Critical oscillations of mass-spring systems due to nonsmooth friction. (English) Zbl 1161.70346
Summary: Critical values of the parameters governing the dynamics of simple systems appear when Coulomb friction is not regularized. We explore such systems using a method based on the fact that under constant or analytical data the trajectory exists, is unique and is also sufficiently regular. In fact these properties justify elementary analytical computations on successive time intervals where the condition used to connect the solution from one interval to the other is due to the regularity. Although the systems are simple the dynamics turn out to be quite complex and thus furnish an interesting benchmark for contact dynamics numerical codes. Among other possible applications we choose to present here how to use a mass-spring chain with Coulomb friction to slow down in a progressive and regular manner an oncoming mass with a given initial velocity.
MSC:
| 70K40 | Forced motions for nonlinear problems in mechanics |
| 70F40 | Problems involving a system of particles with friction |
Keywords:
Coulomb friction; bilateral contact; nonsmooth dynamics; mass-spring systems; shock absorberReferences:
| [1] | Andreaus U., Casini P. (2001) Dynamics of friction oscillators excited by a moving base and/or driving force. J. Sound Vib. 254(4): 685–699 · doi:10.1006/jsvi.2000.3555 |
| [2] | Basseville S., Léger A. (2006) Stability of equilibrium states in a simple system with unilateral contact and Coulomb friction. Arch. Appl. Mech. 76: 403–428 · Zbl 1168.74407 · doi:10.1007/s00419-006-0040-x |
| [3] | Ballard P., Basseville S. (2005) Existence and uniqueness for dynamical unilateral contact with Coulomb friction: a model problem. Math. Model. Numer. Anal. 39(1): 57–77 · Zbl 1089.34010 · doi:10.1051/m2an:2005004 |
| [4] | Basseville S., Léger A., Pratt E. (2003) Investigation of the equilibrium states and their stability for a simple model with unilateral contact and Coulomb friction. Arch. Appl. Mech. 73, 409–420 · Zbl 1068.74585 · doi:10.1007/s00419-003-0300-y |
| [5] | Bastien J., Lamarque C.-H., Schatzman M. (2000) Study of some rheological models with a finite number of degrees of freedom. Eur. J. Mech. A/Solids 19: 277–307 · Zbl 0954.74011 · doi:10.1016/S0997-7538(00)00163-7 |
| [6] | Bouchitte, G., Lidouh, A., Michel, J-C., Suquet, P.: Might boundary homogenization help to understand friction? In: Curnier A. (ed.) Proceedings of Contact Mechanics International Symposium, pp. 175–192, PPUR, Lausanne (1992) |
| [7] | Brezis H. (1973) Operateurs Maximaux Monotones et Semi-Groupes de Contraction Dans les Espaces de Hilbert. North Holland, Amsterdam |
| [8] | Hanbum Cho., Barber J.R. (1998) Dynamics behavior and stability of simple frictional systems. Math. Comput. Model. 28, 37–53 · Zbl 1002.74534 |
| [9] | Jean M. (1999) The nonsmooth contact dynamics method. Comput. Methods Appl. Mech. Eng. 177: 235–257 · Zbl 0959.74046 · doi:10.1016/S0045-7825(98)00383-1 |
| [10] | Jean M., Thoulouze-Pratt E. (1985) A system of rigid bodies with dry friction. Int. J. Eng. Sci. 23(5): 497–513 · Zbl 0566.73091 · doi:10.1016/0020-7225(85)90060-6 |
| [11] | Klarbring A. (1990) Examples of non-uniqueness and non-existenceof solutions to quasistatic contact problems with friction. Ing. Arch. 60: 529–541 |
| [12] | Leine R.I., Glocker Ch., van Campen D.H. (2003) Nonlinear dynamics and modeling of some wooden toys with impact and friction. J. Vib. Control 9: 25–78 · Zbl 1045.70008 |
| [13] | Licht C., Michaille G. (1997) A modelling of elastic adhesive bonded joints. Adv. Math. Sci. Appl. 7: 711–740 · Zbl 0892.73007 |
| [14] | Martins J.A.C., Monteiro Marques M.D.P., Gastaldi F. (1994) On an example of non-existence of solution to a quasistatic frictional contact problem. Eur. J. Mech. A/Solids 13(1): 113–133 · Zbl 0830.73057 |
| [15] | Martins J.A.C., Oden J.T. (1985) Models and computational methods for dynamic friction phenomena. Comput. Methods Appl. Mech. Eng. 52, 527–634 · Zbl 0567.73122 · doi:10.1016/0045-7825(85)90009-X |
| [16] | Monteiro Marques M.D.P. (1994) An existence, uniqueness and regularity study of the dynamics of systems with one-dimensional friction. Eur. J. Mech. A/Solids 13(2): 277–306 · Zbl 0855.70015 |
| [17] | Pratt, E., Léger, A.: Exploring the dynamics of a simple system involving Coulomb friction. In: Mota Soares C.A. et al. (eds.) 3rd European Conference Computational Mechanics Solids, Structures and Coupled Problems in Engineering. Lisbon, Portugal, 5-9 june 2006 |
| [18] | Quinn D.D. (2004) A new regularization of Coulomb friction. J. Vib. Acoust. 126: 391–397 · doi:10.1115/1.1760564 |
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.
