含摩擦滑移铰平面多刚体系统动力学的数值算法
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摘要
基于LuGre摩擦模型和线性互补问题(LCP)的数值算法,给出了具有双边约束含摩擦滑移铰平面多体系统动力学的数值算法.首先,根据滑移铰的特点,当间隙充分小时,将其视为双边约束,给出了滑移铰中滑道作用于滑块上的法向接触力的互补关系;LuGre摩擦模型能有效地描述机械系统中的黏滞与滑移运动,将该模型用于描述滑块与滑道间的摩擦力.其次,结合Baumgarte约束稳定化方法,应用第一类Lagrange方程,建立了该多体系统的动力学方程,给出了Lagrange乘子与滑移铰中作用于滑块上的法向接触力的关系式.然后,将滑块与滑道间多种接触状态的判断以及作用于滑块上的法向接触力的计算转换为线性互补问题的求解,并用常微分方程的数值算法求解该多体系统的动力学方程.最后,通过数值仿真算例揭示了滑移铰中滑块的黏滞与滑移现象,以及滑块在滑道内的多种接触状态;另外,在文中分别采用Coulomb干摩擦模型和LuGre摩擦模型,对算例中的某些工况进行了数值仿真,并且分别用本文方法得到的数值仿真结果与已有方法得到的数值仿真结果对比,表明了本文给出的方法的有效性.
A numerical method for the dynamics of the planar multi-rigid-body system with frictional translational joints is presented in this paper. The multibody system consists of the several rigid bodies which are linked with ideal revolute joints and imperfect translational joints. The frictional forces on the slider in the imperfect translational joint are modeled by the LuGre friction law which can effectively describe stick-slip motion in the mechanical system. The sizes of the clearances and the impacts between the guide and the slider in the translational joints can be neglected when the clearance sizes are very small, so the geometric constraints of the translational joints are treated as bilateral constraints. In this work,firstly, the complementarity conditions and formulations about the normal forces on the slider in the translational joint are given. Secondly, the dynamical equations of the multibody system are obtained by the Lagrange's equations of thefirst kind and the Baumgarte stabilization method for the constrained multibody systems in order to reduce the constraint drift in the numerical simulation of the multibody systems. Thirdly, the problems of determining contact situations of the slider in the translational joint and solving normal forces on the slider in all contact situations are formulated and solved as a linear complementarity problem(LCP). Finally, two numerical examples of the planar multi-rigid-body system with a frictional translational joint are given to illustrate their dynamical behaviors such as stick-slip motion and several contact situations of the slider in the translational joint. The LuGre friction model and the Coulomb friction model are used in two numerical examples to compare the dynamical behaviors of two mechanical systems. The numerical results obtained by our method are compared with that obtained by other method. The numerical results of the examples show the availability of the method presented in this paper.
A numerical method for the dynamics of the planar multi-rigid-body system with frictional translational joints is presented in this paper. The multibody system consists of the several rigid bodies which are linked with ideal revolute joints and imperfect translational joints. The frictional forces on the slider in the imperfect translational joint are modeled by the LuGre friction law which can effectively describe stick-slip motion in the mechanical system. The sizes of the clearances and the impacts between the guide and the slider in the translational joints can be neglected when the clearance sizes are very small, so the geometric constraints of the translational joints are treated as bilateral constraints. In this work,firstly, the complementarity conditions and formulations about the normal forces on the slider in the translational joint are given. Secondly, the dynamical equations of the multibody system are obtained by the Lagrange's equations of thefirst kind and the Baumgarte stabilization method for the constrained multibody systems in order to reduce the constraint drift in the numerical simulation of the multibody systems. Thirdly, the problems of determining contact situations of the slider in the translational joint and solving normal forces on the slider in all contact situations are formulated and solved as a linear complementarity problem(LCP). Finally, two numerical examples of the planar multi-rigid-body system with a frictional translational joint are given to illustrate their dynamical behaviors such as stick-slip motion and several contact situations of the slider in the translational joint. The LuGre friction model and the Coulomb friction model are used in two numerical examples to compare the dynamical behaviors of two mechanical systems. The numerical results obtained by our method are compared with that obtained by other method. The numerical results of the examples show the availability of the method presented in this paper.
引文
1刘丽兰,刘宏昭,吴子英等.机械系统中摩擦模型的研究进展.力学进展,2008,38(2):200-213(Liu Lilan,Liu Hongzhao,Wu Ziying,et al.An overview of friction models in mechanical systems.Advances in Mechanics,2008,38(2):200-213(in Chinese))
2丁千,翟红梅.机械系统摩擦动力学研究进展.力学进展,2013,43(1):112-131(Ding Qian,Zhai Hongmei.The advance in researches of friction dynamics in mechanics system.Advances in Mechanics,2013,43(1):112-131(in Chinese))
3 Marques F,Flores P,Pimenta Claro JC,et al.A survey and comparison of several friction force models for dynamic analysis of multibody mechanical systems.Nonlinear Dynamics,2016,86(3):1407-1443
4 Pennestr`?E,Rossi V,Salvini P,et al.Review and comparison of dry friction force models.Nonlinear Dynamics,2016,83(4):1785-1801
5 Piatkowski T.Dahl and LuGre dynamic friction models-The analysis of selected properties.Mechanism and Machine Theory,2014,73:91-100
6 Muvengei O,Kihiu J,Ikua B.Dynamic analysis of planar multibody systems with LuGre friction at differently located revolute clearance joints.Multibody System Dynamics,2012,28(4):369-393
7曹登庆,初世明,李郑发等.空间可展机构非光滑力学模型和动力学研究.力学学报,2013,45(1):4-14(Cao Dengqing,Chu Shiming,Li Zhengfa,et al.Study on the non-smooth mechanical model and dynamics for space deployable mechanism.Chinese Journal of Theoretical and Applied Mechanics,2013,45(1):4-14(in Chinese))
8王琪,庄方方,郭易圆等.非光滑多体系统动力学数值算法的研究进展.力学进展,2013,43(1):101-111(Wang Qi,Zhuang Fangfang,Guo Yiyuan,et al.Advances in the research on numerical methods for non-smooth dynamics of multibody systems.Advances in Mechanics,2013,43(1):101-111(in Chinese))
9王庚祥,刘宏昭.多体系统动力学中关节效应模型的研究进展.力学学报,2015,47(1):31-50(Wang Gengxiang,Liu Hongzhao.Research progress of joint e ects model in multibody system dynamics.Journal of Theoretical and Applied Mechanics,2015,47(1):31-50(in Chinese))
10田强,刘铖,李培等.多柔体系统动力学研究进展与挑战.动力学与控制学报,2017,15(5):385-405(Tian Qiang,Liu Cheng,Li Pei,et al.Advances and challenges in dynamics of fiexible multibody systems.Journal of Dynamics and Control,2017,15(5):385-405(in Chinese))
11 Tian Q,Flores P,Lankarani Hamid M.A comprehensive survey of the analytical,numerical and experimental methodologies for dynamics of multibody mechanical systems with clearance or imperfect joints.Mechanism and Machine Theory,2018,122:1-57
12赵振,刘才山,陈滨.Painlev’e疑难的理论分析和实验验证.力学学报,2013,45(1):37-44(Zhao Zhen,Liu Caishan,Chen Bin.Theoretical analysis and experimentnal verification for Painlev’e paradox.Chinese Journal of Theoretical and Applied Mechanics,2013,45(1):37-44(in Chinese)
13 Zhao Z,Liu CS.Contact constraints and dynamical equations in Lagrangian systems.Multibody System Dynamics,2016,38(1):77-99
14钱震杰,章定国,金诚谦.柔性多体系统含摩擦碰撞stick-slip过程动力学仿真.振动与冲击,2017,36(23):32-37(Qian Zhenjie,Zhang Dingguo,Jin Chengqian.Dynamic simulation for flexible multibody systems containing frictional impact and stick-slip processes.Journal of Vibration and Shock,2017,36(23):32-37(in Chinese))
15 Wang XJ,L¨u J.Modeling and simulation of dynamics of a planarmotion rigid body with friction and surface contact.International Journal of Modern Physics B,2017,31(16-19):1744021
16 Chen P,Liu JY,Hong JZ.An efficient formulation based on the Lagrangian method for contact-impact analysis of flexible multi-body system.Acta Mechanica Sinica,2016,32(2):326-334
17 Sun Y,Chen T,Wu C,et al.A comprehensive experimental setup for identification of friction model parameters.Mechanism and Machine Theory,2016,100:338-357
18富立,胡洪奎,富腾.多体系统接触碰撞问题的牛顿-欧拉线性互补方法.力学学报,2017,49(5):1115-1125(Fu Li,Hu Hongkui,Fu Teng.Contact-impact analysis in multi-body systems based on Newton-Euler lcp approach.Chinese Journal of Theoretical and Applied Mechanics,2017,49(5):1115-1125(in Chinese))
19薛冰寒,林皋,胡志强.摩擦接触问题的比例边界等几何B可微方程组方法.力学学报,2016,48(3):615-623(Xue Binghan,Lin Gao,Hu Zhiqiang,et al.Analysis of frictional contact problems by SBIGA-BDE method.Chinese Journal of Theoretical and Applied Mechanics,2016,48(3):615-623(in Chinese))
20孟光,周徐斌,苗军.航天重大工程中的力学问题.力学进展,2016,46:267-322(Meng Guang,Zhou Xubin,Miao Jun.Mechanical problems in momentous projects of aerospace engineering.Advances in Mechanics,2016,46:267-322(in Chinese))
21 Zhao Z,Liu CS,Chen T.Docking dynamics between two spacecrafts with APDSes.Multibody System Dynamics,2016,37(3):245-270
22张润森,王琪.浮放物体平面多刚体动力学建模与算法研究.力学学报,2017,49(6):1370-1379(Zhang Runsen,Wang Qi.Research on modeling and numerical method of free standing body on planar rigid multibody dynamics.Chinese Journal of Theoretical and Applied Mechanics,2017,49(6):1370-1379(in Chinese))
23于春宇,张丛发,张朋等.STACER伸展变形的一种计算方法.力学学报,2016,48(6):1398-1405(Yu Chunyu,Zhang Congfa,Zhang Peng,et al.A method for simulating stacer’s deployment deformation.Chinese Journal of Theoretical and Applied Mechanics,2016,48(6):1398-1405(in Chinese))
24 Qi F,Wang TS,Li JF.The elastic contact influences on passive walking gaits.Robotica,2011,29(5):787-796
25秦琅,谢建华.基于Melnikov方法的单自由度干摩擦振子的混沌分析.动力学与控制学报,2018,16(3):239-243(Qin Lang,Xie Jianhua.Chaos analysis of a single-degree-of-freedom dry friction oscillator based on melnikov method.Journal of Dynamics and Control,2018,16(3):239-243(in Chinese))
26 Zheng XD,Wang Q.LCP method for a planar passive walker based on an event-driven scheme.Acta Mechanica Sinica,2018,34(3):578-588
27 Fan XX,Walker Paul D,Wang Q.Modeling and simulation of longitudinal dynamics coupled with clutch engagement dynamics for ground vehicles.Multibody System Dynamics,2018,43(2):153-174
28李文杰,吕晓寅,苏晓雪.固定和浮放条件下文物展柜动力响应分析与实验研究.文物保护与考古科学,2017,29(3):38-44(Li Wenjie,L¨u Xiaoyin,Su Xiaoxue.Dynamic response analysis and experimental study on the fixed and floating cultural showcases.Sciences of Conservation and Archaeology,2017,29(3):38-44(in Chinese))
29朱琳,王琪,张润森.含摩擦阻尼器非光滑振动系统的建模与数值仿真.动力学与控制学报,2017,15(2):172-177(Zhu Lin,Wang Qi,Zhang Runsen.Modelling and simulation of the non-smooth vibration systems with friction dampers.Journal of Dynamics and Control,2017,15(2):172-177(in Chinese))
30 Flores P,mbrosio J.Kinematics and Dynamics of Multibody Systems with Imperfect Joints.Berlin:Springer,2008:47-100
31 Pennestrì E,Valentini PP,Vita L.Multibody dynamics simulation of planar linkages with Dahl friction.Multibody System Dynamics,2007,17(4):321-347
32 Flores P,Leine R,Glocker C.Modeling and analysis of planar rigid multibody systems with translational clearance joints based on the non-smooth dynamics approach.Multibody System Dynamics,2010,23(2):165-190
33 Zhuang FF,Wang Q.Modeling and simulation of the nonsmooth planar rigid multibody systems with frictional translational joints.Multibody System Dynamics,2013,29(4):403-423
34 Muvengei O,Kihiu J,Ikua B.Dynamic analysis of planar rigid-body mechanical systems with two-clearance revolute joints.Nonlinear Dynamics,2013,73(1-2):259-273
35 Wang Z,Tian Q,Hu HY,et al.Nonlinear dynamics and chaotic control of a flexible multibody system with uncertain joint clearance.Nonlinear Dynamics,2016,86(3):1571-1597
36 Flores P,Machado M,Seabra E,et al.A parametric study on the Baumgarte stabilization method for forward dynamics of constrained multibody systems.Journal of Computational and Nonlinear Dynamics,2011,6(1):011019-1-1-9-9
2丁千,翟红梅.机械系统摩擦动力学研究进展.力学进展,2013,43(1):112-131(Ding Qian,Zhai Hongmei.The advance in researches of friction dynamics in mechanics system.Advances in Mechanics,2013,43(1):112-131(in Chinese))
3 Marques F,Flores P,Pimenta Claro JC,et al.A survey and comparison of several friction force models for dynamic analysis of multibody mechanical systems.Nonlinear Dynamics,2016,86(3):1407-1443
4 Pennestr`?E,Rossi V,Salvini P,et al.Review and comparison of dry friction force models.Nonlinear Dynamics,2016,83(4):1785-1801
5 Piatkowski T.Dahl and LuGre dynamic friction models-The analysis of selected properties.Mechanism and Machine Theory,2014,73:91-100
6 Muvengei O,Kihiu J,Ikua B.Dynamic analysis of planar multibody systems with LuGre friction at differently located revolute clearance joints.Multibody System Dynamics,2012,28(4):369-393
7曹登庆,初世明,李郑发等.空间可展机构非光滑力学模型和动力学研究.力学学报,2013,45(1):4-14(Cao Dengqing,Chu Shiming,Li Zhengfa,et al.Study on the non-smooth mechanical model and dynamics for space deployable mechanism.Chinese Journal of Theoretical and Applied Mechanics,2013,45(1):4-14(in Chinese))
8王琪,庄方方,郭易圆等.非光滑多体系统动力学数值算法的研究进展.力学进展,2013,43(1):101-111(Wang Qi,Zhuang Fangfang,Guo Yiyuan,et al.Advances in the research on numerical methods for non-smooth dynamics of multibody systems.Advances in Mechanics,2013,43(1):101-111(in Chinese))
9王庚祥,刘宏昭.多体系统动力学中关节效应模型的研究进展.力学学报,2015,47(1):31-50(Wang Gengxiang,Liu Hongzhao.Research progress of joint e ects model in multibody system dynamics.Journal of Theoretical and Applied Mechanics,2015,47(1):31-50(in Chinese))
10田强,刘铖,李培等.多柔体系统动力学研究进展与挑战.动力学与控制学报,2017,15(5):385-405(Tian Qiang,Liu Cheng,Li Pei,et al.Advances and challenges in dynamics of fiexible multibody systems.Journal of Dynamics and Control,2017,15(5):385-405(in Chinese))
11 Tian Q,Flores P,Lankarani Hamid M.A comprehensive survey of the analytical,numerical and experimental methodologies for dynamics of multibody mechanical systems with clearance or imperfect joints.Mechanism and Machine Theory,2018,122:1-57
12赵振,刘才山,陈滨.Painlev’e疑难的理论分析和实验验证.力学学报,2013,45(1):37-44(Zhao Zhen,Liu Caishan,Chen Bin.Theoretical analysis and experimentnal verification for Painlev’e paradox.Chinese Journal of Theoretical and Applied Mechanics,2013,45(1):37-44(in Chinese)
13 Zhao Z,Liu CS.Contact constraints and dynamical equations in Lagrangian systems.Multibody System Dynamics,2016,38(1):77-99
14钱震杰,章定国,金诚谦.柔性多体系统含摩擦碰撞stick-slip过程动力学仿真.振动与冲击,2017,36(23):32-37(Qian Zhenjie,Zhang Dingguo,Jin Chengqian.Dynamic simulation for flexible multibody systems containing frictional impact and stick-slip processes.Journal of Vibration and Shock,2017,36(23):32-37(in Chinese))
15 Wang XJ,L¨u J.Modeling and simulation of dynamics of a planarmotion rigid body with friction and surface contact.International Journal of Modern Physics B,2017,31(16-19):1744021
16 Chen P,Liu JY,Hong JZ.An efficient formulation based on the Lagrangian method for contact-impact analysis of flexible multi-body system.Acta Mechanica Sinica,2016,32(2):326-334
17 Sun Y,Chen T,Wu C,et al.A comprehensive experimental setup for identification of friction model parameters.Mechanism and Machine Theory,2016,100:338-357
18富立,胡洪奎,富腾.多体系统接触碰撞问题的牛顿-欧拉线性互补方法.力学学报,2017,49(5):1115-1125(Fu Li,Hu Hongkui,Fu Teng.Contact-impact analysis in multi-body systems based on Newton-Euler lcp approach.Chinese Journal of Theoretical and Applied Mechanics,2017,49(5):1115-1125(in Chinese))
19薛冰寒,林皋,胡志强.摩擦接触问题的比例边界等几何B可微方程组方法.力学学报,2016,48(3):615-623(Xue Binghan,Lin Gao,Hu Zhiqiang,et al.Analysis of frictional contact problems by SBIGA-BDE method.Chinese Journal of Theoretical and Applied Mechanics,2016,48(3):615-623(in Chinese))
20孟光,周徐斌,苗军.航天重大工程中的力学问题.力学进展,2016,46:267-322(Meng Guang,Zhou Xubin,Miao Jun.Mechanical problems in momentous projects of aerospace engineering.Advances in Mechanics,2016,46:267-322(in Chinese))
21 Zhao Z,Liu CS,Chen T.Docking dynamics between two spacecrafts with APDSes.Multibody System Dynamics,2016,37(3):245-270
22张润森,王琪.浮放物体平面多刚体动力学建模与算法研究.力学学报,2017,49(6):1370-1379(Zhang Runsen,Wang Qi.Research on modeling and numerical method of free standing body on planar rigid multibody dynamics.Chinese Journal of Theoretical and Applied Mechanics,2017,49(6):1370-1379(in Chinese))
23于春宇,张丛发,张朋等.STACER伸展变形的一种计算方法.力学学报,2016,48(6):1398-1405(Yu Chunyu,Zhang Congfa,Zhang Peng,et al.A method for simulating stacer’s deployment deformation.Chinese Journal of Theoretical and Applied Mechanics,2016,48(6):1398-1405(in Chinese))
24 Qi F,Wang TS,Li JF.The elastic contact influences on passive walking gaits.Robotica,2011,29(5):787-796
25秦琅,谢建华.基于Melnikov方法的单自由度干摩擦振子的混沌分析.动力学与控制学报,2018,16(3):239-243(Qin Lang,Xie Jianhua.Chaos analysis of a single-degree-of-freedom dry friction oscillator based on melnikov method.Journal of Dynamics and Control,2018,16(3):239-243(in Chinese))
26 Zheng XD,Wang Q.LCP method for a planar passive walker based on an event-driven scheme.Acta Mechanica Sinica,2018,34(3):578-588
27 Fan XX,Walker Paul D,Wang Q.Modeling and simulation of longitudinal dynamics coupled with clutch engagement dynamics for ground vehicles.Multibody System Dynamics,2018,43(2):153-174
28李文杰,吕晓寅,苏晓雪.固定和浮放条件下文物展柜动力响应分析与实验研究.文物保护与考古科学,2017,29(3):38-44(Li Wenjie,L¨u Xiaoyin,Su Xiaoxue.Dynamic response analysis and experimental study on the fixed and floating cultural showcases.Sciences of Conservation and Archaeology,2017,29(3):38-44(in Chinese))
29朱琳,王琪,张润森.含摩擦阻尼器非光滑振动系统的建模与数值仿真.动力学与控制学报,2017,15(2):172-177(Zhu Lin,Wang Qi,Zhang Runsen.Modelling and simulation of the non-smooth vibration systems with friction dampers.Journal of Dynamics and Control,2017,15(2):172-177(in Chinese))
30 Flores P,mbrosio J.Kinematics and Dynamics of Multibody Systems with Imperfect Joints.Berlin:Springer,2008:47-100
31 Pennestrì E,Valentini PP,Vita L.Multibody dynamics simulation of planar linkages with Dahl friction.Multibody System Dynamics,2007,17(4):321-347
32 Flores P,Leine R,Glocker C.Modeling and analysis of planar rigid multibody systems with translational clearance joints based on the non-smooth dynamics approach.Multibody System Dynamics,2010,23(2):165-190
33 Zhuang FF,Wang Q.Modeling and simulation of the nonsmooth planar rigid multibody systems with frictional translational joints.Multibody System Dynamics,2013,29(4):403-423
34 Muvengei O,Kihiu J,Ikua B.Dynamic analysis of planar rigid-body mechanical systems with two-clearance revolute joints.Nonlinear Dynamics,2013,73(1-2):259-273
35 Wang Z,Tian Q,Hu HY,et al.Nonlinear dynamics and chaotic control of a flexible multibody system with uncertain joint clearance.Nonlinear Dynamics,2016,86(3):1571-1597
36 Flores P,Machado M,Seabra E,et al.A parametric study on the Baumgarte stabilization method for forward dynamics of constrained multibody systems.Journal of Computational and Nonlinear Dynamics,2011,6(1):011019-1-1-9-9