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

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

邢景点^*,
李向红^{*, †, ,},
申永军^{†, **}

*.
石家庄铁道大学数理系, 石家庄 050043
†.
石家庄铁道大学机械工程学院, 石家庄 050043
**.
石家庄铁道大学省部共建交通工程结构力学行为与系统安全国家重点实验室, 石家庄 050043

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
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出版历程
- 网络出版日期: 2023-09-05

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

Xing Jingdian^*,
Li Xianghong^{*, †
, ,},
Shen Yongjun^{†, **}

*.
Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China
†.
School of Mechanical Engineering, Shijiazhuang Tiedao University, Shijiazhuang 050043, China
**.
State Key Laboratory of Mechanical Behavior and System Safety of Traffic Engineering Structures, Shijiazhuang Tiedao University, Shijiazhuang 050043, China

摘要

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.
- Zener系统 /
- 参外联合激励 /
- 簇发振动 /
- 包络快慢分析法
Abstract: The aim is to reveal the vibration reduction mechanism of nonlinear Zener systems with different scales under the combined parametric and external excitation. With the Duffing system as the main system, low-frequency parametric excitation and external excitation are introduced, and the system is changed into a 1.5-degree-of-freedom nonlinear Zener system by coupling the viscoelastic element. After comparing the time history diagram and phase diagram before and after the change of the system, it is found that the system changes from a single large-amplitude vibration of the excited state to the bursting vibration of the excited state and the silent state, the vibration amplitude is greatly reduced, and the vibration reduction effect is obvious. Then analyze the stability and bifurcation of autonomous systems. The stability of the generalized autonomous system, the close relationship between the imperfect bifurcation and the vibration behavior of the non-autonomous system are analyzed based on the idea that there is a maximum value of the external excitation in the range of the excitation amplitude change by using the method of envelope fast and slow analysis, which defines the parameter excitation term as a slow-varying parameter. It is found that the autonomous system has an obvious regulating effect on the non-autonomous system, which is manifested in the enhanced stability of the equilibrium point of the autonomous system after coupling viscoelastic elements, the type of equilibrium point changes from center to stable focus, the enhanced attraction of the equilibrium line to the system track line, and at the same time, multiple stabilized equilibrium lines limit the vibration region of the non-autonomous system, which are the fundamental reasons for vibration reduction. In addition, based on the analysis of dual parameter bifurcation, it was found that adjusting parameters can control the occurrence of the system imperfect bifurcation, thereby improving the system's vibration reduction performance.
- Zener system /
- combined parametric and external excitations /
- bursting vibrations /
- enveloping slow-fast analysis method

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图 1 非线性Zener模型

Figure 1. Nonlinear Zener model

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图 2 系统(5)振动响应

Figure 2. Vibration response of system (5)

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图 3 系统(6)振动响应

Figure 3. Vibration response of system (6)

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图 4 系统(7)关于$F$的分岔图

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

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图 5 系统(8)关于$F$的分岔图

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

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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)

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图 7 系统(6)时间历程图放大图

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

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图 8 簇发现象产生机理

Figure 8. Generation mechanism of bursting phenomenon

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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 $k = 3$

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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 $k = 3$

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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 $k = 3$

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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 $k = 3$

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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 $k = 3$

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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$\alpha = 40.2$

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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 $\alpha = 40.2$

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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 $\alpha = 40.2$

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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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