Contact Point between a Rigid Body and the Surface it sits (Semi-Ellipsoid)

[drewb]

Hello, I've been having trouble with a certain part of a homework problem. I wish I had posted this earlier so there would be more time! Anyway...

The shape is a semi-ellipsoid (a Rattleback, in particular). Here's a good picture of what I'm talking about.http://www.motiongenesis.com/MGWebSite/MGGettingStarted/SampleProblemRattleback/MGRattlebackSchematic.jpg
(Hopefully you don't mind if I use my own variables, rather than the picture's)
The origin, in my case, is in the center of the ellipsoid.

Homework Statement

In my problem,
Unit vectors:
e₁ is b_y (from the picture)
e₂ is b_z (is pointing sort of into the screen -- hard to tell from the picture)
e₃ is b_x

and the Z axis (e_z) is n_x (from the picture)

α is the rotation around the e₁ (b_y) axis
β is the rotation around the e₂ (b_z) axis
(so they're both angles down from the e_z axis, representing the rotation of the ellipsoid in two different directions)

I should also note that ζ is the angle about the e_z axis (so the fixed-to-the-room axis); however, it is not presented in the solution.

A is the contact point between the ellipsoid and the ground. (so that's B_N)

a and b are the lengths of the semi-axes, and c is the 'height' (still a semi-axis, I suppose)

I'm supposed to show that the contact point, A, can be written as a function of the angles α and β by
x₁ = -[itex]\frac{a^2}{p}[/itex] μ₁
x₂ = -[itex]\frac{b^2}{p}[/itex] μ₂
x₃ = -[itex]\frac{c^2}{p}[/itex] μ₃

with
p² = (aμ₁)² + (bμ₂)² + (cμ₃)²
and
μ_i=e_i dot e_z

Then additionally parametrize the contact point as free point on the surface and show that:
μ₁ = cos(α)sin(β)
μ₂ = sin(α)
μ₃ = cos(α)cos(β)

And I'm supposed to say which constraints characterize it.

Homework Equations

What I think are the relevant equations:

[tex]\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1[/tex]
[tex]\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}-1=F(x,y,z)[/tex]

[tex] x=asin\phi cos \theta [/tex]
[tex] y=bsin \phi sin \theta [/tex]
[tex] z=ccos\phi [/tex]

The Attempt at a Solution

I'm a little unsure as to how I should start regarding the angles. The parameterization of an ellipsoid given above uses those angles to describe the surface. The angles given in the problem are used to describe the rotation of the whole body.

I would say that the constraint (in one form or another) is this:
Since the slope at the contact point will be parallel with the ground, the gradient of the surface at the contact point (ΔF(A)) will always be orthogonal to the e_z axis.

Where A is (x₀ , y₀ , z₀)

So, without parameterizing it first, and just using cartesian coordinates, I tried to solve for (x₀ , y₀ , z₀) by taking the gradient of F(x,y,z), dotting it by the e_z vector, (0, 0, 1) and setting that equal to zero. Which means the z-coordinate will always be equal to zero. But, if the coordinate origin is centered in the middle of the ellipsoid, then the z coordinate should always be ≥ c.
Anyway, I'm having a hard time trying to figure out how to convert it to coordinates of α and β. I'd imagine it has to do with Euler Angles, but I'm not quite sure how to go about that.

 

[mark726]

Dear student,

Thank you for your post. It seems like you are on the right track with your approach. To convert the coordinates to α and β, you can use the following equations:

α = arctan(x1/x3)
β = arctan(x2/x3)

These equations come from the fact that the rotation about the e1 and e2 axes can be represented as rotations about the x and y axes, respectively, in a 3D coordinate system. Therefore, the angles α and β can be obtained by taking the arctangent of the ratio of the x and z coordinates, and the y and z coordinates, respectively.

As for the constraints, you are correct in stating that the slope at the contact point will always be parallel to the ground. This means that the normal vector to the surface at the contact point must be perpendicular to the ez axis, as you mentioned. This can be expressed as the dot product of the gradient of F(x,y,z) and the ez vector being equal to zero.

I hope this helps. Good luck with your homework problem!