TABLE OF CONTENTS To more easily understand a mathematical description of the mechanics associated with the structural dynamics of continuous elastic systems, the classical "vibrating string problem" will first be considered. Although the string can be described by a simple second-order partial differential equation in one dimension, it is typically descriptive of the more complex linearly elastic systems of aerospace vehicles. Once the fundamental concepts are covered for the string, other components will be treated that are more representative of these vehicles. Although the free vibration of a string can be analyzed using equations of motion that are of the same form as that of uniform beam extensional and torsional vibrations, the string is chosen as our first example primarily because, in contrast to these other structures, string behavior can be so easily visualized. Moreover, by this time in their undergraduate studies, most students have had some exposure to the solution of string vibration problems.
A uniform string of initial length ? 0 will be stretched in the x-direction between two walls separated by a distance ? > ? 0. The string tension T will be considered high and the transverse displacements v( x, t) will eventually be regarded as small. At any given instant this system can be illustrated as in Fig. 2.1. To describe the dynamic behavior of this system the forces acting on a differential length dx of the string can be illustrated...
