Abstract
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.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
References
Biggs, N. L., Lloyd, E. K., and Wilson, R. J.,Graph Theory: 1736–1936, Oxford University Press, Oxford, 1976.
Seshu, S. and Reed, M. B.,Linear Graphs and Electrical Networks, Addison-Wesley, London, 1961.
Busacker, R. G. and Saaty, T. L.,Finite Graphs and Networks: An Introduction with Applications, McGraw-Hill, New York, 1965.
Koenig, H. E., Tokad, Y., and Kesavan, H. K.,Analysis of Discrete Physical Systems, McGraw-Hill, New York, 1967.
Trent, H. M., ‘Isomorphisms between oriented linear graphs and lumped physical systems’,Journal of the Acoustic Society of America 27, 1955, 500–527.
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.
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.
Nikravesh, P. E.,Computer-Aided Analysis of Mechanical Systems, Prentice-Hall, New Jersey, 1988.
Haug, E. J.,Computer-Aided Kinematics and Dynamics of Mechanical Systems, Volume 1, Allyn and Bacon, Boston, Massachusetts, 1989.
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.
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.
Shabana, A. A.,Dynamics of Multibody Systems, Wiley, New York, 1989.
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.
Wittenburg, J.,Dynamics of Systems of Rigid Bodies, B. G. Teubner, Suttgart, Germany, 1977.
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.
Li, T. W., ‘Dynamics of rigid body systems: A vector-network approach’, M.A.Sc. Thesis, University of Waterloo, Canada, 1985.
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.
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.
Wittenburg, J., ‘Graph-theoretical methods in multibody dynamics’,Contemporary Mathematics 97, 1989, 459–468.
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.
Huston, R. L. and Passerello, C., ‘On multi-rigid-body system dynamics’,Computers & Structures 10, 1979, 439–446.
Amirouche, F. M. I.,Computational Methods in Multibody Dynamics, Prentice-Hall, New Jersey, 1992.
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.
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.
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.
Hiller, M., Kecskemethy, A., and Woernle, C., ‘A loop-based kinematical analysis of complex mechanisms’,ASME Paper 86-DET-184, 1986.
Hwang, R. S. and Haug, E. J., ‘Topological analysis of multibody systems for recursive dynamics formulations’,Mechanisms, Structures, and Machines 17, 1989, 239–258.
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.
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.
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.
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.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
McPhee, J.J. On the use of linear graph theory in multibody system dynamics. Nonlinear Dyn 9, 73–90 (1996). https://doi.org/10.1007/BF01833294
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01833294