Towards a Prototype of a Spherical Tippe Top

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. 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. Three main different dynamical behaviours are distinguished: tipping, nontipping, hanging, that is, 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. 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. We propose three designs. This “toy” is suitable to validate the mathematical model qualitatively and quantitatively.

1. Introduction

Spinning toys are among the most ancient toys, and there is a great variety of them. It is quite simple to start spinning objects like a top or a gyroscope, and though it is simple to explain their motion in general, it is challenging to write down the detailed equations of motion. Among spinning objects, the tippe top exhibits one of the most bizarre and counterintuitive behaviours. The commercially available tippe tops, patented in Denmark in the 50s, 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). See Figure 1 for an illustration. Remarkably, at the inverted state, the center of mass lies higher than at the initial condition, defying gravity. Experimentally, it is known that such a transition occurs only if the initial spin exceeds a certain critical threshold.

Figure 1 

The tippe top, showing inversion.

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 Jellett invariant (which includes information on the initial angular velocity) six main classes of tops can be identified within three main groups according to the distinguished dynamical behaviours: tipping, nontipping, and hanging; see Figure 3. Note that objects displaying inversion properties such as the tippe top have been known since the 1800s, see, for example, [1]. After the type of tippe top as in Figure 1 was introduced in Denmark, several theoretical articles have been published since then, see, for example, [2–6] for a survey of the literature. Since it was established that sliding friction was necessary to explain the tippe top inversion [2, 7, 8], many studies have been dedicated to the analysis of models for tippe tops, involving linear stability analysis of the relative equilibria, numerical simulations, and so forth. Some studies have addressed the occurrence of transitions between rolling and sliding during the motion, see [3, 4, 9]. In this paper the presented mathematical results mainly reproduce those in [10–15] but our approach is inspired by the hands-on numerical approach as first attempted by Cohen in [2]. 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. We remark that in the mathematical model we stick to the common assumption that the only external force acting on the system consists of a normal reaction force and a frictional force of viscous type opposing the motion of the contact point in the supporting plane. This is the most common assumption in the literature, though in [4] the inclusion of a nonlinear Coulomb-type friction is discussed. It is shown there by numerical simulations that the Coulomb term contributes to the tippe top inversion but the effect is weaker compared to the viscous term. The nonlinear Coulomb term results in algebraic destabilization of the initially spinning top, whereas the viscous friction gives exponential destabilization, see also [5]. This argument motivates our choice of including viscous friction only.

The phase diagram and bifurcation diagrams illustrate the main results that confirm the findings described in [10]; 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. Either the asymptotic state is unique or the system is bistable. 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 [10], 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 [10] 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, nontipping, and hanging that define the three main groups in the classification of spherical tippe tops as mentioned above. As far as we know such a toy does not exist yet. We successfully applied the methodology for efficient use of prototyping during the design process as presented in [16]. To the best of our knowledge this is the first time that 3D printing and rapid prototyping is being applied to design and to produce a “toy” suitable to validate the qualitative and quantitative mathematical model describing the behaviour of a dissipative nonlinear dynamical system. From the bifurcation diagram it was clear that it should be possible to hit three out of the 6 classes of tippe tops (one type for each main group) by keeping one of the characterizing parameters of the system fixed and varying the other. We believe that the realization of an actual toy is a more powerful validation of the model than software simulations which are directly affected by the underlying mathematical idealization assumptions.

Since the two parameters on which the whole classification is based, inertia ratio and eccentricity, are not independent, the challenge was to come up with a feasible prototype which could be easily mechanically driven. After detailed mathematical calculations and the development of 3D animations (see electronic attachments or http://cage.ugent.be/~bm/tippetop/tippetop.html) we used 3D printing to create a functional model giving us a quick and easy hands-on demonstration capability.

2. Mathematical Results

In this paper we consider 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. In [13], Ebenfeld and Scheck presented a detailed analysis of the dynamics of the eccentric spinning sphere on a flat surface where friction is assumed to be only due to sliding, see also [14]. Without making any other assumptions, we show that their results imply a full qualitative understanding of the asymptotic long-term dynamics. Whereas the treatment of [13, 14] is mainly analytical, here we adopt a bifurcation theory point of view leading to a global geometric understanding of the phase diagram. The phase diagram in Figure 2 and bifurcation diagrams in Figure 3 illustrate our main results. Recall that an 𝜔-limit set of a dynamical system is a closed invariant set that is accumulated by a (forward) trajectory [17]. Our main result is summarized in the following theorem.

Figure 2 

Phase diagram for the dynamics of the eccentric sphere with small sliding friction. The regions of the phase diagram correspond to the qualitative types of bifurcation diagrams of 𝑆𝑂(2)×𝑆𝑂(2) relative equilibria in Figure 3.

Figure 3 

Qualitative bifurcation diagrams of 𝑆𝑂(2)×𝑆𝑂(2) relative equilibria, for the different regions in the phase diagram of Figure 2. Solid black branches correspond to stable relative equilibria, while dashed black branches correspond to unstable ones. The vertical states 𝜃=0 and 𝜃=𝜋 always exist, from which intermediate states (with 0<𝜃<𝜋) may branch off.

Theorem 2.1. A spinning eccentric sphere on a flat surface with small slipping friction admits three types of (asymptotically stable) 𝜔-limit sets:(i)vertically spinning top (𝜃=0), which has its center of mass straight below its geometric center,(ii)vertically spinning top (𝜃=𝜋), which has its center of mass straight above its geometric center,(iii)intermediate spinning top (0<𝜃<𝜋), whose center of mass is neither straight below nor straight above its geometric center.These are solutions of constant energy that are purely rolling due to the assumption on sliding friction (i.e., they display no slipping). The vertical states are periodic, whereas, in general, the intermediate states are quasiperiodic.
At most two of the above types of solutions can be stable at the same time. In case the stable solution is unique its basin of attraction consists of almost the entire phase space (subset of full measure). If the system is bistable, the separatrix between the two different domains of attraction for the asymptotically stable states is expected to be formed by the stable manifold of an unstable intermediate spinning top solution.

All the analytical results needed to arrive at this conclusion can in principle be found in [13, 14]. However, these papers stop short of drawing the full global conclusions as formulated in the above theorem, and also crucially they did not present the phase diagram and bifurcation diagrams that we present in Figures 2 and 3.

We note that for the eccentric sphere in regime I, the state 𝜃=0 is always asymptotically stable and thus does not display the tippe top phenomenon. Similarly, no such dynamics arises in regime III since there the inverted state 𝜃=𝜋 is always unstable. Tippe top dynamics may occur in regime II if the Jellett invariant (which is proportional to the initial spin) is sufficiently large, corresponding to the empirical observation that tippe top dynamics requires a sufficiently large initial spin. In the subregimes IIb and III it is also possible to observe incomplete tippe top behaviour, where the top rises but converges to an intermediate state instead of rising all the way to the vertical state 𝜃=𝜋. Note that, in regime I, tipping might occur if the top is initially spun sufficiently fast under an angle not close to 𝜃=0.

It is important to recognize the existence of symmetries. Recall that symmetries are transformations of the phase space that map solutions of a system to other solutions. In our model of the eccentric sphere on a flat surface, symmetries arise due to the homogeneity of the surface on which the sphere moves and the rotational symmetry of the eccentric sphere. The combined symmetries are thus the Euclidean group 𝐸(2) (acting as translations and rotations in the 𝗑𝗒 plane) and the rotation group 𝑆𝑂(2) acting as rotation of the sphere around its axis of symmetry.

It turns out that the 𝜔-limit sets mentioned above are all relative equilibria with respect to the symmetry group 𝑆𝑂(2)×𝑆𝑂(2)<𝐸(2)×𝑆𝑂(2), see Section 4. Recall that relative equilibria with respect to a group Σ are equilibria for the associated flow on a reduced phase space that is obtained from the original phase space by taking the quotient with respect to the action of Σ. The existence and type of such relative equilibria depend solely on the inertia ratio, the eccentricity of the sphere, and the Jellett integral of motion. We identify a number of regimes characterizing the relative equilibria as a function of the Jellett invariant (which is proportional to the initial angular velocity). The vertical states 𝜃=0 and 𝜃=𝜋 are always 𝑆𝑂(2)×𝑆𝑂(2) relative equilibria and their stability depends on the inertia ratio, the eccentricity of the sphere, and the Jellett integral of motion. In addition, intermediate states may exist, which branch off from the vertically spinning solutions. We sketch the phase diagram in Figure 2. For the labeled regions in this phase diagram, the corresponding bifurcation diagrams for the relative equilibria are presented in Figure 3.

The proof of Theorem 2.1, which builds upon the results by [13, 14], can be found in the appendix. In order to present our point of view clearly and in a self-contained way, in Section 3 we also present a derivation of the equations of motion of the eccentric sphere model of the tippe top, including a discussion of the symmetries and their consequences. Here also, one finds a precise description of the assumed nature of the friction and a definition of all the relevant variables that appear as parameters in Figures 2 and 3. In Sections 4 and 5 the relative equilibria of the system and their stability are discussed. The readers who are acquainted with the topic can start reading from Section 6.

We would like to point out that the strategy of proof used here may well be applicable to a large number of similar examples of mechanical systems under the influence of some kind of friction, such as the Rattleback [18] or Hycaro tippe top of Tokieda [19]. The key observation is that for mechanical systems under the influence of friction, in a natural way the energy becomes a Lyapunov function since friction causes energy loss. The next observation is that orbits which do not dissipate energy need to lie entirely on a subvariety of the phase space that is defined by the condition that friction is absent. Equilibria naturally lie on this subset since they have zero velocity and friction is absent at zero velocity. However, one would expect that typically no solution lies entirely on this subvariety, unless the solution lies on the orbit of a symmetry group that leaves the zero friction subvariety invariant. (One can make this precise by constructing a small local perturbation that moves a solution off the zero friction subvariety.) In many cases, the set of such relative equilibria can be accurately analyzed, either analytically or numerically, and the local stability properties can be deduced from a dissipation-induced instability point of view, based on the local stability properties of the relative equilibria in the absence of friction. We note in this respect that the set of 𝜔-limit sets on the zero friction subvariety is independent of the form and size of the friction. The final step is to draw global conclusions from this local information. The latter is within reach if one has a good understanding of the 𝜔-limit sets. We would like to stress that Theorem 2.1 concerns the asymptotic dynamics. In an experiment with small friction, the observation may well be dominated by transient dynamics which bears strong resemblance (on short time scales) to the dynamics of the spherical top without friction. The dynamics of the latter is rather complicated as it is a nonintegrable Hamiltonian system. In Section 7 we present some results from numerical simulations demonstrating explicitly some examples where the transient dynamics does not appear to prevent fast convergence to the asymptotic states (although of course for sufficiently small friction coefficient the transient dynamics would dominate on finite time intervals).

3. The Equations of Motion

We consider an eccentric sphere as in Figure 4, where 𝑂 denotes the center of mass and 𝒞 the center of the sphere. The line joining the center of mass and the geometrical center is an axis of inertial symmetry: in the plane perpendicular to this axis the moment of inertia tensor of the sphere has two equal principal moments of inertia.

Figure 4 

Eccentric sphere version of the tippe top. 𝑅 is the radius of the sphere, the center of mass 𝑂 is off center by 𝜖. The top spins on a horizontal table with point of contact 𝑄. The axis of symmetry is 𝑂𝑧 and the vertical axis is 𝑂𝑍, they define a plane Π (containing 𝐎𝐐) which processes about 𝑂𝑍 with angular velocity 𝜑̇. The height of 𝑂 above the table is ℎ(𝜃).

We describe the motion of the sphere using three reference frames:(I)an inertial (laboratory) frame 𝑀𝗑𝗒𝗓, where 𝑀 is some point on the table and the 𝗓-axis is the vertical.(II)a (noninertial) rotating frame 𝑂𝑋𝑌𝑍 whose origin is in the center of mass 𝑂 and whose 3rd axis is always parallel to the vertical. The 𝑋 and 𝑌 axes are specified below.(III)a principal axis system 𝑂𝑥𝑦𝑧, whose 𝑧-axis is the symmetry axis of the sphere (Note that in [5] the origin of the reference frames (II) and (III) is at the center of the sphere, and not at the center of mass.)

The reference frames (II) and (III) are indicated in Figure 4. The eccentricity 𝜖 is the distance between the center of mass 𝑂 and the geometric center 𝒞 of the sphere, with 0<𝜖<𝑅, where 𝑅 denotes the radius of the sphere. We denote the moments of inertia 𝐼_𝑥=𝐼_𝑦=∶𝐴 and 𝐼_𝑧=∶𝐶. The point of contact with the plane of support is denoted by 𝑄.

Let (𝜃,𝜑,𝜓) be the Euler angles of the body with respect to 𝑂𝑍, see Figure 5 for an illustration. The 𝑂𝑧𝑍-plane Π contains the vector 𝐪=𝐎𝐐 which joins the center of mass to the point of contact. The plane Π is inclined at angle 𝜑 to the vertical 𝑀𝗑𝗓-plane and processes with angular velocity 𝜑̇ around the vertical 𝑂𝑍. We choose the horizontal 𝑂𝑋 in Π so that 𝑂𝑌 is perpendicular to Π. For the rotating frame, 𝑂𝑥 is in Π and perpendicular to the symmetry axis 𝑂𝑧 and the axis 𝑂𝑦 coincides with 𝑂𝑌. (Note that the axes 𝑂𝑥 and 𝑂𝑦 are principle axes, but they are not body fixed.) The angle between the vertical 𝑂𝑍 and the axis 𝑂𝑧 of the top is denoted by 𝜃. The angular velocity 𝜃̇ describes the nutation of the body in the vertical plane Π. The angle 𝜓 describes the orientation of the body with respect to the 𝑂𝑋𝑌𝑍 frame and 𝜓̇ is the spin of the sphere around its symmetry axis. We denote by 𝐞₁,𝐞₂,𝐞₃ the unit vectors along 𝑂𝑋,𝑂𝑌,and𝑂𝑍 and by 𝐢,𝐣,and𝐤 those along 𝑂𝑥,𝑂𝑦, and 𝑂𝑧. Note that 𝐞₂=𝐣. Because of the inherent translational symmetry (of the body on the plane), it is convenient to describe the body in terms of the relative (moving) reference frames (II) and (III), rather than in the absolute reference frame (I). By doing so we thus ignore the translational motion on the plane and focus on the relative motion of the body, which captures the tippe top behaviour.

Figure 5 

Euler angles (𝜑,𝜃,𝜓) to fully determine the body, given the coordinates of the center of mass 𝑂 in an inertial frame with center 𝑀.

The (relative) position vector of the body is𝐱=𝑥𝐢+𝑦𝐣+𝑧𝐤=𝑋𝐞₁+𝑌𝐞₂+𝑍𝐞₃, (3.1) or 𝐱=(𝑥,𝑦,𝑧)_{𝑂𝑥𝑦𝑧}=(𝑋,𝑌,𝑍)_{𝑂𝑋𝑌𝑍}. Note that the coordinates of the reference frames (II) and (III) are related by the relations(𝜃)(𝜃),(𝜃)(𝜃).𝑥=𝑋cos−𝑍sin𝑦=𝑌,𝑧=𝑋sin+𝑍cos (3.2) The reference frames (II) and (III) rotate with respective angular velocities𝛀_𝐼𝐼=0,0,𝜑̇_{𝑂𝑋𝑌𝑍}=−𝜑̇sin(𝜃),0,𝜑̇cos(𝜃)_{𝑂𝑥𝑦𝑧},𝛀_𝐼𝐼𝐼=−𝜑̇sin(𝜃),𝜃̇,𝜑̇cos(𝜃)_{𝑂𝑥𝑦𝑧}. (3.3) The angular velocity of the body 𝝎 involves, in addition, the angular velocity 𝜓̇:̇(𝜃)̇=̇(𝜃)𝐞𝝎=−𝜑sin𝐢+𝜃𝐣+𝑛𝐤𝜓sin₁+𝜃̇𝐞₂+𝜑̇+𝜓̇cos(𝜃)𝐞₃=𝑛−𝜑̇cos(𝜃)sin(𝜃)𝐞₁+𝜃̇𝐞₂+𝜑̇sin²(𝜃)+𝑛cos(𝜃)𝐞₃, (3.4) where 𝑛∶=𝜓̇+𝜑̇cos(𝜃) denotes the component of 𝝎 about 𝑂𝑧, better known as the spin. For later use we introduce the notation 𝝎=(𝜔_𝐢,𝜔_𝐣,𝜔_𝐤). Consequently, with 𝐈 denoting the inertia tensor of the sphere, the angular momentum of the sphere is given bẏ(𝜃)̇=̇(𝜃)(𝜃)𝐞𝐋=𝐈𝜔=−𝐴𝜑sin𝐢+𝐴𝜃𝐣+𝐶𝑛𝐤𝐶𝑛−𝐴𝜑cossin₁+𝐴𝜃̇𝐞₂+𝐴𝜑̇sin²(𝜃)+𝐶𝑛cos(𝜃)𝐞₃. (3.5) The point of contact 𝑄 has coordinates𝑄=−ℎ²(𝜃)𝑑(𝜃)𝑑𝜃cosℎ(𝜃),0,−ℎ²(𝜃)𝑑(𝜃)𝑑𝜃sinℎ(𝜃)_{𝑂𝑥𝑦𝑧}=(𝜖sin𝜃,0,𝜖cos𝜃−𝑅)_{𝑂𝑋𝑌𝑍}=(𝑅sin𝜃,0,𝜖−𝑅cos𝜃)_{𝑂𝑥𝑦𝑧}. (3.6) The velocity of the point of contact 𝑄 is𝐕_𝑄=𝐯_𝑂+𝝎×𝐪, (3.7) where 𝐪=𝐎𝐐 denotes the vector from the center of mass 𝑂 to the point of contact 𝑄 and 𝐯_𝑂 is the velocity of the center of mass. We set 𝐯_𝑂=𝑈𝐞₁+𝑉𝐞₂+𝑊𝐞₃ and use the fact that 𝑄=(ℎ^′(𝜃),0,−ℎ(𝜃))_{𝑂𝑋𝑌𝑍} to obtain𝝎×𝐪=−𝜃̇ℎ(𝜃),sin(𝜃)𝜖𝜑̇+𝑅𝜓̇,−𝜃̇ℎ^′(𝜃)_{𝑂𝑋𝑌𝑍}=−𝜃̇ℎ(𝜃),sin(𝜃)𝜖𝜑̇+𝑅𝑛−𝜑̇cos(𝜃),−𝜃̇ℎ^′(𝜃)_{𝑂𝑋𝑌𝑍}. (3.8) Hence, we obtain𝐕_𝑄=𝑈−𝜃̇ℎ(𝜃),𝑉+sin(𝜃)(𝜖𝜑̇+𝑅(𝑛−𝜑̇cos(𝜃))),𝑊−𝜃̇ℎ^′(𝜃)_{𝑂𝑋𝑌𝑍}. (3.9) The fact that the sphere remains in contact with the table is expressed by the (holonomic) constraint𝗓_𝑂=ℎ(𝜃)=𝑅−𝜖cos(𝜃), (3.10) where ℎ(𝜃) denotes the height of the center of mass 𝑂 above the table, compare Figure 4. From this constraint it follows that the 𝐞₃ component of 𝐯₀ is𝑑𝑊∶=𝗓𝑑𝑡⁰=𝜃̇ℎ^′(𝜃)=𝜖𝜃̇sin(𝜃) (3.11) so that the 𝑍-component of 𝐕_𝑄 vanishes, consistent with the constraint.

We note that the physical interpretation of 𝐕_𝑄 concerns the phenomenon of slipping. In case 𝐕_𝑄≠𝟎 the body slips on the surface. In contrast, a rolling motion of the body is characterized by the fact that 𝐕_𝑄=𝟎.

The equations of motion will be derived, in Newton’s spirit, as a consequence of the action of external forces. We distinguish the following forces acting on the sphere(i)The gravitational force: 𝐆=−𝑚𝑔𝐞₃, where 𝑚 is the total mass of the sphere.(ii)A force 𝐅_𝑄=𝐑_𝑛+𝐑_𝑓 acting on the point of contact 𝑄, where 𝐑_𝑛=𝑅_𝑛𝐞₃, is the normal reaction force at 𝑄 (due to the stiffness of the surface) and 𝐑_𝑓 is a friction force. For completeness we mention [20] where a mathematical model for the tippe top is proposed taking elasticity properties of the table and tippe top into account.

Friction is the resistive force acting between bodies that tends to oppose and damp out motion. Friction is usually distinguished as being either static friction (the frictional force opposing placing a body at rest into motion) or kinetic friction (the frictional force tending to slow a body in motion). Importantly, we assume that the friction force is entirely due to the slipping of the sphere on the surface and neglect all other sources of friction. Friction forces can be complicated, and there are various models in circulation. We adopt the assumption of viscous friction [9, 15, 21] and assume the friction force to be given by𝐑_𝑓=−𝜇𝑅_𝑛𝐕_𝑄, (3.12) where 𝜇 is the coefficient of sliding friction with the dimension of (velocity)⁻¹. 𝐑_𝑓 is proportional to the size of the normal reaction force and vanishes smoothly when 𝐕_𝑄→𝟎. (An alternative model for the friction force is the so-called Coulomb friction 𝐑_𝑓=−𝜇𝑅_𝑛(𝐕_𝑄/|𝐕_𝑄|). This model is not appropriate when 𝐕_𝑄→𝟎 due to the singular nature of this force when 𝐕_𝑄=𝟎.) Euler’s equations of motion for the sphere,𝑑𝐋𝑑𝑡+𝛀×𝐋=𝐪×𝐅_𝑄, (3.13) govern the evolution of the angular momentum 𝐋 in a noninertial reference frame, rotating with frequency Ω, due to the influence of the external torque 𝐪×𝐅_𝑄. The equation of motion for the center of mass 𝑂 in the rotating frame is𝑚𝑑𝐯_𝑂𝑑𝑡+𝛀×𝐯_𝑂=−𝐆+𝐅_𝑄. (3.14) In terms of the coordinates in reference frame (III) the equations of motion (3.13) yield𝐴𝜑̈sin(𝜃)=𝐶𝑛−2𝐴𝜑̇cos(𝜃)𝜃̇+𝑧_𝑄𝐹_𝑌,𝐴𝜃̈=−𝜑̇sin(𝜃)𝐶𝑛−𝐴𝜑̇cos(𝜃)+𝑍_𝑄𝐹_𝑋−𝑋_𝑄𝑅_𝑛,𝐶𝑛̇=𝑥_𝑄𝐹_𝑌, (3.15) where 𝑄=(𝑥_𝑄,𝑦_𝑄,𝑧_𝑄)_{𝑂𝑥𝑦𝑧}=(𝑋_𝑄,𝑌_𝑄,𝑍_𝑄)_{𝑂𝑋𝑌𝑍} and 𝐅_𝑄=(𝐹_𝑋,𝐹_𝑌,𝐹_𝑍)_{𝑂𝑋𝑌𝑍}. From the equation for the motion of the center of mass (3.14), in terms of reference frame (II), we obtain𝑚𝑈̇−𝜑̇𝑉=𝐹_𝑋,𝑚𝑉̇+𝜑̇𝑈=𝐹_𝑌,𝑚𝑊̇=𝑅_𝑛−𝑚𝑔. (3.16) Recalling that 𝑊=𝜃̇ℎ^′(𝜃), from the last of the latter equations we may derive an expression for 𝑅_𝑛:𝑅_𝑛𝑑=𝑚𝑔+𝑚̇𝑑𝑡𝜃ℎ^′(𝜃)=𝑚𝑔+𝑚𝜖𝜃̈sin(𝜃)+𝜃̇²cos(𝜃). (3.17) The equations of motion (3.15) and (3.16) can be written as a system of six coupled first-order nonlinear ordinary differential equations in the variables (𝑢,𝑣,𝛼,𝜑,𝛽,𝜃,𝑛), where 𝛼∶=𝜑̇, 𝛽∶=𝜃̇, 𝑢∶=𝑚𝑈, and 𝑣∶=𝑚𝑉.

Setting 𝑚=1 for simplicity, these may be arranged in the standard form 𝑏̇=𝑓(𝑏) (when sin(𝜃)≠0)1𝛼̇=(𝜃)𝐶sin𝐴𝑛𝛽−2𝛼cos(𝜃)𝛽+𝑧_𝑄𝐹_𝑌𝐴,𝛽̇=1𝐴𝛼sin(𝜃)(𝐴𝛼cos(𝜃)−𝐶𝑛)−𝑅_𝑛𝑋_𝑄+𝑍_𝑄𝐹_𝑋,̇𝑥𝜃=𝛽,𝑛̇=^𝑄𝐹_𝑌𝐶,𝑢̇=𝛼𝑣+𝐹_𝑋,𝑣̇=−𝛼𝑢+𝐹_𝑌,̇̇(𝜃).𝜑=𝛼,𝜓=𝑛−𝛼cos (3.18) It should be remembered that 𝑅_𝑛, 𝐹_𝑋, and 𝐹_𝑌 are still functions of the other variables. For instance, from (3.17) and (3.18) one finds𝑅_𝑛=𝑔+𝛽²ℎ^′′(𝜃)+ℎ^′(𝜃)𝛼sin(𝜃)(𝛼cos(𝜃)−𝐶𝑛/𝐴)1+ℎ^′(𝜃)/𝐴−ℎ(𝜃)𝜇𝑈−𝛽ℎ(𝜃)+ℎ^′(𝜃). (3.19) Recall that we require that 𝑅_𝑛≥0. If this condition fails, the sphere loses contact with the surface. Expressions for 𝐹_𝑋 and 𝐹_𝑌 follow similarly from (3.12).

It is important to recognize that some of the structure of the equations of motion (3.18) is due to symmetry. We recall that the symmetries are the Euclidean group 𝐸(2) (acting as translations, rotations, and reflections in the 𝑀𝗑𝗒 plane) and the rotation group 𝑆𝑂(2) acting as rotation of the sphere around its axis of symmetry. The effect of the Euclidean symmetry is that the right-hand side of the equations of motion contain no reference to the position of the sphere on the surface. In a similar way, due to the rotational symmetry the equations of motion do not depend explicitly on 𝜑. The system can be viewed as three coupled systems, where the coupling is of skew product type: the evolution of 𝛼,𝛽,𝜃, and 𝑛 does not depend on 𝑢,𝑣, and 𝜑, and the evolution of 𝑢, and 𝑣 does not depend on 𝜑. Moreover, note that the position of the center of mass relative to the surface (in 𝑀𝗑𝗒 coordinates) could in principle be obtained by integrating the velocities 𝑢 and 𝑣 over time. Because of the fact that we take friction into account, Noether’s theorem does not apply, so the continuous symmetries we observe need not (and do not) give rise to conserved quantities. However, it was discovered by Jellett [8] by an approximate argument, and later proved by Routh [6], that the system (3.18) has the following conserved quantity:𝐽=−𝐋⋅𝐪. (3.20) Indeed, it follows from (3.13) that (𝑑𝐋/𝑑𝑡+Ω×𝐋)⟂𝐪, so that𝑑𝑑𝑑𝑡𝐽=−𝐋⋅̇(𝜃)ℎ𝑑𝑡𝐪+𝛀×𝐪=−𝐶𝑛−𝐴𝜑cos^′(𝜃)²𝑑(𝜃)𝑑𝑡sinℎ^′(𝜃)₌₀=0. (3.21) Note that the Jellett invariant can be written as ([10, 15, 21])𝐽=𝐶𝑛(𝑅cos(𝜃)−𝜖)+𝐴𝜑̇𝑅sin²(𝜃). (3.22)

4. 𝜔-Limit Sets Are Relative Equilibria

Our aim is to describe the asymptotic dynamics of the eccentric sphere. Recall that a subset of the phase space is an 𝜔-limit set if this set is accumulated by (forward) orbits. While the friction force destroys the Hamiltonian nature of the dynamics, it greatly simplifies the asymptotic dynamics. This follows from the fact that in the presence of friction the energy, which is conserved in the absence of friction, is almost always decreasing along solutions.

The energy is given by 𝐸=𝑇+𝑉, where 𝑇=𝑇_rot+𝑇_tr is the kinetic energy with its rotational and translational part and 𝑉=𝑚𝑔ℎ(𝜃) is the potential energy. With our choice of variables we may write 𝑇_rot=(1/2)(𝐴𝜔²_𝐢+𝐴𝜔²_𝐣+𝐶𝜔²_𝐤) and 𝑇_tr=(1/2)𝑚(𝗑̇²+𝗒̇²+𝗓̇²), where 𝗓̇=𝜖𝑚sin𝜃𝜃̇.

Lemma 4.1 (see [13]). The energy 𝐸 is a Lyapunov function (recall that a Lyapunov function is nonincreasing along orbits) for (3.18). In particular, 𝑑𝑑𝑡𝐸=𝐕_𝑄⋅𝐑_𝑓≤0. (4.1)

As 𝐑_𝑓 is parallel (and opposite) to 𝐕_𝑄, (𝑑/𝑑𝑡)𝐸 vanishes if and only if 𝐕_𝑄 vanishes. Observe that 𝐸(𝑡) decreases monotonically and hence is a suitable Lyapunov function. Moreover, 𝐸(𝑡) is analytic and therefore along orbits it is either strictly decreasing or constant. The energy 𝐸 is constant only if 𝐕_𝑄=0, that is in the absence of friction. Thus, the 𝜔-limit sets must consist of orbits which do not experience friction. We show that such orbits are necessarily relative equilibria.

Proposition 4.2. Solutions have constant energy only if they are relative equilibria with respect to the action of 𝑆𝑂(2)×𝑆𝑂(2).

Proof. We already concluded that 𝐕_𝑄 needs to be equal to 0 along any orbit in an 𝜔-limit set. A straightforward calculation shows that 𝐕_𝑄=(𝑑/𝑑𝑡)𝐕_𝑄=0 indeed implies that 𝜙̇=𝑋̇=0, so that such a solution must be an 𝑆𝑂(2)×𝑆𝑂(2) relative equilibrium.

This observation is in fact what one would generically expect to find. If 𝑀 is a submanifold of the phase space that corresponds to the absence of friction, in general it would be quite unexpected to find a nonequilibrium solution that lies entirely inside 𝑀.

5. Stability and Bifurcations of Relative Equilibria

Having determined that the relative equilibria are the only possible asymptotic states in the presence of friction, we derive in this section these solutions of constant energy using the explicit equations of motion (3.15)-(3.16), see also [5, 13, 15]. With 𝜑̇=constant and 𝐕_𝑄=0,𝐑_𝑓=0, the equations of motion yield 𝛼̇=0, 𝛽̇=0, 𝛽=0, 𝑛̇=0, 𝑈=0, 𝑊=0 and(𝜃(𝜃))(𝜃)[𝑅((𝜃](𝜃)𝛼𝑉=0,𝛼sin)(𝐴𝛼cos−𝐶𝑛−𝑚𝑔𝜖sin=0,𝑉+𝑛−𝛼cos))+𝜖𝛼sin=0. (5.1) These equations have the following three types of solutions. The linear stability analysis differs from [13, 14] in methodology.

Vertical States
(1)Vertical state 𝜃=0: ̇𝑈=𝑉=0,𝜃=0,𝑛=arbitraryconstant,𝛼=𝜑=undefinedconstant. (5.2) The top is spinning about its axle with center of mass straight below the geometric center.(2)Vertical state 𝜃=𝜋:̇𝑈=𝑉=0,𝜃=0,𝑛=arbitraryconstant,𝛼=𝜑=arbitraryconstant. (5.3) The top is spinning about its axle with center of mass straight above the geometric center.

Intermediate States
For these solutions we have 𝑈=𝑉=0, 0<𝜃<𝜋, and 𝑛,𝛼,and𝜃 are related by (𝜃)−𝜖𝑛=𝛼cos𝑅𝛼((𝜃))𝛼,𝐴𝛼cos−𝐶𝑛−𝑚𝑔𝜖=0. (5.4)(5.5) Elimination of 𝑛 from the above yields 𝛼²=𝑚𝑔𝜖(𝐴−𝐶)cos(𝜃)+𝐶(𝜖/𝑅). (5.6) Hence, the condition for the existence of intermediate states is 𝐴𝐶(𝜃)+𝜖−1cos𝑅>0. (5.7)

It is natural to divide the solutions into three groups, according to regimes of the parameters 𝐴/𝐶 and 𝜖/𝑅 [21].Group I: 𝐴/𝐶<1−𝜖/𝑅. Intermediate states exist with 𝜃>𝜃_𝑐1=cos⁻¹𝜖/𝑅1−𝐴/𝐶,with0<𝜃_𝑐1<𝜋2. (5.8)Group II: 1−𝜖/𝑅<𝐴/𝐶<1+𝜖/𝑅. Intermediate states exist with any 0<𝜃<𝜋.Group III: 𝐴/𝐶>1+𝜖/𝑅. Intermediate states exist with 𝜃<𝜃_𝑐2=cos⁻¹𝜖/𝑅𝜋1−𝐴/𝐶,with2<𝜃_𝑐2<𝜋. (5.9)

As in [10] we further refine this classification. Note that the intermediate states discussed here correspond to the tumbling solutions discussed in [13].

The intermediate states are completely determined by (5.4), (5.5), and the Jellett invariant 𝐽. More precisely, combining the square of (3.22) with (5.4) and (5.6), they are obtained by solving𝑓𝐽²𝐽,cos𝜃∶=²𝑚𝑔𝜖𝐶𝑅²𝐴𝐶(𝜃)+𝜖−1cos𝑅−(𝜃)−𝜖cos𝑅²+𝐴𝐶1−cos²(𝜃)²=0. (5.10) The next theorem summarizes the linear stability and local bifurcation results, as depicted in Figure 3 (cf. [21]), and the proof is sketched in the Appendix. We identify six different groups according to how the value of 𝐴/𝐶 is related to the eccentricity 𝜖/𝑅. In the literature, results have previously been expressed in terms of variables 𝑛₀ and 𝑛_𝜋, referring to the spin of an initial condition in a vertical state. We define 𝑛₀∶=𝐽/𝐶(𝑅−𝜖), which is the value of the spin 𝑛 at 𝜃=0 for motion with Jellett invariant 𝐽. Similarly, 𝑛_𝜋∶=−𝐽/𝐶(𝑅+𝜖) denotes the spin of the solution with Jellett invariant 𝐽 at 𝜃=𝜋. Note that, for a fixed value of the Jellett invariant 𝐽, these spins are related by 𝑛₀=−𝑛_𝜋(𝑅+𝜖)/(𝑅−𝜖). We further define𝑛_±∶=±𝑚𝑔𝜖𝐶)]()[(1±𝜖/𝑅−𝐴/𝐶1±𝜖/𝑅,𝑏∶=𝜖/𝑅−(1/3)𝐴/𝐶1−𝐴/𝐶−(𝜖/𝑅)²,1−𝐴/𝐶 (5.11) and 𝜃_𝑏∶=arccos(𝑏). Furthermore we define 𝑐=𝜖/𝑅/1−(𝐴/𝐶) and 𝜃_𝑐∶=arccos(𝑐).

Theorem 5.1. The bifurcation diagrams of the eccentric sphere spinning on a flat surface with small friction fall in one of the following six categories (Figure 3).Group I: 𝐴/𝐶−1<−𝜖/𝑅.
(i)The vertical state 𝜃=0 is stable for any value of 𝐽.(ii)The vertical state 𝜃=𝜋 is stable if |𝑛_𝜋|>𝑛₊ and unstable otherwise.(iii)Intermediate states exist for all values of 𝜃 satisfying 𝜃>𝜃_𝑐. Group Ia: 𝑏<−1. The entire branch of intermediate states is unstable. Group Ib: 𝑏>−1. The branch of intermediate states has a fold point at 𝜃=𝜃_𝑏. The branch with 𝜃>𝜃_𝑏 is stable, while the branch with 𝜃<𝜃_𝑏 is unstable.
Group II: −𝜖/𝑅<(𝐴/𝐶−1)<𝜖/𝑅.
(i)The vertical state 𝜃=0 is stable if |𝑛₀|<𝑛₋ and unstable otherwise.(ii)The vertical state 𝜃=𝜋 is stable if |𝑛_𝜋|>𝑛₊ and unstable otherwise.(iii)Intermediate states exist for all 𝜃. We distinguish the following three subgroups.Group IIa: (𝐴/𝐶−1)<−(𝜖/𝑅)²  and |𝑏|<1. A fold bifurcation of intermediate states occurs.Group IIb: (𝐴/𝐶−1)>−(𝜖/𝑅)²  or 𝑏>1. The entire branch of intermediate states is stable.Group IIc: (𝐴/𝐶−1)<−(𝜖/𝑅)²  and 𝑏<−1. The entire branch of intermediate states is unstable.
Group III: (𝐴/𝐶−1)>𝜖/𝑅.
(i)The vertical state 𝜃=0 is stable if |𝑛₀|<𝑛₋.(ii)The vertical state 𝜃=𝜋 is unstable for all 𝐽.(iii)Intermediate states exist for 𝜃<𝜃_𝑐 and are all stable.

The proof of Theorem 5.1 can be mainly recovered from [10]; for completeness we provide the calculations based on a direct approach in the appendix.

6. Prototype of a Spherical Tippe Top

Rapid prototyping (RP) technologies enable solid models to be obtained from designs generated with CAD applications. Their increasing popularity in industry is due to the reduction in cost and time associated with the use of these models when verifying product development stages and improvements in end quality. These technologies can also be applied to verify the correctness and/or accuracy of mathematical models and last but not least to enhance students’ active learning in the frame of a learning-by-doing approach. Students can bring their designs to fruition and develop a deeper insight in abstract concepts. We made a prototype of a spherical tippe top for educational use in the Product Development Laboratory of Howest.

As pointed out earlier, to realize a 3-in-1 toy an axially symmetric sphere where one has control over 𝐴/𝐶 and 𝜖/𝑅 is needed. We considered three possible designs:(1)a solid sphere with a cylindrical hole through the center where a setscrew can move;(2)a hollow sphere with a cylindrical rod on which a weight can move;(3)a hollow sphere with a toroidal band at the equator fitted with a cylindrical rod on which a weight can be screwed.

From the bifurcation diagram in Figure 2, it is clear that it is possible to hit the three main groups by fixing 𝜖/𝑅 and changing 𝐴/𝐶. Therefore, it is important to understand for the three designs how 𝐴/𝐶 and 𝜖 vary with respect to each other when the weight is moved. We set up a Maple worksheet based on the given mathematical description and calculated 𝐴/𝐶 and 𝜖 in function of the position of the midpoint of the moving weight with respect to the center of the sphere; this will be further on denoted by 𝑍₂. We took into account the physical parameters: dimensions of the different parts (radii, heights, and thickness) and the density of the materials.

From this we realized that for the solid sphere the goal of the three types is within reach, whereas for the hollow sphere the design has to be modified. Our modifications resulted in the third design as given above. We now discuss our findings for the realized prototypes. Our realizations were all printed with the commercial available Dimension SST1200es with printing technology based on the FDM principle (fused deposition modeling) in ABSplus.

6.1. Sphere with Cylindrical Hole and Setscrew

For the first design we realized three different tops, varying the geometrical dimensions. This was done because the calculation showed that for the given materials some zones are hard to achieve or are very narrow, see Figure 7. The prototype consists of a sphere with a cylindrical hole through the center, together with a piece of adjustable cylindrical iron wire (setscrew), see Figure 6. With a caliper, it can be checked how deep the setscrew is set in the hole. The position of the midpoint of the setscrew with respect to the center of the sphere is denoted by 𝑍₂. The hole is suitable for a setscrew M12. The dimensions of the toy were chosen based on the mathematical calculations derived from the model. The diameter of the sphere was chosen so that one can comfortably spin the toy by hand. With a sphere of diameter 50 mm, good values for the chosen design are a hole of radius 5.5 mm, filled with the setscrew of height 15 mm, or a hole of radius 1.5 mm, filled with the setscrew of height 3 mm, see Figure 7. The densities are 1.08 g/cm³ for ABSplus and 7.87 g/cm³ for the setscrew.

(a)

(b)

Figure 6 

First prototype of a spherical tippe top.

(a)

(b)

Figure 7 

Different tippe top regimes in function of the position 𝑍2 of the midpoint of the setscrew. The black lines are ±𝜖(𝑍2)/𝑅, the blue curve is 𝐴(𝑍2)/𝐶−1, and the red curve is −(𝜖(𝑍2)/𝑅)2. On the left, the result for the printed prototype with a M12 setscrew, see Figure 6. On the right, the result for a prototype of the same shape but a setscrew M3.

The prototype is axially symmetric; therefore, only the eccentricity 𝜖 and the moment of inertia 𝐴 are functions of 𝑍₂, they are easily calculated; 𝐶 remains constant when moving the setscrew up and down. In Figure 7 the quantities (𝐴/𝐶)−1, ±𝜖/𝑅, and −(𝜖(𝑍₂)/𝑅)² are plotted as functions of 𝑍₂, (a) for the prototype with a M12 setscrew and (b) for the prototype with a M3 setscrew. The printed prototype in Figure 6, is of the first type and according to the mathematical calculations will exhibit the predicted behaviour as follows: for 𝑍₂ between 0mm and 7.95mm the toy does not show tippe top dynamic no matter what the initial spin is (type I), for 𝑍₂ between 7.95mm and 17.76mm complete tippe top dynamic is observed (type IIc). For 𝑍₂ above 17.76mm the top is of type IIb (incomplete tipping is observed if initial spin is not sufficiently high).

For the different positions of the weight, we launched by hand the toy plenty (≥50) of times and registered each time tipping, nontipping, or hanging. In Table 1 we report the typical results for 5 launches. Note that the tippe top is hand spun, so there will be a deviation from the starting position 𝜃=0. Tipping and nontipping were mostly observed in the setup of type IIa and I, respectively. For the setup of type IIb, the expected behaviour (tipping) was not observed; we always observed the hanging behaviour. This is because we were not able to launch the toy fast enough by hand and also because the setscrew sticking out the toy does not allow ideal launching position. Our observations indicate that the prototype behaves as predicted by the model. In details, tipping and hanging at 𝑍₂=5 can be explained by the presence of a branch of intermediate states and stable 𝜋 position that one could hit if the toy is not launched exactly from the 𝜃=0 position, see Figure 3. The hanging at 𝑍₂=15 is due to the fact that we did not launch the top fast enough.

Table 1 

Hand launched tippe top. Observed occurrences of the different types on 5 launches.

For a prototype fit for a M3 screw, the intervals for 𝑍₂ are as follows: for 𝑍₂ between 0mm and 3.33mm the toy does not show tippe top dynamic no matter what the initial spin is (type I); for 𝑍₂ between 3.33mm and 18.20mm complete tippe top dynamic is observed (type II). For 𝑍₂ above 18.20mm the top is of type III (incomplete tipping is observed). Also, this prototype was spun ≥50 times, and we registered similar observations as for the previous one.

We conclude that this prototype can give a working 3-into-1 toy but has some disadvantages.(i)The setscrew can come loose after intensive use.(ii)When there are three zones present, at least one of the zones is small.(iii)Using a caliper to know if the setscrew is in the center is not practical.

Several attempts were done in the computations to improve the design, for example, by adding holes into the solid sphere. These attempts were not successful, so no other prototype was printed. Instead, we concentrated on the sphere with a cylindrical rod.

6.2. Sphere with Cylindrical Rod

The second design consists of a spherical shell with a cylindrical solid rod through the center along which a symmetric bead is spun; this bead can be put at different heights along the rod. In this design, the user must open the sphere and change the position of the weight by screwing it up or down, after which the sphere can be closed and spun. See Figure 8.

Figure 8 

Design of the second prototype.

The advantage of this design is that different ABS colors can easily be used for both sides making the tipping more visible and that the rod can be marked at the critical positions. Where in the first design setscrews of different length can be used, in this design weights of different lengths and different widths can be considered within one toy. Also, different materials for the rod can be considered.

However, computations show that a 3-into-1 toy was difficult to obtain with the chosen materials: ABS, iron, and nylon. We briefly summarize our findings that form the basis of further improvements that lead to the third prototype. We tried three different possibilities for the rod: iron, nylon, and ABS. Nylon and ABS seem to work best. With an iron rod and physical parameters that allow easy playing with the toy we did not succeed in catching all the three zones. See below for the specific values of the parameters. As Figure 9(a) shows, only zone three is hit, which means that this tippe top never shows complete inversion but may tip up to a certain angle.

(a)

(b)

Figure 9 

(a) Different tippe top regimes in function of the position of the midpoint of the setscrew. The black lines are ±𝜖(𝑍2)/𝑅, the blue curve is 𝐴(𝑍2)/𝐶−1, and the red curve is −(𝜖(𝑍2)/𝑅)2. The physical parameters for the construction are radius of the spherical shell in ABS of 25mm, thickness of the shell of 2.5mm, radius of the iron cylindrical rod of 3mm, radius of the (iron) weight of 10 mm, and height of the weight of 10mm. (b) Different tippe top regimes in function of the position of the midpoint of the setscrew. The black lines are ±𝜖(𝑍2)/𝑅, the blue curve is 𝐴(𝑍2)/𝐶−1, and the red curve is −(𝜖(𝑍2)/𝑅)2. The physical parameters for the construction are radius of the ABS spherical shell of 25mm, thickness of the shell of 2.5mm, radius of the nylon cylindrical rod of 2.5mm, radius of the (iron) weight of 10mm, and height of the weight of 10mm.

The prototype is axially symmetric; therefore, only the eccentricity 𝜖 and the moment of inertia 𝐴 are functions of 𝑍₂, they are easily calculated; 𝐶 remains constant when moving the weight up and down. The physical parameters for the construction are radius of the spherical shell of 25 mm, thickness of the shell of 2.5mm, radius of the cylindrical rod of 3mm, radius of the weight of 10 mm, and height of the weight of 10mm.

In the case of a nylon or ABS rod, it was possible to obtain a tipping top. As illustrated in Figure 9(b), both zones II and III are hit, which means that complete tipping and incomplete tipping may be observed according to the position of the weight. Note that the section of the blue curve in zone III is very small, which makes it very difficult to observe the hanging phenomenon. A similar observation holds for the section of the blue curve in zone I; this is so small that nontipping behaviour cannot in practice be observed. Many launches of this prototype only confirmed the observations above. This seems to be a good type II tippe top, but we were not able to observe the other behaviours.

6.3. Sphere with a Toroidal Band and a Cylindrical Rod

The third design consists of a spherical shell with a toroidal band around the equator and a cylindrical solid rod through the center along which a symmetric bead is spun. Also in this design, the user must open the sphere and change the position of the weight by screwing it up or down, after which the sphere can be closed and spun. See Figure 10.

Figure 10 

Design of the third prototype.

Adding a toroidal band was a way to find a compromise between a spherical shell and a solid sphere. The prototype is still axially symmetric, and the band provides a better click system to open and close the toy. The physical parameters for the construction are radius of the spherical shell of 25mm, thickness of the shell of 3mm, radius of the cylindrical rod of 3mm, radius of the weight of 6mm, and height of the weight of 10mm. The band has the form of a solid of revolution generated by an ellipse rotating around the rod, the semiaxes of the ellipse measure, respectively, 7mm and 4mm. The rod is made of iron. According to the mathematical model, the top should behave as follows: for 𝑍₂ between 0mm and 4.14mm the toy does not show tippe top dynamic (type I); for 𝑍₂ between 4.14mm and 16.8mm complete tippe top dynamic is observed (type II). For 𝑍₂ above 16.8mm the top is of type III (incomplete tipping is observed).

The toy was launched many times, allowing to observe without problems zones I and II predicted by the model. Zone III is, however, difficult to observe at first, the maximum 𝑍₂ that can be obtained with the constructed prototype was only 15 mm (due to the fastening system of the rod), and moreover 𝜃_𝑐>2𝜋/3 which is not so easy to see. Different weights can, however, be used so that zone III becomes visible; for example, this is the case when using an iron weight with radius 8mm and height 5mm. We conclude that this last prototype is a good candidate for the 3-in-1 toy, although some further optimizations of the physical parameters can be considered (Figure 11).

Figure 11 

Different tippe top regimes in function of the position of the midpoint of the setscrew. The black lines are ±𝜖(𝑍2)/𝑅, and the blue curve is 𝐴(𝑍2)/𝐶−1, the red curve is −(𝜖(𝑍2)/𝑅)2. The physical parameters for the construction are here: radius of the ABS spherical shell 25mm, thickness of the shell 3mm, radius of the cylindrical rod 3mm, radius of the (iron) weight 6mm, height of the weight 10mm, semi-axis of the ellipse generating the torus 7mm and 4mm.

7. Numerical Illustrations

7.1. System Trajectories

In this section we present some simulations of (3.15)-(3.16). We focus on the parameter regime of Group II since these are the tops exhibiting “tipping” behaviour. Indeed, if the initial spin |𝑛₀|>max(𝑛₋,𝑛₂∶=𝑛₊(𝑅+𝜖)/(𝑅−𝜖)), then tipping occurs. The trajectories lie in the (reduced) 6-dimensional phase space (we ignore the 𝜑̇ equation); here we show their projections in the 3-dimensional subspace of the variables (𝛼,𝜃,𝑛).

Figure 12 shows a number of trajectories for a tippe top of Group IIb starting from initial conditions 𝑢=0,𝑣=0,𝜃=0.01,𝑛=arbitrary and 𝛼=(1/2𝐴)(𝐶𝑛+𝐶²𝑛²+4𝐴𝑚𝑔𝜖). Other input parameters are 𝑚=𝑔=1, the friction coefficient 𝜇=0.04, the eccentricity 𝜖/𝑅=0.3, and inertia ratio 𝐴/𝐶=0.92. Points with 𝑛<𝑛₋ are stable whereas those for which 𝑛>𝑛₋ are unstable. Let 𝑛_∗=−𝑛₊(𝑅+𝜖)/(𝑅−𝜖) be the value of the initial spin calculated for the angle 𝜃=0 at the Jellett where the change in stability for the inverted position (𝜃=𝜋) occurs. Trajectories originating near an unstable noninverted position are attracted either to one of the intermediate states at an angle 𝜃>0 when 𝑛₋<𝑛<𝑛_∗ or when 𝑛>𝑛_∗ to a steady state for which 𝜃=𝜋; in this case, the ball rises fully to a stable inverted vertical (tippe top) position with a final spin 𝑛 determined by the Jellett. The blowups in the insets show oscillations in the immediate neighborhood of the fixed points; these depend on the precise initial conditions. We note also that changing 𝜇 does not affect the final destination of the trajectories but it does affect the time needed to follow such trajectories in phase space, this at least within our parameter range of computations.

Figure 12 

Trajectories of the system (3.15)-(3.16) projected onto the subspace of variables (𝛼,𝜃,𝑛) for an eccentric ball of Group IIb with friction 𝜇=0.04. The bold-solid curve is that of intermediate states (5.4), (5.5), and (5.10); the line 𝑆 in the (𝛼,𝑛)-plane represents the initial condition 𝛼=(1/2𝐴)(𝐶𝑛+𝐶2𝑛2+4𝐴𝑚𝑔𝜖). For 𝑛>𝑛−, fixed points on 𝜃=0 are unstable. Trajectories are then attracted to a stable intermediate state (a point on the solid curve) for 𝑛−<𝑛<𝑛∗ or to a point on the plane 𝜃=𝜋 (i.e., flipping occurs) for 𝑛>𝑛∗. The two insets are details at the starting position, one for the stable case and one for the unstable case. The parameters are chosen as 𝑚=𝑔=1, 𝜖/𝑅=0.3, and 𝐴/𝐶=0.92.

Figure 14 shows a number of trajectories for a tippe top of Group IIa with friction coefficient 𝜇=0.04, starting now under an angle close to the inverted position 𝜃=𝜋. Physical parameters are 𝑚=𝑔=1, 𝜖/𝑅=0.5, and 𝐴/𝐶=0.55.

Recall from Figure 3 that trajectories starting near the inverted position will, depending on the initial spin, remain in the neighborhood of 𝜃=𝜋 go all the way down to the noninverted position 𝜃=0, or reach a stable intermediate state. For a clearer overview of the possible behaviours we use the symmetry (𝑛,𝛼)→(−𝑛,−𝛼) and sketch in Figure 13 the curve of intermediate states (5.10) in (𝐽,𝜃)-plane also indicating the essential 𝑛’s at which changes in stability type occur. Recall that for a given 𝜃 the relations (5.6) and (5.4) determine 𝛼 and 𝑛.

Figure 13 

Curve of intermediate states (5.10) in (𝐽,𝜃)-plane.

Figure 14 

Trajectories of the system (3.15)-(3.16) projected onto the subspace of variables (𝛼,𝜃,𝑛) for an eccentric ball of Group IIa with friction 𝜇=0.04. The bold curves are the intermediate states (5.10). The curves 𝑠1, 𝑠2, and 𝑠3 in (𝛼,𝑛)-plane represent the curves on which the initial value of 𝛼 is chosen. The starting angle is 𝜃=𝜋−0.01. The arrows indicate the final position. Points on the plane 𝜃=𝜋 are stable if |𝑛|>𝑛+, unstable otherwise. Trajectories are attracted to a noninverted position on the plane 𝜃=0 for |𝑛|<𝑛𝑏, or they reach a point on the red curves otherwise. Two trajectories are added to illustrate the initial wild oscillations when 𝛼 is not chosen on one of the 𝑠𝑖 curves. We used 𝑚=𝑔=1, 𝜖/𝑅=0.5, 𝐴/𝐶=0.55, and 𝜇=0.04.

The depicted trajectories have been obtained with the initial conditions 𝑢=0,𝑣=0,𝜃=𝜋−0.01, 𝑛 is arbitrary, and 𝛼 is one of the following: 𝛼=−(𝐶𝑛−𝐶²𝑛²−4𝐴𝑚𝑔𝜖)/(2𝐴)=∶𝑠₁(𝑛), 𝛼=−(𝐶𝑛+𝐶²𝑛²−4𝐴𝑚𝑔𝜖)/(2𝐴)=∶𝑠₂(𝑛), or 𝛼=−𝐶𝑛/2𝐴=∶𝑠₃(𝑛). These choices were made to reduce to a minimum in the drawings the initial oscillations in the 𝛼-direction. We start near the equilibrium on the eigenvector of the positive eigenvalue(s) so that the motion quickly evolves in the unstable direction away from the initial position. Trajectories originating near an unstable inverted position will either reach a stable intermediate state at 𝜃>𝜃_𝑏 when 𝑛_𝑏<|𝑛|<𝑛₊ or fall in the noninverted vertical position when |𝑛|<𝑛_𝑏. Here 𝑛_𝑏 is the value of the initial spin corresponding to the angle 𝜃=𝜃_𝑏 at which intermediate states change stability type. Points on 𝜃=𝜋 with |𝑛|>𝑛₊ are stable and trajectories starting near the inverted state will be attracted to it.

7.2. 3D Animations

In this section we comment on the 3D animations illustrating the phenomena of “tipping” or “hanging on an intermediate state” for an eccentric sphere (http://cage.ugent.be/~bm/tippetop/tippetop.html). In the films, the eccentric sphere is drawn as a transparent ball with a top in it. We focus on eccentric spheres belonging to Groups IIb and IIa, see Figure 3 for the corresponding bifurcation diagrams. The films have been made using Maple to solve the ODE system and feeding the results to Povray, Imagemagick and ffmpeg. The films are in 5x slow motion for the sake of clarity, so 1 second takes 5 seconds in the animation, with 30 frames for every second. For the sake of clarity the evolution of the nutation angle 𝜃 is shown in each film.

The films for a top of Group IIb show a complete flip (See http://cage.ugent.be/~bm/tippetop/tippetop_IIb_flip.mpg), and the rising to a stable intermediate state (http://cage.ugent.be/~bm/tippetop/tippetop_IIb_IntSt_Comp.mpg) for a top of Group IIa the film shows how the top launched upside-down migrates to a stable intermediate state (http://cage.ugent.be/~bm/tippetop/tippetop_IIaComb.mpg).

In the first film one sees a complete flip (tippe top effect) of the sphere; the physical parameters used are 𝑚=6 gram, 𝑅=1.5cm, 𝑒/𝑅=0.5, 𝐴=0.82mg/m², and 𝐶=0.7mg/m². The friction coefficient 𝜇 is 0.3. We show 90 seconds, which is presented as 7.5 minutes of the film.

The second film shows the motion towards a stable intermediate state from the unstable noninverted or unstable inverted position. The initial data is chosen so that the Jellett coincides with that of a stable intermediate state. It corresponds to 45 seconds of the tippe top movement. The physical parameters are here as in the first film except for the friction coefficient which is now 𝜇=0.1. The initial conditions around 𝜃=0 and 𝜃=𝜋 are chosen to correspond to a Jellett of approximately 0.84⋅10⁶. The intermediate state is at 𝜃=134.5 degrees.

The animations for a top out of Group IIa are meant to illustrate how, depending on the initial conditions, the top started at the inverted position (𝜃=𝜋) can fall either to an intermediate state or to the noninverted position (𝜃=0), see Figure 3. The physical parameters are 𝑚=6 gram, 𝑅=1.5cm, eccentricity 𝑒=50%, 𝐴=0.385mg/(m²), and 𝐶=0.7mg/(m²). The friction coefficient 𝜇 is 0.08. In the left of the animation we see the behaviour in which the tippe top flips towards the noninverted state. However, oscillation of theta occurs. To the right, we see the movement for a slightly different initial state, with motion towards an intermediate state. Also here oscillation of theta occurs.

8. Further Remarks

Global Dynamics
Concerning Theorem 2.1 we wish to stress the importance of the local bifurcation diagrams for the global dynamics. Clearly, if we have a unique asymptotically stable 𝜔-limit set it is clear that the basin of attraction for this set is equal to nearly the full measure set of the phase space defined by the complement to the stable manifolds of all coexisting (unstable) 𝜔-limit sets.
It thus remains to analyze the situation when we have two coexisting asymptotically stable 𝜔-limit sets. From the bifurcation analysis we know that in such case the coexisting stable 𝜔-limit sets are the vertical states.

Theorem 8.1. The 𝜔-limit sets of the eccentric sphere on a flat surface with small friction are asymptotically stable relative equilibria (with respect to 𝑆𝑂(2)×𝑆𝑂(2)). The 𝜔-limit set is a unique relative equilibrium, in which case the basin of attraction is the complement of the stable manifolds of the unstable relative equilibria (and hence dense, and of full measure, in the phase space). Otherwise, there are at most two stable relative equilibria (the vertical states) and the system is bistable. In this case, the union of the basins of attraction of the two vertical states is the complement of the stable manifold of the unstable relative equilibria, which is an intermediate state. This union is dense and of full measure in the phase space. The separatrix between the two basins of attraction (inside a level set of the Jellett invariant) consists of the stable manifold of an unstable intermediate state.

Proof. Most of the above statement is a direct consequence of the existence of the energy as a Lyapunov function (through La Salle’s principle). One readily verifies (from the local bifurcation analysis) that the stable manifold of the intermediate state coexisting with two asymptotically stable vertical states has codimension one (inside the level set of the Jellett invariant) and divides the phase space into two parts.

The regions of bistability (as a function of the Jellett invariant 𝐽) follow from the local bifurcation diagrams discussed in Theorem 5.1.

Remark 8.2. Note that the specific 𝑛’s where the changes in stability type of the steady states occur do not depend on 𝜇. The viscous friction influences the time needed for an orbit to reach such a point. This fact was already clear in [13] and could be proved in advance also in our setup. The result remains true also for more general forms of friction laws proportional to 𝐕_𝑄 such as those proposed in [4]. This suggests that the study of the asymptotic dynamics of other (mechanical) problems, as for example the rattleback, might notably simplify by the introduction of viscous friction in the model.

Rolling Model
A “rolling” eccentric ball does not tip. In this section we give a simple argument showing that, if pure rolling is assumed, then the tippe top phenomenon cannot occur.

Solving the tippe top under the constraint of pure rolling (i.e., when the nonholonomic constraint 𝐕_𝑄=0 is satisfied) allows for complete reduction of the equations of motion to a second-order 𝗈𝖽𝖾. See [3] for a discussion of this approach. In the pure rolling regime the system is not anymore dissipative and admits three conserved quantities: the energy, 𝐸, the Jellett 𝐽 as before, and the Routhian, Routh, given by [3]1ℛ𝑜𝑢𝑡ℎ=21𝐶+2𝑚𝑅²sin²(𝜃)+12𝑚𝐶𝐴(𝑅cos(𝜃)−𝜖)²𝜔²_𝐤. (8.1)

“Tipping” in the rolling model would violate the conservation of ℛ𝑜𝑢𝑡ℎ. Indeed, from the Jellett invariant we know that the sign of 𝜔_𝐤 has to change in a complete inversion since 𝜔_𝐤(𝜃=𝜋)=−((𝑅−𝜖)/(𝑅+𝜖))𝜔_𝐤(𝜃=0) and 𝑅>𝜖>0. But this is not allowed if Routh = constant has to hold.

The motion in the rolling model is governed by a functional relation of the type 𝜃̇²=𝑓(𝜃). Indeed, the conserved quantities give three relations for the components of the angular velocity 𝜔_𝐢,𝜔_𝐣,and𝜔_𝐤. In details, for a given 𝜃, the Routhian (8.1) fixes 𝜔_𝐤, then the Jellett fixes 𝜔_𝐢, and finally the energy 𝐸 fixes the tipping rate 𝜃̇=𝜔_𝐣 (cf. (3.4)), yielding a functional relation of the type 𝜃̇²=𝑓(𝜃). Note further that the constraint 𝐕_𝑄=0 gives 𝑈(𝜃), 𝑉(𝜃) from (3.9). In this approach, as mentioned in [3] one has to check whether a found solution is physically possible, that is, one has to take into account that rolling cannot be sustained if |𝑅_𝑡|<𝜇_𝑠𝑅_𝑛, where 𝜇_𝑠 is the coefficient of static friction. In [3] it is remarked that only a few pure rolling precessional solutions satisfy this condition. The analysis, however, leaves open the possibility to have pure rolling periodic motions around the intermediate states as we discuss below.

Sliding versus Rolling
A debatable issue is whether transitions between sliding and rolling are possible during the motion of the top. As it was pointed out in [9], such transitions must also be considered when setting up a realistic model to describe the dynamics of the tippe top. To the different regimes there correspond different sets of equations. A switch between sliding and rolling occurs as the absolute contact velocity 𝐕_𝑄 vanishes. In the Coulomb-friction model, a switch from rolling to sliding occurs when the tangent reaction force required to maintain rolling exceeds 𝜇_𝑠𝑅_𝑛. We refer to [4] for considerations and simulations on this topic and to [3] for a detailed analysis of the pure rolling model. We consider two test cases, the pendulum motion and the behaviour of the tippe top around a stable intermediate state.

(i)The pendulum motion is easily observed by placing the tippe top on the ground, rotating it under an angle 𝜃_𝑆, and releasing it. We have 𝐽=0=ℛ𝑜𝑢𝑡ℎ in this regime. Normally a pure rolling motion is observed. Only when 𝜃_𝑆 is large, some slipping might be observed initially. Solved under the pure rolling constraint, the solution is pendulum-like. In contrast the sliding equation of motion (3.18), gives a qualitatively different solution. For 𝜇=0, the solution is the pure sliding pendulum, where the center of mass remains fixed, but the tippe top makes a pure slipping periodic pendulum motion. (Recall that for 𝜇=0 periodic solutions are possible, which was also clear from the Hopf bifurcation. However they all disappear when 𝜇>0.) On activation of 𝜇, we have that 𝜃=0 is stable, and the slipping pendulum solution slowly degrades towards the stable point. In this case, the tippe top is best modeled with the pure rolling equations.(ii)Periodic motion around an intermediate state is characterized by the precession of the tippe top axle around the 𝑧-axis combined with a nutation 𝜃(𝑡) where 0<𝜃_𝑚≤𝜃(𝑡)≤𝜃_𝑀<𝜋. In the pure rolling case periodic solutions can be obtained exactly. Using a point on this periodic solution as initial condition for (3.18) allows to investigate the persistence of this solution when friction is added, see Figure 15. For 𝜇=0 a quasiperiodic motion is obtained around the pure rolling solution. Activating 𝜇 makes this motion unstable, and the solution goes towards the intermediate state, this behaviour is already dominant for 𝜇=0.1 (note also the high |𝑣_𝑄| value). However, for ever larger 𝜇 the decay slows down, the solution remaining very long in the neighborhood of the pure rolling solution. The |𝑣_𝑄| value in this case is very small, indicating that the condition for transition from slipping to rolling is satisfied.

Figure 15 

Periodic motion around intermediate state (IS) in (𝜃,𝜃̇) frame, 𝑚=1=𝑔, 𝜖=0.3𝑅, and 𝐴=0.92𝐶. Solution (i) is the pure rolling case for 𝐽=0.63, 𝐸=1.1, and 𝑅𝑜𝑢𝑡ℎ=0.33, (ii) is the pure slipping case for 𝜇=0 and (iii) is for 𝜇=0.1, and (iv) for 𝜇=10, the first 100s. The data is chosen for a stable IS at 𝜃IS=𝜋/6, and the initial condition for (ii), (iii), and (iv) is a point on the solution curve of (i). For (i) a periodic motion is obtained, and for (ii) a quasiperiodic motion is observed as it is sketched in the (𝜃,𝜃̇,𝛼) frame. For 𝜇>0 the trajectories spiral inwards towards the IS. For large friction, this motion is very slow (only the first 100 s is depicted). Also the speed at the contact point, |𝑣𝑄|, is given. In case (iii) this speed is large when far from the IS, and for (iv) this speed is very small corresponding to the very slow spiraling motion. Note that at 𝑡=0, 𝑣𝑄=0 as it is expected.

Appendices

A. Proof of Theorem 5.1

For the interested reader, the following sections contain the calculations needed for a straightforward linear stability analysis of the steady states. These form the proof of Theorem 5.1.

A.1. Stability of the Vertical State 𝜃=0

With the Taylor expansions in 𝜃𝑍_𝑄(𝜃)𝜃=−ℎ≈−𝑅+𝜖−𝜖²2+𝑂𝜃³,𝑋_𝑄=ℎ^′(𝜃)𝜃≈𝜖𝜃−³6+𝑂𝜃⁴, (A.1) linearizing 𝐕_𝑄 in 𝜃𝑉_𝑋,𝑄≈𝑈−(𝑅−𝜖)𝜃̇,𝑉_𝑌,𝑄≈𝑉+𝑛𝑅𝜃−(𝑅−𝜖)𝜑̇𝜃,𝑉_𝑍,𝑄≈0 (A.2) and noting that 𝑅_𝑛≈𝑚𝑔 and 𝑛=const, the linearization of the equations of motion (3.16) and (3.15) at 𝜃=0 yieldṡ̇𝜃̇()̇̇()̇𝐴̈𝜑̇𝜃̇̇()()̇𝐴̈𝜑̇𝑈−𝜑𝑉=−𝜇𝑔𝑈−𝑅−𝜖,𝑉+𝜑𝑈=−𝜇𝑔𝑉+𝑅𝑛𝜃−𝑅−𝜖𝜑𝜃,𝜑𝜃+2−𝐶𝑛𝜃=𝜇𝑚𝑔𝑅−𝜖𝑉−𝑅−𝜖𝜑𝜃+𝑅𝑛𝜃,𝜃−²𝜃+𝐶𝑛𝜑̇𝜃=−𝑚𝑔𝜖𝜃+𝜇𝑚𝑔(𝑅−𝜖)𝑈−𝜃̇(𝑅−𝜖). (A.3)(A.4)(A.5)(A.6) Introducing the complex coordinates𝜉=𝜃𝑒^𝑖𝜑,𝑤=(𝑈+𝑖𝑉)𝑒^𝑖𝜑, (A.7)(A.3)–(A.6) can be reduced to two complex equations. The addition (A.3) +𝑖(A.4) yields𝑤̇=𝜇𝑔−𝑤+(𝑅−𝜖)𝜉̇−𝑖𝑅𝑛𝜉, (A.8) whereas 𝑖(A.5) + (A.6) leads to𝐴𝜉̈−𝑖𝐶𝑛𝜉̇=𝜇𝑚𝑔(𝑅−𝜖)𝑤−(𝑅−𝜖)²𝜉̇+𝑖𝑅(𝑅−𝜖)𝑛𝜉−𝜖𝑚𝑔𝜉. (A.9)

These equations admit a solution of the form (𝜉,𝑤)=(𝝃,𝑤)𝑒^𝜆𝑡 when 𝜆 satisfies the determinant equation𝐷𝜆,𝜇∶=−𝐴𝜆³+−𝜇𝑚𝑔𝜖²−𝜇𝑔𝐴+2𝜇𝑚𝑔𝑅𝜖+𝑖𝐶𝑛−𝜇𝑚𝑔𝑅²𝜆²+𝑖𝜇𝑚𝑔𝑅²𝑛−𝑚𝑔𝜖+𝑖𝜇𝑔𝐶𝑛−𝑖𝜇𝑚𝑔𝑅𝑛𝜖𝜆−𝜇𝑚𝑔²𝜖=0. (A.10) When 𝜇=0 the roots 𝜆(𝜇) of (A.10) are𝜆₁(0)=0,𝜆_2,3(0)=𝑖𝐶𝑛±𝑆2𝐴, (A.11) where we set𝑆²∶=𝐶²𝑛²+4𝐴𝑚𝑔𝜖. (A.12) In the absence of friction, that is, when 𝜇=0, the vertical state 𝜃=0 is marginally stable as 𝜆_2,3(0) are purely imaginary since 𝑛²>−4𝐴𝑚𝑔𝜖/𝐶².

We now analyze the effect of small friction (0<𝜇≪1) by examining how the roots (A.11) are perturbed to first order in 𝜇:𝜆₁𝜇𝜆=−𝜇𝑔,_2,3𝜇=𝜆_2,3(0)()−𝜇𝑚𝑔𝑅−𝜖[𝑆()]±𝐴𝑆𝑅−𝜖∓𝐴𝑅𝑛2𝑚²𝑔²𝜖(𝑅−𝜖)²𝑆().𝐶𝑛±𝑆 (A.13) As Re(𝜆₁(𝜇))<0, the vertical state 𝜃=0 is stable if Re(𝜆_2,3(𝜇))<0. This is the case when𝑛²𝐴𝐶−𝜖1−𝑅<𝑚𝑔𝜖𝐶𝜖1−𝑅². (A.14) It follows that, in Group I (𝐴/𝐶<(1−𝜖/𝑅)), the vertical state 𝜃=0 is always stable, while for Group II and Group III (𝐴/𝐶>(1−𝜖/𝑅)) stability requires that 𝑛₀<𝑛₋.

It remains to be shown how Re(𝜆_2,3(𝜇))<0 yields the relation (A.14). We focus on the inequality Re(𝜆₂(𝜇))<0, the arguments are similar for Re(𝜆₃(𝜇)). The inequality Re(𝜆₂(𝜇))<0 yields−1𝐴[𝑆(𝑅−𝜖)−𝐴𝑅𝑛]<−2𝑚𝑔𝜖𝑅−𝜖⟺()𝑆𝐶𝑛+𝑆𝐶𝑛+𝑆𝑅𝑛−𝐴(𝑅−𝜖)<−2𝑚𝑔𝜖(𝑅−𝜖). (A.15) Using (A.12), this gives𝐶𝑅−𝐴(𝑅−𝜖)𝑛²𝐶+𝑆𝑛<2𝑚𝑔𝜖(𝑅−𝜖). (A.16) Note that if (𝑅−(𝐶/𝐴)(𝑅−𝜖))<0 the above condition is satisfied for all 𝑛. If on the other hand the inequality (𝑅−(𝐶/𝐴)(𝑅−𝜖))>0 holds, then (0<)𝑆𝑛<2𝑚𝑔𝜖(𝑅−𝜖)/(𝑅−(𝐶/𝐴)(𝑅−𝜖))−𝑛²𝐶; squaring both sides yields𝑛²<𝑚𝑔𝜖𝐴𝑅𝒦(1−𝜖𝑅)², (A.17) where 𝒦∶=(𝑅−(𝐶/𝐴)(𝑅−𝜖)). Since we are in the case 𝒦>0, we can rewrite this last condition as (A.14).

Remark A.1. Ignoring translational effects, that is, throwing everything in the variable 𝑤 away (cf. [5]), one is left with ()̇𝐴̈̇()𝑤̇=𝜇𝑅−𝜖𝜉−𝑖𝜇𝑅𝑛𝜉,𝜉−𝑖𝐶𝑛𝜉=−𝜇𝑅−𝜖²𝜉̇−𝜖−𝑖𝜇𝑅𝑛𝜉, (A.18)(A.19) where we set for simplicity 𝑚=1, 𝑔=1. Equation (A.19) is of Maxwell-Bloch type [5] and allows us to recover the analysis carried out in [5]. An analogous result holds when linearizing around 𝜃=𝜋.

A.2. Stability of the Vertical State 𝜃=𝜋

The stability of the vertical state 𝜃=𝜋 is studied in a similar way as in Section A.1. From the equation of motion (3.15)-(3.16), introducing complex coordinates𝜉=(𝜃−𝜋)𝑒^𝑖𝜑,𝑤=(𝑈+𝑖𝑉)𝑒^𝑖𝜑, (A.20) we obtain the coupled complex equations()̇,𝐴̈̇()()𝑤̇=−𝜇𝑔𝑤+𝑅+𝜖𝜉+𝑖𝑅𝑛𝜉𝜉+𝑖𝐶𝑛𝜉=𝜇𝑚𝑔𝑅+𝜖𝑤−𝑅+𝜖²𝜉̇−𝑖𝑅(𝑅+𝜖)𝑛𝜉+𝑚𝑔𝜖𝜉. (A.21) The corresponding determinant equation for eigenvalue 𝜆 is given by𝐷𝜆,𝜇=−𝐴𝜆³+−2𝜇𝑚𝑔𝜖𝑅−𝜇𝑔𝐴−𝜇𝑚𝑔𝜖²−𝑖𝐶𝑛−𝜇𝑚𝑔𝑅²𝜆²+−𝑖𝜇𝑚𝑔𝑅²𝑛+𝑚𝑔𝜖−𝑖𝜇𝑔𝐶𝑛−𝑖𝜇𝑚𝑔𝜖𝑅𝑛𝜆+𝜇𝑔²𝑚𝜖. (A.22) When 𝜇=0, the roots 𝜆(𝜇) of (A.22) are𝜆₁(0)=0,𝜆_2,3(0)=−𝑖𝐶𝑛∓𝑆2𝐴,with𝑆²∶=𝐶²𝑛²−4𝐴𝑚𝑔𝜖. (A.23) Thus, at 𝜇=0, the vertical state 𝜃=𝜋 is at most marginally stable, when 𝜆_2,3(0) is purely imaginary, that is, when|𝑛|>2𝐴𝑚𝑔𝜖𝐶=∶𝑛_⋆. (A.24) This is of course also a necessary condition for stability if 𝜇 is small. Provided 𝑆≠0, corresponding to a resonance 𝜆₂(0)=𝜆₃(0) at |𝑛|=2𝐴𝑚𝑔𝜖/𝐶, the roots of (A.22) are perturbed at order 𝜇 to𝜆₁𝜇𝜆=−𝜇𝑔,_2,3𝜇=𝜆₂(0)()+𝜇𝑚𝑔𝑅+𝜖(())±𝐴𝑆−𝑆𝑅+𝜖∓𝐴𝑛𝑅2𝑚²𝑔²𝑒(𝑅+𝑒)²𝑆().𝐶𝑛∓𝑆 (A.25) Thus, for stability we have to require Re(𝜆_2,3(𝜇))<0, which yields𝑛²𝜖1+𝑅−𝐴𝐶>𝑚𝑔𝜖𝐶𝜖1+𝑅². (A.26) Condition (A.26) is never satisfied for Group III, so 𝜃=𝜋 is unstable. In the case of Groups I and II, when 𝐴/𝐶<(1+𝜖/𝑅), the condition for stability is𝑛_𝜋>𝑚𝑔𝜖𝐶)]𝜖[(1+𝜖/𝑅−𝐴/𝐶1+𝑅=∶𝑛₊. (A.27)

Note that for tippe tops of Group I and II 𝑛²₊≥𝑛²_⋆, with 𝑛_⋆∶=2𝐴𝑚𝑔𝜖/𝐶. The equality 𝑛²_⋆=𝑛²₊ holds when 𝐴/𝐶=(1/2)(1+𝜖/𝑅).

A.3. Stability of Intermediate States 0<𝜃<𝜋

In this section we consider intermediate asymptotic states, which exist if the condition (5.7) is satisfied. Such a state, if it exists, is of the form 𝑣₀=(0,0,𝛼,0,𝜃,𝑛), with 𝛼,𝜃,𝑛 constant and related by (5.4). In order to study the stability properties, we study the eigenvalues of the 𝑆𝑂(2)(𝜑)-reduced equations of motion, obtained from (3.18) by omitting the 𝜑̇ equation.

With 𝐽₀ denoting the corresponding Jacobian of this reduced equation, eigenvalues 𝜆 satisfy the determinant equation𝐷𝜆,𝜇∶=det𝐽₀−𝜆I₆=𝜆𝑝₅(𝜆)=0, (A.28) where I is the (six-by-six) identity matrix and 𝑝₅(𝜆) is a polynomial of degree 5 in 𝜆. So, 0 is always a solution of (A.28).

Remark A.2. It is not possible to reduce the system to a “Maxwell-Bloch” form around an intermediate state as on the contrary was the case around the two vertical spin states.

When 𝜇=0 the six roots of (A.28) are𝜆_1,2=0,𝜆_3,4=±𝑖𝛼₀,𝜆_5,6=±𝑖𝛼₀√𝒜, (A.29) where𝐴𝒜∶=²+2𝐴ℬcos𝜃+ℬ²𝐴𝑚𝜖²sin²𝜃()+𝐴>0,withℬ∶=𝐴−𝐶cos𝜃𝜖+𝐶𝑅, (A.30) and 𝛼 is given by (5.6). Note that ℬ>0 from (5.7). All the eigenvalues 𝜆_𝑗, 𝑗=1,…,6 are on the imaginary axis, and the intermediate states at 𝜇=0 are marginally stable.

Remark A.3. It is worth noting that for 𝜇=0 the resonance 𝜆_3,4=𝜆_5,6=±𝑖𝛼 occurs when 𝒜=1. Using the expressions for 𝛼 and 𝒜 given before, one checks that this equality is satisfied when the equation 𝐴𝐶3𝐴−1𝐶+𝐴−1𝐶²𝑚𝜖²cos²𝜃𝜖+2𝑅2𝐴𝐶−1cos𝜃+𝜖²𝑅²−𝐴𝐶²𝑚𝜖²=0 (A.31) admits a real root between −1 and 1. The resonance disappears when higher-order terms in 𝜇 are added.

When 𝜇 is small, 0<𝜇≪1, we write the first-order perturbation of the roots as𝜆_𝑗𝜇∶=𝜆_𝑗(0)+𝜇𝑞_𝑗,𝑗=1,…,6. (A.32) From (A.28) it follows that𝑞₁=0,𝑞₂=𝑚³𝑔³𝜖²𝑅⁴sin²𝜃𝑔cos𝜃𝑞𝒟ℰ,₃=𝑞₄=−𝑚𝑔,𝑞₅=𝑞₆=𝑎₀𝑎₁𝑚+𝑎₂,with𝑎₀𝑚∶=³𝑔³𝜖²2𝐴²𝑅²ℬ²ℱ²𝛼⁴₀𝒜>0, (A.33)𝛼² and 𝜆₅ are as before and𝑔cos𝜃𝐴∶=𝐶+4)−1(𝐴/𝐶−1cos𝜃+𝜖/𝑅²(𝐴/𝐶)sin²𝜃+cos𝜃−𝜖/𝑅². (A.34) The coefficients 𝑎₁ and 𝑎₂ are given by𝑎₁∶=−𝜖²sin²𝜃₀𝐴𝑅𝑅cos²𝜃₀−2𝜖cos𝜃₀+𝑅−𝐶𝑅cos𝜃₀−𝜖²²𝑎<0,₂∶=𝐴𝐶²𝑅⁴𝑓𝐴𝐶;𝜖𝑅,cos𝜃₀, (A.35) with𝑓𝐴𝐶;𝜖𝑅,cos𝜃₀∶=𝑓₂𝐴²𝐶²+𝑓₁𝐴𝐶+𝑓₀, (A.36) where𝑓₂𝜖∶=−2+6𝑅cos𝜃₀𝜖−5²𝑅²+1cos²𝜃₀+10cos³𝜃₀𝜖𝑅−𝜖1+3²𝑅²cos⁴𝜃₀,𝑓₁𝜖∶=−4𝑅𝜖²𝑅²+2cos𝜃₀𝜖+61+3²𝑅²cos²𝜃₀𝜖−4𝑅𝜖4+²𝑅²cos³𝜃₀𝜖+21+2²𝑅²cos⁴𝜃₀𝜖+2²𝑅²,𝑓₀𝜖∶=−²𝑅²−𝜖⁴𝑅⁴𝜖−2𝑅𝜖−1−3²𝑅²cos𝜃₀+𝜖−1−⁴𝑅⁴𝜖−10²𝑅²cos²𝜃₀𝜖−2𝑅𝜖−3−²𝑅²cos³𝜃₀−𝜖²𝑅²+1cos⁴𝜃₀. (A.37) The coefficients 𝒟,ℰ are calculated with the help of Maple𝒟∶=ℬ²𝜖−𝐴𝐴−𝐶+𝐶²𝑅²,()ℰ∶=−𝐴−𝐶𝐴ℬ²−𝑚𝑅𝜖ℬ−𝜖²𝐶²+𝑅²(𝐶−𝐴)²𝑚𝜖².+𝐴 (A.38) For tippe tops of Group II they have a fixed sign for 𝜃₀ varying in (0,𝜋). Note that 𝑞_3,4<0 and 𝑞_5,6<0. The first inequality is obvious, and the second is less straightforward and is proved below. The friction is stabilizing at 𝑂(𝜇) when 𝑞₂<0.

The sign of 𝑞₂ depends on 𝑔(cos𝜃) only, since the two terms 𝒟, ℰ are never zero for 0<𝜃<𝜋. The zero of 𝑔(cos𝜃) is the bifurcation point where the change in stability type happens. When 𝑞₂>0 the intermediate states are not anymore stable. For the sake of brevity we refer to [10] for the details on the behaviour of 𝑔(cos𝜃), where the same crucial function is encountered in the stability analysis via the Routhian reduction. This analysis completes the proof of Theorem 5.1.

Proof of 𝑞₅<0. To prove that 𝑞₅<0 it is sufficient to prove that 𝑎₂≤0, see (A.33), since 𝑎₀>0 and 𝑎₁<0. The sign(𝑎₂) is determined by sign(𝑓(𝐴/𝐶;𝜖/𝑅,cos(𝜃₀))). Considering 𝑓 as a square polynomial in 𝐴/𝐶, 𝑓(𝐴/𝐶), we have that the discriminant of 𝑓(𝐴/𝐶)=0 is given by Δ_𝑓∶=−sin²𝜃₀𝜖−1+𝑅cos𝜃₀²𝜖𝑅cos𝜃₀−1²𝜖+1−2𝑅cos𝜃₀+𝜖²²_>𝑅₀, (A.39) which is negative for all 𝜃₀, 𝜖/𝑅. Hence, the sign of 𝑓 remains fixed. It is easily verified that for 𝜃₀=0, 𝜋 or 𝜋/2, sign(𝑎₂) is always negative (e.g., 𝑓(𝐴/𝐶;𝜖/𝑅,1)=−2(𝜖/𝑅−1)²(𝜖/𝑅+2(𝐴/𝐶)−1)²), hence 𝑎₂≤0 for all 𝜖/𝑅, 𝐴/𝐶 and 𝜃₀. We conclude that 𝑞₅≤0.