Motion complexity in a non-classically damped system with closely spaced modes: From standing to traveling waves

Abstract

This article addresses the phenomenon of motion complexity in a periodically oscillating system, i.e. the occurrence of non-trivial phase lags among the system's coordinates. Specifically, the steady-state forced response of a linear, weakly damped, self-adjoint system is studied, for which the extent of motion complexity is typically expected to be small. Yet, it is shown that under the condition of closely spaced modes, weak non-classical damping may lead to a transition from standing waves to traveling waves. A system of two oscillators weakly coupled via a linear spring-damper element is considered. The emergence of these motions is related to the distribution of the applied forces and the effect of adding nonlinearity in the form of a cubic spring to the system is investigated. Moreover, it is demonstrated that under certain conditions, the traveling wave response is critical, i.e. it is associated with a resonance or anti-resonance.