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参外联合激励下非线性Zener系统的减振机理研究

邢景点 李向红 申永军

邢景点, 李向红, 申永军. 参外联合激励下非线性Zener系统的减振机理研究. 力学学报, 2023, 55(10): 2269-2280 doi: 10.6052/0459-1879-23-294
引用本文: 邢景点, 李向红, 申永军. 参外联合激励下非线性Zener系统的减振机理研究. 力学学报, 2023, 55(10): 2269-2280 doi: 10.6052/0459-1879-23-294
Xing Jingdian, Li Xianghong, Shen Yongjun. Vibration reduction mechanism of nonlinear zener system under combined parametric and external excitations. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(10): 2269-2280 doi: 10.6052/0459-1879-23-294
Citation: Xing Jingdian, Li Xianghong, Shen Yongjun. Vibration reduction mechanism of nonlinear zener system under combined parametric and external excitations. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(10): 2269-2280 doi: 10.6052/0459-1879-23-294

参外联合激励下非线性Zener系统的减振机理研究

doi: 10.6052/0459-1879-23-294
基金项目: 国家自然科学基金(12172233, U1934201和11672191)资助项目
详细信息
    通讯作者:

    李向红, 教授, 主要研究方向为动力系统及其工程应用、机械系统动力学与控制. Email: lxhll601@163.com

  • 中图分类号: O323, O328

VIBRATION REDUCTION MECHANISM OF NONLINEAR ZENER SYSTEM UNDER COMBINED PARAMETRIC AND EXTERNAL EXCITATIONS

  • 摘要: 旨在揭示参外联合激励下不同尺度非线性Zener系统的减振机理. 以Duffing系统为主系统, 引入周期变化的低频参数激励和外激励, 通过耦合黏弹性元件, 系统变为1.5自由度非线性Zener系统, 经过对比系统变化前后时间历程图、相图, 发现耦合黏弹性元件后, 系统由单一激发态的大幅高频振动转变为激发态和沉寂态共存的簇发振动, 且振动幅值大幅降低, 减振效果明显. 然后分析自治系统的稳定性和分岔情况, 利用包络快慢分析法, 将参数激励项定义为慢变参数, 基于外激励在激励幅值变化范围内存在最值思想, 分析了广义自治系统的稳定性、破缺分岔与非自治系统振动行为的密切关系. 结果发现, 自治系统对非自治系统具有明显的调节作用, 具体表现为耦合黏弹性元件后自治系统平衡点稳定性增强, 平衡点类型由中心变为稳定焦点, 平衡线对系统轨线的吸引力增强, 同时多条稳定平衡线限制了非自治系统的振动区域, 这些因素是减振的根本原因. 另外, 基于双参数分岔分析, 发现通过调节参数可以控制系统破缺分岔的发生, 进而提高系统减振性能.

     

  • 图  1  非线性Zener模型

    Figure  1.  Nonlinear Zener model

    图  2  系统(5)振动响应

    Figure  2.  Vibration response of system (5)

    图  3  系统(6)振动响应

    Figure  3.  Vibration response of system (6)

    图  4  系统(7)关于$F$的分岔图

    Figure  4.  The bifurcation diagram of system (7) with respect to $F$

    图  5  系统(8)关于$F$的分岔图

    Figure  5.  The bifurcation diagram of system (8) with respect to$F$

    图  6  系统(5)转换相图与分岔图叠加

    Figure  6.  The overlap of transformed phase portrait and bifurcation diagram of system (5)

    图  7  系统(6)时间历程图放大图

    Figure  7.  Enlargement of time history diagram of system (6)

    图  8  簇发现象产生机理

    Figure  8.  Generation mechanism of bursting phenomenon

    图  9  自治系统双参数分岔曲面及在$k = 3$时截面图

    Figure  9.  Two-parameter bifurcation surfaces of autonomous systems and cross section at $k = 3$

    图  10  $k = 3$时系统(5)与系统(6)时间历程图

    Figure  10.  Time history diagram of system (5) and system (6) for $k = 3$

    图  11  $k = 3$时系统(6)转换相图与平衡线叠加

    Figure  11.  The overlap of transformed phase portrait and bifurcation diagram of system (6) for $k = 3$

    图  12  $k = 4$时系统(5)与系统(6)时间历程图

    Figure  12.  Time history diagram of system (5) and system (6) for $k = 3$

    图  13  $k = 10$时系统(5)与系统(6)时间历程图

    Figure  13.  Time history diagram of system (5) and system (6) for $k = 3$

    图  14  自治系统双参数分岔曲面及在$\alpha = 40.2$时截面图

    Figure  14.  Two-parameter bifurcation surfaces of autonomous systems and cross section at$\alpha = 40.2$

    图  15  $\alpha = 40.2$时系统(5)与系统(6)时间历程图

    Figure  15.  Time history diagram of system (5) and system (6) for $\alpha = 40.2$

    图  16  $\alpha = 40.2$时系统(6)转换相图与平衡线叠加

    Figure  16.  The overlap of transformed phase portrait and bifurcation diagram of system (6) for $\alpha = 40.2$

    图  17  $\alpha = 50$时系统(5)与系统(6)时间历程图

    Figure  17.  Time history diagram of system (5) and system (6) for $\alpha = 40.2$

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