A geometric theory of rapidly spinning tops, tippe tops, and footballs

S.

Ebenfeld

and

F.

Scheck

, “,” , – ().

This paper uses Eq. (1) for energy considerations.

Such a resolution was introduced in a more detailed form in Ref. 3, where it is called the “Hooke–Newton Resolution Principle.”

H.

Soodak

and

M. S.

Tiersten

, “,” , – ().

The motion considered here is one of a general class of guiding center motions, and the class of multiple time-scale motions. Reference 5 discusses the example of charged particles in a magnetic field. Reference 6 touches on multiple time-scale problems and discusses averaging methods used to obtain the variation on the long time scales.

Theodore G. Northrop, The Adiabatic Motion of Charged Particles (Interscience, New York, 1963), Chap. I.

A. J. Lichtenberg and M. A. Lieberman, Regular and Chaotic Motion (Springer-Verlag, Berlin, 1992), 2nd ed., p. 102.

For planar cycloids, the motion is described exactly as a rotation of s at constant rate, Ω, and constant separation distance, $ψ*,$ around a point, $l*,$ that moves parallel to the path of l at the same speed, where the path of $l*$ is shifted from that of l by the distance $λ=λ(τp),$ in the direction of $p×l.$ The value of $ψ*$ is equal to its initial value. For $ψ*>λ,$ ψ varies between the limits, $ψ*+λ$ and $ψ*−λ,$ with $ψ̄$ very close to $Ψ*$ when λ is small compared to unity; the path of s is a looped cycloid, with $|δψ|≈λ.$ For $ψ*<λ,$ the limits are $λ+ψ*$ and $λ−ψ*,$ with $ψ̄$ very close to λ; the path is a wavy cycloid without loops, and with $|δψ|≈ψ*,$ which is less than λ. For $ψ*=0,$ s parallels l, moving uniformly along the path of $l*,$ displaced from l by the constant distance λ; in this case ψ is constant, and δψ vanishes.

The cycling of $θs$ for tops supported on frictionless surfaces is known to follow exactly from the constants of motion, as described in Ref. 9. A similar analysis applies to the fixed point top. It is further noted that a remark regarding the average torque concept appears on p. 513 of Ref. 17.

C. G.

Gray

and

B. G.

Nickel

, “,” , – (). The cycling of $θs$ is demonstrated in Appendix B of the paper, and in footnote 33, it is pointed out that the cycling is related to the absence of dissipation, so that dissipation is required for inversion of a tippe top.

The values used here assume a gyroscope consisting of a ring of mass m and radius D, attached by massless spokes to the center of a massless perpendicular axle of length $2D.$ The fixed point is one end of the axle.

J. H. Jellett, A Treatise on the Theory of Friction (MacMillan, London, 1872), p. 185. Reference 9 lists references in which the Jellett constant was rediscovered.

Hand-spun tippe tops are almost certainly initially in a slipping mode. A transition from slipping to rolling occurs when the slip speed, |u|, is reduced to zero and is maintained at zero by a static friction constraint force. The sliding friction force acts to reduce the slip speed by increasing $vh$ in the direction opposite to u in Eq. (28), thus promoting the possibility of a transition to rolling.

The ε value conforms with the generally held opinion that it is in the range of about 0.07–0.2. The value of $β=(I/Is)−1$ conforms with the opinion that I is typically within a few to several percent of $Is,$ either larger or smaller, depending on the tippe top construction. The Ω value was obtained from movies looking downward on an energetically hand-spun toy tippe top. By following the frame by frame positions of a radial line marked with whiteout on its flat top, it was found that the initial spin rates varied in the range of $180±60 rad/s.$ The value, 177 s⁻¹, in Eq. (34) was selected to be very close to the value of 8 $(g/R)1/2$ used in simulations.

T. R.

Kane

and

D. A.

Levinson

, “,” , – ().

The possibility of early transitions from slipping to rolling was recognized in this paper by using computer simulations.

s is represented by its angular polar coordinates, $θs,$ $φs,$ with k as the pole. L is represented as $L=Isωss+Iω1e1+Iωpe2,$ where $e1$ is in the direction of $k×s,$ and $e2=s×e1. vh$ is represented as $vh=v1e1+v3e3,$ where $e3=k×e1.$ The variables are, $θs,$ $φs,$ $ω1,$ $ωs,$ $v1,$ $v3.$ The kinematical relation, Eq. (1a), is then represented by the two equations, $θ̇s=ω1$ and $sin(θs)$ $φ̇s=ω2,$ where $ω2$ is given in terms of $θs$ and $ωs$ through the Jellett constant, Eq. (26). The equations are solved using the NDSOLVE program of Mathematica.

N. M.

Hugenholtz

, “,” , – ().

C. M.

Braams

, “,” , – ().

D. G.

Parkyn

, “,” , – ().

W. A.

Pliskin

, “,” , – ().

R. J.

Cohen

, “,” , – ().

It is noted that a tippe top started in a state of uniform fast (gravitational) precession can remain close to such a state throughout the motion, only for unrealistically small values of μ. In such a state, l precesses around the vertical $l̄$ at the angle $θLG≈λ(τ0)sin(θs),$ with $τ0=mgH.$ According to Eq. (17), the resonant component of the friction torque induces a fast precession of l around $l̄$ at an angle $θLF$ equal to the strength parameter of the resonant friction torque. For the tippe top parameters of Eq. (34), $θLG≈0.0047 sin(θs),$ and $θLF≈0.03 μ.$ For $θLF$ to be smaller than $θLG,$ it is required that $μ<≈0.16$ $θs,$ which is equal to 0.008 for a typically small initial value of $θs≈0.05.$

Braams (Ref. 17) and Parkyn (Ref. 18) estimate correction factors for the effect of the horizontal velocity, $vh.$ They take $vh$ to be initially zero, and approximate its evolution due to the friction force.