Need help with force calculation involving a ball bearing
Need help with force calculation involving a ball bearing
(OP)
Hi all,
I'm trying to solve the attached problem in order to understand the system so I can design a related tool. I have to admit I'm a little frustrated because I know at one time I could have solved this no problem, but I guess I've been away from it for too long!
I'm trying to follow the 600N force through the first wedge, through the single ball bearing, to determine the force on the wedge at the other end of the system. I'm pretty sure the normal force acting on the ball should be Fb=600/cos(75)=2318.22, but I'm not sure how to deal with the reactions on the 2nd wedge. Can anyone assist me in getting started?
Thanks,
Tim
I'm trying to solve the attached problem in order to understand the system so I can design a related tool. I have to admit I'm a little frustrated because I know at one time I could have solved this no problem, but I guess I've been away from it for too long!
I'm trying to follow the 600N force through the first wedge, through the single ball bearing, to determine the force on the wedge at the other end of the system. I'm pretty sure the normal force acting on the ball should be Fb=600/cos(75)=2318.22, but I'm not sure how to deal with the reactions on the 2nd wedge. Can anyone assist me in getting started?
Thanks,
Tim





RE: Need help with force calculation involving a ball bearing
Then treat each wedge and the ball as a free body in which forces are in equilibrium. This will give you some simultaneous equations which you solve by elimination.
Don't forget the moment equilibrium in the free body force balance.
Try to get rid of the contact at the circular support at the bottom - I'm not 100% sure but I think the three point contact on the ball makes it unsolvable (Newton's three ball problem)
Don't forget to add gravity M.g to each component.
Hope this helps, maybe someone will do it in detail for you.
gwolf.
RE: Need help with force calculation involving a ball bearing
RE: Need help with force calculation involving a ball bearing
The 3rd support makes it difficult to analyse however if you assumed angle for the bottom support reaction one could solve it graphically with a polygon of forces diagram I think.
Also according to my calculations the force to the left of the ball is 600N * Cos 75 deg = 155.29N.
I have uploaded a sketch of what I think the free body diagram would look like see if it helps.
desertfox
RE: Need help with force calculation involving a ball bearing
The ball free body diagram is not enough for analysis, the left wedge free body diagram should be considered too because it rubs on the left side wall and the friction between the left wedge with the wall will affect the results.
RE: Need help with force calculation involving a ball bearing
The reason I think the force is 600/cos(75) is because it's acting like an inclined plane system, so the force should increase due to the mechanical advantage. (As I write this I double checked myself and I think I had it wrong. Mechanical advantage is length/rise, i.e. hypotenuse/opposite or 600*1/sin(75)=621.2)
RE: Need help with force calculation involving a ball bearing
Yes I was using the assumption that the thing was frictionless as was mentioned in the OP's second post which of course in reality it isn't, I am still scanning my books to see what else I can find.
desertfox
RE: Need help with force calculation involving a ball bearing
I presume what were looking at is a static analysis? the vertical force of 600N acting that acts vertical,if you manage that the wedge didn't have the 15 Degree face and was just vertical then there would be zero force acting on the ball assuming the face touch tangential and forgetting friction. Now as you taper the face a small proportion of that vertical force exerts some load on the ball so I think in this case the 600N * Cos 75 deg is correct.
We also need dimensions from the ball centre to the support points I have assumed in my sketch that the angles of the faces are the same off the centre of the ball they may not be you need to confirm.
I don't know what sort of accuracy your looking for and whether ignoring friction matters assuming it doesn't then I have drawn a rough sketch of what I think the force diagram would look like.
RE: Need help with force calculation involving a ball bearing
First layout the system with reasonable accuracy, like you have done. Now graphically move the left wedge vertically, say delta x=1/4 inch on the drawing and allow the system to expand . Now note the vertical displacement, delta y at the Fy location.
From virtual work
600*delta x =Fy* delta y
so
Fy=600*delta x/delta y
You can also get the delta y analytically, involving some trig.
RE: Need help with force calculation involving a ball bearing
You should not ignore friction effects, for many reasons:
- The wedge surface contacts with the spherical intermediate member don't result in a pure rolling motion, and must allow some relative sliding. Which would imply a high friction loss at those contacts.
- The small included angle (15deg)of the driving wedge member surface would produce a high normal force at the wedge's reaction point, for a given axial force (ie. 600N). The wedge must move vertically to produce a continuous force on the sphere, which means it must be able to slide relative to the outer surface it bears against. If it can't slide, the the driving load will balance out once the elasticity of the mechanism's structure achieves equilibrium with the resisting friction forces.
- The negative 7.5deg wedge angle on the reacted side of the output wedge element would also produce lots of friction losses.
I'm too lazy to do a vector analysis of your mechanism. But at a glance, that device appears that it would be self-locking with your 600N input force, even if you apply a liberal Mu for greasy metal surface contacts (ie. <0.2). So your Fy value would thus be zero.
.....Unless of course, you wish to ignore friction.
Regards,
Terry
RE: Need help with force calculation involving a ball bearing
RE: Need help with force calculation involving a ball bearing
This then solves the problem assuming sliding friction which could be a problematical assumption.
RE: Need help with force calculation involving a ball bearing
delta y/delta x=tan(15)*tan(37.5)=.206.
so
Fy=600/.206=2918 lb
without friction
I also did this with the 3 triangle loops and got
Fy= 3020 lb, more or less confirming the above.
With a friction coefficient of ,2 using the three triangular approach of my previous post, I got
Fy= 1547 lb
RE: Need help with force calculation involving a ball bearing
Well your post is stirring quite a bit of interest.
I post my working out for the vertical force on the righthand side of your wedge diagram.
I get a vertical force of about 219N having resolved the 600N force on the lefthand side wedge to parallel and at right angles to it see what you think.
I have ignored friction by the way.
zekeman your answer's are in lbs I think they should be Newtons.
regards
desertfox
RE: Need help with force calculation involving a ball bearing
Why don't you look at other solutions, before hastily posting and avoid the apologies later.
Your value for F1 is obviously wrong. The frictionless case already has shown F1 be
600/cos(75) , a lot higher than yours.
Sorry about my Newton /lb mixup, but the answers I gave are numerically correct since I started out using lbs.
RE: Need help with force calculation involving a ball bearing
Well all I have is your numbers where is the working out where have I gone wrong?
desertfox
RE: Need help with force calculation involving a ball bearing
See what happens if you are not careful and methodical and include all forces including friction!
There is only one way to do this properly and no-one has yet.
I can't be bothered with this anymore - eject.
RE: Need help with force calculation involving a ball bearing
I have no idea if I can send this but I did the triangles with and without friction
RE: Need help with force calculation involving a ball bearing
I cannot open that can you send it as a pdf.
desertfox
RE: Need help with force calculation involving a ball bearing
With and without friction, Results are close (but not exactly as my previous answers)
RE: Need help with force calculation involving a ball bearing
Many many thanks for all the responses and assistance. Here's what I ended up with from summing the forces with and without friction. My numbers came out close to Zekeman's, and they also are similar to the graphical method mentioned. My results are attached. Think I've got everything covered?
Tim
RE: Need help with force calculation involving a ball bearing
Well I am glad you got where you wanted to be and if I made a error in my post then thats is that, I can see now working out as been posted as opposed to just answers where our figures differ and still looking at it whilst doing other things.
Again if I have made a mistake then zekeman as a fair point that I shouldn't rush to make hasty posting, that said if he had followed his own advice the last line of his post wouldn't be nessecary.
desertfox
RE: Need help with force calculation involving a ball bearing
Just noticed in your calculations that on the left wedge there is no allowance for friction between the wedge and the cross hatched vertical wall, on the right hand side you have labelled it F4 but there is no equivalent on the left side.
desertfox
RE: Need help with force calculation involving a ball bearing
Looking at the case for friction on the lefthand wedge it appears that zekeman and yourself have neglected friction between the angled surface and the ball.
Looking at zekemans triangle of forces he as drawn the 600N
force vertical and the corresponding top vector at 11 degrees from horizontal which I concurr with however the vector reaction from the ball is still drawn at 15 degrees from the horizontal which is incorrect when friction is being considered it should be drawn with an angle of 26 degrees from the horizontal which would yield a force of around 990N and not the 1326N using your figures presented.
desertfox
RE: Need help with force calculation involving a ball bearing
Are you talking about friction between the left wedge and the ball or the left wedge and the grounded vertical plane? for the wedge/ball I think I have them all in there? For the wedge/wall combination I see that I do not. (Also, it's not labeled for the wedge/wall on the right, but it is in the calculations.)
This has been an interesting problem, and I'm thankful for all the help and the opportunity I've gotten on it.
RE: Need help with force calculation involving a ball bearing
In your post dated 23rd you have not accounted for friction between the lefthand wedge and the vertical wall that it slides on, I'll post a solution shortly to show you what I mean exactly.
desertfox
RE: Need help with force calculation involving a ball bearing
I have uploaded a file now showing my details for your wedges using a coefficient of friction of 0.2 throughout.
I believe I have taken into account friction on both wedges and the ball.
I get a vertical force at Fy to be about 640N.
desertfox
RE: Need help with force calculation involving a ball bearing
Consider the result of the vector solution.
If you break the two sets of vector forces on the ball into the normal force set and the friction set, you see that the normal set would have zero moment on the ball since all forces pass thru the center. Now the friction set under the assumption of 0.2 coefficient would yield a triangle geometrically similar to the normal set and thus satisfy the linear force equilibrium condition, BUT, unless the linear sum of the bottom two sides of the triangle is equal the top side, it would not be in moment equilibriume.
There are further problems with this friction model, notably the way they manifest themselves.
So,we need a more sophisticated friction mechanism to get the proper answer.
In that absence I used zero friction,probably not the best assumption but at least it satisfies the moment equation,LOL.
I suppose somebody ought to open another thread to address this ball equilibrium problem.
RE: Need help with force calculation involving a ball bearing
So I can see no reason in the calculations other than an error why on the lefthand wedge at the interface with the ball and the 15 degree face that friction was not included and looking back at zekemans triangle of forces which was posted on the 22nd the only place that included friction was the lefthand wedge between the vertical wall and flat bottom of wedge under the heading of friction coeff 0.2.
Zekeman I can post the moment equilibrium of the ball if you require and unless I am reading your post incorrectly your saying that a ball under three resultant loads acting concurrently cannot be moment balanced when those forces are broken into there respective horizontal and vertical components, if I have misinterpreted your last post then perhaps you can expand on it.
desertfox
RE: Need help with force calculation involving a ball bearing
RE: Need help with force calculation involving a ball bearing
So therefore the friction coefficient of 0.2 should have been carried through all the components including the ball as stated earlier.
desertfox
RE: Need help with force calculation involving a ball bearing
Rather than choosing to call me wrong, show me where the the closure of the force vectors account for the moment equation.
Remember, as I have repeatedly mentioned it DOES NOT include the moment condition.
For examples if you take the ball triangle after you have determined one of the three vectors from the left wedge, you are left with 2 unknowns, namely the absolute values of F2 and F3 and you now have three equations to get these, namely the summation of forces in the x and y directions and the moment equation.3 equations 2 unknowns. How do you do it?
I think we have gone 15 rounds already and should close our private discussion.
Maybe someone else can arbitrate this one. If not...
Its been fun.
Best regards
Zeke
RE: Need help with force calculation involving a ball bearing
ht
It involves a cylinder sat in a vee and given its mass you have to calculate the amount of torque to be applied to rotate it. Due to its mass it as reactions and coefficients of friction are given and from this information you calculate the reactions both on the cylinder and wall.
This is no different to what we have done with the wedge in the original post.
Now if I remove the torque in the example of the cylinder (link above)then the cylinder stops rotating however it still as the reactions acting on it due to its mass and wall contact. As stated by zekeman if you look at the moments generated by the friction then they don't cancel however the cylinder doesn't rotate either and thats because the forces calculated are those that oppose motion and only an active force which which exceeds these values will allow rotation to begin. Not anywhere does the book state that we need a better model for friction as suggested by zekeman. Perhaps zekeman can now show us a site that actually supports his claim we need a better model for friction and that ignoring friction on the ball and on both wedges was not a mistake.
desertfox
RE: Need help with force calculation involving a ball bearing
desertfox
RE: Need help with force calculation involving a ball bearing
Since you brought it up,let's take your cylinder and both of us determine the forces on it under its own weight only. Let's say it weighs 100N.Assume static friction coefficient of 0.3.
My solution, since I claim there are virtually no friction forces on it, is:
left oblique wall normal force : 100/cos(30)
right wall normal force : 100* tan(30)
No friction forces.
What's your solution?
RE: Need help with force calculation involving a ball bearing
"so we need a more sophisticated friction mechanism to get a better answer"
"In that absence I used zero friction,probably not the best assumption but at least it satisfies the moment equation,LOL"
My solution I posted yesterday follows the text book but I'll wait see what you produce in light of your statements.
As far as I can see you also left friction off the tapered face of the left wedge and completely off the right wedge which I believe was an oversight which isn't a problem because we all make mistakes.
So have you any evidence to back your claim or not?
RE: Need help with force calculation involving a ball bearing
I gave you my answer. What is yours?
RE: Need help with force calculation involving a ball bearing
The answer I asked for three times now is to back up your statements as posted in my previous post hopefully you can do this time without the blaher.
resultant force Na=237N
resultant force Nb= 312N
friction force from Na= 71N
friction force from Nb= 93.6N
RE: Need help with force calculation involving a ball bearing
Now, the reason I asked that we both solve your cylinder problem is to show that your answer violates the zero moment condition. That paper correctly shows the amount of torque the cylinder can sustain without slipping . In the absence of that torque you cannot use a friction force because of the moment violation take moments about the center). Moreover, even if you choose friction, how can you possibly use the static coefficient since that yields the maximum value before slippage . As an example suppose you had a weight resting on an inlined plane with an angle of 10 degrees, the friction force would be just sufficient to prevent sliding, an effective coefficient of tan(10)=.18; by your method the friction coefficient would incorrectly be 0.3 and the weight would slide up the ramp.Stop.
Why don't we bring this discussion to a humane conclusion and stop now.
RE: Need help with force calculation involving a ball bearing
Secondly your post of the 21st actually suggests using sliding friction and a coefficient of 0.2,no mention of using zero friction on the ball, that only came when you realised that you hadn't accounted for it in your "with friction calculations".
If you look at your force triangles there is no allowance made on the 15 degree face of the left hand wedge and no allowance for friction made on any part of the righthand wedge irrespective of the ball and I have not stated that you said there was zero friction on the wedges.
Yes your right if you assume a value of friction coefficient, then what you get is a maximum force before slippage but the actual force on the parts could be anywhere from zero to that maximum but again this is a part of engineering where one as to make assumptions.
The reason I posted that problem with the link is that it demonstrates that it is acceptable to use a static coefficient of friction of 0.2 or whatever value you choose on a sphere or ball and follow it through to a conclusion as I did previously with the original problem.
your statement from your post dated 24th:-
There are further problems with this friction model, notably the way they manifest themselves.So,we need a more sophisticated friction mechanism to get the proper answer.In that absence I used zero friction,probably not the best assumption but at least it satisfies the moment equation,LOL.I suppose somebody ought to open another thread to address this ball equilibrium problem.
Your statements imply that the method I used with friction isn't valid and yet your method of using zero friction on the ball is a better method, if thats the case then thats what they would show in text books and teach in college but as the example cylinder shows they don't.
I think we can call it a day now.
RE: Need help with force calculation involving a ball bearing
I believe you are both incorrect. Multiplying the friction coefficient times the normal force gives you the friction during sliding. When the parts are not sliding the friction force may be less.
In tfry200's problem, the ball will clearly slide about two of the interfaces and roll about the third. The friction force at the rolling contact will balance the moments. By looking at the frictionless solution you can determine the point with the highest normal force. This is the interface that will have rolling contact.
zekeman: you're solving by assuming 0 friction and that the ball will move freely. I suppose ignoring the problem counts as a solution of sorts...
desertfox: you are assuming that the friction force is always proportional to the normal force. This is also incorrect. The moment equation will not balance, as zekeman has pointed out. Your ball is magically accellerating.
I believe the groove problem that you've each "solved" is statically indeterminate when you consider friction.
-b
RE: Need help with force calculation involving a ball bearing
Looking at the motion of the ball as the 600 N force pushes up , it appears that the ball is in ROLLING contact with the bottom surface and sliding contact with the other 2 surfaces The F1 vector is previously determined from the force triangle on the left wedge, so the additional forces on the wedge are the normals at F2 and F3, the friction at F3 equal to 0.2*F3 and the unknown friction at F2. We now have 3 unknowns, namely F2, F3, and Xf, the friction at the rolling contact at F2 and can write the three equations for static equilibrium and look at the friction Xf and as long as it is less than the coefficient of STATIC friction I would suggest that it is valid.
I invite your comments .
RE: Need help with force calculation involving a ball bearing
Correct... with the exception that I would pick the interface with the largest normal force as the rolling contact. This may indeed be the bottom interface, but it's unclear to me how you came to that conclusion.
-b
RE: Need help with force calculation involving a ball bearing
My understanding is that the normal force multiplied by coefficient of friction gives the limiting force after which that force is exceeded sliding begins. Further once sliding as begun the friction force actually drops down in magnitude as we move from static to dynamic friction coefficients.
Regarding the groove problem that was solved that was from a book link :-ht
(page 12)
So your saying that the book is wrong? Because that also shows that if you try a moment balance as previously mentioned it also contravenes equilibrium.
If our models are wrong then what is the correct method to solve the problem?
RE: Need help with force calculation involving a ball bearing
My understanding is that the normal force multiplied by coefficient of friction gives the limiting force after which that force is exceeded sliding begins. Further once sliding as begun the friction force actually drops down in magnitude as we move from static to dynamic friction coefficients."
I finally agree with your statement since using the sliding friction presumes that we are pushing upward and have broken away from the static friction case and entered the sliding friction domain;the result we have, further assumes that the motion is upward. If we then decide to push downward on the upper surface I believe we will get a different result owing to the reversal of the friction force vectors.
You make a very good point about static friction and we should first verify that the system can indeed "slide" before we invoke sliding. To do this we should use the static coefficient first and if the resultant force, F5 is positive(downward)then we can choose the sliding condition; otherwise, the system will be static and the resulting F5 will be zero.
RE: Need help with force calculation involving a ball bearing
In Tfry200's problem I was assuming sliding friction and that the parts were moving as a result (and in the direction) of the input force. Obviously if the direction of motion is reversed then all of the friction vectors need to be reversed. Also, as zekeman points out, if the system freezes up then F5=0. Checking for freeze up is basically the same problem except we're using the static coefficient.
I outlined the correct method in my post. Zekeman seems to understand it. Please review.
The problem as posed in the book, calculating the torque required to rotate the cylinder, is solvable and the calculated forces, by definition, satisfy the equilibrium equations. The problem as posed to you by zekeman, calculating the forces of the cylinder resting in the groove, is statically indeterminate.
-b
RE: Need help with force calculation involving a ball bearing
I have to say you are correct and the only reason I chose zero is that it satisfies the momentum eondition and is a solution; of course depending on how the cylinder is placed on the ramp there are an infinity of friction forces possible that will close the force loop but since you must satisfy the moment condition, the two friction forces must have the same amplitude.; the ideterminancy occurs because there are 3 equilibrium equations and 4 variable.( 2 normal forces and 2 friction forces) so you can specify one (within some limits)) arbitrarily to get the remaining 3, making sure that the 2 friction force vectors are of equal length.
RE: Need help with force calculation involving a ball bearing
I have to say you are correct and the only reason I chose zero is that it satisfies the momentum eondition and is a solution; of course depending on how the cylinder is placed in the groove there are an infinity of friction forces possible that will close the force loop but since you must satisfy the moment condition, the two friction force vectors must have the same amplitude.; the ideterminancy occurs because there are 3 equilibrium equations and 4 variables.( 2 normal forces and 2 friction forces) so you can specify one (within some limits)) arbitrarily to get the remaining 3, making sure that the 2 friction force vectors are of equal length.
RE: Need help with force calculation involving a ball bearing
Oh right I see the rolling friction on the face with the highest normal force, okay I thought you might be posting some working out for us all to see.
The reason I asked for some working out is that I cannot see how it will solve the out of balance moment on the ball as you claim, so it would be of interest and help the OP too. Perhaps you would be kind enough to post your full solution and then the thread can be closed.
zekeman
Glad we can agree on something, anyway from our initial figure of 0.2 for sliding friction and the 600N vertical force on the wedge it would seem that motion was just impending however if we have rolling friction at that point I suppose that could change things dramatically.
RE: Need help with force calculation involving a ball bearing
-b
RE: Need help with force calculation involving a ball bearing
RE: Need help with force calculation involving a ball bearing
Why wait, just go ahead and put your solution in.
desertfox
RE: Need help with force calculation involving a ball bearing
"There is only one way to do this properly and no-one has yet.
I can't be bothered with this anymore - eject."
I believe you quit 34 posts ago.
No wonder we haven't nailed it.
Cheers
RE: Need help with force calculation involving a ball bearing
Regards,
gwolf
RE: Need help with force calculation involving a ball bearing
Not so much a question of nailing it as agreeing which solution/method is the correct one.
We seem to be at odds as to whether we have moment equilibruim on the ball?, rolling or sliding friction or both.
So what were looking for if the solutions are wrong why are they wrong and how to put them right and if someone can produce the perfect solution then lets have it.