Abstract
Efficient simulation of dynamical systems becomes more and more important in industry and research. Dynamic modeling of multi-body systems yields highly nonlinear equations of motion. Usually, the accelerations are computed by an explicit inversion of the mass matrix that has the dimension according to the degrees of freedom. This classical foregoing implies high computational effort. In the present contribution, an O(n) formulation is introduced for efficient (recursive) procedure. It is based on the Projection Equation in subsystem representation, structuring the problem into parts and yielding interpretable intermediate solutions. The hereby necessary inversion refers to a reduced mass matrix that has the order of the considered subsystem. Additional constraints like endpoint contact are included via corresponding constraint forces. Avoiding an inversion of the total mass matrix is again successfully applied by a recursive procedure. The impact that occurs in the transition phase between the free system and constrained system is also solved in this sense. Results for the simulation of a plane pendulum motion with changing contact scenarios are presented.
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References
Gattringer, H.: Realisierung, Modellbildung und Regelung einer zweibeinigen Laufmaschine. PhD Thesis, Johannes Kepler University of Linz (2006)
Bremer H.: Elastic Multibody Dynamics—A Direct Ritz Approach. Springer, Berlin (2008)
Attia H.A.: A recursive method for the dynamic analysis of mechanical systems in spatial motion. Acta Mech. 167, 41–55 (2004)
Mohan A., Saha S.K.: A recursive, numerically stable, and efficient simulation algorithm for serial robots. Multibody Syst. Dyn. 17, 291–319 (2007)
Khalil W., Dombre E.: Modeling, Identification and Control of Robots. Kogan Page Science, London (2002)
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: Proceedings of IFAC Symposium Vienna, pp. 365–370 (1986)
Baumgarte J.: Stabilization of constraints and integrals of motion in dynamical systems. Comput. Methods Appl. Mech. Eng. 1, 490–501 (1972)
Gattringer, H., Bremer, H.: A penalty shooting walking machine. In: Proceedings of the IUTAM Symposium on Vibration Control of Nonlinear Mechanisms and Structures, pp. 151–160 (2005)
Brandl, H., Johanni, R., Otter, M.: An algorithm for the simulation of multibody systems with kinematic loops. In: Proceedings of 7th IFToMM World Congress on the Theory of Machines and Mechanisms, Sevilla, pp. 407–411 (1987)
Betsch P., Steinmann P.: A DAE approach to flexible multibody dynamics. Multibody Syst. Dyn. 8, 367–391 (2002)
Eich-Soellner E., Führer C.: Numerical Methods in Multibody Dynamics. B.G. Teubner, Stuttgart (1998)
Gerstmayr J., Ambrosio J.A.C.: Component mode synthesis with constant mass and stiffness matrices applied to flexible multibody systems. Int. J. Numer. Methods Eng. 73(11), 1497–1517 (2008)
Pfeiffer F.: Mechanical System Dynamics. Springer, Berlin (2008)
Glocker C.: Set-Valued Force Laws—Dynamics of Non-Smooth Systems. Springer-Verlag, Berlin (2001)
Bremer H.: Dynamik und Regelung mechanischer Systeme. B.G. Teubner, Stuttgart (1988)
Bremer H., Pfeiffer F.: Elastische Mehrkörpersysteme. B. G. Teubner, Stuttgart (1992)
Bremer H.: On the use of nonholonomic variables in robotics. In: Belayev, A., Guran, A. (eds) Selected Topics in Structronics and Mechatronic Systems, pp. 1–48. World Scientific, Singapore (2003)
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Gattringer, H., Bremer, H. & Kastner, M. Efficient dynamic modeling for rigid multi-body systems with contact and impact. Acta Mech 219, 111–128 (2011). https://doi.org/10.1007/s00707-010-0436-0
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DOI: https://doi.org/10.1007/s00707-010-0436-0