Ebook: Free Vibration Analysis of Rectangular Plates

Elsevier North Holland, Ins., 1982. — 324 p.Although the free vibration analysis of rectangular plates has received the attention of researchers for several centuries, its treatment has left much to be desired. Most of the available solutions are approximate in nature in that they do not satisfy exactly the governing differential equation, the prescribed boundary conditions, or both. Furthermore, it has been the experience of most students of mechanical vibration, as well as design engineers, that there is no book to which they can turn for a clear and orderly exposition of the general subject of rectangular plate free vibration.
Based on a determination to rectify this unacceptable situation, this book was prepared with two objectives: first, to provide the designer with highly accurate frequency and mode shape information for the free vibration analysis of a wide family of rectangular plates and, second, to provide a thorough discussion of the fundamentals of plate free vibration theory. In an extension of the latter objective, complete and detailed solutions have been provided for all plate problems considered. Illustrative problems are solved throughout the book so that the student and designer may obtain a rapid introduction to real-life mechanical vibration problems.
Chapter 1 is devoted to the theory underlying rectangular plate vibration. The governing differential equation is developed from first principles, and the mathematical formulation of the boundary conditions is introduced. It is shown how both the equation and the boundary conditions can be expressed in nondimensional form.
Chapter 2 is devoted to the solution of classical rectangular plate problems in which the plate has at least two opposite edges simply supported. Exact solutions of the Levy type are obtained for each of these problems, and eigenvalues and mode shape information are stored for a wide range of plate aspect ratios.
Chapter 3 is a major departure from the traditional literature. It introduces the method of superposition for obtaining accurate solutions to classical plate vibration problems. It is well known that serious difficulties have been encountered when researchers attempted to analyze plates that do not have at least one pair of opposite edges simply supported. Beginning with the fully clamped plate, Chapter 3 shows how such problems can be analyzed by exploiting the method of superposition as developed by the author. The method consists essentially of judiciously choosing forced vibration problems for which solutions of the Levy type can be obtained. The known solutions are then superimposed and parameters appearing in their boundary formulations are so adjusted that the combined solutions satisfy the precribed boundary conditions of the plate. Each solution satisfies exactly the differential equation, and the boundary conditions are satisfied to any desired degree of accuracy.
In Chapters 4-7 the remaining classical plate free vibration problems are solved. Throughout, eigenvalues are provided to four-digit accuracy. It is characteristic of the method of superposition that modes possessing symmetry and antisymmetry with respect to the coordinate axis are delineated. Many traditional problems are thereby eliminated.
Chapter 8 brings this new method to bear on a number of nonclassical problems, introducing plates subjected to other than the classical clamped, simply supported, or free edge conditions. It is seen that problems involving plates resting on point supports distributed along the edges, or even distributed over the plate lateral surface, are easily solved. Eigenvalues and mode shape information are provided for a number of problems of this type. The reader will find that the foundations have been laid for the resolution of virtually any rectangular plate problem, regardless of the support conditions, provided the supports are linear in nature. In fact, it will be found that Chapter 8 provides a sort of jumping-off spot for the resolution of practically any linear rectangular plate free vibration problem.
It would be virtually impossible to give appropriate recognition to everyone who has contributed to the preparation of this book. Nevertheless, certain individuals have made such substantial contributions that their efforts must be singled out. In particular, the author is indebted to Mrs. L. Sprouse, Mrs. D. Champion-Demers, and Miss Nicole Renaud for the typing of the manuscript and to Mr. D. Seaman, administrative officer of the Department of Mechanical Engineering at the University of Ottawa, for his many acts of assistance. The invaluable work of Dr. K. G. Gupta, who prepared all of the drawings, is gratefully acknowledged.
The author would also like to express his deep appreciation for the cooperation he has always received from the staff of the University Computing Centre. It is obvious that without their many hours of effort the eigenvalue tables contained in this book could not have been produced. Finally the author would like to thank the National Research Council of Canada for its support in providing funds to pay student assistants and making the computer facilities available.