by Antonis Vakis

### Introduction

The Tippe Top is a toy that belongs to the general category of spinning tops. Spinning tops are toys that can be spun around an axis and balance on a point, relying on the gyroscopic effect: once spinning is initiated, the angular momentum causes the toy to resist changes to its orientation. The angular momentum, and consequently the gyroscopic effect, gradually diminishes with time due to frictional effects and the spinning ceases. In the case of the Tippe Top, its geometry and construction allow for a more spectacular effect: when spun on a flat surface, the Tippe Top will turn its stem — i.e. the cylindrical rod that is fitted onto the section of a sphere that constitutes the rest of the toy — towards the surface and, upon touching the surface, will invert and start spinning on the stem, while seamlessly changing its direction of rotation. This behavior is attributed, partly, to the fact that the Tippe Top's center of mass is different from the center of the sphere and primarily to the contribution of frictional forces.

Picture 1: Wolfgang Pauli and Niels Bohr observe the Tippe Top (Erik Gustafson, courtesy AIP Emilio Segre Visual Archives, Margrethe Bohr Collection (www.aip.org/history/esva)).

The first documented reference of the behavior of the Tippe Top is described in a 1890s book by John Perry referring to Sir William Thompson and Hugh Blackburn experimenting by rotating egg-shaped stones found at the beach (Fysikbasen.DK). Perry also describes a small spherical object with a center of mass that does not coincide with the center of the sphere, which, when spun, lifts its center of mass away from the ground (Perry, 1890). A patent for an object called a "Wenderkreisl" was filed by Helene Sperl of Munich, Germany, in 1891 but was withdrawn after a year because the patent fee was not paid. Apparently, none of the models described in this patent seem to work. A working model was developed, patented, and mass-produced by Danish engineer Werner Ostberg in 1950 under the name "tippetop". Two patents of similar spinning tops exist under the name of Oscar Hummel, dating back to 1948 and 1949 (Fysikbasen.DK). The first person to write on the dynamics of the Tippe Top was C. M. Braams of the University of Lund (Braams, 1954). Possibly the most rigorous analysis of the top's mechanics was published by the physicist Richard Cohen at the Massachusetts Institute of Technology in 1977 (Cohen, 1977).

### Discussion

The Tippe Top consists of a section of a sphere connected to a cylindrical rod, as shown in Picture 2, below.

Picture 2: Werner Ostberg's figures of the Tippe Top on the left (Ostberg, 1950), and a Tippe Top manufactured at the University College of Aarhus, Denmark on the right (Fysikbasen.DK).

The behavior of conventional spinning tops is based on the principle of change of angular momentum, as summarized in Figure 1, below. In short, the weight of the top due to gravity causes a torque that is responsible for the change in angular momentum, $\Delta L$. This change in angular momentum, then, offsets the overall vector in such a way that it forces the top to right itself when sufficient rotational speed is provided. The movement of the body-fixed z-axis on the top relative to the global z-axis is termed precession.

Figure 1: Change in angular momentum on the left, and a schematic of the precession of a spinning top on the right (Nave, 2005).

In the case of the Tippe Top, friction plays the most important part. First of all, due to the top's geometry, the angular momentum vector, $L$, points almost entirely along the positive z-axis. During the inversion, the center of mass is elevated, thereby necessitating the existence of potential energy — this potential energy is needed to raise the center of mass. In this case, the angular velocity and, consequently, the angular momentum $L$ decrease. However, this reduction in angular momentum requires the action of a torque, since $\tau = \frac{\Delta L}{\Delta t}$. Since the force of gravity and the reaction at the point of contact cannot produce such a torque along the z-axis, they cannot be responsible for the change in angular momentum. Therefore, there must be a frictional force at the point of contact (Cohen, 1977).

The mass distribution is symmetric about the shaft while the center of mass is epicentric. Since the point of contact does not coicide with the rotational axis, the Tippe Top will slide over the surface in a circle around the z-axis. The sliding friction supplies the torque needed for the inversion. After the inversion, it should be noted that the direction of rotation changes precisely so that the angular momentum direction can remain constant.

Cohen uses geometry to extract the equations of motion for the Tippe Top (Cohen, 1977). However, an easier approach is that taken by Ciocci and Langerock, where the Lagrangian of the system is evaluated and the equations of motion are derived from the canonical Euler-Lagrange equations. Figure 2, below, shows the eccentric sphere approximation used by Ciocci and Langerock along with the definitions of the symbols used in the equation for the Lagrangian.

Figure 2: Eccentric sphere model of the Tippe Top (Ciocci and Langerock, 2007).

The forces acting on the sphere are gravity, $\bf G = -m g \bf e_z$, and the frictional force at point Q, $\bf F$. The holonomic constraint of the system is:

$h(\theta) = R - \epsilon \cos \theta = z$

The Lagrangian of the system is:

$L = \frac{1}{2} (m(\dot x^2 + \dot y^2) + (\epsilon^2 m \sin^2 \theta + A) \dot \theta^2 + A \sin^2 \theta \dot \phi^2 + C (\dot \psi + \dot \phi \cos \theta)^2) - m g (R - \epsilon \cos \theta)$

where $A$ is the moment of inertia about the 1 and 2 body-fixed axes, and $C$ is the moment of inertia about the 3rd body-fixed axis. This function is defined on the tangent space of the configuration manifold $M = \mathbb{R}^2 \times \mathbb{SO}(3)$, where $\mathbb{SO}(3)$ is the group of rigid rotations in $\mathbb{R}^3$.

The Euler-Lagrange equation:

$\frac{d}{dt} (\frac{\partial L}{\partial \dot q^i}) - \frac{\partial L}{\partial q^i} = Q_i^F$

where $Q^F = Q_i^F dq^i$ is a one form on $M$ and represents the generalized force moment of the friction force at the point of contact, can be solved to yield the equations of motion.

Furthermore,

$Q^F = \bf R_f \cdot \bf e_x dx + \bf R_f \cdot \bf e_y dy + (\bf q \times \bf R_f) \cdot (\bf e_y d\theta + \bf e_z d\phi + \bf e_Z d\psi)$

where $\bf e_Z$ is the body-fixed 3-axis. The frictional force, then, is $\bf F = \bf R_f + R_n \bf e_z$ with $\bf R_n = R_n \bf e_z$ being the normal reaction force of the floor at point $Q$ (of order $m g$) and $\bf R_f = F_x \bf e_x + F_y \bf e_y = -\mu R_n \bf V_Q$ (from a viscous friction law) being the sliding friction that opposes the slipping motion. Note that $\bf R_f \cdot \bf V_Q < 0$, where $\bf V_Q$ is the slip velocity at $Q$. The coordinates of the point $Q$ are $Q = (R \sin \theta, 0, \epsilon - R \cos \theta)$. Also, $\bf V_Q = \bf V_O + \bf \omega \times \bf q$ where $\bf q$ is the vector between the points $O$ and $Q$, and $\bf V_O = (\dot x, \dot y, h'(\theta) \dot \theta)$ is the velocity at the center of mass.

Then, the angular velocity can be calculated as $\bf \omega = -\dot \phi \sin \theta \bf e_x + \dot \theta \bf e_y + n \bf e_z$ where $n = \dot \psi + \dot \phi \cos \theta$ is the component of $\bf \omega$ about $\bf e_z$ — i.e. the spin.

Now, the equations of motion can be expressed as generalized force moments as:

$Q_x = -\mu R_n (\dot x - \sin \phi \dot \theta (R - \epsilon \cos \theta) + \cos \phi \sin \theta (R \dot \psi + \epsilon \dot \phi))$

$Q_y = -\mu R_n (\dot y + \cos \phi \dot \theta (R - \epsilon \cos \theta) + \sin \phi \sin \theta (R \dot \psi + \epsilon \dot \phi))$

$Q_\theta = -\mu R_n (R - \epsilon \cos \theta)(\cos \phi \dot y - \sin \phi \dot x + (R - \epsilon \cos \theta) \dot \theta)$

$Q_\phi = -\mu R_n \epsilon \sin \theta(\cos \phi \dot x - \sin \phi \dot y + \sin \theta (\epsilon \dot \phi + R \dot \psi))$

$Q_\psi = -\mu R_n R \sin \theta(\cos \phi \dot x - \sin \phi \dot y + \sin \theta (\epsilon \dot \phi + R \dot \psi))$

(from Ciccio and Langerock, 2007)

Ciccio and Langerock go on to use the so-called *Jellet constant*, defined as $J = -\bf L \cdot \bf q = constant$, which, to some extent controls the motion of the spinning top. The Jellet constant is calculated to be:

$J = C n (R \cos \theta - \epsilon) + A \dot \phi R \sin^2 \theta$

Then, using Routhian reduction — by assumming a simplified approximation of the friction law — the authors proceed to perform stability analysis on the system. It is not the purpose of this overview to focus on these procedures. It should suffice to merely report the stability findings:

*In the approximation of negligible transitional effects, a spinning eccentric sphere on an horizontal (perfectly hard) surface subject to a sliding friction is reducible with a Routhian reduction procedure. The relative equilibria of the reduced system are precisely the steady states of the original system. They are purely rolling solutions and except for the trivial state of rest, they are of three types:*

*(i) (non-inverted) vertically spinning top with center of mass straight below the geometric center;*

*(ii) (inverted) vertically spinning top with center of mass straight above the geometric center;*

*(iii) intermediate spinning top, the top precesses about a vertical while spinning about its axle and rolling over the plane without giding.*

*The existence and stability type of these relative equilibria only depend on the inertia ratio $\frac{A}{B}$, the eccentricity of the sphere $\frac{\epsilon}{R}$ and the Jellet invariant $J$* (Ciocci and Langerock, 2007).

### Conclusion

The Tippe Top is a type of spinning top that, due to its construction and the effects of angular momentum and sliding friction, raises its center of mass until it inverts and its rotating direction is reversed. This very interesting dynamic behavior was described in this brief overview by explaining the laws governing changes in angular momentum and by solving for the equations of motion, as presented in the paper by Ciocci and Langerock. That paper, along with most of the literature on the subject, deals mostly with the stability of the system; however, when combined with the paper by Richard Cohen of 1977, it can serve to understand the basic dynamics of the problem. For more on the issue in terms of stability, refer to the papers: "Tippe Top Inversion as a Dissipation-Induced Instability" by Bou-Rabee, Marsden and Romero, "Phase Space of Rolling Solutions of the Tippe Top" by Glad, Petersson and Rauch-Wojciechowski, and "A New Analysis of the Tippe Top: Asymptotic States and Liapuniv Stability" by Ebenfeld and Scheck, which were not covered in this review.

### References

Braams, C.M.: "The Tippe Top", Am. J. Phys. 22, 568 (1954).

Ciocci, M. C. and Lengerock, B.: "Dynamics of the Tippe Top via Routhian Reduction", eprint arXiv:0704.1221(2007).

Cohen, R. J.: "The Tippe Top Revisited", Am. J. Phys. 45, 12 (1977).

FYSIKBASEN.DK, Denmark: The Tippe Top (http://www.fysikbasen.dk/TippetopENGLISH.php).

Nave, C. R.: HyperPhysics, Georgia State University, 2005 (http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html).

Ostberg, W.: Patent number: 656,540. Spinning tops. OSTBERG, W. Oct. 6, 1950, No. 24495. Convention dates, July 4, and July 22, 1950.

Perry, J.: "Spinning Tops", Society for promoting Christian knowledge (1890).