On a Method of Higher Approximation and Determination of Stability Criterion for Steady Oscillations in Nonlinear Systems

An improved algorithm is proposed to obtain numerically a highly approximated steady-state solution for a nonlinear system with an arbitrary number of degrees of freedom. The so-called harmonic balance method is employed to compute the solution and the final Jacobian matrix obtained in the process of successive approximation is used to examine the stability of the solution. The theory is presented concisely by making use of the complex Fourier series. As a numerical example, the Duffing equation with hard spring is treated. The detained analytical results of the primary resonance are presented and the occurrence of the superharmonic resonances of order 2 to 9 is confirmed. Four regions in which the superharmonic resonances of even order bifurcate (that is, the unstable regions of the odd order harmonic solution) are indicated in terms of three parameters of the system.