Abstract
A computationally efficient explicit integrator is proposed to solve the differential-algebraic equations (DAEs) in multibody system dynamics. Algebraic constraint equations in the DAEs are regularized by a simple stabilization method, yielding a set of first order ordinary differential equations (ODEs), whose large eigenvalues are located at the negative real axis. Those ODEs have specific stiff characters, and are integrated by a class of explicit integrators (the Runge-Kutta-Chebyshev family of ODE integrators) with large stability zones on the negative real axis, so as to achieve large step-sizes at the same requirement of accuracy. The integrator adopted in this work is of fourth order, verified by practical example, and compared to several popular integrators. The high efficiency of the explicit integrator renders it a good option for practical simulations of the dynamics of constraint mechanical systems.
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References
Gear, C.W.: Simultaneous numerical solution of differential algebraic equations. IEEE Trans. Circ. Theory CT–18(1), 89–95 (1971)
Orlandea, N., Chace, M.A., Calahan, D.A.: A sparsity-oriented approach to the design of mechanical systems, Part I and II, Paper No. 76-DET-19 and 76-DET-20. Presented at ASME Mechanical Conferences, Montreal, Quebec, Canada, October 1976
Brenan, K.E., Campbell, S.L., Petzold, L.R.: Numerical Solution of Initial Value Problems in Differential Algebraic Equations, 2nd edn. SIAM Classics in Applied Mathematics, vol. 14. SIAM, Philadelphia (1997)
Gear, C.W., Gupta, G.A., Leimkuhler, B.: Automatic integration of Euler-Lagrange equations with constraints. J. Comput. Appl. Math. 12–13, 77–90 (1985)
Negrut, D., Rampalli, R., Ottarsson, G., Sajdak, A.: On an implementation of the Hilber-Hughes-Taylor method in the context of index 3 differential-algebraic equations of multibody dynamics. J. Comput. Nonlinear Dyn. 2(1), 73–85 (2007)
Arnold, M., Brüls, O.: Convergence of the generalized-alpha scheme for constrained mechanical systems. Multibody Syst. Dyn. 18(2), 185–202 (2007)
Negrut, D., Jay, L.O., Khude, N.: A discussion of low-order numerical integration formulas for rigid and flexible multibody dynamics. J. Comput. Nonlinear Dyn. 4(2), 149–160 (2008)
Hairer, E., Roche, L., Lubich, C.: The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods. Springer, Heidelberg (1989)
Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, 2nd edn. Springer, Heidelberg (1996)
Gear, C.W.: Towards explicit methods for differential algebraic equations. BIT 46, 505–514 (2005)
Verwer, J.H., Hundsdorfer, W.H., Sommeijer, B.P.: Convergence properties of the Runge-Kutta-Chebyshev method. Numer. Math. 57, 157–178 (1990)
Abdulle, A., Medovikov, A.A.: Second order Chebyshev methods based on orthogonal polynomials. Numer. Math. 90, 1–18 (2001)
Abdulle, A.: Fourth order chebyshev methods with recurrence relation. SIAM J. Sci. Comput. 23(6), 2041–2054 (2002)
Ostermeyer, G.P.: On Baumgarte stabilization for differential algebraic equations. In: Haug, E.J., Deyo, R.C. (eds.) Real-Time Integration Methods for Mechanical System Simulation, pp. 193–207 (1990)
Bauchau, O.A., Laulusa, A.: Review of contemporary approaches for constraint enforcement in multibody systems. J. Comput. Nonlinear Dyn. 3(1), 011005 (2008)
Braun, D.J., Goldfarb, M.: Simulation of constrained mechanical systems Part I: an equation of motion, and Part II: explicit numerical integration. ASME J. Appl. Mech. 79(4), 041017–041018 (2012)
Haug, E.J., Yen, J.: Generalized coordinate partitioning methods for numerical integration of differential-algebraic equations of dynamics. In: Haug, E.J., Deyo, R.C. (eds.) Real-Time Integration Methods for Mechanical System Simulation, pp. 97–114 (1990)
Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems, pp. 166–167. Springer, Heidelberg (1987)
Schiehlen, W.: Multibody Systems Handbook, pp. 10–15. Springer, Heidelberg (1990)
Bae, D.S., Lee, J.K., Cho, H.J., Yae, H.: An explicit integration method for real time simulation of multibody vehicle models. Comput. Methods Appl. Mech. Eng. 187(1–2), 337–350 (2000). https://doi.org/10.1016/s0045-7825(99)00138-3
Featherstone, R., Orin, D.: Robot dynamics: equations and algorithms. In: Proceedings of the 2000 IEEE International Conference on Robotics and Automation, San Francisco, CA, vol. 1, pp. 826–834 (2000). https://doi.org/10.1109/ROBOT.2000.844153
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This work is supported by the Nature and Science Foundations of China (NSFC) under grant number 11772101.
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Ren, H., Zhou, P. (2020). A Fast Explicit Integrator for Numerical Simulation of Multibody System Dynamics. In: Kecskeméthy, A., Geu Flores, F. (eds) Multibody Dynamics 2019. ECCOMAS 2019. Computational Methods in Applied Sciences, vol 53. Springer, Cham. https://doi.org/10.1007/978-3-030-23132-3_42
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