TABLE OF CONTENTS This chapter begins the mathematical process of solving the linear, matrix differential equation that describes the small deflection, vibratory motion of a structure subjected to an arbitrary, time-varying loading. In this textbook, the loading is limited to being one whose magnitude is known for any time value, as opposed to being a loading whose magnitude can only be described in probabalistic terms. Using the concept of equivalent viscous damping, that matrix equation is
The overall form of the solution process closely parallels that for a single-DOF dynamical system. First a complementary solution, which contains the constants of integration, is obtained for the unloaded and undamped structural system. Then the response to the applied loading, with or without the effect of damping, is separately obtained. These two solutions can be combined to provide the complete description of the system motion. If required, the values of the constants of integration are then, and only then, determined by the system initial conditions.
This chapter concentrates on the first of the above two steps of the solution process, which is the step dealing with the unloaded and undamped structure. This first step has an important physical basis. To understand that physical basis, consider any lightly damped, multidegree of freedom structure. If such a structure, unloaded with respect to time-varying loads, is set in motion by the application of an arbitrary set of initial conditions, then the resulting vibration will generally appear to be without any discernable pattern. Indeed, the motion of the...
