September  2011, 16(2): 589-605. doi: 10.3934/dcdsb.2011.16.589

A constraint-stabilized method for multibody dynamics with friction-affected translational joints based on HLCP

1. 

Department of Dynamics and Control, Beijing University of Aeronautics and Astronautics, Beijing, 100191, China

2. 

Department of Dynamics and Control, Beijing University of Aeronautics and Astronautics, Beijing 100191, China, China

Received  March 2010 Revised  December 2010 Published  June 2011

In this paper, a constraint-stabilized numerical method is presented for the planar rigid multibody system with friction-affected translational joints, in which the sliders and the guides are treated as particles and bilateral constraints, respectively. The dynamical equations of the non-smooth system are obtained by using the first kind of Lagrange's equations and Baumgarte stabilization method. The normal forces of bilateral constraints are expressed by the Lagrange multipliers and described by complementarity condition, while frictional forces are characterized by a set-valued force law of the type of Coulomb's law for dry friction. Using event-driven scheme, the state transition problem of stick-slip and normal forces of bilateral constraints is formulated and solved as a horizontal linear complementarity problem (HLCP). Finally, the planar rigid multibody system with two translational joints is considered as a illustrative application example. The results obtained also show that the drift of constraints of the system remains bounded.
Citation: Qi Wang, Huilian Peng, Fangfang Zhuang. A constraint-stabilized method for multibody dynamics with friction-affected translational joints based on HLCP. Discrete and Continuous Dynamical Systems - B, 2011, 16 (2) : 589-605. doi: 10.3934/dcdsb.2011.16.589
References:
[1]

Paulo Flores, Jorge Ambrósio, J. C. Pimenta Claro and Hamid M. Lankarani, Kinematics and Dynamics of Multibody Systems with Imperfect Joints, in "Lecture Notes in Applied and Computational Mechanics," Springer-Verlag, Berlin Heidelberg, 34 (2008).

[2]

Anna Maria Cherubini, Giorgio Metafune and Francesco Paparella, On the Stopping Time of a Bouncing Ball, Discrete and Continuous Dynamical Systems-Series B, 10 (2008), 43-72. doi: 10.3934/dcdsb.2008.10.43.

[3]

Friedrich Pfeiffer, On non-smooth Dynamics, Meccanica, 43 (2008), 533-554. doi: 10.1007/s11012-008-9139-1.

[4]

Mihai Anitescu and Gary D. Hart, A Constraint-stabilized Time-stepping Approach for Rigid Multibody Dynamics with Joints, Contact and Friction, International Journal for Numerical Methods in Engineering, 60 (2004), 2335-2371. doi: 10.1002/nme.1047.

[5]

B. Brogliato, A. A. ten Dam, L. Paoli, F. Génot and M. Abadie, Numerical simulation of finite dimensional multibody nonsmooth mechanical systems, Applied Mechanics Reviews, 55 (2002), 107-150. doi: 10.1115/1.1454112.

[6]

R. I. Leine, D. H. Van Campen and C. H. Clocker, Nonlinear Dynamics and Modeling of Various Wooden Toys with Impact and Friction, Journal of Vibration and Control, 9 (2003), 25-78.

[7]

Friedrich Pfeiffer, Martin Foerg and Heinz Ubrlch, Numerical aspects of non-smooth multibody dynamics, Computer Methods in Applied Mechanics and Engineering, 195 (2006), 6891-6908. doi: 10.1016/j.cma.2005.08.012.

[8]

P. Flores, J. Ambrósio, J. C. P. Claro and H. M. Lankarani, Translational joints with clearance in rigid multibody systems, Journal of Computational and Nonlinear Dynamics, 011007(3) (2008), 1-10.

[9]

JY. Han, NH. Xiu and HD. Qi, "Theory and Algorithms of Nonlinear Complementarity," Shanghai Scientific and Technical Publishers, Shanghai, 2006(in Chinese).

[10]

H. J. Klepp, Trial-and-error based method for the investigation of multi-body systems with Friction, Journal of Sound and Vibration, 197 (1996), 629-637. doi: 10.1006/jsvi.1996.0552.

[11]

H. J. Klepp, The existence and uniqueness of solutions for a single-degree-of-freedom system with two frition-affected sliding joints, Journal of Sound and Vibration, 185 (1995), 364-371. doi: 10.1006/jsvi.1995.0385.

[12]

H. J. Klepp, Modes of contact and uniqueness of solutions for systems with friction-affected sliders, Journal of Sound and Vibration, 254 (2002), 987-996. doi: 10.1006/jsvi.2001.4147.

[13]

HL. Peng, SM. Wang, Q. Wang, et al, Modeling and Simulation of Multi-body Systems with Multi-friction and Fixed Bilateral Constraint, Chinese Journal of Theoretial and Applied Mechanics, 41 (2009), 105-112.

[14]

Richard W. Cottle, Jong-Shi Pang and Richard E. Stone, "The Linear Complementarity Problem," Academic Press, Boston, c1992.

show all references

References:
[1]

Paulo Flores, Jorge Ambrósio, J. C. Pimenta Claro and Hamid M. Lankarani, Kinematics and Dynamics of Multibody Systems with Imperfect Joints, in "Lecture Notes in Applied and Computational Mechanics," Springer-Verlag, Berlin Heidelberg, 34 (2008).

[2]

Anna Maria Cherubini, Giorgio Metafune and Francesco Paparella, On the Stopping Time of a Bouncing Ball, Discrete and Continuous Dynamical Systems-Series B, 10 (2008), 43-72. doi: 10.3934/dcdsb.2008.10.43.

[3]

Friedrich Pfeiffer, On non-smooth Dynamics, Meccanica, 43 (2008), 533-554. doi: 10.1007/s11012-008-9139-1.

[4]

Mihai Anitescu and Gary D. Hart, A Constraint-stabilized Time-stepping Approach for Rigid Multibody Dynamics with Joints, Contact and Friction, International Journal for Numerical Methods in Engineering, 60 (2004), 2335-2371. doi: 10.1002/nme.1047.

[5]

B. Brogliato, A. A. ten Dam, L. Paoli, F. Génot and M. Abadie, Numerical simulation of finite dimensional multibody nonsmooth mechanical systems, Applied Mechanics Reviews, 55 (2002), 107-150. doi: 10.1115/1.1454112.

[6]

R. I. Leine, D. H. Van Campen and C. H. Clocker, Nonlinear Dynamics and Modeling of Various Wooden Toys with Impact and Friction, Journal of Vibration and Control, 9 (2003), 25-78.

[7]

Friedrich Pfeiffer, Martin Foerg and Heinz Ubrlch, Numerical aspects of non-smooth multibody dynamics, Computer Methods in Applied Mechanics and Engineering, 195 (2006), 6891-6908. doi: 10.1016/j.cma.2005.08.012.

[8]

P. Flores, J. Ambrósio, J. C. P. Claro and H. M. Lankarani, Translational joints with clearance in rigid multibody systems, Journal of Computational and Nonlinear Dynamics, 011007(3) (2008), 1-10.

[9]

JY. Han, NH. Xiu and HD. Qi, "Theory and Algorithms of Nonlinear Complementarity," Shanghai Scientific and Technical Publishers, Shanghai, 2006(in Chinese).

[10]

H. J. Klepp, Trial-and-error based method for the investigation of multi-body systems with Friction, Journal of Sound and Vibration, 197 (1996), 629-637. doi: 10.1006/jsvi.1996.0552.

[11]

H. J. Klepp, The existence and uniqueness of solutions for a single-degree-of-freedom system with two frition-affected sliding joints, Journal of Sound and Vibration, 185 (1995), 364-371. doi: 10.1006/jsvi.1995.0385.

[12]

H. J. Klepp, Modes of contact and uniqueness of solutions for systems with friction-affected sliders, Journal of Sound and Vibration, 254 (2002), 987-996. doi: 10.1006/jsvi.2001.4147.

[13]

HL. Peng, SM. Wang, Q. Wang, et al, Modeling and Simulation of Multi-body Systems with Multi-friction and Fixed Bilateral Constraint, Chinese Journal of Theoretial and Applied Mechanics, 41 (2009), 105-112.

[14]

Richard W. Cottle, Jong-Shi Pang and Richard E. Stone, "The Linear Complementarity Problem," Academic Press, Boston, c1992.

[1]

Leszek Gasiński, Piotr Kalita. On dynamic contact problem with generalized Coulomb friction, normal compliance and damage. Evolution Equations and Control Theory, 2020, 9 (4) : 1009-1026. doi: 10.3934/eect.2020049

[2]

Jingzhen Liu, Ka-Fai Cedric Yiu, Kok Lay Teo. Optimal investment-consumption problem with constraint. Journal of Industrial and Management Optimization, 2013, 9 (4) : 743-768. doi: 10.3934/jimo.2013.9.743

[3]

Yunan Wu, Guangya Chen, T. C. Edwin Cheng. A vector network equilibrium problem with a unilateral constraint. Journal of Industrial and Management Optimization, 2010, 6 (3) : 453-464. doi: 10.3934/jimo.2010.6.453

[4]

Andaluzia Matei, Mircea Sofonea. Dual formulation of a viscoplastic contact problem with unilateral constraint. Discrete and Continuous Dynamical Systems - S, 2013, 6 (6) : 1587-1598. doi: 10.3934/dcdss.2013.6.1587

[5]

Chonghu Guan, Xun Li, Rui Zhou, Wenxin Zhou. Free boundary problem for an optimal investment problem with a borrowing constraint. Journal of Industrial and Management Optimization, 2022, 18 (3) : 1915-1934. doi: 10.3934/jimo.2021049

[6]

Jiang-Xia Nan, Deng-Feng Li. Linear programming technique for solving interval-valued constraint matrix games. Journal of Industrial and Management Optimization, 2014, 10 (4) : 1059-1070. doi: 10.3934/jimo.2014.10.1059

[7]

Yanjun Wang, Shisen Liu. Relaxation schemes for the joint linear chance constraint based on probability inequalities. Journal of Industrial and Management Optimization, 2022, 18 (5) : 3719-3733. doi: 10.3934/jimo.2021132

[8]

Térence Bayen, Alain Rapaport, Fatima-Zahra Tani. Optimal periodic control for scalar dynamics under integral constraint on the input. Mathematical Control and Related Fields, 2020, 10 (3) : 547-571. doi: 10.3934/mcrf.2020010

[9]

Claudia Lederman, Noemi Wolanski. An optimization problem with volume constraint for an inhomogeneous operator with nonstandard growth. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2907-2946. doi: 10.3934/dcds.2020391

[10]

Alexander Arguchintsev, Vasilisa Poplevko. An optimal control problem by parabolic equation with boundary smooth control and an integral constraint. Numerical Algebra, Control and Optimization, 2018, 8 (2) : 193-202. doi: 10.3934/naco.2018011

[11]

Xiaoliang Li, Cong Wang. An optimization problem in heat conduction with volume constraint and double obstacles. Discrete and Continuous Dynamical Systems, 2022, 42 (10) : 5017-5036. doi: 10.3934/dcds.2022084

[12]

Behrouz Kheirfam. Multi-parametric sensitivity analysis of the constraint matrix in piecewise linear fractional programming. Journal of Industrial and Management Optimization, 2010, 6 (2) : 347-361. doi: 10.3934/jimo.2010.6.347

[13]

Takeshi Fukao. Variational inequality for the Stokes equations with constraint. Conference Publications, 2011, 2011 (Special) : 437-446. doi: 10.3934/proc.2011.2011.437

[14]

Binghai Zhou, Yuanrui Lei, Shi Zong. Lagrangian relaxation algorithm for the truck scheduling problem with products time window constraint in multi-door cross-dock. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021151

[15]

Zhen-Zhen Tao, Bing Sun. Space-time spectral methods for a fourth-order parabolic optimal control problem in three control constraint cases. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022080

[16]

Zhen-Zhen Tao, Bing Sun. Galerkin spectral method for elliptic optimal control problem with $L^2$-norm control constraint. Discrete and Continuous Dynamical Systems - B, 2022, 27 (8) : 4121-4141. doi: 10.3934/dcdsb.2021220

[17]

Ziteng Wang, Shu-Cherng Fang, Wenxun Xing. On constraint qualifications: Motivation, design and inter-relations. Journal of Industrial and Management Optimization, 2013, 9 (4) : 983-1001. doi: 10.3934/jimo.2013.9.983

[18]

Jingzhen Liu, Lihua Bai, Ka-Fai Cedric Yiu. Optimal investment with a value-at-risk constraint. Journal of Industrial and Management Optimization, 2012, 8 (3) : 531-547. doi: 10.3934/jimo.2012.8.531

[19]

Albert Fathi. An Urysohn-type theorem under a dynamical constraint. Journal of Modern Dynamics, 2016, 10: 331-338. doi: 10.3934/jmd.2016.10.331

[20]

Zuo Quan Xu, Fahuai Yi. An optimal consumption-investment model with constraint on consumption. Mathematical Control and Related Fields, 2016, 6 (3) : 517-534. doi: 10.3934/mcrf.2016014

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (117)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]