Towards a Prototype of a Spherical Tippe Top

Abstract

Abstract.Among spinning objects, the tippe top exhibits one of the most bizarre and counterintuitive behaviours. The commercially available tippe tops basically consist of a section of a sphere with a rod. After spinning on its rounded body, the top flips over and continues spinning on the stem. It is the friction with the bottom surface and the position of the center of mass below the centre of curvature that cause the tippe top to rise its centre of mass while continuing rotating around its symmetry axis (through the stem). The commonly used simplified mathematical model for the tippe top is a sphere whose mass distribution is axially but not spherically symmetric, spinning on a flat surface subject to a small friction force that is due to sliding. Adopting a bifurcation theory point of view we reach a global geometric understanding of the phase diagram of this dynamical system. According to the eccentricity of the sphere and the Jellet invariant (which includes information on the initial angular velocity) three main different dynamical behaviours are distinguished: tipping, non-tipping, hanging (i.e. the top rises but converges to an intermediate state instead of rising all the way to the vertical state). Subclasses according to the stability of relative equilibria can further be distinguished. Since our concern is the degree of confidence in the mathematical model predictions, we applied 3D-printing and rapid prototyping to manufacture a '3-in-1 toy' that could catch the three main characteristics defining the three main groups in the classification of spherical tippe tops as mentioned above. This 'toy' is suitable to validate the mathematical model qualitatively and quantitatively.

Theoretical and experimental investigations of the rolling of a ball on a rotating plane (turntable)

“…Hypothesis of viscous rolling friction, first, satisfies the condition of rolling without slipping and, second, guarantees dissipation. The adequacy of the rolling friction model under consideration is borne out by theoretical and experimental studies for mechanical systems such as a homogeneous rolling disk [17] and ball [13,15], a spherical tippe top [18,19], etc. Taking into account viscous rolling friction (depending on the velocity) has made it possible to qualitatively describe dynamical phenomena such as retrograde motion of a rolling disk, overturn of a tippe top, etc.…”

Non‐holonomic rolling of a ball on the surface of a rotating cylinder

“…Rearrangements of the surface (31) are possible in the case of degeneracy (bifurcation) of fixed points of the reduced system (20). They occur at values of first integrals that satisfy (31) together with the equation From (19) we find that the bifurcation of the fixed points occurs only at the value 1 = 0. Therefore, in what follows we will analyse in more detail the dynamics of the system in two cases: when 1 ≠ 0 and when 1 = 0.…”

“…Thus, according to (19), on the level surface of the integrals 1 = 1 , 2 = 2 , = ℎ the evolution of the variable is governed by the equatioṅ2…”

“…The model of viscous friction is also often used to explain the tippe top inversion [19][20][21][22][23] and to provide insight into the dynamical phenomena of many other mechanical systems, for example, the rise of Jellett's egg [24].…”

The influence of the rolling resistance model on tippe top inversion

“…In this paper we analyze the dynamics of the tippe top on a smooth plane in the presence of forces and torques of rolling resistance. Tippe top inversion has attracted the attention of researchers for the last decades [1][2][3][4][5][6][7][8][9][10][11][12]. Reference [1] discusses the development of a spherical prototype of the top which a capable of performing various modes of motion (in particular, complete or partial inversion) by changing the mass-geometric characteristics.…”