Calculating corrected center distance

[gearcutter]

Would anyone be willing to let me know if below is correct/incorrect? It's based on my understanding of the subject.
Operating P.C.D.s can only be used to calculate center distance for spurs and helicals if the amount of correction given to one member is equally inverse to the amount given to the other, eg. +0.50 and -0.50. If the corrections are not equally inverse to each other then centers can only be calculated from tooth thicknesses based on the difference between the "no longer equal" operating pressure angles.

 

[scarecrow55]

I'd go along with that. But although the operating pressure angle will change if correceted gears are used, I think it must be the same for both gears

 

[EnglishMuffin]

It is incorrect. If you truly know what the operating pitch circle diameters are, then you know the center distance. But to find the operating pitch circle diameters, you must either already know the center distance, or everything about both gears, including addendum shift, and the backlash. Also, the operating pressure angle of the two gears must be the same - but again, the angle depends on data from both gears and the center distance.

 

[Philrock]

Operating pressure angle is always the same for both gears.

If pinion and gear profile shifts (a.k.a. addendum shifts) are equal in magnitude and opposite in sign, operating center distance is the same as for standard gears.

EnglishMuffin is correct. If you know the geometry of both gears, you can calculate the operating center distance. The calculation will require the use of the inverse involute function, which can not be solved explicitly - only iteratively. The center distance calc and the inverse involute algorithm are both in Dudley's Gear Handbook.

Excel add-inns are available that perform the inverse involute algorithm internally, so you don't have to sweat the iteration.

There is a free Excel add-in (trigo_8e.exe / trigono.xla / trigo_8e.zip) at

http://www.emip.de/home_e.htm,

that includes inverse involute and many other very helpful trig functions, like functions that take degrees instead of radians as arguments.

 

[EnglishMuffin]

It might be worth adding that:

1. If the center distance is C, the numbers of teeth on the two gears are Z1 and Z2, and the operating pitch diameters are D1 and D2, then :

D1 = 2*C*Z1/(Z1+Z2)
and D2 = 2*C*Z2/(Z1+Z2)

2. According to ISO international standards, I believe the term "Operating Pitch Diameter" is really a tautology - the term "Reference Diameter" should really be used in the case of individual gears and "Pitch Diameter" used only when the operating data in conjunction with another gear is known. But at least in the U.S., I don't think this convention is followed much.

 

[gearcutter]

Thank you all for the valid comments. My wording suggesting that there was more than one operating pressure angle, you have all correctly pointed out, is incorrect. I should have re-worded the statement.
I’ve carefully read all the comments and no one seems to be picking up on where I’m coming from (this is of course my fault for not properly explaining myself).
If any of you have time, can we use an example? To keep it simple; lets remove the backlash component and use for the example a set of spur gears in hard mesh. The data for the pair is:

Pinion
6 module, 20 deg P.A.
Z = 12
X = +0.50

Gear
Z = 24
X = -0.23

These are arbitrary corrections so just for the example ignore any reasons there may be for not using this combination.
If any one is interested; work out the center distance and post your answers along with the equations you used. Based on most of your comments it looks like some of you are going to be surprised that your answers will probably be wrong.

 

[israelkk]

The center distance without backlash is 109.62132 mm (compared to the 108 mm for gears with no corrections)

 

[Philrock]

I get a center distance of 109.540871 mm. More math than I'm willing to dig up and post here.

Backlash does not enter into center distance / profile shift calculations. If you change the backlash by changing the center distance, you are implicitly also changing the profile shifts. I refer you again to Dudley's Gear Handbook. Other good references on the subject are the Maag Gear Book and John Colbourne's The Geometry of Involute Gears .

gearcutter: in reviewing the posts in this thread I see no evidence of misunderstanding. Please explain what you think we're missing.

 

[EnglishMuffin]

Well, using my old Maag gear handbook, and interpolating from the tables in it, (ie not using a computer), I get 109.731. Israelkk probably used a computer program, so maybe I have screwed up, but the equations should be correct (hopefully).

The number was found from the following equations:

(x1+x2)/zm = (inv(alpha)-inv(alpha'))/tan(alpha)

y/zm = cos(alpha)/cos(alpha') - 1

CD change = y*m

cos(alpha') = (rb1 + rb2)/a'

a' = (d1+d2)/2 + m*y


Where x1, x2 are the addendum modification coefficients
zm = (z1+z2)/2 (mean number of teeth)
alpha = manufactured pressure angle (20 degrees)
alpha' = operating pressure angle
m = module
y = CD modification coefficient
rb1, rb2 - base circle radii
a' = working center distance
d1, d2 = reference circle diameters

The important thing, as I am sure you realize, is that you cannot get the new center distance just by adding the addendum modification coefficients to the original center distance.

 

[israelkk]

EnglishMuffin

Yes, I used a computer program.

Actually, as far as I recall, AGMA 370.01 for "fine pitch gears" allows to get the new center distance just by adding the addendum modification coefficients to the original center distance. It mentions that this will gives slighly incorrect (larger) center distance and a slightly larger backlash.

 

[EnglishMuffin]

israelkk: Yes - that would be near enough I guess, if it's not a critical application, or small teeth and small shifts. But I don't know why we all got different answers. I expect yours is right if you used a program - I'm too busy to track down my error - maybe it came from linearly interpolating the tables in the Maag book (which is the method they intend you to use) although it shouldn't have been that big an error.

 

[israelkk]

I double checked the center distance with another computer program that I wrote some time ago and the results are the same as Philrock's 109.540871 mm. I am sure it is correct because it uses the exact formulations as you presented here. Check your calculations again it should give the same as Philrock's.

I need to check why the commercial computer program gave a little higher center distance. They call it "addendum modification" while our calculation is "profile shift" which is the shift of the theoretical generating rack base line.

 

[gearcutter]

Once again, thankyou all for taking time out on this.
I’ve been cutting gears for years now but have never studied the geometry. In attempting to figure things out I have on several occasions referred back to people who have been in the industry for much longer than I have. Unfortunately most of my questioning has been beyond their scope of understanding or perhaps they’ve just never thought that perhaps some of this stuff is important. So I appreciate the access I have to this forum and the very clever people on it. I hope none of you have been offended with my questioning or statements. I’m using you all to better understand this stuff.
EnglishMuffin; thanks for going to the extra effort of posting the equation you used. I figured asking you to do this would ensure we were talking about the same thing.
Referring back to start of the thread; in essence, what people in my industry generally use to calculate centers are the operating P.C.D.s which are calculated from the P.C.D.s and then plus or minus the addendum modifications (this is also how it’s calculated in several books). No one I spoke to allowed for the change in operating pressure angle. This is what I meant when I stated that operating P.C.D.s (when calculated from profile shift alone) can not be used to calculate centers if the corrections are not equally inverse to each other, as in the example. Then several of you said that this was wrong. In EnglishMuffin’s equation (other than the very last item) I see no reference to the operating P.C.D.s and nor do any of the equations I’ve come across. The only way centers can be calculated is by allowing for the difference between the generated/operating pressure angle and the standard pressure angle, as in the posted equation.
Everyone that came up with 109.54mm is correct. Most people in my industry would have come up with a larger figure. Admittedly the difference, for this example, is a relatively small amount, but with the right conditions, the difference can become quite large.
What’s interesting is that several programs I’ve tried out have come up with the incorrect answer as well. The last one I tried out that seemed to be incorrect was called “GearTrax”. For your information the ones that have been correct so far are “GearCad” and “KISSsoft”. We have GearCad here and find this program to be the best value for money in it’s ease of use and available functions.
Philrock; I’m having trouble understanding your comments about backlash and the effects on a hard mesh center distance. The equation I use for calculating centers uses the tooth thicknesses. In manufacturing it is not uncommon to create backlash with out a profile shift. We do it by “side cutting”. I believe this would have no effect on the addendum length or operating P.C.D.
To end; after wading through several gear geometry books to get to the bottom of this problem (one of them was Faydor L. Litvin’s “Gear Geometry and applied Theory”, the best I’ve been able to find so far) I was pleasantly surprised to come across the correct equations, for calculating corrected centers, in the good old “Machinery’s Handbook”.

 

[Philrock]

israelkk, thanks for confirming my number. I was beginning to worry that we might have to put it to a vote.

Profile shift and addendum modification are the same.

Rack shift is a lot more complicated. The tool (rack or hob) can have any or all of the following built into it:

1. thinning of the teeth of the gear being cut, for backlash;
2. thickening of the teeth of the gear being cut, for grind stock allowance;
3. profile shift of the teeth of the gear being cut.

If a "standard" tool is used (tool tooth thickness = space width), and grind stock allowance is not an issue, rack shift is typically slightly less than profile shift, to thin the teeth for backlash.

 

[gearcutter]

Philrock,
I'm trying to work out why you would need to use inverse of the involute to work out corrected centers. If you have time, could you please explain why?

 

[EnglishMuffin]

Gearcutter :
Although you are addressing this question to Philrock, nevertheless, since Isrealkk says the equations I presented are correct, then assuming he is right, you should be able to see it from them. They could all be combined into one large equation, but as presented, what you have to do is solve the first equation for working pressure angle (alpha'), then substitute the result into the second equation to obtain y, and finally substitute that value into the third equation to obtain the change in center distance. But to solve the first equation, you need to find the inverse involute, which is best done with an iterative procedure such as Newton-Raphson, typically using a computer program, although you can do it by interpolating from tables, which in my case did not quite seem to work for some reason.

 

[EnglishMuffin]

OK - found my mistake - looked in the wrong part of the interpolation table. I get 109.541 which agrees with everyone else.

But I do not agree with this earlier statement of Philrock's

"Backlash does not enter into center distance / profile shift calculations. If you change the backlash by changing the center distance, you are implicitly also changing the profile shifts."

Changing the backlash by changing the center distance does not alter the profile shifts (or addendum shifts - whatever you wish to call them) of the individual gears in any way. The addendum shift of a gear may be determined without any reference to a mating gear whatsoever. If all you know are the details of the gears themselves, you most certainly do need to define the backlash if you wish to determine the correct center distance.

 

[Philrock]

Here’s the math for the center distance / profile shift calculation for spur and helical gears.

Maag Gear Book, p.47:

a = operating center distance

alpha = reference normal pressure angle

alphat = reference transverse pressure angle

alphat’ = operating transverse pressure angle

db1 = pinion base diameter

db2 = gear base diameter

x1 = pinion profile shift

x2 = gear profile shift

z1 = pinion no. of teeth

z2 = gear no. of teeth

eq 74: cos(alphat’) = (db1+db2)/(2*a)

eq 75: x1+x2 = (z1+z2)*[inv(alphat’)-inv(alphat)]/[2*tan(alpha)]

Eq. 74 shows that if the operating center distance changes, then the operating pressure angle changes. This changes the right hand side of eq. 75. The left hand side must, of course, change by the same amount, which means one or both profile shifts change.

Backlash, tooth thinning, and tooth thickness do not appear in the equations.

 

[EnglishMuffin]

Philrock :

Well - I think we are a little at cross purposes here. If you design a pair of gears from scratch, for a known center distance, then the backlash is introduced at the end, being incorporated in the over-pin dimensions or base tangent, or whatever, and is not incorporated in the profile shift as normally defined. However, what I am saying is that if you are presented with a pair of gears, with or without profile shift, either physically or on paper, and are deciding on the center distance, there is no particular reason why you have to use any particular center distance, within reason. If you desire lesser or greater amounts of backlash, you can simply change the center distance to achieve it. So in that sense, you need to define the backlash that you will be using, possibly quite independently of the amount that may or may not be already incorporated in the individual gears.

 

[Philrock]

EnglishMuffin,

I guess I was being a bit of a stickler when stating that changing CD changes profile shifts. While it is technically true, the amount of change in profile shift due to CD changes to adjust backlash will be insignificant in terms of tooth strength, power ratings, etc.