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About this book

This fully revised and updated third edition covers the physical and mathematical fundamentals of vibration analysis, including single degree of freedom, multi-degree of freedom, and continuous systems. A new chapter on special topics that include motion control, impact dynamics, and nonlinear dynamics is added to the new edition. In a simple and systematic manner, the book presents techniques that can easily be applied to the analysis of vibration of mechanical and structural systems. Suitable for a one-semester course on vibrations, the book presents the new concepts in simple terms and explains procedures for solving problems in considerable detail. It contains numerous exercises, examples and end-of-chapter problems.

Table of Contents


1. Introduction

The process of change of physical quantities such as displacements, velocities, accelerations, and forces may be grouped into two categories; oscillatory and nonoscillatory. The oscillatory process is characterized by alternate increases or decreases of a physical quantity. A nonoscillatory process does not have this feature. The study of oscillatory motion has a long history, extending back to more than four centuries ago. Such a study of oscillatory motion may be said to have started in 1584 with the work of Galileo (1564–1642) who examined the oscillations of a simple pendulum. Galileo was the first to discover the relationship between the frequency of the simple pendulum and its length. At the age of 26, Galileo discovered the law of falling bodies and wrote the first treatise on modern dynamics. In 1636 he disclosed the idea of the pendulum clock which was later constructed by Huygens in 1656.
Ahmed A. Shabana

2. Solution of the Vibration Equations

It was shown in the preceding chapter that the application of Newton’s second law to study the motion of physical systems leads to second-order ordinary differential equations. The coefficients of the accelerations, velocities, and displacements in these differential equations represent physical parameters such as inertia, damping, and restoring elastic forces. These coefficients not only have a significant effect on the response of the mechanical and structural systems, but they also affect the stability as well as the speed of response of the system to a given excitation. Changes in these coefficients may result in a stable or unstable system, and/or an oscillatory or nonoscillatory system.
Ahmed A. Shabana

3. Free Vibration

The term free vibration is used to indicate that there is no external force causing the motion, and that the motion is primarily the result of initial conditions, such as an initial displacement of the mass element of the system from an equilibrium position and/or an initial velocity. The free vibration is said to be undamped free vibration if there is no loss of energy throughout the motion of the system. This is the case of the simplest vibratory system, which consists of an inertia element and an elastic member which produces a restoring force which tends to restore the inertia element to its equilibrium position. Dissipation of energy may be caused by friction or if the system contains elements such as dampers which remove energy from the system.
Ahmed A. Shabana

4. Forced Vibration

In the preceding chapter, the free undamped and damped vibration of single degree of freedom systems was discussed, and it was shown that the motion of such systems is governed by homogeneous second-order ordinary differential equations. The roots of the characteristic equations, as well as the solutions of the differential equations, strongly depend on the magnitude of the damping, and oscillatory motions are observed only in underdamped systems. In this chapter, we study the undamped and damped motion of single degree of freedom systems subjected to forcing functions which are time-dependent. Our discussion in this chapter will be limited only to the case of harmonic forcing functions. The response of the single degree of freedom system to periodic forcing functions, as well as to general forcing functions, will be discussed in the following chapter.
Ahmed A. Shabana

5. Response to Nonharmonic Forces

The response of damped and undamped single degree of freedom systems to harmonic forcing functions was discussed in the preceding chapter. It was shown that the steady state response of the system to such excitations is also harmonic, with a phase difference between the force and the displacement which depends on the amount of damping. The analysis presented and the concepts introduced in the preceding chapter are fundamental to the study of the theory of vibration, and the use of these concepts and methods of vibration analysis was demonstrated by several applications.
Ahmed A. Shabana

6. Systems with More Than One Degree of Freedom

Thus far, the theory of vibration of damped and undamped single degree of freedom systems was considered. Both free and forced motions of such systems were discussed and the governing differential equations and their solutions were obtained. Basic concepts and definitions, which are fundamental in understanding the vibration of single degree of freedom systems, were introduced. It is the purpose of this chapter to generalize the analytical development presented in the preceding chapters to the case in which the systems have more than one degree of freedom. We will start with the free and forced vibrations of both damped and undamped two degree of freedom systems.
Ahmed A. Shabana

7. Continuous Systems

Mechanical systems in general consist of structural components which have distributed mass and elasticity. Examples of these structural components are rods, beams, plates, and shells. For the most part, our study of vibration thus far has been limited to discrete systems which have a finite number of degrees of freedom. As has been shown in the preceding chapters, the vibration of mechanical systems with lumped masses and discrete elastic elements is governed by a set of second-order ordinary differential equations. Rods, beams, and other structural components on the other hand are considered as continuous systems which have an infinite number of degrees of freedom, and as a consequence, the vibration of such systems is governed by partial differential equations which involve variables that depend on time as well as the spatial coordinates.
Ahmed A. Shabana

8. Special Topics

Ahmed A. Shabana


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