Friction Between Two Curved Surfaces

sorry for double posting, here there is an external link to the first image at a higher resolution (I thought it was visible) :

By classical formula I assume you are referring to F=mu*R. How did you determine the R? Was it the resultant force perpendicular to the curvature? i.e.R=(F- component of weight in that direction). i.e. resolved perpendicular to the direction of frictional force? Where did you assume force F was applied. At the centre of the plate or at some other point?

Thank you for your answer, mathewkyle.

Yes, sorry, I forgot to explain the kind of force which is applied. In normal conditions there is a vertical force F distributed on the whole upper surface of the white plate, pushing on it. This surface is initially in horizontal position (as shown in the second image posted in my first post), then inclinates when the plate slides laterally. During this process, the force stays always distributed on its surface and is always directed perpendicular to it.

In my calculation I strongly simplified the situation: I neglected the radius of curvature, since it is a huge value (332.35 mm) with respect to the plate thickness (2 mm in the center, lower at the sides) and i considered the contact surface as plane for both the plates. In this simplified model, the force is always transmitted perpendicularly to the grey plate while in reality I think this happens only at infinitesimal dimension, at each point of the interface between the two plates. And yes, I used the law Friction = mu*F .

If F is simply the weight of the white plate (computed thanks to its volume and its density) the Friction force is really low; it sounds quite reasonable but I'm sure that the approximation that I made is too strong.

Thanks for the details, but in Friction = mu*F, F is the net perpendicular reaction (to direction of friction). I am not sure you got this well. It is the resultant of the applied force and weight of the plate.

Thank you, actually I got it well. I considered the net force as acting at the baricenter of the body.

In my computation I just wanted to have a first numerical result and, concerning the force, I used the mere weight of the plate by making the aforementioned assumptions. We can reason symbolically, with a net force F composed by the force applied on the plate and the weight of the plate itself. What I'm looking for is a good approach to compute the friction reaction force in presence of this curved surface as interface region.

I have been trying to understand this. Question, does the white plate rotate or slide in a linear direction? If it rotates, there is a moment involved, the force would probably be concentrated around the radius of gyration (ignoring the curvature, that might modify it slightly).

"...vertical force F distributed on the whole upper surface of the white plate."
In this case, this is not a point force like in classical calculations. The force F is distributed evenly over the entire surfeace area of the upper plate in a direction perpendicular to the area. Again, I assume the slipping occurs in opposite directions i.e. no rotation is involved?

To answer to both mathewkyle and passingtongreen (thank you for your questions):

no rotation is occurring. The white plate is "guided" by a worm drive mechanism and it translates in a lateral direction, as shown in the second small image that I posted in my first post. If you look at it from a frontal position, then, you'll see it slide to the left and to the right.

Question: since F is not a point force, am I not allowed to employ the classical formula Friction=mu*F then?

Have a nice day

The equations hold. It is just how you use it that matters. Be sure that the force will not be considered as pressure if distributed over the area.

Yes, but the problem is that I am not sure I can use that formula if the surfaces are not flat but curved, as in my case.

In past years I needed to calculate shear stress between a sphere and a flat plate when the sphere was loaded with a normal force and a shear force. Your problem looks to be a form of this type. There are closed form solutions for this problem in mechanical engineering texts. I think I can find the equations if necessary. I can tell you that the maximum shear stress (contrary to my intuition) is a pair of parallel lines, the maximum stress being lower at the point of initial contact.

Thank you for your answer, welderman.

Actually what makes things a bit messy in my head is that the contact between the two surfaces is complete at the starting position, and during the sliding process it keeps being complete except for the lateral sides of the white plate, that "exit" laterally with respect to the grey plate.

The difference with respect to your example is that I don't have a point contact but a surface sliding phenomenon.