机械多体系统碰撞动力学的对称性和守恒量研究
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摘要
为给复杂机械多体系统碰撞动力学问题的定量和定性分析提供一个强有力新工具,该文将现代分析力学中的对称性理论引入到机械多体外碰撞动力学研究中.首先,基于冲量动量法推导系统碰撞动力学的Euler-Lagrange方程;其次,引进群分析理论,根据不变性原则给出系统存在Noether对称性与Lie对称性的各自条件方程以及得到相应守恒量的形式,为动力学方程的解析积分理论提供了有效途径.最后以一平面开环两连杆机构的碰撞力学为例进行实际分析运用.研究表明,借助对称性和守恒量可以得到机械多体系统动力学更深层次的力学规律和运动特性,可为系统更精确的动态优化设计和先进控制奠定理论基础.
To provide a powerful new tool for quantitative and qualitative analysis of collision dynamics in complex mechanical multibody systems,the symmetry theory in modern analytical mechanics was introduced into the study of mechanical multibody external collision dynamics. Firstly,the EulerLagrange equation of collision dynamics was derived based on the momentum method; secondly,the group theory was introduced,then,according to the invariance principle,the condition equations for the Noether symmetry and the Lie symmetry were obtained and the corresponding conserved quantity form was got,which made possible an effective approach to the analytic integral theory for dynamic equations.Finally,the collision dynamics of a planar open-loop 2-connecting-rod mechanism was taken as an example for application and analysis. Research shows that deeper mechanical laws and motion characteristics of mechanical multibody system collision dynamics can be obtained by means of symmetries and conserved quantities,and the results also lay a theoretical foundation for more precise dynamic optimal design and advanced control.
To provide a powerful new tool for quantitative and qualitative analysis of collision dynamics in complex mechanical multibody systems,the symmetry theory in modern analytical mechanics was introduced into the study of mechanical multibody external collision dynamics. Firstly,the EulerLagrange equation of collision dynamics was derived based on the momentum method; secondly,the group theory was introduced,then,according to the invariance principle,the condition equations for the Noether symmetry and the Lie symmetry were obtained and the corresponding conserved quantity form was got,which made possible an effective approach to the analytic integral theory for dynamic equations.Finally,the collision dynamics of a planar open-loop 2-connecting-rod mechanism was taken as an example for application and analysis. Research shows that deeper mechanical laws and motion characteristics of mechanical multibody system collision dynamics can be obtained by means of symmetries and conserved quantities,and the results also lay a theoretical foundation for more precise dynamic optimal design and advanced control.
引文
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[2]尤超蓝,洪嘉振.空间交会对接过程的动力学模型与仿真[J].动力学与控制学报,2004,2(2):23-28.(YO U C haolan,HO N G Jiazhen. D ynamic model and simulation of space rendezvous and docking procedure[J]. J ournal of Dynamics and Control,2004,2(2):23-28.(in C hinese))
[3]刘锦阳,洪嘉振.卫星太阳能帆板的撞击问题[J].宇航学报,2000,21(3):34-38.(LIU Jingyang,HO N G Jiazhen. C ontact-impact of satellite’s plates[J]. J ournal of Astronautics,2000,21(3):34-38.(in C hinese))
[4]邹元杰,韩增尧,白照广,等.航天器柔性多体结构锁定撞击动力学分析与试验验证[J].强度与环境,2011,38(1):42-51.(ZHOU Yuanjie,HAN Zengyao,BAI Zhaoguang,et al. Impact analysis and test verification of flexible multi-body spacecraft structures in the latching process[J].Structure&Environment Engineering,2011,38(1):42-51.(in C hinese))
[5]杨晓谦.运输过程中堆码包装件碰撞分析[D].硕士学位论文.无锡:江南大学,2011.(YANG X iaoqian. T he transport process of stacking package collision analysis[D]. M aster T hesis. W uxi:Jiangnan U niversity,2011.(in C hinese))
[6]陈钢,贾庆轩,孙汉旭,等.空间机器人目标捕获过程中碰撞运动分析[J].机器人,2010,3(1):432-438.(CHEN Gang,JIA Qingxuan,SUN Hanxu,et al. Analysis on impact motion of space robot in the object capturing process[J]. Robot,2010,3(1):432-438.(in C hinese))
[7]华卫江,章定国.柔性机器人系统碰撞动力学建模[J].机械工程学报,2008,43(12):222-228.(HUA W eijiang,ZHANG Dingguo. M odeling of impact dynamics of flexible robots[J]. Journal of M echanical Engineering,2008,43(12):222-228.(in C hinese))
[8]解江,李翰,周书婷,等.爆炸冲击载荷下航空铝合金平板动态响应数值分析方法[J].应用数学和力学,2017,38(4):410-420.(XIE Jiang,LI Han,ZHOU Shuting,et al. A numerical method for dynamic responses of aviation aluminum alloy plates under blast loads[J]. Applied Mathematics and M echanics,2017,38(4):410-420.(in C hinese))
[9]秦于越,邓子辰,胡伟鹏.偏心冲击荷载作用下薄圆板动力学响应的保结构分析[J].应用数学和力学,2014,35(8):410-420.(QIN Yuyue,DENG Zichen,HU W eipeng. Dynamic analysis of circular thin plates under eccentric impact load w ith the structure-preserving method[J]. Applied M athematics and M echanics,2014,35(8):410-420.(in C hinese))
[10]ST RONGE W J. Impact Mechanics[M]. Cambridge:Cambridge University Press,2000.
[11]SEIFRIED R,SCHIEHLEN W,EBERHARD P. Numerical and experimental evaluation of the coefficient of restitution for repeated impacts[J]. International J ournal of Impact Engineering,2005,32(1):508-524.
[12]AM BROSIO J,POM BO J,RAUT ER F,et al. A memory based communication in the co-simulation of multibody and finite element codes for pantograph-catenary interaction simulation[J]. Multibody Dynamics,2009,12:231-252.
[13]董富祥,洪嘉振.多体系统动力学碰撞问题研究综述[J].力学进展,2009,39(3):352-359.(DONG Fuxiang,HONG Jiazhen. Review of impact problem for dynamics of multibody system[J]. Progress in Mechanics,2009,39(3):352-359.(in Chinese))
[14]LEONE R,GOURIEUX T. Classical Noether’s theory with application to the linearly damped particle[J]. European J ournal of Physics,2015,36(6):065022.
[15]BALSEIRO P. Hamiltonization of solids of revolution through reduction[J]. Journal of Nonlinear Science,2017,27(6):2001-2035.
[16]SINGLA K,GUPT A R K. Conservation laws for certain time fractional nonlinear systems of partial differential equations[J]. Communications in Nonlinear Science and Numerical Simulation,2017,53:10-21.
[17]章定国.多刚体系统的外碰撞动力学方程[J].应用数学和力学,1997,18(6):551-555.(ZHANG Dingguo. Exterior collision dynamic equations of multi rigid body system[J]. Applied M athematics and M echanics,1997,18(6):551-555.(in C hinese))