The dynamical nature of a backlash system with and without fluid friction. (English) Zbl 1177.70029
Summary: We study the dynamics of a simple system with backlash and impacts. Both the presence or the absence of fluid friction is considered. The fluid friction is modeled by a fractional derivative, but it is also shown how an inhomogeneous time scale, although not arising from a fractional differential equation, may lead to some features similar to fractional solutions.
MSC:
| 70K99 | Nonlinear dynamics in mechanics |
| 76D99 | Incompressible viscous fluids |
| 26A33 | Fractional derivatives and integrals |
References:
| [1] | Special issue on Non-Smooth Mechanics, Phil. Trans. of the Royal Soc. of London: Math., Phys. and Eng. Sciences, 359, (December 15) (2001) · Zbl 0991.00014 |
| [2] | Tao, G., Kokotovic, P.V.: Adaptive control of systems with unknown output backlash. IEEE Trans. Autom. Control 40, 326–330 (1995) · Zbl 0825.93351 · doi:10.1109/9.341803 |
| [3] | Stepanenko, Y., Sankar, T.S.: Vibro-impact analysis of control systems with mechanical clearance and its application to robotic actuators. ASME J. Dyn. Syst. Meas. Contr. 108, 9–16 (1986) · Zbl 0591.93053 · doi:10.1115/1.3143750 |
| [4] | Karagiannis, K., Pfeiffer, F.: Theoretical and experimental investigations of gear-rattling. Nonlinear Dyn. 2, 367–387 (1991) · doi:10.1007/BF00045670 |
| [5] | Feng, Q., Pfeiffer, F.: Stochastic model on a rattling system. J. Sound Vib. 215, 439–453 (1998) · doi:10.1006/jsvi.1998.1646 |
| [6] | Özgüven, H.N., Housen, D.R.: Mathematical models used in gear dynamics-a review. J. Sound Vib. 121, 383–411 (1998) |
| [7] | Faria, M., Streit, L., Vilela Mendes, R.: Map dynamics in gearbox models. In: Proceedings of 2nd European Symposium on Mathematics in Industry, H. Neunzert (ed.) B. G. Teubner, Stuttgart (1988) · Zbl 0850.70030 |
| [8] | Hongler, M.O., Streit, L.: On the origin of chaos in gearbox models. Phy. D 29, 402–408 (1988) · Zbl 0657.70026 · doi:10.1016/0167-2789(88)90038-3 |
| [9] | Litak, G., Friswell, M.I.: Vibrations in gear systems. Chaos Solitons Fractals 16, 145–150 (2003) · Zbl 1103.70304 · doi:10.1016/S0960-0779(02)00452-6 |
| [10] | Barbosa, R.S., Machado, J.A.T.: Describing function analysis of systems with impacts and backlash. Nonlinear Dyn. 29, 235–250 (2002) · Zbl 1030.70012 · doi:10.1023/A:1016514000260 |
| [11] | Ma, C., Hori, Y.: Backlash vibration suppression in torsional system based on the fractional order Q-filter of disturbance observer. In: Proceedings of 8th IEEE International Workshop on Advanced Motion Control, Kawasaki, Japan, pp. 577–582 (2004) |
| [12] | Machado, J.A.T., Azenha, A.: Fractional-order hybrid control of robot manipulators. In: Proceedings of IEEE International Conference on Systems, Man and Cybernetics, Hammamet, Tunisia, Vol. 1, pp. 788–793 (1998) |
| [13] | Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000) · Zbl 0998.26002 |
| [14] | Hilfer, R.; Foundations of fractional dynamics. Fractals 3, 449–556 (1995) · Zbl 0870.58041 |
| [15] | Kulish, V.V., Lage, J.L.: Applications of fractional calculus to fluid mechanics. J. Fluids Eng. 124, 803–806 (2002) · doi:10.1115/1.1478062 |
| [16] | Jacek, L.: The calculation of a normal force between multiparticle contacts using fractional operators. In: 2nd MIT Conference on Computational Fluid an Solid Mechanics, MA, arXiv: physics/0209085 (2003) |
| [17] | Tenreiro Machado, J.A. (ed.): Special issue on fractional order calculus and its applications. Nonlinear Dyn. 29 (2002) |
| [18] | Torvik, P.J., Bagley, R.L.: On the appearance of the fractional derivative in the behavior of real materials. Trans. ASME 51, 294–298 (1984) · Zbl 1203.74022 · doi:10.1115/1.3167615 |
| [19] | Bagley, R.L.: A theoretical basis for the application of fractional calculus to viscoelasticity. J. Rheol. 27, 201–210 (1983) · Zbl 0515.76012 · doi:10.1122/1.549724 |
| [20] | Chatterjee, A.: Statistical origins of fractional derivatives in viscoelasticity. J. Sound Vib., in press (2005) |
| [21] | West, B.J.: Fractional calculus and memory in biophysical time series, in Fractals in Biology and Medicine. Vol. III, pp. 221–234, Birkhäuser, Basel (2003) · Zbl 1134.92308 |
| [22] | Leszczynski, J.S.: Using the fractional interaction law to model the impact dynamics of multiparticle collisions in arbitrary form. Phys. Rev. E70, 051315 (2004) |
| [23] | Podlubny, I.: Geometric and physical interpretation of fractional integration and fractional differentiation. arXiv:math.CA/0110241 (2001) · Zbl 1042.26003 |
| [24] | Bullock, G.L.: A geometric interpretation of the Riemann-Stieltjes integral. Am. Math. Monthly 95, 448–455 (1988) · Zbl 0651.26009 · doi:10.2307/2322483 |
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