Geometrical Method

Abstract

This chapter introduces two kinds of geometric methods for analyzing self-excited vibration including the phase plane method and point mapping method. The phase plane method can analyze not only second-order autonomous systems with weak nonlinearity, but also those with strong nonlinearity. Its superiority over the other methods is the ability to complete the global analysis of motion occurring in second-order autonomous dynamic systems. The point mapping method can be applied to studying autonomous systems, particularly piecewise linear autonomous systems. The first five sections of this chapter are designed to introduce some elementary concepts such as the structure of phase plane, the phase diagram of conservative systems, the phase diagram of nonconservative systems, the classification of equilibrium points of dynamic systems, and the existence of limit cycles. The sixth section describes two types of self-excited vibrations respectively due to soft or hard excitations. The seventh section introduces the self-excited vibration occurring in strong nonlinear systems and the features in its phase diagram and time history. The point mapping method and its application are explained in the last section.