Energy scattering in weakly non-linear systems

Abstract

The Chinese Tam-Tam exhibits non-linear behavior in its vibro-acoustic response. The frequency content of the response during free, unforced vibration smoothly changes, with energy being progressively smeared out over a greater bandwidth with time. This is used as a motivating case for the general study of the phenomenon of energy cascading through weak nonlinearity. Numerical models based upon the Fermi-Pasta-Ulam system of non-linearly coupled oscillators, modified with the addition of damping, have been developed. These were used to study the response of ensembles of systems with randomized natural frequencies. Results from simulations will be presented here. For un-damped systems, individual ensemble members exhibit cyclical energy exchange between linear modes, but the ensemble average displays a steady state. For the ensemble response of damped systems, lightly damped modes can exhibit an effective damping which is higher than predicated by linear theory. The presence of a non-linearity provides a path for energy flow to other modes, increasing the apparent damping spectrum at some frequencies and reducing it at others. The target of this work is a model revealing the governing parameters of a generic system of this type and leading to predictions of the ensemble response.

Statistical energy analysis of nonlinear vibrating systems

“…This paper addresses a particular type of nonlinear high-frequency vibration problem. The nonlinearity of interest produces an effect termed 'energy scattering' or an 'energy cascade' [1] which is characterized by a power input to one frequency band, through either impact or random forcing, resulting in a response that contains energy across a wider frequency range than the bandwidth of excitation. This paper contains a possible extension of an established energy balance technique used for linear vibration analysis [2] to the nonlinear energy cascade scenario.…”