The Chaplygin sleigh with friction moving due to periodic oscillations of an internal mass

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.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

References

  1. 1.

    Goldstein, H., Poole Jr., ChP, Safko, J.L.: Classical Mechanics, 3rd edn. Addison-Wesley, Boston (2001)

    Google Scholar 

  2. 2.

    Gantmacher, F.R.: Lectures in Analytical Mechanics. Mir, Moscow (1975)

    Google Scholar 

  3. 3.

    Neimark, J.I., Fufaev, N.A.: Dynamics of Nonholonomic Systems, Translations of Mathematical Monographs, vol. 33. American Mathematical Society, Providence (2004)

    Google Scholar 

  4. 4.

    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)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Borisov, A.V., Mamaev, I.S.: Rigid Body Dynamics. RCD, Izhevsk (2001). (In Russian)

    Google Scholar 

  6. 6.

    Bloch, A.M.: Nonholonomic Mechanics and Control. Springer, Berlin (2015)

    Google Scholar 

  7. 7.

    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)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Li, Z., Canny, J.F. (eds.): Nonholonomic Motion Planning. Springer, Berlin (2012)

    Google Scholar 

  9. 9.

    Fukao, T., Nakagawa, H., Adachi, N.: Adaptive tracking control of a nonholonomic mobile robot. IEEE Trans. Robotics Autom. 16(5), 609–615 (2000)

    Article  Google Scholar 

  10. 10.

    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)

    Article  Google Scholar 

  11. 11.

    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)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    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)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Borisov, A.V., Mamaev, I.S.: Strange attractors in rattleback dynamics. Phys. Uspekhi 46(4), 393–403 (2003)

    Article  Google Scholar 

  14. 14.

    Borisov, A.V., Kazakov, A.O., Kuznetsov, S.P.: Nonlinear dynamics of the rattleback: a nonholonomic model. Phys. Uspekhi 57(5), 453–460 (2014)

    Article  Google Scholar 

  15. 15.

    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)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    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))

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    Caratheodory, C.: Der schlitten. Z. Angew. Math. Mech. 13(2), 71–76 (1933)

    MATH  Article  Google Scholar 

  18. 18.

    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)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Borisov, A.V., Mamaev, I.S.: The dynamics of a Chaplygin sleigh. J. Appl. Math. Mech. 73(2), 156–161 (2009)

    MathSciNet  Article  Google Scholar 

  20. 20.

    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)

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Tallapragada, P., Fedonyuk, V.: Steering a Chaplygin sleigh using periodic impulses. J. Comput. Nonlinear Dyn. 12(5), 054501 (2017)

    Article  Google Scholar 

  22. 22.

    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)

  23. 23.

    Kuznetsov, S.P.: Regular and chaotic motions of the Chaplygin sleigh with periodically switched location of nonholonomic constraint. Europhys. Lett. 118(1), 10007 (2017)

    MathSciNet  Article  Google Scholar 

  24. 24.

    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)

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    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)

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Bizyaev, I.A., Borisov, A.V., Kuznetsov, S.P.: Chaplygin sleigh with periodically oscillating internal mass. Europhys. Lett. 119(6), 60008 (2017)

    Article  Google Scholar 

  27. 27.

    Fedonyuk, V., Tallapragada, P.: Sinusoidal control and limit cycle analysis of the dissipative Chaplygin sleigh. Nonlinear Dyn. 93(2), 835–846 (2018)

    MATH  Article  Google Scholar 

  28. 28.

    Jung, P., Marchegiani, G., Marchesoni, F.: Nonholonomic diffusion of a stochastic sled. Phys. Rev. E 93(1), 012606 (2016)

    Article  Google Scholar 

  29. 29.

    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)

  30. 30.

    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)

    MathSciNet  MATH  Article  Google Scholar 

  31. 31.

    Pollard, B., Tallapragada, P.: An aquatic robot propelled by an internal rotor. IEEE/ASME Trans. Mechatron. 22(2), 931–939 (2017)

    Article  Google Scholar 

  32. 32.

    Fermi, E.: On the origin of the cosmic radiation. Phys. Rev. 75(8), 1169–1174 (1949)

    MATH  Article  Google Scholar 

  33. 33.

    Lichtenberg, A.J., Lieberman, M.A., Cohen, R.H.: Fermi acceleration revisited. Phys. D Nonlinear Phenom. 1(3), 291–305 (1980)

    MathSciNet  MATH  Article  Google Scholar 

  34. 34.

    Rabinovich, M.I., Trubetskov, D.I.: Oscillations and Waves: In Linear and Nonlinear Systems. Springer, Berlin (1989)

    Google Scholar 

  35. 35.

    Akhmanov, S.A., Khokhlov, R.V.: Parametric amplifiers and generators of light. Phys. Uspekhi 9(2), 210–222 (1966)

    Article  Google Scholar 

  36. 36.

    Nikolov, D.V.: Nonlinear and Parametric Phenomena: Theory and Applications in Radiophysical and Mechanical Systems. World Scientific, Singapore (2004)

    Google Scholar 

  37. 37.

    Yudovich, V.I.: The dynamics of vibrations in systems with constraints. Phys. Doklady 42, 322–325 (1997)

    MathSciNet  MATH  Google Scholar 

  38. 38.

    Champneys, A.: Dynamics of parametric excitation. In: Meyers, R.A. (ed.) Encyclopedia of Complexity and Systems Science, pp. 2323–2345. Springer, New York (2009)

    Google Scholar 

  39. 39.

    Fedonyuk, V., Tallapragada, P.: Stick-slip motion of the Chaplygin Sleigh with Piecewise-Smooth nonholonomic constraint. J. Comput. Nonlinear Dyn. 12(3), 031021 (2017)

    Article  Google Scholar 

  40. 40.

    Feller, W.: An Introduction to Probability Theory and Its Applications, vol. 1, 3rd edn. Wiley, New York (1968)

    Google Scholar 

  41. 41.

    Rytov, S.M., Kravtsov, Y.A., Tatarskii, V.I.: Principles of Statistical Radiophysics. 1. Elements of Random Process Theory. Springer, Berlin (1987)

    Google Scholar 

  42. 42.

    Cox, D.R., Miller, H.D.: The Theory of Stochastic Processes. CRC Press, Boca Raton (1973)

    Google Scholar 

  43. 43.

    Schöll, E., Schuster, H.G. (eds.): Handbook of Chaos Control. Wiley, New York (2008)

    Google Scholar 

Download references

Acknowledgements

This work was supported by Grant No. 15-12-20035 of the Russian Science Foundation.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Sergey P. Kuznetsov.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

Keywords

  • Nonholonomic mechanics
  • Chaplygin sleigh
  • Parametric oscillator
  • Strange attractor
  • Lyapunov exponent
  • Chaotic dynamics