The tippedisk: a tippetop without rotational symmetry. (English) Zbl 1475.70006
Summary: The aim of this paper is to introduce the tippedisk to the theoretical mechanics community as a new mechanical-mathematical archetype for friction induced instability phenomena. We discuss the modeling and simulation of the tippedisk, which is an inhomogeneous disk showing an inversion phenomenon similar but more complicated than the tippetop. In particular, several models with different levels of abstraction, parameterizations and force laws are introduced. Moreover, the numerical simulations are compared qualitatively with recordings from a high-speed camera. Unlike the tippetop, the tippedisk has no rotational symmetry, which greatly complicates the three-dimensional nonlinear kinematics. The governing differential equations, which are presented here in full detail, describe all relevant physical effects and serve as a starting point for further research.
MSC:
| 70E18 | Motion of a rigid body in contact with a solid surface |
| 70K20 | Stability for nonlinear problems in mechanics |
| 70E50 | Stability problems in rigid body dynamics |
References:
| [1] | Acary, V.; Brogliato, B., Numerical Methods for Nonsmooth Dynamical Systems: Applications in Mechanics and Electronics (2008), Berlin: Springer, Berlin · Zbl 1173.74001 · doi:10.1007/978-3-540-75392-6 |
| [2] | Ashbaugh, M. S.; Chicone, C. C.; Cushman, R. H., The Twisting Tennis Racket, J. Dynam. Differential Equations, 3, 1, 67-85 (1991) · Zbl 0713.70005 · doi:10.1007/BF01049489 |
| [3] | Borisov, A. V.; Kilin, A. A.; Karavaev, Yu. L., On the Retrograde Motion of a Rolling Disk, Physics-Uspekhi, 60, 9, 931-934 (2017) · doi:10.3367/UFNe.2017.01.038049 |
| [4] | Borisov, A. V.; Mamaev, I. S., Strange Attractors in Rattleback Dynamics, Physics-Uspekhi, 46, 4, 393-403 (2003) · doi:10.1070/PU2003v046n04ABEH001306 |
| [5] | Borisov, A. V.; Mamaev, I. S.; Kilin, A. A., Dynamics of Rolling Disk, Regul. Chaotic Dyn., 8, 2, 201-212 (2003) · Zbl 1112.37320 · doi:10.1070/RD2003v008n02ABEH000237 |
| [6] | Bou-Rabee, N. M.; Marsden, J. E.; Romero, L. A., Tippe Top Inversion As a Dissipation-Induced Instability, SIAM J. Appl. Dyn. Syst., 3, 3, 352-377 (2004) · Zbl 1147.70303 · doi:10.1137/030601351 |
| [7] | Cohen, C. M., The Tippe Top Revisited, Am. J. Phys., 45, 1, 12-17 (1977) · doi:10.1119/1.10926 |
| [8] | Garcia, A.; Hubbard, M., Spin Reversal of the Rattleback: Theory and Experiment, Proc. Roy. Soc. London Ser. A, 418, 1854, 165-197 (1988) |
| [9] | Glocker, Ch., Set-Valued Force Laws: Dynamics of Non-smooth Systems (2013), Berlin: Springer, Berlin · Zbl 0979.70001 |
| [10] | Glocker, Ch.; Johann, A.; Kruse, H.-P.; Rupp, F.; Schmitz, S., Simulation of Hard Contacts with Friction: An Iterative Projection Method, Recent Trends in Dynamical Systems, 493-515 (2013), Basel: Springer, Basel · Zbl 1317.74068 · doi:10.1007/978-3-0348-0451-6_19 |
| [11] | Hemingway, E. G.; O’Reilly, O. M., Perspectives on Euler Angle Singularities, Gimbal Lock, and the Orthogonality of Applied Forces and Applied Moments, Multibody Syst. Dyn., 44, 1, 31-56 (2018) · Zbl 1423.70012 · doi:10.1007/s11044-018-9620-0 |
| [12] | Jachnik, J., Spinning and Rolling of an Unbalanced Disk, Master’s Thesis, London: Imperial College London, 2011. |
| [13] | Karapetyan, A. V.; Zobova, A. A., Tippe-Top on Visco-Elastic Plane: Steady-State Motions, Generalized Smale Diagrams and Overturns, Lobachevskii J. Math., 38, 6, 1007-1013 (2017) · Zbl 1427.70017 · doi:10.1134/S1995080217060051 |
| [14] | Kessler, P.; O’Reilly, O. M., The Ringing of Euler’s Disk, Regul. Chaotic Dyn., 7, 1, 49-60 (2002) · Zbl 1013.70005 · doi:10.1070/RD2002v007n01ABEH000195 |
| [15] | Le Saux, C.; Leine, R. I.; Glocker, Ch., Dynamics of a Rolling Disk in the Presence of Dry Friction, J. Nonlinear Sci., 15, 1, 27-61 (2005) · Zbl 1094.70005 · doi:10.1007/s00332-004-0655-4 |
| [16] | Leine, R. I., Experimental and Theoretical Investigation of the Energy Dissipation of a Rolling Disk during Its Final Stage of Motion, Arch. Appl. Mech., 79, 11, 1063-1082 (2009) · Zbl 1184.70004 · doi:10.1007/s00419-008-0278-6 |
| [17] | Leine, R. I.; Glocker, Ch., A Set-Valued Force Law for Spatial Coulomb - Contensou Friction, Eur. J. Mech. A Solids, 22, 2, 193-216 (2003) · Zbl 1038.74513 · doi:10.1016/S0997-7538(03)00025-1 |
| [18] | Leine, R. I.; Nijmeijer, H., Dynamics and Bifurcations of Non-Smooth Mechanical Systems (2004), Berlin: Springer, Berlin · Zbl 1068.70003 · doi:10.1007/978-3-540-44398-8 |
| [19] | Magnus, K., Kreisel: Theorie und Anwendungen (1971), Berlin: Springer, Berlin · doi:10.1007/978-3-642-52162-1 |
| [20] | Moffatt, H., Euler’s Disk and Its Finite-Time Singularity, Nature, 404, 6780, 833-834 (2000) · doi:10.1038/35009017 |
| [21] | Moffatt, K.; Shimomura, Y., Classical Dynamics: Spinning Eggs — A Paradox Resolved, Nature, 416, 6879, 385-386 (2002) · doi:10.1038/416385a |
| [22] | Moffatt, K.; Shimomura, Y.; Branicki, M., Dynamics of an Axisymmetric Body Spinning on a Horizontal Surface: 1. Stability and the Gyroscopic Approximation, Proc. Roy. Soc. London Ser. A, 460, 2052, 3643-3672 (2004) · Zbl 1105.70005 · doi:10.1098/rspa.2004.1329 |
| [23] | Moreau, J. J.; Moreau, J. J.; Panagiotopoulos, P. D., Unilateral Contact and Dry Friction in Finite Freedom Dynamics, Non-Smooth Mechanics and Applications, 1-82 (1988), Wien: Springer, Wien · Zbl 0703.73070 · doi:10.1007/978-3-7091-2624-0 |
| [24] | Nützi, G. E., Non-Smooth Granular Rigid Body Dynamics with Applications to Chute Flows, PhD Thesis, Zürich: ETH Zürich, 2016. |
| [25] | O’Reilly, O. M., The Dynamics of Rolling Disks and Sliding Disks, Nonlinear Dyn., 10, 3, 287-305 (1996) · doi:10.1007/BF00045108 |
| [26] | Poinsot, L., Théorie nouvelle de la rotation des corps (1834), Paris: Bachelier, Paris |
| [27] | Przybylska, M.; Rauch-Wojciechowski, S., Dynamics of a Rolling and Sliding Disk in a Plane: Asymptotic Solutions, Stability and Numerical Simulations, Regul. Chaotic Dyn., 21, 2, 204-231 (2016) · Zbl 1377.70022 · doi:10.1134/S1560354716020052 |
| [28] | Rauch-Wojciechowski, S., What Does It Mean to Explain the Rising of the Tippe Top?, Regul. Chaotic Dyn., 13, 4, 316-331 (2008) · Zbl 1229.70019 · doi:10.1134/S1560354708040060 |
| [29] | Rockafellar, R. T., Convex Analysis (1970), Princeton, N.J.: Princeton Univ. Press, Princeton, N.J. · Zbl 0932.90001 · doi:10.1515/9781400873173 |
| [30] | Tiaki, M. M.; Hosseini, S. A. A.; Zamanian, M., Nonlinear Forced Vibrations Analysis of Overhung Rotors with Unbalanced Disk, Arch. Appl. Mech., 86, 5, 797-817 (2016) · doi:10.1007/s00419-015-1063-y |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
