Asymptotic analysis of the dither effect in systems with friction. (English) Zbl 1006.93029
In systems with friction the dither signal is used in order to smooth the “discontinuous” effects of friction at low velocities. The dither signal can be used in the normal and tangential directions of motion. But the distinction between these two approaches is essential: the effect of the tangential dither is to modify the influence of friction by averaging its effect, while the normal dither has the effect of changing the friction coefficient.
Here, the authors are concerned with the tangential dither signal and its interaction with the dynamic friction model. To overcome the discontinuity, the authors have used a model where the friction force is described by a differential equation including a continuous function and small parameter. The dither effect is analyzed on the basis of averaging theory and a new variant of the asymptotic approach. The stabilizing property by means of the dither is proved experimentally.
Here, the authors are concerned with the tangential dither signal and its interaction with the dynamic friction model. To overcome the discontinuity, the authors have used a model where the friction force is described by a differential equation including a continuous function and small parameter. The dither effect is analyzed on the basis of averaging theory and a new variant of the asymptotic approach. The stabilizing property by means of the dither is proved experimentally.
Reviewer: Yuri N.Sankin (Ul’yanovsk)
MSC:
| 93B51 | Design techniques (robust design, computer-aided design, etc.) |
| 74M10 | Friction in solid mechanics |
| 93D15 | Stabilization of systems by feedback |
| 70Q05 | Control of mechanical systems |
| 34C29 | Averaging method for ordinary differential equations |
References:
| [1] | Amstrong-Helouvry, B.; Dupont, P.; Canudas de Wit, C., A survey of model, analysis tools and compensation methods for the control of machines with friction, Automatica, 30, 7, 1083-1138 (1994) · Zbl 0800.93424 |
| [2] | Atherton, D. P., Nonlinear control engineering (1975), Van Nostrand Reinhold Co: Van Nostrand Reinhold Co London · Zbl 0568.93032 |
| [3] | Bentsman, J., Oscillations-induced transitions and their applications in control of dynamics systems, Journal of Dynamic Systems, Measurement and Control, 112, 3, 313-319 (1990) · Zbl 0722.93048 |
| [4] | Bliman, P.-A., Mathematical study of the Dahl’s friction model, European Journal of Mechanics A, 11, 6, 835-848, 313-319 (1992) · Zbl 0766.73059 |
| [5] | Bogoliubov, N., & Mitropolski, J. (1963). Asymptotical method in the theory of non-linear oscillations; Bogoliubov, N., & Mitropolski, J. (1963). Asymptotical method in the theory of non-linear oscillations |
| [6] | Canudas-de-Wit, C.; Olsson, H.; Åström, K. J.; Lischinsky, P., A New Model for Control of Systems with Friction, IEEE Transactions on Automatic Control, 40, 3, 419-425 (1995) · Zbl 0821.93007 |
| [7] | Lee, S.; Meerkov, S. M., Generalized dither, IEEE Transactions on Information Theory, 37, 1, 50-56 (1991) |
| [8] | Le Suan, A. (1972). Experimental research of mechanical autooscillations under dry friction. Mechanics of Solid Bodies\(4\); Le Suan, A. (1972). Experimental research of mechanical autooscillations under dry friction. Mechanics of Solid Bodies\(4\) |
| [9] | MacColl, L. A., Fundamental theory of servomechanism (1945), Vand Nostrand: Vand Nostrand Princenton, NJ |
| [10] | Mossaheb, S., Application of a method of averaging to the study of dither in non-linear systems, International Journal of Control, 38, 3, 557-576 (1983) · Zbl 0522.93035 |
| [11] | Khalil, H. K., Nonlinear Systems. (1992), Macmillan: Macmillan New York · Zbl 0626.34052 |
| [12] | Tikhonov, A., Valsiljeva, A., & Volosov, V. (1970). Ordinary differential equations. In E. Roubine (Ed.), Mathematics Applied to Physics; Tikhonov, A., Valsiljeva, A., & Volosov, V. (1970). Ordinary differential equations. In E. Roubine (Ed.), Mathematics Applied to Physics |
| [13] | Volosov, V., & Morgunov, B. (1971). Averaging method in the theory of nonlinear oscillatory systems; Volosov, V., & Morgunov, B. (1971). Averaging method in the theory of nonlinear oscillatory systems · Zbl 0232.70021 |
| [14] | Volgarev, A., Burdakov, S., Katkovnik, V., & Poljmski, V. (1988). Mathematical models of robot TUR-10, under slow speeds. Proceedings of Leningrad Polytechnic InstituteN428; Volgarev, A., Burdakov, S., Katkovnik, V., & Poljmski, V. (1988). Mathematical models of robot TUR-10, under slow speeds. Proceedings of Leningrad Polytechnic InstituteN428 |
| [15] | Zames, G.; Shneydor, N. A., Dither in non-linear systems, IEEE Transactions on Automatic Control, AC-21, 5, 660-667 (1976) · Zbl 0337.93009 |
| [16] | Zames, G.; Shneydor, N. A., Structural stabilization and quenching by dither in nonlinear systems, IEEE Transactions on Automatic control, AC-22, 3, 352-361 (1977) · Zbl 0356.93022 |
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