Advertisement

Multibody System Dynamics

, Volume 41, Issue 3, pp 275–295 | Cite as

Energy-consistent simulation of frictional contact in rigid multibody systems using implicit surfaces and penalty method

  • Roberto Ortega
  • Juan Carlos García Orden
  • Marcela Cruchaga
  • Claudio García
Article

Abstract

We present a conservative formulation for the frictional contact forces developed in the framework of an energy-consistent method. Bilateral constrains, that is, joints and rigid links, are imposed using the augmented Lagrange technique, provided that the constraints are exactly satisfied. We propose a formulation based on a penalty method to deal with the contact problem including damping and friction. The numerical scheme guarantees that the energy is consistently preserved or unconditionally dissipated. The treatment of the contact constraint also constitutes a novel aspect of this investigation. It is based on the description of implicit surfaces given an efficient strategy for detecting and evaluating the contact. The numerical tests exhibit a good energy preservation performance. Moreover, we use the proposed formulation to analyze a mechanism including a revolute joint with frictional clearance.

Keywords

Multibody system Constraint equations Contact Impact Energy consistent method Penalty 

Notes

Acknowledgements

The authors thank the support given by Chilean Council for Scientific and Technological Research (PAI79130042) and the Scientific Research Projects Management Department of the Vice Presidency of Research, Development and Innovation (DICYT-VRID) at Universidad de Santiago de Chile (project “Aporte Basal 061516CSSA”).

References

  1. 1.
    Flores, P., Ambrósio, J., Claro, J.C.P., Lankarani, H.M., Koshy, C.S.: Lubricated revolute joints in rigid multibody systems. Nonlinear Dyn. 56(3), 277–295 (2008) CrossRefzbMATHGoogle Scholar
  2. 2.
    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 (2016) MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Xu, L., Yang, Y.: Modeling a non-ideal rolling ball bearing joint with localized defects in planar multibody systems. Multibody Syst. Dyn. 35(4), 409–426 (2015) MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Pereira, C., Ambrósio, J., Ramalho, A.: Dynamics of chain drives using a generalized revolute clearance joint formulation. Mech. Mach. Theory 92, 64–85 (2015) CrossRefGoogle Scholar
  5. 5.
    Erkaya, S., Dŏgan, S., Ulus, Ş.: Effects of joint clearance on the dynamics of a partly compliant mechanism: numerical and experimental studies. Mech. Mach. Theory 88, 125–140 (2015) CrossRefGoogle Scholar
  6. 6.
    Shabana, A.A.: Flexible multibody dynamics: review of past and recent developments. Multibody Syst. Dyn. 1(2), 189–222 (1997) MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Wasfy, T.M., Noor, A.K.: Computational strategies for flexible multibody systems. Appl. Mech. Rev. 56(6), 553 (2003) CrossRefGoogle Scholar
  8. 8.
    Bauchau, O.A., Laulusa, A.: Review of contemporary approaches for constraint enforcement in multibody systems. J. Comput. Nonlinear Dyn. 3(1), 011005 (2007) CrossRefGoogle Scholar
  9. 9.
    Baumgarte, J.: Stabilization of constraints and integrals of motion in dynamical systems. Comput. Methods Appl. Mech. Eng. 1(1), 1–16 (1972) MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Blajer, W.: Elimination of constraint violation and accuracy aspects in numerical simulation of multibody systems. Multibody Syst. Dyn. 7, 265–284 (2002) MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Arnold, V.I.: Mecánica Clásica. Métodos matemáticos. Paraninfo S.A., Madrid (1983) Google Scholar
  12. 12.
    Negrut, D., Rampalli, R., Ottarsson, G., Sajdak, A.: On an implementation of the Hilber–Hughes–Taylor method in the context of index 3 differential–algebraic equations of multibody dynamics (DETC2005-85096). J. Comput. Nonlinear Dyn. 2(1), 73 (2007) CrossRefGoogle Scholar
  13. 13.
    Arnold, M., Brüls, O.: Convergence of the generalized-\(\alpha\) scheme for constrained mechanical systems. Multibody Syst. Dyn. 18(2), 185–202 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differential–Algebraic Problems, vol. 14, 2nd revised edn. Springer, Berlin (2004) zbMATHGoogle Scholar
  15. 15.
    Bayo, E., García de Jalón, J., Avello, A., Cuadrado, J.: An efficient computational method for real time multibody dynamic simulation in fully Cartesian coordinates. Comput. Methods Appl. Mech. Eng. 92(3), 377–395 (1991) CrossRefzbMATHGoogle Scholar
  16. 16.
    Avello, A., Jiménez, J.M., Bayo, E., García de Jalón, J.: A simple and highly parallelizable method for real-time dynamic simulation based on velocity transformations. Comput. Methods Appl. Mech. Eng. 107(3), 313–339 (1993) CrossRefzbMATHGoogle Scholar
  17. 17.
    Cuadrado, J., Cardenal, J., Bayo, E.: Modeling and solution methods for efficient real-time simulation of multibody dynamics. Multibody Syst. Dyn. 1(3), 259–280 (1997) MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Cuadrado, J., Dopico, D., Naya, M.A., Gonzales, M.: Penalty, semi-recursive and hybrid methods for MBS real-time dynamics in the context of structural integrators. Multibody Syst. Dyn. 12, 117–132 (2004) CrossRefzbMATHGoogle Scholar
  19. 19.
    González, F., Kövecses, J.: Use of penalty formulations in dynamic simulation and analysis of redundantly constrained multibody systems. Multibody Syst. Dyn. 29(1), 57–76 (2013) MathSciNetCrossRefGoogle Scholar
  20. 20.
    Bayo, E., García de Jalón, J., Serna, M.A.: A modified Lagrangian formulation for the dynamic analysis of constrained mechanical systems. Comput. Methods Appl. Mech. Eng. 71(2), 183–195 (1988) MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont (1999) zbMATHGoogle Scholar
  22. 22.
    Blajer, W.: Augmented Lagrangian formulation: geometrical interpretation and application to systems with singularities and redundancy. Multibody Syst. Dyn. 8(2), 141–159 (2002) MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    García Orden, J.C., Ortega, R.A.: A conservative augmented Lagrangian algorithm for the dynamics of constrained mechanical systems. Mech. Based Des. Struct. Mach. 34(4), 449–468 (2006) CrossRefGoogle Scholar
  24. 24.
    Cavalieri, F., Cardona, A.: An augmented Lagrangian technique combined with a mortar algorithm for modelling mechanical contact problems. Int. J. Numer. Methods Eng. 93(4), 420–442 (2013) MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Cavalieri, F., Cardona, A.: Numerical solution of frictional contact problems based on a mortar algorithm with an augmented Lagrangian technique. Multibody Syst. Dyn. 35(4), 353–375 (2015) MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Mashayekhi, M.J., Kövecses, J.: A comparative study between the augmented Lagrangian method and the complementarity approach for modeling the contact problem. Multibody Syst. Dyn. (2016). doi: 10.1007/s11044-016-9510-2 Google Scholar
  27. 27.
    Brenan, K.E., Campbell, S.L., Petzold, L.: Numerical Solution of Initial-Value Problems in Differential–Algebraic Equations. Classics in Applied Mathematics, vol. 14. SIAM, Philadelphia (1996) zbMATHGoogle Scholar
  28. 28.
    Ascher, U.M., Petzold, L.R.: Computer Methods for Ordinary Differential Equations and Differential–Algebraic Equations. SIAM, Philadelphia (1998) CrossRefzbMATHGoogle Scholar
  29. 29.
    García Orden, J.C., Conde Martín, S.: Controllable velocity projection for constraint stabilization in multibody dynamics. Nonlinear Dyn. 68(1), 245–257 (2011) MathSciNetzbMATHGoogle Scholar
  30. 30.
    García Orden, J.C., Ortega, R.: Energy considerations for the stabilization of constrained mechanical systems with velocity projection. In: Arczewski, K., Blajer, W., Fraczek, J., Wojtyra, M. (eds.) Multibody Dynamics. Computational Methods in Applied Sciences, vol. 23, pp. 153–171. Springer, Dordrecht (2011) CrossRefGoogle Scholar
  31. 31.
    Simo, J.C., Wong, K.K.: Unconditionally stable algorithms for rigid body dynamics that exactly preserve energy and momentum. Int. J. Numer. Methods Eng. 31(1), 19–52 (1991) MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Stuart, A.M., Humphries, A.R.: Dynamical Systems and Numerical Analysis. Cambridge University Press, Cambridge (1996) zbMATHGoogle Scholar
  33. 33.
    Simo, J.C., Tarnow, N.: The discrete energy-momentum method. Conserving algorithms for nonlinear elastodynamics. Z. Angew. Math. Phys. 43(5), 757–792 (1992) MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Simo, J.C., Tarnow, N., Wong, K.K.: Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics. Comput. Methods Appl. Mech. Eng. 100(1), 63–116 (1992) MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Simo, J.C., Gonzalez, O.: Assessment of energy-momentum and symplectic schemes for stiff dynamical systems. In: Proc. ASME Winter Annual Meeting, pp. 1–25. American Society of Mechanical Engineers, New York (1993) Google Scholar
  36. 36.
    Simo, J.C., Tarnow, N., Doblare, M.: Non-linear dynamics of three-dimensional rods: exact energy and momentum conserving algorithms. Int. J. Numer. Methods Eng. 38(9), 1431–1473 (1995) MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Gonzalez, O.: Time integration and discrete Hamiltonian systems. J. Nonlinear Sci. 6(5), 449–467 (1996) MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    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) MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Gonzalez, O.: Mechanical systems subject to holonomic constraints: differential–algebraic formulations and conservative integration. Physica D 132(1–2), 165–174 (1999) MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Gonzalez, O.: Exact energy and momentum conserving algorithms for general models in nonlinear elasticity. Comput. Methods Appl. Mech. Eng. 190(13–14), 1763–1783 (2000) MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Betsch, P., Steinmann, P.: Conservation properties of a time FE method. Part I: time-stepping schemes for \(N\)-body problems. Int. J. Numer. Methods Eng. 49(5), 599–638 (2000) CrossRefzbMATHGoogle Scholar
  42. 42.
    García Orden, J.C., Goicolea, J.M.: Conserving properties in constrained dynamics of flexible multibody systems. Multibody Syst. Dyn. 4(2–3), 225–244 (2000) MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Conde Martin, S., Garcia Orden, J.: An energy-consistent integration scheme for flexible multibody systems with dissipation. Proc. Inst. Mech. Eng., Proc., Part K, J. Multi-Body Dyn. 230(3), 268–280 (2016) Google Scholar
  44. 44.
    Bauchau, O.A., Bottasso, C.L.: On the design of energy preserving and decaying schemes for flexible, nonlinear multi-body systems. Comput. Methods Appl. Mech. Eng. 169(1–2), 61–79 (1999) MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Borri, M., Bottasso, C.L., Trainelli, L.: Integration of elastic multibody systems by invariant conserving/dissipating algorithms. I. Formulation. Comput. Methods Appl. Mech. Eng. 190(29–30), 3669–3699 (2001) MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Bottasso, C.L., Borri, M., Trainelli, L.: Integration of elastic multibody systems by invariant conserving/dissipating algorithms. II. Numerical schemes and applications. Comput. Methods Appl. Mech. Eng. 190(29–30), 3701–3733 (2001) MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Armero, F., Romero, I.: On the formulation of high-frequency dissipative time-stepping algorithms for nonlinear dynamics. Part I: low-order methods for two model problems and nonlinear elastodynamics. Comput. Methods Appl. Mech. Eng. 190(20–21), 2603–2649 (2001) CrossRefzbMATHGoogle Scholar
  48. 48.
    Armero, F., Romero, I.: On the formulation of high-frequency dissipative time-stepping algorithms for nonlinear dynamics. Part II: second-order methods. Comput. Methods Appl. Mech. Eng. 190(51–52), 6783–6824 (2001) CrossRefzbMATHGoogle Scholar
  49. 49.
    Betsch, P., Steinmann, P.: Constrained integration of rigid body dynamics. Comput. Methods Appl. Mech. Eng. 191(3–5), 467–488 (2001) MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Betsch, P., Steinmann, P.: Conservation properties of a time FE method. Part III: Mechanical systems with holonomic constraints. Int. J. Numer. Methods Eng. 53(10), 2271–2304 (2002) CrossRefzbMATHGoogle Scholar
  51. 51.
    Lens, E.V., Cardona, A., Géradin, M.: Energy preserving time integration for constrained multibody systems. Multibody Syst. Dyn. 11(1), 41–61 (2004) MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Lens, E., Cardona, A.: An energy preserving/decaying scheme for nonlinearly constrained multibody systems. Multibody Syst. Dyn. 18(3), 435–470 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Marques, F., Flores, P., Pimenta Claro, J.C., Lankarani, H.M.: A survey and comparison of several friction force models for dynamic analysis of multibody mechanical systems. Nonlinear Dyn. 86(3), 1407–1443 (2016) MathSciNetCrossRefGoogle Scholar
  54. 54.
    Rodriguez, A., Bowling, A.: Solution to indeterminate multipoint impact with frictional contact using constraints. Multibody Syst. Dyn. 28(4), 313–330 (2012) MathSciNetCrossRefGoogle Scholar
  55. 55.
    Kikuuwe, R., Brogliato, B.: A new representation of systems with frictional unilateral constraints and its Baumgarte-like relaxation. Multibody Syst. Dyn. 39(3), 267–290 (2017) MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Armero, F., Petőcz, E.: Formulation and analysis of conserving algorithms for frictionless dynamic contact/impact problems. Comput. Methods Appl. Mech. Eng. 158(3–4), 269–300 (1998) MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Armero, F., Petőcz, E.: A new dissipative time-stepping algorithm for frictional contact problems: formulation and analysis. Comput. Methods Appl. Mech. Eng. 179(1–2), 151–178 (1999) MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Laursen, T.A., Chawla, V.: Design of energy conserving algorithms for frictionless dynamic contact problems. Int. J. Numer. Methods Eng. 40(5), 863–886 (1997) MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Chawla, V., Laursen, T.A.: Energy consistent algorithms for frictional contact problems. Int. J. Numer. Methods Eng. 42(5), 799–827 (1998) MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Hesch, C., Betsch, P.: Transient 3d contact problems—NTS method: mixed methods and conserving integration. Comput. Mech. 48(4), 437–449 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Glocker, C.: Energetic consistency conditions for standard impacts. Part I: Newton-type inequality impact laws and Kane’s example. Multibody Syst. Dyn. 29(1), 77–117 (2013) MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Glocker, C.: Energetic consistency conditions for standard impacts. Part II: Poisson-type inequality impact laws. Multibody Syst. Dyn. 32(4), 445–509 (2014) MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    Wriggers, P.: Computational Contact Mechanics. Wiley, New York (2002) zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Departamento de Ingeniería MecánicaUniversidad de Santiago de ChileSantiagoChile
  2. 2.ETSI de Caminos, Canales y PuertosUniversidad Politécnica de MadridMadridSpain

Personalised recommendations