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A nonsmooth contact dynamic algorithm based on the symplectic method for multibody system analysis with unilateral constraints

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Abstract

In multibody systems, there are not only holonomic bilateral constraints, but also unilateral constraints. The existence of unilateral constraints brings nonsmooth contact dynamic problems into multibody dynamic systems. In this paper, we present an approach based on the symplectic method and the linear complementary method to solve multibody dynamic problems with impact contact. As the symplectic method has good energy conservation and no numerical damping, the proposed approach is expected to inherit these properties for solving nonsmooth problems of multibody dynamic systems. We present multiple numerical examples to demonstrate a high accuracy and good energy conservation behavior of the proposed approach even for large time step sizes.

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References

  1. Newmark, N.M.: A method for computation for structural dynamics. J. Eng. Mech. Div. 85(3), 67–94 (1959)

    Google Scholar 

  2. Hilber, H.M., Hughes, T.J.R., Taylor, R.L.: Improved numerical dissipation for time integration algorithms in structural dynamics. Earthq. Eng. Struct. Dyn. 5(3), 283–292 (1977)

    Article  Google Scholar 

  3. Chung, J., Hulbert, G.M.: A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-\(\alpha \) method. J. Appl. Mech. 60(2), 371–375 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bathe, K.J., Baig, M.: On a composition implicit integration procedure for nonlinear dynamics. Comput. Struct. 83, 2513–2524 (2005)

    Article  Google Scholar 

  5. Bathe, K.J.: Conserving energy and momentum in nonlinear dynamics: a simple implicit time integration scheme. Comput. Struct. 85(7–8), 437–445 (2007)

    Article  MathSciNet  Google Scholar 

  6. Flores, P., Lankarani, H.M.: Spatial rigid-multibody systems with lubricated spherical clearance joints: modeling and simulation. Nonlinear Dyn. 60(1–2), 99–114 (2009)

    MATH  Google Scholar 

  7. Flores, P., Ambrósio, J., Claro, J.C.P., Lankarani, H.M.: Translational joints with clearance in rigid multibody systems. J. Comput. Nonlinear Dyn. 3(1), 112–113 (2008)

    Google Scholar 

  8. Cavalieri, F.J., Cardona, A.: Non-smooth model of a frictionless and dry three-dimensional revolute joint with clearance for multibody system dynamics. Mech. Mach. Theory 121, 335–354 (2018)

    Article  Google Scholar 

  9. Flores, P., Ambrósio, J.: Revolute joints with clearance in multibody systems. Comput. Struct. 82(17–19), 1359–1369 (2004)

    Article  Google Scholar 

  10. Tian, Q., Zhang, Y., Chen, L., Yang, J.: Simulation of planar flexible multibody systems with clearance and lubricated revolute joints. Nonlinear Dyn. 60(4), 489–511 (2009)

    Article  MATH  Google Scholar 

  11. Wang, Z., Tian, Q., Hu, H., Flores, P.: Nonlinear dynamics and chaotic control of a flexible multibody system with uncertain joint clearance. Nonlinear Dyn. 86(3), 1571–1597 (2016)

    Article  Google Scholar 

  12. Yaqubi, S., Dardel, M., Daniali, H.M., Ghasemi, M.H.: Modeling and control of crank–slider mechanism with multiple clearance joints. Multibody Syst. Dyn. 36(2), 143–167 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Flores, P., Machado, M., Silva, M.T., Martins, J.M.: On the continuous contact force models for soft materials in multibody dynamics. Multibody Syst. Dyn. 25(3), 357–375 (2010)

    Article  MATH  Google Scholar 

  14. Pereira, C., Ramalho, A., Ambrosio, J.: Applicability domain of internal cylindrical contact force models. Mech. Mach. Theory 78, 141–157 (2014)

    Article  Google Scholar 

  15. Pereira, C.M., Ramalho, A.L., Ambrósio, J.A.: A critical overview of internal and external cylinder contact force models. Nonlinear Dyn. 63(4), 681–697 (2010)

    Article  Google Scholar 

  16. Zhuang, F.F., Wang, Q.: Modeling and simulation of the nonsmooth planar rigid multibody systems with frictional translational joints. Multibody Syst. Dyn. 29(4), 403–423 (2012).

    MathSciNet  Google Scholar 

  17. Zhuang, F.F., Wang, Q.: Modeling and analysis of rigid multibody systems with driving constraints and frictional translation joints. Acta Mech. Sin. 30(3), 437–446 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Acary, V.: Higher order event capturing time-stepping schemes for nonsmooth multibody systems with unilateral constraints and impacts. Appl. Numer. Math. 62(10), 1259–1275 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Studer, C., Leine, R.I., Glocker, C.: Step size adjustment and extrapolation for time-stepping schemes in non-smooth dynamics. Int. J. Numer. Methods Eng. 76(11), 1747–1781 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Schindler, T., Acary, V.: Timestepping schemes for nonsmooth dynamics based on discontinuous Galerkin methods: definition and outlook. Math. Comput. Simul. 95, 180–199 (2014)

    Article  MathSciNet  Google Scholar 

  21. Paoli, L., Schatzman, M.: A numerical scheme for impact problems I: the one-dimensional case. SIAM J. Numer. Anal. 40(2), 702–733 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  22. Paoli, L., Schatzman, M.: A numerical scheme for impact problems II: the multi-dimensional case. SIAM J. Numer. Anal. 40(2), 734–768 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Moreau, J.J.: Unilateral contact and dry friction in finite freedom dynamics. In: Moreau, J.J., Panagiotopoulos, P.D. (eds.) Nonsmooth Mechanics and Applications, CISM Courses and Lectures, vol. 302, pp. 1–82. Springer, Wien–New York (1988)

    Google Scholar 

  24. Jean, M.: The non-smooth contact dynamics method. Comput. Methods Appl. Mech. Eng. 177(3–4), 235–257 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  25. Acary, V.: Projected event-capturing time-stepping schemes for nonsmooth mechanical systems with unilateral contact and Coulomb’s friction. Comput. Methods Appl. Mech. Eng. 256, 224–250 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. 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 Syst. Dyn. 23(2), 165–190 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Anitescu, M., Tasora, A.: An iterative approach for cone complementarity problems for nonsmooth dynamics. Comput. Optim. Appl. 47(2), 207–235 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Chen, Q.z., Acary, V., Virlez, G., Brüls, O.: A nonsmooth generalized-\(\alpha \) scheme for flexible multibody systems with unilateral constraints. Int. J. Numer. Methods Eng. 96(8), 487–511 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  29. Brüls, O., Acary, V., Cardona, A.: Simultaneous enforcement of constraints at position and velocity levels in the nonsmooth generalized-\(\alpha \) scheme. Comput. Methods Appl. Mech. Eng. 281, 131–161 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  30. Charles, A., Casenave, F., Glocker, C.: A catching-up algorithm for multibody dynamics with impacts and dry friction. Comput. Methods Appl. Mech. Eng. 334, 208–237 (2018)

    Article  MathSciNet  Google Scholar 

  31. Moreau, J.J.: Standard inelastic shocks and the dynamics of unilateral constraints. In: Del Piero, G., Maceri, F. (eds.) Unilateral Problems in Structural Analysis, pp. 173–221. Springer, New York (1983)

    Google Scholar 

  32. Charles, A., Ballard, P.: The formulation of dynamical contact problems with friction in the case of systems of rigid bodies and general discrete mechanical systems—Painlevé and Kane paradoxes revisited. Z. Angew. Math. Phys. 67(4), 99 (2016)

    Article  MATH  Google Scholar 

  33. Haddouni, M., Acary, V., Garreau, S., Beley, J.D., Brogliato, B.: Comparison of several formulations and integration methods for the resolution of DAEs formulations in event-driven simulation of nonsmooth frictionless multibody dynamics. Multibody Syst. Dyn. 41(3), 201–231 (2017)

    Article  MathSciNet  Google Scholar 

  34. Acary, V., Brogliato, B.: Energy conservation and dissipation properties of time-integration methods for the nonsmooth elastodynamics with contact. Z. Angew. Math. Mech. 96(5), 585–603 (2015)

    Article  MathSciNet  Google Scholar 

  35. Gao, Q., Tan, S.J., Zhang, H.W., Zhong, W.X.: Symplectic algorithms based on the principle of least action and generating functions. Int. J. Numer. Methods Eng. 89(4), 438–508 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  36. Gonzalez, O., Simo, J.C.: On the stability of symplectic and energy-momentum algorithms for non-linear Hamiltonian systems with symmetry. Comput. Methods Appl. Mech. Eng. 134(3–4), 197–222 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  37. Reich, S.: Backward error analysis for numerical integrators. SIAM J. Numer. Anal. 36(5), 1549–1570 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  38. Reich, S.: Symplectic integration of constrained Hamiltonian systems by composition methods. SIAM J. Numer. Anal. 33(2), 475–491 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  39. Peng, H.J., Jiang, X.: Nonlinear receding horizon guidance for spacecraft formation reconfiguration on libration point orbits using a symplectic numerical method. ISA Trans. 60, 38–52 (2016)

    Article  Google Scholar 

  40. Peng, H.J., Wang, X., Li, M., Chen, B.: An \(\mathit{hp}\) symplectic pseudospectral method for nonlinear optimal control. Commun. Nonlinear Sci. Numer. Simul. 42, 623–644 (2017)

    Article  MathSciNet  Google Scholar 

  41. Peng, H.J., Gao, Q., Wu, Z.G., Zhong, W.X.: Symplectic adaptive algorithm for solving nonlinear two-point boundary value problems in astrodynamics. Celest. Mech. Dyn. Astron. 110(4), 319–342 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  42. Lens, E.V., Cardona, A., Géradin, M.: Energy preserving time integration for constrained multibody systems. Multibody Syst. Dyn. 11, 41–61 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  43. Laursen, T., Chawla, V.: Design of energy conserving algorithms for frictionless dynamic contact problems. Int. J. Numer. Methods Eng. 40, 863–886 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  44. Bauchau, O.A.: Analysis of flexible multibody systems with intermittent contacts. Multibody Syst. Dyn. 4, 23–54 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  45. Alart, P., Curnier, A.: A mixed formulation for frictional contact problems prone to Newton like solution methods. Comput. Methods Appl. Mech. Eng. 92(3), 353–375 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  46. Christensen, P., Klarbring, A., Pang, J., Stromberg, N.: Formulation and comparison of algorithms for frictional contact problems. Int. J. Numer. Methods Eng. 42(1), 145–172 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  47. Acary, V., Brogliato, B.: Numerical Methods for Nonsmooth Dynamical Systems: Applications in Mechanics and Electronics. Lecture Notes in Applied and Computational Mechanics, vol. 35. Springer, Berlin (2008)

    Book  MATH  Google Scholar 

  48. Cottle, R., Pang, J., Stone, R.: The Linear Complementarity Problem. Academic Press, Boston (1992)

    MATH  Google Scholar 

  49. Klarbring, A.: A mathematical programming approach to three-dimensional contact problems with friction. Comput. Methods Appl. Mech. Eng. 58(2), 175–200 (1986)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors are grateful for the financial support of the National Key Research and Development Program of China (2017YFB1301103) and the National Natural Science Foundation of China (11922203, 11772074).

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  1. Department of Engineering Mechanics, State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian, 116024, P.R. China

    Haijun Peng, Ningning Song & Ziyun Kan

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  1. Haijun Peng
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Correspondence to Haijun Peng.

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Peng, H., Song, N. & Kan, Z. A nonsmooth contact dynamic algorithm based on the symplectic method for multibody system analysis with unilateral constraints. Multibody Syst Dyn 49, 119–153 (2020). https://doi.org/10.1007/s11044-019-09719-8

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