gear, calculate, calculation, definition, involute function

Calculation of involute gears

This article provides answers to the following questions, among others:

  • 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

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 \(\alpha\), which is spanned between the straight lines \(\overline{GP}\) and \(\overline{GT}\). The point \(G\) corresponds to the center of the base circle and \(T\) to the tangent point on the base circle.

gear, calculate, calculation, definition, involute function
Figure: Definition of involute function

The length of the distance \(\overline{TP}\) is identical to the radius of curvature \(\rho\) of the involute at point \(P\). Furthermore, the distance \(\overline{TP}\) corresponds to the arc distance \(\overset{\frown}{ST}\) 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 \(\alpha\) clearly describes a point on the involute, for many geometric calculations the angle \(\varphi\) drawn in the figure above is of greater importance. The angle \(\varphi\) describes the “thickness” of the involute tooth, so to speak.

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

\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 \(\text{inv}(\alpha)\). 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 \(\alpha\) is no coincidence! The involute angle \(\alpha\) in the involute function can be interpreted as the operating pressure angle \(\alpha_b\), if the considered point \(P\) is located on the operating pitch circle of the gear and thus corresponds to the pitch point \(C=P\)!

gear, calculate, calculation, pressure angle, line of action, involute function
Figure: Pressure angle and involute function

Since the line of action is defined by the tangent to the base circle, which runs through the pitch point \(C\), the distance \(\overline{TP}\) is thus a part of the line of action. The involute angle \(\alpha\) thus corresponds to the operating pressure angle \(\alpha_b\). If the point \(P\) is located on the reference pitch circle of the gear, then the standard pressure angle \(\alpha_0\) 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: \(s_0\) denotes the tooth thickness on the reference pitch circle and \(r_0\) 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 \(s\).

gear, calculate, calculation, tooth thickness, involute function
Figure: Calculation of tooth thickness

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 \(\delta_0\) and \(\delta\), 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 \(\varphi\) and \(\varphi_0\). Thus the following relationship applies between the angles \(\delta\) and \(\varphi\) or \(\delta_0\) and \(\varphi_0\):

\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 \(\varphi\) and \(\varphi_0\) correspond to the angles that can be determined using the involute function \(\text{inv}(\alpha)\) 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}

gear, calculate, calculation, tooth thickness, involute function
Figure: Involute function for calculating the tooth thickness

When using equation (\ref{ss}), it must be noted that the tooth thickness \(s_0\) on the reference pitch circle depends on a possible profile shift. In the article Profile shift, the relationship between the profile shift coefficient \(x\) and the tooth thickness \(s_0\) 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 \(\alpha\) 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 \(P_0\), which is located on the reference pitch circle, the involute angle \(\alpha_0\) is then identical to the standard pressure angle \(\alpha_0\), which is usually set to \(\alpha_0\)=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 \(\alpha\) can be established. This connection is made by the standard pressure angle \(\alpha_0\) and the corresponding reference pitch circle diameter \(d_0\) and has already been derived in the article “Geometry 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 \(\alpha)\) and when considering the reference pitch circle (with the parameters \(d_0\) and \(\alpha_0\)):

\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 \(\alpha\) in equation (\ref{z}) generally does not correspond to the operating pressure angle \(\alpha_b\)! In this case, the angle \(\alpha\) 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 \(\alpha\) will be identical to the operating pressure angle \(\alpha_b\). When considering the tooth thickness on the reference pitch circle, the angle \(\alpha\) corresponds to the standrad pressure angle \(\alpha_0\).

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 \(\alpha\) in the equations above is to be considered as an “auxiliary quantity” and generally does not correspond to the operating pressure angle \(\alpha_b\)!

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 \(\pi \cdot d\) 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 \(d_0\), the circular pitch \(p_0\) is obtained::

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

gear, calculate, calculation, circumferential pitch, circular, base pitch
Figure: Calculation of the pitch

If equation (\ref{p1}) is divided by equation (\ref{p0}), then the circumferential pitch \(p_0\) 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 \(\tfrac{d}{d_0}\) can also be expressed by the involute angle \(\alpha\) (corresponding to the diameter \(d\)) and the involute angle \(\alpha_0\) (corresponding to the reference pitch diameter \(d_0\) \(\rightarrow\) standard pressure angle \(\alpha_0\)). This ultimately means that a point on the involute is considered which is located on the circle with the diameter \(d\) (\(\rightarrow \alpha\)) or on the reference pitch circle with the diameter \(d_0\) (\(\rightarrow \alpha_0\)).

\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_0}{d} \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 “Geometry of involute gears“, the term \(p_0 \cdot \cos(\alpha_0)\) 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!

gear, calculate, calculation, base pitch
Figure: Calculation of the base pitch

Since the circular pitch \(p_0\) can also be expressed by the module \(m\) and the number \(\pi\) (\(p_0=\pi \cdot m\), see article “Geometry 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 \(s_1\) and \(s_2\) 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}

gear, calculate, calculation, center distance, profile shift
Figure: Calculation of the center distance

The circular pitch \(p\) on the operating pitch circles should not be confused with the circular pitch \(p_0\) 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 \(\text{inv}(\alpha)\) refers to the opersting pitch circles \(d_1\) or (d_2), the involute angle \(\alpha\) corresponds to the operating pressure angle \(\alpha_b\). 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 \(d_1\) or \(d_2\) in equation (\ref{pppp}) can be determined from the definition of the circular pitch \(p\) as the ratio of pitch circle circumference \(\pi \cdot d\) and the number of teeth \(z\) (\(p = \frac{\pi \cdot d}{z}\)). Therefore, for the operating pitch circle diameters of the two gears \(d_1\) and \(d_2\) 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 \(\alpha_b\) in terms of the involute function \(\text{inv}(\alpha_b)\) 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.

gear, calculate, calculation, operating pressure angle, profile shift
Figure: Calculation of 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 \(a\) can be determined, since operating pitch circle diameter \(d\) and reference pitch circle diameter \(d_0\) are related by the operating pressure angle \(\alpha_b\) and the standard pressure angle \(\alpha_0\) 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 \(a\) itself results from the sum of the operating pitch circle radii \(r=\frac{d}{2}\):

\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 \(a_0\)). The operating pressure angle then also corresponds to the standard pressure angle \(\alpha_0\).

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 \(\alpha_b\) 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 \(a\) 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 \(x_1\) and \(x_2\).

gear, calculate, calculation, center distance, profile shift
Animation: Change of the center distance with a profile shift

If the center distance \(a\) is given in advance, then the operating pressure angle \(\alpha_b\) 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 \(a\) 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.

gear, calculate, calculation, tip clearance, profile shift
Figure: Change of the tip clearance with a profile shift

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.

gear, calculate, calculation, manufacturing tip clearance, profile shift
Figure: Manufacturing tip clearance

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 \(d_a^\text{*}\) 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}

gear, calculate, calculation, tip tooth clearance, profile shift, tip shortening
Figure: Operating tip tooth clearance

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 \(d_{a2}^\text{*}\):

\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}

gear, calculate, calculation, tip tooth clearance, profile shift, tip shortening
Figure: Tip shortening to obtain a certain tip tooth clearance

The analog equation applies to the tip diameter \(d_{a1}^\text{*}\):

\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 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 \(\epsilon\).

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

gear, calculate, calculation, contact ratio, profile shift
Figure: Calculation of the contact ratio

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 \(\overline{T_1 E}\) (yellow triangle) and \(\overline{T_2 A}\) (blue triangle) is greater by the amount of the line of contact \(l\) than the distance \(\overline{T_1 T_2}\). 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}

gear, calculate, calculation, contact ratio, profile shift
Figure: Calculation of the contact ratio

The distance \(\overline{T_1 E}\) can be determined from the yellow triangle using the base circle diameter \(d_{b1}\) and the (possibly shortened) tip diameter \(d_{a1}^\text{*}\):

\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 \(\overline{T_2 A}\) can be determined by the blue triangle using the base circle diameter \(d_{b2}\) and the (possibly shortened) tip diameter \(d_{a2}^\text{*}\):

\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 \(\overline{T_1 T_2}=\overline{T_1^\text{*} T_2^\text{*}}\) 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}

gear, calculate, calculation, contact ratio, profile shift
Figure: Calculation of the contact ratio

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 \(\alpha_0\) 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 \(\epsilon\) 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 \(d_a^\text{*}\) 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!