Static equilibrium with friction question

[remettub]

Homework Statement

This is a real world application. I am posting it here to follow the guideline for any "coursework-like" questions, but since I have no formal training, I'm not sure what category this would actually fall under. Any recommendations for a more specific posting location would be welcome!

I have a cantilever beam in static equilibrium, with a known mass and center of gravity. It is supported at one end by contact at three points inside an irregularly shaped cavity in a wall. My task is to find the force(s) present at each of the points of contact, which have known coordinates and a known normal direction. For better visibility, the image below only shows the three relevant surfaces (brown disks) of the cavity, where they contact spherical protrusions on the beam.

Homework Equations

The Attempt at a Solution

If friction at the points of contact is zero, then I can easily solve for the unknown forces (red arrows) at the points of contact by using translation and/or moment equilibrium equations, since the direction of the force at each of the points of contact must be normal to the contact surface, and I therefore only have three unknowns (the magnitudes of the forces).

However, in the actual scenario, friction IS present between the beam and the cavity at the points of contact, and here I run into trouble. I am conceptualizing the frictional forces as a vectors originating from the contact points and tangent to the contact surfaces, but since they could also point in any direction on the contact surface, I now have nine unknowns and only six equations of equilibrium from which to find them.

What am I missing here? It seems like the system must have a unique solution.

 

[Nidum]

This system does not have a unique solution anyway . The only way to deal with a system like this is to base all calculations on the worst case configurations of loading and fixation .

You can use professional judgement about what are the worst case configurations or you can analyse multiple cases and rank the solutions .

 

[haruspex]

It seems like the system must have a unique solution.

In the idealised world of rigid bodies, arrangements with multiple frictional forces do not necessarily have unique solutions. A simple example is an object rammed into V cleft. In the real world, it depends on the elastic properties of the materials and how the arrangement was produced.

I agree with your counts of unknowns and statics equations. To get more equations you could throw in, say, that the normal forces are to be minimised.

 

[remettub]

Thanks for your replies. I came across the example of an object in a V cleft as I was browsing the forum for ideas. It seems highly relevant.

Would minimizing the normal forces provide an approximation of the equilibrium that a real world object would reach due to elasticity?

 

[haruspex]

Thanks for your replies. I came across the example of an object in a V cleft as I was browsing the forum for ideas. It seems highly relevant.

Would minimizing the normal forces provide an approximation of the equilibrium that a real world object would reach due to elasticity?

Not just the elasticity, but also the deformation. Simple example is tightening a nut. The minimal forces solution would be that degree of tightening that just holds things in place for now. That would not be typical.

 

[remettub]

After thinking about this some more, I don't think that considering deformation is necessary. The beam would not be wedged tightly into the opening. The more natural entry method would be to simply insert the beam at an upwards angle, then let gravity bring all the points into contact. Here is what I think is the 2-dimensional analogy:

In the 2-D version there are only four unknowns (the magnitude of the two normal forces, and the magnitude of the two tangent forces due to friction), so it should be simple to find a rigid body solution. Of course in the real world the beam would deform somewhat even if it was lightly placed rather than rammed into the opening, but I still believe a rigid body solution would be an accurate enough representation to serve my purposes.

So, to rephrase my question: would minimizing the normal forces give me the rigid body solution for the 3-D problem?

 

[haruspex]

After thinking about this some more, I don't think that considering deformation is necessary. The beam would not be wedged tightly into the opening. The more natural entry method would be to simply insert the beam at an upwards angle, then let gravity bring all the points into contact. Here is what I think is the 2-dimensional analogy:

View attachment 112713

In the 2-D version there are only four unknowns (the magnitude of the two normal forces, and the magnitude of the two tangent forces due to friction), so it should be simple to find a rigid body solution. Of course in the real world the beam would deform somewhat even if it was lightly placed rather than rammed into the opening, but I still believe a rigid body solution would be an accurate enough representation to serve my purposes.

So, to rephrase my question: would minimizing the normal forces give me the rigid body solution for the 3-D problem?

Good idea to start with a simpler version: 4 unknowns but only 3 equations.
Yes, looking for a solution that minimises a force is reasonable, but it is not obvious whether it matters which force. I suspect that in this 2D example it won't matter, that all forces are minimised simultaneously. Might not be true in general.

 

[Nidum]

As drawn the beam does not have a unique location in the slot and also it could be unstable ..

Can you tell us more completely what the actual problem is ?

 

[haruspex]

As drawn the beam does not have a unique location in the slot

It is drawn at a particular location. Maybe I am missing your point.

it could be unstable ..

As drawn, it would only be unstable if the coefficient of friction is too low. Assume we are to find the minimum for stability.

 

[remettub]

Good idea to start with a simpler version: 4 unknowns but only 3 equations.

Oops, I was mistakenly thinking that there would be four equations of equilibrium for the 2 dimensional version. Of course, there are only three!

As drawn the beam does not have a unique location in the slot and also it could be unstable ..

Can you tell us more completely what the actual problem is ?

My goal is to write some code which would be able to evaluate any specific instance of this problem. So, while I have not provided a specific instance here, the locations of the points of contact are still considered known variables. There may or may not be a stable solution, depending on the inputs and the coefficient of friction