Click a chart to move the playhead there. Drag across a chart to zoom in; double-click to zoom out.
Angular velocity
Body components, rad/s
Tilt of the body axis
Angle between e₃ and the vertical, degrees
Normal reaction
N/mg; dashed line N = mg
Tangential contact force
|F|, newtons
Slip speed of the contact point
mm/s; zero for rolling
Total energy
Joules; never increases
Every run is kept here, newest first, up to twelve. Tick up to six to overlay their spin; the current run is drawn in black.
| Show | Run | Reversal | ω₃ end | Max tilt | Energy lost |
|---|
Spin ω₃ of the selected runs
rad/s
Runs the current setup once for each value of one parameter and plots a result against it. Everything else stays as set in Model setup.
Sweep result
Run a sweep to see how the result depends on the parameter.
| Value | Reversal | ω₃ end | Max tilt | Energy lost | N/mg max |
|---|
What is simulated
A semi-ellipsoid with semi-axes a₁, a₂, a₃ whose principal axes of inertia are turned slightly against its axes of curvature, through the product of inertia I₁₂. Spun one way it keeps spinning; spun the other way it rocks and comes out spinning backwards.
Rolling forbids slip: six equations, no energy loss, and the contact force is recovered after the fact. Coulomb and viscous let the contact point slip: nine equations for the angular velocity, the 1–2–3 Euler angles and the velocity of the centre of mass.
The normal reaction is solved, not assumed
Keeping the centre of mass at the height set by the contact geometry gives
During a rolling reversal it swings between 0.20 mg and 5.86 mg, so N = mg is not a safe shortcut.
Contact laws
- Coulomb: F = −μN vP/√(|vP|² + ε²), after Moffatt and Shimomura (2002).
- Drilling: a torque about the vertical of size up to μNρ, from friction over a small contact patch.
- Viscous: F = −c vP, the second law of Moffatt and Shimomura.
Checked against the paper
This page uses the same equations as the paper's MATLAB code and reproduces its results:
| Case, unstable sense | Paper | This page |
|---|---|---|
| Rolling | 2.83 s | 2.831 s |
| Coulomb, μ = 0.15 | 4.22 s | 4.223 s |
| Coulomb + drilling, ρ = 0.5 mm | 3.98 s | 3.977 s |
| Viscous, c = 500 kg/s | 2.90 s | 2.898 s |
| Coulomb, μ = 1, N fixed at mg | 2.43 s | 2.427 s |
Verification checks
Every run reports whether contact is maintained (the gap rate stays at zero), whether the energy only goes down, whether N stays positive so the body never leaves the table, and whether the denominator of the N formula stays positive.
Reference
V. Itauriya and D. Mukherjee, Slipping-contact friction in rattleback dynamics: extension of a rolling formulation using the Moffatt–Shimomura contact laws, Department of Mechanical Engineering, IIT Kanpur. Body parameters after Kane and Levinson (1982).