Skip to main content
SpringerLink
Log in

Energy Considerations for the Stabilization of Constrained Mechanical Systems with Velocity Projection

  • Chapter
  • First Online:
Multibody Dynamics

Part of the book series: Computational Methods in Applied Sciences ((COMPUTMETHODS,volume 23))

Abstract

There are many difficulties involved in the numerical integration of index-3 Differential Algebraic Equations (DAEs), mainly related to stability, in the context of mechanical systems. An integrator that exactly enforces the constraint at position level may produce a discrete solution that departs from the velocity and/or acceleration constraint manifolds (invariants). This behaviour affects the stability of the numerical scheme, resulting in the use of stabilization techniques based on enforcing the invariants. A coordinate projection is a poststabilization technique where the solution obtained by a suitable DAE integrator is forced back to the invariant manifolds. This paper analyzes the energy balance of a velocity projection, providing an alternative interpretation of its effect on the stability and a practical criterion for the projection matrix selection.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Not necessarily the same employed for the velocity projection.

References

  1. Alishenas T, Ólafsson Ö (1994) Modeling and velocity stabilization of constrained mechanical systems. BIT Numer Math 34:455–483

    Article  MATH  Google Scholar 

  2. Arnold M, Bruls O (2007) Convergence of the generalized-alpha scheme for constrained mechanical systems. Multibody Syst Dyn 18:185–202

    Article  MathSciNet  MATH  Google Scholar 

  3. Ascher UM (1997) Stabilization of invariants of discretized differential systems. Numer Algorithm 14:1–23

    Article  MathSciNet  MATH  Google Scholar 

  4. Ascher UM, Chin H, Petzold LR, Reich S (1995) Stabilization of constrained mechanical systems with DAEs and invariant manifolds. J Mech Struct Mach 23:135–158

    Article  MathSciNet  Google Scholar 

  5. Ascher UM, Petzold LR (1998) Computer methods for ordinary differential equations and differential-algebraic equations. SIAM, Philadelphia, PA

    MATH  Google Scholar 

  6. Bauchau OA, Laulusa A (2008) Review of contemporary approaches for constraint enforcement in multibody systems. J Comput Nonlin Dynam (ASME) 3:1–8

    Google Scholar 

  7. Bayo E, Ledesma R (1996) Augmented Lagrangian and mass-orthogonal projection methods for constrained multibody dynamics. Nonlinear Dynam 9:113–130

    Article  MathSciNet  Google Scholar 

  8. Brenan KE, Campbell SL, Petzold LR (1996) Numerical solution of initial-value problems in differential-algebraic equations, SIAM, Philadelphia, PA, http://www.ec-securehost.com/SIAM/CL14.html

  9. Cuadrado J, Cardenal J, Bayo E (1997) Modeling and solution methods for efficient real-time simulation of multibody dynamics. Multibody Syst Dyn 1:259–280

    Article  MathSciNet  MATH  Google Scholar 

  10. Cuadrado J, Cardenal J, Morer P, Bayo E (2000) Intelligent simulation of multibody dynamics: space-state and descriptor methods in sequential and pararell computing environments. Multibody Syst Dyn 4:55–73

    Article  MATH  Google Scholar 

  11. Cuadrado J, Dopico D, Naya M, González M (2004) Penalty, semi-recursive and hybrid methods for MBS real-time dynamics in the context of structural integrators. Multibody Syst Dyn 12:117–132

    Article  MATH  Google Scholar 

  12. Eich E (1993) Convergence results for a coordinate projection method applied to mechanical systems with algebraic constraints. SIAM J Numer Anal 30(5):1467–1482

    Article  MathSciNet  MATH  Google Scholar 

  13. Eich E, Führer C, Leimkhuler B, Reich S (1990) Stabilization and projection methods for multibody dynamics – Helsinki Institute of Technology. Technical Report A281

    Google Scholar 

  14. García de Jalón J, Bayo E (1994) Kinematic and dynamic simulation of multibody systems. The real time challenge. Springer, New York

    Google Scholar 

  15. García Orden JC (2009) Energy considerations for the stabilization of constrained mechanical systems with velocity projection. Nonlinear Dynam. doi:10.1007/s11071-009-9579-8

    MATH  Google Scholar 

  16. García Orden JC, Dopico Dopico D (2007) On the stabilizing properties of energy-momentum integrators and coordinate projections for constrained mechanical systems. In: Multibody Dynamics. Computational Methods in Applied Sciences. Springer, The Netherlands, pp 49–67

    Google Scholar 

  17. García Orden JC, Goicolea JM (2000) Conserving properties in constrained dynamics of flexible multibody systems. Multibody Syst Dyn 4:225–244

    Article  MathSciNet  MATH  Google Scholar 

  18. García Orden JC, Goicolea JM (2005) Robust analysis of flexible multibody systems and joint clearances in an energy conserving framework. In: Advances in Computational Multibody Dynamics. Computational Methods in Applied. Springer, The Netherlands, pp 205–237

    Google Scholar 

  19. García Orden JC, Ortega R (2006) A conservative augmented Lagrangian algorithm for the dynamics of constrained mechanical systems. Mech Base Des Struct Mach 34(4):449–468

    Article  Google Scholar 

  20. Goicolea JM, García Orden JC (2002) Quadratic and higher-order constraints in energy-conserving formulations in flexible multibody systems. Multibody Syst Dyn 7:3–29

    Article  MathSciNet  MATH  Google Scholar 

  21. González O (1999) Mechanical systems subjected to holonomic constraints: Differential-algebraic formulations and conservative integration. Physica D 132:165–174

    Article  MathSciNet  MATH  Google Scholar 

  22. Hairer E, Wanner G (1991) Solving ordinary differential equations II, stiff and differential-algebraic problems. Springer, Berlin

    MATH  Google Scholar 

  23. Hughes T (1987) The finite element method. Prentice-Hall, Englewood Cliffs, NJ

    MATH  Google Scholar 

  24. Ortiz M (1986) A note on energy conservation and stability of nonlinear time-stepping algorithms. Comput Struct 24(1):167–168

    Article  Google Scholar 

  25. Lubich C (1991) Extrapolation integrators for constrained multibody systems. Impact Comput Sci Eng 3:213–234

    Article  MathSciNet  MATH  Google Scholar 

  26. Stuart A, Humphries A (1996) Dynamical systems and numerical analysis. Cambridge University Press, Cambridge

    MATH  Google Scholar 

Download references

Acknowledgements

The authors wish to acknowledge the financial support of the Spanish Ministry of Science and Innovation as part of project DPI-2006-15613-C03-02 under the name “Modelización numérica eficiente de grandes sistemas flexibles con aplicaciones de impacto”.

Author information

Authors and Affiliations

  1. ETSI Caminos, Canales y Puertos, Universidad Politécnica de Madrid, C/Profesor Aranguren s/n, 28040, Madrid, Spain

    Juan C. García Orden & Roberto A. Ortega Aguilera

Authors
  1. Juan C. García Orden
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Roberto A. Ortega Aguilera
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Juan C. García Orden .

Editor information

Editors and Affiliations

  1. Inst. Techniki Lotniczej, i Mechaniki Stosowanej, Technical University Warszawa, ul. Nowowiejska 24, Warszawa, 00-665, Poland

    Krzysztof Arczewski

  2. Dept. Mechanics, Technical University Radom, Ul. Malczewskiego 29, Radom, 26-600, Poland

    Wojciech Blajer

  3. Inst. Aeronautics &, Applied Mechanics, Warsaw University of Technology, ul. Nowowiejska 24, Warszawa, 00-665, Poland

    Janusz Fraczek

  4. Inst. Aeronautics &, Applied Mechanics, Warsaw University of Technology, ul. Nowowiejska 24, Warszawa, 00-665, Poland

    Marek Wojtyra

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer Science+Business Media B.V.

About this chapter

Cite this chapter

Orden, J.C.G., Aguilera, R.A.O. (2011). Energy Considerations for the Stabilization of Constrained Mechanical Systems with Velocity Projection. In: Arczewski, K., Blajer, W., Fraczek, J., Wojtyra, M. (eds) Multibody Dynamics. Computational Methods in Applied Sciences, vol 23. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-9971-6_8

Download citation

  • DOI: https://doi.org/10.1007/978-90-481-9971-6_8

  • Published:

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-9970-9

  • Online ISBN: 978-90-481-9971-6

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation