TABLE OF CONTENTS The mathematical language which is most convenient for analyzing multiple degree-of-freedom vibratory systems is that of matrices. Matrix notation simplifies the preliminary analytical study, and in situations where particular numerical answers are required, matrices provide a standardized format for organizing the data and the computations. Computations with matrices can be carried out by hand or by digital computers. The availability of programs such as MATLAB makes the solution of many complex problems in vibration analysis a matter of routine.
This chapter describes how matrices are used in vibration analysis. It begins with definitions and rules for operating with matrices. The formulation of vibration problems in matrix notation then is treated. This is followed by general matrix solutions of several important types of vibration problems, including free and forced vibrations of both undamped and damped linear multiple degree-of-freedom systems. Part II of this chapter considers finite element models.
Matrices are mathematical entities which facilitate the handling of simultaneous equations. They are applied to the differential equations of a vibratory system as follows:
A single degree-of-freedom system of the type in Fig. 28.1 has the differential equation
where m is the mass, c is the damping coefficient, k is the stiffness, F is the applied force, x is the displacement coordinate, and dots denote time derivatives. In Fig. 28.2 a similar three degree-of-freedom system is shown. The equations of motion may be obtained by applying Newton's second...
