A hybrid averaging and harmonic balance method for weakly nonlinear asymmetric resonators

Abstract

Models for nonlinear vibrations commonly employ polynomial terms that arise from series expansions about an equilibrium point. The analysis of symmetric systems with cubic terms is very common, and the inclusion of asymmetric quadratic terms is known to modify the effective cubic nonlinearity in weakly nonlinear systems. The net effect is a monotonic dependence of the vibration frequency on the amplitude squared in both cases. However, in many applications, such a monotonic dependence is not observed, necessitating the use of techniques for strongly nonlinear systems or the inclusion of higher-order terms in a weakly nonlinear formulation. In either case, the analysis involves very tedious and/or numerical approaches for determining the system response. In the present work, we propose a method that is a hybrid of the methods of averaging and harmonic balance, which provides, with relatively straightforward calculations, good approximations for the amplitude-frequency dependence and for the steady-state response of damped, driven vibrations, including information about overtone harmonics and stability. The method is described, and general results are obtained for an asymmetric system with up to quintic nonlinear terms. The results are applied to a numerical example and validated using simulations. This approach will be useful for analyzing various systems with higher-order polynomial nonlinearities.