The general solution of time-invariant vibrating systems

Abstract

It has been shown in Chapter 3 that linear time-invariant vibrating systems can be described by the system (3.10) of f second order differential equations

$$ \begin{array}{l} M\ddot y(t) + (D + G) \dot y (t) + (K + N) + (t) = h(t),\\ \begin{array}{*{20}{c}} {}&y \end{array}({t_0}) = {y_0},\begin{array}{*{20}{c}} {}&{\dot y} \end{array}({t_0}) = {{\dot y}_0} \end{array} $$

(4.1)

or, more generally, by the system (3.43) of n first order diferrential equations

$$ \dot x(t) = Ax(t) + b(t),\begin{array}{*{20}{c}} {}&x \end{array}({t_0}) = {x_0}. $$

(4.2)

The general solution

$$ y(t) = y(t; {y_0},{\dot y_0},{t_0};h(t)),\begin{array}{*{20}{c}} {}&t \end{array} \ge {t_0}, $$

(4.3)

respectively

$$ x(t) = x(t;{x_0},{t_0};b(t)),\begin{array}{*{20}{c}} {}&{t \ge {t_0}} \end{array}, $$

(4.4)

of these systems of differential equations yields information about the motion of the vibrating system when it is perturbed from its equilibrium position, or if it is subject to contnuing excitation. The general solution is in principle determined by the fundamental matrix of the system of differential equations. Therefore, this chapter will be devoted to discussing the properties of the fundamental matrix and the possibilities of its calculation.