Abstract
In the process of modelling of the Celtic stone rotating and rolling on a plane surface, different versions of simplified models of contact forces between two contacting bodies are used. The considered contact models take into account coupled dry friction force and torque, as well as rolling resistance. They were developed using Padé approximations, their modifications, and polynomial functions. Before the use of these models, some kinds of regularisations have been made, allowing to avoid singularities in the differential equations. The models were tested both numerically and experimentally, giving some practical guesses of the most essential elements of contact modelling in the Celtic stone numerical simulations. Since the tested contact models do not require the space discretisation, they can find application in relatively fast numerical simulations of rigid bodies with friction contacts.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Awrejcewicz, J., Kudra, G.: Coupled model of dry friction and rolling resistance in the Celtic stone modelling. In: International Conference on Structural Engineering Dynamics (ICEDyn 2011), Tavira, Portugal (2011)
Awrejcewicz J., Kudra G.: Celtic stone dynamics revisited using dry friction and rolling resistance. Shock Vib. 19, 1–9 (2012)
Awrejcewicz J., Kudra G.: Mathematical modelling and simulation of the bifurcational wobblestone dynamics. Discontinuity Nonlinearity Complex. 3, 123–132 (2014). doi:10.5890/DNC.2014.06.002
Bondi S.H.: The rigid body dynamics of unidirectional spin. Proc. R. Soc. Lond. A Math. 405, 265–279 (1986)
Borisov A.V., Kilin A.A., Mamaev I.S.: New effects in dynamics of rattlebacks. Dokl. Phys. 51, 272–275 (2006)
Caughey T.K.: A mathematical model of the “Rattleback”. Int. J. Nonlinear Mech. 15, 293–302 (1980)
Contensou, P.: Couplage entre frottement de glissement et de pivotement dans la téorie de la toupe. Kreiselprobleme Gyrodynamics, In: IUTAM Symp, Calerina, pp. 201–216 (1962)
Garcia A., Hubbard M.: Spin reversal of the rattleback: theory and experiment. P. R. Soc. Lond. A Math. 418, 165–197 (1988)
Goyal S., Ruina A., Papadopoulos J.: Planar sliding with dry friction. Part 1. Limit surface and moment function. Wear 143, 307–330 (1991)
Gray A.: Modern Differential Geometry of Curves and Surfaces with Mathematica, pp. 457–480. CRC Press, Boca Raton (1997)
Howe R.D., Cutkosky M.R.: Practical force-motion models for sliding manipulation. Int. J. Robot. Res. 15, 557–572 (1996)
Johnson K.L.: Contact Mechanics. Cambridge University Press, Cambridge (1985)
Kane T.R., Levinson D.A.: Realistic mathematical modeling of the rattleback. Int. J. Nonlinear Mech. 17, 175–186 (1982)
Kireenkov A.A.: Combined model of sliding and rolling friction in dynamics of bodies on a rough plane. Mech. Solids 43, 412–425 (2008)
Kosenko, I., Aleksandrov, E.: Implementation of the Contensou–Erisman model of friction in frame of the Hertz contact problem on Modelica. In: 7th Modelica Conference, Como, Italy, pp. 288–298 (2009)
Kudra, G., Awrejcewicz, J.: Mathematical modelling and numerical simulations of the celtic stone. In: 10th Conference on Dynamical Systems-Theory and Applications, Lodz, Poland, pp. 919–928 (2009)
Kudra, G., Awrejcewicz, J.: A wobblestone modelling with coupled model of sliding friction and rolling resistance. In: XXIV Symposium ”Vibrations in Physical Systems”, Poznan-Bedlewo, Poland, pp. 245–250 (2010)
Kudra G., Awrejcewicz J.: Tangens hyperbolicus approximations of the spatial model of friction coupled with rolling resistance. Int. J. Bifurc. Chaos 21, 2905–2917 (2011)
Kudra G., Awrejcewicz J.: Bifurcational dynamics of a two-dimensional stick-slip system. Differ. Equ. Dyn. Syst. 20, 301–322 (2012)
Kudra G., Awrejcewicz J.: Approximate modelling of resulting dry friction forces and rolling resistance for elliptic contact shape. Eur. J. Mech. A/Solids 42, 358–375 (2013)
Leine R.I., Glocker Ch.: A set-valued force law for spatial Coulomb–Contensou friction. Eur. J. Mech. A/Solid 22, 193–216 (2003)
Leine, R.I., Le Saux, C., Glocker, Ch.: Friction models for the rolling disk. In: 5th EUROMECH Nonlinear Dynamics Conference (ENOC 2005), Eindhoven, The Netherlands (2005)
Lindberg J., Longman R.W.: On the dynamic behavior of the wobblestone. Acta Mech. 49, 81–94 (1983)
Magnus K.: Zur Theorie der keltischen Wackelsteine. Zeitschrift für angewandte Mathematik und Mechanik 5, 54–55 (1974)
Markeev A.P.: On the dynamics of a solid on an absolutely rough plane. Regul. Chaotic Dyn. 7, 153–160 (2002)
Milne-Thompson, L.M.: Elliptic integrals. In: Abramowitz M., Stegun, I.A. (eds.) Hand-Book of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, Dover Publications Inc., New York (1972)
Möller, M., Leine, R.I., Glocker, Ch.: An efficient approximation of orthotropic set-valued laws of normal cone type. In: 7th EUROMECH Solid Mechanics Conference, Lisbon, Portugal (2009)
Walker G.T.: On a dynamical top. Q. J. Pure Appl. Math. 28, 175–184 (1896)
Wood S.N.: Minimising model fitting objectives that contain spurious local minima by bootstrap restarting. Biometrics 57, 240–244 (2001)
Zhuravlev V.P.: The model of dry friction in the problem of the rolling of rigid bodies. J. Appl. Math. Mech. 62, 705–710 (1998)
Zhuravlev V.P.: Friction laws in the case of combination of slip and spin. Mech. Solids. 38, 52–58 (2003)
Zhuravlev V.P., Klimov D.M.: Global motion of the celt. Mech. Solids 43, 320–327 (2008)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Kudra, G., Awrejcewicz, J. Application and experimental validation of new computational models of friction forces and rolling resistance. Acta Mech 226, 2831–2848 (2015). https://doi.org/10.1007/s00707-015-1353-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-015-1353-z