Higher-order approximate steady-state solutions for strongly nonlinear systems by the improved incremental harmonic balance method

Combining the harmonic balance method with the incremental harmonic balance approach, an improved incremental harmonic balance method is presented to obtain the higher-order approximate steady-state solutions for strongly nonlinear systems, which can simplify the calculation process for high-order nonlinear terms. Taking a strongly nonlinear Duffing oscillator with cubic nonlinearity and a strongly nonlinear Duffing oscillator with quintic nonlinearity as examples, the forced vibrations under harmonic excitation are investigated. Based on the first-order approximate analytical solutions obtained by the harmonic balance method, the higher-order approximate solutions are obtained by the improved incremental harmonic balance method. The correctness of the approximate analytical results is verified by the numerical results. The comparison results show that the approximations obtained by the improved incremental harmonic balance method agree with the numerical solutions well, and the improved method is effective to analyze the dynamical response for strongly nonlinear systems.