Generation of Symbolic Equations of Motion of Multibody Systems

  • E. J. Kreuzer
Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 343)


In applied mathematics and mechanical engineering we normally think of numerical calculations if computers are involved. Although simulations by means of numerical methods are powerful tools for investigations in mechanics, they do have drawbacks, e.g. finite precision, errors generated when evaluating expressions. General global results or proofs of theoretical results cannot be obtained from simulations.

A broader understanding of mechanical phenomena can be gained by means of analytical methods. But even for seemingly simple mathematical models, analytical calculations by paper and pencil may become very time-consuming, may be the source of many errors, and will sometimes be impossible. In such cases, computerized symbolic manipulation is clearly faster as well as safer and therefore preferable. But often, purely symbolical investigations cannot fulfill all of our needs in mechanics. Therefore, a semi-analytical approach, combining the features of analytical and numerical computations, is a most desirable synthesis. This allows the analytic work to be pushed further before numerical computations start.

The aim of the course is to present important software tools, basic concepts, methods, and applications of computerized symbolic manipulation to mechanics problems. A survey on different approaches and possibilities for symbol manipulation is followed by a review of all major general purpose software packages. Lectures on generation of equations of motion, structural mechanics in general, analysis of nonlinear dynamic systems, and perturbation methods represent some of the most advanced applications of symbol manipulation methods. The main objective of this course is to encourage the use of symbolic manipulation in the analysis of mechanical systems.


Rigid Body Multibody System Computer Algebra Computer Algebra System Industrial Robot 
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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  1. Andrews, G. C. (1983), A Brief Survey of “Self-Formulating” Simulation Programs for Multibody Dynamic Systems. In: Proc. IASTED Intern. Symp. Simulation and Modeling.Google Scholar
  2. Bremer, H. (1988), Dynamik und Regelung mechanischer Systeme. Stuttgart: Teubner.CrossRefMATHGoogle Scholar
  3. Buchberger, B., Collins, G.E., and Loos, R. (eds.) (1982), Computer Algebra — Symbolic and Algebraic Computation. Wien, New York: Springer-Verlag.MATHGoogle Scholar
  4. Daberkow, A. (1993), Zur CAD-gestützten Modellierung von Menrkörpersystemen. Fortschritt-Berichte der VDI-Zeitschriften, Reihe 20, Nr. 80, Düsseldorf, VDI-Verlag.Google Scholar
  5. Daberkow, A., and Kreuzer, E. (1994), An Integrated Approach for Computer Aided Design in Multibody System Dynamics. To appear.Google Scholar
  6. Davis, M.S. (1978), Analytical Mathematics on Computers. Applied Mechanics Reviews, 31, 1–9.Google Scholar
  7. Duffek, W., Führer, C, Schwartz, W., and Wallrapp, O. (1986), Analysis and Simulation of Rail and Road Vehicles with the program MEDYNA. In: Proc. 9th IAVSD-Symposium on the Dynamics of Vehicles on Roads and on Tracks, Nordström, O. (ed.), Lisse: Swets and Zeitlinger, 71–85.Google Scholar
  8. Frisch, H. P. (1990), NBOD & DISCOS — Dynamic Interaction Simulation of Controls and Structure. In: Multibody Systems Handbook, Schiehlen, W. (ed.), Berlin/... : Springer-Verlag, 145–160.CrossRefGoogle Scholar
  9. Fitch, J. (1993), Mathematics goes automatic. Physics World, 48–52.Google Scholar
  10. Grossmann, R. (ed.) (1989), Symbolic computation: application to Scientific computing. Philadelphia: Soc. for Ind. and Appl. Math. (SIAM).Google Scholar
  11. Haug, E. J. (1984), Elements and Methods of Computational Dynamics. In: Computer Aided Analysis and Optimization of Mechanical System Dynamics, Haug, E. J. (ed.), Berlin/... : Springer-Verlag, 3–38.CrossRefGoogle Scholar
  12. Jiménez, J. M., Avello, A., Garcia-Alonso, A., and Garcia de Jalon, J. (1990), COM-PAMM — A Simple and Efficient Code for Kinematic and Dynamic Numerical Simulation of 3-D Multibody Systems with Realistic Graphics, In: Multibody Systems Handbook, Schiehlen, W. (ed.), Berlin/... : Springer-Verlag, 285–304. — Simulation and Software Tools, Schiehlen, W. (ed.). Dordrecht/...: Kluwer Academic Publishers, 19–48.CrossRefGoogle Scholar
  13. Kahrimanian, H.G. (1954), Analytical Differentiation by a Digital Computer. Sympos. on Automatic Programm. for Digital Comput., Off. of Naval Res., Dept. of the Navy, 6–14.Google Scholar
  14. Kortiim, W. and Schiehlen, W. (1985), General Purpose Vehicle System Dynamics Software Based on Multibody Formalisms. Vehicle System Dynamics, 14, 229–263.CrossRefGoogle Scholar
  15. Kreuzer, E. (1979), Symbolische Berechnung der Bewegungsgleichungen von Mehrkörpersystemen. Fortschritt-Berichte der VDI-Zeitschriften, Reihe 11, Nr. 32, Düsseldorf: VDI-Verlag.Google Scholar
  16. Kreuzer, E. and Leister, G. (1991), Programmpaket NEWEUL’90. Stuttgart: Universität, Inst. B für Mechanik, Anleitung AN-23.Google Scholar
  17. Kreuzer, E. and Schiehlen, W. (1990), NEWEUL-Software for the Generation of Symbolical Equations of Motion. In: Multibody Systems Handbook, Schiehlen, W. (ed.), Berlin/... : Springer-Verlag, 181–202.CrossRefGoogle Scholar
  18. Leister, G. (1992), Beschreibung und Simulation von Mehrkörpersystemen mit geschlossenen kinematischen Schleifen. Fortschritt-Berichte der VDI-Zeitschriften, Reihe 11, Nr. 167, Düsseldorf, VDI-Verlag.Google Scholar
  19. Leister, G. and Bestie, D. (1992), Symbolic-numerical Solution of Multibody Systems with Closed Loops. International Journal of Vehicle Design.Google Scholar
  20. Levinson, D. A. and Kane, T. R. (1990), AUTOLEV — A New Approach to Multibody Dynamics. In: Multibody Systems Handbook, Schiehlen, W. (ed.), Berlin / ... : Springer-Verlag, 81–102.CrossRefGoogle Scholar
  21. Loos, R. (1982), Introduction. In: Computer Algebra — Symbolic and Algebraic Computation. Buchberger, B., Collins, G.E., and Loos, R. (eds.), Computing Supplementum 4. Wien, New York: Springer-Verlag.Google Scholar
  22. Nolan, J. (1953), Analytical Differentiation on a Digital Computer. Cambridge, Mass.: Inst. of Technol. (MIT), M.A. Thesis.Google Scholar
  23. Noor, A.K. and Andersen, C.M. (1979), Computerized Symbolic Manipulation in Structural Mechanics — Progress and Potential. Computers and Structures, 18, 95–118.CrossRefGoogle Scholar
  24. Orleanda, N. (1973), Node-Analogous Sparsity-Oriented Methods for Simulation of Mechanical Systems. Ph. D. dissertation, University of Michigan.Google Scholar
  25. Otter, M., Hooke, M., Daberkow, A., and Leister, G. (1993), An Object-Oriented Data Modul for Multibody Systems. In: Advanced Multibody System DynamicsGoogle Scholar
  26. Parker, T.S. and Chua, L.O. (1989), Practical Numerical Algorithms for Chaotic Systems. New York/...: Springer-Verlag.CrossRefMATHGoogle Scholar
  27. Pavelle, R., Rothstein, M., and Fitch, J. (1981), Computer Algebra. Scientific American, 136–152.Google Scholar
  28. Rand, R.H. and Armbruster, D. (1987), Perturbation Methods, Bifurcation Theory and Computer Algebra. New York/...: Springer-Verlag.CrossRefMATHGoogle Scholar
  29. Rauh, J. (1987), Ein Beitrag zur Modellierung elastischer Balkensysteme. Fortschritt-Berichte der VDI-Zeitschriften, Reihe 18, Nr. 37, Düsseldorf, VDI-Verlag.Google Scholar
  30. Rosenthal, D.E. and Sherman, M.A. (1986), High performance multibody simulation via symbolic equation manipulation and Kane’s method. J. of Astronautical Sciences, 34, 223–239.Google Scholar
  31. Rulka, W. (1990), SIMPACK — A Computer Program for Simulation of Large-Motion Multibody Systems. In: Multibody Systems Handbook, Schiehlen, W. (ed.), Berlin / ... : Springer-Verlag, 265–284.CrossRefGoogle Scholar
  32. Schiehlen, W. (ed.) (1990), Multibody Systems Handbook. Berlin/...: Springer-Verlag.MATHGoogle Scholar
  33. Schirm, W. (1993), Symbolisch-numerische Behandlung von kinematischen Schleifen in Mehrkörpersystemen. Fortschritt-Berichte der VDI-Zeitschriften, Reihe 11, Nr. 198, Düsseldorf, VDI-Verlag.Google Scholar
  34. Schmoll, K.-P. (1988), Modularer Aufbau von Mehrkörper-Systemen unter Verwendung der Relativkinematik, Fortschritt-Berichte der VDI-Zeitschriften, Reihe 18, Nr. 57, Düsseldorf: VDI-Verlag.Google Scholar
  35. Schramm, D. (1986), Ein Beitrag zur Dynamik reibungsbehafteter Mehrkörpersysteme. Fortschritt-Berichte der VDI-Zeitschriften, Reihe 18, Nr. 32, Düsseldorf, VDI-Verlag.Google Scholar
  36. Sunada, W.H. and Dubowsky, S. (1982), On the Dynamic Analysis and Behavior of Industrial Robotic Manipulators with Elastic Members. ASME Journal of Mechanical Design 1–10.Google Scholar
  37. Truckenbrodt, A. (1980): Bewegungsverhalten und Regelung hybrider Mehrkörpersysteme mit Anwendung auf Industrieroboter. Fortschritt-Berichte der VDI-Zeitschriften, Reihe 8, Nr. 33, Düsseldorf: VDI-Verlag.Google Scholar
  38. Wittenburg, J. and Wolz, U. (1985), MESA VERDE — Ein Computerprogramm zur Simulation der nichtlinearen Dynamik von Vielkörpersystemen. Robotersysteme, 1, 7–18.Google Scholar
  39. Wolfram, S. (1988), Mathematica — A System for Doing Mathematics by Computer. Redwood City/...: Addison-Wesley Publ. Comp.MATHGoogle Scholar
  40. Zeis, E. and Cefola, P. (1980), Computerized Algebraic Utilities for the Construction of Nonsingular Satellite Theories. J. Guidance and Control, 3, 48–54.CrossRefMATHGoogle Scholar
  41. N.N. (1989), ARIES Conceptstation Software Simulation Mechanism Reference. Aries Technology Inc., Lowell, M.A.Google Scholar
  42. N.N. (1987), PATRAN Mechanical System Simulation, P/Mechanism User’s Guide. PDA Engineering, Costa Mesa, CA.Google Scholar

Copyright information

© Springer-Verlag Wien 1994

Authors and Affiliations

  • E. J. Kreuzer
    • 1
  1. 1.Technical University Hamburg-HarburgHamburgGermany

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