偏心旋转环状周期结构参激振动分析

魏振航; 王世宇; 王一凡

doi:10.13224/j.cnki.jasp.20220332

偏心旋转环状周期结构参激振动分析

doi: 10.13224/j.cnki.jasp.20220332

魏振航^1,,
王世宇^{1, 2, 3,*, ,},
王一凡¹

1.
天津大学机械工程学院，天津 300350
2.
天津大学机构理论与装备设计教育部重点实验室，天津 300350
3.
天津大学天津市非线性动力学与控制重点实验室，天津 300350

基金项目: 国家重点研发计划（2018YFB2001300）；国家自然科学基金（52175109，51721003）

详细信息

作者简介:
魏振航（2000-），男，博士生，主要从事机械系统动力学研究

通讯作者:
王世宇（1974-），男，教授，博士，主要从事旋转机械的动力学对称性研究。E-mail：wangshiyu@tju.edu.cn

中图分类号: V214.4
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出版历程
- 收稿日期: 2022-05-13
- 网络出版日期: 2024-06-24

Parametric vibration analyses of eccentric rotational ring-shaped periodic structures

WEI Zhenhang^1
,,
WANG Shiyu^{1, 2, 3,*
, ,},
WANG Yifan¹

1.
School of Mechanical Engineering，Tianjin University，Tianjin 300350，China
2.
Key Laboratory of Mechanism Theory and Equipment Design，Ministry of Education，Tianjin University，Tianjin 300350，China
3.
Tianjin Key Laboratory of Nonlinear Dynamics and Control，Tianjin University，Tianjin 300350，China

摘要

摘要:
针对偏心旋转环状周期结构的参激振动问题，利用Hamilton原理建立了含时变激励的动力学模型，分析了不同附加支撑的拓扑结构及其参数组合对固有频率分裂的影响，应用Floquét理论预测了不稳定域，揭示了固有频率分裂与稳定性的关系，还分析了分组支撑拓扑结构的振动波数、组内夹角及节径夹角对稳定性的影响，并据此提出一种改善稳定性的方法。结果表明：附加支撑的拓扑结构与波数满足一定条件时，固有频率发生分裂，参激振动被激起，参激不稳定域主要出现在固有频率及其线性组合处；参激不稳定域随组内夹角呈现周期性变化，节径夹角对各不稳定域的影响不同。
- 偏心旋转环状周期结构 /
- 旋转支撑 /
- 参激振动 /
- 固有频率分裂 /
- 参激稳定性
Abstract:
A dynamic model with time-varying excitation of eccentric rotating structures was established based on Hamilton principle to solve the parametric vibration problem of eccentric rotational ring-shaped periodic structures. The influences of topological structures with different supports and parameter combinations on natural frequency splitting were analyzed. Floquét theory was used to calculate the instability regions for different parameters, and the relationship between the natural frequency splitting and the instability regions was revealed. In addition, the influences of wavenumbers, intergroup angle, and the angle between the radius and rotary supports of grouping topology on the stability were analyzed, based on which a method for improving stability was proposed. The results showed that natural frequencies splitting and the parameter excitation arose when the topology of the rotary supports and wavenumbers met some specific relationships. The parametric instability mainly appeared at the natural frequency and their linear combinations. The instability regions changed periodically with the intergroup angle, and the angle between the radius and rotary supports had different influences on each instability region.
- eccentric rotational ring-shaped periodic structures /
- rotational supports /
- parametric vibration /
- natural frequency splitting /
- parametric stability

HTML

图 1 偏心旋转环状周期结构几何模型

Figure 1. Geometric model of an eccentric rotating ring-shaped periodic structure

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图 2 离心力产生的内力分布

Figure 2. Distributions of internal forces produced by centrifugal forces

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图 3 固有频率随圆环自转转速变化规律

Figure 3. Natural frequencies versus rotation speed of the ring

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图 4 参激不稳定域

Figure 4. Parametric instability regions

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图 5 不稳定域验证点

Figure 5. Verification points of the instability regions

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图 6 验证点的时域动态响应

Figure 6. Time-domain dynamic responses at the verification points

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图 7 不同波数下的稳定响应

Figure 7. Stable responses for different wavenumbers

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图 8 不同波数下的不稳定响应

Figure 8. Unstable responses for different wavenumbers

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图 9 组内夹角对参数不稳定的影响

Figure 9. Influences of intergroup angle on instability regions

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图 10 节径夹角对参数不稳定的影响

Figure 10. Influences of angle between the radius and rotary supports on parametric instability

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表 1 不同波数和支撑参数下矩阵元素取值

Table 1. Values of matrix elements for different wavenumbers and support parameters

$2n{\text{/}}{N_1} $	n，N₂，φ	${A'_8}$	${A'_9}$
不为整数		0	0
为整数	$\sin n{N_2}\varphi = 0$, $\sin n\varphi \ne 0$	0	0
	$\sin n\varphi = 0$	${N_1}{N_2}\cos 2n{\varOmega _{\text{r}}}t$	${N_1}{N_2}\sin 2n{\varOmega _{\text{r}}}t$
	$\sin n{N_2}\varphi \ne 0$, $\sin n\varphi \ne 0$	$ \dfrac{{{N_1}\sin {N_2}n\varphi \cos \left[ {2n{\varOmega _{\text{r}}}t + （ {{N_2} - 1} ） n\varphi } \right]}}{{\sin n\varphi }} $	$ \dfrac{{{N_1}\sin {N_2}n\varphi \sin \left[ {2n{\varOmega _{\text{r}}}t + （ {{N_2} - 1} ） n\varphi } \right]}}{{\sin n\varphi }} $

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表 2 偏心环状周期结构基本参数

Table 2. Parameters of an eccentric ring-shaped periodic structure

参数	取值
中性圆半径R/10⁻² m	3.00
径向厚度h/10⁻³ m	2.00
轴向厚度b/10⁻³ m	5.00
弹性模量E/10¹¹ Pa	2.06
密度ρ/10³ （kg/m³）	7.85
偏心率η	1/3
节径夹角β	π/8
刚度k₀	0.5

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表 3 固有频率分裂规律

Table 3. Natural frequency splitting rules

支撑刚度拓扑结构	参数组合	固有频率
均布	$2n{\text{/}}N $不为整数	重合
均布	$2n{\text{/}}N $为整数	分裂
分组	$2n{\text{/}}{N_1} $不为整数	重合
	$2n{\text{/}}{N_1} $为整数, $\sin n\varphi \ne 0$, $\sin n{N_2}\varphi = 0$	重合
	$2n{\text{/}}{N_1} $为整数, $\sin n\varphi \ne 0$, $\sin n{N_2}\varphi \ne 0$	分裂
	$2n{\text{/}}{N_1} $为整数, $\sin n\varphi = 0$	分裂

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