# American Institute of Mathematical Sciences

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.

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##### 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.
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