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

doi:10.3901/JME.2024.17.167

机械工程学报 ›› 2024, Vol. 60 ›› Issue (17): 167-178.doi: 10.3901/JME.2024.17.167

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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

王鸿宇¹, 王雪峰², 赵剑¹, 张健¹, 黄毓³

1. 大连理工大学工业装备结构分析国家重点实验室大连 116024;
2. 北京大学先进制造与机器人系北京 100871;
3. 大连海洋大学海洋与土木工程学院大连 116023

收稿日期:2023-09-02 修回日期:2024-05-04 出版日期:2024-09-05 发布日期:2024-10-21
作者简介:王鸿宇，男，1992出生，博士研究生。主要研究方向为非线性动力学、智能材料和超材料、振动控制与衰减。E-mail:wanghy19920102@163.com
王雪峰，男，1989年出生，博士，研究员。主要研究方向为微创手术机器人设计、人机交互及远程控制、基于全息影像的机器人医疗与康复、一般非线性动力学系统稳态分析与控制。E-mail:wang_xf@pku.edu.cn
赵剑(通信作者)，男，1980年出生，博士，教授，博士研究生导师。主要研究方向为非线性传感器、柔性力学、非线性动力学、振动控制与衰减。E-mail:jzhao@dlut.edu.cn
张健，男，1990年出生，博士研究生。主要研究方向为智能材料驱动控制与应用。E-mail:261141338@qq.com
黄毓，女，1984年出生，博士，副教授。主要研究方向为拓扑优化、结构分析、带隙材料和应用力学。E-mail:35325942@qq.com
基金资助:
国家重点研发计划(2022YFB3203600)和国家自然科学基金(U1930206)资助项目。

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

WANG Hongyu¹, WANG Xuefeng², ZHAO Jian¹, ZHANG Jian¹, HUANG Yu³

1. State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024;
2. Department of Advanced Manufacturing and Robotics, Peking University, Beijing 100871;
3. College of Ocean and Civil Engineering, Dalian Ocean University, Dalian 116023

Received:2023-09-02 Revised:2024-05-04 Online:2024-09-05 Published:2024-10-21

摘要/Abstract

摘要： 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。

关键词: 强非线性, 增量谐波平衡法, 波动问题, 时滞微分方程, 带隙性质

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

中图分类号:

O322

王鸿宇, 王雪峰, 赵剑, 张健, 黄毓. 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[J]. 机械工程学报, 2024, 60(17): 167-178.

WANG Hongyu, WANG Xuefeng, ZHAO Jian, ZHANG Jian, HUANG Yu. Research on Algorithm of Strong Nonlinear One-dimensional Multi-degree-of-freedom Lattice Wave Problem Based on Incremental Harmonic Balance Method[J]. Journal of Mechanical Engineering, 2024, 60(17): 167-178.

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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

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