Quantitative Methods

Abstract

Different from the geometry and the stability methods introduced until now, which are all the qualitative methods, the methods arranged in this chapter are quantitative methods that can calculate the magnitude of amplitude and frequency of self-excited vibrations. In general, the analytical solutions of the nonlinear differential equations cannot be found and the dynamic problems of nonlinear systems are much more difficult to be solved, in particular, the problems of high-dimensional nonlinear systems. Starting from how to reduce the order of system equations, the contents of this chapter are divided into five sections: the first is concerned with the basic concepts of the center manifold from the local theory of ordinary differential equations; the second is devoted to Hopf bifurcation theory and its application to calculating amplitude and frequency of self-excited vibration; the last three sections provide three analytical methods, Lindstedt-Poincare method, averaging method, and the method of multiple scales, for seeking approximate solutions of weakly nonlinear autonomous systems with single degree of freedom.