Skip to main content

Appendix B: Numerical Solution of Ordinary Differential Equations of Motion

  • Chapter
  • 1373 Accesses

Part of the Simulation Foundations, Methods and Applications book series (SFMA)

Abstract

This Appendix discusses some of the most common methods used to integrate the differential equations of motion in dynamic simulations. These methods range from simple explicit-Euler, to more sophisticated Runge–Kutta methods, with adaptive time step sizing.

Keywords

  • Angular Momentum
  • Angular Position
  • Kutta Method
  • Truncation Error
  • Taylor Series Expansion

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.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-1-4471-4417-5_7
  • Chapter length: 15 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   59.99
Price excludes VAT (USA)
  • ISBN: 978-1-4471-4417-5
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   79.95
Price excludes VAT (USA)
Hardcover Book
USD   109.99
Price excludes VAT (USA)
Fig. 7.1
Fig. 7.2
Fig. 7.3

Notes

  1. 1.

    In this appendix, the word scene refers to the simulated world containing all bodies being simulated.

  2. 2.

    See Appendix D (Chap. 9) for details on how to compute the inertia tensor I b in body-frame coordinates.

  3. 3.

    They may have different constant values for different time intervals, but their value is constant within the same time interval.

  4. 4.

    Stability analysis of this method indicates that the numerical solution is stable for all time-step sizes.

  5. 5.

    The superscript ∗ is used to differentiate the \(\vec{k}_{3}\) time-derivative estimate from the \(\vec{k}_{2}\) estimate, since both refer to the same time t=(t 0+h/2).

References

  1. Baraff, D., Witkin, A.: Physically based modeling. SIGGRAPH Course Notes 13 (1998)

    Google Scholar 

  2. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C: The Art of Scientific Computing. Cambridge University Press, Cambridge (1996)

    Google Scholar 

  3. Sharp, P.W., Verner, J.H.: Completely embedded Runge–Kutta pairs. SIAM J. Numer. Anal. 31, 1169–1190 (1994)

    MathSciNet  MATH  CrossRef  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and Permissions

Copyright information

© 2013 Springer-Verlag London

About this chapter

Cite this chapter

Coutinho, M.G. (2013). Appendix B: Numerical Solution of Ordinary Differential Equations of Motion. In: Guide to Dynamic Simulations of Rigid Bodies and Particle Systems. Simulation Foundations, Methods and Applications. Springer, London. https://doi.org/10.1007/978-1-4471-4417-5_7

Download citation

  • DOI: https://doi.org/10.1007/978-1-4471-4417-5_7

  • Publisher Name: Springer, London

  • Print ISBN: 978-1-4471-4416-8

  • Online ISBN: 978-1-4471-4417-5

  • eBook Packages: Computer ScienceComputer Science (R0)