Abstract
In this paper we discuss an approximately steady motion of an oscillator as a single whole with a constant “on the average” velocity. For that purpose we analyze the position and stability of some special points of the phase portrait. In the presence of internal excitation and nonsymmetric Coulomb dry friction, a motion of the oscillator with a constant “on the average” velocity is possible. The algebraic equation for this constant velocity is found. For different parameters of the model there exist at most three regimes of motion with a constant velocity, but only one or two of them are stable. The theoretical results obtained can be used for the design of worm-like moving robots.
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References
Andronov A.A., Witt A.A., Chaikin S.E. (1965). Theorie der Schwingungen. Akademie-Verlag, Berlin
Arnold V.I., Kozlov V.V., Neishtadt A.I. (1997). Mathematical Aspects of Classical and Celestial Mechanics. Springer, Berlin Heidelberg New York
Awrejcewicz, J., Delfs, J.: Dynamics of a self-excited stick-slip oscillator with two degrees of freedom. Eur. J. Mech. A. Solids 9, 269–282, 397–418 (1992)
Blekhman I.I. (2000). Vibrational Mechanics: Nonlinear dynamic. Effects, General Approach, Applications. World Scientific, Singapore
Bogoljubow N.N., Mitropolski Yu.A. (1965). Asymptotische Methoden in der Theorie der nichtlinearen Schwingungen. Akademie-Verlag, Berlin
Deimling K., Szilagyi P. (1994). Periodic solutions of dry friction problems. Zeitschr. Angew. Math. Phys. 45: 53–60
Den Hartog J.P. (1930). Forced vibrations with Coulomb and viscous friction. Trans. Am. Soc. Mech. Eng. 53: 107–115
Gantmacher F.R. (1960). The Theory of Matrices. Chelsea, New York
Glocker Ch., Pfeiffer F. (1993). Complementary problems in multibody systems with planar friction. Arch Appl Mech. 63: 452–463
Guckenheimer J., Holmes Ph. (1997). Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, Berlin Heidelberg New York
Hagedorn P. (1978). Nichtlineare Schwingungen. Akademische Verlagsgesellschaft, Wiesbaden
Hayashi C. (1964). Nonlinear Oscillations in Physical Systems. McGraw–Hill, New York
Kauderer H. (1958). Nichtlineare Mechanik. Springer, Berlin Heidelberg New York
Magnus K., Popp K. (1997). Schwingungen. Teubner-Verlag, Stuttgart
Miller G. (1988). The motion dynamics of Snakes and worms. Comput. Graph. 22(4): 169–173
Reissig R. (1954). Erzwungene Schwingungen mit zäher und trockner Reibung. Math. Nachr. 11: 345–384
Shaw S.W. (1986). On the dynamic response of a system with dry friction. J. Sound Vib. 108(2): 305–325
Steigenberger, J.: On a class of biomorphic motion systems. Prep M12 Faculty in Mathematical and Natural Sciences, Technical University of Ilmenau (1999)
Zimmermann, K., Zeidis, I., Steigenberger, J.: Mathematical model of worm-like motion systems with finite and infinite degree of freedom. In: Theory and Practice of Robots and Manipulators. Proceedings of 14th CISM-IFToMM Symposium. Springer, Berlin Heidelberg New York, pp. 507–516 (2002)
Zimmermann, K., Zeidis, I., Steigenberger, J., Pivovarov, M.: An approach to worm-like motion. In: 21st International Congress of Theoretical and Applied Mechanics, Warsaw, Poland, 15–21 August 2004, p. 371 (2004)
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Zimmermann, K., Zeidis, I., Pivovarov, M. et al. Forced nonlinear oscillator with nonsymmetric dry friction. Arch Appl Mech 77, 353–362 (2007). https://doi.org/10.1007/s00419-006-0072-2
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DOI: https://doi.org/10.1007/s00419-006-0072-2