vibrations of thin plate with piezoelectric actuator

161 Pages·2017·1.91 MB·English

Clemson University
TigerPrints
All Theses Theses
12-2009
VIBRATIONS OF THIN PLATE WITH
PIEZOELECTRIC ACTUATOR: THEORY
AND EXPERIMENTS
Parikshit Mehta
Clemson University, [email protected]
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Recommended Citation
Mehta, Parikshit, "VIBRATIONS OF THIN PLATE WITH PIEZOELECTRIC ACTUATOR: THEORY AND EXPERIMENTS"
(2009).All Theses. 707.
https://tigerprints.clemson.edu/all_theses/707
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VIBRATIONS OF THIN PLATE WITH PIEZOELECTRIC ACTUATOR:
THEORY AND EXPERIMENTS
A Thesis
Presented to
the Graduate School of
Clemson University
In Partial Fulfillment
of the Requirements for the Degree
Master of Science
Mechanical Engineering

by
Parikshit Mehta
December 2009

Accepted by:
Dr. Nader Jalili, Committee Chair
Dr. Mohammed Daqaq
Dr. Marian Kennedy
ABSTRACT
Vibrations of flexible structures have been an important engineering study owing to its
both deprecating and complimentary traits. These flexible structures are generally
modeled as strings, bars, shafts and beams (one dimensional), membranes and plate (two
dimensional) or shell (three dimensional). Structures in many engineering applications,
such as building floors, aircraft wings, automobile hoods or pressure vessel end-caps, can
be modeled as plates. Undesirable vibrations of any of these engineering structures can
lead to catastrophic results. It is important to know the fundamental frequencies of these
structures in response to simple or complex excitations or boundary conditions.
After their discovery in 1880, piezoelectric materials have made their mark in
various engineering applications. In aerospace, bioengineering sciences, Micro Electro
Mechanical Systems (MEMS) and NEMS to name a few, piezoelectric materials are used
extensively as sensors and actuators. These piezoelectric materials, when used as sensors
or actuators can help in both generating a particular vibration behavior and controlling
undesirable vibrations. Because of their complex behavior, it is necessary to model them
when they are attached to host structures. The addition of piezoelectric materials to the
host structure introduces extra stiffness and changes the fundamental frequency.
The present study starts with modeling and deriving natural frequencies for
various boundary conditions for circular membranes. Free and forced vibration analyses
along with their solutions are discussed and simulated. After studying vibration of
membranes, vibration of thin plates is discussed using both analytical and approximate
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methods. The method of Boundary Characteristic Orthogonal Polynomials (BCOP) is
presented which helps greatly in simplifying computational analysis. First of all it
eliminates the need of using trigonometric and Bessel functions as admissible functions
for the Raleigh Ritz analysis and the Assumed Mode Method. It produces diagonal or
identity mass matrices that help tremendously in reducing the computational effort. The
BCOPs can be used for variety of geometries including rectangular, triangular, circular
and elliptical plates. The boundary conditions of the problems are taken care of by a
simple change in the first approximating function. Using these polynomials as admissible
functions, frequency parameters for circular and annular plates are found to be accurate
up to fourth decimal point.
A simplified model for piezoelectric actuators is then derived considering the
isotropic properties related to displacement and orthotropic properties of the electric field.
The equations of motion for plate with patch are derived using equilibrium (Newtonian)
approach as well as extended Hamilton’s principle. The solution of equations of motion is
given using BCOPs and fundamental frequencies are then found. In the final chapter, the
experimental verification of the plate vibration frequencies is performed with
electromagnetic inertial actuator and piezoelectric actuator using both circular and
annular plates. The thesis is concluded with a summary of work and discussion about
possible future work.

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DEDICATION
I dedicate this work to Lord Swaminarayan, Swami, my Guru, my family and my soul
mate Shraddha for their love and support.

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ACKNOWLEDGEMENTS
I sincerely thank my advisor, Dr. Nader Jalili, for his continuous support and guidance
throughout my Master’s degree program. The time that I spent doing research under Dr.
Jalili’s guidance helped me learn many professional and interpersonal traits which are
essential for my professional development. I am also grateful to my advisory committee
members Dr. Marian Kennedy and Dr. Mohammed Daqaq, whose timely guidance
helped me stay on the right track for my research.
I am also thankful to my fellow lab mates and researchers, Mr. Siddharth Aphale,
Dr. Reza Saeidpourazar and Mr. Anand Raju with the help in laboratory experiments.
The help from Mechanical Engineering Technicians Mr. Michael Justice, Mr. Jamie Cole
and Mr. Stephen Bass in fabricating the experimental set up is duly acknowledged.
.
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TABLE OF CONTENTS
ABSTRACT ......................................................................................................................... i
DEDICATION ................................................................................................................... iii
ACKNOWLEDGEMENTS ............................................................................................... iv
TABLE OF CONTENTS .................................................................................................... v
LIST OF FIGURES ........................................................................................................... ix
LIST OF TABLES ........................................................................................................... xiii
CHAPTER ONE: INTRODUCTION AND OVERVIEW ................................................. 1
1.1 Introduction ............................................................................................................... 1
1.2 Brief History of Vibration Analysis .......................................................................... 1
1.3 Literature Review and Research Motivation ............................................................ 3
1.4 Thesis Overview ....................................................................................................... 4
1.5 Thesis Contributions ................................................................................................. 7
CHAPTER TWO: VIBRATION OF MEMBRANES ........................................................ 8
2.1 Introduction ............................................................................................................... 8
2.2 Governing Equations of Motion for a Rectangular Membrane ................................ 8
2.3 Rectangular to Circular Coordinate Transformation .............................................. 11
2.4 Circular Membrane Clamped at the Boundary ....................................................... 15
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2.5 Forced Vibrations of Circular Membranes ............................................................. 19
2.5.1 Special Case: Point Load Harmonic Excitation ............................................... 20
2.6 Numerical Simulations of the Forced Vibration ..................................................... 22
CHAPTER THREE: VIBRATION OF THIN PLATES .................................................. 29
3.1 Introduction ............................................................................................................. 29
3.2 Governing Equations for Vibration of a Thin Plate ................................................ 30
3.3 Free Vibrations of a Clamped Circular Plate .......................................................... 34
3.4 Forced Vibration Solution for Clamped Circular Plate: Approximate Analytical
Solution ......................................................................................................................... 42
3.5 Raleigh Ritz Method for Clamped Circular Plate ................................................... 48
3.5.1 Orthogonalization: Concept and Process ......................................................... 53
3.5.2 Axisymmetric Vibration of the Circular Plate using Boundary Characteristic
Orthogonal Polynomials ........................................................................................... 55
3.6 Numerical Simulations............................................................................................ 57
3.6.1 Free Vibrations of Circular Plates.................................................................... 57
3.6.2 Axisymmetric Forced Vibrations of Circular Plate: Assumed Mode Method 60
3.7 Comparison between Thin Plate and Membrane Vibrations .................................. 65
CHAPTER FOUR: VIBRATION OF THIN PLATE WITH PIEZOELECTRIC
ACTUATOR ..................................................................................................................... 69
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4.1 Introduction ............................................................................................................. 69
4.2 Piezoelectricity: A Brief History ............................................................................ 69
4.3 Derivation of Equations of Motion of Plate with Piezoelectric Actuator ............... 70
4.3.1 Derivation of Equations of Motion with Newtonian Approach ....................... 76
4.3.2 Derivation of Equations of Motion with Hamilton’s Principle ....................... 81
4.4 Axisymmetric Vibration of Plate with Piezoelectric Patch .................................... 86
4.4.1 Numerical Simulation: Axisymmetric Plate Vibration with Piezoelectric Patch
................................................................................................................................... 89
4.4.2 Numerical Simulation of Force Vibrations under piezoelectric Excitation:
Axisymmetric Case ................................................................................................... 92
4.5 Non-axisymmetric Vibration of Plate with Piezoelectric Patch ............................. 93
4.5.1 Orthogonal Polynomials Generation for Two Dimensional Problem.............. 94
4.5.2 Domain Transformation of Specific Problem .................................................. 95
4.5.3 Generation of the Function Set “ f (x,y)” ....................................................... 96
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4.5.4 Generating Orthogonal Polynomial Set for the Circular Plate Modes ............ 98
4.6: Raleigh Ritz Analysis for Circular Plate with the Piezoelectric Patch: Free
Vibrations .................................................................................................................... 102
4.6.1 Forced Vibration of the Plate with Piezoelectric Actuator ............................ 105
4.6.2 Numerical Simulation of Vibration of Plate with Piezoelectric Actuator ..... 106
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CHAPTER FIVE: EXPERIMENTAL ANALYSIS OF PLATE VIBRATIONS .......... 108
5.1. Experimental Set up for Vibration of Circular Plate ........................................... 108
5.2 Experimental Analysis of Forced Vibration of Circular Clamped Plates ............. 110
5.2.1 Circular Plates with Partially Clamped Boundary ......................................... 110
5.2.2 Forced Vibration of Circular Clamped Plates with Piezoelectric Inertial
Actuator: Experiments ............................................................................................ 115
5.2.3 Forced Vibration of Circular Clamped Plates with Piezoelectric Patch ........ 117
5.3 Experimental Analysis of Forced Vibration of Circular Annular Plates .............. 121
5.3.1 Annular Plate Forced Vibration with SA 5 Electromagnetic Actuator .......... 125
5.3.2 Annular Plate Forced Vibration with PZT Actuator ...................................... 129
CHAPTER SIX: CONCLUSIONS AND SCOPE FOR FUTURE WORK ................... 134
Conclusions ................................................................................................................. 134
Scope for future work ................................................................................................. 136
APPENDIX A: EQUATIONS OF MOTION FOR RECTANGULAR MEMBRANE . 137
APPENDIX B: DOMAIN TRANSFORMATION FROM RECTANULAR TO
CIRCULAR FOR MEMBRANE ................................................................................... 141
REFERENCES ............................................................................................................... 142
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