0
Research Papers

A Theoretical and Numerical Study of the Dzhanibekov and Tennis Racket Phenomena1

[+] Author and Article Information
Hidenori Murakami

Department of Mechanical
and Aerospace Engineering,
University of California, San Diego,
9500 Gilman Drive,
La Jolla, San Diego, CA 92093-0411
e-mail: hmurakami@ucsd.edu

Oscar Rios

Department of Mechanical
and Aerospace Engineering,
University of California, San Diego,
9500 Gilman Drive,
La Jolla, San Diego, CA 92093-0411
e-mail: osrios@ucsd.edu

Thomas Joseph Impelluso

Department of Mechanical
and Marine Engineering,
Bergen University College,
Bergen 5020, Norway
e-mail: Thomas.J.Impelluso@hib.no

1This paper was presented at the 2015 ASME IMECE in Houston, TX.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received January 8, 2016; final manuscript received July 21, 2016; published online September 8, 2016. Assoc. Editor: Alexander F. Vakakis.

J. Appl. Mech 83(11), 111006 (Sep 08, 2016) (10 pages) Paper No: JAM-16-1017; doi: 10.1115/1.4034318 History: Received January 08, 2016; Revised July 21, 2016

This paper presents a complete explanation of the Dzhanibekov and the tennis racket phenomena. These phenomena are described by Euler's equation for an unconstrained rigid body that has three distinct moment of inertia values. In the two phenomena, the rotations of a body about the principal axes that correspond to the largest and the smallest moments of inertia are stable. However, the rotation about the axis corresponding to the intermediate principal moment of inertia becomes unstable, leading to the unexpected rotations that are the basis of the phenomena. If this unexpected rotation is not explained from a complete perspective which accounts for the relevant physical and mathematical aspects, one might misconstrue the phenomena as a violation of the conservation of angular momenta. To address this, the phenomenon is investigated using more precise mathematical and graphical tools than those employed previously. The torque-free Euler equations are integrated using the fourth-order Runge–Kutta method. Then, a recovery equation is applied to obtain the rotation matrix for the body. By combining the geometrical solutions with numerical simulations, the unexpected rotations observed in the Dzhanibekov and the tennis racket experiments are shown to preserve the conservation of angular momentum.

FIGURES IN THIS ARTICLE
<>
Copyright © 2016 by ASME
Your Session has timed out. Please sign back in to continue.

References

Euler, L. , 1758, “ Du mouvement de rotation des corps solides autour d'un axe variable,” Mémoires de l'académie des sciences de Berlin, Vol. 14, Berlin Academy, Berlin, Germany, pp. 154–193.,
Euler, L. , 1964, “ Du mouvement de rotation des corps solides autour d'un axe variable,” Opera Omnia: Commentationes Mechanicae ad Theoriam Corporum Rigidorum Pertinentes, volumen prius, Orell Füssli, Zürich, Switzerland, pp. 200–235.
Cartan, É. , 1928, Leçons sur la Géométrie des Espaces de Riemann, Gauthiers-Villars, Paris, France.
Cartan, E. , 1986, On Manifolds With an Affine Connection and the Theory of General Relativity (Translated by A. Magnon and A. Ashtekar), Bibiliopolis, Napels, Italy.
Frankel, T. , 2012, The Geometry of Physics: An Introduction, 3rd ed., Cambridge University Press, New York.
Murakami, H. , 2013, “ A Moving Frame Method for Multi-Body Dynamics,” ASME Paper No. IMECE2013-62833.
Synge, J. L. , 1960, “ Classical Dynamics,” Principles of Classical Mechanics and Field Theory (Encyclopedia of Physics), Vol. III/1, S. Flügge , ed., Springer-Verlag, Berlin, Germany.
Landau, L. D. , and Lifshitz, E. M. , 1960, Mechanics (Translated From Russian by J. B. Sykes and J. S. Bell), Pergamon Press, New York.
Goldstein, H. , 1980, Classical Mechanics, 2nd ed., Addison-Wesley Publishing, Reading, MA.
Greenwood, D. T. , 1965, Principle of Dynamics, Prentice-Hall, Englewood Cliffs, NJ.
Arnold, V. I. , 1989, Mathematical Methods of Classical Mechanics, 2nd ed., Springer-Verlag, New York.
Wittenburg, J. , 1977, Dynamics of Rigid Bodies, B. G. Teubner , Stuttgart, Germany.
Wittenburg, J. , 2008, Dynamics of Multibody Systems, 2nd ed., Springer, Berlin, Germany.
Poinsot, L. , 1834, Theorie Nouvelle de la Rotation des Corps, Bachelier, Paris, France.
Brody, H. , 1985, “ The Moment of Inertia of a Tennis Racket,” Phys. Teach., 23(4), pp. 213–216. [CrossRef]
Ashbaugh, M. , Chicone, C. C. , and Cushman, A. H. , 1991, “ The Twisting Tennis Racket,” J. Dyn. Differ. Equations, 3(1), pp. 67–85. [CrossRef]
Haug, E. J. , 1989, Computer-Aided Kinematics and Dynamics of Mechanical Systems, Vol. I: Basic Methods, Allyn and Bacon, Boston, MA.
Nikravesh, P. , 1988, Computer-Aided Analysis of Mechanical Systems, Prentice Hall, Englewood Cliffs, NJ.
Holm, D. D. , 2008, Geometric Mechanics, Part II: Rotating, Translating, and Rolling, Imperial College Press, London.
Noble, B. , and Daniel, J. W. , 1977, Applied Linear Algebra, 2nd ed., Prentice-Hall, Englewood, NJ.
Hughes, T. J. R. , 1987, The Finite Element Method, Linear Static and Dynamics Finite Element Analysis, Prentice-Hall, Englewood Cliffs, NJ.
Impelluso, T. J. , 2016, “ WebGL Animations for: A Theoretical and Numerical Study of the Dzhanibekov and Tennis Racket Phenomena,” accessed Aug. 11, 2016, http://tinyurl.com/ucsdhib
Rios, O. , Ono, T. , Murakami, H. , and Impelluso, J. T. , 2016, “ An Analytical and Geometrical Study of the Dzhanibekov and Tennis Racket Phenomena,” ASME Paper No. IMECE2016-65570.

Figures

Grahic Jump Location
Fig. 1

A body-attached principal coordinate system {s1s2s3} and an inertial coordinate system {x1x2x3}

Grahic Jump Location
Fig. 2

Polhodes on a constant rotational energy ellipsoid

Grahic Jump Location
Fig. 3

(a) A cuboid with an initial spin about the J1-axis; (b) with an initial spin about the J2-axis; and (c) an initial spin about the J3-axis

Grahic Jump Location
Fig. 4

(a) The head plane of a tennis racket, (b) a side view normal to the head plane of the racket, (c) an initial spin with the normal axis to the head plane, (d) an initial spin with the axis normal to the grip axis, and (e) an initial spin with the grip axis

Grahic Jump Location
Fig. 5

(a) Angular momentum trajectories which are intersections between the energy ellipsoid and the angular momentum sphere for J1 C/J2 C=2.0 and J3 C/J2 C=0.5. For d¯= 0.65, 0.8, 0.95, 1.05, 1.20, 1.35, 1.5, 1.65, 1.8, and 1.95, the projection of the trajectories: (b) onto the H¯2 C, H¯3 C-plane, (c) onto the H¯1 C, H¯3 C-plane, and (d) onto the plane H¯1 C, H¯2 C-plane.

Grahic Jump Location
Fig. 6

(a) A sequence of snap shots of the wing nut at times t = 0.00, 4.75, 5.25, 5.50, and 7.00 s. (b) The magnitude and the components of the angular momentum vector about the inertial coordinate axes and (c) about the body-attached coordinate axes.

Grahic Jump Location
Fig. 7

(a) A sequence of snap shots of the tennis racket: t = 0.0, 1.4, 1.8, 3.0, and 4.0 s. (b) The components of the angular momentum vector about the inertial coordinate axes and its magnitude, and (c) about the body-attached coordinate axes and its magnitude.

Grahic Jump Location
Fig. 8

Comparisons between the numerical (dotted) and the analytical (solid line) solutions for the tennis racket

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In