior.
Thus, Larson and Vinson did not solve the natural frequencies of the problem
that is necessary to understand the behavior of composite piezoelectric curved
beams, and Qatu did not include any piezoelectric terms in his dynamic solutions.
Washington, Mitchell and Chattopadhyay modeled the composite piezoelectric
circular curved beams and solved it for quasi-static case only. Moskalik and Brei
modeled the composite piezoelectric circular curved beams for C-block actuator
but they did not model the composite piezoelectric laminated slightly curved
beams. Unfortunately, all previous existing research works of composite
piezoelectric curved beams are not sufficient or immediate used to formulate the
model for the PFA.
However, the work of these researchers provides a useful guidance on which
to base the development of the equations of motion for the PFA, even though
additional work must be done to complete the modeling and analysis.
The only composite non-piezoelectric slightly curved beams by Qatu (1992)
can be modified and extended by adding piezoelectric term on its dynamic
equation to fit the model of the PFA.

14
1.4. Objectives and Scope of the Study
This study is focus on the following five aspects:
(1) To develop a new general piezoelectric laminated slightly curved beam
model with arbitrary boundary conditions that can be used to model the
Piezoelectric Forceps Actuator (PFA). By setting the curvature of the
proposed model to zero (i.e., the radius of curvature goes to infinity) and
neglecting stretch-bend couple, a piezoelectric laminated straight beam
model is derived. Based on the proposed model, the theoretical prediction of
natural frequencies of the piezoelectric laminated curved beam are obtained
and verified with experimental result.
(2) With distributed transfer function formulation for the proposed models, the
effects of the layer number, the layer thickness and the bending stiffness on
natural frequencies, deflection, bending moment and output force of the
system are investigated.
(3) The theoretical predictions of the displacement and grasping force of the
PFA derived based on solid mechanics and Castiliagno’s theorems are
obtained and verified with experimental results.
(4) Finally, the control system formulations of the PFA based on distributed
transfer function method (DTFM) and Rayleigh-Ritz method are provided to
show the usefulness of the proposed models for further used in control
system design and analysis
15
The remainder of this dissertation is organized as follow:
In Chapter 2, the strain-displacement of the shell theory is briefly reviewed.
In Chapter 3, the static and dynamic models of the piezoelectric laminated
beam associated with its boundary conditions are derived based on potential
energy, kinetic energy and with the aid of Hamilton’s principle.
In Chapter 4, the DTFM is extended to obtain a closed-form formulation of
the static and dynamic responses of the piezoelectric laminated beams models.
With the transfer function formulation, the exact solutions in predicting the static
performances of the system in terms of displacement, slope, moment and force
easily can be obtained. For the dynamic case, the natural frequencies of the
system can be obtained in exact and closed form.
In Chapter 5, the radius of curvature displacement and grasping force models
of the PFA derived and discussed.
In Chapter 6, the characteristic performance analysis of the piezoelectric
laminated beam models are presented and discussed.
In Chapter 7, the theoretical prediction of the natural frequencies of the PFA is
obtained using the DTFM and Rayleigh-Ritz method.
In Chapter 8, the verification of theoretical models of the PFA with several
experimental results are presented and discussed.
In Chapter 9, the control system formulation of the PFA based on DTFM and
Rayleigh-Ritz method are presented.
16
CHAPTER 2
REVIEW OF SHELL THEORY

2.1. Derivation of the Basic Shell Equation

The goal in this chapter is to review and describe the shell equations and
strain-displacement relations which will be used in Chapter 3 to derive the
composite piezoelectric laminated thin straight/ curved beam models. The
derivation of the basic shell equations and strain-displacement relations are
classical and can be found in numerous references by Love (1944), Leissa (1973)
and Soedel (1993). Those references provide a basis for modeling many structures
and techniques that will be used when including the effects of piezoceramic
actuators on the shell (Banks (1996) and Tzou (1993)). The resulting equations
are sufficiently general enough to be reduced and simplified into the static and
dynamic equations of motion of piezoelectric laminated slightly curved beam
when judicious choices are made for radii and Lam e parameters and certain
defined assumptions.
The description of shells in a completely general setting involves
geometric rather than material considerations. In describing the initial model, a
shell is considered to be a solid bounded by two curved surfaces that are separated
by a distance h and limit the discussion to elastic materials satisfying linear
17
stress/strain relations. The middle surface will then be defined as the locus of
points lying midway between the outer bounding surfaces. The foundation of the
classical theory of thin shells, described in this manner, were first established by
Love (1944). In his formulation, he made the following four assumptions as
follows:
1. The shell thickness h is very small in comparison with the other dimensions
such as radius of curvature and length. This condition is crucial to thin shell
theory and says that the ratio of shell thickness, h, to the smallest radius of
curvature, R, is small as compared to unity; that is h / R << 1.
2. Shell deformations are sufficiently small so as to allow the second and higher
powers to be neglected with respect to the first powers. This requirement
allows all derived kinematic and equilibrium considerations to the original,
unperturbed, reference state of shell and ensures that the differential equations
of shell deformation are linear.
3. Transverse normal stresses (
z
σ ) are small compared to the other normal
stresses ( ,
α β
σ σ ) in the shell and hence can be neglected. In other words, the
stress in the direction normal to the thin dimension is taken to be negligible.
This assumption, in combination with the fourth, deals with the constitutive
properties of thin shells and allows the three-dimensional elasticity problem to
be transformed into a two-dimensional one.

18
4. A line which is originally normal to the shell reference surface will remain
normal to the deformed reference surface (in notation to follow, this yields the
assumption 0 = =
z z β α
γ γ ) and will remain unstrained ( e ). This
assumption is analogous to the Euler hypothesis in thin beam theory which
states that plane sections remain plane. This is referred to as Kirchhoff’s
hypothesis. From the latter two assumptions above, a three-dimensional
elasticity problem is reduced to two dimensions
0 =
z
When defining shell coordinates, it is convenient to choose the unperturbed
middle surface of the shell as the reference surface. On the reference surface, an
orthogonal curvilinear coordinate system is established which coincides with the
orthogonal lines of principal curvature. The thickness direction normal to the
reference is taken to be the third coordinate direction. From the fourth shell
assumption regarding the preservation of the normal, it follows that the
displacements must be linear in the thickness coordinate and thus the behavior of
any point on the shell can be determined from the behavior of a corresponding
point on the reference surface. Let the reference surface be determined by the
vector (, ) r α β
where α and β are independent parameters as shown in Figure
2.1. For the general case, we can define the infinitesimal distance between a point
that is normal to P and a point
1
P
1
P
′
which is normal to point as shown in
Figure 2.2. is located at a distance z from the neutral surface ( z is defined to
P ′
1
P
19
be along a normal straight line to the neutral surface). P ′ is located at a distance
from the neutral surface. We may therefore express the location as dz z +
1
P
β
→
R d
2 dn
dr
⋅

2
d
+
∂
( , , ) ( , ) ( , ) Rz r zn αβαβ α =+
(2.1)
where is a unit vector normal to the neutral surface. The differential change
as we move from to
→
n
1
P P ′ , is
dR dr z dn n dz =+ +

(2.2)
where
nn
dn d d α β
αβ
∂ ∂
=+
∂∂
(2.3)
The magnitude ds of is obtained by dR
(2.4)
()
()
()
2
2
2
2
2
2
22
ds dR dR
dr dr z dn dn n n dz z dr dz dr n
zdzdnn z dndn dz z dn
=⋅
=⋅ + ⋅ + ⋅ + + ⋅
+⋅+ ⋅+ + ⋅

The terms of the magnitude ds, equation (2.4), can be expanded as follow:
The 1
st
term of the equation (2.4),

() ()
() ()
() ( )
22
2
2
22
22
22
2cos
2
rr r r r r
dr dr d d d d
rr rr
dd
Ad B d
α βα
αα β β αβ
π
β
α βα
αβ αβ
αβ
∂∂ ∂ ∂ ∂ ∂
⋅= ⋅ + ⋅ ⋅
∂∂ ∂ ∂ ∂∂
∂∂ ∂∂
=+ +
∂∂ ∂
=+

(2.5) dβ

20

x

2
x
1
3
x
β
α
z
P
P ′
dr
r
rdr +

Figure 2.1. Coordinate system of reference shell element

dR

β
r
β
R
P
zn
R
dr
1
P
1
P
′
' P
R
α
α

Figure 2.2. Distance between two points with height zn
from the reference
surface element

21
The 7
th
term of equation (2.4),
() ()
22
22
2
nn n n n n
zdn dn z d d d d α βα
αα β β αβ
 ∂∂ ∂ ∂ ∂ ∂
⋅= ⋅ + ⋅ + ⋅

∂∂ ∂ ∂ ∂∂

β

(2.6)
The 2
nd
term of the equation (2.6),
() ()
2
2
2
nn n
zd z
2
d β β
ββ β
∂∂ ∂
⋅=
∂∂ ∂

(2.7)
From the Figure 2.2, the following relationship to the radius of curvature :
β
R

( ) /
/
zn
r
Rz
β
β
β
∂ ∂
∂∂
=
(2.8)
Since
r
B
β
∂
=
∂
, equation (2.8) can be rewritten as:

n z
z
B
R
β
β
∂
=
∂
(2.9)
and therefore
() ( )
2
2
22
2
nn B
zd z
R
β
2
d β β
ββ
∂∂
⋅=
∂∂
(2.10)
Similarly, the 1
st
term of the equation (2.6) becomes,
() ( )
2
2
2 2
2
nn
zd z
R
α
α2
d α α
αα
∂∂
⋅=
∂∂
(2.11)
thus, the equation (2.6) can be rewritten in a more compact form,
() ( )
22
2
22
A B
zdn dn z d d
RR
αβ
α
2
β
 
⋅= +
 
 
 

(2.12)

22
Finally, the last expression of equation (2.4) becomes

() ()
22
22
rn r n
zdr dn z d d
rn rn
dd dd
αβ
αα β β
αβα
αβ β α
∂∂ ∂ ∂
⋅= ⋅ + ⋅

∂∂ ∂ ∂

 ∂∂ ∂∂
+⋅ + ⋅

∂∂ ∂ ∂

β

(2.13)
The 1
st
term of the equation (2.23) may be written
() () ()
2
2 2 rn r n A
d d
R
α
2
d α α
αα α α
∂∂ ∂ ∂
⋅= =
∂∂ ∂ ∂
α (2.14)
Similarly,
() () ( )
2
2 2 rn r n B
dd
R
β
2
d β β
ββ β β
∂∂ ∂ ∂
⋅= =
∂∂ ∂ ∂
β (2.15)
Equation (2.13) therefore becomes
() ( )
22
2
22
A B
zdr dn z d d
RR
αβ
α
2
β
 
⋅= +
 
 
 

(2.16)
Substituting equations (2.5), (2.12) and (2.16) into equation (2.4) gives

() () () ( )
() ( ) ( )
2 2
22
22
22 2
12 3
11
zz
ds A d B d dz
R R
gd g d g dz
α
αβ
β
αβ
 
=+ + + +
 
 
=+ +
22
(2.17)
where and are the radii of curvature in the
α
R
β
R α and β directions and A and
B are the Lam constants which are defined by e
2 2
,
rr r r
A B
ααββ
∂∂∂∂
=⋅ =⋅
∂∂∂∂
(2.18)

23
The coefficients
1 , 1 , 1
3
2
2
2
1
=
















+ =














+ = g
R
z
B g
R
z
A g
β α
(2.19)
are a subset of the metric coefficients which provide a link between the length of
an element and the differentials α d , β d , and . dz

2.2. Strain-Displacement Relations
As shown in Leissa (1973) and Soedel (1993), the strain-displacement
equations in orthogonal coordinates which follow from the three dimensional
theory of elasticity are
3 , 2 , 1 ,
2
1
3
1
=
∂
∂
+








∂
∂
=
∑
=
i
g
U g
g
g
U
e
k
k
k k
i
i
i
i
i
i
α α
(2.20)

j i
j i
g
U
g
g
U
g
g g
j
j
i
j
i
i
j
i
j i
ij
≠
=
















∂
∂
+








∂
∂
=
3 , 2 , 1 ,
,
1
α α
γ (2.21)
where , and
i
U
i
e
ij
γ are the displacements, normal strains and shear strains,
respectively, at an arbitrary point in the material. The equations are posed in shell
coordinates by replacing the strain indices 1, 2 and 3 by α , β, and z,
respectively, and by letting U, V and W replace U , and U as the
displacements in the
1 2
U
3
α , β and z directions. Finally, the substitution of the metric
coefficients in (2.19) yields the general strain displacement equations
24








+
∂
∂
+
∂
∂
+
=
α α
α
β α R
W A
AB
V U
A R z
e
1
/ 1
1
(2.22)








+
∂
∂
+
∂
∂
+
=
β β
β
α β R
W B
AB
U V
B R z
e
1
/ 1
1
(2.23)

z
W
e
z
∂
∂
= (2.24)

( )
() ()
()
() ()








+ ∂
∂
+
+
+






+ ∂
∂
+
+
=
β α
β
α β
α
αβ
α
β
γ
R z B
V
R z A
R z B
R z A
U
R z B
R z A
/ 1 / 1
/ 1
/ 1 / 1
/ 1
(2.25)

()
()
( )






+ ∂
∂
+ +
∂
∂
+
=
α
α
α
α
α
γ
R z A
U
z
R z A
W
R z A
z
/ 1
/ 1
/ 1
1
(2.26)

()
()
( )








+ ∂
∂
+ +
∂
∂
+
=
β
β
β
β
β
γ
R z B
V
z
R z B
W
R z B
z
/ 1
/ 1
/ 1
1
(2.27)
In order to satisfy the Kirchhoff hypothesis, the displacements are assumed to
be linear in the thickness direction, thus yielding
( ) ( ) ( ) β α θ β α β α
α
, , , , z u z U + = (2.28)
( ) ( ) ( ) β α θ β α β α
β
, , , , z v z V + = (2.29)
( ) ( ) β α β α , , , w z W = (2.30)
where u, v and w are the displacements of the middle surface in the α , β and z
directions, respectively. The quantities
α
θ and
β
θ are the rotations of the normal
to the middle surface which occur during deformation.
25
In order to determine
α
θ and
β
θ in terms of the displacements u, v and w, it is
noted that the Kirchhoff hypothesis implies that all strain components in the
direction of the normal to the reference surface vanish; that is
0 = = =
z z z
e
β α
γ γ . (2.31)
When the equations (2.28-2.30) are substituted into equations (2.26 & 2.27), this
constraint on
z α
γ and
z β
γ implies that

α
θ
α
α
∂
∂
− =
w
A R
u 1
(2.32)

β
θ
β
β
∂
∂
− =
w
B R
v 1
(2.33)
To obtain the strain-displacement equations in terms of u, v and w for the case
of thin shells, equations (2.28-2.30, 2.32 and 2.33) are substituted into the
equations (2.22, 2.23 and 2.25) to obtain
(
α α
α
α
κ ε z
R z
e +
+
=
/ 1
1
) (2.34)
(
β β
β
β
κ ε z
R z
e +
+
=
/ 1
1
) (2.35)

()()
















+ + +








−
+ +
= τ ε γ
β α
αβ
β α β α
αβ
R
z
R
z
z
R R
z
R z R z 2 2
1 1
/ 1 / 1
1
2
(2.36)
Here
α
ε ,
β
ε , and
αβ
γ are the normal and shear strains in the middle surface,
α
κ and
β
κ are the midsurface changes in curvature, and τ is the midsurface
twist. As shown in Leissa (1973) these quantities are given by
26

α
α
β α
ε
R
w A
AB
v u
A
+
∂
∂
+
∂
∂
=
1
(2.37)

β
β
α β
ε
R
w B
AB
u v
B
+
∂
∂
+
∂
∂
=
1
(2.38)






∂
∂
+ 





∂
∂
=
B
v
A
B
A
u
B
A
α β
ε
αβ
(2.39)
and

β
θ
α
θ
κ
β
α
α
∂
∂
+
∂
∂
=
A
AB A
1
(2.40)

α
θ
β
θ
κ
α
β
β
∂
∂
+
∂
∂
=
B
AB B
1
(2.41)









∂
∂
−
∂
∂
+








∂
∂
−
∂
∂
+








∂
∂
+ 





∂
∂
=
β α
α β
θ
α
θ
β
τ
β
α
β
α
A
AB
u v
A R
B
AB
v u
B R B A
B
A B
A
1 1
1 1
(2.42)
From the equations (2.34 and 2.35), it can be seen that the total strains at any
point can be represented as the sum of two parts, one due to stretching and one
due to bending.
The existing theories dealing with the deformation of a thin shell differ in
how and when the terms and are to be neglected with respect to
unity in setting up the kinematic (strain-displacement), constitutive (Hooke’s
Law) and equilibrium equations. An overview of the various theories can be
found in Leissa (1973). All the theories agree in the expressions for the middle
α
R z/
β
R z/
27
surface strains
α
ε ,
β
ε , and
αβ
ε and also there is general agreement among the
theories about the expressions for the middle surface curvature changes
α
κ and
β
κ .
R z
α
e
β
e
γ
αβ
α



∂
∂
κ
α
β




∂
∂
κ
β




B
2
1
− =
In the Donnell-Mushtari theory, one neglects the terms and n
equations (2.34-2.36) and the tangential displacements and their derivatives in
equations (2.40-2.42) thus yielding
α
R z/
β
/ i
( )
α α
κ ε z + = (2.43)
( )
β β
κ ε z + = (2.44)
( ) τ ε
αβ
z + = (2.45)
and

β β α ∂
∂
∂
∂
− 


∂
∂
− =
w A
AB
w
A A
2
1 1 1
(2.46)

α α β ∂
∂
∂
∂
−




∂
∂
− =
w B
B A
w
B B
2
1 1 1
(2.47)







∂
∂
∂
∂
−




∂
∂
∂
∂
α β β α
τ
w
A B
A w
A
B
2
1
(2.48)
with the mid-surface strains given by the equations (2.37-2.39).
Let the assumption of constant A and B be imposed now, then the equations
(2.37-2.42) become

α
α
α
ε
R
w u
A
+
∂
∂
=
1
(2.49)
28

β
β
β
ε
R
w v
B
+
∂
∂
=
1
(2.50)
() ( ) v
A
u
B α β
ε
αβ
∂
∂
+
∂
∂
=
1 1
(2.51)

α
θ
κ
α
α
∂
∂
=
A
1
(2.52)

β
θ
κ
β
β
∂
∂
=
B
1
(2.53)
() ()






∂
∂
+








∂
∂
+
∂
∂
+
∂
∂
=
α β
θ
α
θ
β
τ
β α
β α
v
A R
u
B R A B
1 1 1 1 1 1
(2.54)
Now, for shallow shells, the α and β coordinates can be replaced by the
Cartesian coordinates, Ambartsumian (1961) and Wang (1982). Which yields :

y
x
B A
=
=
= =
β
α
1
(2.55)
Another assumption is valid for shallow shells, Ambartsumian (1961), which
states that the in-plane displacements are small compared to radii of curvature, i.e.
0 ... ≈ ≈ ≈
β α
R
v
R
u
(2.56)
If the above assumptions are imposed upon equations (2.32 & 2.33), the
angles become:

x
w
x
∂
∂
− = θ (2.57)
29

y
w
y
∂
∂
− = θ (2.58)
and when the above assumptions are imposed upon equations (2.49-2.54), the
strains and curvature become:

x
x
R
w
x
u
+
∂
∂
= ε (2.59)

y
y
R
w
y
v
+
∂
∂
= ε (2.60)

xy
uv
yx
ε
∂ ∂
=+
∂ ∂
(2.61)

2
2
x
w
x
∂
∂
− = κ (2.62)

2
2
y
w
y
∂
∂
− = κ (2.63)






∂
∂
+








∂
∂
+








∂
∂
∂
∂
−






∂
∂
∂
∂
− =
x
v
R y
u
R y
w
x x
w
y
y x
1 1
τ (2.64)
Utilizing the well-known Kirchhoff hypothesis (normal to the middle surface
remain straight and normal, and unstretched in length during deformation, the
strains of an arbitrary point of the shallow shells are
(
x x
x
x
z
R z
e κ ε +
+
=
/ 1
1
) (2.65)
( )
y y
y
y
z
R z
e κ ε +
+
=
/ 1
1
(2.66)
30
Note that the term and is small in comparison with unity and can
be ignored in the cases of shallow shell and slight curved beams which will be
used to model the PFA.
x
R z /
y
R z /

31
CHAPTER 3
COMPOSITE PIEZOELECTRIC LAMINATED BEAMS

3.1. Review of Piezoelectric Materials
In this section, a brief description of the characteristics behavior of the
piezoelectric materials (Ikeda, 1990; Tzou, 1993; Solecki and Conant, 2003) will
be reviewed herein. The piezoelectric material exhibits interesting behavior
whereby an electric field is generated when the material is strained. This is always
accompanied by the converse behavior in which strain occurs when the material is
placed in an electric field. These are referred to as the direct piezoelectric effect
and the converse piezoelectric effect, respectively, and materials exhibiting these
effects are called piezoelectric materials. The piezoelectric effect was discovered
by Pierre and Jacques Curie in 1880. The piezoelectricity is derived from a Greek
word that means “pressure electricity”. The direct piezoelectric effect consists of
the ability of certain crystalline materials (ceramics) to generate an electrical
charge in proportion of an externally applied force. According to the converse
piezoelectric effect, an electric field parallel to the direction of polarization
induces an expansion of the ceramic. The piezoelectricity effect simply means
that these materials change their geometry or dimensions when an electrical
32
charge (voltage) is applied to them, and conversely, they produce an electrical
charge (voltage) when mechanical pressure is applied to them.
When manufactured, the piezoelectric material has electric dipoles arranged in
random directions. The responses of these dipoles to an externally applied electric
field would tend to cancel one another, producing no gross change in dimensions
of the PZT material. In order to obtain a useful macroscopic response, the dipoles
are permanently aligned with one another through a process called poling.
The piezoelectric material has a characteristic Curie temperature. When it is
heated above this temperature, the dipoles can change their orientation in the solid
phase material. In poling, the material is heated above its Curie temperature and a
strong electric field is applied. The direction of this field is the polarization
direction, and the dipoles shift into alignment with it. The material is then cooled
below its Curie temperature while the poling field is maintained, with the result
that the alignment of the dipoles is permanently fixed. The material is then said to
be poled.
Piezoceramic material is composed of randomly oriented crystals or grains,
each having one or possibly a few domains. With the domain dipoles randomly
oriented as shown in Figure 3.1, the material is isotropic and does not exhibit the
piezoelectric effect. By applying electrodes to the ceramic and a strong dc electric
field (poling), the dipoles will tend to align themselves parallel to the direction of
the electric field, so that the material will have a permanent residual polarization.
The material is then considered to be anisotropic because of the increased number
33
of aligned dipoles. The number of domains that can align their dipoles in ceramic
materials such as piezoceramic is less than that found in single crystal materials,
but enough of the dipoles do align to allow the material to become piezoelectric.
After the poling process, the material has a residual polarization, which means
the domains will proportionally increase their alignment with an applied voltage.
The result is a change of the geometric dimensions (expansion, contraction) of the
PZT material with an applied voltage.
The working temperature of the PZT is usually well below its Curie
temperature. If the material is heated above its Curie temperature when no electric
field is applied, the dipoles will revert to random orientations. Even at lower
temperatures, the application of too strong field can cause the dipoles to shift out
of the preferred alignment established during poling. Once depoled, the
piezoelectric material loses the property of dimensional response to an electric
field. When the material is strained, the resulting electric field manifests itself as a
voltage difference across the electrodes.

Figure 3.1 Dipole alignment in piezoelectric material
34
When the material is strained, the resulting electric field manifests itself as a
voltage difference across the electrodes. The relationship between electric field
intensity and voltage is
V E − ∇ = (3.1)
where the del operator ∇ is defined in Cartesian coordinates as

z
k
y
j
x
i
∂
∂
+
∂
∂
+
∂
∂
= ∇ (3.2)
and i, j and k are mutually orthogonal unit vectors.
Because piezoelectric materials are electromechanical materials, there is an
overlap of notation since certain symbols have one meaning in mechanics and
another in electromagnetics. In order to avoid ambiguity, an adoption of the
notation contained in IEEE Standard on Piezoelectricity (1984) in which the
pertinent symbols and their meanings are shown in Table 3.1.
Consider the piezoelectric material as shown in Figure 3.2. For definiteness
we will assume that the material is a ceramic electroded and poled as shown. For
the sake of simplicity, assume the material is one-dimensional, stresses, strains,
and electric fields all act in the poling direction, and behavior normal to the poling
direction can be ignored.

35

Symbol

Meaning
D Electric displacement
E Electric field
S Strain
T Stress
ε Dielectric permittivity
s Compliance
c Stiffness
d Piezoelectric strain coefficient

Table 3 IEEE Standard on Piezoelectricity Symbolic

36

Figure 3.2 Piezoceramic Material

If a stress is applied, a strain results and, it will cause the material becomes
polarized in proportion to the applied stress;
dT P
T
= (3.3)
where is the polarization resulting from stress and d is the piezoelectric
coefficient or piezoelectric modulus, a constant. d is the amount of polarization
resulting from a unit stress. The dimensions of d are charge per force. Typical
units for d are Coulombs per Newton (C/N) in the SI system.
T
P
In the absence of stress it is known from elementary physics that the material
will become polarized if it is placed in an electric field. The polarization is given
by
E P
o E
χ ε = (3.4)
where is the polarization resulting from the electric field, the constant
E
P χ is the
electric susceptibility, and
o
ε is the permittivity of a vacuum, a universal constant.
Its value is, in Farads per meter (F/m),
(3.5)
12
10 854 . 8
−
× =
o
ε
37
By superposition, if the material is in an electric field and subjected to a stress, the
polarization is the sum of and .
T
P
E
P
E dT P
o
χε + = (3.6)
In the study of piezoelectricity it is customary to replace the polarization with
the electric displacement D, defined as
P E D
o
+ = ε (3.7)
Therefore equation (3.6) becomes
E dT D ε + = (3.8)
where ( ) χ ε ε + = 1
o
is the permittivity. Note that the dimensions of electric
displacement are the same as those of polarization.
Now suppose the material is unconstrained and a voltage is applied, as
indicated in Figure 3.3. The voltage differential is the same direction as the
poling, and therefore the material expands as shown. We can write the strain in
terms of the electric field intensity as
dE S
E
= (3.9)
where is the strain due to the applied electric field and d is piezoelectric strain
coefficient.
E
S
If there is no electric field but the material is subjected to a stress, the resulting
strain is
T
S
sT S
T
= (3.10)
where s is the compliance. For a one-dimensional formulation, . Y s / 1 =
38

+

-

Figure 3.3 A piezoceramic expands as a result of an applied voltage differential

The strain due to both the stress and the electric field is therefore given by
dE sT S + = (3.11)
From equations (3.8) and (3.11) note that, ( ) ( ) E T D D E T S , , , = = S , both the strain
and the electric displacement are functions of the stress and the electric field.
Differentiate equation. (3.11) with respect to T, gives s
T
S
=
∂
∂
, that is, s is the ratio
of strain increment to stress increment at constant electric field, a fact that we
signify by placing a superscript E on s. Then equation (3.11) becomes
(3.12) dE T s S
E
+ =
Similar reasoning shows that the permittivity in equation (3.8) is taken at constant
stress and written as
(3.13) E dT D
T
ε + =
For coupled phenomena such as piezoelectricity it is important to recognize
the conditions under which material constants are measured.

39
For example, suppose a piezoelectric sample is short-circuited (E = 0), as
shown in Figure 3.4, a force is applied and incremented, and the resulting strains
are measured. Then the data are used in equation (3.12), which reduces to
(3.14) T s S
E
=

T = 0

P D
E = 0

Figure 3.4 A short circuited (E = 0) piezoelectric sample

The value of compliance obtained from the experimental data is therefore at
constant electric field, as indicated by equation (3.14). On the other hand, suppose
the same experiment is run, but this time with an open circuit (D = 0), as shown in
Figure 3.5.

T = 0

P P

T
T
D = 0
Figure 3.5 An open circuit (D = 0) piezoelectric sample
40
The resulting electric field is no longer constant, but depends on the stress.
The strain became
(3.15) T s S
D
=
where , the compliance measured at constant electric displacement, is given
by
D
S

T
E D
d
s s
ε
2
− = (3.16)
Thus the value of compliance measured under open-circuit conditions is not the
same as that measured under short-circuit conditions. Using a similar approach, it
will be seen that one can measure either the permittivity at constant stress, , or
the permittivity at the constant strain, . When using the equations of
piezoelectricity, therefore, it is important to understand the conditions under
which the material constants have been measured.
T
ε
S
ε
To extend these ideas to three-dimensional problems, consider the situation
where electrical leads are attached to the electrodes and a voltage differential is
applied. The strain in the 3-direction will be accompanied, in general, by the
strains in the 1- and 2-directions, as well as by shear strains:
(3.17)
, , ,
, , ,
3 36 6 3 35 5 3 34 4
3 33 3 3 32 2 3 31 1
E d S E d S E d S
E d S E d S E d S
= = =
= = =
where the first subscript in d gives the direction of the electric field and the
second the direction of the strain. Assume now that the electrodes have been
attached to the 2-faces and a field of intensity applied, as shown in Figure 3.6.
ij
2
E
41
2
+

3

-
+

-
Figure 3.6 Electrodes attached to the 2-faces and a field
of intensity applied
2
E

This causes a normal strain in the 2-direction, which is accompanied by normal
strains in the 1- and 3-directions as well as shear strains in all three directions.
A similar situation results if electrodes are applied to the 1-faces and a field
is applied. The strain in the 3- direction can therefore be written as
1
E

3 33 2 23 1 13 3
E d E d E d S + + = (3.18)
Thus there is a contribution to the strain in the 3-direction from the applied
electric field in each of the directions.
In general, all the strain components are influenced by the applied electric
field in each of the directions 1, 2, and 3.Therefore equations similar to equation
(3.18) can be constructed for each strain component. In matrix form these are
written as
{ } [ ] { } E d S
T
= (3.19)
where the strain matrix and the electric field intensity matrix are, respectively,
{ } [ ]
6 5 4 3 2 1
S S S S S S S
T
= (3.20)
{ } [ ]
3 2 1
E E E E = (3.21)
42
and the piezoelectric strain matrix (Ikeda, 1990; Solecki and Conant, 2003) is
[] (3.22)










=
36 35 34 33 32 31
26 25 24 23 22 21
16 15 14 13 12 11
d d d d d d
d d d d d d
d d d d d d
d
Note that the superscript T outside the braces or brackets signifies matrix
transposition. In practice, however, some of these are zero owing to crystal
symmetry and the remaining nonzero constants are not all independent.
If the material is subjected to mechanical loads, additional strains will result
and are given by
(3.23)










































=






















xy
zx
yz
zz
yy
xx
xy
zx
yz
zz
yy
xx
S S S S S S
S S S S S S
S S S S S S
S S S S S S
S S S S S S
S S S S S S
σ
σ
σ
σ
σ
σ
γ
γ
γ
ε
ε
ε
66 56 46 36 26 16
56 55 45 35 25 15
46 45 44 34 24 14
36 35 34 33 23 13
26 25 24 23 22 12
16 15 14 13 12 11
or
{ } [ ] { } σ ε S = (3.24)
where
[] (3.25)




















=
66 56 46 36 26 16
56 55 45 35 25 15
46 45 44 34 24 14
36 35 34 33 23 13
26 25 24 23 22 12
16 15 14 13 12 11
S S S S S S
S S S S S S
S S S S S S
S S S S S S
S S S S S S
S S S S S S
S
is the compliance matrix and the elements of [S] are the compliances.
43
The strain due to both the applied electric field and applied mechanical loads
is the sum of the two effects,
{ } [ ] { } [ ] { } E d T s S
T E
+ = (3.26)
where, again, the superscript E indicates that the elements of the compliance
matrix are obtained at constant electric field. On the other hand, if stresses are
applied to a piezoelectric material, the resulting electric displacement is given by
{ } [ ] { } T d D = (3.27)
where the electric displacement matrix is
{ } [ ]
3 2 1
D D D D
T
= (3.28)
In the absence of stress the electric displacement produced by an applied electric
field is
{ } [ ] { } E D
T
ε = (3.29)
where [ ] ε is the 3 × 3 permittivity matrix. The electric displacement due to both
an applied stress and electric field is therefore given by
{ } [ ] { } [ ] { } E T d D
T
ε + = (3.30)
Equations (3.26) and (3.30) give the general constitutive equations for
piezoelectric materials. The specific forms of the [ ]
E
s and [ ] d matrices for all 32
crystals classes can be found in IEEE Standard on Piezoelectricity, 1984.

44
3.2. Piezoelectric Laminated Slightly Curved Beam Model
The static and dynamic models of the piezoelectric laminated slightly curved
beam, as shown in Figure 3.7, are derived using potential and kinetic energies
with the aid of extended Hamilton’s principle, respectively. The thin beams are
studied where effects of shear deformation and rotatory inertia will be neglected.
To use the Hamilton’s principle in deriving the governing equations for the
piezoelectric laminated slightly curved beam model, expressions for the internal
stress and strain within the curved beams are found, and these expressions are
used to define the strain energy. Expression for the kinetic energy is developed for
the dynamic case. The kinetic energy is used along with the strain energy in
Hamilton’s principle to obtain the equations of motion and the associated
boundary conditions. These equations and boundary conditions are used to
develop the dynamic frequency-amplitude model of the Piezoelectric Forceps
Actuator (PFA).
The first step in determining the strain energy is to determine an expression
for the stress within the curved beam. The magnitude of the stress in the
circumferential direction at any point of the curved beam is given by the
piezoelectric constitutive law, (IEEE, 1988), as follow:
( )
p m
Y Y ε ε ε σ + = = (3.31)
where the mechanical strain,
m
ε , and piezoelectric induced strain,
p
ε , are
obtained from the equations (2.65) and (3.19), respectively, and defined as follow:
45
(3.32)
3 31
0
E d
z
p
m
=
+ =
ε
κ ε ε
Note that the term of the mechanical strain, equation (2.65), is small in
comparison with unity and can be ignored in the case of slightly thin curved thin
beams. In equations (3.31) and (3.3 2),
x
R z/
Y is the Young’s modulus of the material,
is the induced piezoelectric strain, is the applied electric field through
the thickness of the piezoelectric elements, is the extensional strain at the
neutral axis, is the distance from the neutral axis and
3 31
E d
3
E
0
ε
z κ is the change in
curvature of the neutral axis during bending. The electric field, , is defined as
positive if it is applied in the same direction as the polarity of the piezoelectric
element and negative if applied in the opposite direction. The piezoelectric strain
constant, , relates the electric field in the radial direction to circumferential
strains. In poled piezoelectric ceramics, is a negative quantity; in non-active
materials, it is assumed to be zero.
3
E
31
d
31
d
The stress of each layer of the shallow curved beam can be found by
substituting equation (3.32) into the piezoelectric constitutive equation, equation
(3.31) to yield
( )
i i i
E d z Y ) (
3 31
0
+ + = κ ε σ (3.33)
where number of layers, i q ,... 2 , 1 =

46

Figure 3.7 Piezoelectric Laminated Slightly Curved Beam

47
The strain and curvature at the mid-plane of the slightly curved beam are
obtained from the equations (2.59) and (2.62), respectively, and expressed in
terms of circumference and radial displacements as follow:

0
u w
x R
ε
∂
= +
∂
(3.34)

2
2
x
w
∂
∂ −
= κ (3.35)
where u and w represent the circumferential an radial displacement of the
structure’s geometrical mid-plane in the x and z directions, respectively and R
represents radius of curvature of the curved beam.
The normal force at any cross-section is defined as the stress integrated over
the cross-sectional area. The existence of the multi layers of the piezoelectric
elements and the beam, as shown in Figure 3.7, the integration must be performed
piecewise over the cross-sectional area. Integrating the normal force, equation
(3.33), across the q layers of the composite cross-sectional area, A, yields,

()
() (
()( )
1
0
31 3
1
0 2
11
1 1
1313
1
()
1
2
i
i
z
q
ii
i
Az
q q
ii i i i i
i i
q
ii i
i i
NdAb Y z dE dz
bYz z bYz z
bYz z dE
σεκ
)
2
ε κ
−
=
−−
= =
−
=
== ++
=− + −
+−
∑
∫∫
∑ ∑
∑
(3.36)
where the subscript i refers to the ith layer, Y is the Young’s modulus of the ith
layer, is the distance from the neutral axis to the outside of the i-th layer and
is the width.
i
i
z
b
48
Calculated in a similar way to the normal force, the moment about the neutral
axis is defined as the stress, equation (3.33), multiplied by a moment arm to the
neutral axis, z, and integrated over the cross-sectional area,

()
() ( )
()()
1
0
31 3
1
22 0 3 3
11
11
22
1313
1
()
1
3
1
2
i
i
z
q
ii
i
Az
q q
ii i i i i
i i
q
ii
i i
M z dA b Y z d E zdz
bYz z bYz z
bz z dE
σεκ
ε κ
−
=
−−
= =
−
=
== ++
=− + −
+−
∑
∫∫
∑ ∑
∑
(3.37)
The normal force and moment, equations (3.36) and (3.37), can be restated in
a matrix form as follow:
(3.38)










+












=






p
p
M
N
D B
B A
M
N
κ
ε
0
where the constants A, B, and D are the extensional, coupling and bending
stiffness, respectively, and are defined as,
( )
∑
=
−
− =
q
i
i i i
z z Y b A
1
1
(3.39a)
( )
∑
=
−
− =
q
i
i i i
z z Y b B
1
2
1
2
2
1
(3.39b)
( )
∑
=
−
− =
q
i
i i i
z z Y b D
1
3
1
3
3
1
(3.39c)
Likewise, the piezoelectric force terms and are defined as,
p
N
p
M
( ) ( )
∑
=
−
− =
q
i
i i i i p
E d z z Y b N
1
3 31 1
(3.40a)
( )()
∑
=
−
− =
q
i
i i i i p
E d z z Y b M
1
3 31
2
1
2
2
1
(3.40b)
49
where in the non-piezoelectric layers.
31
0 d =
The strain energy of the multi-layered curved beam during the deformation
can be expressed by using equation (3.33), as follow:
()
∑
∫∫
∑
∫ = =
+ + = = =
−
q
i
i
Lz
z
i
q
i
i
V
dzdx E d z Y b U dV
Ci
i
1
2
3 31
0
0
1
) (
2
1
2
1
1
κ ε σε U (3.41)
where U is a total internal strain energy, σ is a mechanical stress, ε is a
mechanical strain plus piezoelectric induced strain and V is a volume of a curved
beam bonded with piezoelectric layers.
The Hamilton’s principle uses the variance of the energy in the system; thus,
the objective is to determine the variance of the strain energy rather than the
energy itself. The variance of the strain energy shown in equation (3.41) is
(3.42)
()( )
()
1
1
0002
31 3 31 3
1
0
0
0
Ci
i
Ci
i
Lz
q
i
i
z
Lz
z
U b Y z d E z z zd E dzdx
b z dzdx
δεκδεεκ
σδε σ δκ
−
−
=

=++ +++

=+
∑
∫∫
∫∫
δκ
)
dx
Furthermore, the variance of the strain energy, equation (3.42), can be
expressed in terms of the normal force and moment. Using the definitions of
normal force, equation (3.36), and moment, equation (3.37), the variance of the
strain energy, equation (3.42), can be shown as follow:
(3.43)
(
0
0
C
L
UN M δδε δκ =+
∫
50
To produce the equations of motion, the strain energy variance equation must
be restated in terms of the circumferential, u, and radial, w, displacements rather
than strain and curvature change. Substituting the displacement relations,
equations (3.34) and (3.35), into the variance of the potential energy, equation
(3.43), results in a simplified expression for the variation of the strain energy,
dx
x
w
M w
R
N
x
u
N U
C
L
∫














∂
∂
− + 





∂
∂
=
0
2
2
δ δ δ δ (3.44)
Taking integration by parts of equation (3.44) yields,















∂
∂
−
∂
∂
+ +






∂
∂
− +
∂
∂
− =
∫
C C
C
C
L L
L
L
x
w
M w
x
M
u N
dx w
x
M
w
R
N
u
x
N
U
0
0
0
0
2
2
δ δ δ
δ δ δ δ
(3.45)
Along with an expression for the variance of the strain energy, an expression
for the variance of the kinetic energy must also be calculated to complete the
formulation of the Hamilton’s principle.
Kinetic energy of the curved beam can be expressed as follow:

22
0
1
2
C
L
u w
tt
ρTdx
 
∂∂   
=+
 
  
∂∂
  
 
 
∫
(3.46)
where (
1
1
−
=
− =
∑ i i
q
i
i i
z z b ρ ρ ), u and w are the average mass density of the beams
per unit length or so-called linear density, circumferential displacement and the
radial displacement, respectively.

51
The variation of the kinetic energy, equation (3.46) is

0
C
L
uu w w
Td
tt t t
δρ δ δx
  ∂∂ ∂ ∂      
= +
       
∂∂ ∂ ∂
       
∫
(3.47)
Taking the integration by parts of equation (3.47) yields,

22
22
0
C
L
uw
Tu
tt
δρ δ δwdx
  ∂∂
= +
 
∂∂
 
∫
(3.48)
With the variance of the kinetic energy and potential energy defined, all
quantities necessary to implement the Hamilton’s principle are now determined.
Hamilton’s principle states that the time integral of the variation of the kinetic
energy, T, minus the variation of potential strain energy, U, equals zero,

[]
2
1
0
t
t
TUdt δδ − =
∫
(3.49)
Substituting the variance quantities of potential energy and kinetic energy,
equations (3.45) and (3.48), respectively, into the extended Hamilton’s principle,
equation (3.49) yields
()
22
11
22 2
22 2
0
0
0
0
0
C
C C
C
L tt
tt
L
L
L
NN M u w
UT dt u w w u wdx
xR x t t
Mw
Nu w M dt
xx
δδδρδ
δδ δ

 ∂∂ ∂ ∂
−= − ∂+ − + +


∂∂ ∂ ∂



∂∂ 
++ − = 

∂∂



∫∫∫
ρδ
(3.50)
The Hamilton’s principle, equation (3.50), contains the information required
to determine the equations of motion and the associated boundary conditions for
the curved beam laminated with piezoelectric elements. Thus, the equations of
52
motion of the curved beam laminated with piezoelectric elements can be written
as follow:

2
2
22
22
Nu
x t
NM w
R xt
ρ
ρ
∂∂
=
∂ ∂
∂ ∂
−+ =
∂ ∂
(3.51)
The equations of motion of the slightly curved beam laminated with piezoelectric
elements defined for 0
c
x L ≤≤ can be reinstated in terms of circumferential and
radial displacements by substituting combination equations (3.34), (3.35) and
(3.38) into equation (3.51), shown as follow:

23
2 3
uA w w u
A B
2
2
x Rx x t
ρ
∂ ∂∂ ∂
+− =
∂ ∂∂ ∂
(3.52a)

42 3 2
422 3 2
2
p
N
wB w A u Au w
DwB
x Rx R x R x t R
ρ
∂∂ ∂ ∂ ∂
−+ − + − = +
∂∂ ∂ ∂ ∂
(3.52b)
Next, the boundary conditions at each of the two ends of the system
, one of each pair of the following quantities must be zero: (
C
L x , 0 = )

00
0
00
Noru
M
or w
x
w
Mor
0
x
= =
∂
= =
∂
∂
= =
∂
(3.53)
The boundary conditions derived earlier, equation (3.53), can also be reinstated in
terms of circumferential and radial displacements by substituting combination
equations (3.34), (3.35) and (3.38) into the equation (3.53), shown as follow:

53

2
2
0
p
uA w
AwB N oru
xR x
∂∂
0 + −+ =
∂∂
= (3.54a)

2
2
0
uB w
BwD orw
xR x
∂∂
0 + −=
∂∂
= (3.54b)

23
23
0
uB w w w
BD or
xR x x x
∂∂ ∂ ∂
0 + −=
∂∂ ∂ ∂
= (3.54c)
The static model of composite piezoelectric laminated slightly curved beam
with its associated boundary conditions can be derived through the exact same
process as the dynamic model. It is basically only involved only the potential
energy of the composite piezoelectric laminated thin slightly curved beam. The
only difference is that there is no inertia or mass involving time appear in the
governing equation shown as below:

23
2
0
uA w w
A B
xR x x
∂∂ ∂
3
+ −
∂∂ ∂
= (3.55a)

42 3
422 3
2
p
N
wB w A u Au
DwB
x Rx R x R x R
∂∂ ∂ ∂
−+ − + − =
∂∂ ∂ ∂
(3.55b)
Next, the boundary conditions at each of the two ends of the system
, one of each pair of the following quantities must be zero: (
C
L x , 0 = )

00
0
00
Noru
M
or w
x
w
Mor
0
x
= =
∂
= =
∂
∂
= =
∂
(3.56)

54
3.3. Piezoelectric Laminated Straight Beam Model
The attenuation of vibrations, position control and active stiffening effects are
the problem of primary importance in many engineering fields, among which
aerospace, automobile and biomedical device applications. The performance of
many engineering static or dynamic applications can be significantly enhanced
considering the active stiffening using piezoelectric actuators. Thus, the designer
has more options and freedom to tune the natural frequency; vibration and
position control of the device by using composite piezoelectric laminated thin flat
beams or so-called multimorph with an additional implementation of position or
state feedback control system. The piezoelectric devices laminated only with one
or two layers of piezoelectric elements to actuate the system or so-called
unimorph and bimorph, respectively. The unimorph and bimorph models with its
responses have been long time ago studied by Denkmann (1973), Germano
(1971), Smits (1991) and recently by Clark et al (2004). For the multimorph
cases, Devoe and Pisano (1997) investigated the cantilevered piezoelectric beam
model that described only the deflection of the piezoelectric multimorph.
Weinberg (1999) proposed a close form solution to derive simple working
equations for the cantilevered piezoelectric multimorph. All the cantilevered
multimorph models described above were lumped parameter models. Ko (2005)
derived lumped static and distributed parameter dynamic models of multimorph
limited only to cantilevered symmetric layers case. They investigated optimum
55
number of piezoelectric layers to maximize tip deflection and force. Abramovich
(1998) derived first-order shear deformation theory based model of laminated
composite beams with piezoceramic layers and solved it in closed form solution
which provided only the deflection response.
There is still a little work exist in investigating active stiffening effects, in
terms of various thickness of a single piezoelectric layer, number of layers, total
thickness of the system, on natural frequencies and dynamic responses of a
cantilevered asymmetric multimorph. Therefore, in order to address the before
mentioned issue above, the static and dynamic models of the multimorph will be
derived using the previously derived models of piezoelectric laminated slightly
curved beam from Section 3.2.
The static and dynamic models of composite piezoelectric laminated straight
beams, as shown in Figure 3.8, can be derived. By setting the curvature to zero
(i.e., radius of curvature goes to infinity) and no stretching-bending coupling
involved due to the nature part of the straight thin beams, the composite
piezoelectric laminated straight beams model is easily derived. The thin beams are
studied where effects of shear deformation and rotatory inertia will be neglected.
The governing equation of motion is obtained using extended Hamilton’s
principle.

56

Figure 3.8 Piezoelectric Laminated Straight Beam

57
To use the Hamilton’s principle in deriving the governing equation for the
composite piezoelectric laminated beam, expressions for the internal stress and
strain within the beams are found, and these expressions are used to define the
strain energy. Expressions for the external work and the kinetic energy are also
developed. These are used along with the strain energy in Hamilton’s principle to
obtain the equation of motion and the associated boundary conditions. The first
step in determining the strain energy is to determine an expression for the stress
within the composite beam. The magnitude of the stress, in the x-direction, at any
point of the composite beam is given by the piezoelectric constitutive law, IEEE
(1998), as follow:
( )
p m
Y Y ε ε ε σ + = = (3.57)
where
m
ε ,
p
ε are the mechanical strain and piezoelectric induced strain,
respectively.
The following conditions are assumed for the model:
• According to Kirchhoff and Euler hypothesis, lines originally normal to the
reference or neutral surface remain straight and normal during deformations.
• Linear elastic assumption based on Hooke’s Law.
• The beam undergoes small deflection and pure bending. The neutral axis and
mid-plane are the same if the composite beam is in pure bending without
extension.
58
• The thin beams are studied where effects of shear deformation and rotatory
inertia can be neglected.
• No stretching-bending coupling involved due to the geometry nature part of the
straight thin beams. The layers are assumed perfectly bonded.
Based on assumption above, the linear mechanical strain and piezoelectric
induced strain can be defined, respectively, as follow:

31 3
m
p
z
d E
ε κ
ε
=
=
(3.58)
In equations (3.57) and (3.58), Y is the Young’s modulus of the material, is
the induced piezoelectric strain, is the applied electric field through the
thickness of the piezoelectric elements, is the distance from the neutral axis and
3 31
E d
3
E
z
κ is the change in curvature of the neutral axis during bending. The electric field,
, is defined as positive if it is applied in the same direction as the polarity of
the piezoelectric element and negative if applied in the opposite direction. In
poled piezoelectric ceramics, piezoelectric strain constant, is a negative
quantity; in non-active materials, it is assumed to be zero.
3
E
31
d
The stress of each layer of the composite beams can be found by substituting
equation (3.57) into the piezoelectric constitutive equation, equation (3.58) to
yield
( )
31 3
( )
ii i
Yz d E σκ =+ (3.59)
where number of layers, i q ,... 2 , 1 =
59
The curvature, κ , at the mid-plane of the composite beams, can be expressed
in terms of transverse displacement, w, in the z-direction as follow:

2
2
x
w
∂
∂ −
= κ (3.60)
The moment about the neutral axis is defined as the stress, equation (3.59),
multiplied by a moment arm to the neutral axis, z, and integrated over the cross-
sectional area,

()
() ()()
1
31 3
1
33 2 2
11
11
()
11
32
i
i
z
q
ii
i
Az
qq
ii i i i
i ii
M z dA b Y z d E zdz
bYz z b z z dE
σκ
κ
−
=
−−
==
== +
=−+ −
∑
∫∫
∑∑ 313
(3.61)
To simplify the notations above, denote
( )
∑
=
−
− =
q
i
i i i
z z Y b D
1
3
1
3
3
1
(3.62)

()()
22
1313
1
1
2
q
piii
i i
3
M bYz z dE E β
−
=
=−
∑
= (3.63)
where bending stiffness of the piezoelectric layers,
()()
22
1 31
1
1
2
q
ii i
i i
bYz z d β
−
=
=−
∑
i
z
p
M
,
subscript i refers to the ith layer, Y is the Young’s modulus of the ith layer, is
the distance from the neutral axis, is the width, D is the bending stiffness, is
the piezoelectric induced moment and = 0 in the non-piezoelectric layers.
i
b
31
d

60
Thus, the moment about the neutral axis of the composite beam can be rewritten
as follow:

p
M D M κ = + (3.64)
The strain energy of the composite beam during the deformation can be
expressed by using equation (3.59), as follow:
( )
1
2
31 3
11
0
11
()
22
Ci
i
Lz
qq
ii
ii
Vz
V U b YzdEd σε κ
−
==
== = +
∑∑
∫ ∫∫
i
zdx Ud (3.65)
where U is a total internal strain energy, σ is a mechanical stress, ε is a
mechanical strain plus piezoelectric induced strain and V is a volume of the beam
bonded with piezoelectric layers.
The Hamilton’s principle uses the variance of the energy in the system; thus,
the objective is to determine the variance of the strain energy rather than the
energy itself. The variance of the strain energy shown in equation (3.65) is
(3.66)
()
()
1
1
2
31 3
1
0
1
0
Ci
i
Ci
i
Lz
q
i
i
z
Lz
q
i
z
U b Y z zd E dzdx
b z dzdx
δκ
σδκ
−
−
=
=

=+

=
∑
∫∫
∑
∫∫
δκ
Furthermore, the variance of the strain energy, equation (3.66), can be expressed
in terms of the moment. Using the definition of the moment, equation (3.61), the
variance of the strain energy, equation (3.66), in terms of the moment can be
shown as follow:
61
(3.67) ( )
0
C
L
UM δ δκ =
∫
dx
To produce the equation of motion, the strain energy variance equation must
be restated in term of transverse displacement, w, rather than the curvature
change. Substituting the displacement relation, equation (3.60), into the variance
of the strain energy, equation (3.67), results in a simplified expression for the
variation of the strain energy,
dx
x
w
M U
C
L
∫














∂
∂
− =
0
2
2
δ δ (3.68)
Taking integration by parts of equation (3.68) yields,















∂
∂
−
∂
∂
+






∂
∂
− =
∫
C
C C
L
L L
x
w
M w
x
M
dx w
x
M
U
0 0 0
2
2
δ δ δ δ (3.69)
Along with an expression for the variance of the strain energy, an expression for
the variance of the kinetic energy must also be calculated to complete the
formulation of the Hamilton’s principle.
Kinetic energy of the composite beam can be expressed as follow:

2
0
1
2
C
L
w
t
ρ T dx
 
∂ 
=
 

∂

 
 
∫
(3.70)
where (
1
1
−
=
− =
∑ i i
q
i
i i
z z b ρ ρ ) is the linear density and w is the transverse
displacement.

62
The variation of the kinetic energy, equation (3.70), is

0
C
L
w w
T
tt
δρ δdx
  ∂∂  
=
   
∂∂
   
∫
(3.71)
Taking the integration by parts of equation (3.71) yields,

2
2
0
C
L
w
T
t
δρ δwdx
  ∂
=
 
∂
 
∫
(3.72)
With the variance of the kinetic and potential energy, all quantities necessary
to implement the Hamilton’s principle are now determined. The Hamilton’s
principle states that the time integral of the variation of the kinetic energy, T,
minus the variation of potential strain energy, U, equals zero,

[]
2
1
0
t
t
TUdt δδ − =
∫
(3.73)
Substituting the variance quantities of potential and kinetic energy, into the
extended Hamilton’s principle, equation (3.73) yields

()
22
11
22
22
0
0
0
0
C
C C
L tt
tt
L
L
Mw
U T dt w w dx
xt
Mw
wM dt
xx
δδρδ
δδ

 ∂∂
−= − +


∂∂




∂∂ 
 + −= 

∂∂
 

 
∫∫∫
(3.74)
The Hamilton’s principle, equation (3.74), contains the information required to
determine the equations of motion and the associated boundary conditions for the
composite piezoelectric laminated straight beam.

63
Thus, the equation of motion of the system can be written as follow:

22
22
M w
x t
ρ
∂ ∂
=
∂ ∂
(3.75)
The equation of motion of the composite piezoelectric laminated straight
beam, equation (3.75), can be reinstated in terms of transverse displacement, w,
by substituting combinations of equations (3.60) & (3.64) into equation (3.75),
shown as follow:

4
..
4
0
w
D w
x
ρ
∂
+ =
∂
(3.76)
Next, the boundary conditions at each of the two ends of the system ,
one of each pair of the following quantities must be zero:
()
C
L x , 0 =

00
00
M
or w
x
w
Mor
x
∂
= =
∂
∂
= =
∂
(3.77)
The boundary conditions derived earlier, equation (3.77), can also be reinstated in
term of transverse displacement, w, by substituting combinations of equations
(3.60) and (3.64) into equation (3.77), shown as follow:

3
3
2
'
2
00
00
p
w
Dor
x
w
DM or w
w
x
∂
− ==
∂
∂
−+ = =
∂
(3.78)
The static model of composite piezoelectric laminated straight beam and its
associated boundary conditions are derived through the exact same process as the
64
dynamic model. It is basically only involved only the potential energy of the
composite piezoelectric laminated straight beam. The only difference is that there
is no inertia or mass involving time appear in the governing equation shown as
below:

4
4
0
w
D
x
∂
=
∂
(3.79)
Next, the boundary conditions at each of the two ends of the system
, one of each pair of the following quantities must be zero: (
C
L x , 0 = )

3
3
2
'
2
00
00
p
w
Dor
x
w
DM or w
w
x
∂
− ==
∂
∂
−+ = =
∂
(3.80)

65
CHAPTER 4
TRANSFER FUNCTION FORMULATION OF
PIEZOELECTRIC LAMINATED BEAM

The Distributed Transfer Function Method or DTFM by Yang and Tan (1992)
is extended to carry out the transfer function formulation of the composite
piezoelectric laminated beams. The DTFM has been successfully applied to
compute natural frequencies of the cylindrical shells (Yang and Zhou, 1995) and
circular plates (Yang and Zhou, 1997). Up to date, the DTFM is only applied to
analyze the static and dynamic homogeneous passive distributed components. The
art of the DTFM has not been extended to analyze the static and dynamic
behavior of the composite piezoelectric distributed components. In this chapter,
the DTFM will be extended to analyze the static and dynamic behaviors of the
composite piezoelectric laminated beam. Theorem of the DTFM by Yang and Tan
(1992) can be found in Appendix A.

4.1 Transfer Function Formulation of Piezoelectric Forceps Actuator
In this section, the DTFM, is developed and extended to carry out the transfer
function formulation of the piezoelectric forceps actuator piezoelectric forceps
actuator or known as PFA. This method will be used to solve for the natural
66
frequencies of the PFA based on the proposed composite piezoelectric laminated
thin slightly curved beams model derived earlier in Chapter 3.
For the simplification of the model and due to the dominant material stiffness
(in terms of its thickness and rigidity) in the actuated composite curved beam A-B
section which includes the piezoceramic layer, bonding layer and the substrate
layer of the PFA as shown in Figure 4.1, the model is carried out only for the
lower portion section A-B of the PFA. And also since the PFA is a surgery device
that will be operated within only low bandwidth region (i.e. fine grasping with
slow motion). Furthermore, by modeling lower portion section A-B of the PFA
using DTFM will preserve the exact and closed form solution, and there will be
no approximate numerical solution. And finite element method is unnecessary.
The numerical simulation result is validated with the experimental test bench of
the PFA in Chapter 8.
In the development, the governing equations of the system are cast into a
state-space form. The response and distributed transfer functions of the
cantilevered PFA can be derived in term of the fundamental matrix of the state-
space equation.

67

Figure 4.1 Free body diagram of the cantilevered Piezoelectric Forceps Actuator

68
Based on the derived model of piezoelectric laminated slightly curved beam
from section 3.2, the cantilevered PFA (section A-B) with its boundary
conditions, as shown in Figure 4.1, can be formulated as follow:

23
2 3
uA w w u
A B
2
2
x Rx x t
ρ
∂ ∂∂ ∂
+− =
∂ ∂∂ ∂
(4.1a)

42 3 2
422 3 2
2
p
N
wB w A u Au w
DwB
x Rx R x R x t R
ρ
∂∂ ∂ ∂ ∂
−+ − + − = +
∂∂ ∂ ∂ ∂
(4.1b)

2
2 0
0, 0
c
p
x
xL
uA w
wB N
xR x
=
=
 ∂∂
uA = +− + =

∂∂


(4.2a)

23
2 3 0
0, 0
c
x
xL
uB w w
wB D
xR x x
=
=
 ∂∂ ∂
= +−

∂∂ ∂

=

(4.2b)

2
2
0
0, 0
c
x
xL
wuB w
BwD
xxR x
=
=
 ∂∂ ∂
= +− =

∂∂ ∂


(4.2c)
In the DTFM, the governing equations and its associate boundary conditions of
the cantilevered piezoelectric laminated slightly curved beam, equations (4.1) and
(4.2), respectively, are cast in the equivalent spatial state form and Laplace
transformed with respect to time into an equivalent spatial state form as follow:
() {}[] () ,() , (),0
c
d
xsFs xs Ps x
dx
ηη L = + ≤≤ (4.3)
where
69
2
22
2
00 0 1 0
00 1 0 0
00 0 0 1
1
()
00
00 0 0 0
00
0
0
0
0
1
0
B
Fs
s
AR
AB AB
ss
RA RR
ρ
ρρ
αα α α α





 =
−




−− −


A

() 00000
T
p
N
Ps
R α
−  
=
 
 

2
B
D
A
α=−
23
23
(, ) [ ]
T
wu w w
xs u w
x xx x
η
∂∂ ∂ ∂
=
∂∂∂ ∂
.
The boundary conditions, equation (4.2), can also be written in terms of {} () s x, η :

[ ] ( ) { } [ ] ( ) { } { } 0, , ( )
bbc b
M sN Ls s ηη + =γ
] ]
(4.4)
where and [ are six-by-six matrices and [
b
M
b
N { } ) (s
b
γ is a vector of boundary
piezoelectric induced strain force and moment control,
()
()
2
2
0000 0 0
0000 0 0
0 100000
0000 0 0
0 010000
0 001000
00 0
,,
000000
0 000000
00 0 0
000000
00 0
bb
p
p
A
AB
MN
R
Ns
BB
sD
AA
M s
B
BD
R
γ
ρ


 

 

 

 

−
==
 

−
 

 

−
 

−
 
  
− 


=
70
Hence, the dynamic model of the PFA and its boundary conditions as described
by equations (4.1) and (4.2), respectively, have been reduced to the state equation.
Write the solution of equation (4.3) by assuming no external or internal input
of piezoelectric induced strain force or moment as
[( )]
(, ) (0)
Fs x
xs e η =η (4.5)
where is the fundamental matrix. Substituting equation (4.5) into the
boundary conditions, equation (4.4), gives the homogeneous equation (assuming
no piezoelectric induced strain force or boundary moment control):
x s F
e
) (

[] [ ]
()
( )
() 0, 0
c
Fs L
bb
MNe s η


+ = (4.6)
The characteristic equation of the PFA, equation (4.6), is
() [] [ ]
()
( )
det 0
c
Fs L
bb
MNe ω


∆≡ + = (4.7)
where ( ) ω ∆ is a transcendental function with an infinite number of roots. The
characteristic equation can be accurately and easily solved by standard root-
searching techniques. Its roots are of the form
, 1, 1, 2,...
k
sj j k ω==− = (4.8)
where
k
ω are the kth natural frequency of the PFA. The mode shapes
corresponding to
k
ω can be obtained by solving equation (4.6) for a nonzero
at (0, s η )
k
sj ω = and substituting the result into equation (4.5).
71
The theoretical predictions of the natural frequencies of the PFA using DTFM
are validated with both numerical approximation Rayleigh-Ritz method and
experimental results as shown in Chapter 7 and Chapter 8, respectively.

4.2. Transfer Function of Piezoelectric Straight Beams: Static
Last decade has experienced rapid developments in intelligent biomedical
instruments and devices. Several new actuators using piezoelectric actuators in
controlling precision movement and force have been proposed for biomedical
device applications. One of the critical issues in developing such devices is the
compensation between the different geometric aspects and the desired
performance requirements. Therefore, the characteristic performance analysis of
the various thickness of a single piezoelectric layer, number of layers, total
thickness of the system in generating the desired tip displacement, slope, bending
moment and force will be studied and simulated.
The DTFM will be used in this section to obtain the static performance
characteristic of the composite piezoelectric laminated thin straight beam or so-
called multimorph. In the development, the governing equations of the system
with its associated boundary conditions derived from Section 3.3 are cast into a
state-space form. The characteristic performance study of the multimorph with
different kinds of the boundary conditions are demonstrated and formulated using
DTFM as follow:
72
Case (a) cantilevered multimorph with one end subjected to a piezoelectric
induced moment, as shown in Figure 4.2a,

() ()
() ()
3 2
32
00 00
0
CC
w
w
x
ww
p
D LD L
xx
∂
==
∂
∂∂
−=
∂∂
M=
(4.9)
Case (b) simply supported multimorph with both ends subjected to both
piezoelectric induced moment and an external point-wise force,
3
() 10 (
22
c
z
L
f δ
−
= − )
c
L
x , as shown in Figure 4.2b,

() ()
() ()
2
2
2
2
00 0
0
p
CC
w
wD
x
w
wL D L M
p
M
x
∂
=− =
∂
∂
==
∂
(4.10)
Case (c) clamped-elastic constrained multimorph with one end subjected to a
piezoelectric induced moment and the other end elastically supported by a spring
of coefficient, k, as shown in Figure 4.2c,

() ()
() ()
2 3
23
00 00
()
Cp C
w
w
x
ww
DL M D L kwL
xx
∂
==
∂
∂∂
==
∂∂
C
(4.11)
Case (d) clamped-clamped multimorph with both ends subjected to neither
piezoelectric induced moment nor external axial force, as shown in Figure 4.2d,

() ()
() ()
00 0 0
00
CC
w
w
x
w
wL L
x
∂
= =
∂
∂
= =
∂
(4.12)
73
The cases of (a)-(c) present different kind of boundary conditions and
disturbances. And as for the case (d) presents homogeneous boundary condition
due to neither piezoelectric induced moment nor external axial force.
The multimorph model, equation (3.79), is cast into an equivalent spatial state
form as follow:
() {} [] () {}
C z
L x e x f
D
x F x
dx
d
≤ ≤ + = 0 , ). (
1
4
η η (4.13)
where
{} ()
( )
()
()
()
[] {}














=












=














=
1
0
0
0
,
0 0 0 0
1 0 0 0
0 1 0 0
0 0 1 0
,
4
' ' '
' '
'
e F
x w
x w
x w
x w
x η (4.14)
The boundary conditions can also be written in terms of ( ) { } x η :
[ ] ( ) { } [ ] ( ) { } { }
b C b b
L N M γ η η = + 0 (4.15)
where and [ are four-by-four matrices and [
b
M] ]
b
N { }
b
γ is a boundary
disturbance vector or piezoelectric induced moment.

74

(a)

(b)

(c)

(d)

Figure 4.2 Schematic of the multimorph with (a) clamped-free ends; (b) simply-
supported ends; (c) clamped-elastically constrained; (d) clamped-clamped ends.

75
The matrices and [
b
M] [ ]
b
N and the vector { }
b
γ in equation (4.15) are:
Case (a):
[] (4.16) [ ] {}














=












−
=












=
0
0
0
,
0 0 0
0 0 0
0 0 0 0
0 0 0 0
,
0 0 0 0
0 0 0 0
0 0 1 0
0 0 0 1
p
b b b
M
D
D
N M γ
Case (b):
[] (4.17) [ ] {}














=












=












−
=
p
p
b b b
M
M
D
N
D
M
0
0
,
0 0 0
0 0 0 1
0 0 0 0
0 0 0 0
,
0 0 0 0
0 0 0 0
0 0 0
0 0 0 1
γ
Case (c):
[] (4.18) [ ] {}














=












−
=












=
0
0
0
,
0 0
0 0 0
0 0 0 0
0 0 0 0
,
0 0 0 0
0 0 0 0
0 0 1 0
0 0 0 1
p
b b b
M
D k
D
N M γ
Case (d):
[] (4.19) [ ] {}














=












=












=
0
0
0
0
,
0 0 1 0
0 0 0 1
0 0 0 0
0 0 0 0
,
0 0 0 0
0 0 0 0
0 0 1 0
0 0 0 1
b b b
N M γ
Hence, the multimorph model as described in equation (3.79) has been reduced to
the state equation, equation (4.13), and subject to the arbitrary boundary
conditions and disturbances as shown in equations (4.15). By the DTFM, the
solution of the state-space equations (4.13) and (4.15) is
76
() () () {} (){ } (
4
0
1
,0
C
L
z b
),
C
x Gx f d e H x x L
D
ηξξξ γ    = +
   ∫
∈ (4.20)
where the Green’s function [ ( , )] Gx ξ and the boundary influence function [
or distributed transfer functions are
( )] Hx

[]
[][ ]
[]
[][ ]
[]
()
()
1
[] []
() ,
(, )
() ,
[()] [ ] [ ] ,
c
c
F
b
FL
b
FL Fx
bb
Hx M e x
Gx
Hx N e x
Hx e M N e
ξ
ξ
ξ
ξ
ξ
−
−
−

≥

=

− ≤


=+
(4.21)
and
[ ] F x
e is the fundamental matrix of equation (4.13).
The characteristic performance analysis of the case (a), cantilevered
multimorph, in terms the tip displacement, slope, bending moment and force can
be obtained using equations (4.14), (4.16), (4.20) and (4.21) as follow, with
[( , )] Gx ξ = 0 due to no external force:

( )
()
()
()
[]
2
2
3
3
0
0
()
0
p
wx
w
D isplacem ent
x
x
Slope
Hx
w
Moment M Dx
x
Force
w
Dx
x


∂
 
  ∂
 
==
∂
 
 ∂ 


∂


∂







(4.22)
The requirement of piezoelectric induced moment for the case (b), simply-
supported multimorph associated with its boundary conditions, in maintaining its
equilibrium position at
2
c
L
x = when it is subjected to an external point-wise
force,
3
() 10 (
22
c
z
L
f δ
−
= − )
c
L
x can be found as follow:
77
14 12 14
0
14
0
12 14
1
() 0 (, ) () ( ) ( ( ) ( ))
2222 2
1
(, ) ( ) ( )
22 2
(( ) ( ))
22
C
C
L
cccc c
2
c
z p
L
cc c
z
p
cc
LLLL L
wgxfxdhh
D
LL L
gx f x d
D
M
LL
hh
δξ
δξ

== − + +



−


=−
+
∫
∫
L
M
(4.23)
Similarly, the characteristic performance analysis of the case (c), clamped-elastic
constrained multimorph, in terms the tip displacement, slope, bending moment
and force can be obtained through the same steps as the both cases (a) and (b),
except for its different boundary conditions.
There is no characteristic performance analysis for the case (d), clamped-
clamped constrained multimorph, due to its both end boundary conditions
subjected to neither piezoelectric induced moment nor external axial force as
indicated in equation (4.19).
The characteristic performance simulations of all the mentioned cases above
will be presented and discussed in Chapter 6.

4.3. Transfer Function of Piezoelectric Straight Beam: Dynamic
The transfer function of multimorph for frequency-domain vibration and
control analysis will be formulated in this section using DTFM. The dynamic
multimorph model derived earlier from Chapter 3.3 is cast into a state-space form.
Its eigensolutions are obtained in exact and closed form solution. With the
transfer function formulation, the active stiffening effects, in terms of various
78
thickness of a single piezoelectric layer, number of layers, total thickness of the
multimorph on natural frequencies and dynamic responses of the system, can be
investigated. The effects of increasing the layer numbers, a single thickness of the
piezoelectric layer and the total thickness of the multimorph on dynamic
performances of the multimorph will be presented and discussed in Chapter 6.
The governing equation of the cantilevered multimorph, equation (3.76) and
its associated boundary conditions, equation (3.78), as shown in Figure 4.3, can
be formulated as follow (assuming no external force and zero initial conditions):

42
4 2
0
w w
D
xt
ρ
∂ ∂
+ =
∂∂
(4.24)

() ()
() ()
32
32
0, 0 0, 0
,0 ,
cc
w
wt t
x
ww
p
D Lt D L t M
xx
∂
==
∂
∂∂
−=
∂∂
=
(4.25)

Figure 4.3 Cantilevered Multimorph

79
In the DTFM, the governing equation and its associated boundary conditions
of the multimorph are cast into the equivalent spatial state form and Laplace
transformed with respect to time into an equivalent spatial state form as follow:
() {}[]() L x s x s F s x
dx
d
≤ ≤ = 0 , , ) ( , η η (4.26)
where
{} ()
()
()
()
()
[]














=














=
0 0 0
1 0 0 0
0 1 0 0
0 0 1 0
) ( ,
,
,
,
,
,
2
' ' '
' '
'
s
D
s F
s x w
s x w
s x w
s x w
s x
ρ
η
The boundary conditions, equation (4.25), can also be written in terms of
() {} s x, η :
[ ] ( ) { } [ ] ( ) { } { } ) ( , , 0 s s L N s M
b b b
γ η η = + (4.27)
where and are four-by-four matrices and [
b
M] ] [
b
N { } ) (s
b
γ is a boundary
piezoelectric induced moment control,

[]
(4.28)
[ ] {}
1000 00 0 0 0
0100 00 0 0 0
, ,(
0000 00 0 ()
0000 00 0 0
bb b
p
MN s
DM
D
γ
   
   
   
==
   
   

−
   
)
s


=


η
Hence, the governing equation of the cantilevered piezoelectric laminated straight
beam and its boundary conditions have been reduced to the state equation.
Write the solution of equation (4.26) as
(4.29)
[( )]
(, ) (0)
Fs x
xs e η =
80
where is the fundamental matrix of equation. Substituting equation (4.29)
into the boundary conditions, equation (4.27), gives the homogeneous equation
(assuming no piezoelectric induced boundary moment control):
x s F
e
) (

[] [ ]
()
( )
() 0, 0
c
Fs L
bb
MNe s η


+ = (4.30)
The characteristic equation of the multimorph, equation (4.30), is
() [] [ ]
()
( )
det 0
c
Fs L
bb
MNe ω


∆≡+ = (4.31)
where ( ) ω ∆ is a transcendental function with an infinite number of roots. The
characteristic equation can be accurately and easily solved by standard root-
searching techniques. Its roots are of the form
, 1, 1, 2,...
k
sj j k ω==− = (4.32)
where
k
ω are the kth natural frequency of the multimorph.
The mode shapes corresponding to
k
ω can be obtained by solving equation
(4.30) for a nonzero at (0, s η )
k
sj ω = and substituting the result into equation
(4.29). Furthermore, the roots of equation (4.31) are the eigenvalues or the poles
of the multimorph.
The solution of the state-space equations (4.26) and (4.27) is in exact and
closed form
( ) ( ) ( ) ,,() 0
b
,
c
xsHxs s x L ηγ = ∈ (4.33)
81
where is called the distributed transfer function of the multimorph, and is
given by
( , Hxs)

( )
1
() ( )
(, )
Fs x F s
bb
Hxs e M Ne
−
= + (4.34)
The distributed transfer function formulation of the multimorph provides a new
way to determine the dynamic response of the system in exact and closed form,
no approximation or series truncation is made.

82
CHAPTER 5
MODELS OF PIEZOELECTRIC FORCEPS ACTUATOR
In this chapter, the radius of curvature displacement and grasping force
models of the PFA based on solid mechanics and Castiliagno’s theorems are
derived and verified with experimental results.
5.1. Radial Displacement of the PFA
As shown in Figure 1.1, the PFA consists of a pair of curved beams laminated
with piezoelectric patches. The PFA opens and closes its jaws according to the
polarity of the applied voltage input. The PFA closes its jaws, as shown in Figure
5.1a, when a positive voltage (same direction as the polarity of the PZT) is
applied across the thickness of the PZT layer. The PFA opens its jaws, as shown
in Figure 5.1b, when a negative voltage (opposite direction as the polarity of the
PZT) is applied across the thickness of the PZT layer. Due to the geometric
symmetry of the PFA, only one-half of the structure, namely one composite
piezoelectric curved beams with clamped-free boundary conditions, is considered
herein.
The PFA is a thin curved beam that divided into two substructures. One
substructure part is the actuated composite curved beam A-B section which
includes the piezoceramic layer, bonding layer and the substrate layer with r
represents the radius of each layer and the subscripts c, b, s, and i refer to the
83
piezoceramic layer, the bonding layer, the substrate layer, and the inner radius,
respectively, as shown in Figure 5.2.

(a)

(b)
Figure 5.1 The Piezoelectric Forceps Actuator (PFA)
(a) closed its jaws, (b) opened its jaws.
84

Figure 5.2 Free body diagram of the cantilevered PFA for radial deflection and
grasping force analysis.

85
The piezoceramic, bonding and substrate layers are assumed perfectly bonded
and no slip between layers. The other substructure part is the deflected end tip jaw
B-C section made of only the substrate layer. The thickness of electrode layers is
assumed to be negligible. The shape of the composite piezoelectric curved beam
A-B section is considered initially curved. The assumption of the beam undergoes
small deformation in bending is valid only if the jaw of the PFA open and closed
a considerable distance at low drive level. Consider an infinitesimal beam element
of extended angle φ d , whose neutral axis is an arc of a circle with radius and
center O. After deformation, the two faces of the element move through an
additional angle
n
R
θ d relative to each other; the neutral axis becomes an arc of a
new circle with R’ and center O ′ , as shown in Figure 5.3.
The normal strain of any arbitrary fiber at a distance of r from the center O, in
the circumferential direction, is

( )
1
n n
x
rR
R d
rd r d
−
d θ θ  
ε= = −
 
φ φ

(5.1)
The normal stress in the fiber is expressed as

31 3 xm
Yd E σ =σ − (5.2)
where is the mechanical stress, Y is the Young’s modulus of each
different layers of piezoceramic, bonding, and substrate materials material, is
the applied electric field through the thickness of the piezoceramic layer and the
piezoelectric strain constant, , relates the applied electric field in the radial
m
Y σ= ε
x
3
E
31
d
86

Figure 5.3 Lower part portion of the PFA (A-B segment) under mechanical and
piezoelectric moments.

87
direction to circumferential strains. In piezoelectric materials, is a negative
quantity and in non-active layers it is assumed to be zero. By equations (5.1) and
(5.2), the mechanical stress is
31
d

31 3
1
n
m
R d
Y
dr
θ
σ= − +

φ

YdE (5.3)
The radius
n
R of the neutral axis of the beam element of multiple layers is
determined by considering the balance of normal stress on a cross-section A of the
beam
0
m
A
dA σ =
∫
(5.4)
which leads to

( ) ( )








+








+








− + − + −
=
i
s
s s
s
b
b b
b
c
c c
i s s s s b b b b c c c
n
r
r
b Y
r
r
b Y
r
r
b Y
r r b Y r r b Y r r b Y
R
ln ln ln
) (
(5.5)
where represents the Young’s modulus, b represents the width of each layer, r
represents the radius of each layer and the subscripts c, b, s, and i refer to the
piezoceramic layer, the bonding layer, the substrate layer, and the inner radius,
respectively.
Y
At a cross-section of the beam, act a mechanical moment
m
M and internal
piezoelectric moment
e
M . These moments can be derived by multiplying the
stress, equation (5.3), with a moment arm measured to the neutral axis,
88
n
R r z − = , and integrating the result over the cross sectional area of the beam; see
Figure 5.3.
G
( )
me m n
A
MM rR +=σ −
∫
dA (5.6)
Substitution of equation (5.3) into equation (5.6) yields the moment-angular strain
relation

me
n
M M d
R
dG
+ θ
=
φ
(5.7)
Where
() ()
22 2 2 2 2
1
(
2
ncc c b bb b s ss s i
R Yb zz Yb zz Ybzz

=−+ −+

with Y represents
the Young’s modulus of each material, b represents the width of each layer, z
represents the distance to the neutral axis (or moment arm) and the subscripts c, b,
s, and i refer to the piezoceramic layer, the bonding layer, the substrate layer, and
the inner layer, respectively, as shown in Figure 5.3.
)−
It can be shown that internal piezoelectric moment, , is derived by
multiplying the piezoelectric stress, Yd , by the layer arm to the neutral axis, z,
and an element of area, dA
e
M
3 31
E
dr db* = , then integrating over each active layer of the
structure in a piecewise fashion. Thus, the internal piezoelectric moment, , can
be written specifically for the piezoelectric moment of the PFA as follow:
e
M

( )
2
2 2
3 31
b c
c c e
z z
b E d Y M
−
= (5.8)
89
The mechanical moment , arises from the application of the reaction
external MiliNewton  force sensor, , varies along the length of the beam as
the distance to the applied force changes. The mechanical bending moment varies
throughout the composite curved beam A-B section,
m
M
y
F
f
θ and deflected end tip jaw
B-C section, D, as shown in Figure 5.2.
( ) D Sin R F M
f n y m
+ − = θ (5.9)
It is assumed that the shape of the curved beam, which its radius of curvature of
the undeformed beam, , is an arc of a circle, remains as an arc of different
curvature in a bending deformation, and the neutral surface in the beam does not
change before and after the deformation, as shown in Figure 5.3. According to the
above-assumption, the radius
n
R
' R of curvature of the beam after deformation, or so
called radial deflection, is expressed as

n
R
d
d
R








+
=
φ
θ
1
1
'
(5.10)
where the angular strain, / d d θ φ, has been given in equation (5.7). As a measure
of the radial deflection of the PFA, ' R depends on the geometric and physical
parameters of the layers of the composite beam, the characteristic piezoelectric
strain constant, and the applied electric field, . The theoretical prediction
of radial deflection of the PFA is verified with experimental measurements using
fiber optic curvature sensor shown and discussed in Chapter 8.
31
d
3
E
90
5.2. Grasping Force Model of the PFA
In this section, the force-deflection relation of the PFA is derived according to
Castigliano’s second theorem (Budynas (1999)), will be derived. To this end,
write the normal stress as

G
M M
r
R
YR
m e n
n x
+






− = 1 σ (5.11)
where the equations (5.2), (5.3), (5.7) and (5.9) have been used. The
complimentary strain energy, U , of the curved beam is
*

2
*
0
2
f
x
A
d
Y
θ
σ
UrAd = θ
∫∫
(5.12)
where
f
θ is the angular span of the composite curved beam, A-B section, as
shown in Figure 5.2.
Substituting equation (5.11) into equation (5.12) leads to

( )
θ
θ
d
G
M M R
f
m e n
∫
+
=
0
2
*
2
U (5.13)
According to Castigliagno’s second theorem, the deflection y ∆ of the tip jaw, as
shown in Figure 5.2,

() ()
()
*
2
0
sin 2 sin
sin
f
n
n n
y
ne
R U
yDRDR
FG
DR M d
θ
θθ
θθ
∂

∆= = + +

∂
 −+

∫
F
(5.14)

91
The composite curved beam A-B section, as shown in Figure 5.2, is assumed
to have the angular span,
4
π
θ =
f
. Thus, the deflection y ∆ of the tip jaw, equation
(14), can be written as follow:
[]
e y
n
M F
G
R
y β α + = ∆ (5.15)
where ( )
n n
R D R D 14 . 0 58 . 0 78 . 0
2
+ + = α and D R
n
78 . 0 3 . 0 − = β .
The tip jaw deflection, equation (15) can easily inverted to find the grasping
force, , as follow:
y
F

y
n
G
F y
R
e
M
β
αα
=∆− (5.16)
The theoretical prediction of the grasping force for the PFA is validated with
the experimental grasping force using external MiliNewton  force sensor and
discussed in Chapter 8.

92
CHAPTER 6
CHARACTERISTIC PERFORMANCE ANALYSIS OF
PIEZOELECTRIC LAMINATED STRAIGHT BEAM

In this chapter, the characteristic performance analysis of the piezoelectric
laminated straight beam or so-called multimorph under both case I and case II as
shown in Figure 6.1 with different geometric configurations are presented and
discussed. The simulation studies of the characteristic performance of the
multimorph will be done using proposed Distributed Transfer Function Method or
known as DTFM. Even though, some of the cases above can be solved by other
existing numerical approximation methods like Rayleigh-Ritz method, Galerkin
method, finite difference method and the finite element method. But the analytical
solutions (exact and closed form) are always desirable as they are accurate and
numerically efficient, and provide deep physical insight into the problem.
Therefore, the DTFM is proposed to study the performance characteristics of the
multimorph since this method provides exact and closed form solution.
The parameters for the piezoelectric and substrate materials used in these
studies are formulated in Table 6.1.

93

Figure 6.1 Schematic diagram of the cantilevered multimorph; case I: EI-varies
for arbitrary number of layers; case II: EI-constant for arbitrary number of layers.

94

Property Piezoelectric Substrate
Young’s modulii
2
() Nm
−

piezoelectric
Y
,
substrate
Y

9
2.0 10 ×

9
2.7 10 ×
Piezoelectric strain constant
(C/N)
31
d

12
20 10
−
−×

____
Width (m): b 0.013 0.013
Length (m):
C
L
0.03 0.03
Thickness (m):
piezoelectric
h ∆ ,
substrate
h ∆

65
8.67 10 7.803 10
− −
×− ×

6
137 10
−
×
Number of layers 1-9 1

Table 6.1 Material properties of multimorph

95

6.1. Performance Analysis Piezoelectric Straight Beams: Static
In this section, the static characteristic performance analysis of the multimorph
under both cases I and II with different boundary conditions as shown in Figure
6.1 and Figure 4.2, respectively, will be presented and discussed. Define the total
thickness of the multimorph, as shown in Figure 6.1:
( 1)
substrate piezoelectric
Hh q h =∆+−×∆ (6.1)
where H is the total thickness of the multimorph,
substrate
h ∆ is the thickness of a
single substrate, is the thickness of a single piezoelectric layer and q is
the total number layers of the multimorph.
piezoelectric
h ∆
Figure 6.2 shows the spatial distribution of cantilevered multimorph response
subject to a boundary disturbance of piezoelectric induced moment with input
50V. The characteristic performance of the multimorph can be obtained all in one
transfer function formulation in term of its maximum tip displacement, slope,
moment and output force. The simulation is shown with increasing layer numbers,
q and total thickness, H while
piezoelectric
h ∆ and
substrate
h ∆ are fixed.
Figure 6.3 shows the spatial distribution of cantilevered multimorph response
subject to a boundary control of piezoelectric induced moment with input 50V.
The simulation is shown with increasing layer numbers, q and varying
while
piezoelectric
h ∆
H and
substrate
h ∆ are fixed.
96

Figure 6.2 Case I: The spatial distribution of cantilevered multimorph

97

Figure 6.3 Case II: The spatial distribution of cantilevered multimorph

98
The comparison studies of the characteristic performances of the cantilevered
multimorph between both cases I and II as shown in Figure 6.2 and Figure 6.3,
respectively, particularly of its tip displacement reveals that there is an efficient
way to increase the tip displacement. It can be achieved by increasing its layer
numbers, q and total thickness, H while
piezoelectric
h ∆ and
substrate
h ∆ are fixed.
Another words, case I is preferred to the case II for increasing tip displacement of
the cantilevered multimorph.
Furthermore, it is observed that a laminate that consists of several single thin
piezoelectric layers has advantage of improving the tip displacement compared to
a laminate that consists of only a single thick piezoelectric layer. As expected
from one of the fundamental theories of piezoelectricity that
VV
Ei j k
V
x yz
∂∂ ∂
=+ +
∂∂ ∂
, equation (3.1), we knew that the thicker piezoelectric layer
demanded more voltage input compared to the thinner piezoelectric layer in order
to deflect the tip of the cantilevered multimorph at the same quantity of the
displacement.
Figure 6.4 shows the bending stiffness of the cantilevered multimorph is hold
constant with increasing layer number, n and varying
piezoelectric
h ∆ while the total
thickness, H and
substrate
h ∆ are fixed.

99
0.00E+00
5.00E-06
1.00E-05
1.50E-05
2.00E-05
2.50E-05
3.00E-05
01 234 5678 9 10
Number of Layers
EI [Nm^2]
Single Layer Thickness Constant
Total Thickness Constant

Figure 6.4 Cases I and II: Bending stiffness of the cantilevered multimorph

100
However, the linear relationship exists between increasing the bending stiffness of
the piezoelectric beams with increasing layer numbers, n and total thickness, H
while and
piezoelectric
h ∆
substrate
h ∆ are fixed, as shown in Figure 6.4.
Figure 6.5 shows a non-linear relationship between increasing tip displacement
of the cantilevered multimorph with increasing layer number, n and total
thickness, H while and
piezoelectric
h ∆
substrate
h ∆ are fixed. But the tip displacement will
eventually reach to a saturated region.
0.00E+00
2.00E-04
4.00E-04
6.00E-04
8.00E-04
1.00E-03
1.20E-03
1.40E-03
01 2 3 45 67 8 9 10
Number of Layers
Maximum Displacement [m]
Single Layer Thickness Constant [100V]
Single Layer Thickness Constant [200V]
Single Layer Thickness Constant [50V]

Figure 6.5 Case I: Maximum tip displacement and number of layers
101
Figure 6.6 shows a linear relationship between increasing maximum tip
displacement of the cantilevered multimorph with increasing layer number, n and
varying while H and
piezoelectric
h ∆
substrate
h ∆ are fixed.

0.00E+00
2.00E-04
4.00E-04
6.00E-04
8.00E-04
1.00E-03
1.20E-03
1.40E-03
012 34567 89 10
Number of Layers
Maximum Displacement [m
Total Thickness Constant [100V]
Total Thickness Constant [200V]
Total Thickness Constant [50V]

Figure 6.6 Case II: Maximum tip displacement and number of layers

102
Figure 6.7 shows the linear relationship exists between increasing bending
moment of the cantilevered multimorph with increasing layer number, q and total
thickness, H while ∆ and
piezoelectric
h
substrate
h ∆ are fixed. Similar results are obtained
by increasing the layer numbers, q and varying
piezoelectric
h ∆ while the total
thickness, H and
substrate
h ∆ are fixed. Both of cases I and II yield almost the same
bending moment but not for the end tip displacement as shown earlier in Figures
6.2 and 6.3.

0.00E+00
1.00E-05
2.00E-05
3.00E-05
4.00E-05
5.00E-05
6.00E-05
7.00E-05
12 345 678 9 10
Number of Layers
Moment [Nm]
Single Layer Thickness Constant [100V] Total Thickness Constant [100V]
Single Layer Thickness Constant [200V] Single Layer Thickness Constant [50V]
Total Thickness Constant [200V] Total Thickness Constant [50V]

Figure 6.7 Cases I and II: Bending moment of cantilevered multimorph
103
Figure 6.8(a) shows the bending moment of the cantilevered multimorph can
be increased at 50 V by increasing
piezoelectric
h ∆ but it will simultaneously decrease
its tip displacement as shown in Figure 6.8(b).

(a)

(b)
Figure 6.8 The effect of each piezoelectric layer thickness: (a) bending moment;
(b) tip displacement
104
As a result of the contribution of increased thickness of each piezoelectric
layer to the system, it increased overall the total stiffness of the system. Thus, the
tip displacement will decreased as expected due to stiffer of the system.
Figure 6.9 shows the effects of the ON-OFF activation of the constant piezoelectric
induced moment,
p
M towards the simply supported bimorph (q = 2 layers) in
counteracting the external downward point wise force
3
10
z
f N
−
= in the middle span of
the bimorph and trying to achieve an equilibrium position.

Figure 6.9 The effects of ON-OFF activation of piezoelectric induced moment of
simply supported bimorph.

105
Figure 6.10 shows the spatial distribution of clamped-elastic constrained
multimorph response subjected to a boundary control of piezoelectric induced
moment under 50V and constrained end spring constant, k = 1 N/m. The
simulation is shown with increasing layer numbers, n and total thickness, H
while and
piezoelectric
h ∆
substrate
h ∆ are fixed.

Figure 6.10 The spatial distribution of clamped-elastic constrained multimorph

106
Figure 6.11 shows the clamped-elastic constrained multimorph increases its
grasping force linearly by increasing the piezoelectric layer numbers up to
optimum 5 layers and total thickness, H while
piezoelectric
h ∆ and
substrate
h ∆ are fixed.
Beyond the optimum layers, the force of the multimorph increases non-linearly
and eventually reaches the saturated region.

0.00E+00
1.00E-04
2.00E-04
3.00E-04
4.00E-04
5.00E-04
6.00E-04
7.00E-04
8.00E-04
9.00E-04
1.00E-03
02 46 8
Number of Layers
Force [N]
10
50 V 100 V 200 V

Figure 6.11 Case I: Force and number of layers

107
Figure 6.12 shows the clamped-elastic constrained of the multimorph increases
linearly its grasping force towards arbitrary elastic coefficients by increasing input
voltage and number of layers, q while
piezoelectric
h ∆ and
substrate
h ∆ are fixed.

0
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
0.0014
0 50 100 150 200
Input Voltage [V]
Force [N]
k = 2 N/m, N = 5 Layers k = 4 N/m, N = 5 Layers
k = 6 N/m, N = 5 Layers k = 8 N/m, N = 5 Layers
k = 10 N/m, N = 5 Layers

Figure 6.12 Case I: Force and input voltage

108
6.2. Performance Analysis Piezoelectric Beams: Natural Frequencies
In this section, the dynamic performance analysis of the cantilevered
multimorph under both cases I and II, as shown in Figure 6.1, will be presented
and discussed. Define the total thickness of the multimorph,
(1)
substrate piezoelectric
Hh q h =∆ + − × ∆ , as shown in equation (6.1). With the
distributed transfer function formulation of the dynamic model of the multimorph
derived earlier in Chapter 4, the numerical simulation studies of the effects of
both cases I and II on natural frequencies of the multimorph can be obtained in
exact and closed form solution.
Figure 6.13 shows that for the case I, there is an approximate linear
relationship effects of increasing number of the piezoelectric layers on natural
frequencies if one increases layer numbers, q and total thickness, H while
and
piezoelectric
h ∆
substrate
h ∆ are fixed. But for the case II, the effects of increasing
number of the piezoelectric layers on natural frequencies are just steady constant
relationship if one increases layer numbers, q and varying ∆ while
piezoelectri
h
c
H
and
substr
h ∆
ate
are fixed.
6.3. Conclusion
The effects of the layer number, the layer thickness and bending stiffness of
the multimorph on tip displacement, bending moment, force and natural
frequencies are easily simulated using the DTFM. All the solutions are obtained
in exact and closed form solution.
109

Figure 6.13 Cases I and II: Natural frequencies and number of layers

110
CHAPTER 7
CHARACTERISTIC PERFORMANCE ANALYSIS OF
PIEZOELECTRIC FORCEPS ACTUATOR

7.1. Exact Natural Frequencies by DTFM
For readability and conveniency, the characteristic equation of the cantilevered
piezoelectric forceps actuator or PFA from Chapter 4 will be repeated as follow:
() [] [ ]
()
( )
det 0
c
Fj L
bb
MNe
ω
ω


∆≡ + = (7.1)
where ( ) ω ∆ is a transcendental function with an infinite number of roots,
2
22
2
00 0 1 0
00 1 0 0
00 0 0 1
1
()
00
00 0 0 0
00
0
0
0
0
1
0
B
Fj
A RA
AB AB
RA RR
ρ
ω
ω
ρρ
ωω
αα α α α





 =
−−




−− −



100000
010000
001000
000000
000000
000000
b
M




=






,
2
2
0000 0 0
0000 0 0
0000 0 0
00 0
00 0 0
00 0
b
A
AB
N
R
BB
D
AA
B
BD
R
ρω
 
 
 
 
 
 
−
=
 
 
 
−−
 
 
 −
 
 

111
The characteristic equation can be accurately and easily solved by standard root-
searching techniques. Its roots are of the form
, 1, 1, 2,...
k
sj j k ω==− = (7.2)
where
k
ω are the kth natural frequencies of the PFA.
The comparison between the theoretical predictions of the natural frequencies
of the PFA based on DTFM and Rayleigh-Ritz method are tabulated in section
6.2. The parameters for the Piezoelectric Forceps Actuator (PFA) materials are
tabulated in Table 7.1.
The theoretical predictions of the natural frequencies of the PFA are validated
by the experimental results, which will be presented and discussed in Chapter 8.

Material Property PZT 3195 HD LaRC-SI 302 Stainless Steel
Young’s modulii
2
() Nm
−

10
6.7 10 ×

9
4.0 10 ×

9
2.62 10 ×
Piezoelectric strain
constant (C/N):
31
d

12
190 10
−
−×
____ ____
Width (m): b
3
5.97 10
−
×
3
5.97 10
−
×
3
5.97 10
−
×
Length (m):
C
L
2
3.26 10
−
×
2
3.26 10
−
×
2
3.26 10
−
×
Thickness (m):
4
2.03 10
−
×
5
2.54 10
−
×
4
1.52 10
−
×
Number of layers 1 1 1

Table 7.1 Material Properties of Piezoelectric Forceps Actuator
112
7.2. Approximate Natural Frequencies by Rayleigh-Ritz Method
In this section, the Rayleigh-Ritz method (Meirovitch, 1997) will be used to
calculate the natural frequencies of the Piezoelectric Forceps Actuator (PFA)
approximately based on energy functionals derived earlier in Chapter 3. For
convenience, the potential and kinetic energies of the cantilevered PFA from the
Chapter 3 are rewritten as follow:
For the total potential energy,

12 3
Π = Π +Π +Π (7.3)
where
j
ϕ and
j
φ are the chosen polynomial shape or admissible functions that
satisfy the essential boundary conditions,

2
2
1
2
0
12 1
2
L
uu
Aw w
xR xR
dx
  
∂∂ 
Π= + +   


∂∂

 

 
∫
(7.4a)

2
2
2
2
0
1
2
L
w
D
x
dx
 
 ∂
  Π=

∂
  
 
∫
(7.4b)

2
3
2
0
1
2
2
L
uw w
B dx
xR x
  ∂ ∂ 
Π= − +
  
∂∂

 
∫
(7.4c)

1
()
N
j j
j
ux a ϕ
=
=
∑
(7.4d)

1
()
N
j j
j
wx b φ
=
=
∑
(7.4e)

113
For the total kinetic energy,

22 2
0
() ()
2
L
xwx
ρ
ω Tu dx   = +
  ∫
(7.5)
where its displacement fields, u and w have been defined in equations (7.4d) and
(7.4e), respectively.
The Rayleigh-Ritz method requires minimization of the Lagrangian energy
functional, . The Lagrangian is then minimized by taking its derivatives
with respect to the undetermined coefficients of the displacement fields and
making them equal to zero, shown as follow:
LT =−Π

0, 1,2,...,
0, 1,2,...,
i
i
L
iN
a
L
iN
b
∂
==
∂
∂
==
∂
(7.6)
This yields equations that can be written in a matrix form and give the natural
frequencies as follow:
(7.7)
[] [ ] ()()
2 2
0
ua ub ua ub N
wa wb wa wb N
kk m m a
KM
kk m m b
ωγ ω

  
−= −

   
  

=
The derivation of equation (7.6) and the elements of the matrix, equation (7.7) can
be found details in Appendix B.
The comparison between theoretical prediction of the natural frequencies
based on DTFM and Rayleigh-Ritz Method are tabulated in Table 7.2 as follow:

114

Distributed Transfer Function
Method
Rayleigh-Ritz
Method
1
st
Natural Frequency
[rad/s]
0.85 0.87
2
nd
Natural Frequency
[rad/s]
5.38 5.39
3
rd
Natural Frequency
[rad/s]
15.09 15.51

Table 7.2 Theoretical Predictions of the Natural Frequencies of the PFA

7.3. Conclusion
The computation of the natural frequencies of the PFA is obtained easily with
DTFM compared to the Rayleigh-Ritz method. Furthermore, the DTFM solutions
are exact and closed-form solutions.

115
CHAPTER 8
PIEZOELECTRIC FORCEPS ACTUATOR
EXPERIMENTAL TESTING

8.1. Radial Deflection Measurements
To validate the radial deflection model (Susanto and Yang, 2007), equation
(5.10), the PFA was rigidly clamped at one end in the fixture and the other end
was free to make a radial deflection, ' R . The curvature sensing fiber optic sensor,
Measurand , was bonded on top of an initial curved beam of the PFA as shown
in Figure 8.1.
The curvature sensor measured instantaneous change of radius of curvature
bending, ' R of the composite curved beam section A-B when the PFA was
subjected to the ranges of voltage input.
The experimental results were based on low and high drive regimes. Low
drive regime was tested by varying the voltage inputs from 0 V to 100 V in one
cycle as shown in Figure 8.2a. High drive regime was tested by varying the
voltage inputs from –300 V to +300 V in one cycle as shown in Figure 8.2b.

116

FIGURE 8.1 Schematic diagram of the experimental set-up in which a curvature
fiber optic sensor, Measurand , mounted on the PFA.

117

(a)

(b)
Figure 8.2 Theoretical radial deflection of the PFA verified with experimental
measurements: (a) low drive from 0 V to 100 V; (b) high drive from -300 V to
300 V.

118
In Figure 8.2a, the PFA predictably behaved approximately linear in voltage-
radial deflection relationship when it was subjected to the input voltage ranges
from 0 V to 100 V in one cycle. Physically, it indicated that the composite curved
beam A-B section of the PFA geometrically flattened and closed its jaw. This
could be seen from its initial measured radius of the curvature, 95 mm, increased
its value when the PFA was subjected to the input voltage ranges from 0 V to 100
V. Though, the PFA behaved approximately linear and predictably within a
bounded region of hysteresis loop, it came with the expense of lower performance
in achieving bigger opening jaws.
In Figure 8.2b, the PFA predictably behaved approximately linear in voltage-
radial deflection when it was subjected to the input voltage ranges from 0 V to
100 V in one cycle. But a nonlinear behavior in voltage-radial deflection, also
known as “Butterfly Hysteresis Loop” found in Ounaies (2000) and Li (2001),
occurred when the PFA was subjected to the input ranges from –300 V to 300 V
in one cycle. As shown in Figure 8.2b, the tracing point A to point B and from
point B to point A in one cycle physically indicated that the PFA was closing its
jaws. Then, the tracing from the point A to point C physically indicated that the
PFA geometrically arched and opened its jaw. This could be seen from its initial
measured radius of the curvature, 95 mm, decreased its value when the PFA was
subjected to the input voltage ranges from 0 V to -100 V. When the PFA was
subjected to the input voltage ranges from –100 V to –300 V, the tracing point C
to point D physically indicated that the PFA was basically going back to closed its
119
jaw by geometrically flattening the curved beam. Finally, when the PFA was
subjected to the input voltage ranges from –300 V to –100 V, the tracing point D
to point A physically indicated that the PFA basically was going back to open its
jaw by arching the curved beam. The butterfly hysteresis loop occurred due to the
non-linearity characteristic found in the ferroelectric ceramics material. Its non-
linearity behavior was a consequence of energy dissipation associated with the
domain wall motion switch.

8.2. Grasping Force Measurements
To validate the grasping force-deflection relation model (Susanto and Yang,
2007), equation (5.16), the PFA was rigidly clamped at one end in the fixture and
the other end pushed on the miniature Milinewton  cantilever force sensor.
Consequently, the reaction static force or preload of the force sensor deflected the
end tip of the jaw a considerate small displacement. At this position, the PFA was
in equilibrium with no applied input voltage. The force sensor was locally
connected to a microcontroller with an A/D converter. The normal force
measurement of the tip jaw, , was located on the force centering ball being
made through a bending moment of the cantilever beam sensor as shown in
Figures 8.3a and 8.3b.
y
F

120

(a)

(b)
Figure 8.3 Experimental grasping force of the PFA measured by
Milinewton  force sensor, (a) Schematic diagram of the PFA grasps a cantilever
beam force sensor, (b) Top view, tip jaw of the PFA is ready to grasp the force
sensor.

121
The theoretical grasping force and experimental measurement started at 49
mN due to the existence of initial reaction static force or preload of the force
sensor that deflected the end tip of the jaw a considerate small displacement. The
theoretical grasping force and experimental measurement yield 124.9 mN and
124.1 mN, respectively at the maximum voltage input, as shown in Figure 8.4.
Observe that there was a spike occurred in the grasping force measurement as
shown in Figure 8.4 due to the PFA was subjected to holding the constant
maximum input voltage for few seconds. The spike showed that the grasping
force had reached to its steady state and saturated at 124.1 mN. Simply stated that
the spike as shown in Figure 8.4 indicated that PFA has “squeezed to the limit” at
the maximum voltage input and hold it for a few seconds.

Figure 8.4 Theoretical grasping force model of the PFA verified with
experimental result.
122
8.3. Natural Frequencies Measurements
To validate the theoretical natural frequencies prediction of the PFA,
equations (7.1) and (7.8), the frequency response measurements of the PFA were
obtained using Tektronix Spectrum Analyzer. The measured natural frequencies
up to the 3
rd
natural frequencies are shown in Figure 8.5. And the theoretical
predicted natural frequencies of the PFA is shown in Figure 8.6

Figure 8.5 Measured Frequency Response of the PFA

123

Figure 8.6 Theoretical Predicted Frequency Response of the PFA

124
The comparison between theoretical prediction of the natural frequencies
based on the DTFM and experimental values are tabulated in Table 8 as follow:

Distributed Transfer Function
Method
Experimental
Testing
Percentage
Error
1
st
Natural Frequency
[rad/s]
0.85 0.87 2 %
2
nd
Natural Frequency
[rad/s]
5.38 6.40 15.9 %
3
rd
Natural Frequency
[rad/s]
15.09 12.8 17.8 %

Table 8. Natural Frequencies of the PFA

8.5. Conclusion
The theoretical predictions of the radial deflection and grasping force follow
closely with the experimental results within low drive regime. The difference
between the numerical and experimental results especially in the 2
nd
and 3
rd

natural frequencies are due to the unmodeled dynamic of the B-C section part of
the PFA (see Figure 5.2). Since the PFA is designed to be a surgery device and it
will be operated only within low bandwidth region (i.e. fine grasping with slow
motion) so the proposed theoretical modeling lower portion section A-B of the
PFA is adequate.
125
CHAPTER 9
CONTROL SYSTEM FORMULATION OF
PIEZOELECTRIC FORCEPS ACTUATOR

In this chapter, the open and closed-loop control system formulation of the
section A-B part of the Piezoelectric Forceps Actuator, as shown in Figure 9.1,
will be presented. The plant model is based on the model derived earlier in
Chapter 4. The open and closed-loop control system formulation of the PFA can
be presented either based on DTFM or Rayleigh-Ritz methods for frequency
response and time domain analysis, respectively.

Figure 9.1 Schematic of the PFA under position control system
126
9.1. Open-Loop Transfer Function: DTFM Model
DTFM model derived earlier in Chapter 4 is readily to be used for open-loop
transfer function formulation. Following is the theorem of the transfer function of
a distributed component (Yang and Tan, 1992).
Theorem:
Let the boundary value problem

(, ) ( ) ( , ) (, ) (0, )
(0, ) ( , ) ( )
bb
d
xsFs xs Pxs x L
dx
MsN Ls rs
ηη
ηη
=+ ∈
+=
(9.1)
with and (, ) 0 Px s = ( ) 0 rs = has only the null solution. Then there exists the
unique solution of equation (9.1) which is formulated as:

0
(, ) ( , , ) ( , ) (, ) ( ) (0, )
L
xsGx sP sd Hxsrs x ηζζζ =+
∫
L∈
x
(9.2)
where

() ( ) 1 ( )
() () 1 ( )( )
() ,
(, , )
() ,
Fs x F s L Fs
bb b
Fs x F s L Fs L
bb b
eM Ne Me
Gx s
e M Ne Ne x
ζ
ζ
ζ
ζ
ζ
−−
−−
 + ≤
=

−+≥

(9.3)
and is the fundamental matrix of equation (9.1). Gx
() Fs x
e ( , , ) s ζ and are
called the distributed transfer functions of the component,
( , ) Hxs

11 1
1
(, , ) (, , )
(, , )
(, , ) (, , )
n
nnn
gx s g x s
Gx s
gx s g x s
ζζ
ζ
ζζ
 
 
=
 
 
 
…

…

127

11 1
1
(, ) ( , )
(, )
(, ) ( , )
n
nnn
hxs h xs
Hxs
hxs h xs
 
 
=
 
 
 
…
…
. (9.4)
By the equations (4.3), (4.4), (9.2) and (9.4), the colocated open-loop transfer
function of the PFA from pointwise piezoelectric force, to the pointwise
radial displacement sensor output located at
p
N
s c
x L = , can be formulated:

31
26 1
1
(, )
() ( , , ) ( )
()
q
OL c i i i
i
bd wx s
g xLs Yz z
Vs R α
−
=
==− −
∑
Gs
(9.5)
where (, ) wxs
is the Laplace transformed of the transverse displacement and Vs
is the input voltage. All other variables have been defined in Section 4.1. The
closed-loop formulation for feedback control of DTFM based model will be
presented in Section 9.3.
( )

9.2. Open-Loop Transfer Function: Rayleigh-Ritz Model
In this section, the Rayleigh-Ritz or assumed method (Meirovitch, 1997) is
used to discretize the composite piezoelectric laminated slightly curved beam
model derived in Chapter 3. Based on this method, the former partial differential
equations model can be discretized into discrete equations of motion directly from
the kinetic and potential energies making use of Lagrange’s equation.
Consider the composite piezoelectric laminated slightly curved beam model
and approximate the radial displacement and circumference displacement
by the finite series
( , ) wx t
(, ) uxt
128
(9.6)
1
(, ) ( ) ()
N
j j
j
wx t x q t φ
=
=
∑
(9.7)
1
(, ) ( ) ()
N
j j
j
uxt x q t ϕ
=
=
∑
where ()
j
x φ , ()
j
x ϕ are admissible functions defined similarly in Chapter 7 and
are unknown generalized coordinates. ( )
j
q t
To this end, substitute the series equations (9.6) and (9.7) into the energy
expressions (7.3) and (7.5), to obtain
{} {
1
() [ ] ()
2
T
t Mqt = }
Tq (9.8)
{} {
1
() [ ] ()
2
T
t Kqt = } Uq (9.9)
where the elements of the [ ] M and [ matrices are shown in Appendix A. The
mass and stiffness matrices are positive definite and positive semi definite,
respectively.
] K
The kinetic and strain energies of the system are functions of
{ }
()
j
qt . The
equations governing these generalized coordinates can be formulated by
Lagrange’s equation. For this, consider the virtual work done by the piezoelectric
force, in equation (4.1b) is ( ,)
pc
NL t

(,)
(,)
pc
c
NL t
W w
R
δ δ = Lt (9.10)
where w δ is the virtual radial displacement of the PFA.
129
With the series equations (9.6) and (9.7), the virtual work, equation (9.10), can be
rewritten as follow:
(9.11)
1
() ( )
N
j j
j
Wptq δ
=
=
∑
tδ
where

(,)
() ( )
pc
j
NL t
jc
pt
R
φ = L (9.12)
Substitute equations energy functionals, equations (9.8) and (9.9) into Lagrange’s
equations as follow:
1,2,...
j
jj j
dT T U
pj
dt q q q

∂∂ ∂
−+ = =


∂∂ ∂

(9.13)
to yield the equations of the motion of the discretized model
{ } { } { } [ ] () [ ] () () M qt K q t p t + = (9.14)
where {}
(,)
() 0000 ( )
T
pc
Nc
NL t
pt L
R
φ

=


… .
The proportionality of damping (Meirovitch, 1997) is added to the equations
of the motion, equation (9.14), for analytical convenience. This approach is
justified by the fact that the nature of damping is not known exactly, that its
values are rather roughly approximated. Thus, the co-located actuating and
sensing M-DOF system can be written as follow:
{ } { } { } { } [ ] () [ ] () [ ] () ( )
a
M qt C q t K qt B v t ++ = (9.15)
{ } { } () ()
T
s
yt C q t = (9.16)
130
where is the radial displacement output, vt is the input control voltage,
is the vector of generalized coordinates,
( ) yt ( )
( ) qt

31
1
1
00 ( ) ( )
T
q
a ii
i
bd
B Yzz
R
φ
α
−
=

= −


∑
…
iNc
L (9.17)
{ } 0000 ( )
T
s
CL φ = …
Nc
. (9.18)
By defining the state vector
{}
{ }
{ }
()
()
()
qt
zt
qt

=



(9.19)
Equations (9.15) and (9.16) are converted to the state equation
{ } [ ] { } [ ]
() () ( ) zt A z t B v t = + (9.20)
and the output equation
{ } { } () ()
T
yt C z t = (9.21)
where

[]
[ ] [ ]
[] [ ] [] [ ]
[]
[]
[] {}
{} { } []
11
1
0
,
0
,0
s
a
I
A
MK M C
BC
MB
−−
−

=
−−



.C   == 
 


(9.22)
with
[ ]
0 and
[ ]
I being zero matrix and identity matrix of appropriate order.
Using the Laplace transformation of equations (9.20) and (9.21) for the zero
initial condition, the colocated open-loop transfer function from input control
131
voltage Vs to output response Ys (or radial displacement ) in terms
of the state parameters ( can be expressed as follow:
() ( )
)
( ,
c
wL s)
,, AB C
( )
1 ()
()
()
OL
Ys
Gs CsI A B
Vs
−
== − (9.23)
The is known as the open-loop transfer function. The closed-loop
formulation for feedback control of Rayleigh-Ritz based model will be presented
in Section 9.4.
()
OL
G s

9.3. Feedback Control Formulation of the PFA: DTFM Model
In this section, position feedback control of the PFA as shown in Figure 9.1
will be formulated based on the theorem of the transfer function of a distributed
component described in Section 9.1 and the DTFM based model derived earlier in
Chapter 4. Let the actuating piezoelectric force, , located at ( )
p
N t
a c
x L = (refer
to equations (4.1b) and (4.3)) and the radial displacement sensor located at
s c
x L = . The radial displacement response of the PFA with control is described by

0
(, ) ( , , ) ( , )
c
L
wxs G x s f s d ζ ζ =
∫
ζ
(9.24)
where
(,)( )()( )()(() (,)
aa
)
s
f sxxVsxxCsrswxs ζδ δ =− =− −
(9.25)

132
( ) Vs is the voltage control input, Cs is the transfer function of the feedback
controller, is the reference input and
( )
() rs ( , )
s
wx s
is the radial displacement
sensor output.
Solve equation (9.24) for wx ( , )
s
s
by letting
s
x x = and substitute for ( , ) wxs

in equation (9.24). The closed-loop system response is

(, , ) ( ) (, )
(, , )
() 1 ( ) ( , , )
a
CL a
sa
Gx x s C s wx s
Gxx s
rs Cs G x x s
∆
==
+∆
(9.26)
where

()
2
31 1
1
22
31 1
1
;;
();
1
,
2
q
ii i
i
q
ii i
i
B
D
RA
bd Y z z
bd Y z z
γ
α
α
χ
β
−
=
−
=
∆= − = −
=−
=−
∑
∑

() () () 1
(, , ) ( ) ,
ca
Fs L F s x Fs x
abb b
Gx x s e M N e M e
− −
=+

() ( ) () 1
(, , ) ( ) ,
s ca
Fs x F s L Fs x
sa b b b
x s e M N e M e
− −
=+ Gx

2
22
2
0 0 010
00 1 0 0
00 0 0 1
1
() ,
00
00 0 0 0
00
0
0
0
0
1
0
B
Fs
s
AR
AB AB
ss
RA RR
ρ
ρρ
αα α α α





 =
−




−− −


A

133

100000
010000
00 1 0 00
,
000000
000000
000000
b
M
 
 
 
 
=
 
 
 
 
 
 

2
2
00 00 0 0
00 00 0 0
00 00 0 0
()
0 0
,
() () ()
00 0 0
()
00
() () ()
b
AR Cs A B
N
RCs Cs Cs
BB
sD
AA
BR Cs A D
RCs Cs Cs
χ
χχχ
ρ
β
βββ





−+

−
=


−


−+

−


0
0

0
0
0
,
()
0
()
rs
rs
γ
 
 
 
 
=
 
 
 
 
 
 

1
01 1
1
11
()
mm
mm
nn
nn
ks ks k s k
sds d s d
µ
−
−
−
−
Cs
+ ++ +
=
+++ +
…
…

The numerical frequency response of the system described above will be
presented in Section 9.5.

134
9.4. Feedback Control Formulation of the PFA: Rayleigh-Ritz Model
A position control of the PFA is shown in Figure 9.1, where the function of
the feedback controller is to assure that the output of the system tracks the
reference input rt with zero or minimum error. The output of the system is
( ) yt
( ) ( ) yt
{ } { } () ()
T
yt C z t = (9.25)
where vector { } C has been defined in equations (9.18) and (9.22).
The sensor output (measured system output) is given by
{ } { } () () ()
T
ss s
yt k yt k C zt == (9.26)
where
s
k is a constant sensor gain. The controller input is the error signal
( ) ( ) ( )
s
et r t y t = − (9.27)
Consider PD feedback control law as follow:

()
()
()
c p
Vs
Gs K Ks
Es
== +
d
(9.28)
which indicates that

[ ] { } () () () () () ( )
pd c p d
vt K e t K e t K z t K r t K r t =+ =− + + (9.29)
where

[] {} {}
T
cs p a d a
KkK B K B
T
 
=
 
(9.30)
Substitute equation (9.29) into equation (9.20) to obtain the state equation of the
closed-loop
{ } [ ] [ ] [ ] () { } [ ] [ ]
() () () ( )
cp d
z t A B K z t K Br t K Br t =− + + (9.31)
135
The step response of the system described above under PD feedback control will
be demonstrated in Section 9.5.

9.5. Numerical Simulations
In this section, the simulations of the PD feedback control system of the PFA
model based on DTFM and Rayleigh-Ritz method will be presented and
discussed. The material properties of the PFA as shown in Table 7.1 will be used
for the simulation parameters. The proportional gain, and derivative gain,
used in the simulation are 320 and 3, respectively.
p
K
d
K
Figure 9.2 shows the comparison of the frequency responses of the PFA based
on DTFM model with and without feedback controller. The amplitude response of
the system can be reduced under PD feedback control.
Figure 9.3 shows the step responses of the PFA with and without feedback
control based on Rayleigh-Ritz model. It is shown that the PFA followed the
reference step input pretty well under PD feedback control and simultaneously
damped out the response nicely in allowable period of time.
Figure 9.4 shows the error position tracking of the PFA under the PD
feedback control. The error approached to zero or minimal error in allowable
period of time.

136
9.6. Conclusion
The control system formulation of the PFA based on DTFM and Rayleigh-
Ritz method are presented. The frequency response of the position feedback
control of the PFA is easily formulated by the DTFM. The position feedback
control of the PFA in time domain can be formulated by Rayleigh-Ritz method.

Figure 9.2 Frequency responses of the PFA.

137

Figure 9.3 Step responses of the PFA with and without feedback control

Figure 9.4 Error position tracking of the PFA under PD feedback control
138
CHAPTER 10
CONCLUDING REMARKS

The innovative design of the Piezoelectric Forceps Actuator or PFA has
demonstrated this forceps is very potential to be one of the surgery tool used
during the minimum invasive surgery (MIS) procedures. Compared to the existing
surgery tools used in the MIS procedures, the PFA is simple in design,
inexpensive in manufacture, small size, light weight, being able to achieve
precision deflection and grasping force with computer interface controlled data
glove. Furthermore, the PFA does not have moving parts such as mechanical
gears, hinges, bearings, racks and pinions, and thus avoids problems in operation
like friction, backlash, lubrication, leakage and sterilization.
In the course of designing and conducting research on the PFA, a number of
accomplishments have been achieved as following:
• Successfully design a simple, inexpensive, miniature surgery tool that is
addressing the drawbacks of existing surgery tools used in minimum invasive
surgery procedures.
• A new general model of the composite piezoelectric laminated thin slightly
curved beam has been derived to study its characteristics. Although this proposed
model in this research was originally derived for the PFA, but it is also applicable
to any type of homogeneous or composite non-piezoelectric laminated thin
139
slightly curved beams. Furthermore, the proposed model can be reduced to the
composite piezoelectric laminated thin straight beams model by setting the
curvature to zero (i.e., the radius of curvature goes to infinity) and assuming no
stretching-bending coupling due to the nature geometry of the flat beams
• Characteristic performance static and dynamic analysis of the composite
piezoelectric laminated thin straight beams in terms of the various thickness of a
single piezoelectric layer, number of layers, total thickness of the system in
generating the maximum tip displacement, slope, bending moment, force and
natural frequencies. The characteristic performance studies provide engineers and
researchers the ability to choose the design parameters from the wide design space
so the performance meets application requirements.
• Based on solid mechanics and Castigliano’s theorems, the radius curvature
deflection and grasping force models of the PFA have been derived in details and
validated with the experimental results.
• Based on the proposed composite piezoelectric laminated thin slightly curved
beam model, the predicted natural frequencies of the PFA are validated with the
experimental results.
• Finally, the control system formulations of the PFA based on distributed transfer
function method (DTFM) and Rayleigh-Ritz method are provided to show the
usefulness of the proposed models for further used in control system design and
analysis
140
CHAPTER 11
FUTURE RESEARCH

In this research, the design, modeling and experimental testing of the
Piezoelectric Forceps Actuator have demonstrated that this actuator is potentially
to be used as a surgical tool in minimum invasive surgery or a miniature robotic
gripper in IC semiconductor manufacturing plant. However, additional could be
undertaken that would extend the studies accomplished already on this subject.
Some future research topics includes:
• Implementation of real time data glove fuzzy controlled the PFA as shown in
Figure 11.1.
• The PFA is integrated with micro-machined tactiled force feedback sensing to
achieve the desired grasping force as shown in Figure 9.2.
• Grasping force and radial displacement of the Piezoelectric Forceps Actuators
probably can be improved by laminating several thin layers of piezoelectric
elements on the host structure.
• Fabricates the forceps from the shape memory alloy or other smart materials to
increase its displacement and grasping force.
• Improved the proposed model by including the non-linear hysteresis effects and
voltage non-linearities for large displacement and grasping force analysis.
141

Figure 11.1 Data glove fuzzy controlled the PFA

Micro-machined
Force Sensor
Fiber Optics Camera
Flexible Snake
Type Robotic

Figure 11.2 Future PFA attached to the snake-type robot and its tip jaws
embedded with micro-machined force sensor

142
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149
APPENDIX A
In this appendix, the steps of calculating the natural frequencies of the
Piezoelectric Forceps Actuator approximately using Rayleigh-Ritz method
(Meirovitch, 1997) will be presented. The energy functionals, potential and
kinetic energies of the composite piezoelectric laminated thin curved beams,
derived earlier in Chapter 3 will be used during this process. Its results will be
used to compare with the exact natural frequencies predicted by DTFM as shown
in Chapter 7.
For readability and conveniency, the potential and kinetic energies of the
cantilevered PFA from the Chapter 3 are rewritten as follow:
For the total potential energy,

12 3
Π = Π +Π +Π (A.1)
where
j
ϕ and
j
φ are the chosen polynomial shape functions that satisfy only the
essential boundary condtions,

2
2
1
2
0
12 1
2
L
uu
Aw w
xR xR
dx
  
∂∂ 
Π= + +   


∂∂

 

 
∫
(A.2)

2
2
2
2
0
1
2
L
w
D
x
dx
 
 ∂
  Π=

∂
  
 
∫
(A.3)

2
3
2
0
1
2
2
L
uw w
B dx
xR x
  ∂ ∂ 
Π= − +
  
∂∂

 
∫
(A.4)
150

1
()
N
j j
j
ux a ϕ
=
=
∑
(A.5)

1
()
N
j j
j
wx b φ
=
=
∑
(A.6)
For the total kinetic energy,

22 2
0
() ()
2
L
xwx
ρ
ω Tu dx   = +
  ∫
(A.7)
where its displacement fields, u and w are defined in equations (A.5) and (A.6),
respectively.
The Rayleigh-Ritz method requires minimization of the Lagrangian energy
functional, . The Lagrangian is then minimized by taking its derivatives
with respect to the undetermined coefficients of the displacement fields and
making them equal to zero, shown as follow:
LT =−Π

( )
()
12 3
12 3
0, 1, 2,...,
0, 1, 2,...,
ii
ii
T
L
iN
aa
T
L
iN
bb
∂−Π−Π−Π
∂
==
∂∂
∂−Π−Π−Π
∂
==
∂∂
=
=
(A.8)

1
11
0
0
L
NN
jj
i i
j j
jj i
dd
d d A
Aa b
a dx dx R dx dx
ϕφ
ϕ ϕ
==
dx
 
 
∂Π
== +

   
∂

 
  
 
∑ ∑
∫
(A.9)

1
2
11
0
0
L
NN
j
ij i jj
jj i
d
A A
a b
bR dx R
ϕ
φ φ
==
dxφ
 
 
∂Π
== +

   
∂

 
  
 
∑ ∑
∫
(A.10)

2
0
i
a
∂ Π
=
∂
(A.11)
151

2
2
2
2 2
1
0
0
L
N
j
i
j
j i
d
d
Db
bdx dx
φ
φ
=
dx
 

∂Π
==   

∂


 

 
∑
∫
(A.12)

2
3
2
1
0
0
L
N
j
i
j
j i
d
d
B b
adx dx
φ
ϕ
=
dx
 

∂Π
==−   

∂


 

 
∑
∫
(A.13)

2
2
3
22
11
0
0
L
NN
j
i
ji j
jj i
dd
d B
j
B a b
bdx dxR dx
ϕφ
φ
φ
==

 
∂Π
==− +   

∂
  

∑ ∑
∫
dx


(A.14)

2
1
0
0
L
N
ijj
j i
T
a
a
ωρ ϕ ϕ
=
dx
 

∂
==
   
∂
 

 
∑
∫
(A.15)

2
1
0
0
L
N
ijj
j i
T
b
b
ωρ φ φ
=
dx
 

∂
==
   
∂
 

 
∑
∫
(A.16)

This yields equations that can be written in a matrix form and give the natural
frequencies as follow:
(A.17)
[] [ ] ()()
2 2
0
ua ub ua ub N
wa wb wa wb N
kk m m a
KM
kk m m b
ωγ ω

  
−= −

   
  

=
where

[ ]
12 1 2
, ,..., , , ,...,
N
aa a b b b γ =
N
(A.18)

0
L
ua i j
m ρϕϕdx   =
  ∫
(A.19)

00
00
ub
NN
m
×
 
 
=
 
 
 
(A.20)
152

00
00
wa
NN
m
×
 
 
=
 
 
 
(A.20)

0
L
wb i j
m ρφφdx   =
  ∫
(A.21)

0
L
j
i
ua
d
d
kA d
dx dx
ϕ
ϕ
x
 
=
 
 
∫
(A.22)

2
2
0
L
j
ii
ub j
d
dd A
k B
Rdx dx dx
φ
ϕϕ
φ dx
 
=−
 
 
 
∫
(A.23)

2
2
0
L
j
i
wa i
dd
d A
k B
Rdx dx dx
ϕϕ
φ
φ
j
dx
 
=−
 
 
∫
(A.24)

22
2
222 2
0
L
j j
i
wb i j i
dd
d A B
kD
RdxdxRdx
φφ
φ
φφ φ


=+ −   
 



∫
dx (A.25)

153
APPENDIX B

Following is the review theorem of the transfer function of a distributed
component or so-called DTFM (Yang and Tan, 1992).
Theorem:
Let the boundary value problem

(, ) ( ) ( , ) (, ) (0, )
(0, ) ( , ) ( )
bb
d
xsFs xs Pxs x L
dx
MsN Ls rs
ηη
ηη
=+ ∈
+=
(B.1)
with and (, ) 0 Px s = ( ) 0 rs = has only the null solution. Then there exists the
unique solution of equation (9.1) which is formulated as:

0
(, ) ( , , ) ( , ) (, ) ( ) (0, )
L
xsGx sP sd Hxsrs x ηζζζ =+
∫
L∈
x
(B.2)
where

() ( ) 1 ( )
() () 1 ( )( )
() ,
(, , )
() ,
Fs x F s L Fs
bb b
Fs x F s L Fs L
bb b
eM Ne Me
Gx s
e M Ne Ne x
ζ
ζ
ζ
ζ
ζ
−−
−−
 + ≤
=

−+≥

(B.3)
and is the fundamental matrix of equation (9.1). Gx
() Fs x
e ( , , ) s ζ and are
called the distributed transfer functions of the component,
( , ) Hxs
11 1 11 1
11
(, , ) (, , ) (, ) ( , )
( , ,) , ( ,) .
(, , ) (, , ) (, ) ( , )
nn
nnn n nn
gx s g x s h xs h xs
Gx s H x s
gx s g x s h xs h xs
ζζ
ζ
ζζ


==



……
……






154

Abstract (if available)

Abstract Meso/micro grasping of tiny soft objects such as biological tissues or biopsy, which ranges from hundreds to thousands micro-millimeters in dimension, plays a significant role in the fields of tele-surgery, minimally invasive surgery (MIS), and biomedical instrumentation. In this dissertation, a proposed novel piezoelectric forceps actuator (PFA) is fabricated to improve one of the existing surgery tools used in MIS. One of the advantages of the PFA over conventional MIS forceps lies in that it can be remotely controlled to achieve precision deflection and grasping force. Furthermore, it does not have any moving parts such as gears and hinges, and hence avoids problems in operation like friction, backlash, lubrication, leakage and sterilization. A new general piezoelectric laminated slightly curved beam model is derived to predict the natural frequencies of the PFA, and verified with experimental results. By setting the curvature of the proposed model to zero (i.e., the radius of curvature goes to infinity) and neglecting stretch-bend couple, a piezoelectric laminated straight beam model can be derived. With a distributed transfer function formulation for the proposed models, the effects of the layer number, the layer thickness and the bending stiffness on natural frequencies, deflection, bending moment and output force of the system can be investigated. Based on solid mechanics and Castiliagno's theorems, radius of curvature displacement and grasping force models of the PFA is derived and verified with experimental results. Finally, the control system formulations of the PFA based on the distributed transfer function method (DTFM) and Rayleigh-Ritz method are provided to show the usefulness of the proposed models for further used in control system design and analysis.

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