A novel adaptive harmonic balance method with an asymptotic harmonic selection

The harmonic balance method (HBM) is one of the most widely used methods in solving nonlinear vibration problems, and its accuracy and computational efficiency largely depend on the number of the harmonics selected. The adaptive harmonic balance (AHB) method is an improved HBM method. This paper presents a modified AHB method with the asymptotic harmonic selection (AHS) procedure. This new harmonic selection procedure selects harmonics from the frequency spectra of nonlinear terms instead of estimating the contribution of each harmonic to the whole nonlinear response, by which the additional calculation is avoided. A modified continuation method is proposed to deal with the variable size of nonlinear algebraic equations at different values of path parameters, and then all solution branches of the amplitude-frequency response are obtained. Numerical experiments are carried out to verify the performance of the AHB-AHS method. Five typical nonlinear dynamic equations with different types of nonlinearities and excitations are chosen as the illustrative examples. Compared with the classical HBM and Runge-Kutta methods, the proposed AHB-AHS method is of higher accuracy and better convergence. The AHB-AHS method proposed in this paper has the potential to investigate the nonlinear vibrations of complex high-dimensional nonlinear systems.