Tippe-top on visco-elastic plane: steady-state motions, generalized smale diagrams and overturns

2021

Nonlinear Dyn

The tippedisk is a mechanical-mathematical archetype for friction-induced instability phenomena that exhibits an interesting inversion phenomenon when spun rapidly. The inversion phenomenon of the tippedisk can be modeled by a rigid eccentric disk in permanent contact with a flat support, and the dynamics of the system can therefore be formulated as a set of ordinary differential equations. The qualitative behavior of the nonlinear system can be analyzed, leading to slow–fast dynamics. Since even a freely rotating rigid body with six degrees of freedom already leads to highly nonlinear system equations, a general analysis for the full system equations is not feasible. In a first step the full system equations are linearized around the inverted spinning solution with the aim to obtain a local stability analysis. However, it turns out that the linear dynamics of the full system cannot properly describe the qualitative behavior of the tippedisk. Therefore, we simplify the equations of motion of the tippedisk in such a way that the qualitative dynamics are preserved in order to obtain a reduced model that will serve as the basis for a following nonlinear stability analysis. The reduced equations are presented here in full detail and are compared numerically with the full model. Furthermore, using the reduced equations we give approximate closed form results for the critical spinning speed of the tippedisk.

Section: Introductionmentioning

confidence: 99%

Model reduction of the tippedisk: a path to the full analysis

2021

Nonlinear Dyn

“…Together, they form a mathematical playground to explain, develop and test novel methods in nonlinear dynamics without losing touch with the real world. This special feature of such types of systems explains that the research on the tippetop, which originated in the 1950s, is a topic of increased current research [15][16][17].…”

mentioning

confidence: 99%

Singularly perturbed dynamics of the tippedisk

2021

Proc. R. Soc. A.

The tippedisk is a mathematical-mechanical archetype for a peculiar friction-induced instability phenomenon leading to the inversion of an unbalanced spinning disc, being reminiscent of (but different from) the well-known inversion of the tippetop. A reduced model of the tippedisk, in the form of a three-dimensional ordinary differential equation, has been derived recently, followed by a preliminary local stability analysis of stationary spinning solutions. In the current paper, a global analysis of the reduced system is pursued using the framework of singular perturbation theory. It is shown how the presence of friction leads to slow–fast dynamics and the creation of a two-dimensional slow manifold. Furthermore, it is revealed that a bifurcation scenario involving a homoclinic bifurcation and a Hopf bifurcation leads to an explanation of the inversion phenomenon. In particular, a closed-form condition for the critical spinning speed for the inversion phenomenon is derived. Hence, the tippedisk forms an excellent mathematical-mechanical problem for the analysis of global bifurcations in singularly perturbed dynamics.

“…The Euler disk is an archetype for finite time singularities. And the tippetop describes a friction-induced inversion phenomenon of a mushroom-shaped wooden toy top, a topic receiving much scientific interest [10][11][12].…”

Section: Introductionmentioning

confidence: 99%

Heteroclinic bifurcation analysis of the tippedisk through the use of Melnikov theory

2023

Proc. R. Soc. A.

The tippedisk is a mechanical–mathematical archetype for friction-induced instabilities caused by geometry and interaction with a frictional support. The instability leads to a counterintuitive rise of the centre of gravity when an unbalanced disc is spun rapidly about an in-plane axis. To understand the qualitative behaviour of the tippedisk, a nonlinear analysis is performed, revealing the singularly perturbed structure of the system equations. Application of singular perturbation theory shows that the long-term behaviour is dominated by a two-dimensional slow manifold, on which the asymptotic dynamics takes place. Moreover, Melnikov theory is used to derive a closed-form approximation of a heteroclinic bifurcation, which allows general statements to be made about the dynamic behaviour of the tippedisk.