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Acta Mechanica Sinica

, Volume 27, Issue 4, pp 587–592 | Cite as

A linear complementarity model for multibody systems with frictional unilateral and bilateral constraints

  • Hai-Ping GaoEmail author
  • Qi Wang
  • Shi-Min Wang
  • Li Fu
Research Paper

Abstract

The Lagrange-I equations and measure differential equations for multibody systems with unilateral and bilateral constraints are constructed. For bilateral constraints, frictional forces and their impulses contain the products of the filled-in relay function induced by Coulomb friction and the absolute values of normal constraint reactions. With the time-stepping impulse-velocity scheme, the measure differential equations are discretized. The equations of horizontal linear complementarity problems (HLCPs), which are used to compute the impulses, are constructed by decomposing the absolute function and the filled-in relay function. These HLCP equations degenerate into equations of LCPs for frictional unilateral constraints, or HLCPs for frictional bilateral constraints. Finally, a numerical simulation for multibody systems with both unilateral and bilateral constraints is presented.

Keywords

Coulomb friction Bilateral constraint Unilateral constraint Horizontal linear complementarity problem (HLCP) Time-stepping impulse-velocity algorithm 

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References

  1. 1.
    Pfeiffer, F.: On non-smooth dynamics. Meccanica 43, 533–554 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Ferris, M.C., Pang, J.S.: Engineering and economic applications of complementarity problems. Siam Rev. 39(4), 669–713 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Pfeiffer, F.: Unilateral multibody dynamics. Meccanica 34(6), 437–451 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Pfeiffer, F.: The idea of complementarity in multibody dynamics. Archive of Applied Mechanics 72(11–12), 807–816 (2003)zbMATHGoogle Scholar
  5. 5.
    Pfeiffer, F.G.: Applications of unilateral multibody dynamics. Phil. Trans. R. Soc. Lond. A 359, 2609–2628 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Pfeiffer, F.: Multibody systems with unilateral constraints. J. Appl. Maths Mechs. 65(4), 665–670 (2001)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Leine, R.I., Van, Campen D.H., Glocker CH.: Nonlinear dynamics and modeling of various wooden toys with impact and friction. Journal of Vibration and Control 9, 25–78 (2003)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Moreau, J.J.: Unilateral contact and dry friction in finite freedom dynamics. In: Moreau, J.J., Panagiotopoulos, P.D., Nonsmooth mechanics and applications. CISM courses and lectures No.302, 1–82, Springer Verlag, Wien (1988)Google Scholar
  9. 9.
    Brogliato, B., Dam, A.A. ten, Paoli, L., et al.: Numerical simulation of finite dimensional multibody nonsmooth mechanical systems. Appl. Mech. Rev. 55(2), 107–150 (2002)CrossRefGoogle Scholar
  10. 10.
    Pfeiffer, F., Foerg, M., Ulbrich, H.: Numerical aspects of nonsmooth multibody dynamics. Computer Methods in Applied Mechanics and Engineering 195(50-51), 6891–6908 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Anitescu, M., Potra, F.A.: Formulating dynamic multi-rigidbody contact problems with friction as solvable linear complementarity problems. Nonlinear Dynamics 14, 231–247 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Gao, H.P., Wang, Q., Wang, S.M., et al.: Linear complementarity model and numerical integration scheme of multibody system with unilateral and bilateral constraints. Journal of Vibration and Shock 27(8), 38–41 (2008)Google Scholar
  13. 13.
    Glocker, C.H., Studer, C.: Formulation and preparation for numerical evaluation of linear complementarity systems in dynamics. Multibody System Dynamics 13(4), 447–463 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Glocker, C.H., Pfeiffer, F.: Complementarity problems in multibody systems with planar friction. Archives of Applied Mechanics 63, 452–463 (1993)zbMATHGoogle Scholar
  15. 15.
    Pfeiffer, F.: Unilateral problems of dynamics. Archive of Applied Mechanics 69, 503–527 (1999)zbMATHCrossRefGoogle Scholar
  16. 16.
    Glocker, C.H.: Set-valued Force Laws: Dynamics of Nonsmooth Systems. Springer-Verlag, Berlin, Heidelberg (2001)zbMATHGoogle Scholar
  17. 17.
    Glocker, C.H.: On frictionless impact models in rigid-body systems. Phil. Trans. R. Soc. Lond. A 359, 2385–2404 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Studer, C.W.: Augmented time-stepping integration of nonsmooth dynamical systems. Diss. ETH No.17597. ETH Zurich (2008)Google Scholar
  19. 19.
    Moreau, J.J.: Numerical aspects of the sweeping process. Comput. Methods Appl. Mech. Engrg. 177, 329–349 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Cottle, R.W., Pang, J.S., Stone, R.E.: The Linear Complementarity Problem. CA: Academic Press, San Diego (1992)zbMATHGoogle Scholar
  21. 21.
    Zhang, P.A., He, S.Y., Li, X.S.: Smoothing iterative algorithms for complementarity problems. Journal of Dalian University of Technology 43(1), 16–19 (2003) (in Chinese)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Xiu, N.H., Zhang, J.Z.: A smoothing Gauss-Newton method for the generalized HLCP. Journal of Computational and Applied Mathematics 129, 195–208 (2001)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Department of dynamics and controlBeihang UniversityBeijingChina

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