Why must gears have at least 17 teeth? What happens if the number is less than 17?
Gears are widely used components in everyday life, found in aviation, cargo ships, automobiles, and more. However, gears have specific requirements regarding tooth counts during design and manufacturing. Some say gears with fewer than 17 teeth cannot rotate, while others disagree, citing the prevalence of such gears. Actually, both statements are correct. Do you know why? Feel free to leave a comment and discuss.
Why 17 teeth?
Why 17, and not some other number? The reason for 17 stems from gear manufacturing methods. As shown in the Image below, a widely used method involves cutting with a hob.
When the number of teeth is low, undercutting occurs, affecting the strength of the manufactured gear. What is undercutting? It’s when the tooth root is cut away. Note the red box in the Image:
When the intersection of the gear’s tooth tip and the line of action exceeds the limit of engagement of the gear being cut, a portion of the involute tooth profile at the root of the gear being cut is removed. This phenomenon is called undercutting.
So, under what circumstances can undercutting be avoided? The answer is 17 (when the addendum coefficient is 1 and the pressure angle is 20 degrees).
First, a gear can rotate because the upper and lower gears must form a good transmission relationship. Only when the two are properly engaged can its operation be smooth. Take involute gears as an example: the two gears must mesh well for the gears to function. Specifically, there are two types: spur gears and helical gears.
A standard spur gear has an addendum coefficient of 1, a kerf height coefficient of 1.25, and a pressure angle of 20 degrees. During gear machining, the gear blank and the cutting tool act like two gears.
If the number of teeth on the blank is less than a specific value, part of the tooth root will be removed, a process called undercutting. If the undercut is too slight, it will affect the gear’s strength and stability. The 17 mentioned here refers to gears in general. Without considering gear efficiency, the gear will still function and run regardless of the number of teeth.
Furthermore, 17 is a prime number, meaning that a particular tooth of a gear has the fewest overlaps with another gear within a given number of rotations, preventing it from being stressed for extended periods. Gears are precision instruments, and while each gear has some degree of error, 17 teeth significantly increase the likelihood of axle wear. Therefore, a gear with 17 teeth might work temporarily, but not in the long run.
However, here’s the problem! Many gears on the market have fewer than 17 teeth and still rotate perfectly fine—proof is in the pictures!
Some netizens have pointed out that it’s possible to manufacture standard involute gears with fewer than 17 teeth using a different manufacturing method. Of course, such gears are also prone to jamming (due to gear interference, no image available, please imagine), making them truly unable to rotate. There are many solutions; modified gears are the most common (simply put, moving the cutting tool slightly during cutting), and there are also helical gears, cycloidal gears, and even hypercycloidal gears.
Another netizen’s opinion: Everyone still trusts books too much. How many people have thoroughly studied gears in their work? The derivation in the course of mechanical principles that involute spur gears with more than 17 teeth do not experience undercutting is based on the fact that the rake face radius (R) of the gear cutting tool is 0. But in reality, how could cutting tools in industrial production not have a radius? (Without a radius, the sharp parts of the tool are prone to stress concentration and cracking during heat treatment, and are more prone to wear or breakage during use.) Moreover, even if the tool has no radius, the maximum number of teeth at which undercutting occurs may not be 17. Therefore, the claim that 17 teeth is the condition for undercutting is actually debatable! Please take a look at the following pictures.
As shown in the pictures, when machining gears with a tool whose rake face radius is 0, the tooth root transition curve from 15 teeth to 18 teeth changes little. So why is it said that 17 teeth is the number of teeth at which undercutting begins in involute spur gears?
This Image, likely drawn by mechanical engineering students using a gear generating apparatus, illustrates the impact of the cutter’s radius (R-angle) on gear undercut.
In the Image above, the equidistant curves of the purple extended epicycloid at the tooth root represent the tooth profile after undercutting. To what extent will undercutting the gear tooth root affect its usability? This is determined by the relative motion of the other gear’s tooth tip and the strength reserve of the gear’s tooth root. If the mating gear tooth tip does not mesh with the undercut, the two gears can still rotate normally. (Note: The undercut portion is a non-involute tooth profile. In non-specific designs, the meshing of an involute tooth profile and a non-involute tooth profile is usually not conjugate, meaning they will interfere.)
This Image shows that the meshing line of these two gears barely touches the maximum diameter circle corresponding to the transition curves of the two gears (Note: the purple part is the involute tooth profile, the yellow part is the undercut; the meshing line cannot extend below the base circle because there can be no involute below the base circle, and the meshing point of the two gears at any position is on this line). In other words, these two gears can mesh normally, although this is not allowed in engineering. The mesh line length is 142.2, and this value/base pitch equals the contact ratio.
Some people say: First, the premise of this question is incorrect; gears with fewer than 17 teeth will not affect their use (the description in the first answer is wrong; the number of teeth is irrelevant to the three conditions for correct gear meshing). However, 17 teeth can cause machining difficulties in certain specific situations. This is more about supplementing some knowledge about gears.
First, let’s talk about the involute. The involute is the most widely used type of gear tooth profile. So why an involute? What is the difference between this line and a straight line or a circular arc? The Image below shows an involute (only half a tooth’s width).
In short, an involute is an imaginary straight line and a fixed point on it. As the line rolls along a circle, the trajectory of that fixed point is the path it follows. Its advantages are obvious. When two involutes mesh, as shown in the Image below…
When the two gears rotate, the direction of the force at the contact point (e.g., M, M’) remains on the same straight line, and this line is perpendicular to the contact surface (cut surface) of the two involutes. Because of this perpendicularity, there is no “slippage” or “friction” between them, which objectively reduces the frictional force of gear meshing, improving efficiency and extending gear life.
Of course, the involute, as the most widely used tooth profile form, is not our only choice.
Regarding “undercutting,” as engineers, we must not only consider its theoretical feasibility and effectiveness, but also find ways to realize it. This involves material selection, manufacturing, precision, testing, and other aspects. Gear machining methods are generally divided into forming and generating methods. Forming involves directly cutting the tooth shape using a tool that corresponds to the gap shape between the teeth. This typically includes milling cutters and butterfly grinding wheels. Generating is more complex. You can think of it as two gears meshing: one is very hard (the cutting tool), while the other is still rough. The meshing process involves gradually moving from a distance to a normal meshing state, and cutting creates a new gear. Those interested can study this in detail in “Principles of Machinery.”
Gear generating is widely used, but when the gear has a small number of teeth, the intersection of the tool’s tooth tip line and the line of engagement may exceed the gear’s meshing limit. In this case, the gear root will be overcut. Since the overcut portion exceeds the meshing limit, it does not affect the gear’s regular meshing. However, this weakens the teeth. When such gears are used in heavy-duty applications such as gearboxes, tooth breakage is likely. The figure shows a model of a 2-module 8-tooth gear after standard machining (with overcutting).
The 17-tooth count is the limit calculated under Chinese gear standards. Gears with fewer than 17 teeth will experience undercutting when machined using the generating method. In this case, the machining method must be adjusted, such as by modification. The Image shows a modified 2-module 8-tooth gear (slight undercut).
Of course, this description is incomplete. There are many more interesting parts in mechanics, and manufacturing these parts in engineering presents even more challenges. Those interested can follow for more information.
Conclusion: The 17-tooth count depends on the machining method. If the gear-machining method is changed or improved, for example, by forming or modifying (specifically for spur gears), undercutting will not occur, and the 17-tooth limit will not apply.
Furthermore, this question and its answer demonstrate a characteristic of mechanical engineering—a high degree of integration between theory and practice.
Netizen Opinion: First, the statement that gears with fewer than 17 teeth cannot rotate is incorrect. Below, we briefly explain how the number 17 teeth is derived.
Gears are mechanical components that transmit motion and power through the continuous meshing of gears on their rims. Gear tooth profiles include involute and circular arc shapes, with involute gears being more widely used.
Involute gears are further divided into spur gears and helical gears. For a standard spur gear, the addendum coefficient is 1, the dedendum coefficient is 1.25, and the pressure angle is 20°. Gear machining generally uses the generating method, meaning that the motion of the cutting tool and the gear blank during machining is similar to that of a pair of meshing gears. For standard gear machining, if the number of teeth is less than a specific value, a portion of the involute profile at the root of the gear blank will be removed. This is called an undercut, as shown in the left figure below. Undercut will severely affect gear strength and the smoothness of transmission. The minimum value at which undercut does not occur is 2*1/sin(20)^2 (1 is the addendum coefficient, and 20 is the pressure angle).
The 17 teeth mentioned here refer to standard spur gears. There are many ways to avoid undercutting, such as gear modification, which involves moving the cutting tool toward or away from the gear blank’s center of rotation. To avoid undercutting, it’s necessary to choose a position away from the center of rotation, as shown in the right Image below. The complete involute profile reappears.
After gear modification, the gear can rotate without being affected. With appropriate modification, even a 5-tooth gear can rotate.
In fact, helical gears can also avoid undercutting or reduce the minimum number of teeth required to do so.
The number 17 is calculated. It doesn’t mean a gear with fewer than 17 teeth won’t rotate; instead, it means that if there are fewer than 17 teeth, a portion of the gear root may be cut away during machining, resulting in undercutting and weakening the gear’s strength. The calculation method is purely mathematical. Referring to the formula above, with a meshing angle α = 20 degrees, the minimum number of teeth required to avoid undercutting is 17.
A netizen’s opinion: Whether the number of teeth on a gear can be less than 17 is a question worth considering. For standard gears, the number of teeth really shouldn’t be less than 17. Why? Because when the number of teeth is less than 17, undercutting will occur.
Undercutting refers to the phenomenon where, under certain conditions, the cutting tool’s tip cuts too deeply into the root of the tooth during generating gear cutting, thus removing a portion of the involute tooth profile.
Gear Generating Method and Undercutting
Gear Generating Method
The generating method (or generating method) is a method of machining gears using the geometric principle of envelope. Given the involute tooth profiles of two gears and the angular velocity w1 of the driving gear, the angular velocity w2 of the driven gear can be obtained through the meshing of the two tooth profiles, and i12 = w1/w2 = a constant. Because during the meshing of two tooth profiles, the two pitch circles undergo pure rolling. During the pure rolling process of pitch circle 1 on pitch circle 2, the tooth profile of gear 1 occupies a series of relative positions with respect to gear 2. The envelope of this series of relative positions is the tooth profile of gear 2. That is, when the two pitch circles roll without slip, the two involute tooth profiles can be considered each other’s envelopes.
Undercutting Phenomenon
Cause of Undercutting: When the intersection of the cutter’s tooth tip line and the line of action exceeds the meshing limit point N1, as the cutter continues to move from position II, it further cuts away a portion of the involute tooth profile that has already been cut at the root.
Consequences of Undercutting: Gears with severe undercutting, on the one hand, weaken the bending strength of the teeth; on the other hand, they reduce the engagement ratio of the gear transmission, which is very detrimental to the transmission. Cause of Undercutting: When the intersection of the cutter’s tooth tip line and the line of action exceeds the meshing limit point N1, as the cutter continues to move from position II, it further cuts away a portion of the involute tooth profile that has already been cut at the root.
For non-standard gears, fewer than 17 teeth is acceptable.