Abstract
For a Chaplygin sleigh moving in the presence of weak friction, we present and investigate two mechanisms of arising acceleration due to oscillations of an internal mass. In certain parameter regions, the mechanism induced by small oscillations determines acceleration which is on average one-directional. The role of friction is that the velocity reached in the process of the acceleration is stabilized at a certain level. The second mechanism is due to the effect of the developing oscillatory parametric instability in the motion of the sleigh. It occurs when the internal oscillating particle is comparable in mass with the main platform and the oscillations are of a sufficiently large amplitude. In the nonholonomic model the magnitude of the parametric oscillations and the level of mean energy achieved by the system turn out to be bounded if the line of the oscillations of the moving particle is displaced from the center of mass; the observed sustained motion is in many cases associated with a chaotic attractor. Then, the motion of the sleigh appears to be similar to the process of two-dimensional random walk on the plane.
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Goldstein, H., Poole Jr., ChP, Safko, J.L.: Classical Mechanics, 3rd edn. Addison-Wesley, Boston (2001)
Gantmacher, F.R.: Lectures in Analytical Mechanics. Mir, Moscow (1975)
Neimark, J.I., Fufaev, N.A.: Dynamics of Nonholonomic Systems, Translations of Mathematical Monographs, vol. 33. American Mathematical Society, Providence (2004)
Borisov, A.V., Mamaev, I.S., Bizyaev, I.A.: Historical and critical review of the development of nonholonomic mechanics: the classical period. Regul. Chaotic Dyn. 21(4), 455–476 (2016)
Borisov, A.V., Mamaev, I.S.: Rigid Body Dynamics. RCD, Izhevsk (2001). (In Russian)
Bloch, A.M.: Nonholonomic Mechanics and Control. Springer, Berlin (2015)
Borisov, A.V., Mamaev, I.S., Bizyaev, I.A.: Historical and critical review of the development of nonholonomic mechanics: the classical period. Regul. Chaotic Dyn. 21(4), 455–476 (2016)
Li, Z., Canny, J.F. (eds.): Nonholonomic Motion Planning. Springer, Berlin (2012)
Fukao, T., Nakagawa, H., Adachi, N.: Adaptive tracking control of a nonholonomic mobile robot. IEEE Trans. Robotics Autom. 16(5), 609–615 (2000)
Alves, J., Dias, J.: Design and control of a spherical mobile robot. Proc. Inst. Mech. Eng. Part I J. Syst. Control Eng. 217(6), 457–467 (2003)
Borisov, A.V., Kilin, A.A., Mamaev, I.S.: How to control Chaplygin’s sphere using rotors. Regul. Chaotic Dyn. 17(3), 258–272 (2012)
Borisov, A.V., Mamaev, I.S.: Conservation laws, hierarchy of dynamics and explicit integration of nonholonomic systems. Regul. Chaotic Dyn. 13(5), 443–490 (2008)
Borisov, A.V., Mamaev, I.S.: Strange attractors in rattleback dynamics. Phys. Uspekhi 46(4), 393–403 (2003)
Borisov, A.V., Kazakov, A.O., Kuznetsov, S.P.: Nonlinear dynamics of the rattleback: a nonholonomic model. Phys. Uspekhi 57(5), 453–460 (2014)
Borisov, A.V., Jalnine, A.Y., Kuznetsov, S.P., Sataev, I.R., Sedova, J.V.: Dynamical phenomena occurring due to phase volume compression in nonholonomic model of the rattleback. Regul. Chaotic Dyn. 17(6), 512–532 (2012)
Chaplygin, S.A.: On the theory of motion of nonholonomic systems. The reducing-multiplier theorem. Regul. Chaotic Dyn. 13(4), 369–376 (2008). (see also: Mat. Sb., 28(2), 303-314 (1912))
Caratheodory, C.: Der schlitten. Z. Angew. Math. Mech. 13(2), 71–76 (1933)
Borisov, A.V., Kilin, A.A., Mamaev, I.S.: On the Hadamard–Hamel problem and the dynamics of wheeled vehicles. Regul. Chaotic Dyn. 20(6), 752–766 (2015)
Borisov, A.V., Mamaev, I.S.: The dynamics of a Chaplygin sleigh. J. Appl. Math. Mech. 73(2), 156–161 (2009)
Borisov, A.V., Kuznetsov, S.P.: Regular and chaotic motions of a Chaplygin Sleigh under periodic pulsed torque impacts. Regul. Chaotic Dyn. 21(7–8), 792–803 (2016)
Tallapragada, P., Fedonyuk, V.: Steering a Chaplygin sleigh using periodic impulses. J. Comput. Nonlinear Dyn. 12(5), 054501 (2017)
Fedonyuk, V., Tallapragada, P.: The dynamics of a two link Chaplygin sleigh driven by an internal momentum wheel. In: American Control Conference (ACC), pp. 2171–2175. IEEE (2017)
Kuznetsov, S.P.: Regular and chaotic motions of the Chaplygin sleigh with periodically switched location of nonholonomic constraint. Europhys. Lett. 118(1), 10007 (2017)
Kuznetsov, S.P.: Regular and chaotic dynamics of a Chaplygin Sleigh due to periodic switch of the nonholonomic constraint. Regul. Chaotic Dyn. 23(2), 178–192 (2018)
Bizyaev, I.A., Borisov, A.V., Mamaev, I.S.: The Chaplygin Sleigh with parametric excitation: chaotic dynamics and nonholonomic acceleration. Regul. Chaotic Dyn. 22(8), 957–977 (2017)
Bizyaev, I.A., Borisov, A.V., Kuznetsov, S.P.: Chaplygin sleigh with periodically oscillating internal mass. Europhys. Lett. 119(6), 60008 (2017)
Fedonyuk, V., Tallapragada, P.: Sinusoidal control and limit cycle analysis of the dissipative Chaplygin sleigh. Nonlinear Dyn. 93(2), 835–846 (2018)
Jung, P., Marchegiani, G., Marchesoni, F.: Nonholonomic diffusion of a stochastic sled. Phys. Rev. E 93(1), 012606 (2016)
Halme, A., Schonberg, T., Wang, Y.: Motion control of a spherical mobile robot. In: Proceedings of the 4th International Workshop on Advanced Motion Control, pp. 259–264. IEEE (1996)
Borisov, A.V., Kilin, A.A., Mamaev, I.S.: How to control Chaplygin’s sphere using rotors. Regul. Chaotic Dyn. 17(3), 258–272 (2012)
Pollard, B., Tallapragada, P.: An aquatic robot propelled by an internal rotor. IEEE/ASME Trans. Mechatron. 22(2), 931–939 (2017)
Fermi, E.: On the origin of the cosmic radiation. Phys. Rev. 75(8), 1169–1174 (1949)
Lichtenberg, A.J., Lieberman, M.A., Cohen, R.H.: Fermi acceleration revisited. Phys. D Nonlinear Phenom. 1(3), 291–305 (1980)
Rabinovich, M.I., Trubetskov, D.I.: Oscillations and Waves: In Linear and Nonlinear Systems. Springer, Berlin (1989)
Akhmanov, S.A., Khokhlov, R.V.: Parametric amplifiers and generators of light. Phys. Uspekhi 9(2), 210–222 (1966)
Nikolov, D.V.: Nonlinear and Parametric Phenomena: Theory and Applications in Radiophysical and Mechanical Systems. World Scientific, Singapore (2004)
Yudovich, V.I.: The dynamics of vibrations in systems with constraints. Phys. Doklady 42, 322–325 (1997)
Champneys, A.: Dynamics of parametric excitation. In: Meyers, R.A. (ed.) Encyclopedia of Complexity and Systems Science, pp. 2323–2345. Springer, New York (2009)
Fedonyuk, V., Tallapragada, P.: Stick-slip motion of the Chaplygin Sleigh with Piecewise-Smooth nonholonomic constraint. J. Comput. Nonlinear Dyn. 12(3), 031021 (2017)
Feller, W.: An Introduction to Probability Theory and Its Applications, vol. 1, 3rd edn. Wiley, New York (1968)
Rytov, S.M., Kravtsov, Y.A., Tatarskii, V.I.: Principles of Statistical Radiophysics. 1. Elements of Random Process Theory. Springer, Berlin (1987)
Cox, D.R., Miller, H.D.: The Theory of Stochastic Processes. CRC Press, Boca Raton (1973)
Schöll, E., Schuster, H.G. (eds.): Handbook of Chaos Control. Wiley, New York (2008)
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This work was supported by Grant No. 15-12-20035 of the Russian Science Foundation.
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Bizyaev, I.A., Borisov, A.V. & Kuznetsov, S.P. The Chaplygin sleigh with friction moving due to periodic oscillations of an internal mass. Nonlinear Dyn 95, 699–714 (2019). https://doi.org/10.1007/s11071-018-4591-5
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DOI: https://doi.org/10.1007/s11071-018-4591-5