Abstract: To address the problem that the incremental harmonic balance (IHB) method using higher-order basis functions is computationally intensive and not easy to converge when analyzing the bandgap properties of multi-degree-of-freedom (multi-DoF) nonlinear media, an improved IHB method is proposed for solving nonlinear wave problems with known frequencies and unknown wave vectors when the external excitation is determined. In the method, the original wave equation is transformed into a delay differential equation (DDE) with any degree of freedom and any order of basis function, and the analytical formula of Jacobian matrix is constructed. The fast Fourier transform (FFT) is used to replace the numerical integration, and the minimum expansion order is determined by convergence analysis, so as to obtain the steady-state solution efficiently. Two classical models, fractional nonlinearity (lattice model with Hertzian contact law) and cubic nonlinearity (lattice model with cubic stiffness spring), are used as examples to analyze the bandgap properties of the lattice structure. The results show that when the system is in strong nonlinearity, the converged steady-state solutions are obtained only for the higher order basis functions, and the computational efficiency is increased by more than 220 times. The convergence of the steady-state solution when the wave vector is calculated at known frequencies is higher than that of the case where the wave vector is known. The nonlinear intensity can regulate the frequency-band and width of the bandgap, and the stronger the nonlinear intensity, the larger the regulation range of the bandgap.

Key words: strong nonlinearity, incremental harmonic balance, wave problem, delay differential equation, bandgap property