基于增量谐波平衡法的一维强非线性多自由度波动问题算法研究

Research on Algorithm of Strong Nonlinear One-dimensional Multi-degree-of-freedom Lattice Wave Problem Based on Incremental Harmonic Balance Method

针对采用高阶基函数的增量谐波平衡法(Incremental harmonic balance,IHB)在分析多自由度非线性介质的带隙性质时存在计算量大且不易收敛的问题,提出一种改进的IHB方法以高效求解外部激励确定情况下频率已知而波矢未知的非线性波动问题。该方法将原始波动方程转化为具有任意自由度和任意阶数基函数的时滞微分方程(Delay differential equation,DDE),并构造Jacobian矩阵的解析式,利用快速傅里叶变换(Fast Fourier transform,FFT)代替数值积分,并通过收敛性分析确定最小展开阶数及稳态解。以分数非线性(基于赫兹接触定律的晶格)和立方非线性(基于立方弹簧的晶格)模型为例,分析晶格结构的带隙性质。结果表明,当系统处于强非线性时,采用高阶基函数才可获得收敛的稳态解,且计算效率提高220余倍。已知频率计算波矢时的稳态解的收敛性高于已知波矢的情况。非线性强度可以调控带隙的频段和宽度,且非线性越强,带隙的调控范围越大。

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.