Calculating corrected center distance
Calculating corrected center distance
(OP)
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.
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.





RE: Calculating corrected center distance
RE: Calculating corrected center distance
RE: Calculating corrected center distance
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.
RE: Calculating corrected center distance
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.
RE: Calculating corrected center distance
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.
RE: Calculating corrected center distance
RE: Calculating corrected center distance
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.
RE: Calculating corrected center distance
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.
RE: Calculating corrected center distance
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.
RE: Calculating corrected center distance
RE: Calculating corrected center distance
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.
RE: Calculating corrected center distance
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”.
RE: Calculating corrected center distance
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.
RE: Calculating corrected center distance
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?
RE: Calculating corrected center distance
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.
RE: Calculating corrected center distance
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.
RE: Calculating corrected center distance
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.
RE: Calculating corrected center distance
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.
RE: Calculating corrected center distance
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.
RE: Calculating corrected center distance
Changing the CD does not change the profile shift of either gear if you have already defined them. The profile shift of a gear can be defined and measured quite independently of any center distance you intend to use. In addition, when measuring a gear, you cannot distinguish between the amount of basic rack infeed introduced to provide backlash and the design amount of the profile shift, unless you happen to know what either of them were intended to be in advance. And you might, for example, have a first gear that meshes with one or more other gears. I hope you are not suggesting that whatever you might do with the other gears would alter the design properties of the first gear.
RE: Calculating corrected center distance
Your 07Feb 06 10:34 posting has a simple mistake;
You posted
(x1+x2)/zm = (inv(alpha)-inv(alpha'))/tan(alpha)
I think that needs to be
(x1+x2)/zm = (inv(alpha')-inv(alpha))/tan(alpha)
Otherwise everything looked good.
I've been struggling through this issue on a set of gears and found this posting very helpful - you guys really know your stuff. I've learned a lot.
Where I'm now stuck is applying backlash to a gear set. Most Hobs are only a 2.157/Diametral Pitch deep thus if you make an enlarged gear you still can only cut it so deep. IE the middle of the tooth profile doesn't occur at the theoretical pitch diameter anymore.
This whole exercise (Enlarged pinions/smaller gears) is an effort to lower slip ratios which essentially lowers the sliding velocity at the pinion root. Sliding velocities are bad from what I've recently read (Drago).
RE: Calculating corrected center distance
undercutting as well as strengthen the teeth
and to balance the strengths of the two members.
Center distance changes are a sin function
while the tan function is used to calculate
the change in tooth thickness.
If gears are cut with 20 degree pressure tools,
the pressure line at the theoretical pitch line
remains constant at 20 degrees although the tooth
thickness varies as the cutter is sunk deeper
into the part.
Gears do not have operating pressure angles
until they are assembled together at a defined
center distance.
What makes things even more confusing is the
fact that some use the term corrected pitch
line or corrected pressure angle. The corrected
pitch line is the pitchline of the part where
the tooth thickness equals the tooth space.
Just to add a little fun to this serious topic.