Numerical scheme for period-m motion of second-order nonlinear dynamical systems based on generalized harmonic balance method

Prediction of periodic motion plays key roles in identifying bifurcations and chaos for nonlinear dynamical systems. In this paper, a semi-analytical and semi-numerical scheme is developed as a combination of the analytical generalized harmonic balance method and the Newton–Raphson iteration for period-m solution of second-order nonlinear systems. The nonlinear external loading is approximated by the Taylor’s expansion of displacement and velocity, and is expressed as summations of many orders of Fourier harmonics pairs. A set of nonlinear algebraic equations are solved iteratively for the coefficients of harmonic pairs until the convergence of solution is achieved. The periodic solutions for period-2 motion in a periodically forced Duffing oscillator and period-3 motion in a buckled, nonlinear Jeffcott rotor system are obtained from the present scheme, and the corresponding stability and bifurcation are evaluated through eigenvalue analysis. The results from the present scheme are found in good agreement with the existent analytical solutions. The present scheme can be used as a general purpose numerical realization of the generalized harmonic balance method in evaluating periodical nonlinear dynamical systems since it is not involved with analytical derivation of Fourier expansion of external loading.