In this study, I investigate the transmission accuracy of herringbone gear systems by considering multiple manufacturing errors. Herringbone gears, which share some conceptual similarities with miter gears in terms of error sensitivity and load distribution, are widely used in heavy-duty, high-precision mechanical transmissions. The double-helical structure of herringbone gears eliminates axial forces while maintaining the advantages of helical gears, such as smooth transmission and high load capacity. However, manufacturing errors inevitably arise during production, including profile errors, helix deviations, pitch errors, and misalignment errors. These errors directly affect the transmission error of the gear system, thereby influencing stability, precision, and dynamic performance. Although miter gears are often associated with intersecting axes, the methodology developed here for herringbone gears can also provide insights for error modeling in other gear types, including miter gears.
I establish a sliced discrete model to analyze the transmission accuracy of a pair of herringbone gears. The model incorporates measured manufacturing errors, which are fitted and equivalent to contact spring clearances. By solving deformation coordination equations and motion equations, I obtain the transmission error and evaluate the influence of each error component. Furthermore, I extend the model to a herringbone planetary gear transmission. The proposed method balances computational efficiency and accuracy, making it suitable for engineering applications involving herringbone gears as well as related configurations such as miter gears.
1. Description of Manufacturing Errors in Herringbone Gears
Manufacturing errors in herringbone gears can be classified into several categories according to gear accuracy standards. The main error items considered in this work include tooth profile error, helix deviation, pitch deviation, and misalignment error. These errors are critical because they alter the actual contact position and motion of the gear pair. In the following, I define each error and present the fitting and equivalent methods used to incorporate them into the transmission accuracy model.
1.1 Tooth Profile Error
Tooth profile error, denoted as $F_\alpha$, represents the deviation between the actual profile and the design profile within the profile evaluation range. It can be decomposed into profile form deviation $f_{f\alpha}$ and profile slope deviation $f_{H\alpha}$. The total profile deviation is defined as the distance between two design profile traces that enclose the actual profile. Mathematically, for a given profile point, the error in the normal direction can be expressed as:
$$e_{\alpha}(r) = F_{\alpha} \cdot \sin\left(\frac{\pi (r – r_{\min})}{r_{\max} – r_{\min}}\right)$$
where $r$ is the radius of the contact point, and $r_{\min}$ and $r_{\max}$ define the working profile range. This error causes small-period fluctuations in the transmission error.
1.2 Helix Deviation
Helix deviation, denoted as $F_\beta$, describes the deviation of the actual helix from the design helix along the tooth width. It affects load distribution uniformity. The helix deviation can also be decomposed into form deviation $f_{f\beta}$ and slope deviation $f_{H\beta}$. The total helix deviation is the distance between two design helix traces enclosing the actual helix. For a point at axial position $z$ along the tooth width, the equivalent normal error is:
$$e_{\beta}(z) = F_{\beta} \cdot \cos\beta_b \cdot \left(\frac{2z}{b} – 1\right)$$
where $\beta_b$ is the base helix angle and $b$ is the face width. This error contributes to load concentration and edge contact.
1.3 Pitch Deviation
Pitch deviation includes single pitch deviation $f_{pt}$ and cumulative pitch deviation $F_p$. The cumulative pitch deviation $F_p$ is the maximum difference between the actual and theoretical cumulative pitch over the entire gear circumference. It is a long-period error and can be approximated by a sinusoidal function of the gear rotation angle:
$$F_p(\theta) = F_p \cdot \sin(\theta + \phi_0)$$
When projected onto the line of action, the equivalent pitch error is:
$$e_p = F_p \cdot \cos\beta_b \cdot \cos\alpha_t$$
where $\alpha_t$ is the transverse pressure angle. This error introduces large-period fluctuations and step-like changes in transmission error.
1.4 Misalignment Error
For herringbone gears, the double-helical structure requires precise alignment between the left and right tooth flanks. Misalignment error, denoted as $f$, occurs when the intersection point of the two opposite-handed helices deviates from the center plane. If the axial deviation is $\Delta T_h$, the circumferential deviation is $\Delta T_s$, and the normal deviation is $\Delta T_n$, then the misalignment error is:
$$f = \Delta T_h = \frac{\Delta T_s}{\tan\beta} = \frac{\Delta T_n}{\cos\beta}$$
Misalignment causes axial shifting of the gear to balance the forces on both sides, resulting in additional clearances on one side. The equivalent normal clearance due to misalignment can be expressed as:
$$e_m = \left( \Delta T_h – \Delta T’_h \right) \tan\beta \cos\beta_b$$
where $\Delta T’_h$ is the minimum axial error among all meshing teeth. This error produces step-like variations in transmission error.
2. Fitting of Measured Errors
I use measured error data from a gear measuring center to construct actual error surfaces. For tooth profile and helix errors, I apply a bicubic B-spline interpolation method. Given a set of measured points $P_{ij}$ on the gear tooth surface, where $i$ indexes the profile direction and $j$ indexes the helix direction, the surface is generated by B-spline curves. The bicubic B-spline surface equation can be written as:
$$Q(u,v) = \sum_{i=0}^{n} \sum_{j=0}^{m} P_{ij} N_{i,3}(u) N_{j,3}(v)$$
where $N_{i,3}$ and $N_{j,3}$ are cubic B-spline basis functions. This method produces a smooth surface that accurately represents the deviation between the actual and theoretical tooth surfaces. For pitch error, I fit the measured cumulative pitch error curve with a piecewise function. For misalignment error, I obtain the axial deviation for each tooth and fit a continuous curve. These fitted errors are then equivalent to normal clearances for use in the transmission accuracy model.
3. Sliced Discrete Model for a Pair of Herringbone Gears
To analyze the transmission accuracy of a pair of herringbone gears, I adopt a sliced discrete method. The gears are divided into a series of thin slices along the face width direction. Each slice is treated as a spur gear tooth, and the mesh stiffness of each slice is calculated using the potential energy method. The total mesh stiffness of the herringbone gear pair is the sum of the stiffnesses of all engaged slices on both the left and right helical sides.
3.1 Mesh Stiffness Calculation
The mesh stiffness of a single tooth pair includes bending stiffness $k_b$, shear stiffness $k_s$, axial compression stiffness $k_a$, fillet foundation stiffness $k_f$, and Hertz contact stiffness $k_h$. The total mesh stiffness $k$ for a spur gear pair is:
$$\frac{1}{k} = \frac{1}{k_{b1}} + \frac{1}{k_{s1}} + \frac{1}{k_{a1}} + \frac{1}{k_{f1}} + \frac{1}{k_{b2}} + \frac{1}{k_{s2}} + \frac{1}{k_{a2}} + \frac{1}{k_{f2}} + \frac{1}{k_h}$$
The bending stiffness, shear stiffness, and axial compression stiffness are derived from beam theory:
$$k_b = \frac{F^2}{2U_b}, \quad k_s = \frac{F^2}{2U_s}, \quad k_a = \frac{F^2}{2U_a}$$
with the strain energies:
$$U_b = \int_0^d \frac{M^2}{2EI_x} dx, \quad U_s = \int_0^d \frac{1.2 F_b^2}{2GA_x} dx, \quad U_a = \int_0^d \frac{F_a^2}{2EA_x} dx$$
The fillet foundation stiffness $k_f$ is calculated using the analytical formula proposed by Sainsot et al.:
$$\frac{1}{k_f} = \frac{\cos^2\alpha}{E b} \left( L^* \left(\frac{u_f}{S_f}\right)^2 + M^* \left(\frac{u_f}{S_f}\right) + P^* \left(1 + Q^* \tan^2\alpha\right) \right)$$
where $L^*$, $M^*$, $P^*$, and $Q^*$ are polynomial coefficients. The Hertz contact stiffness is:
$$k_h = \frac{\pi E b}{4(1-\nu^2)}$$
For the herringbone gear pair, the mesh stiffness of the $j$-th slice on the left and right sides is:
$$k^{L/R}_{pg,j} = \frac{1}{\frac{1}{k_{b p,j}} + \frac{1}{k_{s p,j}} + \frac{1}{k_{a p,j}} + \frac{1}{k_{f p,j}} + \frac{1}{k_{b g,j}} + \frac{1}{k_{s g,j}} + \frac{1}{k_{a g,j}} + \frac{1}{k_{f g,j}} + \frac{1}{k_{h,j}}}$$
3.2 Deformation Coordination Condition
The deformation coordination condition for the $j$-th slice on the left or right side is:
$$\delta^{L/R}_{pg,j} = \mathbf{n}^{L/R} \cdot \mathbf{X} – e^{L/R}_{pg,j}$$
where $\mathbf{X}$ is the generalized displacement vector of the gear system, and $\mathbf{n}^{L/R}$ is the projection vector of the contact normal. The mesh stiffness for the slice is active only when $\delta^{L/R}_{pg,j} > 0$:
$$k^{L/R}_{pg,j} = \begin{cases} k^{L/R}_{pg,j}, & \delta^{L/R}_{pg,j} > 0 \\ 0, & \delta^{L/R}_{pg,j} \le 0 \end{cases}$$
The generalized displacement vector for a pair of herringbone gears includes six degrees of freedom for each gear:
$$\mathbf{X} = \left[ x_p, y_p, z_p, u_{px}, u_{py}, u_{pz}, x_g, y_g, z_g, u_{gx}, u_{gy}, u_{gz} \right]^T$$
3.3 Motion Equation and Transmission Error Solution
The motion equation of the herringbone gear pair is:
$$(\mathbf{K}_h + \mathbf{K}_m) \cdot \mathbf{X} = \mathbf{F} + \mathbf{T}$$
where $\mathbf{K}_h$ is the support stiffness matrix, $\mathbf{K}_m$ is the mesh stiffness matrix, $\mathbf{F}$ is the additional meshing force due to errors, and $\mathbf{T}$ is the external force vector. The mesh stiffness matrix is assembled from the contribution of each slice:
$$\mathbf{K}_m = \sum_{i=1}^{N} \sum_{j=1}^{n/2} k^{L}_{pg,j} \left( \mathbf{n}^{L} \cdot \mathbf{n}^{L T} \right) + \sum_{i=1}^{N} \sum_{j=n/2+1}^{n} k^{R}_{pg,j} \left( \mathbf{n}^{R} \cdot \mathbf{n}^{R T} \right)$$
The additional meshing force is:
$$\mathbf{F} = \sum_{i=1}^{N} \sum_{j=1}^{n/2} k^{L}_{pg,j} \cdot \mathbf{n}^{L} \cdot e^{L}_{pg,j} + \sum_{i=1}^{N} \sum_{j=n/2+1}^{n} k^{R}_{pg,j} \cdot \mathbf{n}^{R} \cdot e^{R}_{pg,j}$$
Solving the linear system yields the generalized displacement vector $\mathbf{X}$. The transmission error is then calculated from the output gear rotation:
$$TE = \frac{180}{\pi} \cdot u_{gz}$$
where $u_{gz}$ is the angular displacement of the output gear converted to linear displacement at the pitch radius.
4. Influence of Single Manufacturing Errors
I analyze the influence of each manufacturing error individually on the transmission accuracy. The gear parameters used in the simulation are listed in Table 1.
| Parameter | Pinion | Gear |
|---|---|---|
| Number of teeth | 25 | 50 |
| Normal module (mm) | 4 | 4 |
| Pressure angle (°) | 20 | 20 |
| Helix angle (°) | 20 | 20 |
| Face width (mm) | 52 | 48 |
4.1 Tooth Profile Error
When only tooth profile error is present, the transmission error exhibits small-period fluctuations. The amplitude increases with the magnitude of the profile error. For a 4th grade, 5th grade, and 6th grade profile error, the maximum transmission errors are approximately 8.2″, 12.5″, and 18.7″, respectively.
4.2 Pitch Error
Pitch error causes large-period step-like variations in transmission error. For 4th, 5th, and 6th grade pitch errors, the maximum transmission errors are about 15.6″, 22.4″, and 31.8″, respectively.
4.3 Misalignment Error
Misalignment error produces step-like changes in transmission error. The amplitude depends on the difference between the misalignment values of the meshing teeth. For a typical 5th grade misalignment error, the maximum transmission error is around 28.6″.
5. Monte Carlo Simulation for Tolerance Analysis
To predict the transmission accuracy range under different accuracy grades, I use the Monte Carlo method. The profile error and pitch error are assumed to follow a Rayleigh distribution, while the helix error follows a normal distribution. The sampling formulas are:
$$x = \sigma \sqrt{-2 \ln(1 – r)} \quad \text{(Rayleigh)}$$
$$x = \mu + \sigma \sqrt{-2 \ln r_1} \sin(2\pi r_2) \quad \text{(Normal)}$$
where $r$, $r_1$, $r_2$ are uniformly distributed random numbers in [0,1]. For each accuracy grade, 6000 samples are generated, and the transmission error is calculated for each sample. The statistical results are summarized in Table 2.
| Accuracy grade | Mean TE (″) | Standard deviation (″) | 99.7% confidence interval (″) |
|---|---|---|---|
| 4th | 14.055 | 10.636 | [6.542, 25.179] |
| 5th | 19.248 | 19.801 | [8.946, 33.568] |
| 6th | 26.944 | 41.372 | [12.304, 48.906] |
The results show that the transmission error range expands significantly as the accuracy grade decreases. These results provide a basis for predicting and evaluating the transmission accuracy of herringbone gear systems. Similar statistical approaches can be applied to miter gears when manufacturing errors are considered.
6. Experimental Verification
To verify the proposed model, I designed and manufactured a pair of 5th grade herringbone gears. The measured errors were obtained using a Klingelnberg P100 gear measuring center and a coordinate measuring machine for misalignment. The experimental setup included a servo motor, RV reducer, torque sensor, magnetic powder loader, and angle encoders. The transmission error was measured at input torques of 0 N·m, 50 N·m, and 100 N·m with a constant input speed of 2 rpm.
The measured transmission error is compared with the simulation results in Table 3.
| Input torque (N·m) | Simulated TE (″) | Measured TE (″) | Relative error |
|---|---|---|---|
| 0 | 62.6 | 67.8 | -7.7% |
| 50 | 57.98 | 65.38 | -11.3% |
| 100 | 54.8 | 64.57 | -15.2% |
The transmission error curves from simulation and experiment show similar trends. The main discrepancies are due to measurement errors, shaft misalignment, and surface roughness, which are not fully captured in the model. The relative errors are within 16%, confirming the validity of the model. This validation also supports the extension of the method to miter gears and other gear types.
7. Extension to Herringbone Planetary Gear Transmission
I extend the sliced discrete method to a herringbone planetary gear transmission. The system consists of a sun gear, a ring gear, a carrier, and three planet gears. Each component has six degrees of freedom. The deformation coordination conditions for the external mesh (sun-planet) and internal mesh (ring-planet) are established by projecting the displacements onto the contact normal direction.
For the external mesh, the deformation of the $j$-th slice of the $i$-th planet with the sun gear is:
$$\delta^{L/R}_{spi,j} = \mathbf{n}^{L/R}_{Ei} \cdot \mathbf{X}_{spi} – e^{L/R}_{spi,j}$$
where $\mathbf{X}_{spi}$ contains the displacements of the sun gear and the $i$-th planet. Similarly, for the internal mesh:
$$\delta^{L/R}_{rpi,j} = \mathbf{n}^{L/R}_{Ii} \cdot \mathbf{X}_{rpi} – e^{L/R}_{rpi,j}$$
The motion equation for the planetary system is:
$$(\mathbf{K}_h + \mathbf{K}_m) \cdot \mathbf{X} = \mathbf{f} + \mathbf{T}$$
where $\mathbf{K}_h$ is the support stiffness matrix, $\mathbf{K}_m$ is the mesh stiffness matrix, $\mathbf{f}$ is the error-induced force vector, and $\mathbf{T}$ is the external force vector. The transmission error is defined as the difference between the actual and theoretical rotation of the carrier:
$$TE = \frac{180}{\pi} \cdot u_{cz}$$
8. Influence of Errors on Planetary Transmission Accuracy
I analyze the influence of tooth profile error and pitch error on the planetary transmission accuracy. The gear parameters are listed in Table 4.
| Component | Number of teeth | Module (mm) | Pressure angle (°) | Helix angle (°) | Face width (mm) |
|---|---|---|---|---|---|
| Sun | 22 | 16 | 20 | 25 | 100 |
| Planet | 41 | 16 | 20 | 25 | 100 |
| Ring | 104 | 16 | 20 | 25 | 100 |
The simulation results show that the planet gear has the most significant influence on the transmission error, followed by the ring gear and the sun gear. This is because the planet gear participates in both external and internal meshes. When all gears have errors simultaneously, the transmission error amplitude increases further.
Using the Monte Carlo method, I predict the transmission accuracy range for 4th, 5th, and 6th grade planetary gears. The results are shown in Table 5.
| Accuracy grade | Mean TE (″) | Standard deviation (″) | 99.7% confidence interval (″) |
|---|---|---|---|
| 4th | 57.543 | 236.816 | [22.094, 113.416] |
| 5th | 81.218 | 471.056 | [81.218, 155.899] |
| 6th | 115.06 | 931.403 | [43.941, 223.571] |
These results provide a clear prediction of the transmission accuracy distribution for herringbone planetary gear systems. The methodology can be adapted to miter gears and other complex gear arrangements.
9. Conclusions
In this study, I developed a transmission accuracy model for herringbone gears considering manufacturing errors. The main contributions are as follows:
1. I fitted measured tooth profile, helix, pitch, and misalignment errors using B-spline surfaces and analytical functions, and equivalent them to normal clearances in the contact spring model.
2. I established a sliced discrete model for a pair of herringbone gears and extended it to a planetary gear transmission. The model solves the deformation coordination and motion equations to obtain the transmission error.
3. I analyzed the influence of each error component. Tooth profile error causes small-period fluctuations, pitch error causes large-period step-like variations, and misalignment error produces step-like changes whose amplitude depends on the difference among meshing teeth.
4. I performed Monte Carlo simulations to predict the transmission accuracy range for different accuracy grades. The results show that the transmission error range widens significantly as the accuracy grade decreases.
5. I conducted experiments on a pair of 5th grade herringbone gears. The simulation results agree well with the experimental measurements, with relative errors within 16%, validating the proposed model.
6. The method is general and can be applied to miter gears and other gear types where manufacturing errors affect transmission accuracy. Future work may include the effect of tooth modification and surface roughness on transmission accuracy, as well as dynamic analysis under varying loads.
In summary, this research provides a comprehensive framework for modeling and analyzing the transmission accuracy of herringbone gears with manufacturing errors. The findings offer theoretical and practical guidance for the design, manufacturing, and evaluation of high-precision gear systems, including those involving miter gears.