A Matrix-Based Computational Scheme of Generalized Harmonic BalanceMethod for Periodic Solutions of Nonlinear Vibratory Systems

Abstract

A matrix-based computational scheme is developed based on the Generalized Harmonic Balance method for periodic solutions of nonlinear dynamical systems. The nonlinear external loading is expanded into a Taylor’s series as a function of displacement and velocity, and is then expressed as a combination of Fourier harmonics through the Generalized Harmonic Balance method. Using the Newton-Raphson’s approach, an iteration scheme is formulated to obtain the solution of harmonic coefficients for the displacement. The present scheme is a general purpose realization of the Generalized Harmonic Balance method in the sense that it does not need an analytical Fourier expansion of loadings, and all of the coefficient matrices involved with the scheme are created in a standard way. An example of a periodically forced Duffing oscillator is provided to demonstrate the performance of the present scheme. Numerical solutions of period-1 motion from the present scheme are compared with numerical results given by the Runge-Kutta method. The numerical results agree well with analytical predictions by Luo et al.