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Numerical Methods in Multibody System Dynamics

  • Paulo FloresEmail author
  • Hamid M. Lankarani
Chapter
Part of the Solid Mechanics and Its Applications book series (SMIA, volume 226)

Abstract

This chapter includes the main numerical methods commonly utilized in multibody systems, namely those necessary to solve the dynamic equations of motion for constrained multibody systems. In this process, the fundamental aspects associated with the use of direct integration method together with the use of Baumgarte stabilization technique are described. In addition, several numerical algorithms for the integration process of the dynamics equations of motion are presented. An algorithm on contact detection for multibody systems encountering contact-impact events is discussed. Finally, numerical methods to systems of linear and nonlinear equations are analyzed.

Keywords

Dynamic analysis Direct integration method Baumgarte stabilization method Systems of linear equations Systems of nonlinear equations 

References

  1. Acary V, Brogliato B (2008) Numerical methods for nonsmooth dynamical systems: applications in mechanics and electronics. In: Lecture notes in applied and computational mechanics, vol. 35. Springer, Berlin, Heidelberg, New YorkGoogle Scholar
  2. Amirouche FML (1992) Computational methods for multibody dynamics. Prentice Hall, Englewood Cliffs, New JerseyzbMATHGoogle Scholar
  3. Arabyan A, Wu F (1998) An improved formulation for constrained mechanical systems. Multibody Sys Dyn 2(1):49–69CrossRefzbMATHGoogle Scholar
  4. Atkinson KA (1989) An introduction to numerical analysis, 2nd edn. Wiley, New YorkzbMATHGoogle Scholar
  5. Baumgarte J (1972) Stabilization of constraints and integrals of motion in dynamical systems. Comput Methods Appl Mech Eng 1:1–16MathSciNetCrossRefzbMATHGoogle Scholar
  6. Bayo E, Jálon JG, Serna AA (1988) Modified Lagrangian formulation for the dynamic analysis of constrained mechanical systems. Comput Methods Appl Mech Eng 71:183–195MathSciNetCrossRefzbMATHGoogle Scholar
  7. Blajer W (1995) An orthonormal tangent space method for constrained multibody systems. Comput Methods Appl Mech Eng 121:45–57MathSciNetCrossRefzbMATHGoogle Scholar
  8. Blajer W (1999) Elimination of constraint violation and accuracy improvement in numerical simulation of multibody systems. In: Ambrósio J, Schiehlen W (ed) Proceedings of EUROMECH Colloquium 404, advances in computational multibody dynamics, IDMEC/IST, Lisbon, Portugal, 20–23 Sept, pp 769–787Google Scholar
  9. Brenan KE, Campbell SL, Petzold LR (1989) Numerical solution of initial-value problems in differential-algebraic equations. Elsevier Science Pub. Co., New YorkzbMATHGoogle Scholar
  10. Carsten H, Wriggers P (2003) An explicit multi-body contact algorithm. Proc Appl Math Mech 3:280–281CrossRefGoogle Scholar
  11. Cochin I, Cadwallender W (1997) Analysis and design of dynamic systems, 3rd edn. Addison Wesley, New JerseyGoogle Scholar
  12. Conte SD, Boor C (1981) Elementary numerical analysis: an algorithmic approach, 3rd edn. McGraw-Hill, SingaporezbMATHGoogle Scholar
  13. Dahlquist G, Björck A (1974) Numerical methods. Prentice-Hall, New JerseyzbMATHGoogle Scholar
  14. Ebrahimi S, Eberhard P (2006) A linear complementarity formulation on position level for frictionless impact of planar deformable bodies. ZAMM Z Angew Math Mech 86(10):807–817MathSciNetCrossRefzbMATHGoogle Scholar
  15. Ebrahimi S, Hippmann G, Eberhard P (2005) Extension of polygonal contact model for flexible multibody systems. Int J Appl Math Mech 1:33–50Google Scholar
  16. Eich-Soellner E, Führer C (1998) Numerical methods in multibody dynamics. Teubner-Verlag Stuttgart, GermanyCrossRefzbMATHGoogle Scholar
  17. Erickson D, Weber M, Sharf I (2003) Contact stiffness and damping estimation for robotic systems. Int J Robot Res 22(1):41–57CrossRefGoogle Scholar
  18. Fisette P, Vaneghem B (1996) Numerical integration of multibody system dynamic equations using the coordinate method in an implicit Newmark scheme. Comput Methods Appl Mech Eng 135:85–105MathSciNetCrossRefzbMATHGoogle Scholar
  19. Flores P, Ambrósio J (2010) On the contact detection for contact-impact analysis in multibody systems. Multibody Sys Dyn 24(1):103–122MathSciNetCrossRefzbMATHGoogle Scholar
  20. Flores P, Seabra E (2009) Influence of the Baumgarte parameters on the dynamics response of multibody mechanical systems. Dyn Continuous Discrete Impulsive Sys Ser B Appl Algorithms 16(3):415–432MathSciNetzbMATHGoogle Scholar
  21. Flores P, Ambrósio J, Claro JCP, Lankarani HM (2008) Kinematics and dynamics of multibody systems with imperfect joints: models and case studies. In: Lecture notes in applied and computational mechanics, vol 34. Springer, Berlin, Heidelberg, New YorkGoogle Scholar
  22. Flores P, Machado M, Seabra E, da Silva MT (2011) A parametric study on the Baumgarte stabilization method for forward dynamics of constrained multibody systems. J Comput Nonlinear Dyn 6(1):011019, 9 pGoogle Scholar
  23. Gear CW (1981) Numerical solution of differential-algebraic equations. IEEE Trans Circuit Theory (CT) 18:89–95Google Scholar
  24. Haug EJ (1989) Computer-aided kinematics and dynamics of mechanical systems—volume I: basic methods. Allyn & Bacon, Boston, MassachusettsGoogle Scholar
  25. He K, Dong S, Zhou Z (2007) Multigrid contact detection method. Phys Rev 75(3):036710CrossRefGoogle Scholar
  26. Hildebrand FB (1974) Introduction to numerical analysis, 2nd edn. McGraw-Hill, SingaporeGoogle Scholar
  27. Hippmann G (2004) An algorithm for compliant contact between complexly shaped bodies. Multibody Sys Dyn 12:345–362MathSciNetCrossRefzbMATHGoogle Scholar
  28. Jálon JG, Bayo E (1994) Kinematic and dynamic simulations of multibody systems: the real-time challenge. Springer, New YorkCrossRefGoogle Scholar
  29. Leader JJ (2004) Numerical analysis and scientific computation. Addison Wesley, New JerseyGoogle Scholar
  30. Neto MA, Ambrósio J (2003) Stabilization methods for the integration of differential-algebraic equations in the presence of redundant constraints. Multibody Sys Dyn 10(1):81–105CrossRefzbMATHGoogle Scholar
  31. Nikravesh PE (1984) Some methods for dynamic analysis of constrained mechanical systems: a survey. In: Haug EJ (ed) Computer-aided analysis and optimization of mechanical system dynamics. Springer, Berlin, Germany, pp 351–368Google Scholar
  32. Nikravesh P (1988) Computer-aided analysis of mechanical systems. Prentice Hall, Englewood Cliffs, New JerseyGoogle Scholar
  33. Nikravesh PE (2008) Planar multibody dynamics: formulation, programming, and applications. CRC Press, LondonzbMATHGoogle Scholar
  34. Petzold LR (1983) A description of DASSL: a differential/algebraic system solver. In: Stepleman R et al (ed) Scientific computing. North-Holland Pub. Co., pp 65–68Google Scholar
  35. Pina H (1995) Métodos numéricos. McGraw-Hill, Lisboa, PortugalGoogle Scholar
  36. Polyanin AD, Zaitsev VF (2003) Handbook of exact solutions for ordinary differential equations, 2nd edn. Chapman & Hall/CRC Press, Boca RatonzbMATHGoogle Scholar
  37. Portal RJF, Dias JMP, Sousa LAG (2009) Contact detection between convex superquadric surfaces on multibody dynamics. In: Arczewski K, Frączek J, Wojtyra M (eds) Proceedings of the multibody dynamics 2009, ECCOMAS thematic conference, Warsaw, Poland, 29 June–2 July 2009, 14 pGoogle Scholar
  38. Shampine L, Gordon M (1975) Computer solution of ordinary differential equations: the initial value problem. Freeman, San Francisco, CaliforniazbMATHGoogle Scholar
  39. Sousa L, Veríssimo P, Ambrósio J (2008) Development of generic multibody road vehicle models for crashworthiness. Multibody Sys Dyn 19:133–158CrossRefzbMATHGoogle Scholar
  40. Studer C, Leine RI, Glocker C (2008) Step size adjustment and extrapolation for time-stepping schemes in non-smooth dynamics. Int J Numer Meth Eng 76(11):1747–1781MathSciNetCrossRefzbMATHGoogle Scholar
  41. Tseng F-C, Ma Z-D, Hulbert GM (2003) Efficient numerical solution of constrained multibody dynamics systems. Comput Methods Appl Mech Eng 192:439–472MathSciNetCrossRefzbMATHGoogle Scholar
  42. Wehage RA, Haug EJ (1982) Generalized coordinate partitioning for dimension reduction in analysis of constrained systems. J Mech Des 104:247–255CrossRefGoogle Scholar
  43. Weijia Z, Zhenkuan P, Yibing W (2000) An automatic constraint violation stabilization method for differential/algebraic equations of motion in multibody system dynamics. Appl Math Mech 21(1):103–108CrossRefzbMATHGoogle Scholar
  44. Wellmann C, Lillie C, Wriggers P (2008) A contact detection algorithm for superellipsoids based on the common-normal concept. Eng Computations Int J Comput Aided Eng Softw 25(5):432–442CrossRefzbMATHGoogle Scholar
  45. Yoon S, Howe RM, Greenwood DT (1994) Geometric elimination of constraint violations in numerical simulation of Lagrangian equations. J Mech Des 116:1058–1064CrossRefGoogle Scholar
  46. Zwillinger D (1997) Handbook of differential equations, 3rd edn. Academic Press, BostonzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of MinhoGuimaraesPortugal
  2. 2.Department of Mechanical EngineeringWichita State UniversityWichitaUSA

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