Machining Bevel Gears

I found the MEW article "Design and CNC Machining of Straight Tooth Bevel Gears" very interesting, but one statement defeated me.

On page 14 the author states...

"This particular differential has 3 pinions equi-spaced around the gear. It is important that the number of teeth on the gear is divisible by three, with no remainder. Otherwise the three pinions will not sit equally spaced around the gear, each in the same orientation, and will not therefore mate with the second gear wheel."

I agree that the number of gear teeth has to be an integral multiple of three in order for the pinions to have the same orientation. But why do they need to have the same orientation? The second gear will mate whatever the orientation of the pinions.

If you image straightening the gears to make two racks, then it is clear that they will mate with any number of equal sized pinions, whatever their spacing. Or have I missed something?

Gary

Hi Gary,

You are right, but there is still a relationship between the angle of the pinions and the number of teeth. With pinions at 90-degrees, for example, you can use a gear whose number of teeth is a multiple of four (e.g. Fordson Model F differential).

The pinions have to be symmetrically arranged and number of teeth has to be a multiple of the number of pinions. (obviously you don't have to fit every pinion, but with a 360-tooth gears you could fit a pinion in any of 360 degrees around the wheel).

The rack has an 'infinite' number of teeth, which means you can fit an infinite number of pinions...

Neil

I was also a little confused about that statement. so in my own words: -

The pinions are sandwiched between the 2 gears, for both gear teeth to mesh with the pinion teeth the pinion teeth for each pinion will need to be orientated to the same angular orientation and therefore the ratio of teeth between the gears and pinions must be divisible by 3.

If the number of teeth on the gear is not divisible by 3, then the angular orientation of each pinions teeth will be at different angular positions therefore, the second gear cannot mesh with all the pinions.

I think!

Nigel

What you say is true only if it's a requirement that the pinions have the same orientation, but I don't see why that is a requirement. Equally spaced, yes, but why same orientation? All gears are based on rolling cylinders or cones, the teeth are simply to guarantee no slipping. Where does the equal orientation come in? No matter what the orientation of the pinions happen to be, the second gear is bound to mate properly.

The point I was trying to make with the rack is that the two racks will mate with the pinions no matter how they are spaced. Random or equi-spacing has no effect on the mating.

Gary

You're making my brain hurt, Gary! But your infinite rack thought gives me the answer.

I agree with Nigel, but that doesn't tell us WHERE you CAN put pinions.

But there are limits on where you can put pinions along a rack. Once you have a pinion at position 1, you cant have another at, say 3.5, it has to be at 3 or 4. The proof of this is that if you have a pinion between two racks, you can't roll it along without moving the racks. The pinions can only mesh between the racks at points 1-tooth space apart, not just anywhere.

This means the infinity of pinion positions is just like the infinity of integers (1,2,3,4..) not the infinity of real numbers (brain really throbbing now...).

So take a pair of bevel gears with a finite number of teeth, say 12.

There are 12 places where you can fit a pinion (OK the first one can go anywhere, but once it's there it sets the possible locations for the eleven other pinions).

With 32-tooth gears, for, example there are 32 possible relative locations for the pinions, and you could have a slightly assymetrical arrangement of pinions.

<cold shower>

Neil

I think you are missing the point Andrew stated equally spaced and same orientation in the same sentence.

You can't change one without the other.

It also depends on how many teeth in the pinions, many have odd numbers of teeth to equalise wear and if you move one pinion out of it's 120,120,120 orientation it will affect where the tops of the pinions sit on the mating gear.

It's a situation that is also present in epicyclic gears and there is even software that tells you what will fit and what won't.

Humble pie tastes horrible - arghh.

I decided to try it out with a CAD program. The following *LINK* illustrates that randomly spaced pinions cannot be fitted between two racks.

Gary

P)erhaps this might put it in words...

The first pinion locks the tooth relationship between the gears or racks. Any extra pinions have to have the same amount of rotation as the first pinion. If a pinion is at a different rotation -say 1/4 tooth clockwise then the gear or rack needs to be 1/4 tooth around to fit one side of the pinion but the other side is then also 1/4 tooth clockwise whereas the pinion has been rotated so is 1/4 tooth anticlockwise i.e. the pinion cannot mesh both sides when at a diffferent relative position to the first pinion.

Billy.

Gary, you need Gravy with that pie, goes down easier.

There is actually nothing wrong with your theory only your drawing skills

Now if you had your two racks vertically in line and the gears on a vertical plane iy would work out correct.

However using a rack which is a gear with an infinite number of teeth isn't the answer because it can't be divided by 3

Now if we use say two 30 T racks bent round we get a 30 T gear and if each of the three pinions is dropped on every 10 teeth then it all works out correct and the spacing is 120 degrees.

Now if we had 31 teeth then the whole lot goes to pot because we can't have the factor "Equally spaced"

Now if we had a spider spaced at 116.129 degrees, 116.129 degrees and 127.741 degrees this would add up to 360 degrees and space the pinions out at 10 teeth, 10 teeth and 11 teeth.

This answers the second part of the statement but cancels the first "equally Spaced part"

So it's easier to do the equally spaced bit than unequally spaced.

Sorry for the mind numbing maths

Typo changed to "can't be divided by 3"

Edited By John Stevenson on 05/05/2014 00:45:38

It's not the same module or DP.

It reduces from the OD to the ID. I think Andrew covered this quite well in his article. The bit that actually matters is than for any given diameter on the gear what ever DP it is matches it's mating gear.

In theory a bevel with say a pitch of 12 DP on it's outer edge will have a DP at it's centre of an infinite number where it disappears up it's own, well we wont get into that.

This is why bevel gears usually have a 1:3 ratio over the part used for the teeth so they are working with min and max acceptable numbers.

My first reaction when I saw this thread was 'Oh fudge (or words to that effect) I have missed an error'. Being busy yesterday with glider paperwork and inspections for the issue of an ARC (equivalent of the MOT) I then thought I'd let everyone sort it out. Which I think has now been done very satisfactorily. I did do a thought experiment which convinced me that I was correct, although it could possibly have been worded more clearly.

Regards,

Andrew