A split-frequency harmonic balance method for nonlinear oscillators with multi-harmonic forcing

Abstract

A new harmonic balance method (HBM) is presented for accurately computing the periodic responses of a nonlinear sdof oscillator with multi-harmonic forcing and non-expansible nonlinearities. The presence of multi-harmonic forcing requires a large number of solution harmonics with a substantial increase in computational demand for either the conventional or the incremental HBM. In this method, the oscillator equation-error is first defined in terms of two functions (originally proposed for obtaining free-vibration periods in: R.E. Mickens, Iteration procedure for determining approximate solutions to nonlinear oscillator equations, Journal of Sound and Vibration 116 (1987) 185–187; and more recently: R.E. Mickens, A Generalised iteration procedure for calculating approximations to periodic solutions of “truly nonlinear oscillations”, Journal of Sound and Vibration 287 (2005) 1045–1051). A Fourier series solution is assumed, in which the total number of harmonics is fixed by the chosen discrete-time interval—this series is split into two partial sums nominally associated with either low-frequency or high-frequency harmonics. By exploiting a convergence property of the equation-error functions, the total solution is obtained in a new iterative scheme in which the low-frequency components are computed via a conventional HBM using a small number of algebraic equations, whereas the high frequency components are obtained in a separate step by updating. By gradually increasing the number of harmonics in the low-frequency group, the equation-error can be progressively reduced. Efficient use is made of FFT-based algebraic equation generation which allows an important class of non-expansible nonlinearities to be handled. The proposed method is tested on a Duffing-type oscillator, and an oscillator with a non-expansible 7th power stiffness term, where in both cases up to 24 component multi-harmonic forcing is applied. As a comparison, a conventional HBM is also used on the Duffing model in which the algebraic equations are generated in symbolic form to totally avoid errors from entering the formulation through complicated expansion of the cubic stiffness term (as in: I. Senjanović, Harmonic analysis of nonlinear oscillations of cubic dynamical systems, Journal of Ship Research 38 (3) (1994) 225–238; and in: A. Raghothama, S. Narayanan, Periodic response and chaos in nonlinear systems with parametric excitation and time delay, Nonlinear dynamics 27 (2002) 341–365). The paper shows that in obtaining period-1 solutions, the computational accuracy and efficiency of the proposed method is very good.