基于传递矩阵法的柔性微定位平台动态特性研究

徐伟胜; 许添旗; 马慧欣; 曹毅

doi:10.13700/j.bh.1001-5965.2022.0845

基于传递矩阵法的柔性微定位平台动态特性研究

doi: 10.13700/j.bh.1001-5965.2022.0845

徐伟胜¹,
许添旗¹,
马慧欣¹,
曹毅^{1, 2, ,}

1.
江南大学机械工程学院，无锡 214122
2.
江苏省食品先进制造装备技术重点实验室，无锡 214122

基金项目: 国家自然科学基金(51375209)；江苏省“六大人才高峰”计划(ZBZZ-012)；高等学校学科创新引智计划(B18027)；江苏省优秀科技创新团队基金(2019SK07)

详细信息

通讯作者:
E-mail：caoyi@jiangnan.edu.cn

中图分类号: V414；TH122
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- 被引次数: 0
出版历程
- 收稿日期: 2022-10-10
- 录用日期: 2022-12-02
- 网络出版日期: 2022-12-15
- 整期出版日期: 2024-11-30

Dynamic characteristics of flexible micro-positioning platforms based on transfer matrix method

XU Weisheng¹,
XU Tianqi¹,
MA Huixin¹,
CAO Yi^{1, 2
, ,}

1.
School of Mechanical Engineering，Jiangnan University，Wuxi 214122，China
2.
Jiangsu Key Laboratory of Advanced Food Manufacturing Equipment & Technology，Wuxi 214122，China

Funds: National Natural Science Foundation of China (51375209); The Six Talent Peaks Project in Jiangsu Province (ZBZZ-012); Program of Introducing Talents of Discipline to Universities (B18027); The Excellent Technology Innovation Team Fundation of Jiangsu Province (2019SK07)

More Information

Corresponding author: E-mail：caoyi@jiangnan.edu.cn

摘要

摘要:
随着精密微定位技术的发展，对柔性微定位平台的动态特性研究越发重要。传递矩阵法(TMM)作为分析多体系统动力学的一种有效方法，具有建模方便、计算精度高等优势。为此，基于传递矩阵法建立微定位平台的动力学模型，以分析其动态特性。介绍传递矩阵法的基本思想并建立柔性机构的传递方程，推导了其在任意激励下的瞬时动力响应；基于柔性铰链和柔性梁设计了一种XY柔性微定位平台并推导出其支链及平台整体的传递矩阵；基于振动理论推导了平台主要特征单元的传递矩阵；最后，为验证理论模型的有效性，分别基于传递矩阵法、有限元法(FEM)及等效质量法(EMM)开展了对该平台固有频率和瞬时动力响应的研究。结果表明：基于传递矩阵法程式化地构建传递矩阵即可分析平台动力学特性，无须建立烦琐的动力学方程；相较于等效质量法，传递矩阵法可计算平台的高阶固有频率；平台固有频率理论值同仿真值最大相对误差仅为2.4%，动力响应理论值与仿真值基本吻合，证明了基于传递矩阵法所建模型的有效性。
- 传递矩阵法 /
- 柔性机构 /
- 微定位平台 /
- 固有频率 /
- 瞬时动力响应
Abstract:
With the development of precision micro-positioning technology, the research on dynamic characteristics of flexible micro-positioning platforms is necessary. As an effective method to analyze the dynamics of multi-body systems, the transfer matrix method (TMM) has the advantages of convenient modeling and high calculation accuracy. Therefore, a dynamics model of the micro-positioning platform was established based on TMM to analyze its dynamic characteristics. The basic idea of TMM was introduced, and the transfer equation of flexible mechanism was established. Its instantaneous dynamic response under arbitrary excitation was deduced. An XY flexible micro-positioning platform based on the flexible hinge and flexible beam was designed, and the transfer matrix of its branch chain and the whole platform was derived. Based on vibration theory, the transfer matrix of the main characteristic elements of the platform was derived. Finally, in order to verify the validity of the theoretical model, the natural frequency and instantaneous dynamic response of the platform were studied based on the TMM, finite element method (FEM), and equivalent mass method (EMM), respectively. The results show that the transfer matrix can be programmatically constructed based on the TMM to analyze the dynamic characteristics of the platform without establishing complicated dynamics equations. Compared with EMM, TMM can calculate the higher-order natural frequency of the platform. The maximum relative error between theoretical and simulated natural frequencies of the platform is just 2.4%, and the theoretical and simulated dynamic responses basically agree with each other, which proves the validity of the model based on TMM.
- transfer matrix method /
- flexible mechanism /
- micro-positioning platform /
- natural frequency /
- instantaneous dynamic response

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图 1 并联式微定位平台示意

Figure 1. Schematic diagram of Parallel micro-positioning platform

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图 2 单元编号及状态矢量

Figure 2. Element number and state vector

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图 3 PPP型柔性支链示意

Figure 3. Diagram of PPP-type flexible branch chain

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图 4 二自由度柔性微定位平台

Figure 4. 2-DOF flexible micro-positioning platform

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图 5 驱动副及单元编号

Figure 5. Drive pair and element number

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图 6 被动副及单元编号

Figure 6. Passive pair and element number

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图 7 横向与纵向振动的柔性梁模型

Figure 7. Flexible beam model of lateral and longitudinal vibrations

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图 8 圆弧形铰链及简化模型

Figure 8. Simplified model of circular hinges

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图 9 具有N输入M输出的平面刚性单元

Figure 9. Rigid element with N input and M output

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图 10 柔性微定位平台的网格划分

Figure 10. Meshing of flexible micro-positioning platform

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图 11 柔性微定位平台1～6阶模态振型

Figure 11. Modes of flexible micro-positioning platform at 1～6 orders

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图 12 Ω=100 Hz时平台的动力响应

Figure 12. Dynamic response of platform at Ω = 100 Hz

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图 13 Ω=200 Hz时平台的动力响应

Figure 13. Dynamic response of platform at Ω = 200 Hz

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表 1 微定位平台结构参数

Table 1. Structure parameters of micro-positioning platform mm

参数	数值	参数	数值
l₁	15	u₁	2
l₂	26.8	u₂	4
l₃	24	u₃	2.5
s₁	21.2	t₁	0.3
s₂	15	t₂	1.2
s₃	5.7	r	0.7
s₄	27	h	5

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表 2 平台固有频率的理论值、仿真值及相对误差

Table 2. Theoretical and simulated natural frequencies and relative error

模态	有限元仿真值/Hz	传递矩阵法/Hz	等效质量法/Hz	相对误差/%
1阶	196.54	192.67	192.27	2.0
2阶	196.54	192.67	192.27	2.0
3阶	487.03	475.62		2.4
4阶	530.04
5阶	581.25
6阶	597.68
注：相对误差=｜(理论值−仿真值)/理论值｜×100%。

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