Rotation of a ring pivoted at several points

[davidpascu]

Homework Statement

A rigid ring is fixed with three bearings evenly spaced around its circunference, which allow it to rotate but not to displace in the radial direction. An external force is applied in a fixed point P of the ring which moves with the ring. The bearings have friction.

These parameters from the problem are known: ring mass, radium and inertia momentum, external force (value F and angle phi) and friction coefficient. Obtain the expression for the angular acceleration as a function of these parameters and the angular position of the point P.

Homework Equations

For the friction forces: F_fr = μN

The Attempt at a Solution

I have tried projecting all the forces on the horizontal and vertical axis, and using Newtons Equations in the center of mass: ΣF = 0 and ΣM = Iα, but I end up with more unknown variables than equations.

 

[BvU]

Namely ? Please post what you have so far, perhaps we can find another relationship between the variables you should have listed in full...

 

[davidpascu]

First, projecting the force in the tangential and radial direction, with the angle phi:

F_TAN = F cos Φ and F_RAD = F sin Φ

Then, projecting this two again in the vertical and horizontal axis:

F_X = -F_TAN cos θ - F_RAD sin θ
F_Y = -F_TAN sin θ + F_RAD cos θ

where the vertical axis Y is taken as the origin of the angle θ.

Considering that all normal reactions point towards the ring center, I apply Newtons laws in the center of mass. The normal reactions and friction forces are numbered 1,2,3 following counterclockwise order starting from the one in the top.

In X: N₂ cos 30 - N₃ cos 30 + F_FRICT1 - F_FRICT2 sin 30 - F_FRICT3 sin 30 + F_X= 0
In Y: -N₁ + N₂ sin30 + N₃ sin 30 + F_FRICT2 cos30 - F_FRICT3 cos 30 -mg + F_Y = 0

Momentum around CM: R x (F_TAN - F_FRICT1 - F_FRICT2 - F_FRICT3) = Iα

And F_FRICT = μ N for all 3 bearings.

So as I need to know the angular acceleration as α = f(m, I, μ, F, Φ, θ) I need to get first the expression of the normal forces (and with them the friction forces). And with this set of equations only I am not able to do it.

 

[BvU]

Apart from ##I = mR^2## I don't have much to contribute, I'm afraid.
Perhaps an extra assumption is required ? Because I see that increasing all three normal forces by the same amount of Newton does not change the problem description but it does change the outcome. (I.e. how tightly is the ring clamped in ?) A sensible assumption to reduce the number of degrees of freedom would then be to put ##F_{N,1} = 0## -- but then: why isn't it in the problem statement ?

 

[haruspex]

I see that increasing all three normal forces by the same amount of Newton does not change the problem description but it does change the outcome.

Quite so.
It would be reasonable to look for the solution which minimises their magnitude. Since there are three, it is not obvious what that means. I suggest finding the maximum possible acceleration.

 

[BvU]

I completely forgot: Hello David,
I'm glad Haru joined us: I felt lonely wanting to help but unable to do so. Goes to show that even at PF we don't know everything

Can you make headway assuming ##
F_{N,1} = 0## ?

 

[haruspex]

I completely forgot: Hello David,
I'm glad Haru joined us: I felt lonely wanting to help but unable to do so. Goes to show that even at PF we don't know everything

Can you make headway assuming ##
F_{N,1} = 0## ?

Not sure that would correspond to max acceleration. Min Σ|F_N| would.
@davidpascu, I forgot to ask... The description does not say whether the plane is horizontal or vertical. The diagram shows mg. is that in the original or did you add it?

 

[davidpascu]

Hi all,

The ring is in the vertical plane.

Actually this is not a fixed problem, but more one case of a more general model I need to do. The issue is this: I have a ring "pivoted" in several points with bearings so that it can rotate around it axis but not translate. The number of bearings is not fixed. The ring is in the vertical plane. The ring´s rotation is driven by a force such as the one in the example.

And I want to create a simple dynamics model that allows me to get the loads on the bearings. The problem I put in the heading is just the particular case of this model with 3 bearings. But I guessed it was easier to post one particular case than the global problem, and that if I managed to solve this one I could extend the solution to the other cases.

Thanks for your help

 

[BvU]

Ah, explains the problem statement. Is it wise then, to use ##F_{\rm friction} = \mu N## ? (We think it makes the problem underdetermined).

 

[davidpascu]

I am nor really sure, it is just an assumption... I guess maybe it is not correct and that is why it does not lead to a solution. Then I would ask on ideas on how to model the problem.

I guess the heart of the problem is how to model correctly the bearings. My first feeling was to put the reaction forces in the radial direction, because the bearings in essence work as sliders (prismatic joint, see picture below), but that leads to inconsistencies also, because if that, how can the reaction counteract the force in every moment to prevent the ring from moving if for example there is only one bearing, or there are two of them situated radially opposite?

Again thanks for your ideas.

PD: I guess maybe with this new formulation of the problem it may not longer belong to this section of the forum. I don´t know if it can be moved to the correct one.

 

[Nidum]

(1) If the three bearings are just in nominal contact with the ring then only two of them are carrying load at anyone time ?

A solution may be possible using a piecewise method where forces are analysed in separate zones as different bearing combinations come into play .

or

(2) Ignore the fact that there are actually three separate bearings and just use a rule relating combined bearing friction drag torque to radial load .

The calculation of friction drag torque for the case of radial load on a dry plain or segmented sleeve bearing / shaft combination is quite simple and may be useful .

 

[Nidum]

What are you trying to do overall with this work ?

Bearing technology is quite well understood out in the real world . If you have any particular area of interest let us know . You may get some useful information back .

 

[davidpascu]

I am involved in the conceptual design of some mechanisms. For my part I need to design this mechanism for rotating a mobile part. The project is still in an early step, so the only things I know for sure are the ring's dimensions and the external force. I have to come up with an idea for a guiding system for the ring's rotation (something such as a set of bearings or some kind of rail-wheel system), and for that I want to first know what forces and momenta I can expect on them.

So that is why I wanted to make a simple model that can allow me to get some quick estimations without having to go to FEM modelling or similar.

I will try taking a look at what I can find on bearing forces and friction. Thanks!

 

[Nidum]

How big is the ring and what sort of speeds and loads are involved ?

 

[davidpascu]

The ring has around 2 meter diameter. It rotates at low speed (below 100 rpm) and the load applied to it is variable, but its always around 200-300 N

 

[Nidum]

Problem with big thin rings is that they distort under their own weight so any support system really has to be multi contact or ideally continuous .

Ball races are manufactured in large diameter small cross section configurations for some types of medical scanners . May provide a ready made solution .

 

[Dr.D]

Regarding the wording of the original question, the word "pivot" suggests a pin joint, a fixed point (a revolute joint to be precise) Everything discussed seems to indicate that you intend either a rolling or sliding support (a prismatic joint). The word pivot can easily misdirect the discussion.