Examples of inertia (possibly using the inertia ellipsoid)

[moorepants]

A student requested some visuals to connect to the numbers/entries in an inertia matrix. We've discussed how to do this. One way is to show the inertia ellipsoid for different shapes or to show the inertia matrix with some numbers for a cylinder and then change the basis of the inertia matrix.

[moorepants]

I realized I have a GUI tool that could do the trick. Our yeadon program shows the inertia ellipsoid of a human:

and then with a skewed inertia ellipsoid:

[moorepants]

A bit on the inertia ellipsoid. The inertia ellipsoid has axes that align with the principal axes of the rigid body. The radius of the ellipsoid along the 1st principal axis is equal to $r_1 = \frac{1}{\sqrt{I_1}}$ where $I_1$ is the principal moment of inertia about the 1st principal axis. This makes an ellipsoid that gives the inertia shape of the object.

[Peter230655]

Is this not somehow related to the eigenvectors and eigenvalues of the inertia tensor, or is my memory failing me?

[chris-konrad]

Uh really cool! I think that is already pretty ideal to get an understand of the inertia matrix.

The cherry on top of the cake would be if the inertia tensor where a little more readable (or even highlighting larger numbers with colors?). After all, that was the primary concern of the question: How do the numbers in the inertia tensor map to a shape of an object.

[moorepants]

Here is an example of how the inertia ellipsoid is not the same thing as a uniform dense ellipsoid with the same inertia of the object:

This comes from this method: https://github.com/moorepants/BicycleParameters/blob/master/bicycleparameters/parameter_sets.py#L738

The four colored ellipses are generated by finding a constant density ellipsoid that has the same inertia as the rigid body.

The four black ellipses are generated from radii = 1/sqrt(I). Notice how the ellipse size varies largely among the bodies because the mass of the fork and wheels are much smaller than the rear frame + rider inertia. Also, the rear frame/rider inertia ellipsoid has one dimension that matches the constant density ellipsoid, but is less tall.

I find this confusing and did so 15 years ago when I first made these ellipse shapes for the bicycle rider inertia.

In the yeadon package uses 1/sqrt(I) for the single inertia ellipsoid.