Nonlinear Dynamics of Continuously Tunable Planetary Gear Metamaterial

doi:10.3969/j.issn.1004-132X.2025.11.005

Abstract

Abstract:

A continuous steady-state metamaterial with tunable mechanics was designed based on planetary gears. The excellent tunable mechanics properties of metamaterials were studied， which showed the designed metamaterials had a wide range of programmable stiffness and a significantly tunning band gap. The configuration relationship between tunability and structural parameters was analyzed. Then， a rigid-flexible coupling nonlinear dynamics model of planetary gear metamaterials was established and the dynamics characteristics of planetary gear metamaterials were explored by numerical simulation in the processes of dynamic tunning. Moreover， the influences of the system dynamic parameters on the bifurcation characteristics and stability were explored under coupling excitations， and the distribution characteristics of the system vibration response with the coupling parameter plane were disclosed. Finally， an improved non-iterative cell mapping method was used to analyze the dependence of the vibration response of the system on the initial conditions. The results show that the planetary gear metamaterials have a wide range of stiffness tunability and significantly variable band gap interval. Under the influences of internal and external excitations， the system exhibits various dynamics characteristics. The dynamics characteristics of flexible construction are mainly affected by structural parameters， and the dynamics characteristics of rigid construction are affected by multiple sets of coupling parameters and initial conditions.

Key words: mechanics metamaterial, planetary gear, nonlinear dynamics, attractive domain

摘要：

设计了一种基于行星齿轮的连续稳态可调力学超材料。研究了超材料优异的力学可调特性，表明所设计的超材料具有广泛的可编程刚度和显著可调的带隙，分析了可调性与结构参数的配置关系；建立行星齿轮超材料的刚柔耦合非线性动力学模型，通过数值模拟研究行星齿轮超材料在动态调节过程中的动力学特性，探讨耦合激励下系统动力学参数对其分岔特性和稳定性的影响，揭示系统振动响应在耦合参数平面上的分布特征；最后使用改进的非迭代胞映射方法来分析系统的振动响应对初始条件的依赖性。研究结果表明，行星齿轮超材料具有广泛的刚度可调谐性和显著可变的带隙间隔，在内外激励作用下系统表现出丰富的动力学特性，柔性结构的动力特性主要受结构参数的影响，刚性齿轮的动力学特性受多组耦合参数和初始条件的影响。

关键词: 力学超材料, 行星齿轮, 非线性动力学, 吸引域

CLC Number:

TB34

Shuai MO, Xuan HUANG, Wenbin LIU, Wei ZHANG. Nonlinear Dynamics of Continuously Tunable Planetary Gear Metamaterial[J]. China Mechanical Engineering, 2025, 36(11): 2509-2516.

莫帅, 黄轩, 刘文斌, 张伟. 连续可调行星齿轮超材料非线性动力学[J]. 中国机械工程, 2025, 36(11): 2509-2516.

Figures/Tables 22

Fig.1 Planetary gear metamaterial and its adjustable mechanical properties

Tab.1 Comparison of natural frequencies of flexible ring gear

方法	第7阶	第8阶	第9阶	第10阶
所提方法	4972.3	13 881	22 442	27 717
ANSYS	4852.1	13 755	21 186	27 247

Fig.2 Nonlinear dynamic model of planetary gear metamaterials

Tab.2 Basic parameters of planetary gear metamaterials

参数	行星轮	太阳轮	齿圈
模数/mm	0.5	0.5	0.5
齿数	23	34	80
齿宽/mm	3	3	3

Fig.3 Vibration displacement of ring gear under different external excitations

Fig.4 Influence of structural parameters on the vibration of ring gear

Fig.5 Influence of external excitation on gear vibration

Fig.6 Bifurcation diagram of gear vibration with external excitation

Fig.7 Evolution process of vibration phase trajectory of planet gear with torsional damping

Fig.8 Bifurcation diagram of planet gear vibration with torsional damping

Fig.9 LLE of planet gear vibration with torsional damping

Fig.10 System vibration time domain and phase trajectory

Fig.11 Bifurcation diagram of planet gear vibration with support damping

Fig.12 LLE of planet gear vibration with support damping

Fig.13 Global bifurcation of system vibration with torsional damping and support damping

Fig.14 Vibration distribution characteristics in coupling parameters plane

Fig.15 Phase trajectory in coupling parameter plane

Fig.16 Global bifurcation characteristics of the system withtorsional damping

Fig.17 Evolution of attraction basin and response

Fig.18 Planetary gear vibration test bench

Fig.19 Vibration acceleration time domain signal on planet gear

Fig.20 Vibration acceleration time domain signal on sun gear

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