Undercut

Undercut due to manufacturing process

The animation below shows schematically the manufacturing process of three gears with different numbers of teeth by hobbing. It can be seen that if the number of teeth is too small, the hob obviously undercuts the tooth root. This is due to the fact that in the case of small gears, the cutting edges of the hob cutter engage relatively far into the gear (in the case of the red gear, up to about half the radius). This causes the tooth to be very strongly undercut during the rotation of the gear.

Therefore, undercuts must always be avoided, i.e. the number of teeth must not fall below a minimum.

Undercut occurs when the number of teeth of a gear is too small. An undercut leads to a weakening of the strength of the tooth!

Undercut due to meshing

An undercut during gear cutting occurs not only with hobbing but with shaping or planing as well. Although an undercut could be avoided by other manufacturing processes such as form cutting or broaching, the undercut is also absolutely necessary for functional reasons. If an undercut were not present with small gears, the teeth would interfere! As the animation below shows, the teeth of the red gear must be undercut by the teeth of the green gear for meshing.

An undercut not only weakens the respective tooth but also shortens the line of contact. The undercut cuts off part of the involute tooth flank. The tooth flanks thus lose contact with each other (already at point E) well before the actual end of engagement (point E’). The enlarged figure shows that the flank contact after point E is already no longer present. The line of action is shortened accordingly.

An undercut leads not only to a weakening of the tooth but also to a shortening of the line of contact!

Minimum number of teeth to avoid undercut

To avoid an undercut, the gear must have a minimum number of teeth. The animation below shows the reference profile of the hob as it meshes with a gear with 6 teeth. This situation can be looked at analog to the meshing of a driving rack with a gear (the basics are explained in detail in the chapter on racks). The line of action results as a tangent to the base circle and runs perpendicular to the flank of the reference profile. The meshing begins at the point of intersection A between the line of action and the tip circle of the gear and ends at the point of intersection E between the line of action and the tip line of the reference profile (the shortening of the line of contact by the undercut is not taken into account in the figure).

As the animation shows, the tooth is undercut from the point B. This corresponds to the point from which the corner of the reference profile moves over the radial line of the gear, thus undercutting the tooth. Between the beginning of undercutting in point B and the end of meshing in point E, the tooth is undercut within the green area.

Note: The radial line corresponds to the tangent to the tooth flank at the the base circle. In the case of an undercut, however, part of the involute tooth flank is cut off, leaving a small “gap” between the radial line and the actual tooth flank.

The point B at which an undercut occurs generally corresponds to the point of contact between the base circle and the line of action. At this point, the flank of the reference profile coincides with the radial line of the gear. Beyond this point, the reference profile will then cross the radial line and undercut the tooth.

An undercut occurs at the point where the base circle touches the contact line!

In comparison to the above example, the animation below shows the meshing of the reference profile with a gear with 20 teeth. The point B at which an undercut theoretically occurs is outside the line of contact AE. The profile corner is therefore already out of mesh before it could have undercut the tooth. The teeth of the gear are therefore not undercut. An undercut will always occur if the contact point B of the base circle and the line of action lies within the line of contact AE.

An undercut always occurs when the base circle touches the line of action within the line of contact!

For the limiting case in which the teeth of a gear are not yet undercut, the beginning of the undercutting in point B coincides with the end of engagement in point E. As the animation below shows, this is the case for a gear with 17 teeth.

No undercut occurs for gears with a number of teeth above 17!

Calculation of the minimum number of teeth

The minimum number of 17 teeth mentioned in the previous section is independent of the module (or diametral pitch) and thus applies to all tooth sizes! This will be shown mathematically in the following. For this purpose, the geometric conditions resulting in the limiting case are examined more closely, i.e. if the points B and E coincide theoretically exactly.

The distance between the center line and the tip line of the reference profile generally corresponds to the module m and the inclination of the flanks to the standard pressure angle α₀ (see also the chapter on gear cutting). If the orange triangle shown in the figure below is considered, it can be seen that the opposite side of the standard pressure angle α₀ corresponds to the module m of the gear. Thus, the following relationship applies to the distance CB.

\begin{align}
\label{1}
& \overline{CB} =\frac{m}{\sin(\alpha_0)} \\[5px]
\end{align}

The distance CB can also be determined by the pitch circle radius r₀ or the pitch circle diameter d₀ (see yellow triangle). The pitch circle diameter d₀ results from the product of module m and (minimum) number of teeth z_min (see also the article Construction and design of involute gears):

\begin{align}
\label{2}
& \overline{CB} = r_0 \cdot sin(\alpha_0) = \frac{d_0}{2} \cdot \sin(\alpha_0) = \frac{m \cdot z_{min}}{2} \cdot \sin(\alpha_0)  \\[5px]
\end{align}

The two equations (\ref{1}) and (\ref{2}) can now be equated and solved for the minimum number of teeth z_min:

\begin{align}
&\overline{CB} = \overline{CB} \\[5px]
&\frac{m}{\sin(\alpha_0)} = \frac{m \cdot z_{min}}{2} \cdot \sin(\alpha_0) \\[5px]
&\boxed{z_{min} = \frac{2}{\sin^2(\alpha_0)} } \\[5px]
\end{align}

For a standard pressure angle of α₀ = 20 °, a theoretical minimum number of teeth of z_min = 17 results. In practice, however, a minimum number of teeth of 14 is assumed, at which an undercut then actually has a negative effect.

The theoretical minimum number of teeth above which no undercut occurs is 17 for a standard pressure angle of 20°. In practice, a minimum number of teeth of 14 is usually assumed!

In fact, however, it is also possible to produce gears below the minimum number of 17 teeth, without an undercut! For this, the manufacturing process must be specially adapted with a so-called profile shift. Such profile shifted gears will be discussed in more detail in the next article.

Introduction

The fundamentals when two gears are meshing have already been explained in detail in the article Engaging of involute gears (meshing). In this article a special “gear” will be examined: the rack. The animation below shows the engagement of a driving gear with a driven rack.

Line of contact

A rack is basically a special case of a gear with an infinitely large diameter. The curved flanks finally change into straight flanks. The flanks are just inclined by the amount of the standard pressure angle α₀ against the vertical. The line of action is also inclined by this angle α₀ (= operating pressure angle), since the line of action is perpendicular to the rack flanks. As usual, the line of action is a tangent to the base circle of the gear and is thus clearly defined.

The line of contact begins analogous to the pairing of two gears at the intersection A between the line of action and the addendum circle of the driven gear, which in the case of a rack ist an addendum line. Likewise, the end of the line of contact E is again defined by the intersection between the line of action and the addendum circle of the driving gear.

Pitch point

Analogous to two gears, the pitch point C is located at the intersection between the line of action and the center line. Since such a center line always points in the radial direction of a gear and is thus always perpendicular to the “surface” of the gear, in the case of a rack it is thus perpendicular to the racks “surface”. Thus the pitch point C is also cleary defined.

With backlash-free meshing, the rack sits firmly with both flanks on the tooth flanks of the gear. Shifting the rack in radial direction does not change the position of the line of action, which is always the flank normal of the rack and the tangent of the gear’s base circle. Neither the base circle nor the flank angle change in this case, so that the line of action remains identical. This also does not result in any changes in the position of the pitch point, since the center line is also retained! The only effect of a distance change is on the length of the line of contact, which is shortened by a radial displacement of the rack away from the gear.

The position of the pitch point always remains constant with racks! The line of contact is shortened if the distance between rack and gear is increased!

The unchangeable position of the pitch point is also immediately apparent from its definition. The pitch point is defined as the point at which the circumferential speeds of both “gears” are identical. In the case of the rack, however, it is a translatory motion. Each point on the rack thus always moves at the same speed, regardless of the distance to the gear. This also does not change the pitch circle on the gear whose points move at the same speed as the rack. The pitch circle remains identical and so does the pitch point!

Definition of the standard reference pitch circle

As explained in the section before, the pitch point is independent of the distance between gear and rack! The pitch line of the rack and the pitch circle of the gear (both of which run through the pitch point) do not change either! This fact plays a special role especially when manufacturing gears with the help of rack-shaped tools (hob cutters). This means that even in the case of so-called profile shifts (which are equivalent to the radial displacement of the rack), identical pitch circles always occur on a gear. These are then also referred to as manufacturing pitch circles.

In contrast to a gear, rack-shaped tools (or racks) have identical tooth spacings p₀ at every point. In the case of gear, however, these tooth spacings depend on the circumference of the circle under consideration. With increasing diameter, which is considered, the teeth are also distributed on an ever larger circumference and the tooth distance consequently increases. Only where the teeth of the rack-shaped tool (or rack) “roll” directly on the gear are identical tooth spacings found (circumferential pitches or circular pitches p₀). By definition, this corresponds to the manufacturing pitch circles.

For this reason, the manufacturing pitch circles are also called standard reference pitch circles, since the circumferential pitches p₀ are identical for all gears manufactured with the same rack-shaped tool. The standard reference pitch circle is therefore a fixed (unchangeable) parameter of a gear, which is determined solely by the rack-shaped cutting tool! Therefore, a rack can actually be used as a definition of the standard reference pitch circle of a gear:

The standard reference pitch circle of a gear corresponds to the manufacturing pitch circle when manufacturing the gear with a rack-shaped tool! Or in case of an actual rack: The standard reference pitch circle of a gear corresponds to the operating pitch circle when meshing with a rack!

The always identical circular pitches on the manufacturing pitch circles make it possible not only to pair “normal” gears with profile-shifted gears, but also to combine gears of any size, i.e. any number of teeth, if these were manufactured with the same rack-shaped tool!

Line of action

Due to the special design of the tooth shape of an involute gear (rolling a straight line on a circle), the intersection of two involutes rolling off each other describes a straight line. This situation occurs when two gears are meshing. The involute tooth flanks then slide along a straight line (black line in the animation below). This straight line is also referred to as the line of action or line of engagement. This line of action corresponds in principle to the rolling line for the construction of the involute tooth flanks.

The line of action corresponds to the tangent applied on the base circles of the gears.

The distance actually covered on the line of action is then called line of contact (red line in the animation above). The line of contact begins at the intersection A between the line of action and the addendum circle of the driven gear and ends at the intersection E between the line of action and the addendum circle of the driving gear.

Pressure angle

The so-called pressure angle α_b refers to the angle between the normal of the line of centers and the line of action. For a standard gear, this pressure angle is set to 20° with backlash-free pairing of the teeth. In this standard state, the pressure angle is also referred to as the standard pressure angle α₀ (=20°). In such a case, the center distance is called standard center distance a₀ and the resulting operating pitch circles are called standard (reference) pitch circles.

This standardization of the pressure angle α₀ is particularly important for the tool geometry in gear manufacturing, since the flank angle of the rack-shaped cutting tool (hob) depends on it. For example, a standard pressure angle of 20° means that the flanks of the hob must also be inclined by 20° to produce the tooth flanks. The tooth shape is thus decisively determined by the standard pressure angle. For more information see article Geometry of involute gears.

The standard pressure angle will be automatically obtained for non-profile shifted gears during backlash-free pairing. However, if the center distance changes or gears with profile shifts are used, the pressure angle will change and then differ from the standard pressure angle. The pressure angle actually resulting during operation is called the operating pressure angle α_b (the index b is used because the line of action to which the operating pressure angle refers results as a tangent to the base circles of the gears).

Changing the pressure angle is ultimately associated with a direct change in the line of action and thus in the line of contact. If the center distance differs from the standard center distance, then there is no backlash-free pairing of the gears and the line of contact is shortened (see dark blue gear in the animation below).

Base pitch and contact ratio

In order to ensure continuous power transmission between the flanks of two gears, it is important to ensure that at least one pair of teeth is always meshing with each other on the line of contact. This is not always the case if the center distances are, for example, too large and the line of contact ist thus shortened (see animation above)!

Ideally, the second pair of teeth already engages as long as the first pair of teeth has not yet left the line of contact, i.e. there are even two or more pairs of teeth in contact at the same time. Accordingly, the circumferential forces are distributed over several teeth, which means a reduction of the individual tooth loads. This reduces the risk of tooth breakage.

The distance between two adjacent contact points B₁ and B₂ on the line of contact l is called the base pitch p_b.

The base pitch p_b is the distance between two flanks on the line of contact.

For continuous power transmission, the base pitch must therefore always be smaller than the line of contact (p_b < l). The ratio of the line of contact and the base pitch is also called the contact ratio ε:

\begin{align}
&\boxed{\epsilon = \frac{l}{p_b}} \ge 1 \\[5px]
\end{align}

The contact ratio must therefore always be greater than 1; usually in the range of 1.2 for spur gears. With a large number of teeth and a small module, the contact ratio becomes particularly large, resulting in low noise emissions!

The contact ratio indicates how many teeth are in mesh simultaneously on the line of contact. The greater the contact ratio, the greater the forces that can be transmitted and the lower the noise level!

Influence of the center distance on the base pitch and the contact ratio

Due to the special construction of the involute for the shape of the tooth flanks, these are always perpendicular to the line of action. One can simply imagine the line of action as the rolling line with which the involute shape is constructed. Then it becomes immediately clear that the line of action is always perpendicular to the involute and thus perpendicular to the tooth flank.

In general, the base pitch corresponds to the perpendicular distance between two adjacent tooth flanks of a gear or the perpendicular distance between adjacent involutes!

The base pitch corresponds to the perpendicular distance between two adjacent tooth flanks of a gear!

Since the distance between the tooth flanks and thus the base pitch is a constant parameter of a gear, the base pitch does not change even if the center distance changes. Only the line of contact is shortened when the center distance is increased, so that the contact ratio is reduced. If the center distance changes too much, the contact ratio can be less than 1 and the tooth flanks can therefore partially lose their contact with each other.

The base pitch does not change when the center distance is changed, only the line of contact and thus the contact ratio is shortened when the center distance is increased.

The term “base pitch”

Of course, there is a reason why the distance between the flanks on the line of contact is called the base pitch. Because of the special design of the involute (rolling off the line of action (as the rolling line) on the base circle), this base pitch ultimately corresponds to the pitch of the teeth on the base circle.

The base pitch corresponds to the arc-shaped distance between two adjacent flanks on the base circle!

Influence of the center distance on the transmission ratio

When two gears mesh, the direction of the force corresponds to the normal at the contact point of two tooth flanks. In the case of involute gears, this corresponds exactly to the line of action. No matter how the line of contact changes with a change of the center distance, the force therefore always remains tangential to the base circle and is therefore always perpendicular to the base circle radius.

The base circle itself does not change for gears, even with changes in center distance, since the base circle determines the shape of the involute tooth flank. Conversely, this means that the base circle is clearly predetermined by the shape of the tooth. Since the tooth shape does not change when the center distance changes, the base circle or the base circle radius always remains constant.

Since the force is always perpendicular to the base circle radius even when the center distance is changed, the torque does not change. This also means that the transmission ratio of involute gears is independent of the center distance (for example, this is not the case with cycloidal gears). For this reason – and because of the relatively simple manufacturing process – involute gears are mainly used in mechanical engineering. This statement must of course be limited if the center distance becomes so large that the flanks lose contact with each other.

The transmission ratio of involute gears is independent of the center distance!

The constant transmission ratio applies not only to the torques but also to the speeds. This means that if the center distance changes, there is no change in the speed at the output wheel! Because the mechanical power results from the product of rotational speed (angular velocity) and torque. If the torque does not change when the supplied power is constant, then the speed must also remain constant. Otherwise, this would contradict the principle of energy conservation.

Pitch point and operating pitch circle

The tooth flanks of involute gears generally slide onto each other during meshing, only at the so-called pitch point C there is no sliding but pure rolling. This means that the circumferential speeds of both gears are identical at this pitch point. One can imagine the gears at this point as pitch cylinders that roll onto each other.

The corresponding diameters are then called operating pitch diameters. Before and after the pitch point, relative motions take place between the tooth flanks. These sliding motions are also the reason why gears generally have to be lubricated in order to minimize wear on the flanks.

The tooth flanks of the gears generally slide onto each other; pure rolling occurs only at the pitch point! The corresponding operating pitch diameters indicate the diameters of imaginary (pitch) cylinders that roll onto each other without sliding!

The pitch point does not lie at the center of the line of action, as is often claimed, but is located at the intersection between the line of action and the line of centers.

In contrast to the standard (reference) pitch diameter, the operating pitch diameter is not a constant parameter of a gear, but depends on the center distance. If the center distance is changed, the inclination of the line of action (i.e. the pressure angle) and thus the position of the pitch point also changes. This also changes the operating pitch circles. The operating pitch diameters increase with increasing center distance.

The operating pitch circles of two meshing gears depend on the center distance: The larger the center distance, the larger the operating pitch diameter!

Calculation of the operating pitch diameter

As already explained in detail in the article Construction and design of involute gears, the standard reference pitch circles of gears are specially defined circles to which the circular pitch is related. The standard pitch circles are only identical with the operating pitch circles if the gears are paired backlash-free and the operating pressure angle α_b thus corresponds to the standard pressure angle α₀.

If, however, the pressure angle changes due to a change in center distance, the operating pitch circles differ from the standard pitch circles. The operating pitch circles can be determined using the operating pressure angle α_b, as shown below.

As already explained in the previous section, the base circles do not change even if the center distance or the pressure angle is changed. Thus the same base circle radius r_b applies to the standard pressure angle α₀ with the corresponding standard pitch circle radius r₀ as well as to any operating pressure angle α_b with the corresponding operating pitch circle radius r.

Both radii and diameters are related by the cosine of the pressure angle (see yellow and blue triangles in the figure above). In this way, the standard pitch diameter d₀ can be used to determine the operating pitch diameter d by means of the operating pressure angle α_b as follows:

\begin{align}
&r_b(\alpha_b)=r_b(\alpha_0)  \\[5px]
&r \cdot \cos(\alpha_b) = r_0 \cdot \cos(\alpha_0)  \\[5px]
&\tfrac{d}{2} \cdot \cos(\alpha_b) = \tfrac{d_0}{2} \cdot \cos(\alpha_0)  \\[5px]
&\boxed{d = d_0 \cdot \frac{\cos(\alpha_0)}{\cos(\alpha_b)}} ~~~\text{with } \alpha_0 = 20° ~\text{for a standard gear} \\[5px]
\end{align}

Note that the standard pitch diameter d₀ and the standard pressure angle α₀ are fixed parameters of a gear and do not change during operation! In a nutshell, the standard pressure angle describes the shape of a tooth flank and the standard pitch diameter the size of the gear. Since these parameters of the gear are known in advance, the corresponding operating pitch diameter d can be determined directly using the operating pressure angle α_b.

At this point it becomes clear once again that the operating pitch circle obviously only corresponds to the standard pitch circle if the operating pressure angle is equal to the standard pressure angle (α_b=α₀). In many cases, however, this is not the case in practice, so that the operating pitch circle differs from the standard pitch circle. This applies in particular to corrected gears (gears with a profile shift).

Fundamental law of gearing

It has already been explained in the section “Influence of the center distance on the transmission ratio” that the transmission ratio of involute gears is independent of the center distance. Such an independence of the transmission ratio should not only apply to changes in center distance, but should also be present in principle.

If, for example, the tooth flanks of an involute gear differ from the ideal involute shape, the force direction of the tooth flanks can change during meshing on the line of action. This also changes the lever arm perpendicular to the force and thus the torque. This in turn leads to torque and speed fluctuations. In such a case, no constant transmission ratio is obtained.

The flank shape thus has a decisive influence on the constancy of the transmission ratio. For this reason, the first question to be asked with all tooth forms for gears is how they must be formed so that a constant transmission ratio is present during operation. The answer to this question can be reduced to the following fact:

For a constant transmission ratio, the normal at the point of contact of two tooth flanks must run at any time through the pitch point (fundamental law of gearing)!

This statement is also referred to as the fundamental law of gearing. If the direction of the force were not to run constantly through the pitch point, this would lead to a permanently changing lever arm and thus cause torque fluctuations. The transmission ratio would therefore not be constant.

Involute function

For the calculation of involute gears, the involute tooth flank must first be described mathematically. The figure below shows the involute belonging to the base circle with the radius r_b. A point P on this involute can be described by the angle α, which is spanned between the straight lines GP and GT. The point G corresponds to the center of the base circle and T to the tangent point on the base circle.

The length of the distance TP is identical to the radius of curvature ϱ of the involute at point P. Furthermore, the distance TP corresponds to the arc distance ST on the base circle, because the rolling line rolls without gliding on the base circle during the construction of the involute:

\begin{align}
\label{1}
\overset{\frown}{ST} &= \overline{TP} \\[5px]
\end{align}

Definition of the involute function

Although the angle α clearly describes a point on the involute, for many geometric calculations the angle φ drawn in the figure above is of greater importance. The angle φ describes the “thickness” of the involute tooth, so to speak.

Using equation (\ref{1}), the following relationship can be established between the angles φ and α:

\begin{align}
\overset{\frown}{ST} &= \overline{TP}  \\[5px]
r_b \cdot \left(\varphi + \alpha \right) &= r_b \cdot \tan(\alpha)  \\[5px]
\end{align}

\begin{align}
\label{p}
&\boxed{\varphi = \tan(\alpha)-\alpha} \\[5px]
\end{align}

The function resulting from the equation (\ref{p}) is called involute function inv(α). With the involute function many geometric gear parameters can be calculated.

\begin{align}
\label{involute}
&\boxed{\text{inv}(\alpha) = \tan(\alpha)-\alpha}  = \varphi ~~~\text{involute function}  \\[5px]
\end{align}

All angles for the involute function must always be given in radians!

Operating pressure angle

That the involute angle as well as the pressure angle are denoted by the same greek letter α is no coincidence! The involute angle α in the involute function can be interpreted as the operating pressure angle α_b, if the considered point P is located on the operating pitch circle of the gear and thus corresponds to the pitch point C=P!

Since the line of action is defined by the tangent to the base circle, which runs through the pitch point C, the distance TP is thus a part of the line of action. The involute angle α thus corresponds to the operating pressure angle α_b. If the point P is located on the reference pitch circle of the gear, then the standard pressure angle α₀ is obtained!

Calculation of the tooth thickness

The involute function explained in the previous section can be used to determine the tooth thickness s on an arbitrary diameter d of a gear. The figure below shows the geometric relationships: s₀ denotes the tooth thickness on the reference pitch circle and r₀ the corresponding reference pitch circle radius. The tooth thickness at the considered distance r to the center of the base circle G is denoted by s.

The derivation for the formula to calculate the tooth thickness s is done by the yellow triangle marked in the figure above. The acute angle of the yellow triangle can be determined by the difference between the angles δ₀ and δ, whereby the angles are determined according to the definition of radians as “ratio of the arc length to the length of the radius” as follows:

\begin{align}
\label{delta}
\underline{\delta_0} =\frac{\tfrac{s_0}{2}}{r_0}=\frac{s_0}{2 r_0} =\underline{\frac{s_0}{d_0}}  ~~~~\text{and}~~~~ \underline{\delta} =\frac{\tfrac{s}{2}}{r}=\frac{s}{2 r} = \underline{\frac{s}{d}}  \\[5px]
\end{align}

On the other hand, the acute angle of the yellow triangle can also be determined by the difference between the angles φ and φ₀. Thus the following relationship applies between the angles δ and φ or δ₀ and φ₀:

\begin{align}
 \delta –  \delta_0 &= \varphi_0 – \varphi \\[5px]
 \frac{s}{d} –  \frac{s_0}{d_0} &= \varphi_0 – \varphi \\[5px]
\end{align}

This equation can now be solved for to the tooth thickness s as a function of the considered diameter d:

\begin{align}
s &= d \left( \frac{s_0}{d_0} + \varphi_0 – \varphi \right) \\[5px]
\end{align}

The angles φ and φ₀ correspond to the angles that can be determined using the involute function inv(α) according to equation (\ref{involute}).

\begin{align}
\label{ss}
\underline{s = d \left( \frac{s_0}{d_0} + \text{inv}(\alpha_0) – \text{inv}(\alpha) \right)} \\[5px]
\end{align}

When using equation (\ref{ss}), it must be noted that the tooth thickness s₀ on the reference pitch circle depends on a possible profile shift. In the article Profile shift of gears, the relationship between the profile shift coefficient x and the tooth thickness s₀ has already been derived (with m as module of the gear):

\begin{align}
&\underline{s_0 = m \cdot \left(\frac{\pi}{2} +2 \cdot x \cdot \tan(\alpha_0)  \right) }  \\[5px]
\end{align}

In the previous section it was explained that the involute angle α in equation (\ref{ss}) corresponds to the operating pressure angle if the considered point P is located on the operating pitch circle. In the case of the point P₀, which is located on the reference pitch circle, the involute angle α₀ is then identical to the standard pressure angle α₀, which is usually set to α₀=0.349 rad (=20°).

Even if the considered point P on the circle on which the tooth thickness s is to be determined does not necessarily correspond to the actual operating pitch circle, any point P can always be regarded as being located on a operating pitch circle. This is because in the end the operating pitch circle only results from the center distance between the two gears in mesh. Since the center distance can be chosen arbitrarily, the operating pitch circle can theoretically always be adjusted so that it runs through the point P.

Due to this consideration a connection between the (operating pitch circle) diameter d of the circle on which the tooth thickness s is to be determined and the (operating pressure) angle α can be established. This connection is made by the standard pressure angle α₀ and the corresponding reference pitch circle diameter d₀ and has already been derived in the article Construction and design of involute gears. The basis of this relationship is the identical basic circle diameter d_b which is identical both when considering the operating pitch circle (with the parameters d and α and when considering the reference pitch circle (with the parameters d₀ and α₀):

\begin{align}
\label{base}
&\overbrace{d \cdot \cos(\alpha)}^{\text{base circle diameter } d_b} = \overbrace{d_0 \cdot \cos(\alpha_0)}^{\text{base circle diameter }d_b} \\[5px]
\label{z}
&\underline{\alpha = \arccos \left(\frac{d_0}{d} \cdot \cos(\alpha_0)\right)} \\[5px]
\end{align}

Note: The angle α in equation (\ref{z}) generally does not correspond to the operating pressure angle α_b! In this case, the angle α merely represents an “auxiliary quantity“, which results from the considered diameter d. Only if the diameter d actually corresponds to the operating pitch circle diameter, the angle α will be identical to the operating pressure angle α_b. When considering the tooth thickness on the reference pitch circle, the angle α corresponds to the standard pressure angle α₀.

With the involute function according to equation (\ref{involute}), the tooth thickness s at an considered circle with the diameter d is completely determined. The necessary equations are summarized again below:

\begin{align}
\label{tooth}
&\boxed{s = d \left( \frac{s_0}{d_0} + \text{inv}(\alpha_0) – \text{inv}(\alpha) \right)} \\[5px]
&\text{with} \\[5px]
\label{tooth0}
&\boxed{s_0 = m \cdot \left( \frac{\pi}{2} + 2 \cdot x \cdot \tan(\alpha_0) \right)}  \\[5px]
&\boxed{\text{inv}(\alpha_0) = \tan(\alpha_0)-\alpha_0} ~~~~~\text{with}~~~~~ \boxed{\alpha_0 =0,349 \text{ rad } (=20°)} \\[5px]
&\boxed{\text{inv}(\alpha) = \tan(\alpha)-\alpha} ~~~~~\text{with}~~~~~ \boxed{\alpha = \arccos \left(\frac{d_0}{d} \cdot \cos(\alpha_0)\right) } \\[5px]
\end{align}

The angle α in the equations above is to be considered as an “auxiliary quantity” and generally does not correspond to the operating pressure angle α_b!

Calculation of the circular and the base pitch

The circumferential pitch (circular pitch) is the arc distance between two adjacent tooth flanks of the same direction. On an arbitrary circle with a diameter d, the circular pitch p results from the ratio of the circumferential length π⋅d and the number of teeth z:

\begin{align}
\label{p1}
&\underline{p = \frac{\pi \cdot d}{z}} \\[5px]
\end{align}

In case of the reference pitch circle with the diameter d₀, the circular pitch p₀ is obtained::

\begin{align}
\label{p0}
&\underline{p_0 = \frac{\pi \cdot d_0}{z}} \\[5px]
\end{align}

If equation (\ref{p1}) is divided by equation (\ref{p0}), then the circumferential pitch p₀ can be used to establish a connection between an arbitrary diameter d and the resulting circular pitch p:

\begin{align}
&\frac{p}{p_0} = \frac{d}{d_0} \\[5px]
\label{pitch}
&\boxed{p = \frac{d}{d_0} \cdot p_0} \\[5px]
\end{align}

According to equation (\ref{base}), the ratio d/d₀ can also be expressed by the involute angle α (corresponding to the diameter d) and the involute angle α₀ (corresponding to the reference pitch diameter d₀ → standard pressure angle α₀). This ultimately means that a point on the involute is considered which is located on the circle with the diameter d (→ α) or on the reference pitch circle with the diameter d₀ (→ α₀).

\begin{align}
\label{kap}
&\overbrace{d \cdot \cos(\alpha)}^{\text{base circle diameter } d_b} = \overbrace{d_0 \cdot \cos(\alpha_0)}^{\text{base circle diameter }d_b} \\[5px]
\label{d}
&\underline{   \frac{d}{d_0}=\frac{\cos(\alpha_0)}{\cos(\alpha)}    } \\[5px]
\end{align}

If equation (\ref{d}) is applied in equation (\ref{pitch}), the pitch p can also be determined as follows:

\begin{align}
\label{pp}
&p = \frac{d}{d_0} \cdot p_0= \frac{\cos(\alpha_0)}{\cos(\alpha)} \cdot p_0 = \frac{\overbrace{\cos(\alpha_0) \cdot p_0}^{p_b}}{\cos(\alpha)} = \frac{p_b}{\cos(\alpha)}   \\[5px]
\end{align}

As shown in the article Construction and design of involute gears, the term p₀⋅cos(α₀) occurring in equation (\ref{pp}) corresponds to the base pitch p_b and therefore corresponds the distance between two tooth flanks in contact on the line of action when meshing (see figure below). 

The pitch on the base circle corresponds to the distance between two tooth flanks in contact on the line of action!

Since the circular pitch p₀ can also be expressed by the module m and the number π (p₀=π⋅m, see article Construction and design of involute gears), for the base pitch applies finally:

\begin{align}
\label{pb}
&\boxed{p_b = \pi \cdot m \cdot \cos(\alpha_0)} \\[5px]
\end{align}

Calculation of the center distance

In this section, the center distance of two corrected gears will be determined as a function of the respective profile shift coefficients x.

The starting point is the backlash-free meshing of the two gears, so that the tooth thickness on the operating pitch circle of one gear fits exactly into the tooth space on the operating pitch circle of the mating gear. The sum of the respective tooth thicknesses s₁ and s₂ thus corresponds to the circumferential pitch p on the operating pitch circles of the gears, which must be identical for both, otherwise the teeth could not mesh.

\begin{align}
\label{ppp}
& \underline{p = s_1 + s_2}  \\[5px]
\end{align}

The circular pitch p on the operating pitch circles should not be confused with the circular pitch p₀ on the reference pitch circles!

The tooth thickness s on an arbitrary circle with a diameter d can be determined from equations (\ref{tooth}) and (\ref{tooth0}) (parameters with the index “0” refer to the reference pitch circle):

\begin{align}
\label{s}
&s = d \left( \tfrac{s_{0}}{d_{0}} + \text{inv}(\alpha_0) – \text{inv}(\alpha) \right) ~~~\text{with} ~~~ s_0 = m \left( \tfrac{\pi}{2} + 2 x \cdot \tan(\alpha_0) \right) ~~~\text{applies}: \\[5px]
&s = d \left(\tfrac{m}{d_0} \left( \tfrac{\pi}{2} + 2 x \cdot \tan(\alpha_0) \right) + \text{inv}(\alpha_0) – \text{inv}(\alpha) \right) ~~~\text{
with}~~~m=\tfrac{d_0}{z} ~~~\text{applies:}  \\[5px]
&\underline{s = d \left(\tfrac{1}{z} \left( \tfrac{\pi}{2} + 2 x \cdot \tan(\alpha_0) \right) + \text{inv}(\alpha_0) – \text{inv}(\alpha) \right)}  \\[5px]
\end{align}

Since in this case the involute function inv(α) refers to the operating pitch circles d₁ or d₂, the involute angle α corresponds to the operating pressure angle α_b. For equation (\ref{ppp}) applies finally:

\begin{align}
\notag
p =  &d_1 \left(\tfrac{1}{z_1} \left( \tfrac{\pi}{2} + 2 x_1 \cdot \tan(\alpha_0) \right) + \text{inv}(\alpha_0) – \text{inv}(\alpha_b) \right) \\[5px]
\label{pppp}
&+  d_2 \left(\tfrac{1}{z_2} \left( \tfrac{\pi}{2} + 2 x_2 \cdot \tan(\alpha_0) \right) + \text{inv}(\alpha_0) – \text{inv}(\alpha_b) \right) \\[5px]
\end{align}

The operating pitch circle diameters d₁ or d₂ in equation (\ref{pppp}) can be determined from the definition of the circular pitch p as the ratio of pitch circle circumference π⋅d and the number of teeth z (p = π⋅d/z). Therefore, for the operating pitch circle diameters of the two gears d₁ and d₂ applies:

\begin{align}
\label{dd}
&d_1 = \frac{z_1 \cdot p}{\pi} ~~~~~\text{and}~~~~~d_2 = \frac{z_2 \cdot p}{\pi}  \\[5px]
\end{align}

The equations (\ref{dd}) can now be applied in equation (\ref{ppp}):

\begin{align}
\notag
p =  & \tfrac{z_1 \cdot p}{\pi} \left(\tfrac{1}{z_1} \left( \tfrac{\pi}{2} + 2 x_1 \cdot \tan(\alpha_0) \right) + \text{inv}(\alpha_0) – \text{inv}(\alpha_b) \right) \\[5px]
&+  \tfrac{z_2 \cdot p}{\pi} \left(\tfrac{1}{z_2} \left( \tfrac{\pi}{2} + 2 x_2 \cdot \tan(\alpha_0) \right) + \text{inv}(\alpha_0) – \text{inv}(\alpha_b) \right) \\[5px]
\end{align}

Solving this equation for the operating pressure angle α_b in terms of the involute function inv(α_b) finally leads to:

\begin{align}
\notag
\boxed{\text{inv}(\alpha_b) = 2 \frac{x_1+x_2}{z_1+z_2} \cdot \tan(\alpha_0) + \text{inv}(\alpha_0)} ~~~\text{and} ~~~\boxed{\text{inv}(\alpha_0) = \tan(\alpha_0)-\alpha_0} \\[5px]
\end{align}

Note that the involute function is not an algebraic function and therefore a reverse function cannot be derived. In such a case, the iterative Newton’s method offers a possibility for determining the operating pressure angle.

If the operating pressure angle is determined by such an approximation method, then not only the operating pitch circle diameter but also the center distance can be determined, since operating pitch circle diameter d and reference pitch circle diameter d₀ are related by the operating pressure angle α_b and the standard pressure angle α₀ according to equation (\ref{d}):

\begin{align}
&\boxed{d  = d_0 \cdot \frac{\cos(\alpha_0)}{\cos(\alpha_b)}} ~~~\text{operating pitch circle diameter}  \\[5px]
\end{align}

The center distance “a” results from the sum of the operating pitch circle radii r=d/2:

\begin{align}
a &= r_1+r_2 \\[5 px]
&= \frac{d_1}{2} + \frac{d_2}{2} \\[5px]
& = \frac{d_{0,1}}{2}  \cdot \frac{\cos(\alpha_0)}{\cos(\alpha_b)} + \frac{d_{0,2}}{2} \cdot \frac{\cos(\alpha_0)}{\cos(\alpha_b)} \\[5px]
& = (d_{0,1}+d_{0,2}) \cdot \frac{\cos(\alpha_0)}{2 \cdot \cos(\alpha_b)}  \\[5px]
& = (m \cdot z_1 + m \cdot z_2) \cdot \frac{\cos(\alpha_0)}{2 \cdot \cos(\alpha_b)}  \\[5px]
\end{align}

\begin{align}
\boxed{a = m \cdot( z_1 + z_2) \cdot \frac{\cos(\alpha_0)}{2 \cdot \cos(\alpha_b)}}  \\[5px]
\end{align}

Note that gears can also be manufactured with negative profile coefficients. If the sum of the profile shift factors is zero, the same center distance is obtained as in the case of non-corrected gears (called standard center distance a₀). The operating pressure angle then also corresponds to the standard pressure angle α₀.

Calculation of the profile shift coefficients

The previous section derived the formula for calculating the center distance “a” of two corrected gears:

\begin{align}
\label{a}
&a = m \cdot( z_1 + z_2) \cdot \frac{\cos(\alpha_0)}{2 \cdot \cos(\alpha_b)} \\[5px]
\end{align}

The operating pressure angle α_b has to be determined from an approximation method by the involute function:

\begin{align}
\label{inv}
\text{inv}(\alpha_b) = 2 \frac{x_1+x_2}{z_1+z_2} \cdot \tan(\alpha_0) + \text{inv}(\alpha_0) \\[5px]
\end{align}

In some cases, however, the center distance to be achieved is fixed by the gearbox. Then the center distance must be adjusted by a specific profile shift. The animation below shows the change of the center distance by a profile shift of both gears with the profile shift coefficients x₁ and x₁.

If the center distance “a” is given in advance, then the operating pressure angle α_b can first be determined by solving equation (\ref{a}):

\begin{align}
\label{alpha}
&\boxed{\alpha_b= \arccos \left(m \cdot( z_1 + z_2) \cdot \frac{\cos(\alpha_0)}{2 a} \right)} \\[5px]
\end{align}

Equation (\ref{inv}) can then be solved for the profile shift coefficients:

\begin{align}
\text{inv}(\alpha_b) &= 2 \frac{x_1+x_2}{z_1+z_2} \cdot \tan(\alpha_0) + \text{inv}(\alpha_0) \\[5px]
2 \frac{x_1+x_2}{z_1+z_2} \cdot \tan(\alpha_0) &= \text{inv}(\alpha_b) – \text{inv}(\alpha_0)  \\[5px]
\frac{x_1+x_2}{z_1+z_2} &= \frac{\text{inv}(\alpha_b) – \text{inv}(\alpha_0)}{2 \cdot \tan(\alpha_0) }  \\[5px]
\end{align}

\begin{align}
\label{x}
\boxed{x_1+x_2 = \frac{\text{inv}(\alpha_b) – \text{inv}(\alpha_0)}{2 \cdot \tan(\alpha_0)} \cdot (z_1+z_2)} \\[5px]
\end{align}

If a certain center distance is to be achieved by a profile shift, then the sum of the profile shift coefficients must satisfy the equation (\ref{x}). As long as this is fulfilled, the coefficients can in principle be chosen arbitrarily. However, it makes sense to assign the profile shift
coefficients evenly over the two gears, whereby the sum should not be much larger or smaller than one (i.e. the sum of the profile shifts should be in the range of the module).

The sum of the profile shifts should be in the order of the module of the gears!

The distribution of the profile shift coefficients over the two gears also depends on how pointed the tip of the tooth become with a profile shift. As explained in the article Profile shift, the tip tooth thickness should be at least 0.2 times the module. If this is no longer the case with a profile shift, the tip circle must be shortened. Such a tip shortening will be discussed in more detail in the next section.

Calculation of the tip shortening

In the article Profile shift it was shown that a profile shift is connected with an increase in the tip diameter d_a and the root diameter d_d by the amount of the (positive) profile shift:

\begin{align}
&\boxed{d_a =  m \cdot (z+2x+2) }   \\[5px]
\label{f}
&\boxed{d_d =  m \cdot (z+2x-2) -2c }   \\[5px]
\end{align}

In equation (\ref{f}), z denotes the number of teeth, m the module, x the profile shift coefficient and c) the manufacturing tip tooth clearance. The manufacturing tip tooth clearance results from the tool profile during gear cutting.

The manufacturing tip tooth clearance c must not be confused with the operating tip tooth clearance c_b, which actually results in operation when two gears are in mesh! It has already been explained in the article Profile shift that the backlash-free meshing of corrected gears results in a reduction of the tip tooth clearance in comparison to the
backlash-free meshing of standard gears, since the change in centre distance is smaller than the sum of the profile shifts.

The manufacturing tip tooth clearance c given in equation (\ref{f}) therefore refers only to the clearance between tool and gear during gear cutting (see figure below). The operating tip tooth clearance c_b, on the other hand, refers to the actual clearance in operation between the tip of the tooth of one gear and the root of the tooth of the mating gear. Only in the case of non-corrected standard gears are both manufacturing clearance  and operating clearance identical.

The reduction of the operating tip tooth clearance when corrected gears are engaging therefore makes it necessary to shorten the tip circles if the required tip tooth clearance c is to be maintained during operation. The amounts d_a^* to which the tip circles must be shortened are shown in the following sections.

The figure below shows that the operating tip tooth clearance c_b is generally determined by the center distance “a”, the root diameter d_d1 of one gear and the tip diameter d_a2 of the mating gear:

\begin{align}
\label{cb}
&c_b = a – \frac{d_{d1}}{2} – \frac{d_{a2}}{2} \\[5px]
\end{align}

If equation (\ref{f}) is applied in equation (\ref{cb}), then the operating tip tooth clearance c_b can be determined from the manufacturing tip tooth clearance c as follows:

\begin{align}
&c_b = a – \frac{\overbrace{    m \cdot (z_1+2x_1-2) -2c      }^{d_{d1}}}{2} – \frac{d_{a2}}{2} \\[5px]
\label{cc}
&\boxed{c_b = a – m \cdot \left( \frac{z_1}{2} + x_1 – 1 \right) – \frac{d_{a2}}{2}  + c } \\[5px]
\end{align}

If the tip diameter is to be adjusted so that the operating tip tooth clearance c_b equals the manufacturing tip tooth clearance c=c_b, then equation (\ref{cc}) can be solved for the shortened tip diameter d_a2^*:

\begin{align}
&c_b = a – m \cdot \left( \frac{z_1}{2} + x_1 – 1 \right) – \frac{d_{a2}^\text{*}}{2}  + c \overset{!}{=} c \\[5px]
&a – m \cdot \left( \frac{z_1}{2} + x_1 – 1 \right) – \frac{d_{a2^\text{*}}}{2} = 0   \\[5px]
\label{da2}
&\boxed{d_{a2}^\text{*} = 2 a – m \cdot \left(z_1 + 2 x_1 – 2 \right)   }  \\[5px]
\end{align}

The analog equation applies to the tip diameter d_a1^*:

\begin{align}
\label{da1}
&\boxed{d_{a1}^\text{*} = 2 a – m \cdot \left(z_2 +2 x_2 – 2 \right)   }  \\[5px]
\end{align}

Note that the tip shortenings according to the equations (\ref{da2}) and (\ref{da1}) are not dependent on the tip tooth clearance itself!

Calculation of the contact ratio

In the article Meshing of in volute gears it has already been explained that the line of action results as a tangent to the base circles of the meshing gears. The actual line of contact runs from the intersection point A between the line of action and the tip circle of the driven gear to the intersection point E between the line of action and the tip circle of the driving gear. The ratio of the line of contact l to the base pitch p_b (distance between two adjacent contact points) is called contact ratio ε.

\begin{align}
&\boxed{\epsilon = \frac{l}{p_b}} >1 \\[5px]
\end{align}

For continuous power transmission, a new tooth must be engaged before the preceding tooth leaves the line of contact. The contact ratio must therefore always be greater than one. The determination of this contact ratio of two profile shifted gears will be shown in the following sections.

Since the base pitch p_b is already be determined by equation (\ref{pb}), only the line of contact l has to be calculated. For the derivation of the formula to calculate the line of contact l, the figure below is used. The figure shows that the sum of the distances T₁E (yellow triangle) and T₂A (blue triangle) is greater by the amount of the line of contact l than the distance T₁T₂. For the line of contact l therefore applies:

\begin{align}
& \overline{T_1 E} + \overline{T_2 A} – l = \overline{T_1 T_2} \\[5px]
\label{0}
& \underline{ l = \overline{T_1 E} + \overline{T_2 A} – \overline{T_1 T_2} } \\[5px]
\end{align}

The distance T₁E can be determined from the yellow triangle using the base circle diameter d_b1 and the (possibly shortened) tip diameter d_a1^*:

\begin{align}
& \left( \frac{d_{a1}^\text{*}}{2} \right)^2 = \overline{T_1 E}^2 + \left( \frac{d_{b1}}{2} \right)^2 \\[5px]
\label{11}
&\underline{   \overline{T_1 E}  = \sqrt{ \left( \frac{d_{a1}^\text{*}}{2} \right)^2 – \left( \frac{d_{b1}}{2} \right)^2} }\\[5px]
\end{align}

The distance T₂A can be determined by the blue triangle using the base circle diameter d_b2 and the (possibly shortened) tip diameter d_a2^*:

\begin{align}
& \left( \frac{d_{a2}^\text{*}}{2} \right)^2 = \overline{T_2 A}^2 + \left( \frac{d_{b2}}{2} \right)^2 \\[5px]
\label{2}
&\underline{   \overline{T_2 A}  = \sqrt{ \left( \frac{d_{a2}^\text{*}}{2} \right)^2 – \left( \frac{d_{b2}}{2} \right)^2} }\\[5px]
\end{align}

The distance T₁T₂=T₁^* T₂^* results from the red marked triangle in the figure below on the basis of the center distance “a” as well as the base circle diameter d_b1 resp. d_b2:

\begin{align}
& a^2 =\overline{T_1 T_2}^2 + \left( \frac{d_{b1}}{2} + \frac{d_{b2}}{2} \right)^2 \\[5px]
\label{3}
& \underline{\overline{T_1 T_2} = \sqrt{a^2 – \left( \frac{d_{b1}}{2} + \frac{d_{b2}}{2} \right)^2}    } \\[5px]
\end{align}

The equations (\ref{11}), (\ref{2}) and (\ref{3}) can now be applied to equation (\ref{0}):

\begin{align}
& l = \overline{T_1 E} + \overline{T_2 A} – \overline{T_1 T_2}  \\[5px]
& l = \sqrt{ \left( \frac{d_{a1}^\text{*}}{2} \right)^2 – \left( \frac{d_{b1}}{2} \right)^2}  +  \sqrt{ \left( \frac{d_{a2}^\text{*}}{2} \right)^2 – \left( \frac{d_{b2}}{2} \right)^2} – \sqrt{a^2 – \left( \frac{d_{b1}}{2} + \frac{d_{b2}}{2} \right)^2}     \\[5px]
\label{l}
&  \boxed{l = \frac{1}{2}  \left[ \sqrt{ d_{a1}^\text{* 2}  – d_{b1}^2}   +  \sqrt{ d_{a2}^\text{* 2} – d_{b2}^2  }  –  \sqrt{   4 a^2 – \left( d_{b1} + d_{b2} \right)^2}  \right]}   \\[5px]
\end{align}

The base circle diameters d_b in equation (\ref{l}) can be determined by the module m, the standard pressure angle α₀ and the respective number of teeth z:

\begin{align}
&d_b = \overbrace{d_0}^{= m \cdot z} \cdot \cos(\alpha_0) \\[5px]
&\boxed{d_b = m \cdot z \cdot \cos(\alpha_0) } \\[5px]
\end{align}

Finally, all parameters for determing the contact ratio ε can be calculated:

\begin{align}
&\boxed{\epsilon = \frac{l}{p_b}} \\[5px]
\text{with} \\[5px]
&\boxed{p_b= \pi \cdot m \cdot \cos(\alpha_0)}  \\[5px]
\text{and} \\[5px]
&\boxed{l = \frac{1}{2}  \left[ \sqrt{ d_{a1}^\text{* 2}  – d_{b1}^2}   +  \sqrt{ d_{a2}^\text{* 2} – d_{b2}^2  }  –  \sqrt{   4 a^2 – \left( d_{b1} + d_{b2} \right)^2}  \right]}  \\[5px]
&\boxed{d_{b1} = m \cdot z_1 \cdot \cos(\alpha_0) } \\[5px]
&\boxed{d_{b2} = m \cdot z_2 \cdot \cos(\alpha_0) } \\[5px]
\end{align}

The tip diameters d_a^* correspond to the shortened tip circles, if a tip shortening was carried out.

The formulas given for the calculation of the contact ratio only apply to gears without undercut! In the case of undercutted gears, the line of contact is shortened and the contact ratio is thus reduced!

Excel spreadsheet for calculating involute gears

The following Excel spreadsheet for gear calculation offers the possibility to calculate different geometric parameters of involute gears:

• center distance
• tip shortening
• sum of the profile shift coefficients to obtain a certain center distance
• operating pressure angle
• transmission ratio
• standard reference pitch circle
• operating pitch circle
• tip, root and base circle diameter
• tip tooth clearance
• (tip) tooth thickness
• contact ratio

The calculations have not yet been completely checked for correctness!

Gear hobbing

Involute gears are often manufactured by hobbing. The cutting edges of a hob are straight-flanked and wrap spirally around the milling tool (similar to the thread of a screw). The cross-sectional profile of a hob is equal to that of a rack!

As the gear rotates, the teeth of the hob, which also rotate, continue to mill inwards into the gear over time until the final depth is reached. During one rotation of the hob, the gear moves forward by one tooth. In this respect, the pitch of the cutters corresponds exactly to the tooth pitch of the gear. Gear and hob form a kind of “worm gear” in their motion sequence.

The rotational speed between gear and hob must therefore be matched to each other so that the teeth can form correctly. The number of teeth on the gear reflects the ratio of the rotational speeds. When manufacturing a gear with 18 teeth, the rotational speed of the hob must therefore be 18 times higher than the speed of the gear.

The geometry of the rack-shaped tool profile depends on the desired flank profile of the gear tooth. The tool flanks are inclined against the vertical by the standard pressure angle α₀. The actual tool profile is created from the so-called standard reference profile after consideration of the root fillet and the clearance c. The center line of the profile corresponds to the pitch line.

The cutting performance of gear hobbing is very high, so that especially very thick gears can be produced in a relatively short time. However, this process cannot be used to produce internal gears. For this purpose, for example, gear shaping can be used, which will be described in more detail in the next section.

The basic relationship between a gear and a rack (“tool profile”) is shown in the figure below. It is shown that the perpendicular distance between the involutes for all gears with the same module always corresponds to the perpendicular distance between two adjacent flanks. This distance ist called base pitch p_b and corresponds to the distance between two contacting flanks in mesh with a mating gear (or rack).

Furthermore the figure shows, that the base pitch p_b is directly related to the circular pitch p₀ on the reference pitch circle by the standard pressure angle α₀. Note that the circular pitch p₀ is identical for all gears with the same module, otherwise they could not mesh. For this reason, the circular pitch p₀ of the rack also corresponds to the circular pitch of the gears.

Therefore, one single rack-type hob can be used to produce gears with any number of teeth and then paired with each other. Thus, a 6-tooth gear (red) is produced with the same tool as an 18-tooth gear (green) or a 9-tooth gear. This applies not only to gear hobbing, but also to gear shaping and gear planning.

Note: As the above animation shows, if the number of teeth is too low, the teeth are “undercut” during the manufacturing process and thus weakened. Such an undercut – and how it can be avoided even with a small number of teeth – is described in more detail in the article Profile shift.

Gear shaping

Gears can be produced not only by hobbing but also by shaping. In contrast to hobbing, however, shaping is performed by a translational reciprocating motion of the tool.

With shaping the cutting edges are arranged like the teeth of a gear on the circumference of the tool. The backward tapered cutting edges have involute flanks. The tool reciprocates in axial direction during the cutting process. During the cutting stroke, the cutting edges push into the gear blank and remove material. During the return stroke (back stroke), the tool is retracted slightly to prevent collision with the gear blank. The reciprocating motion of the tool is a rotating motion of the tool and workpiece superimposed. The tool rolls on the gear blank in a cutting manner. The tool can actually be regarded as a “gear with cutting edges”.

In contrast to gear hobbing, gear shaping can also be used to manufacture internal gears. Shaping can also be used to produce helical gears. For this purpose, the tool is simply inclined by the amount of the helix angle.

With gear shaping internal gears can be produced!

Hobbing, shaping and planing (described in the next section) are ultimately based on the same principle, which can be illustrated using the example of gear shaping: The cutting tool roll on the pitch circle of the gear blank perpendicular to the cutting motion. The envelope of the cutting edges then creates an involute tooth shape.

Gear planning with a rack type cutter

Instead of a “pinion-shaped” tool as is the case with gear shaping, a rack-shaped tools can also be used. In terms of kinematics, the tool and workpiece form a kind of “rack and pinion gear“. One speacks of planning with a rack type cutter (straight-flanked cutting edges). The required clearance angle at the cutting edges is again achieved by the fact that they taper to the rear.

During the cutting stroke, the cutting edges removes material from the gear blank and is then lifted slightly off the workpiece during the return stroke to avoid collision. As is usual with a “rack and pinion gear”, both the gear blank and the rack type cutter continue to move during planing. Since the rack type cutter  usually has fewer cutting edges than the teeth on the gear are to be produced, the cutter must be reset after one pass.

The rack shaped cutter can also be used to produce helical gears. For this purpose, the cutter only has to be set at an angle corresponding to the helix angle. In contrast to gear shaping with a pinion type cutter, whose cutting edges are involute-shaped, the rack type cutter has straight flank cutting edges. Rack type cutters can therefore be produced much more easily and thus more cost-effectively. However, internal gears cannot be produced. Planing with a rack type cutter is mainly used for very large gears. The cutting performance is relatively low compared to gear hobbing.

Gear form cutting with a disc cutter

With form cutting, the cutting edges on the milling tool have the shape of the tooth space. The milling tool mills each tooth space individually before the gear blank is rotated one tooth space at a time. The teeth of the gear are produced exclusively by the rotational motion of the tool. During the machining process, there is therefore no superimposed rotational motion of the gear blank as is the case with gear hobbing, shaping or planing. In the latter three manufacturing processes, the shape of the tooth is created by the envelope of the cutting edges; with form cutting or broaching (explained in the next section), however, the tooth shape is created directly by the cutting edge geometry of the tool.

In contrast to hobbing, shaping or planing, the cutting motion of the tool does not have to be matched to the motion of the workpiece. Therefore, form cutting can also be carried out on ordinary milling machines. However, since the tooth spaces have different geometries depending on the size of the gear, a single form cutter can only be used to produce a specific gear. For each module (or diametral pitch) and each number of teeth, separate disc cutters are required, which are relatively expensive due to the individual shape.

Form cutting of gears (milling) can be performed on ordinary milling machines. Special disc cutters are required for each module (diametral pitch) and each number of teeth!

Gear broaching

Broaching is similar to shaping or planing in terms of motion. However, a broaching tool has several cutting edges arranged one behind the other. Each cutting edge takes material and thus successively creates the final shape. If the broaching tool has been completely moved through the workpiece, the final shape is usually already reached. Broaching thus has a high cutting efficiency and is used above all for internal gears.

In contrast to gear shaping or planing, the tooth geometry of the gear is produced directly by the involute shape of the cutting edges of the broaching tool and not by an envelope of the cutting edges. Therefore, only one specific gear can be produced with one broaching tool. For each number of teeth or module, a specific broaching tool is required. Gear broaching is therefore only used in mass production.

Introduction

In the article Undercut of gears it was shown that avoiding an undercut with a standard gear (standard pressure angle of 20°) requires a minimum number of teeth of 17. If gears are nevertheless to be manufactured below the limit number of teeth (e.g. because a certain transmission ratio is to be achieved), the undercut must be avoided in another way. For this purpose, a so-called profile shift can be used. Such a profile shift is described in more detail in this article.

Profile shift

With a profile shift, the tool profile is shifted outwards by a certain amount during gear cutting. The animation below shows the effects of a profile shift on the tooth form of a gear with 8 teeth. It becomes clear that as the profile shift increases, the undercut becomes smaller and can even be completely avoided.

The figure below shows again the comparison of the tooth shapes with increasing profile shift (from left to right). Even if the tooth shapes differ from each other, the teeth can still mesh with each other. Profile shifted gears (also called corrected gears) can therefore be easily paired with non-profile shifted gears (so-called standard gears) as long as they are manufactured with the same tool and therefore have the same module.

Undercuts can be avoided with a profile shift. For this purpose, the tool profile is shifted outwards during gear manufacturing. Gears with different profile shifts can mesh with each other without further ado!

Influence of the profile shift on the shape of the tooth flank

Even if this may not seem so at first glance, a profile shift has no influence on the shape of the tooth flank itself. All profile shifted gears use the same involute for the tooth shape compared to their corresponding standard gears. Only another part of the same involute is used. This becomes clear when the tooth flanks of the gears with different profile shifts are placed on top of each other.

Note that the base circle for constructing the involute is determined solely by the flank angle of the tool profile (= standard pressure angle) during gear cutting. And since the angle of the cutting edges does not change with a profile shift, the base circle and thus the involute do not change either.

For profile-shifted gears, the same involute is used for the shape of the tooth flank. The base circle therefore does not change with a profile shift, since the base circle is determined solely by the flank angle of the cutting tool (standard pressure angle)!

As already explained in the article Construction and design of involute gears, the radius of curvature of the involute increases with increasing length, i.e. the further away the involute is from the base circle, the larger the radius of curvature is and the less strongly it is therefore curved. The flank shape at this more distant area is rather “flat” than “pointed”. The smaller curvature leads to a larger contact surface of the flanks, which reduces the pressure accordingly (less Hertzian contact stress). This reduces the stress on the flanks and thus increases the flank load-bearing capacity.

The flank load-bearing capacity can be increased by a profile shift!

Influence of a profile shift on the standard reference pitch circle

The reason why profile-shifted gears can engage with other profile-shifted gears is that all gears have the same circular pitch on their manufacturing pitch circles. This will be explained in more detail in this section.

First of all, it should be noted that the pitch point C between the rack-shaped cutting tool and the gear does not change during gear cutting, even if the profile is shifted (see article rack for further information on this). This is due to the fact that the pitch point is defined as the intersection between the center line and the line of action. The line of action in turn always results as a normal to the tool flank, which is tangent to the base circle of the gear.

As the figure above shows, a profile shift has no effect on the flank angle of the tool or on the base circle of the gear as mentioned in the previous section. Thus, even with a profile shift, the line of action and thus the pitch point always remain unchanged.

The unchangeable position of the pitch point is already apparent from its meaning. The pitch point describes the point in which the speeds of the rack-shaped tool profile and the gear are identical (“sliding-free rolling”). A radial shift of the tool profile, however, does not change the speed ratios and thus the position of the pitch point.

The pitch line of the tool and the (manufacturing) pitch circle of the gear run through the pitch point. Regardless of the profile shift, the unchangeable pitch point therefore always leads to the same manufacturing pitch circles on the gears. Finally, the pitch of the teeth is referenced to these manufacturing pitch circles (circular pitch p₀). This means that the manufacturing pitch circle corresponds to the standard reference pitch circle of the gear (see also the article on racks). A profile shift therefore has no effect on the resulting reference pitch circle and the related circumferential pitch, so that profile-shifted gears can mesh with non-profile-shifted gears!

The reference pitch circle (manufacturing pitch circle) of a gear does not change with a profile shift, so that corrected gears can mesh with standard gears!

Profile shift coefficient

The profile shift V for gears is usually indicated by a profile shift coefficient x in relation to the module m. With positive coefficients (x>0) the tool profile is shifted outwards and with negative factors (x<0) it is shifted inwards (applies to external gears).

\begin{align}
&\boxed{V = x \cdot m} \\[5px]
\end{align}

For example, a profile shift coefficient of x=+0.25 means that the tool profile is shifted outwards by 0.25 times the module m. In general, both the root circle radius (dedendum circle) and the tip circle radius (addendum circle) increase by the amount of the profile shift.

The calculation of the root diameter d_d,0 and the tip diameter d_a,0 for standard gears has already been explained in the article Construction and design of involute gears. These diameters result from the module m and the number of teeth z, whereby a clearance c is also taken into account for the root diameter:

\begin{align}
d_{a,0} &= m \cdot (z+2)   ~~~&&\text{applies only to standard gears}\\[5px]
d_{d,0} &= m \cdot (z-2) – 2 \cdot c   ~~~&&\text{applies only to standard gears}\\[5px]
\end{align}

In case with a profile shift, however, the tip circle radii and the root circle radii are increased by the amount of the (positive) profile shift V with corrected gears. For the corresponding diameters applies:

\begin{align}
&d_a = d_{a,0} + 2 \cdot V = m \cdot (z+2) + 2 \cdot V = m \cdot (z+2) + 2 \cdot m \cdot x     \\[5px]
\label{a}
&\boxed{d_a =  m \cdot (z+2x+2)}   ~~~\text{applies in general (without tip shortening)}\\[5px]
&d_d = d_{d,0} + 2 \cdot V = m \cdot (z-2) – 2 \cdot c + 2 \cdot V =  m \cdot (z-2) – 2 \cdot c + 2 \cdot m \cdot x   \\[5px]
&\boxed{d_d =  m \cdot (z+2x-2) -2c }   ~~~\text{applies in general}\\[5px]
\end{align}

Increase of tooth thickness

A profile shift also has an effect on the circular tooth thickness and the tooth space width. As the tooth thickness s₀ increases on the pitch circle, the tooth space e₀ decreases accordingly. In the following, the tooth thickness s₀ on the reference pitch circle is determined as a function of the profile shift coefficient x.

The figure below shows the increase in the distance of the tool flanks on the pitch line (width Δs of the triangle marked blue) when the tool profile is shifted outwards by the profile shift V=x⋅m. This distance of the tool flanks on the pitch line corresponds to the tooth thickness s₀ on the manufacturing pitch circle of the gear (= reference pitch circle).

In comparison to a standard gear, whose tooth thickness corresponds to half the circular pitch p₀ (p₀/2), with a corrected gear the tooth thickness increases by an amount Δs. The figure above shows the following relationship between the profile shift coefficient x and the resulting tooth thickness s₀ on the pitch circle (with α₀ as the standard pressure angle):

\begin{align}
&s_0 = \frac{p_0}{2} + \Delta s \\[5px]
&s_0 = \frac{p_0}{2} + 2 \cdot V \cdot \tan(\alpha_0) \\[5px]
&\underline{s_0 = \frac{p_0}{2} + 2 \cdot m \cdot x \cdot \tan(\alpha_0)} \\[5px]
\end{align}

The circular pitch p₀ can also be expressed by the module m (see article Construction and geometry of involute gears):

\begin{align}
&\underline{p_0 = \pi \cdot m} \\[5px]
\end{align}

Finally, for the tooth thickness s₀ on the reference pitch circle of a profile shifted gear applies:

\begin{align}
&s_0 = \frac{\pi \cdot m}{2} + 2 \cdot m \cdot x \cdot \tan(\alpha_0) \\[5px]
&\boxed{s_0 = m \cdot \left(\frac{\pi}{2} +2 \cdot x \cdot \tan(\alpha_0)  \right) } ~~~\text{with } \alpha_0 = 20° \\[5px]
\end{align}

By the same amount Δs as the tooth thickness increases, the tooth space width e₀ decreases:

\begin{align}
&\boxed{e_0 = m \cdot \left(\frac{\pi}{2} – 2 \cdot x \cdot \tan(\alpha_0)  \right) } \\[5px]
\end{align}

Tip shortening

In the previous section it was shown that a (positive) profile shift increases the tooth thickness on the reference pitch circle and thus increases tooth strength. At the same time, however, the width of the tooth tip s_a on the addendum circle is reduced.

If the tip of the tooth is too small, however, there is a risk of teeth breaking out. In order to prevent this, the tip diameter must then be shortened so that a certain thickness is maintained. The tip diameter should be shortened in such a way that the tooth thickness at the addendum circle is at least 0.2 times the module (such a tip shortening shortening is not yet considered in the equation (\ref{a})!). Note that with a shortend tip diameter the line of contact is reduced accordingly! The detailed calculations for determining the line of contact are shown in the article Calculation Of Involute Gears.

The thickness of the tip tooth at the addendum circle should be at least 20 % of the module. To achieve this, a tip shortening may be necessary! This reduces the line of contact!

The animation below shows the profile shift of a gear with 6 teeth to avoid an undercut. In this case, the thickness of the tip tooth even decreases so much that the involutes taper before the shifted tip diameter is reached. The increase of the tip circle radius by the amount of the profile shift cannot therefore be maintained in this case – the tip diameter is inevitably shortened.

In addition, the tip circle would have to be shortened again to at least 0.2 times the module in order to increase the thickness of the tip tooth. However, such a large reduction of the tip circle would also result in a correspondingly large reduction of the line of action. Involute gears with fewer than 7 teeth should therefore be avoided by any means.

Involute gears with less than 7 teeth should be avoided due to excessive tip shortening!

Center distance and operating pressure angle

In the previous section it was shown that with corrected gears (green gear in the figure below) an extended part of the involute is used as tooth flank compared to standard gears (shown in red). When meshing with another gear, this further curved part of the involute is used for power transmission. Therefore, there is a clearance between the tooth flanks when the center distance a is increased by the amount of the (positive) profile shift V=x⋅m. The back flank bends away before it touches the flank of the other gear, so to speak.

For a backlash-free engaging, the gears must therefore be moved together a little bit. This reduces the centre distance slightly, but it is still larger compared to the a standard gear. At the same time, however, the clearance c between the tip of the tooth and the tooth root of the other gear is reduced (see figure  below). This may require a tip shortening in order to maintain a certain clearance.

The center distance is therefore larger with a positive profile shift and smaller with a negative profile shift compared to standard gears without a profile shift. This means that the center distance can be adjusted by means of a profile shift. This is another reason why a profile shift is frequently used.

Profile shifts are often used to adjust the centre distance!

In addition to the effects already mentioned, a profile shift also results in a change in the operating pressure angle α. The operating pressure angle α is not to be confused with the standard pressure angle α₀, which ultimately determines the flank angle of the rack-shaped cutting tool and of course does not change with a profile shift!

Detailed calculations for determining the operating pressure angle and the centre distance are shown in the article Calculation of involute gears.

Calculation of the profile shift coefficient to avoid undercutting

In the article Undercut it was shown that below the minimum number of z_min=17 teeth an undercut occurs which weakens the teeth. A profile shift now provides the possibility to completely compensate such an undercut. This raises the question of how the profile shift coefficient x must be chosen to avoid an undercut with a given number of teeth z<z_min.

As also explained in detail in the article Undercut, in order to avoid an undercut, the intersection point B between the base circle and the line of action must be located outside the line of contact AE. In the limiting case where an undercut is to be avoided, the beginning of the undercut coincides with the end of the line of action. Then the reference profile (“tool profile”) exits the gear before it undercuts the tooth.

As the animation above shows, the displacement of the tangent point B to the end of the line of contact in point E is achieved by a positive profile shift. Note that the end of contact is determined by the intersection of the line of action and the tip line of the reference profile and can therefore be influenced by a profile shift, whereas the line of action does not change if the profile is shifted!

In the limiting case, the tangent point B coincides with the end of engagement E, as it is the case for the green gear. From the resulting geometrics, the profile shift coefficient x can be determined for a given standard pressure angle α₀.

The orange triangle in the figure above is now considered more closely. It is shown that the opposite side of the standard pressure angle α₀ corresponds to the difference between the module m and the profile shift V=x⋅m. Therefore, the following relationship applies to the distance CE between the pitch point C and the end of engagement E:

\begin{align}
&\sin(\alpha_0) = \frac{m-V}{\overline{CE}}= \frac{m-m \cdot x}{\overline{CE}} = \frac{m\cdot (1-x)}{\overline{CE}} \\[5px]
\label{1}
&\underline{\overline{CE} = \frac{m \cdot (1-x)}{\sin(\alpha_0)}} \\[5px]
\end{align}

Now the blue triangle is considered more closely. It is shown that the distance CE can also be determined from the pitch circle radius r₀ or the pitch circle diameter d₀. The pitch circle diameter results from the module m and the number of teeth z.

\begin{align}
&\overline{CE} = r_0 \cdot \sin(\alpha_0) = \frac{d_0}{2} \cdot \sin(\alpha_0) = \frac{m \cdot z }{2} \cdot \sin(\alpha_0) \\[5px]
\label{2}
&\underline{\overline{CE} = \frac{m \cdot z}{2} \cdot \sin(\alpha_0)} \\[5px]
\end{align}

The two equations (\ref{1}) and (\ref{2}) can now be equated and solved for the profile shift coefficient x:

\begin{align}
&\overline{CE} = \overline{CE}  \\[5px]
&\frac{m \cdot (1-x)}{\sin(\alpha_0)} = \frac{m \cdot z}{2} \cdot \sin(\alpha_0) \\[5px]
&\underline{x = 1-z \cdot \frac{\sin^2(\alpha_0)}{2}  } \\[5px]
\end{align}

Furthermore, the term sin²(α₀)/2 in the equation above corresponds to the reciprocal of the minimum number of teeth z_min, above which an undercut would occur without a profile shift (for the derivation of the formula, see article Undercut). For a gear with a standard pressure angle of α₀=20°, the minimum number of teeth is theoretically z_min=17. The profile shift coefficient x required to avoid an undercut can thus be determined as follows:

\begin{align}
&\boxed{x = 1-\frac{z}{z_{min}} } ~~~ \text{with } z_{min}=17 \\[5px]
\end{align}

For a gear with z = 8 teeth, the profile shift coefficient is x=0.53. In practice, a slight undercut can often be accepted without major negative effects. In these cases the minimum number of teeth is assumed to be z_min=14.

Note that for a number of teeth greater than z_min, the profile shift coefficient x becomes negative. This means that theoretically a negative profile shift can be made without undercutting.

Summary

In summary, it can be stated that a (positive) profile shift is always applied if …

• an undercut is be avoided,
• the tooth strength must be increased,
• the surface pressure at the flanks is to be decreased, or
• the center distance must be adjusted.

With external gears, a positive profile shift leads to …

• an increase of the root circle,
• an increase of the tip circle,
• an increase of the tooth root thickness (increased strength) and thus to
• a reduction of the undercut,
• a decrease of the tip tooth thickness (tip shortening may be necessary),
• an increase of the tooth thickness at the pitch circle and to
• a decrease of the tooth space width at the pitch circle,
• a decrease of the Hertzian contact stress on the flanks (increased flank load-bearing capacity) and to
• an increase of the centre distance when meshing with a standard gear.

Neither the base circle diameter nor the reference pitch circle diameter change with a profile shift! In the case of a negative profile shift, the effects listed above are just the opposite. Due to the many positive effects, a profile shift is generally recommended.

Introduction

In mechanical engineering, the involute is used almost exclusively as a tooth form for gears. Such gears are called involute gears. The use of involute toothing is due on the one hand to the favorable meshing (engagement of two gearwheels). On the other hand, involute gears can be manufactured cost-effectively due to the relatively simple tool geometry.

Construction of an involute

In the case of involute toothing, the shape of the tooth flanks consists of two involutes of circles (called involutes for short). An involute is constructed by rolling a so-called rolling line around a base circle. The resulting trajectory curve describes the shape of the involute. Two mirror-inverted involutes then form the basic shape of a tooth.

The involute can also be constructed by unwinding a string from a circle. The string is always kept taut during unwinding. The end of the string then also describes the shape of the involute.

An involute is constructed by unwinding a rolling line on a base circle or by unwinding a string from the base circle!

The longer the involute, the greater the curvature radius, i.e. the smaller the curvature of the involute. From the construction of the involute it becomes clear that the radius of curvature corresponds exactly to the arc length on the base circle.

The longer the involute, the less curvature it has!

However, the flank shape cannot only be influenced by changing the base circle diameter. By means of a so-called profile shift, the tooth flanks of gearwheels are composed of the more distant part of the involute, while the base circle diameter remains the same. The tooth flank is therefore less curved and “flatter”. During meshing with another gear, the contact forces can thus be distributed more evenly. This reduces the tooth load and thus the tooth wear.

Profile shifted gears use the less curved part of the involute for the tooth shape!

Nomenclature

In order to avoid contact between the tip and the root of the tooth of two meshing gears, the root of the tooth is rounded out (called fillet). The resulting diameter at the root of the tooth is called the root diameter (dedendum circle). Analogous to the root diameter, a tip diameter can be assigned to the tip of the gear (addendum circle).

Note, that there is no contact between the tooth flanks of two meshing gears within the base circle! For further information see article meshing.

There is no contact of the tooth flanks within the base circle!

The size of a gearwheel is defined by the so-called standard reference pitch diameter. Strongly simplified, this diameter corresponds to the diameters of imaginary cylinders that roll on each other. The tooth spacing is related to this diameter and is referred to as the circumferential pitch or circular pitch. The circular pitch is the arc distance between two tooth flanks of the same direction on the pitch circle. This circular pitch must be identical for all gears so that the teeth can mesh without interfering.

The spacing of the teeth can basically also be related to the base circle. In this case, the pitch is then referred to as the base pitch. More detailed information on this can be found in the next section.

Tooth size: the module

In order to characterize gears and above all to ensure that the teeth of two gears can mesh properly, one of the most important parameters is the so-called module m. The module is a measure of the tooth size of a gear and is usually given in millimeters (if the unit is in inches, the module is also called “English Module”.). Only if the teeth of gears are of the same size and thus have the same module can they be paired with each other!

The module corresponds directly to the addendum h_a. The dedendum h_d also results from the module, whereby a clearance c is also taken into account. The clearance corresponds to the amount by which the tooth root is additionally deepened to avoid contact between the tip and root of two gearwheels. Depending on the application, the clearance is typically 10% to 30% of the module (often 0.167⋅m). The whole depth of the tooth h thus results from twice the value of the module m, plus the clearance c.

\begin{align}
&\boxed{h_a = m} ~~~\text{addendum} \\[5px]
&\boxed{h_d = m + c} ~~~\text{dedendum} \\[5px]
&\boxed{c = 0,167 \cdot m} ~~~\text{clearance } \\[5px]
&\boxed{h = h_a + h_d = 2 \cdot m + c} ~~~\text{whole depth} \\[5px]
\end{align}

The module is a measure of the tooth size: the larger the module, the larger the tooth! Only gears with the same module can be paired!

The figure below shows three gears of the same size (i.e. identical reference pitch circles), but manufactured with different modules. The next section deals with the reference pitch circle in more detail.

Gear size: the standard reference pitch diameter

If the clearance is neglected for now, the tooth is “divided” by the standard reference pitch diameter at half depth, so to speak. The standard reference pitch diameter d₀ results from the product of module m and number of teeth z and is a measure for the size of the gear:

\begin{align}
&\boxed{d_0 = m \cdot z} ~~~\text{standard reference pitch diameter} \\[5px]
\end{align}

Note that, in contrast to the so-called operating pitch diameter, the reference pitch diameter is a fixed size of a gearwheel which is determined solely by the product of the module and the number of teeth.

The standard reference pitch diameter serves only as a reference circle for specifying the circular pitch p₀, i.e. the tooth spacing which must be identical for all gears if they are to mesh. This arc length p₀ between two identical tooth flanks can be determined from the quotient of the pitch circle circumference u₀=π⋅d₀ and the number of teeth z:

\begin{align}
&p_0 = \frac{u_0}{z} = \frac{\pi \cdot d_0}{z} = \frac{\pi \cdot m \cdot z}{z} = m \cdot \pi \\[5px]
&\boxed{p_0 = m \cdot \pi} ~~~\text{circular pitch} \\[5px]
\end{align}

At this point it becomes again clear that only gears with identical modules can be paired with each other, because obviously only then the tooth spacing p₀ for all gears are identical and the teeth can mesh properly.

The standard reference pitch diameter is a measure for the size of a gearwheel. The circular pitch of the teeth is related to this diameter. All gears to be paired must have identical circular pitches on their standard reference pitch circles and thus identical modules!

The standard reference pitch diameter d₀ can now be used to determine the tip diameter d_a (addendum circle) and the root diameter d_d (dedendum circle):

\begin{align}
&d_a =d_0 + 2 \cdot h_a = m \cdot z + 2 \cdot m   \\[5px]
&\boxed{d_a =m \cdot (z+2)}  ~~~\text{tip diameter (addendum circle diameter)} \\[5px]
&d_d = d_0 – 2 \cdot h_d = m \cdot z – 2 \cdot (m+c)   \\[5px]
&\boxed{d_d = m \cdot (z-2) – 2 \cdot c}  ~~~\text{root diameter (dedendum circle diameter)} \\[5px]
\end{align}

For gears that are not profile-shifted, the circular tooth thickness s₀ and the tooth space width e₀ on the reference pitch circle are identical in length and thus correspond to half the circular pitch p₀. Two gears can thus be paired without backlash. Thus no “rattling” occurs during the change of the direction of rotation.

\begin{align}
&\boxed{s_0 = e_0 = \frac{p_0}{2} = \frac{m}{2} \cdot \pi} ~~~\text{circular tooth thickness , tooth space width} \\[5px]
\end{align}

The center distance of the two non-profile shifted gears for backlash-free pairing is referred to as the standard center distance a₀. The standard center distance results from the sum of the two reference pitch radii or half the sum of the reference pitch diameters:

\begin{align}
&\boxed{a_0 = \frac{d_{0,1}+ d_{0,2} }{2} = \frac{m}{2} \cdot (z_1+z_2)} ~~~\text{standard center distance} \\[5px]
\end{align}

Diametral pitch

Instead of the module, the so-called diametral pitch is often used as a measure of the tooth size. The diametral pitch DP corresponds to the inverse of the module m:

\begin{align}
&\boxed{DP = \frac{1}{m} = \frac{z}{d_0} } ~~~\text{diametral pitch} \\[5px]
\end{align}

Therefore, the diametral pitch indicates how many teeth per unit length of pitch diameter a gearwheel has.

Author’s note: The term diametral pitch ist somewhat a little bit confusing, since in contrast to the terms base pitch oder circular pitch, the diametral pitch doesn’t state a “distance per tooth”, but just the opposite: “tooth per distance”. To be consistent, the module m should therefore be the “real diametral pitch”. This inconsistence is the reason why I’m not using the term diametral pitch any further, but only the module.

Tooth shape: the pressure angle

If the teeth of gears are to mesh perfectly, not only the tooth sizes must match (described by the module m), but also the tooth flank shape must match. This flank shape is described by the so-called pressure angle. The pressure angle is described in more detail in this section.

The shape of the involute and thus the tooth flank is only dependent on the base circle diameter. Each base circle therefore always has a specific involute. All involutes are geometrically similar, i.e. they can be “scaled” (enlarged or reduced in size) to the same shape.

The flank shape of a gear is determined by the base circle diameter, whereby all involutes of any base circles are geometrically similar to each other!

If larger base circle diameters are used to construct the tooth flanks for a certain gear size (i.e. for identical pitch diameters), the teeth will tend to look “square”. With smaller base diameters, however, the tip of the teeth tend to look “pointed”. The figure below shows the tooth shapes for different base diameters with identical reference pitch circles. The tooth size, i.e. the module, was chosen the same for all gears!

Two gears can only mesh properly if they have the same tooth shape. This is only the case if the ratio of base circle diameter to reference pitch diameter is identical for all gears. When increasing the number of teeth and thus the reference pitch diameter, the base circle diameter must increase by the same amount so that the tooth shape remains identical (this corresponds to the “scaling” of the involutes mentioned above).

In addition to the module, a further parameter must therefore be defined which describes the ratio of the base circle diameter to the reference pitch diameter and thus the tooth shape. The relationship between the base circle and the pitch circle becomes apparent when two gears with no profile shift are paired without backlash. In this case, the center distance corresponds to the standard center distance a₀ and the pitch circles of both gears touch each other at the so-called pitch point C.

If a tangent is now applied to the two base circles of the gears, this so-called line of action encloses a certain angle with the normal of the center line. This angle is called the pressure angle α₀. As the yellow and blue triangles in the figure above show, the base circle radii r_b and the pitch circle radii r₀ are related by this pressure angle α₀:

\begin{align}
&\cos(\alpha_0) = \frac{r_b}{r_0}  \\[5px]
&\boxed{\cos(\alpha_0) = \frac{d_b}{d_0}} ~~~\alpha_0 \text{ : pressure angle} \\[5px]
\end{align}

In practice, the diameter ratio of the base circle to the pitch circle, and thus the flank shape, is determined by this pressure angle. Independent of the module, a pressure angle of α₀ = 20° is usually used for all gears. However, there are also gears with pressure angles of 14.5° or 25°.

In the figure “Flank shape for different base circle diameters”, a pressure angle of 8° was selected for the left gear and an angle of 20° for the center gear and an angle of 32° for the right gear.

The fact that the pressure angle directly determines the flank shape of a gear is also immediately obvious from an manufacturing point of view. In general, gears are manufactured by gear hobbing. In this case, the flank shape of the gear is determined by the inclination of the cutting edges of the rack-shaped tool profile. This tool angle corresponds directly to the pressure angle α₀!

Note:
– the module is a measure for the tooth size!
– The standard reference pitch diameter is a measure for the gear size!
– The pressure angle is a measure for the shape of the tooth flank!

Base pitch

All involutes of a gear are equidistant to each other, i.e. the right-angled distances of two adjacent involutes are identical for all points on an involute. This right-angled distance also corresponds to the arc section on the base circle, since the involutes are constructed by rolling a rolling line on the base circle without sliding. This curved distance of the involutes on the base circle (= curved distance of the tooth flanks!) is therefore also called base pitch p_b.

The base pitch p_b on the base circle is not to be confused with the circular pitch p₀ on the pitch circle. However, both pitches are not independent of each other. Just as the base circle diameter and the reference pitch diameter are proportional to each other by the pressure angle α₀ (see previous section), the same ratio also applies to the corresponding pitches on the base circle and the pitch circle:

\begin{align}
&\boxed{\cos(\alpha_0) = \frac{p_b}{p_0}} \\[5px]
\end{align}

The rectangular distance between two involutes (i.e. the base pitch!) also corresponds to the distance between two tooth flanks touching each other during meshing (“meshing pitch”)!

The base pitch corresponds to the distance between two meshing flanks and results from the right-angled distance between two adjacent involutes (“meshing pitch”)!

More information on the engagement of two involute gears can be found in the article Meshing of involute gears.