Date of Award


Document Type


Degree Name

Master of Science (MS)

Legacy Department

Mechanical Engineering

Committee Chair/Advisor

Jalili, Nader

Committee Member

Daqaq , Mohammed

Committee Member

Kennedy , Marian


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 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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