On the dynamics of rotating rigid tube and its interaction with air

Yifan Liu¹

¹School of Physical Science and Technology, Inner Mongolia University, Inner Mongolia, Hohhot 010021, PR China

Mathematical modelling of natural phenomena, Tome 18 (2023), article no. 31

Résumé

Rotating an axially symmetric rigid body on a horizontal plane is rather a common and simple experience, but this experience has attracted a great deal of interests due to it exhibiting novel features and containing fairly complicated mechanics. This paper is concerned with the threedimensional rotational motion of a rigid tube on a plane.We present the governing dynamical equations of this motion and give a numerical treatment, based on which we discuss the nutation of tube and simulate the trajectory of tube end. We also discuss how fast the angular velocity should be in order to initiate an uninterrupted, steady rotational motion. Then the air lift related to such a three-dimensional rotation of tube is modeled by using Kutta-Joukowski law. By employing this model, we show that the air lift indeed “lift” the tube head during rotating.

Détail

DOI : 10.1051/mmnp/2023035

Bibliographie

[1] M. Batista Steady motion of a rigid disk of finite thickness on a horizontal plane Int. J. Non-Linear Mech 2006 605

[2] L. Bildsten Viscous dissipation for Euler’s disk Phys. Rev. E 2002 056309

[3] A.V. Borisov, I.S. Mamaev The rolling of rigid body on a plane and sphere Reg. Chaotic Dyn 2002 177

[4] A.V. Borisov, A.A. Kilin, Y.L. Karavaev Retrograde motion of a rolling disk Physics-Uspekhi 2017 931

[5] A.V. Borisov, I.S. Mamaev, A.A. Kilin Dynamics of rolling disk Reg. Chaotic Dyn 2003 201

[6] E. Bormashenko Rotating and rolling rigid bodies and the “hairy ball” theorem Am. J. Phys 2017 447

[7] C.M. Braams On the influence of friction on the motion of a top Physica 1952 503 514

[8] M. Branicki, H.K. Moffatt, Y. Shimomura Dynamics of an axisymmetric body spinning on a horizontal surface. III. Geometry of steady state structures for convex bodies Proc. R. Soc. Lond. A 2006 371

[9] H. Caps, S. Dorbolo, S. Ponte, H. Croisier, N. Vandewalle Rolling and slipping motion of Euler’s disk Phys. Rev. E 2004 056610

[10] N. Cheesman, S.J. Hogan, K.U. Kristiansen The geometry of the Painlevé paradox SIAM J. Appl. Dyn. Syst 2022 1798

[11] Y.A. Çengel and J.M. Cimbala, Fluid Mechanics: Fundamentals and Applications, 3rd edn. McGraw Hill (2013).

[12] K. Easwar, F. Rouyer, N. Menon Speeding to a stop: the finite-time singularity of a spinning disk Phys. Rev. E 2002 045102

[13] R.A. Granger, Fluid Mechanics, 1st edn. Dover Publications (1995).

[14] C.G. Gray, B.G. Nickel Constants of the motion for nonslipping tippe tops and other tops with round pegs Am. J. Phys 2000 821

[15] F. Génot, B. Brogliato New results on Painlevé paradoxes Eur. J. Mech. A/Solids 1999 653

[16] S.J. Hogan, K.U. Kristiansen On the regularization of impact without collision: the Painlevé paradox and compliance Proc. R. Soc. A 2017 20160773

[17] N.M. Hugenholtz On tops rising by friction Physica 1952 515 527

[18] M. Inarrea, V. Lanchares, V.M. Rothos, J.P. Salas Chaotic rotations of an asymmetric body with time-dependent moments of inertia and viscous drag Int. J. Bifurcat. Chaos 2003 393

[19] D.P. Jackson, J. Huddy, A. Baldoni, W. Boyes The mysterious spinning cylinder — rigid-body motion that is full of surprises Am. J. Phys 2019 85

[20] M.A. Jalali, M.S. Sarebangholi, M.-R. Alam Terminal retrograde turn of rolling rings Phys. Rev. E 2015 032913

[21] P. Kessler, O.M. O’Reilly The ringing of Euler’s disk Reg. Chaotic Dyn 2002 49

[22] A.M. Kuethe and J.D. Schetzer, Foundations of Aerodynamics, 2nd edn. John Wiley Sons (1959).

[23] C. Le Saux, R.I. Leine, C. Glocker Dynamics of a rolling disk in the presence of dry friction J. Nonlinear Sci 2005 27

[24] R.I. Leine Experimental and theoretical investigation of the energy dissipation of a rolling disk during its final stage of motion Arch. Appl. Mech 2009 1063

[25] D. Ma, C. Liu, Z. Zhao, H. Zhang Rolling friction and energy dissipation in a spinning disc Proc. R. Soc. A 2014 20140191

[26] D. Ma, C. Liu Dynamics of a spinning disk J. Appl. Mech 2016 061003

[27] H.K. Moffatt Euler’s disk and its finite-time singularity Nature 2000 833

[28] H.K. Moffatt, Y. Shimomura Spinning eggs — a paradox resolved Nature 2002 385

[29] H.K. Moffatt, Y. Shimomura, M. Branicki Dynamics of an axisymmetric body spinning on a horizontal surface. I. Stability and the gyroscopic approximation Proc. R. Soc. Lond. A 2004 3643

[30] H. Rouse, Elementary Mechanics of Fluid, original edn. Dover Publications (1946).

[31] M.R.A. Shegelski, I. Kellett, H. Friesen, C. Lind Motion of a circular cylinder on a smooth surface Can. J. Phys 2009 607

[32] Y. Shimomura, M. Branicki, H.K. Moffatt Dynamics of an axisymmetric body spinning on a horizontal surface. II. Selfinduced jumping Proc. R. Soc. Lond. A 2005 1753

[33] A.E. Sikkema, S.D. Steenwyk, J.W. Zwart Spinning tubes: an authentic research experience in a three-hour laboratory Am. J. Phys 2010 467

[34] W.Z. Stepniewski and C.N. Keys, Rotary-Wing Aerodynamics, reprint edn. Dover Publications (1984).

[35] D.E. Stewart Rigid-body dynamics with friction and impact SIAM Rev 2000 3

[36] R. Villanueva, M. Epstein Vibrations of Euler’s disk Phys. Rev. E 2005 066609

[37] F.M. White, Fluid Mechanics, 8th edn. McGraw Hill (2015).

[38] L.A. Whitehead, F.L. Curzon Spinning objects on horizontal planes Am. J. Phys 1983 449

[39] https://www.youtube.com/watch?v=wQTVcaA3PQw.

[40] https://www.youtube.com/watch?v=7rAiZR_zasg.