Generalized Coordinate Partitioning Methods for Numerical Integration of Differential-Algebraic Equations of Dynamics

  • Edward J. Haug
  • Jeng Yen
Conference paper
Part of the NATO ASI Series book series (volume 69)


Explicit and implicit, stiffly stable numerical integration algorithms based on generalized coordinate partitioning are developed and demonstrated for automated simulation of multibody dynamic systems. The concept of generalized coordinate partitioning is used to parameterize the constraint set with independent generalized coordinates. A stiffly stable, backward differentiation formulas are applied to determine independent generalized coordinates and velocities. Dependent generalized coordinates, velocities, and accelerations, as well as Lagrange multipliers that account for constraints, are explicitly retained in the formulation to satisfy all of the governing kinematic and dynamic equations. The algorithm is shown to be valid and accurate, both theoretically and through solution of a numerical example.


Multibody System Newton Iteration Differentiation Formula Algebraic Constraint Multibody Dynamic System 
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. 1.
    ADAMS users’ guide: Mechanical Dynamics, Inc., 555 South Forest, Ann Arbor, Mich: USA 1981.Google Scholar
  2. 2.
    DADS users’ manual: Rev. 4.0, Computer Aided Design Software, Inc., Oakdale, Iowa: USA 1987.Google Scholar
  3. 3.
    Roberson, R.E.: The path matrix of a graph, its construction and its use in evaluating certain products: Computer Methods in Applied Mechanics and Engineering, Vol. 24: pp. 47–56 1984.CrossRefGoogle Scholar
  4. 4.
    Wittenburg, J.: Dynamics of systems of rigid bodies B.G. Teubner: Stuttgart, 1977.Google Scholar
  5. 5.
    Wittenburg, J., and Wolz, U.: Mesa Verde: A symbolic program for nonlinear articulated rigid body dynamics: ASME: Paper No. 85-DET-141 1985.Google Scholar
  6. 6.
    Bae, D.S., Hwang, R.S., and Haug, E.J.: A recursive formulation for real-time dynamic simulation: Advances in Design Automation ASME: New York, NY, USA pp. 499–508 1988.Google Scholar
  7. 7.
    Hwang, R.S., Bae, D.S., Haug, E.J., and Kuhl, J.G.: Parallel processing for real-time dynamic system simulation: Advances in Design Automation ASME: New York, NY, USA pp. 509–518 1988.Google Scholar
  8. 8.
    Petzold, L.: Differential/algebraic equations are not ODEs: SIAM J. Sci. Stat. Comput. Vol. 3: No. 3: pp. 367–384 1982.MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Gear, C.W., and Petzold, L.R.: ODE methods for the solution of differential/algebraic systems: SIAM J. Num. Anal. Vol. 21: No. 4: pp. 716–728 1984.MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Wehage, R.A., and Haug, E.J.: Generalized coordinates partitioning for dimension reduction in analysis of constrained dynamic system: J. of Mechanical Design Vol. 104: pp. 247–255 1982.CrossRefGoogle Scholar
  11. 11.
    Mani, N.K., and Haug, E.J.: Singular value decomposition for dynamic system design sensitivity analysis: Engineering with Computers Vol. 1: pp. 103–109 1985.MathSciNetCrossRefGoogle Scholar
  12. 12.
    Liang, C.G., and Lance, G.M.: A differentiate null space method for constrained dynamic analysis J. of Mechanisms, Transmissions, and Automation in Design, Vol. 109: pp. 405–411 1987.CrossRefGoogle Scholar
  13. 13.
    Park, T.: A hybrid constraint stabilization-generalized coordinate partitioning method for machine dynamics: J. of Mechanisms, Transmissions, and Automation in Design Vol. 108: No. 2: pp. 211–216 1986.CrossRefGoogle Scholar
  14. 14.
    Brenan, K.E., Campbell, S.L., and Petzold, L.R.: Numerical solution of initial-value problems in differential-algebraic equations North-Holland: New York, NY, USA 1989.Google Scholar
  15. 15.
    Fuhrer, C.: On the description of constrained mechanical systems by differential/algebraic equations German Aerospace Research Establishment ( DFVLR ): Institute for Flight System Dynamics: 1987.Google Scholar
  16. 16.
    Chace, M.A.: Methods and experience in computer aided design of large-displacement mechanical systems: (E.J. Haug, ed.). NATO ASI: Vol. F9: Computer Aided Analysis and Optimization of Mechanical System Dynamics, Berlin-Heidelberg: Springer- Verlag 1984.Google Scholar
  17. 17.
    Baumgarte, J.: Stabilization of constraints and integrals of motion in dynamical systems: Computer Methods in Applied Mechanics and Engineering, Vol. 1: pp. 1–16 1972MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Petzold, L., and Ldtstedt, P.: Numerical solution of nonlinear differential equations with algebraic constraints. II: Practical Implementations SIAM J. Sci. Stat. Comput.: Vol. 7: pp. 720–733 1986.MATHCrossRefGoogle Scholar
  19. 19.
    Haug, E.J.: Computer aided kinematics and dynamics of mechanical systems, Vol I: basic methods: Allyn and Bacon: Boston, MA USA 1989.Google Scholar
  20. 20.
    Strang, G.: Linear algebra and its applications, 2nd ed.: Academic Press: New York, NY USA 1980.Google Scholar
  21. 21.
    Gear, G.W.: Numerical initial value problems in ordinary differential equations: Prentice-Hall: New York, NY USA 1969.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1990

Authors and Affiliations

  • Edward J. Haug
    • 1
  • Jeng Yen
    • 1
  1. 1.Center for Computer Aided Design College of Engineering 208 Engineering Research FacilityThe University of IowaIowa CityUSA

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