Periodic and transient nonlinear dynamics under discontinuous loading

Abstract

In this chapter, two variable expansions are introduced, where the fast temporal scale is represented by the triangular wave. In contrast to the conventional two variable procedure, the differential equations for slow dynamics emerge from the boundary conditions eliminating discontinuities rather than resonance terms. For illustrating purposes, an impulsively loaded single degree-of-freedom model with qubic damping and no elastic force is considered. Further, the method is applied to the Duffing’s oscillator under the periodic impulsive excitation whose principal frequency is close to the linear resonance frequency. Note that the illustrating models are weakly nonlinear, nevertheless the triangular wave temporal argument adequately captures specifics of the impulsive loading and, as a result, provides closed form asymptotic solutions. Following the conventional quasi-harmonic approaches, in such cases, would face generalized Fourier expansions for the external forcing function with no certain leading term.