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Numerical Integration of Second Order Differential—Algebraic Systems in Flexible Mechanism Dynamics

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Computer-Aided Analysis of Rigid and Flexible Mechanical Systems

Part of the book series: NATO ASI Series ((NSSE,volume 268))

Abstract

This paper studies second order accurate methods to numerically time-integrate the equations of motion for flexible mechanism dynamics. The aspects of stability, accuracy, conditioning of equations and time step control are discussed for the implicit scheme of Hilber, Hughes and Taylor (HHT).

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References

  1. Baumgarte J. (1972), ‘Stabilization of constraints and integrals of motion in dynamic systems’, Comp. Meth. Appl. Mech Engng., Vol.1, pp.1–16.

    Article  MathSciNet  MATH  Google Scholar 

  2. Gear C.W. and Petzold L.R. (1984), ‘ODE methods for the solution of differential/algebraic systems’, SIAM J. Numer. Anal. Vol.21, pp. 716–728.

    Article  MathSciNet  MATH  Google Scholar 

  3. Gear C.W. (1984), ‘Differential-algebraic equations’, in Haug E.J. (ed.), ‘Computer aided analysis and optimization of mechanical system dynamics’, NATO ASI Series Vol. F9, Springer-Verlag, pp. 323–334.

    Google Scholar 

  4. P. Lotstedt and L. Petzold (1986), ‘Numerical solution of nonlinear differential equations with algebraic constraints I: convergence results for backward differentiation formulas’, Math. of Computation, Vol.46, pp. 491–516.

    Article  MathSciNet  Google Scholar 

  5. P. Lotstedt and L. Pet zold (1986), ‘Numerical solution of nonlinear differential equations with algebraic constraints II: practical implications’ SIAM J. Sci. Stat. Comput., Vol.7, pp. 720–733.

    Article  MathSciNet  Google Scholar 

  6. K.E. Brenan, S.L. Campbell and L.R. Petzold (1989), ‘Numerical solution of initial-value problems in differential-algebraic equations’, North-Holland.

    Google Scholar 

  7. T.J.R. Hughes (1987), ‘Algorithms for hyperbolic and parabolic-hyperbolic problems’ in “The Finite Element Method. Linear Static and Dynamic Finite Element Analysis”, chap. 9, pp 490–569. Prentice - Hall, Englewood Cliffs.

    Google Scholar 

  8. Newmark N.M. (1959), ‘A method of computation for structural dynamics’, Journal of the Engineering Mechanics Division, ASCE, pp. 67–94.

    Google Scholar 

  9. H.M. Hilber, T.J.R. Hughes and R.L. Taylor (1977), ‘Improved numerical dissipation for time integration algorithms in structural dynamics’, Earthquake engineering and structural dynamics, vol. 5, 283–292.

    Article  Google Scholar 

  10. H.M. Hilber and T.J.R. Hughes (1978), ‘Collocation, dissipation and ‘overshoot’ for time integration schemes in structural dynamics’, Earthquake engineering and structural dynamics, vol. 6, 99–117.

    Article  Google Scholar 

  11. Wood W.L., Bossak M. and Zienkiewicz O.C. (1980), ‘An alpha modification of Newmark’s method’, Int. J. Num. Meth. Engng., Vol.15, pp. 1562–1566.

    Article  MathSciNet  MATH  Google Scholar 

  12. C. Hoff and P.J. Pahl (1988), ‘Development of an implicit method with numerical dissipation from a generalized single-step algorithm for structural dynamics’, Comp. Meth. Appl. Mech. Engng., Vol 67, pp. 367–385.

    Article  MathSciNet  MATH  Google Scholar 

  13. J.C. Simo and K.K. Wong (1991), ‘Unconditionally stable algorithms for rigid body dynamics that exactly preserve energy and momentum’, Int. J. Numer. Meth. Engng., Vol.31, pp. 19–52.

    Article  MathSciNet  MATH  Google Scholar 

  14. J.C. Simo, N. Tarnow and K.K. Wong (1992), ‘Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics’, Comp. Meth. Appl. Mech. Engng., Vol 100, pp. 63–116.

    Article  MathSciNet  MATH  Google Scholar 

  15. A. Cardona (1989), ‘An Integrated approach to mechanism analysis’, PhD. Thesis, Applied Sciences Faculty, University of Liège, Belgium.

    Google Scholar 

  16. A. Cardona, M. Géradin (1989) ‘Time integration of the equations of motion in mechanism analysis’, Computers and Structures, vol 33, 801–820.

    Article  MATH  Google Scholar 

  17. J. García de Jalón and E. Bayo (1992), ‘Computer assisted kinematic and dynamic analysis of multibody systems, Chapter 7, Centro de Estudios e Investigaciones Técnicas de Guipuzcoa, San Sebastián, España.

    Google Scholar 

  18. Gantmacher F.R. (1959), ‘The theory of matrices’, Vol.2, Chelsea Publishing Company.

    Google Scholar 

  19. R.M. Thomas, I. Gladwell (1988), ‘Variable-Order Variable-Steps Algorithms for Second-Order Systems. Part 1: The Methods’, Int. J. Num. Meth. Engng. vol 26, 39–53.

    Article  MathSciNet  MATH  Google Scholar 

  20. I. Gladwell, R.M. Thomas (1988), ‘Variable-Order Variable-Steps Algorithms for Second-Order Systems. Part 2: The Codes’, Int. J. Num. Meth. Engng. vol 26, 55–80.

    Article  MathSciNet  MATH  Google Scholar 

  21. L.F. Shampine, M.K. Gordon (1975), ‘Computer solution of ordinary differential equations. The initial value problem’, Freeman and Company.

    MATH  Google Scholar 

  22. S.H. Lee, S.S. Hsieh (1990), ‘Expedient implicit integration with adaptive time stepping algorithm for nonlinear transient analysis’, Comp. Meth. Appl. Mech. Engng., vol 81, 151–172.

    Article  MATH  Google Scholar 

  23. C. Hoff, R. L. Taylor (1990), ‘Step-by-step integration methods and time step control for systems with arbitrary stiffness’, in Proc. of ‘Second World Congress on Computational Mechanics’, Stuttgart.

    Google Scholar 

  24. O.C. Zienkiewicz, W.L. Wood, N.W. Hine, R.L. Taylor (1984), ‘A unified set of single step algorithms; part 1: General formulation and applications’, International Journal for Numerical Methods in Engineering, vol 20, 1529–1552.

    Article  MathSciNet  MATH  Google Scholar 

  25. O.C. Zienkiewicz and Y.M. Xie (1991), ‘A simple error estimator and adaptive time stepping procedure for dynamic analysis, Earthquake engineering and structural dynamics, vol. 20, pp. 871–887.

    Article  Google Scholar 

  26. A. Cardona and A. Cassano (1992),‘Integrador temporal de paso variable para análisis dinámico de estructuras y mecanismos’, Revista Internacional de Métodos Numéricos Para Cálculo y Diseño en Ingeniería (Barcelona, Spain), Vol.8, pp. 439–461.

    Google Scholar 

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© 1994 Springer Science+Business Media Dordrecht

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Cardona, A., Géradin, M. (1994). Numerical Integration of Second Order Differential—Algebraic Systems in Flexible Mechanism Dynamics. In: Seabra Pereira, M.F.O., Ambrósio, J.A.C. (eds) Computer-Aided Analysis of Rigid and Flexible Mechanical Systems. NATO ASI Series, vol 268. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-1166-9_16

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  • DOI: https://doi.org/10.1007/978-94-011-1166-9_16

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-4508-7

  • Online ISBN: 978-94-011-1166-9

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