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I am designing gears in FreeCad, and printing then on a FlashForge Finder. I want to know if the pitch diameter is the same as the distance between the centers of the two gears. Both gears are external spur gears.

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migrated from 3dprinting.stackexchange.com Dec 11 '16 at 18:43

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    $\begingroup$ Hi, and welcome to 3D Printing SE! I find your question is a bit unclear: what are we discussing here? Is there a "pitch diameter" in FreeCad, on your FlashForge Finder, in some theoretical mathematical model of gears, or some CAD model that you are currently designing? When writing your question, plese try to be as specific as possible. If the question is not obvious, you might want to include some sketch, or give and example to make it easier for other users to understand. I have put your question on hold for now, so that you may do your edits. $\endgroup$ – Tormod Haugene Dec 9 '16 at 15:35
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    $\begingroup$ It's not unclear, but doubtful if it's really on topic. Easily answered with a web search. $\endgroup$ – Sean Houlihane Dec 10 '16 at 21:00
  • $\begingroup$ @Greenonline, you seem know this topic. Do you think we should reopen it, or perhaps migrate it to Mechanical Engineering SE? $\endgroup$ – Tormod Haugene Dec 11 '16 at 15:27
  • $\begingroup$ Alright, thanks @Greenonline. :) I will migrate it to the ME site then, since it is not directly relevant to 3D printing. Perhaps you could post your answer there. ;) $\endgroup$ – Tormod Haugene Dec 11 '16 at 18:41
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Pitch diameter is the diameter of the pitch circle described by the mid point of the length of the teeth around the gear, as shown in this diagram:

Pitch diameter

The pitch circle defines the point where the teeth of two gears meet:

The pitch circles of two spur gears

Let's say you have two gears, each with a respective pitch circle diameter of $d_1$ and $d_2$. The distance between the two gear centers, $C$, is given by,

$$C = \frac{d_1 + d_2} { 2 }$$

If the gears are identical ($d_1 = d_2$), then the pitch diameter is, indeed, the same as the distance between the two gear centers, $C$. Otherwise, is it not.

Another method of calculating the gear center distance is,

$$C = \frac{N_1 + N_2}{2P_d}$$

Where $P_d$ is the diametrical pitch, and $N_1$ and $N_2$ are the number of teeth of the respective gears. From Gear design equations and formula


You may find the following useful - from Wikipedia - Standard Pitch Diameter:

The standard reference pitch diameter is the diameter of the standard pitch circle. In spur and helical gears, unless otherwise specified, the standard pitch diameter is related to the number of teeth and the standard transverse pitch. The diameter can be roughly estimated by taking the average of the diameter measuring the tips of the gear teeth and the base of the gear teeth.

The pitch diameter is useful in determining the spacing between gear centers because proper spacing of gears implies tangent pitch circles. The pitch diameters of two gears may be used to calculate the gear ratio in the same way the number of teeth is used.

$$d = \frac{N}{ P_d } = \frac {pN}{\pi}$$

Where $N$ is the total number of teeth, $p$ is the circular pitch, and $P_d$ is the diametrical pitch.

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As indeed answered by Greenonline, the centre distance is the average of the pitch diameters of the two meshing gears, but this is only strictly true when the gears are operating at standard centre distances, i.e. where the pitch circles are tangent to one another. There are indeed cases where two gears can be operating at non-standard centre distances, and therefore the standard pitch circles are no longer tangent to one another.

Sometimes non-standard centre distances are accidental and occur due to difficulty in exactly aligning two gears to have a standard centre distance. Other times it may be a deliberate part of the design. For example, slightly increasing the centre distance above the standard value will implement some backlash (as shown below) which helps to prevent any jamming from occurring.

Illustration of backlash

Here are two gears operating at standard centre distance:

two gears operating at standard centre distance

And operating at non-standard centre distance:

two gears operating at non-standard centre distance

Most gears in industry have teeth that are involute (the notable exception to this is in the clockmaking industry, where teeth are frequently cycloidal instead), and the advantage of involute teeth is that gears can still operate smoothly at non-standard centre distances.

When dealing with non-standard centre distances, we have to be careful by what is meant by pitch circle and pitch diameter. The diameter of the pitch circle, the pitch diameter, is a dimension that belongs to a single gear on its own, regardless of how and with what it mesh. However, the pitch circles for a pair of meshing gears also represent the diameters of two equivalent disks that are rolling without slip; this means the pitch circles should be tangent. For operation at non-standard centre distance, there appears to be a contradiction in the definition of a pitch circle: therefore, the two following terms should be observed:

Standard pitch diameter: this is a dimension on a gear that is independent of how or with what it meshes, and defined for any two meshing gears as follows:

$$d_1 = N_1m = \frac{N_1}{P_D}$$ $$d_2 = N_2m = \frac{N_2}{P_D}$$

Where $N$ is the number of teeth, $m$ is the module (a measure of tooth size in millimetres), $P_D$ is the diametral pitch (a measure of tooth fineness in teeth per inches) and the subscripts 1 and 2 indicate which gear the variable refers to.

This is the dimension that is provided when buying gears "off the shelf". Taking the average of these diameters for both gears will give you the standard centre distance, which may or may not be equal to the actual centre distance.

$$d_1 = \frac{2C}{\frac{N_1}{N_2}+1}$$

$$d_2 = \frac{2C}{\frac{N_2}{N_1}+1}$$

$$C = \frac{d_1+d_2}{2}$$

Where $C$ is the standard operating distance.

Operating Pitch Diameter: this is a dimension that only exists when two gears are meshing together, and it represents the diameters of the equivalent rolling disks. The operating pitch circles are tangent to one another, and taking the average of these will indeed give you the actual centre distance, i.e. operating centre distance. If the centre distance is larger than standard, then the operating pitch circles will be larger than the standard pitch circles. The operating pitch diameters are defined as follows:

$$d'_1 = \frac{2C'}{\frac{N_1}{N_2}+1}$$

$$d'_2 = \frac{2C'}{\frac{N_2}{N_1}+1}$$

And so...

$$C' = \frac{d'_1+d'_2}{2}$$

Where $d'$ is the operating pitch diameter, and $C'$ is the (actual) operating centre distance.

Operating at standard center distance

Operating at non-standard center distance

A final point worth noting: there are limits to how much you can vary the centre distance from standard: there is a minimum centre distance below which the gear teeth will jam with one another, and there is a maximum above which the teeth will no longer reach and make contact with each other:

Two gears beyond maximum limit of non-standard center distance

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    $\begingroup$ +1 - A very interesting and most detailed explanation. I had avoided dealing with the backlash, and had gone as far as editing one of my images to remove the backlash reference/detail. $\endgroup$ – Greenonline Dec 13 '16 at 14:22
  • $\begingroup$ +1 Good stuff and more. Also to note again, depending on the tip relief and other design adjustments the pitch circle will not pass through the centre of the tooth as such. At first glance it is always going to look close but by design it may be placed above or below the centre of the tooth. $\endgroup$ – KalleMP Feb 2 '17 at 9:22
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Yes they should, in theory, and for most practical purposes be the same.

The pitch circle diameter is the effective radius of a gear in terms of torque, although for some tooth profiles this may be a mean value.

In practice gear design often uses the module system which defines gear ratios by number of teeth for a given tooth profile and ensures that you always have an integer number of teeth and eliminates pi, making calculations more convenient.

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