State equations of linear vibrating systems

Abstract

When arbitrarily large displacements are considered, the equations of vibrating systems are, as a rule, nonlinear. However in mechanical engineering large displacements in vibrations are seldom desirable, indeed, just the opposite holds true. The engineer makes efforts to arrange the vibrating system so that its motion is kept within the vicinity of a prescribed reference motion. When this is the case, the equations of motion relative to this reference motion can often be linearized. The existing linearization possibilities will be illustrated in examples of mechanical systems. The linearization can be carried out either subsequently to generating the equations of motion or can be introduced into the kinematical relations prior to generating the equations. In both cases one obtains the same linear equations of motion, i.e. a second order differential equation for the position vector. The equations of motion can be transformed into the state equation, i.e. into a first order differential equation for the state vector of the system. Ordinary mechanical vibrating systems and the analogous electrical systems may be equally represented by their equations of motion or their state equations. This is however not the case for general vibrating systems whose state equations are preferred since the equations of motion do not offer any special advantage for the analysis of vibrations.