In my research, I focused on the modeling and measurement of herringbone gears, which are widely used in high-speed and heavy-load transmission systems. Herringbone gears consist of two helical gears with opposite helix directions mounted on the same axis. This special configuration can cancel axial forces generated during meshing, making them superior to conventional helical gears in many industrial applications. However, the complex geometry of herringbone gears poses considerable challenges in mathematical modeling, manufacturing, and precision metrology. In this work, I derived the tooth surface equations from first principles, developed a three-dimensional simulation program, and proposed a systematic method for evaluating tooth profile deviations, helix deviations, and pitch deviations. I also designed a LabVIEW-based error evaluation system and verified its feasibility using a coordinate measuring machine (CMM).
To begin with, I analyzed the basic geometrical parameters of herringbone gears. The gear can be considered as two identical helical gears with helix angles of equal magnitude but opposite signs. Therefore, the fundamental parameters follow the definitions of involute cylindrical gears. Table 1 summarizes the key parameters and their formulas used throughout my study.
| Parameter | Symbol | Formula |
|---|---|---|
| Normal module | \(m_n\) | Standard value |
| Transverse module | \(m_t\) | \(m_t = m_n / \cos\beta\) |
| Normal pressure angle | \(\alpha_n\) | \(\alpha_n = 20^\circ\) (standard) |
| Transverse pressure angle | \(\alpha_t\) | \(\sin\alpha_t = \sin\alpha_n / \cos\beta\) |
| Pitch circle diameter | \(d\) | \(d = m_t z\) |
| Base circle diameter | \(d_b\) | \(d_b = d \cos\alpha_t\) |
| Helix angle at pitch cylinder | \(\beta\) | Usually \(8^\circ\)–\(20^\circ\) |
| Base helix angle | \(\beta_b\) | \(\tan\beta_b = \tan\beta \cos\alpha_t\) |
| Addendum | \(h_a\) | \(h_a = h_a^* m_n\) |
| Dedendum | \(h_f\) | \(h_f = (h_a^* + c^*) m_n\) |
| Tip diameter | \(d_a\) | \(d_a = d + 2h_a\) |
| Root diameter | \(d_f\) | \(d_f = d – 2h_f\) |
| Transverse pitch | \(p_t\) | \(p_t = \pi m_t\) |
| Normal pitch | \(p_n\) | \(p_n = \pi m_n\) |
The derivation of the tooth surface equation starts from the involute curve. As shown in Figure 1, when a straight line rolls without slipping on a base circle, any point on the line traces an involute. I can express the involute in polar coordinates using the radius \(r_k\) and the polar angle \(\theta_k\). The fundamental equation of the involute is given by
$$ \text{inv}\,\alpha_k = \tan\alpha_k – \alpha_k . $$
For a point \(K\) on the involute, the radius vector \(r_k\) and the pressure angle \(\alpha_k\) satisfy
$$ r_k = \frac{r_b}{\cos\alpha_k}, $$
and the polar angle \(\theta_k\) is
$$ \theta_k = \tan\alpha_k – \alpha_k. $$
To facilitate parametric modeling, I converted the polar representation into rectangular coordinates. Let \(\varphi_k = \theta_k + \alpha_k\), which is the roll angle of the generating line. Then the coordinates of a point on the involute in the transverse plane are
$$ \begin{aligned}
x_k &= r_b (\cos\varphi_k + \varphi_k \sin\varphi_k), \\
y_k &= r_b (\sin\varphi_k – \varphi_k \cos\varphi_k).
\end{aligned} $$
For helical gears, the planar involute is extended along the axial direction. The generating plane rolls on the base cylinder, and the contact line \(KK\) is inclined by the base helix angle \(\beta_b\). This produces an involute helicoid. I established a workpiece coordinate system whose \(Z\)-axis coincides with the gear axis. The first tooth right flank is taken as the reference. For a point \(P_{R1j}\) on the transverse plane \(z=0\), the coordinates are directly given by the involute equations. When the axial coordinate \(z_{R1j}\) is not zero, the point is obtained by rotating the transverse involute according to the helix angle. I derived the following parametric equation for the right flank of the first tooth:
$$ \begin{aligned}
x_{R1j_z} &= r_b \cos(\varphi_{R1j} – \theta_z) + r_b \varphi_{R1j} \sin(\varphi_{R1j} – \theta_z), \\
y_{R1j_z} &= r_b \sin(\varphi_{R1j} – \theta_z) – r_b \varphi_{R1j} \cos(\varphi_{R1j} – \theta_z), \\
z_{R1j_z} &= \pm r_b \theta_z \cot\beta_b,
\end{aligned} $$
where \(\theta_z\) is the angular rotation of the transverse involute along the helix, and the sign depends on the direction of the helix side. For the upper half of the herringbone gear, the positive sign is used; for the lower half, the negative sign is used. This expression forms the mathematical model of one flank of one tooth.
For the \(i\)-th tooth, I applied a rotation matrix around the \(Z\)-axis. The rotation angle for the \(i\)-th tooth is
$$ \theta_i = \frac{2\pi (i-1)}{Z_0}, \quad i = 1,2,\ldots,Z_0, $$
where \(Z_0\) is the number of teeth. Thus, the right flank point of the \(i\)-th tooth is
$$ P_{Rij_z} = A_z(\theta_i) P_{R1j_z}, $$
with the rotation matrix
$$ A_z(\theta_i) = \begin{bmatrix}
\cos\theta_i & -\sin\theta_i & 0 \\
\sin\theta_i & \cos\theta_i & 0 \\
0 & 0 & 1
\end{bmatrix}. $$
Next, I derived the left flank equation using symmetry. In the transverse plane, the left and right tooth flanks of a gear tooth are symmetric about a line passing through the tooth center. Let the slope of this symmetry line be
$$ k = -\tan(\omega + \alpha_f), $$
where
$$ \omega = \frac{S_f}{2 r_f}, \quad S_f = m Z_0 \sin\left(\frac{\pi}{2 Z_0}\right), \quad r_f = \frac{m Z_0}{2}, $$
and \(\alpha_f\) is the pressure angle at the pitch circle. I derived the mirror transformation matrix \(V\) that maps a right flank point to the corresponding left flank point:
$$ V = \begin{bmatrix}
\frac{1-k^2}{1+k^2} & \frac{2k}{1+k^2} & 0 \\
\frac{2k}{1+k^2} & \frac{k^2-1}{1+k^2} & 0 \\
0 & 0 & 1
\end{bmatrix}. $$
Therefore, the left flank of the first tooth is obtained by
$$ P_{L1j_z} = V P_{R1j_z}. $$
For the \(i\)-th tooth left flank, the same rotation matrix is used:
$$ P_{Lij_z} = A_z(\theta_i) P_{L1j_z}. $$
Combining the upper and lower halves with opposite helix angles gives the complete herringbone gear tooth surface. This mathematical model is the foundation for subsequent simulation and error analysis.
To verify the correctness of my derivation, I implemented the equations in MATLAB. I used a set of gear parameters, which are listed in Table 2.
| Parameter | Symbol | Value |
|---|---|---|
| Normal module | \(m_n\) | 3.456811 mm |
| Number of teeth | \(z\) | 19 |
| Addendum coefficient | \(h_a^*\) | 1 |
| Clearance coefficient | \(c^*\) | 0.25 |
| Helix angle | \(\beta\) | 16.6772° |
| Pressure angle | \(\alpha\) | 22.5° |
| Face width (one half) | \(b\) | 30 mm |
I wrote a MATLAB program to calculate the coordinates of points on the tooth surface for a given step size. The flowchart of the computation is:
- Define all base parameters.
- Compute the transverse module, transverse pressure angle, pitch diameter, base diameter, tip diameter, and root diameter.
- Determine the range of the involute roll angle \(\varphi\) from the base circle to the tip circle.
- Determine the range of the helix rotation angle \(\theta_z\) from the face width.
- Loop over the angular increments and compute the three-dimensional coordinates.
- Use the
surffunction in MATLAB to render the spiral surfaces.
The simulation results showed that the right flank and left flank of a single tooth form a smooth involute helicoid. By mirroring the upper half to the lower half, I obtained the complete herringbone tooth shape. Repeating this around the circumference produced the full gear model. The MATLAB simulation confirmed that the derived tooth surface equations are geometrically consistent and can be used for further analysis.
In the next part of my research, I analyzed the tooth flank errors. According to national standards for involute cylindrical gears, the essential error items are tooth profile deviation, helix deviation, and pitch deviation. Each of these errors affects gear performance differently. Tooth profile deviation affects transmission smoothness, helix deviation affects load distribution along the face width, and pitch deviation affects kinematic accuracy.
Tooth profile deviation is defined in the transverse plane as the distance between two design tooth profiles that exactly contain the actual tooth profile over the usable profile length. I developed a computational method based on the principle of involute sliding. Figure 2 shows the relationship between the theoretical involute and the actual measured points. For a measured point \(P_i(x_i,y_i)\) on the right flank, the polar radius and pressure angle are
$$ r_i = \sqrt{x_i^2 + y_i^2}, \quad \alpha_i = \arccos\left(\frac{r_b}{r_i}\right). $$
The polar angle of the point is
$$ \phi_i = \arctan\left(\frac{y_i}{x_i}\right). $$
For the right flank, the theoretical involute angle is
$$ \theta_i = \tan\alpha_i – \alpha_i. $$
I then defined the angular deviation of each measured point relative to the theoretical tooth profile by
$$ \Delta\omega_i = \phi_i – \theta_i – \varepsilon, $$
where \(\varepsilon\) is a constant for all points on the same tooth flank. The maximum and minimum values of \(\Delta\omega_i\) correspond to the two extreme positions of the theoretical involute that just enclose the actual profile. The tooth profile deviation for that flank is
$$ \Delta f_\alpha = r_b (\omega_{\max} – \omega_{\min}) \cos\beta. $$
By substituting the measured coordinates, I obtained the explicit formula
$$ \Delta f_\alpha = r_b \cos\beta \left[ \arctan\left(\frac{y_1}{x_1}\right) – \arctan\left(\frac{y_2}{x_2}\right) + \arccos\left(\frac{r_b}{r_2}\right) – \arctan\left(\arccos\left(\frac{r_b}{r_2}\right)\right) – \arccos\left(\frac{r_b}{r_1}\right) + \arctan\left(\arccos\left(\frac{r_b}{r_1}\right)\right) \right]. $$
Here I used the fact that the extreme angular deviations occur at the two end points of the measurement range. For a left flank, the sign before \(\theta_i\) changes because the involute curves in the opposite direction.
I measured four teeth uniformly distributed on the circumference. Each tooth profile was sampled at five points radially. Table 3 gives a partial set of the raw measurement data.
| Tooth | Flank | Point | X (mm) | Y (mm) |
|---|---|---|---|---|
| 1 | Left | 1 | 32.1436 | 3.0128 |
| 2 | 33.3346 | 2.7680 | ||
| 3 | 34.5112 | 2.3989 | ||
| 4 | 35.6879 | 1.9035 | ||
| 5 | 36.8559 | 1.2953 | ||
| 1 | Right | 1 | 31.9313 | -4.7107 |
| 2 | 33.1312 | -4.5197 | ||
| 3 | 34.3294 | -4.2110 | ||
| 4 | 35.5334 | -3.7736 | ||
| 5 | 36.7355 | -3.2377 | ||
| 4 | Left | 1 | -5.6283 | 31.7911 |
| 2 | -5.4898 | 32.9902 | ||
| 3 | -5.2167 | 34.1941 | ||
| 4 | -4.8191 | 35.4059 | ||
| 5 | -4.3066 | 36.6160 | ||
| 4 | Right | 1 | 2.0721 | 32.2061 |
| 2 | 1.7852 | 33.3906 | ||
| 3 | 1.3681 | 34.5670 | ||
| 4 | 0.8391 | 35.7239 | ||
| 5 | 0.1977 | 36.8827 |
After processing the data with my proposed algorithm, I obtained the tooth profile deviations for each flank. Table 4 shows the calculated results.
| Tooth | Flank | \(\Delta f_\alpha\) (mm) |
|---|---|---|
| 1 | Left | 0.0139 |
| 1 | Right | 0.0157 |
| 2 | Left | 0.0143 |
| 2 | Right | 0.0162 |
| 3 | Left | 0.0174 |
| 3 | Right | 0.0128 |
| 4 | Left | 0.0174 |
| 4 | Right | 0.0081 |
Therefore, the total tooth profile deviation of the gear is
$$ F_\alpha = \max\{\Delta f_{\alpha,1}, \ldots, \Delta f_{\alpha,8}\} = 0.0174 \ \text{mm}. $$
To verify the tooth surface model, I converted the measured coordinates into the same coordinate system as the theoretical model. I plotted the measured points together with the theoretical tooth profile at the measured axial plane. The measured points were very close to the theoretical involute, which confirmed the correctness of my mathematical model.
The next error item is the helix deviation. In herringbone gears, the tooth trace is a helix on the pitch cylinder. The helix deviation is defined in the transverse plane as the distance between two design helix lines that exactly contain the actual helix line over the face width. To measure the helix deviation, I used the coordinate measuring machine to trace points along the pitch cylinder helix at five equally spaced axial positions. Table 5 shows a subset of the measured helix points.
| Tooth | Flank | Point | X (mm) | Y (mm) | Z (mm) |
|---|---|---|---|---|---|
| 1 | Right | 1 | 34.4913 | 0.6245 | -1.5005 |
| 2 | 34.4754 | 1.3504 | -3.9027 | ||
| 3 | 34.4401 | 2.0770 | -6.3024 | ||
| 4 | 34.3877 | 2.8002 | -8.7026 | ||
| 5 | 34.2921 | 3.5119 | -11.1013 | ||
| 1 | Left | 1 | 33.9568 | -6.0122 | -1.5009 |
| 2 | 34.0888 | -5.3099 | -3.9042 | ||
| 3 | 34.1901 | -4.5981 | -6.3011 | ||
| 4 | 34.2786 | -3.8708 | -8.7024 | ||
| 5 | 34.3640 | -3.1484 | -11.1017 |
For each pair of consecutive points on the same helix, I computed the angular increment
$$ \Delta\theta_i = \theta_i – \theta_{i-1}, $$
and converted it to the arc length on the pitch cylinder:
$$ \Delta W_i = r \Delta\theta_i. $$
The theoretical tangential increment corresponding to the axial displacement \(\Delta Z\) is
$$ \Delta W_t = \Delta Z \tan\beta. $$
Thus, the local helix deviation between two points is
$$ \Delta f_i = (\Delta W_i – \Delta W_t) \cos\beta. $$
For a complete helix, if the deviations have different signs, the total helix deviation is
$$ F_\beta = \max(\Delta f_i) – \min(\Delta f_i). $$
If all deviations have the same sign, the total deviation is the maximum absolute value. Table 6 lists the calculated helix deviations for the four measured teeth.
| Tooth | Flank | \(F_\beta\) (mm) |
|---|---|---|
| 1 | Right | 0.0092 |
| 1 | Left | 0.0172 |
| 2 | Right | 0.0180 |
| 2 | Left | 0.0124 |
| 3 | Right | 0.0086 |
| 3 | Left | 0.0151 |
| 4 | Right | 0.0144 |
| 4 | Left | 0.0105 |
The total helix deviation of the gear is
$$ F_\beta = 0.0180 \ \text{mm}. $$
The third important error item is the pitch deviation. I measured the same-side tooth flanks around the entire circumference at a constant axial section. For each adjacent pair of teeth, I obtained one point on the corresponding involute flanks. The calculation method converts the measured point to the intersection of the involute with the pitch circle. For a measured point \(P_1(x_1,y_1)\), the polar radius is \(r_1 = \sqrt{x_1^2 + y_1^2}\), and the polar angle is
$$ \phi_1 = \arccos\left(\frac{x_1}{r_1}\right). $$
The pressure angle at \(P_1\) is
$$ \alpha_1 = \arccos\left(\frac{r_b}{r_1}\right). $$
Then the angle of the involute point on the pitch circle is
$$ \phi_{f1} = \phi_1 + \tan\alpha_1 – \alpha_1 – \tan\alpha_f + \alpha_f, $$
where \(\alpha_f\) is the pressure angle at the pitch circle. The actual pitch angle between two adjacent teeth is
$$ \Delta\phi = \phi_{f2} – \phi_{f1}. $$
The actual circular pitch is
$$ p_t’ = r \Delta\phi. $$
The theoretical circular pitch is
$$ p_t = \pi m. $$
Therefore, the single pitch deviation is
$$ f_{pt} = p_t’ – p_t. $$
I measured all 19 teeth on both left and right flanks. Table 7 gives the calculated single pitch deviations for a few teeth.
| Tooth interval | Left flank \(f_{pt}\) (mm) | Right flank \(f_{pt}\) (mm) |
|---|---|---|
| 1–2 | -0.0078 | -0.0031 |
| 2–3 | -0.0104 | -0.0122 |
| 3–4 | -0.0020 | -0.0082 |
| 4–5 | -0.0022 | -0.0096 |
| 5–6 | -0.0064 | -0.0018 |
| 6–7 | 0.0008 | 0.0005 |
| 7–8 | 0.0032 | 0.0031 |
| 8–9 | 0.0015 | 0.0042 |
| 9–10 | 0.0141 | 0.0141 |
| 10–11 | 0.0068 | 0.0114 |
| 11–12 | -0.0009 | 0.0024 |
| 12–13 | -0.0110 | -0.0069 |
| 13–14 | -0.0038 | -0.0068 |
| 14–15 | -0.0030 | 0.0032 |
| 15–16 | -0.0010 | 0.0025 |
| 16–17 | -0.0008 | 0.0007 |
| 17–18 | -0.0067 | 0.0002 |
| 18–19 | 0.0002 | -0.0030 |
I plotted the single pitch deviations as a line chart to visualize the changing trend around the gear. The plot clearly showed that the deviations fluctuate within a small range, indicating that the manufacturing error is uniform. The cumulative pitch deviation for a side is the algebraic sum of the single pitch deviations over a given number of pitches. The total cumulative pitch deviation is the maximum minus the minimum of the cumulative sum. From Table 7, I found:
- Left flank: \(f_{pt\max} = 0.0141\) mm, \(f_{pt\min} = -0.0110\) mm.
- Right flank: \(f_{pt\max} = 0.0141\) mm, \(f_{pt\min} = -0.0122\) mm.
Therefore, the total cumulative pitch deviations are
$$ F_{pL} = 0.0141 – (-0.0110) = 0.0251 \ \text{mm}, $$
$$ F_{pR} = 0.0141 – (-0.0122) = 0.0263 \ \text{mm}. $$
The overall gear cumulative pitch deviation is
$$ F_p = 0.0263 \ \text{mm}. $$
All calculated deviations are within the tolerance range for the specified gear accuracy grade, which validates my error formulas.
To make the evaluation process more efficient and user-friendly, I developed a graphical software tool based on LabVIEW. LabVIEW is a graphical programming environment that is widely used for data acquisition, instrument control, and industrial automation. In my software, the tooth surface simulation module calls MATLAB scripts through the MATLAB script node. The user can input the gear parameters on the front panel, and the program calculates the tooth surface coordinate points. The front panel also provides a display for the three-dimensional tooth surface plot. I implemented two modes: single-tooth display and full-gear display. The case structure in the block diagram selects which MATLAB script to execute. The calculated coordinates are saved to a text file and displayed in a table on the front panel.
The error evaluation system consists of three modules: tooth profile deviation evaluation, helix deviation evaluation, and pitch deviation evaluation. The main interface provides a navigation menu for the user to select the desired evaluation module. Each module reads the measured point coordinates from a text file, applies the corresponding algorithm, and displays the results. For the tooth profile deviation module, the user enters the gear basic parameters and clicks the “Calculate Left Flank” or “Calculate Right Flank” button. The program computes the deviation for each tooth flank and shows the total profile deviation. The helix deviation module similarly processes the measured helix points and outputs the helix deviations. The pitch deviation module calculates the single pitch deviations for all teeth, displays them in a table and a waveform chart, and computes the cumulative pitch deviation.
I verified the software by importing the same set of CMM measurement data that I had already processed manually. The software outputs exactly matched the manually calculated deviations. This confirms the correctness and feasibility of the LabVIEW-based evaluation system.
In summary, my research achieved the following results. First, I established a complete mathematical model of herringbone gear tooth surfaces based on involute and helix theory. The model correctly describes the left and right flanks of any tooth on both the upper and lower halves of the herringbone gear. Second, I derived practical formulas for evaluating the tooth profile deviation, helix deviation, and pitch deviation from coordinate measurement data. These formulas are suitable for the data processing of CMM measurements. Third, I developed a LabVIEW-based graphical user interface for tooth surface simulation and error evaluation. The software was successfully applied to actual measurement data. The experimental results demonstrate that the theoretical tooth surface model is accurate and the error evaluation methods are reliable.
There are several directions for future work. The current model can be extended to include tooth modifications such as profile crowning and helix crowning, which are often used in high-performance gear transmissions. The measurement method can be improved by using non-contact measurement techniques for small-module gears where the tooth space is too narrow for a tactile probe. Also, the LabVIEW software can be integrated with a CAD system to generate a solid model for finite element analysis. These extensions would further advance the design and quality control of herringbone gears.