Bifurcations of nonlinear normal modes via the configuration domain and the time domain shooting methods

• Described how to use the configuration domain shooting method to calculate nonlinear normal modes.
• Described how to use the time domain shooting method to calculate nonlinear normal modes.
• Discussed the advantages and disadvantages of both shooting approaches.
• Applied both shooting methods to a spring-mass-pendulum system.
• The global bifurcations of nonlinear normal modes and periodic solutions of the spring-mass-pendulum are studied.

This work discusses the two different approaches of applying the shooting method to calculate nonlinear normal modes (NNMs) – the configuration domain integration shooting method and the time domain integration shooting method. The problems and concerns are itemized and analyzed in the work for both of the configuration domain integration and the time domain integration shooting methods. The advantages and disadvantages of these two different approaches of shooting methods are compared. Different algorithms which are able to handle different problems of each approach are developed, respectively. The NNMs are calculated for an autonomous conservative system with inertially coupled quadratic nonlinearities via these two shooting technics. The bifurcations of the NNMs and periodic motions (PMs) are investigated via the combination of the numerical continuation method and the shooting technique. The stability of the NNMs and PMs are determined by the Floquet multiplier.