1st Edition

Structural Vibration
Exact Solutions for Strings, Membranes, Beams, and Plates

ISBN 9781466576841
Published August 13, 2013 by CRC Press
307 Pages 115 B/W Illustrations

USD $215.00

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Book Description

Structural Vibration: Exact Solutions for Strings, Membranes, Beams, and Plates offers an introduction to structural vibration and highlights the importance of the natural frequencies in design. It focuses on free vibrations for analysis and design of structures and machine and presents the exact vibration solutions for strings, membranes, beams, and plates.

This book emphasizes the exact solutions for free transverse vibration of strings, membranes, beams, and plates. It explains the intrinsic, fundamental, and unexpected features of the solutions in terms of known functions as well as solutions determined from exact characteristic equations. The book provides:

  • A single-volume resource for exact solutions of vibration problems in strings, membranes, beams, and plates
  • A reference for checking vibration frequency values and mode shapes of structural problems
  • Governing equations and boundary conditions for vibration of structural elements
  • Analogies of vibration problems

Structural Vibration: Exact Solutions for Strings, Membranes, Beams, and Plates provides practicing engineers, academics, and researchers with a reference for data on a specific structural member as well as a benchmark standard for numerical or approximate analytical methods.

Table of Contents

Introduction to Structural Vibration

What is Vibration?

Brief Historical Review on Vibration of Strings, Membranes, Beams, and Plates

Importance of Vibration Analysis in Structural Design

Scope of Book


Vibration of Strings


Assumptions and Governing Equations for Strings

Boundary Conditions

Constant Property String

Two-Segment Constant Property String

Transformation for Nonuniform Tension and Density

Constant Tension and Variable Density

Variable Tension and Constant Density

Free-Hanging Nonuniform String

Other Combinations


Vibration of Membranes


Assumptions and Governing Equations

Constant Uniform Normal Stress and Constant Density

Two-Piece Constant-Property Membranes

Nonhomogeneous Membranes

Hanging Membranes



Vibration of Beams


Assumptions and Governing Equations

Single-Span Constant-Property Beam

Two-Segment Uniform Beam

Nonuniform Beam



Vibration of Isotropic Plates


Governing Equations and Boundary Conditions for Vibrating Thin Plates

Exact Vibration Solutions for Thin Plates

Governing Equations and Boundary Conditions for Vibrating Thick Plates

Exact Vibration Solutions for Thick Plates

Vibration of Thick Rectangular Plates Based on 3-D Elasticity Theory


Vibration of Plates with Complicating Effects


Plates with In-Plane Forces

Plates with Internal Spring Support

Plates with Internal Rotational Hinge

Plates with Partial Elastic Foundation

Stepped Plates

Variable-Thickness Plates



Vibration of Nonisotropic Plates


Orthotropic Plates

Sandwich Plates

Laminated Plates

Functionally Graded Plates

Concluding Remarks



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C.Y. Wang, Michigan State University

C.M. Wang, National University of Singapore


"The book is arranged in a way from simple to complex, enabling an easy walk into the whole subject. The exact solutions are illustrated not only by the analytical expressions, but also by numerical results in table or figure form. Thus, they can be directly used as benchmarks for comparison with numerical or approximate analytical solutions, or they can be simply taken over in evaluating the dynamic behavior of a practical structure. … The materials presented are very solid. This book presents a collection of exact solutions for vibration of strings, beams (bars), and plates, which are very common both in engineering and natural science. Complicating effects can be involved, however, including non-uniform geometry, non-homogeneous material property, internal support, elastic foundation, etc. The mathematical techniques used to obtain the solutions are very attractive and mathematically strict, and will provide a base for the study of more involved problems. The book should be very useful for researchers, engineers, and students in various engineering areas such as aerospace, civil engineering, mechanical engineering, ocean engineering, chemical engineering, etc. It can also be used as a reference for those who are working in physics, biology, geology, material science, and nanotechnology."
––W. Q. Chen, Department of Engineering Mechanics, Zhejiang University, Hangzhou, China

"The book provides an excellent comprehensive treatment of analytical vibration analysis of continuous structural systems. It provides the exact solutions for strings, membranes, beams, and plates along with tabulated numerical frequencies which makes this book essential not only for academics but also for practicing engineers and designers. … The authors have done an admirable job in the organization and the flow of the book. Direct explanations tell the story of structural vibrations, moving from basic string model to more complex non-uniform plates. In each chapter, the authors present the current research development based upon the latest research articles."
––Huseyin Yuce, New York City College of Technology, Brooklyn, USA

"This book is a new reference on a special topic (exact eigensolutions) for certain structural components, which can be quite useful for people in R&D of structural systems, educators of engineering vibrations, and developers of numerical algorithms for structural vibration problems."
––Bingen Yang, Dept. of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA, U.S.A