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DAE Aspects of Multibody System Dynamics

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Surveys in Differential-Algebraic Equations IV

Part of the book series: Differential-Algebraic Equations Forum ((DAEF))

Abstract

The dynamical simulation of mechanical multibody systems has stimulated the development of theory and numerical methods for higher index differential-algebraic equations (DAEs) for more than three decades. The equations of motion are linearly implicit second order differential equations. For constrained systems, they form an index-3 DAE with a specific structure that is exploited in theoretical investigations as well as in the numerical solution. In the present survey paper, we give an introduction to this field of research with focus on classical and more recent solution techniques for the time integration of constrained mechanical systems in multibody system dynamics. Part of the material is devoted to topics of current research like multibody system models with nonlinear configuration spaces or systems with redundant constraints.

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References

  1. Arnold, V.I.: Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics, vol. 60, 2nd edn. Springer, New York, Berlin, Heidelberg (1989)

    Google Scholar 

  2. Arnold, M.: A perturbation analysis for the dynamical simulation of mechanical multibody systems. Appl. Numer. Math. 18, 37–56 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arnold, M.: Numerical problems in the dynamical simulation of wheel-rail systems. Z. Angew. Math. Mech. 76 (S3), 151–154 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  4. Arnold, M.: Half-explicit Runge–Kutta methods with explicit stages for differential-algebraic systems of index 2. BIT Numer. Math. 38, 415–438 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. Arnold, M.: Zur Theorie und zur numerischen Lösung von Anfangswertproblemen für differentiell-algebraische Systeme von höherem Index. Fortschritt-Berichte VDI Reihe 20, vol. 264. VDI, Düsseldorf (1998)

    Google Scholar 

  6. Arnold, M.: Numerical methods for simulation in applied dynamics. In: Arnold, M., Schiehlen, W. (eds.) Simulation Techniques for Applied Dynamics. CISM Courses and Lectures, vol. 507, pp. 191–246. Springer, Wien, New York (2009)

    Chapter  Google Scholar 

  7. Arnold, M.: A recursive multibody formalism for systems with small mass and inertia terms. Mech. Sci. 4, 221–231 (2013)

    Article  Google Scholar 

  8. Arnold, M.: Algorithmic aspects of singularly perturbed multibody system models. GACM Report. Summer 2013, 10–16 (2013)

    Google Scholar 

  9. Arnold, M., Brüls, O.: Convergence of the generalized-α scheme for constrained mechanical systems. Multibody Sys. Dyn. 18, 185–202 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Arnold, M., Murua, A.: Non-stiff integrators for differential-algebraic systems of index 2. Numer. Algorithms 19, 25–41 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  11. Arnold, M., Schiehlen, W. (eds.) Simulation Techniques for Applied Dynamics. CISM Courses and Lectures, vol. 507. Springer, Wien, New York (2009)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Arnold, M., Burgermeister, B., Eichberger, A.: Linearly implicit time integration methods in real-time applications: DAEs and stiff ODEs. Multibody Sys. Dyn. 17, 99–117 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Arnold, M., Burgermeister, B., Führer, C., Hippmann, G., Rill, G.: Numerical methods in vehicle system dynamics: state of the art and current developments. Veh. Syst. Dyn. 49, 1159–1207 (2011)

    Article  Google Scholar 

  15. Arnold, M., Brüls, O., Cardona, A.: Error analysis of generalized-α Lie group time integration methods for constrained mechanical systems. Numer. Math. 129, 149–179 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Arnold, M., Cardona, A., Brüls, O.: Order reduction in time integration caused by velocity projection. In: Proceedings of the 3rd Joint International Conference on Multibody System Dynamics and the 7th Asian Conference on Multibody Dynamics, June 30–July 3, 2014, BEXCO, Busan (2014). In revised version online available as Technical Report 02-2015, Martin Luther University Halle-Wittenberg, Institute of Mathematics (2015)

    Google Scholar 

  17. Arnold, M., Cardona, A., Brüls, O.: A Lie algebra approach to Lie group time integration of constrained systems. In: Betsch, P. (ed.) Structure-Preserving Integrators in Nonlinear Structural Dynamics and Flexible Multibody Dynamics, vol. 565. CISM Courses and Lectures, pp. 91–158. Springer International Publishing, Cham (2016)

    Chapter  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Bae, D.S., Haug, E.J.: A recursive formulation for constrained mechanical system dynamics: part II. Closed loop systems. Mech. Struct. Mach. 15, 481–506 (1987)

    Article  Google Scholar 

  20. Bauchau, O.A.: Flexible Multibody Dynamics. Springer, Dordrecht, Heidelberg, London, New York (2011)

    Book  MATH  Google Scholar 

  21. Baumgarte, J.: Stabilization of constraints and integrals of motion in dynamical systems. Comput. Methods Appl. Mech. Eng. 1, 1–16 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  22. Bottasso, C.L., Borri, M.: Integrating finite rotations. Comput. Methods Appl. Mech. Eng. 164, 307–331 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  23. Bottasso, C.L., Bauchau, O.A., Cardona, A.: Time-step-size-independent conditioning and sensitivity to perturbations in the numerical solution of index three differential algebraic equations. SIAM J. Sci. Comp. 29, 397–414 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  25. Brasey, V.: A half-explicit method of order 5 for solving constrained mechanical systems. Computing 48, 191–201 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  26. Brenan, K.E., Campbell, S.L., Petzold, L.R.: Numerical Solution of Initial–Value Problems in Differential–Algebraic Equations, 2nd edn. SIAM, Philadelphia (1996)

    MATH  Google Scholar 

  27. Brüls, O., Cardona, A.: On the use of Lie group time integrators in multibody dynamics. J. Comput. Nonlinear Dyn. 5, 031002 (2010)

    Article  Google Scholar 

  28. Brüls, O., Golinval, J.C.: The generalized-α method in mechatronic applications. J. Appl. Math. Mech./Z. Angew. Math. Mech. 86, 748–758 (2006)

    Google Scholar 

  29. Brüls, O., Arnold, M., Cardona, A.: Two Lie group formulations for dynamic multibody systems with large rotations. In: Proceedings of IDETC/MSNDC 2011, ASME 2011 International Design Engineering Technical Conferences, Washington (2011)

    Google Scholar 

  30. Campbell, S.L., Gear, C.W.: The index of general nonlinear DAEs. Numer. Math. 72, 173–196 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  31. Cardona, A., Géradin, M.: Time integration of the equations of motion in mechanism analysis. Comput. Struct. 33, 801–820 (1989)

    Article  MATH  Google Scholar 

  32. Celledoni, E., Owren, B.: Lie group methods for rigid body dynamics and time integration on manifolds. Comput. Methods Appl. Mech. Eng. 192, 421–438 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  33. Chung, J., Hulbert, G.: A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-α method. ASME J. Appl. Mech. 60, 371–375 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  34. Dormand, J.R., Prince, P.J.: A family of embedded Runge–Kutta formulae. J. Comp. Appl. Math. 6, 19–26 (1980)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  36. Eichberger, A.: Transputer–based multibody system dynamic simulation: part I. The residual algorithm – a modified inverse dynamic formulation. Mech. Struct. Mach. 22, 211–237 (1994)

    Google Scholar 

  37. Eich-Soellner, E., Führer, C.: Numerical Methods in Multibody Dynamics. Teubner, Stuttgart (1998)

    Book  MATH  Google Scholar 

  38. Featherstone, R.: The calculation of robot dynamics using articulated-body inertias. Int. J. Robot. Res. 2, 13–30 (1983)

    Article  Google Scholar 

  39. Frączek, J., Wojtyra, M.: On the unique solvability of a direct dynamics problem for mechanisms with redundant constraints and Coulomb friction in joints. Mech. Mach. Theory 46, 312–334 (2011)

    Article  MATH  Google Scholar 

  40. Führer, C.: Differential-algebraische Gleichungssysteme in mechanischen Mehrkörpersystemen. Theorie, numerische Ansätze und Anwendungen. Ph.D. thesis, TU München, Mathematisches Institut und Institut für Informatik (1988)

    Google Scholar 

  41. Führer, C., Leimkuhler, B.J.: Numerical solution of differential-algebraic equations for constrained mechanical motion. Numer. Math. 59, 55–69 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  42. García de Jalón, J., Gutiérrez-López, M.D.: Multibody dynamics with redundant constraints and singular mass matrix: existence, uniqueness, and determination of solutions for accelerations and constraint forces. Multibody Sys. Dyn. 30, 311–341 (2013)

    Google Scholar 

  43. Gear, C.W.: Maintaining solution invariants in the numerical solution of ODEs. SIAM J. Sci. Stat. Comput. 7, 734–743 (1986)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  45. Géradin, M.D., Cardona, A.: Flexible Multibody Dynamics: A Finite Element Approach. Wiley, Chichester (2001)

    Google Scholar 

  46. Golub, G.H., van Loan, Ch.F.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore (1996)

    MATH  Google Scholar 

  47. Hairer, E., Lubich, Ch., Roche, M.: The Numerical Solution of Differential-Algebraic Systems by Runge–Kutta Methods. Lecture Notes in Mathematics, vol. 1409. Springer, Berlin, Heidelberg, New York (1989)

    Google Scholar 

  48. Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations. I. Nonstiff Problems, 2nd edn. Springer, Berlin, Heidelberg, New York (1993)

    Google Scholar 

  49. Hairer, E., Lubich, Ch., Wanner, G.: Geometric Numerical Integration. Structure–Preserving Algorithms for Ordinary Differential Equations, 2nd edn. Springer, Berlin, Heidelberg, New York (2006)

    Google Scholar 

  50. Hairer, E., Wanner, G.: Solving Ordinary Differential Equations. II. Stiff and Differential-Algebraic Problems, 2nd edn. Springer, Berlin, Heidelberg, New York (1996)

    Google Scholar 

  51. Hilber, H.M., Hughes, T.J.R., Taylor, R.L.: Improved numerical dissipation for time integration algorithms in structural dynamics. Earthq. Eng. Struct. Dyn. 5, 283–292 (1977)

    Article  Google Scholar 

  52. Hoschek, M., Rentrop, P., Wagner, Y.: Network approach and differential-algebraic systems in technical applications. Surv. Math. Ind. 9, 49–75 (1999)

    MathSciNet  MATH  Google Scholar 

  53. Iserles, A., Munthe-Kaas, H.Z., Nørsett, S., Zanna, A.: Lie-group methods. Acta Numer. 9, 215–365 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  54. Jansen, K.E., Whiting, C.H., Hulbert, G.M.: A generalized-α method for integrating the filtered Navier-Stokes equations with a stabilized finite element method. Comput. Methods Appl. Mech. Eng. 190, 305–319 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  55. Jay, L.O., Negrut, D.: Extensions of the HHT-method to differential-algebraic equations in mechanics. Electron. Trans. Numer. Anal. 26, 190–208 (2007)

    MathSciNet  MATH  Google Scholar 

  56. Jay, L.O., Negrut, D.: A second order extension of the generalized-α method for constrained systems in mechanics. In: Bottasso, C. (ed.) Multibody Dynamics. Computational Methods and Applications. Computational Methods in Applied Sciences, vol. 12, pp. 143–158. Springer, Dordrecht (2008)

    Google Scholar 

  57. Kalker, J.J.: Three-Dimensional Elastic Bodies in Rolling Contact. Kluwer, Dordrecht, Boston, London (1990)

    Book  MATH  Google Scholar 

  58. Kunkel, P., Mehrmann, V.: Differential-Algebraic Equations. Analysis and Numerical Solution. EMS Textbooks in Mathematics. European Mathematical Society, Zurich (2006)

    Book  MATH  Google Scholar 

  59. Lötstedt, P.: Mechanical systems of rigid bodies subject to unilateral constraints. SIAM J. Appl. Math. 42, 281–296 (1982)

    Article  MathSciNet  Google Scholar 

  60. Lötstedt, P., Petzold, L.R.: Numerical solution of nonlinear differential equations with algebraic constraints I: convergence results for backward differentiation formulas. Math. Comp. 46, 491–516 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  61. Lubich, Ch., Nowak, U., Pöhle, U., Engstler, Ch.: MEXX – Numerical software for the integration of constrained mechanical multibody systems. Technical Report SC 92–12, ZIB Berlin (1992)

    Google Scholar 

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

    Article  Google Scholar 

  63. Lunk, C., Simeon, B.: Solving constrained mechanical systems by the family of Newmark and α-methods. Z. Angew. Math. Mech. 86, 772–784 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  64. Möller, M., Glocker, C.: Rigid body dynamics with a scalable body, quaternions and perfect constraints. Multibody Sys. Dyn. 27, 437–454 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  65. Müller, A., Terze, Z.: The significance of the configuration space Lie group for the constraint satisfaction in numerical time integration of multibody systems. Mech. Mach. Theory 82, 173–202 (2014)

    Article  Google Scholar 

  66. Negrut, D., Rampalli, R., Ottarsson, G., Sajdak, A.: On the use of the HHT method in the context of index 3 differential algebraic equations of multi-body dynamics. In: Goicolea, J.M., Cuadrado, J., García Orden, J.C. (eds.) Proceedings of Multibody Dynamics 2005 (ECCOMAS Thematic Conference), Madrid (2005)

    Google Scholar 

  67. Orlandea, N.: Development and application of node–analogous sparsity–oriented methods for simulation of mechanical dynamic systems. Ph.D. thesis, University of Michigan (1973)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  69. Petzold, L.R.: Order results for implicit Runge–Kutta methods applied to differential/algebraic systems. SIAM J. Numer. Anal. 23, 837–852 (1986)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  71. Rheinboldt, W.C.: Differential-algebraic systems as differential equations on manifolds. Math. Comp. 43, 473–482 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  72. Rheinboldt, W.C.: On the existence and uniqueness of solutions of nonlinear semi-implicit differential-algebraic equations. Nonlinear Anal. Theory Methods Appl. 16, 647–661 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  73. Rulka, W.: Effiziente Simulation der Dynamik mechatronischer Systeme für industrielle Anwendungen. PhD Thesis, Vienna University of Technology, Department of Mechanical Engineering (1998)

    Google Scholar 

  74. Schiehlen, W.O. (ed.): Multibody Systems Handbook. Springer, Berlin, Heidelberg, New York (1990)

    MATH  Google Scholar 

  75. Schiehlen, W.: Multibody system dynamics: roots and perspectives. Multibody Sys. Dyn. 1, 149–188 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  76. Schiehlen, W., Eberhard, P.: Applied Dynamics. Springer, Dordrecht (2014)

    Book  MATH  Google Scholar 

  77. Schwertassek, R., Wallrapp, O.: Dynamik Flexibler MehrKÖrperSysteme. Vieweg, Wiesbaden (1999)

    Book  Google Scholar 

  78. Shabana, A.A.: Flexible multibody dynamics: Review of past and recent developments. Multibody Sys. Dyn. 1, 189–222 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  79. Shabana, A.A.: Dynamics of Multibody Systems, 2nd edn. Cambridge University Press, Cambridge (1998)

    MATH  Google Scholar 

  80. Simeon, B.: MBSPACK – Numerical integration software for constrained mechanical motion. Surv. Math. Ind. 5, 169–202 (1995)

    MathSciNet  MATH  Google Scholar 

  81. Simeon, B.: On the numerical solution of a wheel suspension benchmark problem. Comp. Appl. Math. 66, 443–456 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  82. Simeon, B.: Computational Flexible Multibody Dynamics: A Differential-Algebraic Approach. Differential-Algebraic Equations Forum. Springer, Berlin Heidelberg (2013)

    Book  MATH  Google Scholar 

  83. Simeon, B.: On the history of differential-algebraic equations. In: Ilchmann, A., Reis, T. (eds.) Surveys in Differential-Algebraic Equations, vol. III. Springer, Cham (2016)

    Google Scholar 

  84. Simeon, B., Führer, C., Rentrop, P.: Differential-algebraic equations in vehicle system dynamics. Surv. Math. Ind. 1, 1–37 (1991)

    MathSciNet  MATH  Google Scholar 

  85. Simo, J.C., Vu-Quoc, L.: On the dynamics in space of rods undergoing large motions – A geometrically exact approach. Comput. Methods Appl. Mech. Eng. 66, 125–161 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  86. Udwadia, F.E., Phohomsiri, P.: Explicit equations of motion for constrained mechanical systems with singular mass matrices and applications to multi-body dynamics. Proc. R. Soc. A 462, 2097–2117 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  87. Valášek, M., Šika, Z., Vaculín, O.: Multibody formalism for real-time application using natural coordinates and modified state space. Multibody Sys. Dyn. 17, 209–227 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  88. von Schwerin, R.: MultiBody System SIMulation – Numerical Methods, Algorithms, and Software. Lecture Notes in Computational Science and Engineering, vol. 7. Springer, Berlin, Heidelberg (1999)

    Google Scholar 

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

    Article  Google Scholar 

  90. Wehage, R.A., Shabana, A.A., Hwang, Y.L.: Projection methods in flexible multibody dynamics. part II: Dynamics and recursive projection methods. Int. J. Numer. Methods Eng. 35, 1941–1966 (1992)

    Google Scholar 

  91. Yen, J., Petzold, L.R., Raha, S.: A time integration algorithm for flexible mechanism dynamics: The DAE α-method. Comput. Methods Appl. Mech. Eng. 158, 341–355 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Institute of Mathematics, Martin Luther University Halle-Wittenberg, 06099, Halle (Saale), Germany

    Martin Arnold

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  1. Institut für Mathematik, Technische Universität Ilmenau, Ilmenau, Germany

    Achim Ilchmann

  2. Fachbereich Mathematik, Universität Hamburg, Hamburg, Germany

    Timo Reis

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Arnold, M. (2017). DAE Aspects of Multibody System Dynamics. In: Ilchmann, A., Reis, T. (eds) Surveys in Differential-Algebraic Equations IV. Differential-Algebraic Equations Forum. Springer, Cham. https://doi.org/10.1007/978-3-319-46618-7_2

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