A discrete model of a stochastic friction system. (English) Zbl 1040.70013
Summary: To investigate a random friction system must cost a large of time on computer. So, appropriate modeling of this system is of significance in practice. In this paper, an average model of simple random system with friction is firstly developed, which is a discrete model represented by a two-dimensional mean Poincaré map. It is applied to describe random stick–slip motion. A numerical example shows that external noise can change the system behavior. That model is extended to a MDOF system only with a friction interface. An example of 3-DOF system exhibits an interesting behavior due to the influence of external noise.
MSC:
| 70L05 | Random vibrations in mechanics of particles and systems |
| 70F40 | Problems involving a system of particles with friction |
| 37N05 | Dynamical systems in classical and celestial mechanics |
Keywords:
average model; random stick-slip motion; non-smooth system; random chaos; two-dimensional mean Poincaré map; external noiseReferences:
| [1] | Bengisu, M. T.; Akay, A., Stability of friction-induced vibrations in multi-degree-of-freedom systems, (Ibrahim, R. A.; Soom, A., Friction-Induced Vibration, Chatter, Squeal, and Chaos, vol. 49 (1992), ASME: ASME DE), 57-64 · Zbl 1064.70508 |
| [2] | Bowden, F. P.; Leben, L., The nature of sliding and analysis of friction, Proc. R. Soc. London, Ser. A, 169, 371-391 (1939) |
| [3] | Bristow, J. R., Kinetic boundary friction, Proc. R. Soc. London, Ser. A, 189, 88-102 (1947) |
| [4] | Feeny, B. F.; Moon, F. C., Autocorrelation on symbol dynamics for a chaotic dry-friction oscllator, Phys. Lett. A, 141, 397-400 (1989) |
| [5] | Feeny, B. F.; Liang, J. W., Phase-space reconstructions of stick-slip systems, (Proc. of 1995 Design Engineering Tech. Conf. Vol. 3A, vol. 84-1 (1995), ASME: ASME DE), 1049-1060 · Zbl 0880.70014 |
| [6] | Feeny, B. F.; Liang, J. W., A decrement method for the simultaneous estimation of Coulomb and viscous damping, J. Sound Vib., 195, 1, 149-154 (1996) |
| [7] | Feeny, B. F., The nonlinear dynamics of oscillators with stick-slip friction, (Guran, A.; Pfeiffer, F.; Popp, K., Dynamics with Friction (1996), World Scientific: World Scientific Singapore) · Zbl 0880.70014 |
| [8] | Hendriks, F., Bounce and chaotic motion in impact print hammers, IBM J. Res. Dev., 27 (1983) |
| [9] | Hsu, C. S., Cell-to-Cell Mapping (1987), Spring: Spring New York · Zbl 0855.70018 |
| [10] | Ibrahim, R. A., Parametric Random Vibration (1985), John Wiley & Sons: John Wiley & Sons New York · Zbl 0652.70002 |
| [11] | Chr. Kaas-Petersen, H. True, Periodic, biperiodic and chaotic dynamical behaviour of railway vehicles, in: Proc. 9th IAVSD-Symp, Linkoping, 1985, pp. 208-221; Chr. Kaas-Petersen, H. True, Periodic, biperiodic and chaotic dynamical behaviour of railway vehicles, in: Proc. 9th IAVSD-Symp, Linkoping, 1985, pp. 208-221 |
| [12] | Kapitaniak, T., Chaos in System with Noise (1988), World Scientific: World Scientific Singapore · Zbl 1235.70186 |
| [13] | Klotter, K., Technische Schwingungslehre, Erster Band: Einfache Schwingunger, Teil B: Nichtlineare Schwingungen (1980), Springer-verlag: Springer-verlag Berlin · Zbl 0447.70001 |
| [14] | May, R. M., Simple mathematical models with very complicated dynamics, Nature, 261, 459-467 (1976) · Zbl 1369.37088 |
| [15] | Moon, F. C.; Shaw, S. W., Chaotic vibrations of a beam with non-linear boundary conditions, Int. J. Non-Linear Mech., 18, 465-477 (1983) |
| [16] | Moss, F.; McVlintock, P. V.E., Noise in Nonlinear Dynamical Systems, vol. 2 (1985), Cambridge University press: Cambridge University press Cambridge |
| [17] | Nagarajaiah, S.; Reinhorn, A. M.; Constatinou, M. C., Torional coupling in sliding base-isolated structure, ASCE J. Struct. Engrg., 119, 1, 130-149 (1993) |
| [18] | Narayanan, S.; Jayaraman, K., Chaotic vibrations in a nonlinear oscillator with Coulomb damping, J. Sound Vib., 146, 17-31 (1991) |
| [19] | Paidoussis, M. P.; Moon, F. C., Nonlinear and fluidelastic vibrations of a flexible pipe conveying fluid, J. Fluids Struct., 2, 567-591 (1988) |
| [20] | Mueller, P. C., Calculation of Lyapunov exponents for dynamic systems with discontinuities, Chaos Solitons Fractals, 5, 9, 1671-1681 (1995) · Zbl 1080.34540 |
| [21] | K. Popp, P. Stelter, Nonlinear oscillations of structures induced by dry friction, in: Proceedings of IUTAM Symposium on Nonlinear Dynamics in Engineering Systems, Stuttgart, 1989; K. Popp, P. Stelter, Nonlinear oscillations of structures induced by dry friction, in: Proceedings of IUTAM Symposium on Nonlinear Dynamics in Engineering Systems, Stuttgart, 1989 |
| [22] | Popp, K.; Stelter, P., Stick-slip vibrations and chaos, Proc. Philos. Trans. R. Soc. London, Ser. A, 332, 89-105 (1990) · Zbl 0709.70019 |
| [23] | Popp, K., Some model problems showing stick-slip motion and chaos, (Ibrahim, R. A.; Soom, A., Friction-Induced Vibration, Chatter, Squeal, and Chaos, vol. 49 (1992), ASME: ASME DE), 1-12 |
| [24] | Popp, K., Nichtlineare Schwingungen mechanischer Strukturen mit Fuge- und Kontaktstellen, ZAMM, 74, 3, 147-165 (1994) |
| [25] | Popp, K.; Hinrichs, N.; Oestreich, M., Analysis of a self-excited friction oscillator with external excitation, (Guran, A.; Pfeiffer, F.; Popp, K., Dynamics with Friction (1996), World Scientific: World Scientific Singapore) · Zbl 1048.70503 |
| [26] | W.M. Szczygielski, Dynamisches Verhalten eines schnell drehenden Rotors bei Anstrifvorgangen, Diss. ETH Zurich Nr. 8094, 1986; W.M. Szczygielski, Dynamisches Verhalten eines schnell drehenden Rotors bei Anstrifvorgangen, Diss. ETH Zurich Nr. 8094, 1986 |
| [27] | Tolstoi, D. M., Significance of the normal degree of freedom and natural normal vibrations in contact friction, Wear, 10, 199-213 (1967) |
| [28] | Vielsack, P.; Hartung, A., An example for the orbital stability of permanently disturbed non-smooth motions, ZAMM, 79, 6, 389-397 (1999) · Zbl 0943.70006 |
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.
