Dynamics of the tippe top via Routhian reduction

Abstract

We consider a tippe top modeled as an eccentric sphere, spinning on a horizontal table and subject to a sliding friction. Ignoring translational effects, we show that the system is reducible using a Routhian reduction technique. The reduced system is a two dimensional system of second order differential equations, that allows an elegant and compact way to retrieve the classification of tippe tops in six groups according to the existence and stability type of the steady states.

Towards a Prototype of a Spherical Tippe Top

“…The phase diagram and bifurcation diagrams illustrate the main results that confirm the findings described in [8], the type of asymptotic dynamics is a function of the Jellett invariant (which includes information on the initial angular velocity) and eccentricity of the sphere. The asymptotic state is either unique or the system is bistable.…”

“…Some studies have addressed the occurrence of transitions between rolling and sliding during the motion, see [13,15,18]. In this paper the presented mathematical results mainly reproduce those in [8,6,7,10,20,16] but our approach is inspired by the hands-on numerical approach as first attempted by Cohen in [9]. We believe this approach is the best choice in giving a clear view of the role of the different parameters that is necessary during the design process of an actual three-dimensional object that effectively demonstrates the model.…”

“…As in [8] we further refine this classification. Note that the intermediate states discussed here correspond to the tumbling solutions discussed in [10].…”

“…Three main different regimes are distinguished: no tippe top phenomenon occurs no matter what the initial spin is, tippe top dynamics may occur if the Jellett invariant (which is proportional to the initial spin) is sufficiently large, or incomplete tippe top behaviour occurs, where the top rises but converges to an intermediate state instead of rising all the way to the vertical state. We underline that though the classification results can be obtained in a less cumbersome way by using the Routhian reduction as in [8], the approach used in this paper is standard and straightforward to implement from a prototyping point of view. Also, it is amenable for extensions to include for example transitions from sliding to rolling and vice versa.…”

“…Our concern in this paper is the degree of confidence in the mathematical model predictions. We wanted to be sure that the mathematical model as presented here and in [8] reflected the reality. Our goal was to investigate if it was possible to make a '3-in-1 toy' that could catch the three main characteristics tipping, non-tipping, hanging that defines the three main groups in the classification of spherical tippe tops as mentioned above.…”