Principal curvature used in a contact problem

[MechEng2010]

Hi,

I am trying to understand a theoretical problem involving the contact between two surfaces. I have uploaded a screen shot of the mathematical formulations of the solution.

I understand most of the solution, except the principal curvatures. I have tried to look up principal curvature, but still not sure how is applied to this problem.

I would really appreciate any help for the mathematicians out there.

https://dl.dropbox.com/u/47274064/Prin_curve_1.JPG

Thanks.

 

[Mandlebra]

Curvature is the reciprocal of the radius. Is that what is confusing you?

 

[MechEng2010]

Curvature is the reciprocal of the radius. Is that what is confusing you?

Thanks Mandlebra, that much I understood.

I am just trying to understand how and why R₁, R₂, the planes of R₁, R₂ and the angle [itex]\Phi[/itex] come into this. I guess my questions are more specifically:

1) How do you determine what R₁ and R₂ are?
2) How do you determine the plane of R₁ and R₂?

Would really appreciate some advice or explanation on this.

Thanks.

 

[haruspex]

Not sure if this helps...
Take the normal N to the surface at point P. Now take any plane containing the normal. The intersection of the plane with the surface produces a line with some curvature at P. As you rotate the plane about N, the curvature reaches a minimum and a maximum, possibly of opposite sign. These are the principal curvatures. The planes containing them will be orthogonal. I believe the curvature in any intermediate plane through N can be computed from the principal curvatures and the angle this plane makes to the planes of principal curvature, but I don't the formula.

 

[haruspex]

Forgot to mention phi. The other body will have the same tangent plane at P and thus the same normal. But the planes through it which give the min and max curvature for that body will not in general be the same. Phi is the angle between the two bodies' max curvature planes (and thus also the angle between their min curvature planes).