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Calculate maximum number of balls for a Conrad ball bearing
2

Calculate maximum number of balls for a Conrad ball bearing

Calculate maximum number of balls for a Conrad ball bearing

(OP)
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?

RE: Calculate maximum number of balls for a Conrad ball bearing

2
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

RE: Calculate maximum number of balls for a Conrad ball bearing

(OP)
Thank you tbuelna.

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

RE: Calculate maximum number of balls for a Conrad ball bearing

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.

 

RE: Calculate maximum number of balls for a Conrad ball bearing

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.

RE: Calculate maximum number of balls for a Conrad ball bearing

(OP)
dinjin

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

RE: Calculate maximum number of balls for a Conrad ball bearing

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).   

=====================================
(2B)+(2B)'  ?

RE: Calculate maximum number of balls for a Conrad ball bearing

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.

=====================================
(2B)+(2B)'  ?

RE: Calculate maximum number of balls for a Conrad ball bearing

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.

=====================================
(2B)+(2B)'  ?

RE: Calculate maximum number of balls for a Conrad ball bearing

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.

=====================================
(2B)+(2B)'  ?

RE: Calculate maximum number of balls for a Conrad ball bearing

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

=====================================
(2B)+(2B)'  ?

RE: Calculate maximum number of balls for a Conrad ball bearing

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.  

=====================================
(2B)+(2B)'  ?

RE: Calculate maximum number of balls for a Conrad ball bearing

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

Quote (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}

=====================================
(2B)+(2B)'  ?

RE: Calculate maximum number of balls for a Conrad ball bearing

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

=====================================
(2B)+(2B)'  ?

RE: Calculate maximum number of balls for a Conrad ball bearing

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

RE: Calculate maximum number of balls for a Conrad ball bearing

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...

=====================================
(2B)+(2B)'  ?

RE: Calculate maximum number of balls for a Conrad ball bearing

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.   

RE: Calculate maximum number of balls for a Conrad ball bearing

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

Quote:

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.

Quote:

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

Quote:

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.

Quote:

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?

=====================================
(2B)+(2B)'  ?

RE: Calculate maximum number of balls for a Conrad ball bearing

[quote]I think the illustration is misleading or simply wrong if it relates to the equation.[quote]
Can you please clarify to which illustration you refer.
Thanks.

=====================================
(2B)+(2B)'  ?

RE: Calculate maximum number of balls for a Conrad ball bearing

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.

=====================================
(2B)+(2B)'  ?

RE: Calculate maximum number of balls for a Conrad ball bearing

Clarication in bold:
"I have provided in my attachment 9 Jan 11 0:02 a means to estimate Phi which I believe is correct subject to.."  

=====================================
(2B)+(2B)'  ?

RE: Calculate maximum number of balls for a Conrad ball bearing

Electricpete,
I have T.A. Harris book Rolling Bearing Analysis, 2nd edition.  Page 7 shows a similar illustration that is in
the url above by tbuelna except it has 7 balls instead of 5.

It also shows the equation similar to what tbuelna listed.
The illustration is titled "Diagram illustrating the method of assembly of a Conrad type deep-groove ball bearings."

Phi in this illustration is slightly greater than 180 degrees.  It does not say the phi in the illustration is the same phi in the calculation.

I have conrad designed fill in my old porch glider but it did not have a cage to keep the balls spaced properly but was grease filled and worked fine and quiet for 50 years or more without a cage.  I finally got rid of the glider as it was too expensive to replace the cushions.  

I made up an excel sheet with the basic equation and found that pi seemed to work in the equation for approximately one half the number of balls for a full complement bearing.

The angular pitch of tight balls is 2 times the arcsine of
Ball Diameter divided by the mean Raceway Diameter.  If this angle is Ta, then (Z-1) times Ta will be approximately 180 degrees.

The centerline shift of the eccentric rings is 1/2 the ball diamter.  That might help.

  

RE: Calculate maximum number of balls for a Conrad ball bearing

Working with Dor and Dir as outer and inner race radius, we know Dor = Dir + 2*Dball

If we have inner race centered in outer race, then we have uniform gap Dball all  around (12:00 / 3:00 / 6:00 and 9:00 positions.)

If we temporarily ignore the presence of a land/shoulder on each side of the ring, we could push the inner race upward into full contact with the outer race.  Now the gap between inner ring and outer ring moving around the bearing is:
0 at 12:00 position
1*Dball at 3:00 position
2*Dball at 6:00 position
1*Dball at the 9:00 position.
In this case, the angle Phi would be Pi because the ball can be inserted only to the 3:00 and 9:00 positions.

However, we ignored the presence of a land/shoulder to get there.  To correct that we recognize we can no longer push the inner ring to contact with the outer ring.  They are prevented from contact by the shoulders.  So to adjust the picture we move the inner ring down slightly from previous positon (and we know the manufacturer still gave us enough room to install ball across the shoulder at the 6:00 position).   This should have the effect of slightly increasing the clearance at 3:00 and 6:00 position compared to the previous case, so we have slightly more than 1*Dball gap at the 3:00 and 9:00 positions, and Phi would be slightly larger than Pi.  But I'll admit it might only be a small amount and probably not worth worrying about.   I am happy to take your suggestion to use Pi as close enough.  

Quote:

The centerline shift of the eccentric rings is 1/2 the ball diamter
Where does this come from?
 

=====================================
(2B)+(2B)'  ?

RE: Calculate maximum number of balls for a Conrad ball bearing

The radial distance between the id of the outer ring and the od of inner ring is 1/2 the ball diameter.  To shift the rings together at the extreme point requires that you shift the ring this distance.
I think it does make a differce whether the number of balls are odd or even for a formula to work.  
For an odd number of balls you can enter 1/2 of the (Z-1) balls to each side and still have room for the last ball.  However when an even number of balls are required, you may not be able to get the last ball in at that point.

RE: Calculate maximum number of balls for a Conrad ball bearing

Quote:

The radial distance between the id of the outer ring and the od of inner ring is 1/2 the ball diameter.
Why not 1x ball diameter?

=====================================
(2B)+(2B)'  ?

RE: Calculate maximum number of balls for a Conrad ball bearing

I would say the amount we shift is given by:
Offset = (Dor-Dir)/2 - OffsetOR - OffsetIR

Also Dor-Dir = 2*BallDiameter, so we could rewrite this as:
Offset = BallDiameter - OffsetOR - OffsetIR

This number will end up somewhere less than BallDiameter, but I see no reason to suspect exactly half BallDiameter unless it is just a general approximate thumbrule.

=====================================
(2B)+(2B)'  ?

RE: Calculate maximum number of balls for a Conrad ball bearing

I layed out the 20 and 10 inch diameter raceways in acad and conclude that the 32 and 16 one inch balls were ok.
I was surprised at the phi values. I assumed that the farmost position of any ball centerline would be at the equivalence of 1/4 the ball dia. to the right of the bearing centerline when the inner race was shifted 1/2 the ball diameter to the right of the bearing centerline.

phi      brg  Z         
max.     dia.
182.8651  20  32
185.7320  10  16    
186.3695   9  
187.1666   8  
188.1921   7  
189.5604   6  
191.4783   5      
194.3615   4
199.1881   3
 
From this I concluded that phi max. was equal to 180
plus 2 times the (arcsin (D/2 divided by d) times 180/pi)
 
 

RE: Calculate maximum number of balls for a Conrad ball bearing

Electricpete,
Tbuelna's sketch should be turned 90 degrees to have a better understanding of how the balls are loaded in a Conrad bearing. You put the first ball in the top where the max lands are the greatest distance apart and shove the ball either to the left and then the next one to the right and they fall down into the bearing raceway.  You keep inserting balls in the same manner until no more will go in the top.  The first and second balls fall to the max possible position in the raceway before the rings are shifted to their final position.  It is this max angle of phi that I was calculating in my analysis.  I was surprised to find out that phi keeps increasing as the mean raceway gets smaller.  When I was designing Conrad type bearings, I almost always used mean raceway diameter having a ratio
to the ball diameter above 16 to 1.  I used the formula
of 1/2 the balls in a full complement bearing and used a plastic snap in cage design to keep the balls spaced almost evenly around the bearing raceway.  We have not heard back from    wzwqx2 (Automotive)  so he may not understand what we were going thru as to analysis and only wanted a final answer or final equation.  If max phi is known, you could divide phi by the tight ball pitch angle and it would give you the nuber of (Z-1) balls.  Adding 1 to that number would be the max balls that could go into the raceways.  In some cases, it is not possible to add 1 and still be able to get that last ball in at the gap.  The formula was true except for one of the cases where the next to the last ball would not allow the last ball to be entered at the gap.     

RE: Calculate maximum number of balls for a Conrad ball bearing

dinjin,

As you point out, the max number of balls that can be loaded between the offset races must take into account the fact that the races must eventually be shifted back to concentric positions with the complement of balls within the race grooves.  So the max phi angle must allow for some minimum clearance between the balls and races.

I made another sketch to illustrate the condition.

Terry

RE: Calculate maximum number of balls for a Conrad ball bearing

In the sketch linked to my previous post, please pretend that the phi angle is centered on the outer race. blush

Terry

RE: Calculate maximum number of balls for a Conrad ball bearing

tbuelna,
Your first illustration was actually more typical of a conrad type bearing.  The second one is typical of what I was talking about in that you could not assemble the last ball in the gap between the lands and have it move to its extreme position.  Although you can draw it there, you cannot put it in the assembly. I wish I knew how to post an acad drawing here or post it to another site so it could be opened or shared.  To maximize the axial load of the bearing you want to minimize the id of the outer ring and maximize the od of the inner ring.  This allows you to have a greater contact angle and not edge load the raceways.  If it is strictly a radial bearing, it is not that critical.
If you subtract the od of the inner ring from the id of the outer ring it should be not less than the ball diameter.  Your first illustration had these in the right proportions.
When the rings are shifted apart, the intersect of outer race nominal ball path and the inner race nominal ball path determines ball centerline at the max phi centerline.  It is hard to define this without showing you a correct illustration.  Thanks for posting the first illustration.

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