Skip to main content

Numerical methods for simulation in applied dynamics

  • Chapter
Simulation Techniques for Applied Dynamics

Part of the book series: CISM International Centre for Mechanical Sciences ((CISM,volume 507))

Abstract

Applied dynamics may be considered as integration platform for simulation in various fields of engineering like vehicle system dynamics, dynamics of machines and mechanisms and robotics. In the present contribution we discuss classical numerical simulation techniques of nonlinear system dynamics, their use in multibody system simulation and extensions to typical problems of applied dynamics like continuous and discrete controllers and multidisciplinary applications. A frequently used alternative approach to the analysis of multidisciplinary problems is based on the coupling of two or more monodisciplinary simulation packages. Typical numerical problems of such co-simulation techniques will be considered and illustrated by numerical tests.

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

Access this chapter

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

Bibliography

  • M. Arnold. A perturbation analysis for the dynamical simulation of mechanical multibody systems. Applied Numerical Mathematics, 18:37–56, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  • M. Arnold. Numerical problems in the dynamical simulation of wheel-rail systems. Z. Angew. Math. Mech., Proceedings of ICIAM 95, Issue 3:151–154, 1996.

    Google Scholar 

  • M. Arnold. Multi-rate time integration for large scale multibody system models. In P. Eberhard, editor, IUTAM Symposium on Multiscale Problems in Multibody System Contacts, pages 1–10. Springer, 2007.

    Google Scholar 

  • M. Arnold and M. Günther. Preconditioned dynamic iteration for coupled differential-algebraic systems. BIT Numerical Mathematics, 41:1–25, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  • M. Arnold and H. Netter. Wear profiles and the dynamical simulation of wheel-rail systems. In M. Brøns, M.P. Bendsøe, and M.P. Sørensen, editors, Progress in Industrial Mathematics at ECMI 96, pages 77–84. Teubner, Stuttgart, 1997.

    Google Scholar 

  • M. Arnold, V. Mehrmann, and A. Steinbrecher. Index reduction in industrial multibody system simulation. Technical Report IB 532-01-01, DLR German Aerospace Center, Institute of Aeroelasticity, Vehicle System Dynamics Group, 2001.

    Google Scholar 

  • M. Arnold, A. Fuchs, and C. Führer. Efficient corrector iteration for DAE time integration in multibody dynamics. Comp. Meth. Appl. Mech. Eng., 195:6958–6973, 2006.

    Article  MATH  Google Scholar 

  • M. Arnold, B. Burgermeister, and A. Eichberger. Linearly implicit time integration methods in real-time applications: DAEs and stiff ODEs. Multibody System Dynamics, 17:99–117, 2007a.

    Article  MATH  MathSciNet  Google Scholar 

  • M. Arnold, Ch. Weidemann, and L. Mauer. Stepsize control versus fixed stepsize time integration: Theory and practical experience. In C.L. Bottasso, P. Masarati, and L. Trainelli, editors, Proc. of Multibody Dynamics 2007 (ECCOMAS Thematic Conference), Milan, Italy, 2007b.

    Google Scholar 

  • U. Ascher and L.R. Petzold. Computer Methods for Ordinary Differential Equations and Differential—Algebraic Equations. SIAM, Philadelphia, 1998.

    MATH  Google Scholar 

  • U.M. Ascher, H. Chin, and S. Reich. Stabilization of DAEs and invariant manifolds. Numer. Math., 67:131–149, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  • J. Baumgarte. Stabilization of constraints and integrals of motion in dynamical systems. Computer Methods in Applied Mechanics and Engineering, 1:1–16, 1972.

    Article  MATH  MathSciNet  Google Scholar 

  • H. Brandl, R. Johanni, and M. Otter. A very efficient algorithm for the simulation of robots and similar multibody systems without inversion of the mass matrix. In P. Kopacek, I. Troch, and K. Desoyer, editors, Theory of Robots, pages 95–100, Oxford, 1988. Pergamon Press.

    Google Scholar 

  • P.C. Breedveld. Port based modelling of multidomain physical systems in terms of bond graphs. — Published in this volume, 2008.

    Google Scholar 

  • K.E. Brenan, S.L. Campbell, and L.R. Petzold. Numerical solution of initial-value problems in differential-algebraic equations. SIAM, Philadelphia, 2nd edition, 1996.

    MATH  Google Scholar 

  • O. Brüls, A. Cardona, and M. Géradin. Modelling, simulation and control of flexible multibody systems. — Published in this volume, 2008.

    Google Scholar 

  • M. Busch. Entwicklung einer SIMPACK-Modelica/Dymola-Schnittstelle. Diploma Thesis, Martin Luther University Halle-Wittenberg, Institute of Computer Science, 2007.

    Google Scholar 

  • M. Busch, M. Arnold, A. Heckmann, and S. Dronka. Interfacing SIMPACK to Modelica / Dymola for multi-domain vehicle system simulations. SIMPACK News (INTEC GmbH, Wessling, Germany), 11(2):01–03, 2007.

    Google Scholar 

  • T.F. Coleman, B.S. Garbow, and J.J. Moré. Software for estimating sparse Jacobian matrices. ACM Transactions on Mathematical Software, 10: 329–345, 1984.

    Article  MATH  Google Scholar 

  • P. Deuflhard, E. Hairer, and J. Zugck. One-step and extrapolation methods for differential-algebraic systems. Numer. Math., 51:501–516, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  • S. Dietz. Vibration and Fatigue Analysis of Vehicle Systems using Component Modes. Fortschritt-Berichte VDI Reihe 12, Nr. 401. VDI-Verlag, Düsseldorf, 1999.

    Google Scholar 

  • S. Dietz. The new powerful linear subsystem solver for flexible bodies in multi-body systems. In J.M. Goicolea, J. Cuadrado, and J.C. García Orden, editors, Proc. of Multibody Dynamics 2005 (ECCOMAS Thematic Conference), Madrid, Spain, 2005.

    Google Scholar 

  • S. Dietz, H. Netter, and D. Sachau. Fatigue life prediction by coupling finite-element and multibody systems calculations. In Proceedings of DETC’97, ASME Design Engineering Technical Conferences DETC/VIB-4229, Sacramento, California, 1997.

    Google Scholar 

  • J.R. Dormand and P.J. Prince. A family of embedded Runge-Kutta formulae. J. Comp. Appl. Math., 6:19–26, 1980.

    Article  MATH  MathSciNet  Google Scholar 

  • S. Dronka. Die Simulation gekoppelter Mehrkörper-und Hydraulik-Modelle mit Erweiterung für Echtzeitsimulation. Shaker Verlag, Aachen, 2004.

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  • E. Eich-Soellner and C. Führer. Numerical Methods in Multibody Dynamics. Teubner-Verlag, Stuttgart, 1998.

    MATH  Google Scholar 

  • A. Eichberger. Simulation von Mehrkörpersystemen auf parallelen Rechnerarchitekturen. Fortschritt-Berichte VDI Reihe 8, Nr. 332. VDI-Verlag, Düsseldorf, 1993.

    Google Scholar 

  • H. Elmqvist, M. Otter, and F.E. Cellier. Inline integration: A new mixed symbolic / numeric approach for solving differential-algebraic equation systems. In Proc. ESM’95, European Simulation Multiconference, Prague, Czech Republic, June 5–8, 1995, pages xxiii–xxxiv, 1995.

    Google Scholar 

  • C. Führer. Differential-algebraische Gleichungssysteme in mechanischen Mehrkörpersystemen. Theorie, numerische Ansätze und Anwendungen. Technical report, TU München, Mathematisches Institut und Institut für Informatik, 1988.

    Google Scholar 

  • C. Führer and B. Leimkuhler. Numerical solution of differential-algebraic equations for constrained mechanical motion. Numer. Math., 59:5–69, 1991.

    Article  Google Scholar 

  • C.W. Gear and R.R. Wells. Multirate linear multistep methods. BIT, 24: 484–502, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  • C.W. Gear, B. Leimkuhler, and G.K. Gupta. Automatic integration of Euler-Lagrange equations with constraints. J. Comp. Appl. Math., 12 & 13:77–90, 1985.

    Article  MathSciNet  Google Scholar 

  • E. Hairer and G. Wanner. Solving Ordinary Differential Equations. II. Stiff and Differential-Algebraic Problems. Springer-Verlag, Berlin Heidelberg New York, 2nd edition, 1996.

    MATH  Google Scholar 

  • E. Hairer, S.P. Nørsett, and G. Wanner. Solving Ordinary Differential Equations. I. Nonstiff Problems. Springer-Verlag, Berlin Heidelberg New York, 2nd edition, 1993.

    MATH  Google Scholar 

  • A. Heckmann, M. Arnold, and O. Vaculín. A modal multifield approach for an extended flexible body description in multibody dynamics. Multibody System Dynamics, 13:299–322, 2005.

    Article  MATH  Google Scholar 

  • G. Hippmann, M. Arnold, and M. Schittenhelm. Efficient simulation of bush and roller chain drives. In J. M. Goicolea, J. Cuadrado, and J.C. García Orden, editors, Proc. of Multibody Dynamics 2005 (ECCOMAS Thematic Conference), Madrid, Spain, 2005.

    Google Scholar 

  • S. Iwnicki, editor. The Manchester Benchmarks for Rail Vehicle Simulation. Supplement to Vehicle System Dynamics, Vol. 31, Swets & Zeitlinger, Lisse, 1999.

    Google Scholar 

  • K. Jackson. A survey of parallel numerical methods for initial value problems for ordinary differential equations. IEEE. Transactions on Magnetics, 27:3792–3797, 1991.

    Article  Google Scholar 

  • J.J. Kalker. Three-Dimensional Elastic Bodies in Rolling Contact. Kluwer Academic Publishers, Dordrecht Boston London, 1990.

    MATH  Google Scholar 

  • C.T. Kelley. Iterative Methods for Optimization. SIAM, Philadelphia, 1999.

    MATH  Google Scholar 

  • C.T. Kelley. Solving Nonlinear Equations with Newton’s Method. SIAM, Philadelphia, 2003.

    MATH  Google Scholar 

  • W. Kortüm and P. Lugner. Systemdynamik und Regelung von Fahrzeugen. Springer-Verlag, Berlin Heidelberg New York, 1994.

    Google Scholar 

  • W. Kortüm and W. Schiehlen. General purpose vehicle system dynamics software based on multibody formalisms. Vehicle System Dynamics, 14: 229–263, 1985.

    Article  Google Scholar 

  • W. Kortüm, W.O. Schiehlen, and M. Arnold. Software tools: From multibody system analysis to vehicle system dynamics. In H. Aref and J.W. Phillips, editors, Mechanics for a New Millennium, pages 225–238, Dordrecht, 2001. Kluwer Academic Publishers.

    Google Scholar 

  • W.R. Krüger and M. Spieck. Aeroelastic effects in multibody dynamics. Vehicle System Dynamics, 41:383–399, 2004.

    Article  Google Scholar 

  • R. Kübler. Modulare Modellierung und Simulation mechatronischer Systeme. Fortschritt-Berichte VDI Reihe 20, Nr. 327. VDI-Verlag GmbH, Düsseldorf, 2000.

    Google Scholar 

  • R. Kübler and W. Schiehlen. Two methods of simulator coupling. Mathematical and Computer Modelling of Dynamical Systems, 6:93–113, 2000.

    Article  MATH  Google Scholar 

  • P. Kunkel, V. Mehrmann, W. Rath, and J. Weickert. GELDA: A software package for the solution of general linear differential algebraic equations. SIAM J. Sci. Comp., 18:115–138, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  • S. Liebig, S. Helduser, M. Stüwing, and S. Dronka. Die Modellierung und Simulation gekoppelter mechanischer und hydraulischer Systeme (http://vwitme011.vkw.tu-dresden.de/TrafficForum/vwt_2001/beitraege/VWT18proceedings_pages505-524.pdf). In Proc. of the 18th Dresden Conference on Traffic and Transportation Sciences, September 17–18, 2001, pages 505–524, 2001.

    Google Scholar 

  • Ch. Lubich, Ch. Engstler, U. Nowak, and U. Pöhle. Numerical integration of constrained mechanical systems using MEXX. Mech. Struct. Mach., 23:473–495, 1995.

    Article  Google Scholar 

  • P. Lugner and M. Plöchl, editors. Tyre Models for Vehicle Dynamics Analysis. Supplement 1 to Vol. 43 of Vehicle System Dynamics, 2005.

    Google Scholar 

  • C.B. Moler. Numerical Computing with MATLAB. SIAM, Philadelphia, 2004.

    MATH  Google Scholar 

  • J.J. Moré, D.C. Sorensen, B.S. Garbow, and K.E. Hillstrom. The MINPACK project. In W.J. Cowell, editor, Sources and Development of Mathematical Software, pages 88–111. Prentice-Hall, Englewood Cliffs, N.J., 1984.

    Google Scholar 

  • N. Orlandea. Development and Application of Node-Analogous Sparsity-Oriented Methods for Simulation of Mechanical Dynamic Systems. PhD thesis, University of Michigan, 1973.

    Google Scholar 

  • L.R. Petzold. Differential/algebraic equations are not ODEs. SIAM J. Sci. Stat. Comput., 3:367–384, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  • L.R. Petzold and P. Lötstedt. Numerical solution of nonlinear differential equations with algebraic constraints II: practical implications. SIAM J. Sci. Stat. Comput., 7:720–733, 1986.

    Article  MATH  Google Scholar 

  • A. Pfeiffer. Numerische Sensitivitätsanalyse unstetiger multidisziplinärer Modelle mit Anwendungen in der gradientenbasierten Optimierung. Fortschritt-Berichte VDI Reihe 20, Nr. 417. VDI-Verlag, Düsseldorf, 2008.

    Google Scholar 

  • F. Pfeiffer and Ch. Glocker. Multibody Dynamics with Unilateral Contacts. Wiley & Sons, New York, 1996.

    Book  MATH  Google Scholar 

  • G. Rill. Simulation von Kraftfahrzeugen. Fundamentals and Advances in the Engineering Sciences. Vieweg, Braunschweig Wiesbaden, 1994.

    Google Scholar 

  • R.E. Roberson and R. Schwertassek. Dynamics of Multibody Systems. Springer-Verlag, Berlin Heidelberg New York, 1988.

    MATH  Google Scholar 

  • W. Rulka. Effiziente Simulation der Dynamik mechatronischer Systeme für industrielle Anwendungen. PhD thesis, Vienna University of Technology, Department of Mechanical Engineering, 1998.

    Google Scholar 

  • W. Schiehlen. Multibody system dynamics: roots and perspectives. Multibody System Dynamics, 1:149–188, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  • W. Schiehlen and P. Eberhard. Multibody systems and applied dynamics. — Published in this volume, 2008.

    Google Scholar 

  • R. Schwertassek and O. Wallrapp. Dynamik flexibler Mehrkörpersysteme. Vieweg, 1999.

    Google Scholar 

  • A.A. Shabana. Dynamics of Multibody Systems. Cambridge University Press, Cambridge, 2nd edition, 1998.

    MATH  Google Scholar 

  • A.A. Shabana. Computational Dynamics. John Wiley & Sons, Inc., New York, 2nd edition, 2001.

    Google Scholar 

  • B. Simeon. Numerische Simulation gekoppelter Systeme von partiellen und differential-algebraischen Gleichungen in der Mehrkörperdynamik. Fortschritt-Berichte VDI Reihe 20, Nr. 325. VDI-Verlag, Düsseldorf, 2000.

    Google Scholar 

  • B. Simeon, C. Führer, and P. Rentrop. Differential-algebraic equations in vehicle system dynamics. Surveys on Mathematics for Industry, 1:1–37, 1991.

    MATH  MathSciNet  Google Scholar 

  • S. Skelboe and P.U. Andersen. Stability properties of backward Euler multirate formulas. SIAM J. Sci. Stat. Comp., 10:1000–1009, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  • M.M. Tiller. Introduction to Physical Modeling with Modelica. Kluwer Academic Publishers, Boston Dordrecht London, 2001.

    Google Scholar 

  • W. Trautenberg. Bidirektionale Kopplung zwischen CAD und Mehrkörpersimulationssystemen. PhD thesis, Munich University of Technology, Department of Mechanical Engineering, 1999.

    Google Scholar 

  • F.C. Tseng and G.M. Hulbert. Network-distributed multibody dynamics simulation — gluing algorithm. In J.A.C. Ambrósio and W.O. Schiehlen, editors, Advances in Computational Multibody Dynamics, pages 521–540, IDMEC/IST Lisbon, Portugal, 1999.

    Google Scholar 

  • O. Vaculín, M. Valášek, and W.R. Krüger. Overview of coupling of multibody and control engineering tools. Vehicle System Dynamics, 41:415–429, 2004.

    Article  Google Scholar 

  • M. Valášek. Modelling, simulation and control of mechatronical systems. — Published in this volume, 2008.

    Google Scholar 

  • M. Valášek, Z. Šika, and O. Vaculín. Multibody formalism for real-time application using natural coordinates and modified state space. Multibody System Dynamics, 17:209–227, 2007.

    Article  MATH  MathSciNet  Google Scholar 

  • A. Veitl. Integrierter Entwurf innovativer Stromabnehmer. Fortschritt-Berichte VDI Reihe 12, Nr. 449. VDI-Verlag, Düsseldorf, 2001.

    Google Scholar 

  • A. Veitl, T. Gordon, A. van de Sand, M. Howell, M. Valášek, O. Vaculín, and P. Steinbauer. Methodologies for coupling simulation models and codes in mechatronic system analysis and design. In Proc. of the 16th IAVSD-Symposium on Dynamics of Vehicles on Roads and Tracks, pages 231–243. Supplement to Vehicle System Dynamics, Vol. 33, Swets & Zeitlinger, 1999.

    Google Scholar 

  • R. von Schwerin. MultiBody System SIMulation — Numerical Methods, Algorithms, and Software, volume 7 of Lecture Notes in Computational Science and Engineering. Springer, Berlin Heidelberg, 1999.

    Google Scholar 

  • O. Wallrapp. Linearized flexible multibody dynamics including geometric stiffening effects. Mechanics of Structures and Machines, 19:385–409, 1991.

    Article  MathSciNet  Google Scholar 

  • O. Wallrapp. Standardization of flexible body modeling in multibody system codes, Part I: Definition of standard input data. Mechanics of Structures and Machines, 22:283–304, 1994.

    Article  Google Scholar 

  • R.A. Wehage and E.J. Haug. Generalized coordinate partitioning for dimension reduction in analysis of constrained dynamic systems. J. Mech. Design, 104:247–255, 1982.

    Article  Google Scholar 

  • Ch. Weidemann, L. Mauer, and M. Arnold. Improving the calculation speed for time domain integration of complex railway vehicles. In Poceedings of ACMD2006, The Third Asian Conference on Multibody Dynamics, August 1–4, 2006, Komaba, Tokyo, Japan, 2006.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2008 CISM, Udine

About this chapter

Cite this chapter

Arnold, M. (2008). Numerical methods for simulation in applied dynamics. In: Arnold, M., Schiehlen, W. (eds) Simulation Techniques for Applied Dynamics. CISM International Centre for Mechanical Sciences, vol 507. Springer, Vienna. https://doi.org/10.1007/978-3-211-89548-1_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-211-89548-1_5

  • Publisher Name: Springer, Vienna

  • Print ISBN: 978-3-211-89547-4

  • Online ISBN: 978-3-211-89548-1

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics