In my master’s research, I focused on the vibration and noise reduction of herringbone gear pairs used in marine transmission systems. The herringbone gear, also known as a double helical gear, is widely adopted in naval propulsion due to its high load capacity, excellent axial force cancellation, and smooth meshing. However, as underwater stealth requirements become more stringent, reducing gear-induced noise is critical. Tooth surface modification is one of the most effective ways to improve meshing performance. Among various modification methods, triangular end relief can lower the meshing impact force and loaded transmission error fluctuation, thereby reducing vibration. In this thesis, I systematically investigated the optimum design of triangular end relief for herringbone gears, proposed a three-section grinding wheel form grinding method to realize the modified tooth surface, and verified the performance through dynamics analysis and experiments.
My dissertation is organized as follows: first, I derived the tooth surface geometry of the herringbone gear with triangular end relief and used a multi-objective genetic algorithm to optimize the modification parameters. Second, I proposed a novel form grinding process that employs three different grinding wheel cross-sections to approximate the modified tooth surface. Third, I established a twelve-degree-of-freedom dynamic model of the herringbone gear transmission system and analyzed the dynamic responses under mesh stiffness and impact excitations. Finally, I built test benches to measure loaded transmission error and housing vibration for both standard and modified gear pairs, confirming the effectiveness of triangular end relief.
Design of Triangular End Relief for Herringbone Gear
The standard tooth surface of a herringbone gear can be generated by a hypothetical rack cutter. I used the coordinate systems shown in Figure (from the paper) to describe the generation process. For a right-handed helical flank, the position vector and unit normal vector are expressed in the moving coordinate system as
$$ \mathbf{r}_{ti}(u_i, l_i, \theta_i) = \mathbf{M}_{i,ti} \mathbf{r}_{ti}^{(c)}(u_i, l_i), \quad \mathbf{n}_{ti}(u_i, l_i, \theta_i) = \mathbf{L}_{i,ti} \mathbf{n}_{ti}^{(c)}(u_i, l_i) $$
with the meshing equation
$$ f(u_i, l_i, \theta_i) = \frac{\partial \mathbf{r}_{ti}}{\partial \theta_i} \cdot \mathbf{n}_{ti} = 0 $$
where \(u_i\) and \(l_i\) are the rack cutter surface parameters, \(\theta_i\) is the gear rotation angle, and \(\mathbf{M}_{i,ti}\) is the transformation matrix from the rack cutter to the gear. Similarly, the tooth root fillet surface is obtained by considering the rounded corner of the rack cutter.
Triangular end relief is defined on the rotating projection plane. The modification regions are the tooth tip and tooth root areas near the engaging end. The starting lines of modification are spatial helices, which are approximately straight lines on the projection plane. I calculated the helix angles of these starting lines from the tooth geometry. For a given maximum modification amount \(C_{Ea}\) at the tooth tip and \(C_{Ef}\) at the tooth root, the modification amount at an arbitrary point \(P\) in the modification region is given by:
$$
\delta(z, x’) =
\begin{cases}
C_{Ea} \left(\frac{l_p}{l_a}\right)^{k_a}, & P \in \triangle ABC \\
C_{Ef} \left(\frac{l_p}{l_f}\right)^{k_f}, & P \in \triangle DEF \\
0, & \text{otherwise}
\end{cases}
$$
where \(l_p\) is the distance from the starting line, \(l_a\) and \(l_f\) are the lengths of the triangular modification regions, and \(k_a\) and \(k_f\) are the modification curve orders. For linear modification \(k=1\), and for quadratic parabola modification \(k=2\). The modified tooth surface is then constructed by superposing the modification amount along the normal direction of the standard flank:
$$ \mathbf{r}_{1m}(u_1, l_1, \theta_1) = \mathbf{r}_1(u_1, l_1, \theta_1) + \delta(z, x’) \, \mathbf{n}_1(u_1, l_1, \theta_1) $$
where \(\mathbf{r}_1\) and \(\mathbf{n}_1\) are the position and normal vectors of the standard tooth surface.
To evaluate the meshing performance, I used tooth contact analysis (TCA) and loaded tooth contact analysis (LTCA). The loaded transmission error is obtained from the normal displacement along the line of action. The meshing impact force occurs when the driven gear tooth enters contact outside the theoretical line due to tooth deflection. I calculated the actual impact point using the LTCA method and then derived the impact velocity and impact force based on energy conservation.
I defined the optimization problem with two objectives: minimizing the loaded transmission error fluctuation \(\Delta T_e\) and minimizing the meshing impact force \(F_s\). The design variables are the maximum modification amounts and modification heights at the tooth tip and root:
$$
\min \; F = w_1 \frac{\Delta T_e}{\Delta T_{e0}} + w_2 \frac{F_s}{F_{s0}}
$$
$$
\text{s.t. } Q_{a\min} \le y_1 \le Q_{a\max}, \quad Q_{f\min} \le y_2 \le Q_{f\max}, \quad h_{a\min} \le y_3 \le h_{a\max}, \quad h_{f\min} \le y_4 \le h_{f\max}
$$
where \(w_1\) and \(w_2\) are weighting factors (both taken as 0.5). I used the NSGA-II genetic algorithm to solve this multi-objective problem. The basic parameters of the herringbone gear pair are listed in the following table.
| Parameter | Pinion | Wheel |
|---|---|---|
| Number of teeth | 30 | 72 |
| Normal module (mm) | 5 | |
| Pressure angle (°) | 20 | |
| Helix angle (°) | 33.273 | |
| Face width (mm) | 44×2 | 40×2 |
| Root clearance (mm) | 46 | 50 |
| Input speed (r/min) | 2500 | — |
| Rated torque (Nm) | — | 3000 |
For the linear triangular end relief, the optimized parameters are summarized below.
| Parameter | Range | Optimum |
|---|---|---|
| Tip maximum modification amount (μm) | (8,20) | 16.10 |
| Tip modification height (mm) | (2.01,5) | 2.72 |
| Root maximum modification amount (μm) | (8,20) | 9.31 |
| Root modification height (mm) | (1.27,4.04) | 2.81 |
For the quadratic parabola triangular end relief, the optimized values are:
| Parameter | Range | Optimum |
|---|---|---|
| Tip maximum modification amount (μm) | (8,20) | 16.53 |
| Tip modification height (mm) | (2.01,5) | 4.52 |
| Root maximum modification amount (μm) | (8,20) | 11.63 |
| Root modification height (mm) | (1.27,4.03) | 2.44 |
At the rated torque of 3000 Nm, the linear modification reduced the loaded transmission error fluctuation by 75.55% and the meshing impact force by 93.60% compared with the standard flank. The quadratic parabola modification achieved even larger reductions: 83.7% in transmission error fluctuation and 94.37% in impact force. These results confirm that the quadratic parabola form is superior to the linear form for improving mesh behavior.
Three-Section Grinding Wheel Form Grinding of Triangular End Relief
To manufacture the modified tooth surface with high precision and efficiency, I proposed a new form grinding method using three grinding wheel cross-sections. The linear triangular end relief can be approximated by three involute helical surfaces with slightly different basic parameters: the tip modification surface is obtained by increasing the pressure angle and helix angle, while the root modification surface is obtained by decreasing them. For the quadratic parabola modification, an additional parabolic tooth profile modification is superimposed.
I first established the coordinate transformation from the grinding wheel to the workpiece gear. The wheel axial profile is derived by applying the condition that the normal of the contact point must pass through the wheel axis. The transformation matrix from the gear coordinate system \(S_1\) to the wheel coordinate system \(S_t\) is given by:
$$ \mathbf{M}_{t1} = \mathbf{M}_{tc} \mathbf{M}_{ca} \mathbf{M}_{a1} $$
with the corresponding matrix elements for the machine movements. Using the actual YK7332A CNC form grinding machine, the machine axis parameters were derived as:
$$
\psi_a = \theta_1, \quad \psi_b = \gamma_m, \quad C_x = -L_t, \quad C_y = E_t
$$
where \(\theta_1\) is the gear rotation, \(\gamma_m\) is the wheel setting angle, \(L_t\) is the axial feed, and \(E_t\) is the center distance.
The target modified tooth surface was discretized into a set of points. For the linear case, the design variables were the normal module \(m_n\), normal pressure angle \(\alpha_n\), and helix angle \(\beta\) of each section. I minimized the sum of squared normal deviations between the actual three-section surfaces and the target modified surfaces. The optimization model is:
$$ \min f_j(\mathbf{d}) = \mathbf{H}_j^T(\mathbf{d}) \mathbf{H}_j(\mathbf{d}), \quad j = a, f $$
where \(\mathbf{H}_j\) is the vector of point-wise normal deviations. The NSGA-II algorithm was used to solve the nonlinear least-squares problem. The optimized results are listed below.
| Parameter | Tip section | Root section |
|---|---|---|
| Module (mm) | 5.00053 | 5.00383 |
| Pressure angle (°) | 20.5342 | 19.9219 |
| Helix angle (°) | 33.4012 | 33.2534 |
| Sum of squared deviations (mm²) | 6.12×10⁻⁷ | 1.30×10⁻⁶ |
For the quadratic parabola case, two additional variables were included: the profile modification height \(h\) and the modification amount \(\varsigma\). The optimized values for the tip and root sections are:
| Parameter | Tip section | Root section |
|---|---|---|
| Module (mm) | 4.9995 | 5.00432 |
| Pressure angle (°) | 20.0423 | 19.9641 |
| Helix angle (°) | 33.3354 | 33.2498 |
| Profile modification height (mm) | 6.44 | 3.89 |
| Profile modification amount (μm) | 16.25 | 3.5 |
| Sum of squared deviations (mm²) | 1.12×10⁻⁴ | 5.43×10⁻⁵ |
The maximum normal deviation was below 1 μm for the linear case and below 3 μm for the quadratic parabola case, which indicates that the three-section method can accurately realize the desired triangular end relief. I then machined a pinion with linear triangular end relief on the YK7332A grinder. The tooth flank quality was inspected with a CLINBERG P40 gear measuring machine. The measured deviations were mostly within the ISO 4 grade accuracy. A rolling test with red lead powder showed satisfactory contact patterns, verifying the feasibility of the proposed grinding method.
Dynamic Characteristics of Triangular End Relieved Herringbone Gear
To assess the reduction in vibration, I developed a twelve-degree-of-freedom lumped-parameter dynamic model for the herringbone gear pair, considering bending, torsional, and axial motions. The gear body is supported by sliding bearings. The dynamic equations are written in matrix form:
$$ \mathbf{M}\ddot{\mathbf{q}} + \mathbf{C}\dot{\mathbf{q}} + \mathbf{K}(t)\mathbf{q} = \mathbf{F}(t) $$
where \(\mathbf{q}\) is the generalized displacement vector containing the translational and rotational coordinates of the two helical gear sections. The time-varying mesh stiffness \(K_m(t)\) was obtained from LTCA and then fitted by a Fourier series. The mesh impact force \(F_s(t)\) was included as an external excitation at the mesh entrance.
The tooth mesh force has components in the radial, axial, and tangential directions. The dynamic equations for the left and right hands of the pinion and wheel are coupled through the axial displacement of the floating pinion. I assumed perfect symmetry between the left and right helical gears, so the excitations on both sides are identical.
I calculated the time-varying mesh stiffness for the standard and modified tooth surfaces. The standard gear has a mesh stiffness fluctuation of \(1.16\times10^8\,\text{N/m}\), whereas the linear triangular end relief reduces it to \(2.4\times10^7\,\text{N/m}\), and the quadratic parabola modification to \(1.6\times10^7\,\text{N/m}\). The mesh impact force is also drastically reduced as mentioned in the previous section. The vibration acceleration in the direction of the line of action was obtained from the dynamic model. At 3000 Nm load and 2500 r/min, the root mean square (RMS) value of the line-of-action vibration acceleration is 70.24 m/s² for the standard gear, 11.83 m/s² for the linear modification, and 8.43 m/s² for the quadratic parabola modification. The corresponding reductions are 83.15% and 88%.
The dynamic load fluctuation is also decreased by 66.98% for the linear modification and 74.2% for the quadratic parabola modification. The frequency spectra show that the dominant vibration component occurs at mesh frequency and its harmonics. The results reveal that triangular end relief effectively weakens the internal excitations, thus reducing gear-borne vibration.
Experimental Verification
To validate the simulation results, I designed and conducted two types of experiments: a loaded transmission error test at very low speed and a vibration test at high speed. The test gears included a standard pinion P01, a triangular end relief pinion P02 (linear modification), and a standard wheel G01.
For the loaded transmission error measurement, I used a low-speed high-torque open-loop test rig. The pinion speed was 4.18 r/min, and wheel torque was varied from 1000 Nm to 4000 Nm. Each gear pair was tested under the same conditions. The transmission error was measured with high-precision rotary encoders and processed to extract the mesh-frequency component.
| Torque (Nm) | P01G01 test | P01G01 sim | P02G01 test | P02G01 sim |
|---|---|---|---|---|
| 1000 | 0.73 | 0.26 | 0.91 | 0.48 |
| 1500 | 0.80 | 0.39 | 0.85 | 0.45 |
| 2000 | 0.86 | 0.53 | 0.74 | 0.36 |
| 2500 | 1.02 | 0.67 | 0.69 | 0.27 |
| 3000 | 1.16 | 0.81 | 0.76 | 0.20 |
| 3500 | 1.25 | 0.95 | 0.86 | 0.26 |
| 4000 | 1.33 | 1.09 | 1.01 | 0.36 |
At 3000 Nm, the modified gear pair had a measured fluctuation of 0.76″ compared with 1.16″ for the standard pair, a reduction of 34%. The simulation shows a 75% reduction, and the discrepancy is mainly due to manufacturing and assembly errors. The general tendency of the measured curves matches well with the simulated curves, confirming the effectiveness of the modification.
For the vibration test, I used a mechanical closed-loop test rig. The pinion speed was 2500 r/min, and the torque was set to 1000, 2000, 3000, and 3600 Nm. Accelerometers were mounted on the housing foot, and the vertical vibration acceleration was recorded. The RMS values over 100 mesh cycles were calculated.
| Torque (Nm) | P01G01 test | P02G01 test | P01G01 sim (line of action) | P02G01 sim (line of action) |
|---|---|---|---|---|
| 1000 | 3.34 | 2.93 | 22.72 | 26.87 |
| 2000 | 3.51 | 2.33 | 44.55 | 20.21 |
| 3000 | 4.22 | 3.36 | 70.24 | 11.83 |
| 3600 | 4.76 | 4.07 | 80.12 | 14.86 |
At 3000 Nm, the housing vibration of the modified pair was 3.36 m/s², 20% lower than the 4.22 m/s² of the standard pair. The simulated line-of-action acceleration was 83% lower. The discrepancy between absolute values is attributed to the attenuation through bearings and housing. The correlation between transmission error fluctuation and vibration acceleration is evident in both measurement and simulation.
In summary, my experimental results confirm that the triangular end relief optimization can significantly reduce the loaded transmission error fluctuation and the housing vibration level of herringbone gear pairs. The proposed three-section grinding wheel method successfully manufactured the modified pinion with acceptable accuracy. Therefore, triangular end relief is a valuable technique for improving the acoustic stealth of marine herringbone gear transmissions.
Conclusion
In this study, I have presented a systematic approach for designing, manufacturing, and evaluating triangular end relief on herringbone gears. The main conclusions are:
- The multi-objective optimization based on TCA and LTCA can effectively determine the optimal modification parameters. The quadratic parabola modification outperforms the linear one in reducing transmission error fluctuation and meshing impact force.
- The three-section grinding wheel form grinding method provides a practical way to produce triangular end relief with high precision. The maximum theoretical deviation is less than 1 μm for linear and less than 3 μm for quadratic parabola modifications.
- The dynamic analysis demonstrates that triangular end relief reduces the primary internal excitations, resulting in a substantial drop in vibration acceleration along the line of action.
- Experiments on loaded transmission error and housing vibration validate the simulation trends and confirm the effectiveness of triangular end relief in reducing vibration.
Future work could extend the method to consider manufacturing errors and perform full system finite element analysis for more accurate housing vibration prediction.