Calculate maximum number of balls for a Conrad ball bearing 2

[wzwqx2]

Does anyone know how to calculate the maximum number of balls that can be assembled into a deep groove ball bearing using a Conrad style assembly?

 

[tbuelna]

wzwqx2,

Harris gives the following formula on page 10 of his book, for the "assembly angle" of a conrad type ball bearing: phi=2(Z-1)D/d where Z=number of balls, D=ball diameter, and d=pitch diameter.

I made a sketch for you describing the approach.

Hope that helps.
Terry

 

[wzwqx2]

Thank you tbuelna.

Does he mention anything about what the maximum phi value can be?

 

[zekeman]

I looked at the geometry and I think that formula is only approximately true. I get

phi=2(Z-1)*inverse sine (D/d)

The approximation is
sine (D/d) approximately D/d

is used when D/d is very small.

I don't understand why any modern text would use it in that context.

 

[dinjin]

phi from the illustration must be 180 or less.
A rough approximation for Z is pi/2 times the mean raceway diameter divided by the ball diameter i.e. 1/2 the number of balls that would be a full complement of balls.

 

[wzwqx2]

dinjin

There are many designs thaat have phi values greater than 180 degrees.

 

[electricpete]

I don't think dinjin was making a comment about bearings but about the illustration and the given formula...small angle approximation.

Even without the small angle approximation it's still somewhat crude. For a complete analysis, along with information about the ball diameter cited one would also need to consider two diameters for each ring: one for the land (which determines how closely together we can push the ring) and one slightly larger diameter for the race (which determines where the balls will sit after inserted).

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[electricpete]

Actually let me back up to see if we can derive the posted expression
The length of the arc which has Z balls is approximately the sum of Z-1 diameters. i.e. Z-ball-arc-length = (Z-1)D
The mean distance around the circumference of bearing is approx Circumference = pi*d.
To the extent the first item is some fraction of the 2nd, the angle is that same fraction of 2*pi.

Phi = [(Z-1)*D / (Pi*d)] * (2*pi) = 2*(Z-1)*D/d

So maybe there is no small angle approximation in the original equation (there are other approximations). But we still only have an angle, not a number of balls.

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[electricpete]

Here is my attempt to find Z (rather than Phi... because I assume there is no easy way to figure out Phi without knowing Z that we're after).

Define Inner Race Diamter is Dir
Define Outer Race Diameter is Dor
Define Dmean = (Dir + Dor)/2

Define ball diameter as Dball
Let's say the radial distance between inner ring land and inner race is OffsetIR
Let's say the radial between outer ring land and outer race is OffsetOR
Define Offset = OffsetIR + OffsetOR

Now when we push the lands together, the center of the circle formed by the inner ring if offset by a distance Offset (defned above) from the center of the circle formed by the outer ring.

(If the bearing was centered, the gap available for ball sitting between between inner race and outer race would be gap = Douter-Dinner. But they are not centered, they are pushed together until the lands contact each other, resulting in a distance Offset between the center of the circles describing the inner ring and outer ring.

The gap in this situation is approximated by:
Gap = (Douter-Dinner) * (1 + Offset/Dmean * cos(theta))
where theta is 0 at the location of the largest gap.
This approximation is used for electric motor airgaps. I forget the exact derivation but I think it may be based on the assumption that Douter and Dinner are close and offset is much smaller than either of them.

NOW, we know Gap as a function of theta.
We want to solve where theta=Dball because that forms the (approximate) limt for which the gap between races can no longer hold a ball.

Solve Gap = (Douter-Dinner) * (1 + Offset/Dmean * cos(theta)) = Dball for max theta that still allows insertion of a ball and we find that:
cos(thetamax) = (Dball/(Douter-Dinner) –1) * Dmean/Doffset
thetamax = arccos{(Dball/(Douter-Dinner) –1) * Dmean/Doffset}

What is relationship to Phi?
Phi = 2*thetamax = 2*arccos{(Dball/(Douter-Dinner) –1) * Dmean/Doffset}


What is Z? I'll use the simple expression since we already have approximations
Solve phi~2(Z-1)Dball/Dmean for Z to get:
Z = Dmean/(2*Dball*Z-1) * Phi
Z = Dmean/(2*Dball*Z-1) * 2*arccos{(Dball/(Douter-Dinner) –1) * Dmean/Doffset}

You might want to double-check that one. I think it's right but I've been known to make a typo or three here and there.

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[electricpete]

Whoops, I have defined offset wrong. I need to account also for difference in diameters. Don't use my formula until I give a revised expression for Offset.

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[electricpete]

The corrected formula for Offset (distance between centers of inner race and outer race when the lands are pushed together) should be:

Offset = ((Douter-Dinner)/2) - OffsetIR - OffsetOR

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[electricpete]

Also there is a factor of 2 error in expression for gap... Everywhere above where it says (Douter-Dinner) or (Dor-Dir) should be replaced by (Dor-Dir)/2
Sorry.

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[electricpete]

There was also an error in solving for Z at the end (corrected in bold below). And I used Dinner and Dir interchangeably so I added that to the definitons. Here is whole thing with all the items identified above corrected

Define Inner Race Diamter is Dir = Dinner
Define Outer Race Diameter is Dor = Douter
Define Dmean = (Dir + Dor)/2

Define ball diameter as Dball
Let's say the radial distance between inner ring land and inner race is OffsetIR
Let's say the radial between outer ring land and outer race is OffsetOR
Define Offset = (Dor – Dir)/2 – OffsetIR – OffsetOR

Now when we push the lands together, the center of the circle formed by the inner ring if offset by a distance Offset (defned above) from the center of the circle formed by the outer ring.

If the bearing was centered, the gap available for ball sitting between between inner race and outer race would be gap = (Dor-Dir)/2. But they are not centered, they are pushed together until the lands contact each other, resulting in a distance Offset between the center of the circles describing the inner ring and outer ring.

The gap in this situation is approximated by:
Gap = (Dor-Dir)/2 * (1 + Offset/Dmean * cos(theta))
where theta is 0 at the location of the largest gap.

NOW, we know Gap as a function of theta.
We want to solve where theta=Dball because that forms the (approximate) limt for which the gap between races can no longer hold a ball.

Solve Gap = (Dor-Dir)/2 * (1 + Offset/Dmean * cos(theta)) = Dball for max theta that still allows insertion of a ball and we find that:
cos(thetamax) = (2*Dball/(Douter-Dinner) –1) * Dmean/Doffset
thetamax = arccos{(2*Dball/(Douter-Dinner) –1) * Dmean/Doffset}

What is relationship to Phi?
Phi = 2*thetamax = 2*arccos{(2*Dball/(Douter-Dinner) –1) * Dmean/Doffset}


What is Z? I'll use the simple expression since we already have approximations
Solve phi~2(Z-1)Dball/Dmean for Z to get:
Z = 1 + Dmean/(2*Dball) * Phi
Z = 1 + Dmean/(2*Dball) * 2*arccos{(2*Dball/(Douter-Dinner) –1) * Dmean/Doffset}

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[electricpete]

No, I still don't have right expression for Gap. Sorry, will try again and maybe post back tomorrow night.

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[tbuelna]

electricpete,

You made a good point about the shoulder diameters limiting how far the inner race can be offset. I must admit that I simply posted what I read in Harris without checking it.

The shoulders (or race groove depths) are normally set by the ball/race contact width. That is you want to make sure that the available race surface is always wide enough to accommodate the full contact width under any loads or deflections.

As a point of reference, here's some data from a standard SBB 50x90x20 conrad bearing:

number of balls- 10
diameter of balls- .500 inch
min outer shoulder dia- 3.33
min inner shoulder dia- 2.16

Terry

 

[electricpete]

I wasn't really complaining. But still we are left with Z as function of Phi which is unknown... we need to compare Dball to the available gap to see how far around we can push the balls which is what led to my discussion.

The correct expression for my Gap should have been:
Gap = Gmean * (1 + cos(theta)*Offset/Gmean)
where Gmean = (Douter-Dinner)/2.

It is still an approximation which depends on some questionable assumptions: Offset, Gmean << Dor, Dir

I have posted attached a sketch and solution which I think is now correct if we accept that questionable assumption. Otherwise you could just plot it to see if it works...

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[dinjin]

Conrad design offsets or lands were designed such that the clearance when the rings are shifted would allow a ball to pass into the opening. The clearance then is equal to 1/2 the ball diameter as a minimum. I think phi equal to pi would give you the right results in the original equation.
I think the illustration is misleading or simply wrong if it relates to the equation. I agree it is a poor equation in the form stated. Nowhere is it stated that phi is in radians rather than angular degrees.

 

[electricpete]

Hi dinjin. Thanks for your comments. I'd like to explore the differences in viewpoint.

Conrad design offsets or lands were designed such that the clearance when the rings are shifted would allow a ball to pass into the opening. The clearance then is equal to 1/2 the ball diameter as a minimum.

Agreed. It is shown in my figure there is enough room to insert a ball at the 6:00 position. Without that the number of balls that can be inserted is 0. With that, we still need to know phi to know how many balls can be inserted.

I think phi equal to pi would give you the right results in the original equation.

On what basis?

I think the illustration is misleading or simply wrong if it relates to the equation.

In what way? To find the number of balls that can be inserted we need to determine phi. Phi is limited by how far around we can insert balls into the offset gap... limiting position in approximatly where gap distance decreases to ball diameter.

Nowhere is it stated that phi is in radians rather than angular degrees

I have derived the equation above (8 Jan 11 14:04) by taking the ratio of the arc of Z balls compared to the circumference and multiplying by 2*pi radians.... resulting in an angle in radians. If I multiplied by 360 degrees then I would have gotten Phi = 360*(Z-1)*D/(d*pi) and Phi would have been in degrees. Do you have some disagreement with that derivation or some alternate derivation that will result in angle in degrees?

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[electricpete]

I think the illustration is misleading or simply wrong if it relates to the equation.

Can you please clarify to which illustration you refer.
Thanks.

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[electricpete]

I would maintain that the illustration posted 6 Jan 11 23:00 related to the diagram posted at the same time, based on the the derivation that I posted 8 Jan 11 14:04 (which gives Phi in radians). However that diagram/equation leaves us no way to determine Phi. I have provided a means to estimate Phi which I believe is correct subject to the limits of some approximations made along the way, particularly the expression for gap.

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