Computation of the normal forms for general M-DOF systems using multiple time scales. Part Ⅱ: non-autonomous systems

摘要

A perturbation technique has been developed in Part Ⅰ to consider the computation of the normal forms for general multiple-degree-of-freedom autonomous systems. In this paper, the perturbation approach is extended to study general non-autonomous systems and is focused on systems with external forcing. With the aid of multiple time scales, efficient recursive algorithms are developed for systematically computing the normal forms. General solutions are obtained for solving ordered perturbation equations. In particular, the following cases are considered in detail: the non-resonance, internal resonances (including general resonance, resonant case involving 1:1 primary resonance, and combination of resonant case with non-resonance), and external resonances (including general resonance, and combination of internal resonance and external resonance). User-friendly Maple programs have been coded which can be "automatically" executed on various computer systems. Examples are given to demonstrate the computational efficiency of the method and the convenience of using computer algebra systems.