# Does gear diametral pitch have to be an integer? (teeth/diameter)

I'm doing a gear design project, and I'm using a multivariate mixed integer non-linear optimization suite known as Couenne distributed by COIN-OR. Within Python, I use this solver to solve for all aspects of the gears in a 2 ratio reverted gear-train given 2 target gear train ratios (4.3 fwd, 9.1 rev). My classmates are telling me that the diametral pitch for each gear pair must be an integer value, or a value with a well known fraction in decimal form. Is this true? I've found that placing this constraint makes it much harder to approach the 2 target ratios. Perhaps it was a consideration about manufacturing, because It would be exceedingly hard to machine a diameter that is an irrational number. However, couldn't the same be said for any diameter? Manufacturing shouldn't depend on the dimension because for any given manufacturing precision we will never know what is after the decimal place. For both irrational and integer diametral pitches, the manufacturing error would be the same. So why would gear diametral pitches have to be integer values?

Without this constraint, I'm able to achieve a total gear train ratio error of 1.3*10^-8 percent from the target ratios.

EDIT: when I refer to "diametral pitch" I am referring to the gear module, number of teeth per inch

• If not a duplicate, strongly related: engineering.stackexchange.com/q/7080/16 – user16 Apr 29 '17 at 18:50
• Almost! But this a design of a meshed gear system. I could remove teeth from the gears as suggested, but this would increase the wear on the teeth and decrease the reliability factor of the gearbox. I'm more concerned about the diametral pitch, ie. # teeth divided by pitch diameter. – denbjornen505 Apr 29 '17 at 19:03
• well now if the "teeth per inch" must be an integer what would that mean to folks working in metric? – agentp Apr 29 '17 at 20:53
• They manufacture according to 0.1 and 0.2s – denbjornen505 Apr 29 '17 at 20:55

## 2 Answers

That's why it's a STANDARD.

Using reasonably available tools you can measure part X, round the results to nearest standard-allowed values, and pick a matching part from catalogue or manufacture following a simple set of standard-defined parameters, and it will fit. With weirdo sizes you have a weirdo system where every element needs to be custom-calculated and custom-made because it fits nothing in the world except what it was made for.

It's an arbitrary restriction to curb anarchy of a billion custom-purpose standards that don't match each other, allow various manufacturers to provide standarized parts that match each other and fulfill all reasonable expectations.

So, the diametral pitch doesn't have to be integer. Engineers will hate you for it, and people will criticize you for vendor lock-in practices, plus you'll need to manufacture all the gears on your own, but there's no law (legal, or of physics) that forbids it.

• As long as there are an integer number of teeth equally spaced on the gear then it should work... – Solar Mike May 31 '17 at 13:43
• @SolarMike: The gears will not mesh well if they are spaced differently on two meshing cogs - there will be at least slippage, increased wear and load, at worst they will just lock or break. The standard doesn't define maximum number of teeth per inch, but reason does - too small teeth on too big gear will cause skipping and transfer torque poorly. By restricting the number per inch to an integer it restricts the number of possible spacings, creating a finite set of meshing pitches without restricting diameters of the gear. – SF. May 31 '17 at 14:12

Obviously gears need an integer number of teeth and spur gears need to have the same tooth profile for all meshing gears and this does impose some practical constraints on what ratios can be achieved.

Many gears and most 'off the shelf' gears are designed with the module system. which defines a ratio between tooth pitch and pitch diameter which also eliminates pi from the calculation.

• I'm concerned about the diametral pitch being a non-integer: number of teeth divided by diameter. Can this module, as you've said, be irrational? – denbjornen505 Apr 29 '17 at 19:03
• "Non-integer" is not the same as "irrational". agentp already answered this in an earlier comment. (I have no idea what your reply about "according to 0.1 and 0.2s" meant.) Theoretically you could design gears with irrational diameters, but in real life you can't measure any irrational number exactly, so it's not a question about practical engineering. – alephzero Apr 29 '17 at 21:14
• alephzero - I'm simply wondering why standards are placed on the module, like the integer constraint. My reasoning is such: if we have a machine that can create parts within 4 decimal accuracy, a diametral pitch of 2 could actually look like 2.0000469273 because we do not know how accurate we are after the 4th place. With this reasoning, it would be similar to machine a part with PI as a diametral pitch, however we would have 3.1415 (measured) and could possibly have (in reality) 3.14150000002. I don't understand, thus, how irrational numbers (or long decimals) are not practically feasible. – denbjornen505 Apr 30 '17 at 0:30
• "by 0.1s and 0.2s" - is what metric gear diametral pitches are standardized by according to my textbook - Fundamentals of Machine Component Design. Similarly how the english system standardizes to integers – denbjornen505 Apr 30 '17 at 0:36