Analytical solutions of periodic motions in a first-order quadratic nonlinear system

Abstract

In this paper, analytical solutions of period-1 motion of a first-order quadratic nonlinear system with both parametric excitation and external excitation are investigated through the generalized harmonic balance method. The analytical solutions of the system are obtained via solving the coefficients of all harmonic terms at the equilibrium position. The precision of analytical solutions is guaranteed via convergence study of harmonic balance terms. Stability analysis is carried out via eigenvalue analysis. The analytical solutions are different from the perturbation analysis solutions. Moreover, the trajectories of periodic motions obtained from analytical solutions can better explain the dynamics of the system. To verify the accuracy of analytical solutions, numerical simulation are performed and the simulation results are compared with the analytical solutions. The harmonic spectra are also presented to illustrate of the contribution of each harmonic term on a periodic motion.