﻿ 基于非线性模态的复杂系统动力学特性分析方法<sup>*</sup>
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1. 北京航空航天大学 中法工程师学院, 北京 100083;
2. 中国运载火箭技术研究院, 北京 100076;
3. 北京航空航天大学 能源与动力工程学院, 北京 100083

Dynamic characteristics analysis method of complex systems based on nonlinear mode
HUANG Xingrong1, LIU Jiuzhou2, LI Lin3
1. L'école Centrale de Pékin, Beihang University, Beijing 100083, China;
2. China Academy of Launch Vehicle Technology, Beijing 100076, China;
3. School of Energy and Power Engineering, Beihang University, Beijing 100083, China
Received: 2018-11-07; Accepted: 2019-02-02; Published online: 2019-03-18:11:35
Corresponding author. LI Lin, E-mail: feililin@buaa.edu.cn
Abstract: Nonlinear problem has always been an obstacle in dynamic analysis domain due to its complexity and high computational cost. This paper aims to present a simple, accurate and efficient nonlinear modal analysis method which can be applied to some common nonlinear systems, including Duffing system, dry friction, nonlinear material and so on. The kernel technique of this numerical method lies in establishing the variation law of the nonlinear modal parameters in function of modal amplitude:on the one hand, the steady-state problem is simplified into one-dimensional algebraic nonlinear problem, resulting in a significant simplification in numerical computation; on the other hand, the analysis of nonlinear modal parameters in function of modal amplitude provides a modal overview for the comprehension of system's nonlinear dynamic behavior. Following a description of the theoretical aspects and numerical simulation process of this method, it has been proven to be efficient in analyzing a Duffing system with real nonlinear mode, a dry friction system with complex nonlinear mode and a multi-physics system integrating piezoelectric material. A reduction method based on the proposed strategy is then presented, which is simple in mathematical form and efficient in numerical computations for analyzing large complex nonlinear systems. It has significant advantages in computational efficiency when combined with the mode synthesis method to solve the dynamic behavior of large complex nonlinear systems.
Keywords: nonlinear modal analysis     mode synthesis     Duffing     dry friction     nonlinear material     reduced model

20世纪60年代初，Rosenberg[6]在其非线性模态奠基之作中指出，系统在共振区的响应可用该阶非线性模态来近似；随后，为了解决非线性模态不具叠加性问题，Szemplinska-Stupnicka[7]提出了单模态共振理论，建立了自由振动状态下的系统非线性模态和响应幅值间的关系；Jézéquel[8]和Setio[9]等分别从理论和实验的角度对该理论进行了完善和发展，将线性模态分析方法推广到求解非线性系统响应。该理论已被众多学者认可和应用[10-11]，如郑兆昌[12]指出表征线性系统特性的主模态是沟通线性振动和非线性振动之间的桥梁。因此，主共振模态的计算与分析是大型复杂非线性结构动力学特性分析中至关重要的一步。

1 非线性系统的模态分析方法理论

1.1 非线性模态理论

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1.2 非线性模态参数

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1.3 基于单模态共振理论求解系统强迫响应

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 图 1 非线性模态分析求解稳态响应流程 Fig. 1 Flowchart of steady-state response solved by nonlinear modal analysis

2 在非线性实模态域的应用

2.1 多自由度杜芬系统的求解

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2.2 二自由度杜芬系统的计算与分析

 图 2 二自由度非线性参数模型 Fig. 2 2-DOF nonlinear parametric model

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 图 3 针对杜芬系统采用时域积分法和非线性模态分析方法计算m1的频响曲线 Fig. 3 Frequency response curves of m1computed by time-domain integration method and nonlinear modal analysis method for Duffing system
 图 4 针对杜芬系统采用时域积分法和非线性模态分析方法计算m2的频响曲线 Fig. 4 Frequency response curves of m2 computed by time-domain integration method and nonlinear modal analysis method for Duffing system

 图 5 杜芬系统的两阶非线性模态频率随模态幅值的变化 Fig. 5 Variation of nonlinear modal frequency with modal amplitude for the two modes of Duffing system
 图 6 杜芬系统的两阶模态参与系数随模态幅值的变化 Fig. 6 Variation of modal participation factor with modal amplitude for the two modes of Duffing system

3 在非线性复模态域的应用 3.1 具有干摩擦阻尼的振动系统

 图 7 双线性迟滞干摩擦模型示意图[11] Fig. 7 Schematic diagram of bilinear hysteresis dry friction model[11]

 图 8 基于Masing法则建立的干摩擦迟滞环[17] Fig. 8 Hysteresis loop of dry friction based on Masing's rule[17]
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Masing法则的表述为：若初始加载过程中摩擦力和相对位移满足如下函数关系：

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3.2 二自由度干摩擦系统的计算与分析

 图 9 针对干摩擦系统采用时域积分法和非线性模态分析方法计算m1的频响曲线 Fig. 9 Frequency response curves of m1computed by time-domain integration method and nonlinear modal analysis method for dry friction system
 图 10 针对干摩擦系统采用时域积分法和非线性模态分析方法计算m2的频响曲线 Fig. 10 Frequency response curves of m2computed by time-domain integration method and nonlinear modal analysis method for dry friction system

 图 11 不同正压力下m1的频响曲线 Fig. 11 Frequency response curves of m1 under different normal pressure
 图 12 不同正压力下m2的频响曲线 Fig. 12 Frequency response curves of m2 under different normal pressure

 图 13 第一阶共振频率附近，不同正压力下的非线性共振频率和干摩擦模态阻尼比随模态幅值的变化 Fig. 13 Variation of nonlinear resonance frequency and dry friction modal damping with modal amplitude under different normal pressure for the first order mode
 图 14 第二阶共振频率附近，不同正压力下的非线性共振频率和干摩擦模态阻尼比随模态幅值的变化 Fig. 14 Variation of nonlinear resonance frequency and dry friction modal damping with modal amplitude under different normal pressure for the second order mode
4 在多场耦合域的应用 4.1 非线性压电阻尼

 图 15 SSDNC电路单元示意图[5] Fig. 15 Schematic diagram of SSDNC circuit unit[5]

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4.2 非线性压电阻尼系统的计算与分析

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 图 16 谐波平衡法和非线性模态分析方法计算m1的频响曲线 Fig. 16 Frequency response curves of m1 computed by harmonic balance method and nonlinear modal analysis method
 图 17 谐波平衡法和非线性模态分析方法计算m2的频响曲线 Fig. 17 Frequency response curves of m2 computed by harmonic balance method and nonlinear modal analysis method

 图 18 第一阶共振频率附近，不同电容比下的非线性共振频率和模态阻尼比随模态幅值的变化 Fig. 18 Variation of nonlinear resonance frequency and modal damping with modal amplitude under different capacitance ratios for the first order mode
 图 19 第二阶共振频率附近，不同电容比下的非线性共振频率和模态阻尼随模态幅值的变化 Fig. 19 Variation of nonlinear resonance frequency and modal damping with modal amplitude under different capacitance ratios for the second order mode
5 大型复杂非线性系统的减缩分析

 图 20 由两个子结构和非线性连接面组成的系统简图 Fig. 20 Illustration of an assembled system with a nonlinear interface and two substructures

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6 结论

1) 本文基于非线性模态理论的研究进展，针对大型复杂非线性结构系统的分析问题提炼了一套基于单模态共振理论和非线性模态理论的计算方法。

2) 给出的杜芬系统和干摩擦系统的算例表明了本文方法对于求解强非线性系统的适用性——对于模态间弱耦合(主共振模态相对稀疏)的系统具有通用性；通过对具有非线性压电阻尼系统的分析算例进一步表明了本文方法对于求解非线性系统具有普适性。

3) 针对大型复杂结构系统，本文提出将上述论证的方法与模态综合法结合的思想，并给出了具体实施步骤，其特点在于：基于此方法、步骤不仅可以快速求得系统的稳态响应，还能得到一组表征系统非线性动力学特性的模态参数(如模态频率、模态阻尼和模态参与系数)，通过分析这些模态参数随系统模态幅值的变化规律，有助于理解系统的非线性动力学行为。

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#### 文章信息

HUANG Xingrong, LIU Jiuzhou, LI Lin

Dynamic characteristics analysis method of complex systems based on nonlinear mode

Journal of Beijing University of Aeronautics and Astronsutics, 2019, 45(7): 1337-1348
http://dx.doi.org/10.13700/j.bh.1001-5965.2018.0643