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Guide to Dynamic Simulations of Rigid Bodies and Particle Systems pp 291–305Cite as

Appendix B: Numerical Solution of Ordinary Differential Equations of Motion

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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

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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)

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  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)

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

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    1. Murilo G. Coutinho
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    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

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

    • Publisher Name: Springer, London

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