Nonlinear Dynamics

, Volume 9, Issue 1–2, pp 73–90 | Cite as

On the use of linear graph theory in multibody system dynamics

  • J. J. McPhee


Multibody dynamics involves the generation and solution of the equations of motion for a system of connected material bodies. The subject of this paper is the use of graph-theoretical methods to represent multibody system topologies and to formulate the desired set of motion equations; a discussion of the methods available for solving these differential-algebraic equations is beyond the scope of this work. After a brief introduction to the topic, a review of linear graphs and their associated topological arrays is presented, followed in turn by the use of these matrices in generating various graph-theoretic equations. The appearance of linear graph theory in a number of existing multibody formulations is then discussed, distinguishing between approaches that use absolute (Cartesian) coordinates and those that employ relative (joint) coordinates. These formulations are then contrasted with formal graph-theoretic approaches, in which both the kinematic and dynamic equations are automatically generated from a single linear graph representation of the system. The paper concludes with a summary of results and suggestions for further research on the graph-theoretical modelling of mechanical systems.

Key words

Multibody dynamics linear graph theory absolute and joint coordinates 


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  1. 1.
    Biggs, N. L., Lloyd, E. K., and Wilson, R. J.,Graph Theory: 1736–1936, Oxford University Press, Oxford, 1976.Google Scholar
  2. 2.
    Seshu, S. and Reed, M. B.,Linear Graphs and Electrical Networks, Addison-Wesley, London, 1961.Google Scholar
  3. 3.
    Busacker, R. G. and Saaty, T. L.,Finite Graphs and Networks: An Introduction with Applications, McGraw-Hill, New York, 1965.Google Scholar
  4. 4.
    Koenig, H. E., Tokad, Y., and Kesavan, H. K.,Analysis of Discrete Physical Systems, McGraw-Hill, New York, 1967.Google Scholar
  5. 5.
    Trent, H. M., ‘Isomorphisms between oriented linear graphs and lumped physical systems’,Journal of the Acoustic Society of America 27, 1955, 500–527.Google Scholar
  6. 6.
    Andrews, G. C., ‘A general restatement of the laws of dynamics based on graph theory’, inProblem Analysis in Science and Engineering, F. H. Branin, Jr. and K. Huseyin (eds.), Academic Press, New York, 1977, pp. 1–40.Google Scholar
  7. 7.
    Nikravesh, P. E. and Haug, E. J., ‘Generalized coordinate partitioning for analysis of mechanical systems with nonholonomic constraints’,ASME Journal of Mechanisms, Transmissions, and Automation in Design 105, 1983, 379–384.Google Scholar
  8. 8.
    Nikravesh, P. E.,Computer-Aided Analysis of Mechanical Systems, Prentice-Hall, New Jersey, 1988.Google Scholar
  9. 9.
    Haug, E. J.,Computer-Aided Kinematics and Dynamics of Mechanical Systems, Volume 1, Allyn and Bacon, Boston, Massachusetts, 1989.Google Scholar
  10. 10.
    Orlandea, N., Chace, M. A., and Calahan, D. A., ‘A sparsity-oriented approach to the dynamic analysis and design of mechanical systems — Parts 1 and 2’,ASME Journal of Engineering for Industry 99, 1977, 773–784.Google Scholar
  11. 11.
    Géradin, M., ‘Computational aspects of the finite element approach to flexible multibody systems’, inAdvanced Multibody System Dynamics, W. Schiehlen (ed.), Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993, pp. 337–354.Google Scholar
  12. 12.
    Shabana, A. A.,Dynamics of Multibody Systems, Wiley, New York, 1989.Google Scholar
  13. 13.
    Avello, A. and García de Jalón, J., ‘Dynamics of flexible multibody systems using cartesian co-ordinates and large displacement theory’,International Journal for Numerical Methods in Engineering 32, 1991, 1543–1563.Google Scholar
  14. 14.
    Wittenburg, J.,Dynamics of Systems of Rigid Bodies, B. G. Teubner, Suttgart, Germany, 1977.Google Scholar
  15. 15.
    Sheth, P. N. and Uicker, Jr., J. J., ‘IMP (Integrated Mechanisms Program), A computer-aided design analysis system for mechanisms and linkage’,ASME Journal of Engineering for Industry 94, 1972, 454–464.Google Scholar
  16. 16.
    Li, T. W., ‘Dynamics of rigid body systems: A vector-network approach’, M.A.Sc. Thesis, University of Waterloo, Canada, 1985.Google Scholar
  17. 17.
    Branin, Jr., F. H., ‘The relation between Kron's method and the classical methods of network analysis’,Matrix and Tensor Quarterly 12, 1962, 69–105.Google Scholar
  18. 18.
    Roberson, R. E. and Wittenburg, J., ‘A dynamical formalism for an arbitrary number of interconnected rigid bodies, with reference to the problem of satellite attitude control’, inProceedings of 3rd IFAC Congress, Vol. 1, Book 3, Paper 46D, Butterworth, London, England, 1966.Google Scholar
  19. 19.
    Wittenburg, J., ‘Graph-theoretical methods in multibody dynamics’,Contemporary Mathematics 97, 1989, 459–468.Google Scholar
  20. 20.
    McPhee, J. J., Ishac, M., and Andrews, G. C., ‘Wittenburg's formulation of multibody dynamics equations from a graph-theoretic perspective’,Mechanism and Machine Theory, accepted for publication, May 1995.Google Scholar
  21. 21.
    Huston, R. L. and Passerello, C., ‘On multi-rigid-body system dynamics’,Computers & Structures 10, 1979, 439–446.Google Scholar
  22. 22.
    Amirouche, F. M. I.,Computational Methods in Multibody Dynamics, Prentice-Hall, New Jersey, 1992.Google Scholar
  23. 23.
    Nikravesh, P. E. and Gim, G., ‘Systematic construction of the equations of motion for multibody systems containing closed kinematic loops’, inProceedings of ASME Design Automation Conference, Montreal, Canada, 1989, pp. 27–33.Google Scholar
  24. 24.
    Kim, S. S. and Vanderploeg, M. J., ‘A general and efficient method for dynamic analysis of mechanical systems using velocity transformations’,ASME Journal of Mechanisms, Transmissions, and Automation in Design 108, 1986, 176–182.Google Scholar
  25. 25.
    Pereira, M. S. and Proença, P. L., ‘Dynamic analysis of spatial flexible multibody systems using joint co-ordinates’,International Journal for Numerical Methods in Engineering 32, 1991, 1799–1812.Google Scholar
  26. 26.
    Hiller, M., Kecskemethy, A., and Woernle, C., ‘A loop-based kinematical analysis of complex mechanisms’,ASME Paper 86-DET-184, 1986.Google Scholar
  27. 27.
    Hwang, R. S. and Haug, E. J., ‘Topological analysis of multibody systems for recursive dynamics formulations’,Mechanisms, Structures, and Machines 17, 1989, 239–258.Google Scholar
  28. 28.
    Lai, H. J., Haug, E. J., Kim, S. S., and Bae, D. S., ‘A decoupled flexible-relative co-ordinate recursive approach for flexible multibody dynamics’,International Journal for Numerical Methods in Engineering 32, 1991, 1669–1689.Google Scholar
  29. 29.
    Andrews, G. C. and Kesavan, H. K., ‘The vector-network model: A new approach to vector dynamics’,Mechanisms and Machine Theory 10, 1975, 57–75.CrossRefGoogle Scholar
  30. 30.
    Andrews, G. C., Richard, M. J., and Anderson, R. J., ‘A general vector-network formulation for dynamic systems with kinematic constraints’,Mechanism and Machine Theory 23, 1988, 243–256.CrossRefGoogle Scholar
  31. 31.
    McPhee, J. J., ‘Formulation of multibody dynamics equations in absolute or relative coordinates using the vector-network method’,Machine Elements and Machine Dynamics, ASME DE-Vol. 71, September 1994, pp. 361–368.Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • J. J. McPhee
    • 1
  1. 1.Systems Design EngineeringUniversity of WaterlooOntarioCanada

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