Clemson University TigerPrints All Theses Theses 12-2009 VIBRATIONS OF THIN PLATE WITH PIEZOELECTRIC ACTUATOR: THEORY AND EXPERIMENTS Parikshit Mehta Clemson University, [email protected] Follow this and additional works at: https://tigerprints.clemson.edu/all_theses Part of the Engineering Mechanics Commons 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 This Thesis is brought to you for free and open access by the Theses at TigerPrints. It has been accepted for inclusion in All Theses by an authorized administrator of TigerPrints. For more information, please contact [email protected]. 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 i 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. ii DEDICATION I dedicate this work to Lord Swaminarayan, Swami, my Guru, my family and my soul mate Shraddha for their love and support. iii 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. iv 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 v 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 vi 4.1 Introduction ............................................................................................................. 69 4.2 Piezoelectricity: A Brief History ............................................................................ 69 4.3 Derivation of Equations of Motion of Plate with Piezoelectric