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Independent Variable Formulations

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

Abstract

In Chapter 5 the dependent variable formulations for simulation of multibody systems were described. The derivation of equations of motion in terms of dependent states, though conceptually simple and easy to handle, leads to large-dimension governing DAEs, which results in computationally inefficient algorithms, burdened further with the constraint violation problem. An old and legitimate approach to the dynamics formulation is therefore to use a minimal number of independent state variables for a unique representation of motion by means of pure reduced-dimension ODEs. The numerical integration of the ODEs is usually by far more efficient compared to the integration of the governing DAEs, and the numerical solution is released from the problem of constraint violation. On the other hand, the minimal-form ODE formulations require often more modeling effort and skill. Effective and computer-oriented modeling procedures of this type are thus highly desirable.

Keywords

Multibody System Nonholonomic System Multibody System Dynamic Constraint Reaction Generalize Coordinate Partitioning 
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.

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References

  1. Arczewski, K., and Blajer, W. A unified approach to the modelling of holonomic and nonholonomic mechanical systems. Mathematical Modelling of Systems, 2, 157–174, 1996.zbMATHCrossRefMathSciNetGoogle Scholar
  2. Blajer, W. A geometric unification of constrained system dynamics. Multibody System Dynamics, 1, 3–21, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  3. Blajer, W. A geometrical interpretation and uniform matrix formulation of multibody system dynamics. ZAMM, 81, 247–259, 2001.zbMATHCrossRefMathSciNetGoogle Scholar
  4. Blajer, W., and Kolodziejczyk, K. Modeling and simulation of a sphere rolling on the inside of a rough vertical cylinder. Mathematical and Computer Modelling of Dynamical Systems, 12, 543–553, 2006.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. Desloge, E.A. The Gibbs-Appell equations of motion. American Journal of Physics, 56, 841–846, 1998.CrossRefMathSciNetGoogle Scholar
  7. Essén, H. On the geometry of nonholonomic dynamics. Journal of Applied Mechanics, 61, 689–694, 1994.zbMATHCrossRefGoogle Scholar
  8. 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
  9. Kane, T.R. Dynamics of nonholonomic systems. Journal of Applied Mechanics, 28, 69–87, 1961.MathSciNetGoogle Scholar
  10. Kane, T.R., and Levinson, D.A. Formulation of equations of motion for complex spacecraft. Journal of Guidance and Control, 3, 99–112, 1980.zbMATHCrossRefMathSciNetGoogle Scholar
  11. Kane, T.R., and Levinson, D.A. Dynamics: Theory and Applications. McGraw-Hill, New York, New York, 1985.Google Scholar
  12. Kane, T.R., and Wang, C.F. On the derivation of equations of motion. Journal of the Society for Industrial and Applied Mathematics, 13, 487–492, 1965.zbMATHCrossRefMathSciNetGoogle Scholar
  13. Lesser, M. A geometrical interpretation of Kane’s equations. Proceedings of the Royal Society in London, A436, 69–87, 1992.zbMATHCrossRefMathSciNetGoogle Scholar
  14. Neimark, J.I., and Fufaev, N.A. Dynamics of Nonholonomic Systems. Translations of Mathematical Monographs, No. 33, American Mathematical Society, Providence, 1972.zbMATHGoogle Scholar
  15. Nikravesh, P.E. Computer-Aided Analysis of Mechanical Systems. Prentice-Hall, Englewood Cliffs, New Jersey, 1988.Google Scholar
  16. Nikravesh, P.E. Systematic reduction of multibody equations of motion to a minimal set. International Journal of Non-Linear Mechanics, 25, 143–151, 1990.zbMATHCrossRefGoogle Scholar
  17. Nikravesh, P.E., and Ambrosio J. Systematic construction of equations of motion for rigid-flexible multibody systems containing open and closed kinematic loops. International Journal for Numerical Methods in Engineering, 32, 1749–1766, 1991.zbMATHCrossRefGoogle Scholar
  18. Papastavridis, J.G. Analytical Mechanics: A Comprehensive Treatise on the Dynamics of Constrained Systems; for Engineers, Physicists and Mathematicians. Oxford University Press, New York, New York, 2002.zbMATHGoogle Scholar
  19. Schiehlen, W. Multibody system dynamics: roots and perspectives. Multibody System Dynamics, 1, 149–188, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  20. Schiehlen, W. Research trends in multibody system dynamics. Multibody System Dynamics, 18, 3–13, 2007.zbMATHCrossRefMathSciNetGoogle Scholar
  21. 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.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

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