Numerical method for dynamics of multi-body systems with two-dimensional Coulomb dry friction and nonholonomic constraints. (English) Zbl 1380.70038
Summary: Based on the dynamical theory of multi-body systems with nonholonomic constraints and an algorithm for complementarity problems, a numerical method for the multi-body systems with two-dimensional Coulomb dry friction and nonholonomic constraints is presented. In particular, a wheeled multi-body system is considered. Here, the state transition of stick-slip between wheel and ground is transformed into a nonlinear complementarity problem (NCP). An iterative algorithm for solving the NCP is then presented using an event-driven method. Dynamical equations of the multi-body system with holonomic and nonholonomic constraints are given using Routh equations and a constraint stabilization method. Finally, an example is used to test the proposed numerical method. The results show some dynamical behaviors of the wheeled multi-body system and its constraint stabilization effects.
MSC:
| 70F25 | Nonholonomic systems related to the dynamics of a system of particles |
| 70E18 | Motion of a rigid body in contact with a solid surface |
| 70F40 | Problems involving a system of particles with friction |
Keywords:
non-smooth dynamics; nonholonomic constraint; Coulomb dry friction; two-dimensional friction; nonlinear complementarity problem (NCP)References:
| [1] | Flores, P. and Ambrósio, J. On the contact detection for contact-impact analysis in multi-body systems. Multibody System Dynamics, 24, 103-122 (2010) · Zbl 1375.70021 · doi:10.1007/s11044-010-9209-8 |
| [2] | Flores, P., Leine, R., and Glocker, C. Application of the non-smooth dynamics approach to model and analysis of the contact-impact events in cam-follower systems. Nonlinear Dynamics, 69, 2117-2133 (2012) · doi:10.1007/s11071-012-0413-3 |
| [3] | Glocker, C. and Studer, C. Formulation and preparation for numerical evaluation of linear complementarity systems in dynamics. Multibody System Dynamics, 13, 447-163 (2005) · Zbl 1114.70008 · doi:10.1007/s11044-005-2519-6 |
| [4] | Forg, M., Pfeiffer, F., and Ulbrich, H. Simulation of unilateral constrained systems with many bodies. Multibody System Dynamics, 14, 137-154 (2005) · Zbl 1146.70317 · doi:10.1007/s11044-005-0725-x |
| [5] | Leine, R. I. and van de Wouw, N. Stability properties of equilibrium sets of non-linear mechanical systems with dry friction and impact. Multibody System Dynamics, 51, 551-583 (2008) · Zbl 1170.70340 |
| [6] | Wang, Q., Tian, Q., and Hu, H. Dynamic simulation of frictional multi-zone contacts of thin beams. Nonlinear Dynamics, 83, 1-19 (2015) |
| [7] | Zhao, Z. and Liu, C. Contact constraints and dynamical equations in Lagrangian systems. Multibody System Dynamics, 38, 77-99 (2016) · Zbl 1372.70051 · doi:10.1007/s11044-016-9503-1 |
| [8] | Angelov, T. A. Variational analysis of thermomechanically coupled steady-state rolling problem. Applied Mathematics and Mechanics (English Edition), 34, 1361-1372 (2013) DOI 10.1007/s10483-013-1751-6 · Zbl 1284.74024 · doi:10.1007/s10483-013-1751-6 |
| [9] | Schiehlen, W. Research trends in multi-body system dynamics. Multibody System Dynamics, 18, 3-13 (2007) · Zbl 1118.70007 · doi:10.1007/s11044-007-9064-4 |
| [10] | Pfeiffer, F. and Glocker, C. Multibody Dynamics with Unilateral Contacts, Wiley-VCH, New Jersey (2004) · Zbl 0922.70001 |
| [11] | Zhuang, F. and Wang, Q. Modeling and simulation of the non-smooth planar rigid multi-body systems with frictional translational joints. Multibody System Dynamics, 29, 403-423 (2013) |
| [12] | Chiu, C. H. Self-tuning output recurrent cerebellar model articulation controller for a wheeled inverted pendulum control. Neural Computing and Applications, 29, 1153-1164 (2010) · doi:10.1007/s00521-009-0335-2 |
| [13] | Horin, P. B., Djerassi, S., Shoham, M., and Horin, R. B. Dynamics of a six degrees-of-freedom parallel robot actuated by three two-wheel carts. Multibody System Dynamics, 16, 105-121 (2006) · Zbl 1147.70305 · doi:10.1007/s11044-006-9016-4 |
| [14] | Sankaranarayanan, V. and Mahindrakar, A. D. Switched control of a nonholonomic mobile robot. Communications in Nonlinear Science and Numerical Simulation, 14, 2319-2327 (2009) · Zbl 1221.70009 · doi:10.1016/j.cnsns.2008.06.002 |
| [15] | Terze, Z. and Naudet, J. Structure of optimized generalized coordinates partitioned vectors for holonomic and non-holonomic systems. Multibody System Dynamics, 24, 1-16 (2010) · Zbl 1376.70029 · doi:10.1007/s11044-010-9195-x |
| [16] | Tasora, A. and Anitescu, M. A complementarity-based rolling friction model for rigid contacts. Meccanica, 48, 1643-1659 (2013) · Zbl 1293.70052 · doi:10.1007/s11012-013-9694-y |
| [17] | Saux, C. L., Leine, R. I., and Glocker, C. Dynamics of a rolling disk in the presence of dry friction. Journal of Nonlinear Science, 15, 27-61 (2005) · Zbl 1094.70005 · doi:10.1007/s00332-004-0655-4 |
| [18] | Marques, F., Flores, P., Claro, J. C. P., and Lankarani, H. M. A survey and comparison of several friction force models for dynamic analysis of multi-body mechanical systems. Nonlinear Dynamics, 86, 1407-1443 (2016) · doi:10.1007/s11071-016-2999-3 |
| [19] | Flores, P., Sio, J., Claro, J., and Lankarani, H. Influence of the contact-impact force model on the dynamic response of multi-body systems. Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics, 220, 21-34 (2006) |
| [20] | Bagci, C. Dynamic motion analysis of plane mechanisms with Coulomb and viscous damping via the joint force analysis. Journal of Engineering for Industry, 97, 551-560 (1975) · doi:10.1115/1.3438619 |
| [21] | Rooney, G. T. and Deravi, P. Coulomb friction in mechanism sliding joints. Mechanism and Machine Theory, 17, 207-211 (1982) · doi:10.1016/0094-114X(82)90006-4 |
| [22] | Haug, E. J., Wu, S. C., and Yang, S. M. Dynamics of mechanical systems with Coulomb friction, stiction, impact and constraint addition-deletion I: theory. Mechanism and Machine Theory, 21, 401-406 (1986) · doi:10.1016/0094-114X(86)90088-1 |
| [23] | Pfeiffer, F., Foerg, M., and Ulbrich, H. Numerical aspects of non-smooth multi-body dynamics. Computer Methods in Applied Mechanics and Engineering, 195, 6891-6908 (2006) · Zbl 1120.70305 · doi:10.1016/j.cma.2005.08.012 |
| [24] | Lemke, C. E. Some pivot schemes for the linear complementarity problem. Mathematical Programming Studies, 7, 15-35 (1978) · Zbl 0381.90073 · doi:10.1007/BFb0120779 |
| [25] | Eaves, B. C. The linear complementarity problem. Management Science, 17, 612-634 (1971) · Zbl 0228.15004 · doi:10.1287/mnsc.17.9.612 |
| [26] | Pfeiffer, F. On non-smooth dynamics. Meccanica, 42, 533-554 (2008) · Zbl 1163.70305 · doi:10.1007/s11012-008-9139-1 |
| [27] | Blajer, W. Methods for constraint violation suppression in the numerical simulation of constrained multi-body systems—-a comparative study. Computer Methods in Applied Mechanics and Engineering, 200, 1568-1576 (2011) · Zbl 1228.70003 · doi:10.1016/j.cma.2011.01.007 |
| [28] | Wang, Q., Peng, H., and Zhuang, F. A constraint-stabilized method for multi-body dynamics with friction-affected translational joints based on HLCP. Discrete and Continuous Dynamical Systems Series B, 2, 589-605 (2011) · Zbl 1385.70031 |
| [29] | Flores, P., Machado, M., Seabra, E., and Silva, M. T. D. A parametric study on the Baumgarte stabilization method for forward dynamics of constrained multibody systems. Journal of Computational and Nonlinear Dynamics, 6, 73-82 (2009) |
| [30] | Neto, M. A. and Ambrósio, J. Stabilization methods for the integration of DAE in the presence of redundant constraints. Multibody System Dynamics, 10, 81-105 (2003) · Zbl 1137.70303 · doi:10.1023/A:1024567523268 |
| [31] | Braun, D. J. and Goldfarb, M. Eliminating constraint drift in the numerical simulation of constrained dynamical systems. Computer Methods in Applied Mechanics and Engineering, 198, 3151-3160 (2009) · Zbl 1229.70003 · doi:10.1016/j.cma.2009.05.013 |
| [32] | Marsden, J. E., Ratiu, T. S., and Scheurle, J. Reduction theory and the Lagrange-Routh equations. Journal of Mathematical Physics, 41, 3379-3429 (2000) · Zbl 1044.37043 · doi:10.1063/1.533317 |
| [33] | Isac, G. Complementarity problems. Journal of Computational and Applied Mathematics, 124, 303-318 (2000) · Zbl 0982.90058 · doi:10.1016/S0377-0427(00)00432-5 |
| [34] | Harker, P. T. and Pang, J. S. Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications. Mathematical Programming, 48, 161-220 (1990) · Zbl 0734.90098 · doi:10.1007/BF01582255 |
| [35] | Leine, R. I., Campen, D. H. V., and Glocker, C. H. Nonlinear dynamics and modeling of various wooden toys with impact and friction. Journal of Vibration and Control, 9, 25-78 (2003) · Zbl 1045.70008 · doi:10.1177/1077546303009001741 |
| [36] | Kwak, B. M. Complementarity problem formulation of three-dimensional frictional contact. Journal of Applied Mechanics, 58, 134-140 (1989) · Zbl 0756.73086 · doi:10.1115/1.2897140 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
