Advertisement

Dependent Variable Formulations

  • Wojciech Blajer
Chapter
Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 511)

Abstract

The multibody dynamics formulations fall into two main categories. In the first group formulations the equations of motion are derived in terms of dependent state variables, and the implicitly given equations of constraints on the system defined this way need to be involved. The price for usual conceptual simplicity and ease of manipulation of these governing equations is their large dimension and the fact that they form mixed sets of differential-algebraic equations (DAEs), both resulting in computationally inefficient algorithms. In the other formulations independent state variables are used, which leads to the minimal-form governing ordinary differential equations (ODEs). The increased pre-processing modeling effort is then repaid by much more efficient numerical integration of the ODEs.

Keywords

Multibody System Constraint Violation Multibody System Dynamics Constraint Manifold Energy Input Rate 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Amirouche, F.M.L. Computational Methods for Multibody Dynamics. Prince-Hall, Englewood Cliffs, New York, 1992.Google Scholar
  2. Ascher, U.M., and Petzold, L.R. Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. SIAM, Philadelphia. 1988.Google Scholar
  3. Bae, D.-S., and Yang, S.-M. A Stabilization method for kinematic and kinetic constraint equations. In: Real-Time Integration Methods for Mechanical System Simulation, Haug, E.J., and Deyo, R. C. (Eds.), NATO ASI Series, Vol. F 69, Springer, Berlin, Germany, 209–232, 1990.Google Scholar
  4. Baumgarte, J. Stabilization of constraints and integrals of motion in dynamical systems. Computer Methods in Applied Mechanics and Engineering, 1, 1–16, 1972.zbMATHCrossRefMathSciNetGoogle Scholar
  5. Blajer, W., Schiehlen, W., and Schirm, W. A projective criterion for the coordinate partitioning method for multibody dynamics. Archive of Applied Mechanics, 64, 86–98, 1994.zbMATHGoogle Scholar
  6. Blajer W. An effective solver for absolute variable formulation of multibody dynamics, Computational Mechanics, 15, 460–472, 1995.zbMATHCrossRefGoogle Scholar
  7. Blajer, W. A geometric unification of constrained system dynamics. Multibody System Dynamics, 1, 3–21, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  8. Blajer, W. A geometrical interpretation and uniform matrix formulation of multibody system dynamics. ZAMM, 81, 247–259, 2001.zbMATHCrossRefMathSciNetGoogle Scholar
  9. Blajer, W. Elimination of constraint violation and accuracy aspects in numerical simulation of multibody systems. Multibody Systems Dynamics, 7, 265–284, 2002.zbMATHCrossRefMathSciNetGoogle Scholar
  10. Bottasso, C.L., Dopico, D., and Trainelli L. On the optimal scaling of index three DAEs in multibody dynamics. Multibody System Dynamics, 19, 3–20, 2008.zbMATHCrossRefMathSciNetGoogle Scholar
  11. Brenan, K.E., Campbell, S.L., and Petzold, L.R. Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, Elsevier, New York, New York, 1989.zbMATHGoogle Scholar
  12. Campbell, S.L., and Gear, C.W. The index of general nonlinear DAEs. Numerische Mathematik, 72, 173–196, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  13. Campbell, S.L., and Meyer, C.D. Generalized Inverses of Linear Transformations, Pitman, London, United Kingdom, 1979.zbMATHGoogle Scholar
  14. Chang, C.O., and Nikravesh, P.E. An adaptive constraint violation stabilization method for dynamic analysis of mechanical systems. Journal of Mechanisms, Transmissions, and Automation in Design, 107, 488–492, 1985.Google Scholar
  15. Chen, S., Hansen, J.M., and Tortorelli, A. Unconditionally energy stable implicit time integration: application to multibody system analysis and design. International Journal for Numerical Methods in Engineering, 48, 791–822, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  16. Chung, J., and Hulbert, G.M. A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-α method. Journal of Applied Mechanics, 60, 371–375, 1993.zbMATHCrossRefMathSciNetGoogle Scholar
  17. Duff, I.S., and Gear C.W. Computing the structural index. SIAM Journal on Algebraic and Discrete Methods, 7, 594–603, 1986.zbMATHCrossRefMathSciNetGoogle Scholar
  18. Eich-Soellner, E., and Führer, C. Numerical Methods in Multibody Dynamics, B.G. Teubner, Stuttgart. (1988).Google Scholar
  19. García de Jalón, J., and Bayo, E. Kinematic and Dynamic Simulation of Multibody Systems: the Real-Time Challenge. Springer-Verlag, New York, New York, 1994.Google Scholar
  20. Gear, C.W. The simultaneous numerical solution of differential-algebraic equations. IEEE Transactions on Circuit Theory, TC-18, 89–95, 1971.Google Scholar
  21. Gear, C.W. Differential-algebraic equation index transformations. SIAM Journal on Scientific and Statistical Computing, 9, 39–47, 1988.zbMATHCrossRefMathSciNetGoogle Scholar
  22. Gear, C.W. An introduction to numerical methods for ODEs and DAEs. In: Real-Time Integration Methods for Mechanical System Simulations, Haug, E.J., and Deyo, R.C. (Eds.), NATO ASI Series, Vol. F69, Springer, Berlin, Germany, 115–126, 1990.Google Scholar
  23. Gear, C.W., Leimkuhler, B., and Gupta, G.K. Automatic integration of Euler-Lagrange equations with constraints. Journal of Computational and Applied Mathematics, 12(13), 77–90, 1985.CrossRefMathSciNetGoogle Scholar
  24. Gear, C.W., and Petzold, L.R. ODE methods for the solution of differential/algebraic systems. SIAM Journal on Numerical Analysis, 21, 716–728, 1984.zbMATHCrossRefMathSciNetGoogle Scholar
  25. Haug, E.J. Computer-Aided Kinematics and Dynamics of Mechanical Systems. Allyn and Bacon, Boston, Massachusetts, 1989.Google Scholar
  26. Kim, S.S., and Vanderploeg, M.J. A general and efficient method for dynamic analysis of mechanical systems using velocity transformations. Journal of Mechanisms, Transmissions, and Automation in Design, 108, 176–182, 1986.CrossRefGoogle Scholar
  27. Liang, C.G., and Lance, G.M. A differentiable null space method for constrained dynamic analysis. Journal of Mechanisms, Transmissions, and Automation in Design, 109, 405–411, 1987.Google Scholar
  28. Lin, S.-T., and Hong, M.-C. Stabilization method for numerical integration of multibody mechanical systems. Journal of Mechanical Design, 120, 565–572, 1998.CrossRefGoogle Scholar
  29. Mani, N.K., Haug, E.J., and Atkinson K.E. Application of singular value decomposition for analysis of mechanical system dynamics. Journal of Mechanisms, Transmissions, and Automation in Design, 107, 82–87, 1985.Google Scholar
  30. Nikravesh, P.E. Computer-Aided Analysis of Mechanical Systems. Prentice-Hall, Englewood Cliffs, New Jersey, 1988.Google Scholar
  31. Nikravesh, P.E. Initial condition correction in multibody dynamics. Multibody System Dynamics, 18, 107–115, 2007.zbMATHCrossRefMathSciNetGoogle Scholar
  32. Ostermeyer, G.-P. On Baumgarte stabilization for differential algebraic equations. In: Real-Time Integration Methods for Mechanical System Simulations, Haug, E.J., and Deyo, R.C. (Eds.), NATO ASI Series, Vol. F69, Springer, Berlin, Germany, 193–207, 1990.Google Scholar
  33. Rabier, P.J., and Rheinboldt, W.C. Theoretical and Numerical Analysis of Differential-Algebraic Equations. In: Handbook of Numerical Analysis, Vol. VIII, Ciarlet, P.G., and Lions, J.L. (Eds.), Elsevier, Amsterdam, The Netherlands, 2002.Google Scholar
  34. Schiehlen, W. (ed.) Multibody System Handbook. Springer-Verlag, Berlin, Germany, 1990.Google Scholar
  35. Simo, J.C., and Wong, K.K. Unconditionally stable algorithms for rigid body dynamics that exactly preserve energy and momentum. International Journal for Numerical Methods in Engineering, 31, 19–52, 1991.zbMATHCrossRefMathSciNetGoogle Scholar
  36. Singh, R.P., and Likins P.W. Singular value decomposition for constrained dynamical systems. Journal of Applied Mechanics, 52, 943–948, 1985.zbMATHCrossRefMathSciNetGoogle Scholar
  37. Wehage, R.A., and Haug, E.J. Generalized coordinate partitioning for dimension reduction in analysis of constrained dynamic systems. Journal of Mechanical Design, 104, 247–255, 1982.Google Scholar
  38. Yoon, S., Howe, R.M., and Greenwood, D.T. Geometric elimination of constraint violations in numerical simulation of Lagrangian equations. Journal of Mechanical Design, 116, 1058–1064, 1994.CrossRefGoogle Scholar
  39. Yoon, S., Howe, R.M., and Greenwood, D.T. Stability and accuracy analysis of Baumgarte’s constraint violation stabilization method. Journal of Mechanical Design, 117, 446–453, 1995.CrossRefGoogle Scholar

Copyright information

© CISM, Udine 2009

Authors and Affiliations

  • Wojciech Blajer
    • 1
  1. 1.Institute of Applied Mechanics, Faculty of Mechanical EngineeringTechnical University of RadomRadomPoland

Personalised recommendations