Solving Steady-State Wave Responses of One-Dimensional Multi-Degree-of-Freedom Lattices with Strong Nonlinearities Based on an Improved Incremental Harmonic Balance Method

Abstract The nonlinearity plays an important role in modulation of wave propagation characteristics, such as the adjustment of band structures by media nonlinearities. When strong nonlinearities are involved in a multi-degree-of-freedom (multi-DoF) wave propagation problem, traditional methods have difficulties to solve it accurately and efficiently due to the growth of dimensions. An algorithm based on an improved incremental harmonic balance (IHB) method is developed in this work to obtain steady-state solutions of a general wave problem with complex nonlinearities. A multi-DoF wave problem is firstly transformed to a delay differential equation (DDE), and Jacobian matrices of a harmonic balanced residual of the DDE can be directly constructed by conducting fast Fourier transform of the residual without any numerical integration. The algorithm decouples wave vectors from basis functions to significantly reduce the number of the functions, and it is more efficient than that based on a traditional IHB method of solving a wave problem, especially for the case that high-order basis functions are involved to handle strong nonlinearities. Moreover, the nonlinear wave problem can be solved when frequencies are given and wave vectors unknown, which is a challenge of a traditional IHB method. In this work, one-dimensional diatomic lattices with Hertz contact law and cubic spring models are studied as examples, where dispersion curves and bandgap properties are generated. Results show that it is necessary to use high-order Fourier functions to obtain accurate steady-state responses when strong nonlinearities exist. Results also show that the convergence rate of the method with use of a fixed frequency is faster than that of a fixed wave vector.