Transient dynamics of a dispersive elastic wave guide weakly coupled to an essentially nonlinear end attachment

Abstract

We consider the dynamics of a dispersive semi-infinite linear rod weakly connected to an essentially nonlinear end attachment. Using a Green’s function formulation we reduce the dynamics to an integro-differential equation, which we express in the form of an infinite set of ordinary differential equations using Neumann expansions. We examine resonant interactions of the attachment with incident traveling waves propagating in the pass band of the rod, as well as resonance capture phenomena where the attachment engages in transient 1:1 internal resonance with the in-phase mode of the rod, at the bounding frequency of its pass and stop bands. The studied resonant interactions of the nonlinear attachment with traveling waves are similar to resonance capture cascades occurring in finite-chain nonlinear attachment configurations; these are the transient dynamics of the attachment as it engages in a series of instantaneous resonances with a continuous spectrum of frequencies in the pass band of the medium. As the energy of the attachment decreases due to damping and energy radiation back to the rod, the attachment engages in 1:1 resonance capture with the in-phase mode of the rod; this resonance capture occurs due to the dispersion property in the linear medium. It is shown that the transient dynamics of resonance capture can be reduced to a set of two ‘slow flow’ amplitude and phase modulation equations. The analytical results are in agreement with direct numerical simulations.