Why Can’t the Number of Gears Be Less Than 17?

Gear is a kind of spare part widely used in life, no matter in aviation, freighter, automobile, etc.

However, there is a requirement for the number of teeth when designing and processing gear.

Some people say that if it is less than 17 teeth, it cannot rotate.

Others say that it is wrong. There are many gears below 17 teeth.

In fact, these statements are correct. Do you know why? Welcome to leave a message for discussion.

Why is the number of teeth 17?

Then why is it 17? Not other numbers?

As for 17, it starts from the gear processing method, as shown in the figure below. A widely used method is to use a hob to cut.

Why Can’t the Number of Gears Be Less Than 17? 1

When gears are manufactured in this way, undercutting will occur when the number of teeth is small, which will affect the strength of the manufactured gears.

What is root cutting? Note the red box in the figure:

Why Can’t the Number of Gears Be Less Than 17? 2

When the intersection point of the tooth top and the meshing line of the gear exceeds the limit meshing point of the gear to be cut, a part of the involute tooth profile of the tooth root of the gear to be cut is cut.

This phenomenon is called undercutting.

Undercutting of tooth profile:

When cutting gears with the generating method, sometimes the cutter will cut a part of the involute tooth profile that has been cut at the root of the gear tooth.

This phenomenon is called undercutting.

Why Can’t the Number of Gears Be Less Than 17? 3

Reasons for undercutting:

When the intersection point of the tool tooth top line and the meshing line exceeds the meshing limit point N1, and the tool continues to move from position II, a part of the involute tooth profile that has been cut at the root will be cut again.

Under what circumstances can undercutting be avoided? The answer is 17 (when the addendum height coefficient is 1 and the pressure angle is 20 degrees).

First of all, the gear can rotate because a good transmission relationship should be formed between the upper gear and the lower gear.

Only when the connection between them is in place, can its operation be stable.

Take involute gears as an example.

Only when two gears mesh well can they play their role.

Specifically, they are divided into straight cylindrical gears and helical cylindrical gears.

For standard spur gears, the coefficient of tooth top height is one, the coefficient of tooth heel height is 1.25, and the degree of its pressure angle must reach 20 degrees.

If the tooth germ and the cutter are like two gears during gear processing.

If the number of teeth of the embryo is less than a certain value, part of the root of the tooth root will be removed, which is called undercutting.

If the undercutting is small, the strength and stability of the gear will be affected.

The 17 mentioned here are for gears. If we don’t talk about the working efficiency of gears, they can work and operate no matter how many teeth.

In addition, 17 is a prime number, that is to say, the number of overlapping times of a gear tooth and other gears is the least under a certain number of cycles, so it will not be at this point for a long time when the force is applied.

Gears are precision instruments. Although errors will occur on each gear, the probability of axle wear is too high for 17.

Therefore, if it is 17, it will be OK for a short time, but not for a long time.

But here comes the problem! There are many gears with less than 17 teeth on the market. They still turn well, see image below!

Why Can’t the Number of Gears Be Less Than 17? 4

Some netizens pointed out that, in fact, if we change the processing method, it is possible to manufacture standard involute gears with a number of teeth less than 17.

Of course, such gears are also easy to get stuck in use (due to gear interference, we can’t find the picture, please fill it up), so they really can’t rotate.

There are also many corresponding solutions, and the modified gear is the most commonly used one (generally speaking, move the cutter a bit when cutting).

In addition, there can be helical gears, cycloidal gears, etc.

There is also the hypocycloid gear.

Another netizen’s point of view: everyone seems to have too much faith in the book. I don’t know how many people have thoroughly studied gears in their work.

In the lesson of Mechanical Principle, the deduction that the number of teeth of involute spur gears is greater than 17 and no undercut occurs is based on the fact that the top fillet R of the rake face of the rack cutter for machining gears is 0, but in fact, how can the cutters in industrial production have no R angle?

(If there is no R angle tool for heat treatment, the sharp part is prone to crack due to stress concentration, and it is easy to wear or crack during use.)

Moreover, even if the tool has no R angle, the maximum number of teeth for undercutting may not be 17 teeth, so the statement that 17 teeth are used as undercutting conditions is still open to discussion!

Let’s have a look at the pictures below.

Why Can’t the Number of Gears Be Less Than 17? 5

It can be seen from the figure that there is no obvious change in the tooth root transition curve from 15 teeth to 18 teeth when the cutter with R angle of 0 at the top of the rake face is used to process the gear.

Then why is 17 teeth the number of teeth with involute straight teeth starting to undercut?

Why Can’t the Number of Gears Be Less Than 17? 6

This picture must have been drawn by the mechanical engineering professional.

It can be seen that the size of the cutter’s R angle affects the gear undercutting.

Why Can’t the Number of Gears Be Less Than 17? 7

The equidistant curve of the purple extended epicycloid of the tooth root in the above figure is the profile of the tooth root after undercutting.

How far does the undercutting of the tooth root of a gear affect its use?

This is determined by the relative movement of the top of the other gear and the strength reserve of the gear root.

If the top of the paired gear does not mesh with the undercutting part, the two gears can rotate normally (Note: the undercutting part is a non involute profile, and the meshing of an involute profile and a non involute profile is usually not conjugate in the case of non special design, that is, interference).

Why Can’t the Number of Gears Be Less Than 17? 8

It can be seen from this figure that the meshing line of the two gears just grasps the maximum diameter circle opposite to the transition curve of the two gears (Note: the purple part is the involute profile, the yellow part is the undercut part, and the meshing line cannot enter below the base circle, because there is no involute below the base circle, and the meshing points of the two gears at any position are on this line), that is, the two gears can just be engaged normally.

Of course, this is not allowed in engineering.

The length of the meshing line is 142.2, and this value/base pitch=coincidence.

Others said that: first of all, this problem is wrong, and the use of gears with less than 17 teeth will not be affected (the description of this point in the first answer is wrong, and the three conditions for correct meshing of gears have nothing to do with the number of teeth).

However, 17 teeth may be inconvenient to process under certain circumstances, and more knowledge about gears is added here.

Involute is the most widely used type of gear tooth profile. Then why is it an involute?

What is the difference between this line and straight line and arc?

As shown in the figure below, it is an involute (here there is only an involute of half a tooth).

Why Can’t the Number of Gears Be Less Than 17? 9

In a word, involute is the track of a straight line and its fixed point when the line rolls along a circle.

Its advantages are obvious.

When two involutes mesh with each other, as shown in the figure below.

Why Can’t the Number of Gears Be Less Than 17? 10

When two wheels rotate, the action direction of the force on the contact point (such as M, M ‘) is always on the same straight line, and this straight line is perpendicular to the contact surface (section) of the two involutes.

Because of the perpendicularity, there will be no “slip” and “friction” between them, which objectively reduces the friction of gear mesh, not only improves efficiency, but also extends the life of the gear.

Of course, as the most widely used form of tooth profile – involute, is not our only choice.

Besides, as engineers, we should not only consider whether the theoretical level is feasible and the effect is good, but also find a way to show theoretical things, which involves material selection, manufacturing, precision, testing and other links.

The commonly used processing methods of gears are generally divided into forming method and generating cutting.

The forming method is to directly cut the tooth profile by manufacturing the cutter corresponding to the gap shape between the teeth, which generally includes milling cutter, butterfly wheel, etc;

Generating cutting is relatively complicated. It can be understood that two gears are engaged, one of which is very hard (tool), and the other is still in the rough state.

The process of engagement is from far away to gradually moving to the normal meshing state.

In this process, new gears are generated by cutting.

If you are interested, you can find the Mechanical Principle for specific study.

Generating cutting is widely used.

However, when the number of gear teeth is small, the intersection of the cutter’s top line and the meshing line will occur, which will exceed the meshing limit point of the gear to be cut.

At this time, the root of the gear to be machined will be cut off.

Since the undercut part exceeds the meshing limit point, it will not affect the normal meshing of the gear, but this disadvantage is that it will weaken the strength of the gear teeth.

When the such gear is used in heavy load occasions such as gearbox, it is easy to break the gear teeth.

As shown in the figure, it is a model (with undercut) of a 2-die 8-tooth gear after normal machining.

Why Can’t the Number of Gears Be Less Than 17? 11

However, 17 is the limited number of teeth calculated under the gear standard in China.

When the number of teeth is less than 17, “undercutting” will occur when the gear is normally processed using the generating cutting.

At this time, it is necessary to adjust the processing method, such as the modification, as shown in the figure, it is a 2-mode 8-tooth gear (small undercutting) processed by the modification.

Why Can’t the Number of Gears Be Less Than 17? 12

Of course, many of the contents described here are not comprehensive.

There are many more interesting parts in machinery, and there are more problems in manufacturing these parts in engineering.

Conclusion: 17 teeth come from the processing method and also depend on the processing method.

If the processing methods of gears are replaced or improved, such as forming method and modification processing (here specially refers to straight cylindrical gears), there will be no undercutting, and there will be no problem of the limit number of 17 teeth.

In addition, from this question and its answer, we can see one of the characteristics of mechanical science – the high combination of theory and practice.

Netizens’ opinion: First of all, it is incorrect to say that a gear cannot rotate if it has less than 17 teeth. Let’s briefly introduce how the number of 17 teeth came into being.

Why Can’t the Number of Gears Be Less Than 17? 13

Gear refers to a mechanical element with gears on the rim to continuously mesh and transfer motion and power.

The gear profile includes involute, circular arc, etc., and involute gears are widely used.

Involute gears are also divided into straight cylindrical gears/helical cylindrical gears, etc.

For standard straight cylindrical gears, the addendum height coefficient is 1, the dedendum height coefficient is 1.25, and the pressure angle is 20 °.

The gear is generally processed by generating cutting, that is, the motion of the cutter and the gear blank is like a pair of meshing gears.

For standard gear processing, if the number of teeth is less than a certain value, a part of the involute profile at the root of the gear blank will be excavated, which is called undercutting.

As shown in the left figure below, undercutting will seriously affect the strength and transmission stability of the gear.

The minimum value without undercutting is 2 * 1/sin (20) ^ 2 (1 is the addendum coefficient, and 20 is the pressure angle).

The 17 teeth here are for standard straight cylindrical gears.

There are many ways to avoid undercutting, such as gear deflection, that is, the tool is far away from or near the wheel blank rotation center.

Here, in order to avoid undercutting, it is necessary to choose to be far away from the contour rotation center.

As shown in the following figure on the right, is the complete involute contour coming out again.

Why Can’t the Number of Gears Be Less Than 17? 14

After the gear is modified, the gear can rotate without being affected. The gear with five teeth can also rotate through proper modification.

In fact, helical gears can also avoid undercutting, or reduce the minimum number of teeth for undercutting.

Why Can’t the Number of Gears Be Less Than 17? 15

This figure is calculated.

It is not to say that a few 17 gears cannot rotate, but if there are less than 17 teeth, it is easy to cut a part of the gear root with the processed split line during gear processing, that is, undercutting, which will weaken the gear strength

As for how to calculate it, it is entirely a mathematical problem.

With reference to the above formula, the kneading angle a=20 degrees is, and the minimum number of teeth without undercutting is 17.

Netizen’s opinion: Whether the number of gears can be less than 17 is a problem worth considering.

For standard gears, the number of teeth cannot be less than 17. Why.

Because when the number of teeth is less than 17, the gear will undercut.

The so-called undercutting refers to that when using the generating method to cut teeth, under certain conditions, the tooth tip of the cutter cuts too much into the root of the gear tooth, and cuts a part of the involute tooth profile of the tooth root.

Generating cutting and Radical Cutting

Generating cutting

The generating cutting is a method of machining gears using the envelope principle of geometry.

Given the involute tooth profile of two gears and the angular velocity w1 of the driving gear, the angular velocity w2 of the driven gear can be obtained by meshing the two tooth profiles, and i12=w1/w2=a fixed value.

Because in the engagement of two tooth profiles, the two pitch circles make pure rolling.

During the pure rolling of pitch circle 1 on pitch circle 2, the tooth profile of gear 1 will occupy a series of relative positions for gear 2, and the envelope of this series of relative positions is the tooth profile of gear 2, that is, when the two pitch circles make pure rolling, the two involute tooth profiles can be regarded as mutually enveloping lines.

Undercutting phenomenon

Reasons for undercutting: 

When the intersection point of the tool tooth top line and the meshing line exceeds the meshing limit point N1, and the tool continues to move from position II, a part of the involute tooth profile that has been cut at the root will be cut again.

Consequences of undercutting:

Gear with serious undercutting, on the one hand, weakens the bending strength of gear teeth;

On the other hand, it will reduce the engagement of gear transmission, which is very unfavorable to transmission.

Reasons for undercutting: 

When the intersection point of the tool tooth top line and the meshing line exceeds the meshing limit point N1, and the tool continues to move from position II, a part of the involute tooth profile that has been cut at the root will be cut again.

For non-standard gears, the number of teeth is less than 17. 

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