Abstract
The dynamics of disks which move with one point in contact with a fixed horizontal surface are investigated. Two cases are considered: one where the disk rolls without slipping and a second where the disk slides. A two-parameter family of integrable second order differential equations is established for both of these systems by exploiting classical integrability results. Using these families of dynamical systems, complete descriptions of the bifurcations and stability of the steady motions of the disks are presented. One of these families is also used to establish the existence of motions which result in the rolling disk colliding with the horizontal surface. These motions occur for arbitrarily large angular velocities and in the absence of dissipation. The paper closes with some comments on non-integrability.
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References
Poisson S. D., A Treatise of Mechanics, Vol. 2, Longman, London, 1842. (English transiation by H. H. Harte of Traite de Mécanique, Bachelier, Paris, 1833.)
Whittaker E. T., A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 4th reprinted edition, Dover Publications, New York, 1944.
Vierkandt A., ‘On sliding and rolling motion’, Monatshefte der Mathematik and Physik 3, 1892, 31–54 and 97–134 (in German).
Appell P., ‘On the integration of the equations of motion of a heavy body of revolution rolling on a circular edge on a horizontal plane: Particular case of a hoop’, Rendiconti del Circolo Mathematico di Palermo 14, 1900, 1–6 (in French).
Korteweg D. J., ‘Excerpt from a letter to Mr. Appell’, Rendiconti del Circolo Mathematico di Palermo 14, 1900, 7–8 (in French).
Appell P., A Treatise of Rational Mechanics, Volume 2, 2nd Edition, Gauthier-Villars, Paris, 1904 (in French).
Webster A. G., The Dynamics of Particles and of Rigid, Elastic, and Fluid Bodies: Lectures on Mathematical Physics, 2nd Edition, Dover Publications, New York, 1959.
Gallop E. A., ‘On the rise of a spinning top’, Transactions of the Cambridge Philosophical Society 19, 1904, 356–373.
Routh E. J., The Advanced Part of a Treatise on the Dynamics of a System of Rigid Bodies, 6th Edition, Macmillan, London, 1905.
Pars L. A., A Treatise on Analytical Dynamics, Corrected 1st Edition, J. Wiley and Sons, New York, 1968.
Carvallo M. E., ‘Theory of motion of a monocycle and a bicycle: Part 1. Hoop and monocycle’, Journal de L'École Polytechnique IIe Série 5, 1900, 119–188 (in French).
Routh E. J., The Advanced Part of a Treatise on the Dynamics of a System of Rigid Bodies, 4th Edition, Macmillan, London, 1882.
Greenwood D. T., Principles of Dynamics, 2nd Edition, Prentice-Hall, Englewood Cliffs, New Jersey, 1988.
Synge J. L. and Griffith B. A., Principles of Mechanics, McGraw-Hill, New York, 1942.
Mindlin I. M., ‘On the stability of motion of a heavy body of revolution on a horizontal plane’, Inzhenernyi Zhurnal 4, 1964, 225–230 (in Russian).
Duvakin A. P., ‘On the stability of motions of a disk’, Inzhenernyi Zhurnal 5, 1965, 3–9 (in Russian).
Mindlin I. M. and Pozharitskii G. K., ‘On the stability of steady motions of a heavy body of revolution on an absolutely rough horizontal plane’, Journal of Applied Mathematics and Mechanics (English Translation of PMM) 29, 1965, 879–883.
Karapetyan A. V. and Rumyantsev V. V., ‘Stability of conservative and dissipative systems’, in Applied Mechanics: Soviet Reviews, Vol. 1, G. K. Mikhailov and V. Z. Parton (eds.), Hemisphere, New York, 1990. pp. 3–144.
Arnol'd V. I., Methematical Methods of Classical Mechanics, Springer-Verlag, Berlin, Heidelberg, New York, 1978.
Holmes P. J. and Marsden J. E., ‘Horseshoes and Arnold diffusion for Hamiltonian systems on Lie groups’, Indiana University Mathematics Journal 32, 1983, 273–310.
Hermans J., ‘A symmetric sphere rolling on a surface’, Preprint No. 873, Department of Mathematics, University of Utrecht, Holland, 1994. (Nonlinearity 8, 1995, 493–515.)
Bloch, A., Krishnaprasad, P. S., Marsden, J. E., and Murray, R., ‘Nonholonomic mechanical systems with symmetry’, Archive for Rational Mechanics and Analysis 1996, to appear.
Casey, J., ‘On the advantages of a geometric viewpoint in the derivation of Lagrange's equations for a rigid continuum’, in Theoretical, Experimental, and Numerical Contributions to the Mechanics of Fluids and Solids, J. Casey and M. J. Crochet (eds.), Journal of Applied Mathematics and Physics (ZAMP) 46 (Special Issue), 1995, 805–847.
Casey J. and Lam V. C., ‘On the relative angular velocity tensor’, ASME Journal of Mechanisms, Transmission and Automation in Design 108, 1986, 399–400.
Casey J. and Lam V. C., ‘On the reduction of the rotational equations of motion’, Meccanica 22, 1987, 41–42.
Arnol'd V. I., Ordinary Differential Equations, 3rd Edition, Springer-Verlag, Berlin, Heidelberg, New York, 1992.
Erdélyi A., Magnus W., Oberhettinger F., and Tricomi F. G., Higher Transcendental Funotions, Vol. 1, McGraw-Hill, New York, 1953.
Zhurina M. I. and Karmazina L. N., Tables and Formulae for the Spherical Functions P -(1/2)+ir(z), Pergamon Press, New York, 1966. (English translation by E. L. Albasiny of Tablitsy i Formuly dlya Sphericheskikh Funktsii P -(1/2)+ir (z)x. Computing Center of the Academy of Sciences of the USSR, Moscow, 1960).
Hobson E. W., The Theory of Spherical and Ellipsoidal Harmonics, Cambridge University Press, Cambridge, 1931.
Zhurina M. I. and Karmazina L. N., Tables of the Legendre Functions P -(1/2)+ir (x), Part 1, Pergamon Press, New York, 1964. (English translation by D. E. Brown of Tablitsy funktsii Lezhandra P -(1/2)+ir (x), Academy of Sciences of the USSR, Moscow, 1960).
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O'Reilly, O.M. The dynamies of rolling disks and sliding disks. Nonlinear Dyn 10, 287–305 (1996). https://doi.org/10.1007/BF00045108
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DOI: https://doi.org/10.1007/BF00045108