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 stiffness terms is very common, and the inclusion of asymmetric quadratic terms is known to modify the effective cubic nonlinearity in weakly nonlinear systems. When using low (second, in this case)-order perturbation methods, the net effect in these cases is found to be a monotonic dependence of the free vibration frequency on the amplitude squared, with a single term that depends on the coefficients of the quadratic and cubic terms. 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 and perturbation methods in weakly nonlinear formulations. 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 free and forced vibration response of weakly nonlinear asymmetric systems. For free vibration, it captures the correct amplitude–frequency dependence, including cases of non-monoticity. The method can also be used to determine the steady-state response of damped, harmonically driven vibrations, including information about 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 a variety of system models with polynomial nonlinearities.