Bearing Assembly Contact Loads & Distribution - Free Body Diagram 1

[jrootatendurance]

Hey folks,
I'm designing an assembly that looks and behaves similarly to a cylindrical roller bearing and I'd like to get your help on the best approach to take in calculating some force loads. The "rollers" are cam followers that mount to a rotating center assembly. There is a stationary assembly that has an OD "race" for the cam followers to react loads against. A singular force is applied to the center of the assembly, radially loading the "bearing". I am trying to calculate the loads that pass through each of the cam followers for given rotational positions of the rotating assembly. Im assuming the assembly is stationary and frictionless, all bodies are rigid, and that the cam followers can only react forces in a radial direction. I attached a pdf that shows the simplified 2D representation for two different cases.

I first started with a free body diagram, summing forces and moments to identify loads on each of the cam followers. I came to the conclusion that the system is indeterminate because of the 5x support locations, and there are multiple solutions that meet the constraints of the sum of moments and forces equations. I proved it to myself with a simplified FBD for a rigid beam simply supported in 3 locations. Based on sum of forces and moments I could get any number of solutions that satisfied all of the constraints. Does this conclusion make sense? Or could I be missing something?

I think the equation that I am missing that could be the last piece in this puzzle would be a method to estimate/determine the load distribution across the multiple supports. Id guess this could be done possibly as a function of proximity to the applied load? Or possibly by making the assumption that the load is evenly divided among the contact points? I could imagine running across a similar problem in determining the loads on each axle of a vehicle with more than 2 axles, or as I mentioned above, determining the compressive loads on each roller or ball in a radially loaded bearing.

Any thoughts on how to estimate the load distribution across the 5x supports?
Thanks,
Jeff

 

[hendersdc]

You can't assume that the cam follower positioning will be uniform enough to load all 5 of them at the same time. Worst case will be contact with only 2 cam followers to fully constrain the system. Worst case of loading only 2 will be one roller located almost at 90 degrees to the load and the other one just off center to the path of the load. This basically will drive you to design it for one roller supporting the full vertical load * safety factor.

At least this should really simplify your equations!

Doug

 

[jrootatendurance]

Thanks for the input Doug. I agree that the case you described certainly simplifies that calculations.

 

[Dougt115]

As stated worst case is a single bearing supporting the load, and for stability at least two will likely be in contact. But due to the ductility of the metal it is likely that the load will distribute over more bearings.

You have a very complex problem that I would solve as Doug before me stated. Solve for the worst case.

 

[jrootatendurance]

Thanks Dougs.
For cam follower sizing and FOS considerations I will take the conservative route and assume the entire load passes through a single point in my 2D representation.

For the sake on discussion, if someone were trying to predict the real world loading case, does anyone know of a method to at least estimate what the distribution may be over the multiple contact points? This would require the big assumption that all 5 cam followers are in simultaneous contact with the body that reacts the forces. What I would love to find is any text book reference to a method for defining this type of load plot, shown in red here: [URL unfurl="true"]http://qualitybearings.files.wordpress.com/2010/08/load.jpg[/url]

Like I said above, it seems like this might be a common calculation done in bearing design, or on other systems that have redundant supports and are indeterminate.

Would any FEA users be able to comment on the feasibility and confidence/accuracy level of analyzing a model like this and getting simulated loads on the contact points?

Its interesting to me that the freebody diagram method is contradictory in the sense that it assumes, and actually requires a perfectly rigid body, but cannot accept the assumption of perfect geometry of more than 2x support points contacting the body.

 

[tbuelna]

The condition you describe is actually not much different than that existing in a cylindrical roller bearing. So in order to get an accurate result you must take into account the structural stiffness of each element, rather than assuming the elements are rigid. Since the load distribution between your rollers will likely vary greatly with very small radial displacement of the shaft, it is important to accurately represent the structural stiffness of each component.

Once you have an accurate analysis result of the deflections at each roller contact under load, you can adjust the installed position of each roller to give a better load distribution under operating conditions.

Hope that helps.
Terry

 

[rb1957]

do the 5 rollers contact a significant piece of metal, or a wimpy little piece of sheet metal (pissibly stiffened by frames) ?

for me if the substrate is a significant, stiff, and reasonably uniform hunk of metal i'd assume 5 equal radial forces react the normal component of the load. the side component would be reacted by the two rollers on the right side, and finally you've got the off-set couple (of the side forces) increasing the vertical loads on one side, reducing them on the other.

another choice for the horizontal component (possibly the same, but a different way of thinking about it) would be to assume four rollers have radial loads (two positive, two negative) so that their resultant balances the horizontal component with no nett vertical. you could try the same magnitude for these four, or 1/3:2/3, ... the "problems" for the rollers is when the outer one starts to lift (if the applied force is outside the rollers).

Quando Omni Flunkus Moritati

 

[jrootatendurance]

Terry, thanks for the suggestion. The loading cases vary on this system, both in terms of magnitude and direction. Sorry, I didn't include that in my original description. I don't think I have an option to bias the roller position based on anticipated deflections, because each load case would expect different deflections among different rollers.

rb1957, The rollers contact a very significant & uniform seamless forged-then-turned ring that acts as an outer race. When I first started on this design I took a similar approach to the one you've outlined - I'd started with the assumption that the vertical component was evenly distributed among the 5 rollers. This even 1/5 reaction load was applied to each roller as the vertical component. I then found the radial resultant on each of the rollers (assuming no friction, they can only react loads radially). I noticed 2 problems with these results:
1. The forces in the "horizontal" direction did not sum to zero in cases where the rollers were positioned asymmetrically.
2. The radial loads on the outermost rollers were highest, the load on the bottom-most roller were lowest. By playing with the geometry, and sending the outermost rollers to near-90° locations, the radial loads on these rollers approached infinity.

I attached a pdf with figures that show this approach. The large orange/brown arrow is the input force. Ignore the yellow arrow. The red construction lines show the "vertical" and "horizontal" load components for each roller. Its plain to see that the horizontal loads do not sum to zero in these figures.

Your suggestion of assuming the radial forces on each roller are equal sounds like a good one. I will try it out and see what it looks like.

BTW this is all just for discussion. The rollers have been sized to carry the entire load through one set, along with a generous FOS.

Thanks for the input.

 

[chicopee]

Both free body diagrams are missing the tangential frictional force that develops when there is rotation.

 

[rb1957]

i agree with your logic, that the inner rollers should be more highly loaded.

can you resolve the applied force along (and normal to) the CL on the middle fastener. now you'll have two force components to balance. the radial one ... maybe 5 equal radial loads produces an unlikely result ... maybe 4:2:1 distribution (ie the CL fastener reacts 40&, the inner two 20% each, the outer two 10% each; a more numberical way to describe this would be to say the radial forces are proportional to cos(theta), the angle off the CL of the group. The normal component would be reacted by 4 radial forces.



Quando Omni Flunkus Moritati

 

[dwallace1971]

I think this is a good problem for FE analysis.

"On the human scale, the laws of Newtonian Physics are non-negotiable"

 

[rb1957]

if you assume all the roller reactions are radial, then will they all intersect at a point (yes, if the mating surface is circular). if the line of action of the applied load is through this point, then you have only two equations of equilibrium. With 5 unknowns, you can do a little math ... find the minimum error solution for a set of under-defined equations. same if you have a constant amount of friction. if you have generalised reactions (then you have no relationship between radial and tangential components of the reactions, and 10 unknowns, but the same method applies.
if the line of action is not through the centre of the radial forces, or if the radial forces don't converge to a center, then you've got three equations.

Quando Omni Flunkus Moritati

 

[Tmoose]

The manufacturing tolerances of the "bolt circle" on the rotating center assembly will likely mean a few cam followers will stand "proud" and do all the work for light loads. Even the tolerances on the cam follower stems will tend to do something similar.
The fit and bending stiffness of the cantilevered cam followers means heavy loads will tend to distribute the loading on a few followers at the bottom. The followers' rollers may even be edge loaded
The stiffness of the outer race support may modify the distribution significantly. I'm picturing a hammock.

One of the things FEA has "taught" me is there aren't many structures or components that are remotely resemble "rigid".

 

[Engdoitbetter]

AFAIK you should use a method similar to that used to obtain the Stribeck formula, which is employed to calculate the load distribution in a rolling bearing.

Stefano