The tippedisk is a mathematical–mechanical archetype, showing a non-intuitive inversion behavior, if the disk is spun fast enough. By introducing a full 3D mechanical model and assuming set-valued force laws to account for normal and frictional contact forces, the dynamics of the tippedisk can be studied numerically. In addition, the model dimension can be reduced through model reduction techniques, yielding a lower dimensional model, being perfectly suited for the closed-form analysis of the qualitative dynamics. The previous work of the authors has shown that the bifurcation scenario contains a heteroclinic bifurcation, followed by a subcritical Hopf bifurcation. In this chapter, we derive a complete stability chart that characterizes various bifurcation scenarios in closed-form, which allows us to understand the qualitative dynamics of the tippedisk.

Based on more than three decades of rod finite element theory, this publication unifies all the successful contributions found in literature and eradicates the arising drawbacks like loss of objectivity, locking, path-dependence and redundant coordinates. Specifically, the idea of interpolating the nodal orientations using relative rotation vectors, proposed by Crisfield and Jelenić in 1999, is extended to the interpolation of nodal Euclidean transformation matrices with the aid of relative twists; a strategy that arises from the SE(3)-structure of the Cosserat rod kinematics. Applying a Petrov-Galerkin projection method, we propose a novel rod finite element formulation where the virtual displacements and rotations as well as the translational and angular velocities are interpolated instead of using the consistent variations and time-derivatives of the introduced interpolation formula. Properties such as the intrinsic absence of locking, preservation of objectivity after discretization and parametrization in terms of a minimal number of nodal unknowns are demonstrated by conclusive numerical examples in both statics and dynamics.

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.

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 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.

The tippedisk is a new mechanical-mathematical archetype for friction induced insta-bility phenomena, showing an inversion similar to the inversion of the tippetop. Un-like the tippetop, the tippedisk has no rotational symmetry, which greatly complicates its analysis. Since the system cannot be reduced to a planar one, one has to consider the full three-dimensional kinematics, being intrinsically nonlinear. In this work a new minimal model is derived that contains the main relevant physical effects so that the inversion phenomenon can be described qualitatively. The in-depth analysis leads to slow-fast systems with homoclinic connections and global bifurcations.

The aim of this paper is to introduce the tippedisk to the theoretical mechanics community as a new mechanical-mathematical archetype for friction induced instability phenomena. We discuss the modeling and simulation of the tippedisk, which is an inhomogeneous disk showing an inversion phenomenon similar but more complicated than the tippetop. In particular, several models with different levels of abstraction, parameterizations and force laws are introduced. Moreover, the numerical simulations are compared qualitatively with recordings from a high-speed camera. Unlike the tippetop, the tippedisk has no rotational symmetry, which greatly complicates the three-dimensional nonlinear kinematics. The governing differential equations, which are presented here in full detail, describe all relevant physical effects and serve as a starting point for further research.