Bifurcations and Chaotic Motions in Resonantly Excited Structures

The focus of this chapter is a discussion of the behavior of multi-degree-of-freedom models of structures with nonlinearities. While an overview of the research conducted in this area is given, the latter part of the chapter is devoted to a study of the response of weakly nonlinear multi-degree-of-freedom models under harmonic excitation. These models were derived from the von Karman equations that describe the behavior of a thin rectangular plate under initial tension. The types of behavior that result from internal resonances are of particular interest, whereby one mode is driven directly but other modes are excited by the nonlinear coupling between the modes. Energy sharing between the directly driven mode and the other modes leads to an amplitude-modulated coupled-mode response that can become chaotic. The approach is to develop models of the slowly varying amplitude and phase of the nonlinear response of the interacting modes through averaging. These equations are studied using the local bifurcation theory for their steady-state solutions. Various bifurcation points are identified in order to understand which types of solutions are possible for a given set of excitation conditions and model parameter values. It is shown that the response of the plate is qualitatively distinct and depends on the mode which is directly excited by the external loading.

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