Analysis of an oscillatory Painlevé-Klein apparatus with a nonholonomic constraint. (English) Zbl 1333.70013
Summary: In dynamics, both the concepts of rigid body and Coulomb’s law of friction are well established, although it is known at least since Painlevé’s time that they may lead to irregularities and contradictions, such as loss of uniqueness or existence of the solution of the equations of motion. The problem is still of very actual interest, since it can be of practical significance also for the industrially used rigid body codes. One of the simplest mechanical systems in which these difficulties can be well described is the Painlevé-Klein apparatus. As most other systems discussed in this context in the literature, this is a holonomic system. In the present note, we briefly examine a nonholonomic oscillatory system which is an extension of the classical Painlevé-Klein apparatus and we study its dynamics with respect to the Painlevé paradox. Both the borders of paradoxical regions and their reachability are addressed.
MSC:
| 70E55 | Dynamics of multibody systems |
| 70F25 | Nonholonomic systems related to the dynamics of a system of particles |
References:
| [1] | Awrejcewicz J., Olejnik P.: Analysis of dynamic systems with various friction laws. Trans. ASME, Appl. Mech. Rev. 58(6), 389–411 (2005) · doi:10.1115/1.2048687 |
| [2] | Fischer F.: Berichte aus dem Maschinenbau, Behandlung des Painlevé’schen Paradoxons in der modernen Mehrkörperdynamik. Shaker, Aachen (2009) |
| [3] | Ivanov A.P.: Singularities in the dynamics of systems with non-ideal constraints. J. Appl. Math. Mech. 67(2), 185–192 (2003) · Zbl 1067.70503 · doi:10.1016/S0021-8928(03)90004-9 |
| [4] | Neimark Y.I., Fufayev N.A.: The Painlevé paradoxes and the dynamics of a brake shoe. J. Appl. Math. Mech. 59(3), 342–352 (1995) |
| [5] | Liu C., Zhao Z., Chen B.: The bouncing motion appearing in a robotic system with unilateral constraint. Nonlinear Dyn. 49, 217–232 (2007) · Zbl 1181.70012 · doi:10.1007/s11071-006-9123-z |
| [6] | Brommundt E.: Reibung, Selbsthemmung, Selbsterregung: Überlegungen zum Painlevé’schen Paradoxon. Int. J. Non-Linear Mech. 38, 691–703 (2003) · Zbl 1346.74137 · doi:10.1016/S0020-7462(01)00127-5 |
| [7] | Zhao Z., Liu C., Chen W.M.: Experimental investigation of the Painlevé paradox in a robotic system. Trans. ASME, J. Appl. Mech 75, 0410061–04100611 (2008) |
| [8] | Acary V., Brogliato B.: Numerical Methods for Nonsmooth Dynamical Systems, Applications in Mechanics and Electronics. Springer, Berlin (2008) · Zbl 1173.74001 |
| [9] | Leine R.I., Nijmeijer H.: Dynamics and bifurcations of non-smooth mechanical systems. Lecture Notes in Applied and Computational Mechanics, vol. 18. Springer, Berlin (2004) · Zbl 1068.70003 |
| [10] | Heffel, E., Wagner, A., Arrieta, A.F., Fischer, F., Spelsberg-Korspeter, G., Hagedorn, P.: On the Dynamics of an Oscillatory Painlevé–Klein Apparatus. In: Proceedings of the 11th Conference on Dynamical Systems Theory and Applications, Lodz, 5–8 December, 2011 |
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.
