Computing smooth solutions of DAEs for elastic multibody systems. (English) Zbl 0954.74027
Summary: We consider multibody systems which involve combinations of rigid and elastic bodies. Discretizations of the PDEs, describing the elastic members, lead to a semidiscrete system of ODEs or DAEs. Asymptotic methods are introduced which provide a theoretical basis of various known engineering results for the ODE case. These results are then extended to the DAE case by means of suitable local ODE representations. The recently developed MANPAK algorithms for computations on implicitly defined manifolds form the basic tools for a computational method, which provides consistent approximate solutions of the semidiscrete DAE that satisfy all constraints and are close to the smooth motion and an average solution. Several numerical examples indicate the effectiveness of this asymptotic method for elastic multibody systems.
MSC:
| 74H15 | Numerical approximation of solutions of dynamical problems in solid mechanics |
| 70-08 | Computational methods for problems pertaining to mechanics of particles and systems |
| 70E55 | Dynamics of multibody systems |
| 74K99 | Thin bodies, structures |
| 65L80 | Numerical methods for differential-algebraic equations |
| 74H10 | Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of dynamical problems in solid mechanics |
Keywords:
differential-algebraic equations; rigid mechanical systems; MANPAK algorithms; implicitly defined manifolds; approximate solutions; smooth motion; average solution; asymptotic method; elastic multibody systemsReferences:
| [1] | (Schiehlen, E. O., Multibody System Handbook (1990), Springer: Springer Berlin) · Zbl 0703.70002 |
| [2] | Géradin, M., Computational aspects of the finite element approach to flexible multibody systems, (Schiehlen, W., Advanced Multibody System Dynamics (1993), Kluwer Academic: Kluwer Academic Stuttgart), 337-354 · Zbl 0800.93136 |
| [3] | Shabana, A., Computer implementation of flexible multibody equations, (Pereia, M.; Ambrosio, J., Computer Aided Analysis of Rigid and Flexible Mechanical Systems (1994), Kluwer Academic: Kluwer Academic Dordrecht), 325-349 · Zbl 0874.73067 |
| [4] | Rheinboldt, W. C., Solving algebraically explicit DAEs with the MANPAK manifold algorithms, Computers Math. Applic., 33, 3, 31-43 (1996) · Zbl 0871.65066 |
| [5] | Jahnke, M.; Popp, K.; Dirr, B., Approximate analysis of flexible parts in multibody systems using the finite element method, (Schiehlen, W., Advanced Multibody System Dynamics (1993), Kluwer Academic: Kluwer Academic Stuttgart), 237-256 · Zbl 0798.73033 |
| [6] | Wallrapp, O., Standardization of flexible body modelling in MBS codes, Mech. Struct. Mach., 22, 283-304 (1994) |
| [7] | B. Simeon, DAEs and PDEs in elastic multibody systems, Annals of Numer. Math.; B. Simeon, DAEs and PDEs in elastic multibody systems, Annals of Numer. Math. · Zbl 0927.70004 |
| [8] | Simeon, B., Modelling a flexible slider crank mechanism by a mixed system of DAEs and PDEs, Math. Modelling of Systems, 2, 1-18 (1996) · Zbl 0871.73028 |
| [9] | Hughes, T. J., The Finite Element Method (1987), Prentice Hall: Prentice Hall Englewood Cliffs, NJ · Zbl 0634.73056 |
| [10] | Lubich, C., Integration of stiff mechanical systems by Runge-Kutta methods, ZAMP, 44, 1022-1053 (1993) · Zbl 0784.70002 |
| [11] | Bornemann, F. A., Homogenization in Time of Singularly Perturbed Conservative Mechanical Systems (1997), Habilitationsschrift: Habilitationsschrift ZIB Berlin |
| [12] | Sachau, D., Berücksichtigung von fiexiblen Körpern und Fügestellen in Mehrkörpersystemen zur Simulation aktiver Raumfahrtstrukturen, (Diss. (1996), Universität Stuttgart, Institut A für Mechanik) |
| [13] | Kim, S.; Haug, E., Selection of deformation modes for flexible multibody dynamics, Mech. Struct. Mach., 18, 565-586 (1990) |
| [14] | Walter, W., Differential and Integral Inequalities (1970), Springer · Zbl 0252.35005 |
| [15] | Cartmell, M., Introduction to Linear, Parametric, and Nonlinear Vibrations (1990), Chapman and Hall · Zbl 0790.70002 |
| [16] | Yan, X., Singularly perturbed differential-algebraic equations, (Techn. Report ICMA 94-192 (1994), University of Pittsburgh) |
| [17] | Lion, W.; de Swart, J.; van der Ween, W., Test set for IVP solvers, (Report NM-R9615 (1996), CWI Amsterdam) |
| [18] | Lubich, C., Extrapolation integrators for constrained multibody systems, Impact Comp. Sci. Eng., 3, 213-234 (1991) · Zbl 0742.70011 |
| [19] | Simeon, B.; Grupp, F.; Führer, C.; Rentrop, P., A nonlinear truck model and its treatment as a multibody system, J. Comp. Appl. Math., 50, 523-532 (1994) · Zbl 0800.70023 |
| [20] | Nordmark, A., The Matlab PDE Toolbox, The Mathworks (1995) |
| [21] | Hairer, E.; Wanner, G., Solving Ordinary Differential Equations II (1996), Springer: Springer Berlin · Zbl 0859.65067 |
| [22] | Rheinboldt, W. C., MANPAK: A set of algorithms for computations on implicitly defined manifolds, Computers Math. Applic., 32, 12, 15-28 (1996) · Zbl 0883.65044 |
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