Rolling Coin on the Floor
|
Tilt Angles
| $\theta$ |
|
0 degrees |
| $\phi$ |
|
0 degrees |
| $\psi$ |
|
0 degrees |
Angular Velocity (in unit of $\sqrt{g/a}$)
| $\omega_Z$ |
|
0 |
| $\omega_Y$ |
|
0 |
| $\omega_X$ |
|
0 |
|
- The yellow mark shows the contact point between the coin and
the floor.
- Turning the "Dissipation" button OFF results in no energy
loss, thus the coin will keep rolling indefinitely.
- In the case of no dissipation, the stability condition for upright
rolling is given by the critical angular velocity
$\omega_c :=\sqrt{g/ 3a}\approx 0.577\sqrt{g/a}$ as
- If $|\omega_Z| < \omega_c$, unstable.
Even a infinitesimally small initial tilt angle
$\theta$ causes wobbling.
- If $|\omega_Z| \ge \omega_c$, neutral.
It rolls steadily for a short time, but due to a
mode lacking restoring force in the linear regime, the
direction $\phi$ slowly drifts even if the initial tilt
$\theta$ is small, and over time the trajectory forms a
large fluctuating circle.
- The equations of motion used for numerical integration are:
\[
\begin{align*}
(I_X + Ma^2) \left(\frac{d\omega_X}{dt}
\textcolor{red}{+ \alpha|\dot\psi|\omega_X} \right)
&= - (I_Z + Ma^2) \omega_Y\omega_Z - I_X \omega_Y^2 \tan \theta + Ma^2 \frac{g}{a} \sin \theta \\
I_X \left(\frac{d\omega_Y}{dt}
\textcolor{red}{+ \alpha|\dot\psi|\omega_Y} \right)
&= I_X \omega_X \omega_Y \tan \theta + I_Z \omega_Z \omega_X \\
(I_Z + Ma^2)\left(\frac{d\omega_Z}{dt}
\textcolor{red}{ + \alpha|\dot\psi|\omega_Z} \right)
&= Ma^2 \omega_X \omega_Y \\
\frac{d\theta}{dt} &= \omega_X \\
\frac{d\phi}{dt} &= \frac{1}{\cos\theta}\,\omega_Y \\
\frac{d\psi}{dt} &= \omega_Z + \tan\theta\,\omega_Y
\end{align*}
\]
These equations were numerically integrated using the 4th-order
Runge-Kutta method.
Their derivation is found in this note. Red terms represent
phenomenologically introduced dissipation.
Dissipation is off when $\alpha = 0$.
Rolling Motion of a Disk, Ring, or Hollow Cylinder
Other Simulators