I have focused my research on the tooth surface modification and NVH performance optimization of hypoid bevel gears used in automobile drive axles. The hypoid bevel gear is a critical transmission component in drive axle systems because it offers large contact ratio, stable transmission, and high load capacity. However, under actual working conditions, elastic deformation of shafts, bearings, and housing causes mesh misalignment, which degrades the meshing performance and leads to high-speed whine. Therefore, I developed a systematic method that combines numerical tooth surface generation, finite element loaded contact analysis, Ease-off topology modification, and drive axle NVH simulation. I validated the proposed method through grinding experiments, end-of-line bench tests, and road tests.
The core motivation of my work is that the tooth surface meshing performance of the hypoid bevel gear directly determines the NVH performance of the drive axle. In real vehicle operation, the hypoid bevel gear pair experiences load-dependent misalignment. This misalignment changes the contact pattern, increases transmission error, and excites gear whine. I found that conventional unmodified tooth surfaces often cannot meet the NVH requirements. Therefore, I proposed a quantitative pre-control method for tooth surface mismatch and a topology modification approach that can freely adjust the Ease-off surface coefficients.
I structured my research into four main parts. First, I constructed a numerical tooth surface and three-dimensional modeling method for hypoid bevel gears manufactured by the HFT method. Second, I established a finite element loaded contact analysis and NVH simulation method. Third, I proposed a tooth surface mismatch pre-control and topology modification method, including reverse adjustment of machining parameters. Fourth, I investigated the influence of loaded meshing performance on drive axle NVH before and after modification. I also completed machining experiments, EOL bench tests, and road tests.
1. Numerical Tooth Surface Generation and 3D Modeling of Hypoid Bevel Gear
I began by establishing a mathematical model for the HFT grinding process of the hypoid bevel gear. The HFT method means that the gear is generated by the forming method, while the pinion is generated by the tilt method. I derived the cutter head coordinate system and the theoretical tooth surface equations. For the pinion, the cutter blade surface can be expressed as:
$$
\mathbf{r}_k(u_k,\theta_k) =
\begin{bmatrix}
(r_k + u_k \sin\alpha_k)\cos\theta_k \\
(r_k + u_k \sin\alpha_k)\sin\theta_k \\
-u_k \cos\alpha_k \\
1
\end{bmatrix}
$$
where \(k=g\) denotes the inner blade and \(k=p\) denotes the outer blade. The unit normal vector is:
$$
\mathbf{n}_k =
\frac{\mathbf{N}_k}{|\mathbf{N}_k|}
=
\begin{bmatrix}
\cos\alpha_k \cos\theta_k \\
\cos\alpha_k \sin\theta_k \\
-\sin\alpha_k
\end{bmatrix}
$$
I transformed the cutter surface into the machine coordinate system through a series of coordinate transformations. The meshing equation for the pinion generated by the tilt method is:
$$
\mathbf{n}_m \cdot \mathbf{v}_m^{(c1)} = 0
$$
where \(\mathbf{v}_m^{(c1)}\) is the relative velocity between the cutter and the workpiece. After substituting the relative velocity and the normal vector, I obtained the equation of meshing:
$$
\phi = \phi(u_k,\theta_k)
$$
By eliminating \(\phi\), I obtained the pinion tooth surface and its unit normal in the workpiece coordinate system:
$$
\begin{aligned}
\mathbf{r}_1(u_k,\theta_k) &= \mathbf{M}_{1h}\mathbf{M}_{hs}\mathbf{M}_{sm}\mathbf{r}_m(u_k,\theta_k) \\
\mathbf{n}_1(u_k,\theta_k) &= \mathbf{L}_{1h}\mathbf{L}_{hs}\mathbf{L}_{sm}\mathbf{n}_m(u_k,\theta_k)
\end{aligned}
$$
For the gear, I used the forming method. The gear tooth surface was obtained directly from the cutter surface without a generating motion. The gear tooth surface equation is:
$$
\begin{aligned}
\mathbf{r}_2(u_w,\theta_w) &= \mathbf{M}_{2n}\mathbf{M}_{ng}\mathbf{r}_g(u_w,\theta_w) \\
\mathbf{n}_2(u_w,\theta_w) &= \mathbf{L}_{2n}\mathbf{L}_{ng}\mathbf{n}_g(u_w,\theta_w)
\end{aligned}
$$
I also derived the fillet transition surface to capture the root bending stress accurately. The fillet equation for the gear is:
$$
\mathbf{r}_g'(u_w,\theta_w,\beta) =
\begin{bmatrix}
(r_w + u_w \sin\alpha_w – r_0 \cos\alpha_w \pm r_0 \sin\beta)\cos\theta_w \\
(r_w + u_w \sin\alpha_w – r_0 \cos\alpha_w \pm r_0 \sin\beta)\sin\theta_w \\
-u_w \cos\alpha_w + r_0 \sin\alpha_w – r_0 \cos\beta \\
1
\end{bmatrix}
$$
where \(\beta\) is the fillet parameter. The corresponding normal is:
$$
\mathbf{n}_g’ = \frac{\mathbf{N}_g’}{|\mathbf{N}_g’|}
$$
To compute the numerical tooth surface, I used the spatial rotation projection relationship. I divided the tooth surface into a grid of \(5 \times 9\) points. The boundary points \(A_1, A_2, A_3, A_4\) were calculated from the gear blank geometry. The shrunk boundary lines were obtained by offsetting the original boundaries by the shrink amounts \(d_1, d_2, d_3, d_4\). For example, the line \(A_1’A_2’\) is:
$$
y = k_{A_1A_2}(x – X_{A_1}) + Y_{A_1} – \frac{d_1}{\cos(\arctan(k_{A_1A_2}))}
$$
I then solved for the intersection points and generated the grid points \(P_{ij}\). For each grid point, the tooth surface coordinates were obtained by solving the following system:
$$
\begin{cases}
x_P(u_P,\theta_P) = X_{ij} \\
\sqrt{y_P^2(u_P,\theta_P) + z_P^2(u_P,\theta_P)} = Y_{ij}
\end{cases}
$$
I implemented this procedure in MATLAB. The geometric parameters and machining parameters of the hypoid bevel gear pair are listed in Table 1 and Table 2. I used these parameters to generate the numerical tooth surface and build the 3D model in UG software.
| Parameter | Pinion | Gear |
|---|---|---|
| Number of teeth | 8 | 39 |
| Hand of spiral | Left | Right |
| Shaft angle (deg) | 90 | 90 |
| Offset (mm) | 35 (down) | 35 (down) |
| Module (mm) | 6.283 | 6.283 |
| Spiral angle at reference (deg) | 50.24 | 31.37 |
| Pressure angle (deg) | 22.5 | 22.5 |
| Face width (mm) | 44.4 | 38 |
| Whole tooth depth (mm) | 12.15 | 11.89 |
| Addendum (mm) | 8.99 | 1.59 |
| Pitch cone angle (deg) | 13.1 | 76.2 |
| Face cone angle (deg) | 18.3 | 77.13 |
| Root cone angle (deg) | 12.2 | 70.75 |
| Parameter | Pinion convex | Pinion concave | Gear |
|---|---|---|---|
| Cutter radius (mm) | 108.075 | 120.395 | 114.3 (cutter top distance 3.12) |
| Cutter edge radius (mm) | 1.297 | 1.297 | 1.397 |
| Blade profile angle (deg) | -31 | 14 | 22.5 / -22.5 |
| Tilt angle (deg) | 15.57 | 12.02 | 0 |
| Swivel angle (deg) | 320.32 | 331.28 | 0 |
| Radial cutter position (mm) | 104.4274 | 103.64879 | 109.058 |
| Angular cutter position (deg) | 82.25 | 83.3 | 67.845 |
| Installation angle (deg) | 355.96 | 359.4 | 69.79 |
| Vertical offset (mm) | 35.86444 | 26.43132 | 0 |
| Axial offset (mm) | -1.54323 | 4.04764 | 0.8075 |
| Machine bed (mm) | 23.62655 | 15.18808 | 0 |
| Roll ratio | 4.6933 | 4.666443 | 0 |
Using the above parameters, I computed the tooth surface point coordinates. Table 3 shows a part of the pinion concave tooth surface points. Table 4 shows a part of the pinion convex tooth surface points. I then built the 3D model of the hypoid bevel gear pair in UG software. The modeling process consisted of creating the blank, importing the point cloud, creating the tooth surfaces, and arraying the tooth slots. The final assembly of the hypoid bevel gear pair was used for finite element analysis.
| Index | X (mm) | Y (mm) | Z (mm) |
|---|---|---|---|
| (1,1) | 76.021369 | 26.656459 | -9.226197 |
| (2,1) | 76.385425 | 24.745287 | -9.876669 |
| (3,1) | 76.749481 | 22.871881 | -10.287764 |
| (4,1) | 77.113537 | 21.066037 | -10.448025 |
| (5,1) | 77.477593 | 19.367708 | -10.329767 |
| (1,5) | 97.187224 | 8.593851 | -34.135786 |
| (5,5) | 99.178903 | 1.909270 | -26.573710 |
| (1,9) | 118.353079 | -24.906947 | -34.058334 |
| (5,9) | 120.880213 | -22.784585 | -21.510340 |
| Index | X (mm) | Y (mm) | Z (mm) |
|---|---|---|---|
| (1,1) | 76.021369 | 20.887011 | -18.958436 |
| (2,1) | 76.385425 | 20.878646 | -16.551736 |
| (3,1) | 76.749481 | 20.545983 | -14.381363 |
| (4,1) | 77.113537 | 19.924228 | -12.488566 |
| (5,1) | 77.477593 | 19.035947 | -10.929087 |
| (1,5) | 97.187224 | -7.062521 | -34.485171 |
| (5,5) | 104.604231 | 2.000862 | -26.566970 |
| (1,9) | 104.072845 | -38.210165 | -17.897192 |
| (5,9) | 108.899855 | -22.581429 | -21.723515 |
2. Finite Element Loaded Contact Analysis and NVH Simulation
After building the 3D model of the hypoid bevel gear, I established the tooth contact analysis (TCA) mathematical model. TCA simulates the meshing process without load and provides the contact pattern and transmission error. The hypoid bevel gear pair meshing model is based on the relative position and motion between the pinion and the gear. The position vectors and normal vectors of the two tooth surfaces must be equal at the contact point:
$$
\begin{aligned}
\mathbf{r}_a^{(1)}(u_1,\theta_1,\phi_1) &= \mathbf{r}_a^{(2)}(u_2,\theta_2,\phi_2) \\
\mathbf{n}_a^{(1)}(u_1,\theta_1,\phi_1) &= \mathbf{n}_a^{(2)}(u_2,\theta_2,\phi_2)
\end{aligned}
$$
These vector equations yield five independent scalar equations. With six unknowns, I fixed the pinion rotation angle \(\phi_1\) and solved for the remaining five unknowns. The theoretical transmission error is defined as:
$$
\Delta\phi_2 = (\phi_2 – \phi_2^{(0)}) – \frac{z_1}{z_2}(\phi_1 – \phi_1^{(0)})
$$
where \(\phi_1^{(0)}\) and \(\phi_2^{(0)}\) are the initial rotation angles at the reference point. I implemented the TCA algorithm in MATLAB and obtained the contact pattern and transmission error curves.
For loaded contact analysis, I used ABAQUS software. The finite element simulation process includes pre-processing, solution setup, and post-processing. I divided the full gear into single tooth models to improve mesh quality. I used hexahedral elements with C3D8R type. The material properties are listed in Table 5. The friction coefficient was set to 0.06, and the contact type was hard contact. The analysis type was dynamic implicit with geometric nonlinearity enabled.
| Parameter | Value |
|---|---|
| Elastic modulus (MPa) | 210000 |
| Density (kg/m³) | 7850 |
| Poisson’s ratio | 0.3 |
| Element type | C3D8R |
| Number of elements (gear) | 92352 |
| Number of elements (pinion) | 97400 |
| Friction coefficient | 0.06 |
| Contact type | Hard contact |
| Analysis type | Dynamic implicit |
I applied a torque to the pinion and extracted the contact stress, transmission error, and root bending stress. To obtain the complete loaded contact area, I wrote a Python script to extract the maximum contact stress for each element over all time increments. The script opens the odb file, reads the field output, stores the maximum contact stress for each element, and writes the results to a file. The extracted loaded contact area for the gear concave and convex surfaces showed an elliptical shape with an internal diagonal contact pattern. The maximum contact stress was \(25.83 \times 10\) MPa for the gear concave surface and \(27.815 \times 10\) MPa for the gear convex surface.
I also extracted the root bending stress. The root bending stress curve showed a parabolic trend. The loaded transmission error was obtained by comparing the theoretical and actual rotation angles. The loaded transmission error amplitude decreased as the load increased.
To evaluate the NVH performance of the drive axle, I built a drive axle model in MASTA software. The hypoid bevel gear pair was generated using the same geometric and machining parameters as in the finite element model. The drive axle model included the main reducer, differential, shafts, and housing. I imported the housing stiffness matrix and mass matrix from ABAQUS into MASTA. The NVH analysis module in MASTA uses the loaded transmission error as the excitation to calculate the system response. I placed a measurement point on the pinion outer bearing housing, consistent with the actual test position. The vibration and noise curves in the Z direction (vertical to the road) were used as the evaluation criterion. Since the pinion has 8 teeth, the 8th order vibration and noise curves were analyzed.
3. Tooth Surface Mismatch Modification and Machining Parameter Reverse Adjustment
I proposed a quantitative pre-control and topology modification method for the hypoid bevel gear tooth surface. The method is based on the complete conjugate principle. I first constructed the pinion reference tooth surface that is completely conjugate to the gear tooth surface. The conjugate meshing mathematical model is similar to the TCA model but with the condition that the two surfaces are conjugate. The equation of meshing for the conjugate pair is:
$$
\mathbf{n}_a^{(2)} \cdot \mathbf{v}_a^{(21)} = 0
$$
where the relative velocity is:
$$
\mathbf{v}_a^{(21)} = \boldsymbol{\omega}_a^{(2)} \times \mathbf{r}_a^{(2)} – \boldsymbol{\omega}_a^{(1)} \times \mathbf{r}_a^{(1)} + \boldsymbol{\omega}_a^{(2)} \times (\mathbf{O}_a^{(1)} – \mathbf{O}_a^{(2)})
$$
After solving the equation of meshing, I transformed the gear tooth surface into the pinion coordinate system to obtain the pinion reference tooth surface \(\mathbf{r}_1’\) and its normal \(\mathbf{n}_1’\).
I then computed the deviation between the actual pinion tooth surface and the reference tooth surface. The actual pinion tooth surface was obtained from the HFT machining model. I rotated the actual tooth surface around its axis so that its midpoint coincides with the reference tooth surface midpoint. The rotation angle \(\theta\) was found from:
$$
\begin{bmatrix}
x_{p0}’ \\ y_{p0}’ \\ z_{p0}’
\end{bmatrix}
=
\begin{bmatrix}
1 & 0 & 0 \\
0 & \cos\theta & \sin\theta \\
0 & -\sin\theta & \cos\theta
\end{bmatrix}
\begin{bmatrix}
x_{p1} \\ y_{p1} \\ z_{p1}
\end{bmatrix}
$$
For any point \(M_1\) on the actual tooth surface, the deviation \(\Delta\delta\) from the reference tooth surface is defined by:
$$
\begin{aligned}
x_{M1} – x_{M0} &= \Delta\delta \, n_{xM0} \\
y_{M1} – y_{M0} &= \Delta\delta \, n_{yM0} \\
z_{M1} – z_{M0} &= \Delta\delta \, n_{zM0}
\end{aligned}
$$
The resulting deviation surface is the Ease-off topology. I approximated the Ease-off topology using a second-order surface polynomial:
$$
\Delta\delta = a_0 + a_1 X + a_2 Y + a_3 X^2 + a_4 Y^2 + a_5 XY
$$
where \(X\) and \(Y\) represent the tooth length and tooth height directions. The coefficients have the following meanings: \(a_0\) is a constant term (zero after midpoint coincidence), \(a_1\) is the spiral angle correction coefficient, \(a_2\) is the pressure angle correction coefficient, \(a_3\) is the tooth length crowning coefficient, \(a_4\) is the profile crowning coefficient, and \(a_5\) is the tooth surface deflection coefficient. I solved these coefficients using the least squares method from the computed deviations.
I found that different combinations of these coefficients produce different Ease-off topologies, which correspond to different contact patterns and transmission errors. Therefore, I can pre-control the tooth surface mismatch by modifying these coefficients. For example, if only \(a_1\) is nonzero, the Ease-off is linear in \(X\), which changes the spiral angle. If only \(a_3\) is nonzero, the Ease-off is parabolic in \(X\), which creates tooth length crowning. By adjusting all five coefficients, I can design a target Ease-off topology that meets the desired meshing performance.
I then constructed the pinion modification target tooth surface using the modified Ease-off topology:
$$
\begin{aligned}
x_{T1} &= x_{M0} + \Delta\delta \, n_{xM0} \\
y_{T1} &= y_{M0} + \Delta\delta \, n_{yM0} \\
z_{T1} &= z_{M0} + \Delta\delta \, n_{zM0}
\end{aligned}
$$
where \(\Delta\delta\) is the modified Ease-off value. The target tooth surface is the pinion surface that would produce the desired mismatch relative to the gear.
To realize the target tooth surface, I needed to reverse-adjust the pinion machining parameters. I established the sensitivity matrix between the tooth surface deviation and the machining parameters. The deviation equation is:
$$
\Delta h_i = \sum_{j=1}^{k} \eta_{ij} \Delta \xi_j
$$
where \(\Delta h_i\) is the tooth surface deviation at the \(i\)-th grid point, \(\eta_{ij}\) is the sensitivity coefficient of the \(j\)-th machining parameter on the \(i\)-th grid point, and \(\Delta \xi_j\) is the correction of the \(j\)-th machining parameter. In matrix form:
$$
\{\Delta h\} = [J] \{\Delta \xi\}
$$
where \([J]\) is the sensitivity matrix of size \(m \times k\), with \(m\) grid points and \(k\) machining parameters. Since \(m > k\), this is an overdetermined system. I solved it using sequential quadratic programming (SQP) with constraints on the parameter adjustments. The optimization objective is:
$$
\min_{\Delta \xi} \max_i \left| \sum_{j=1}^{k} J_{ij} \Delta \xi_j – \Delta h_i \right|
$$
subject to:
$$
\Delta \xi_j^{\min} < \Delta \xi_j < \Delta \xi_j^{\max}
$$
I selected ten machining parameters for correction: cutter radius, tilt angle, swivel angle, radial cutter position, angular cutter position, installation angle, vertical offset, axial offset, machine bed, and roll ratio. The allowable adjustment ranges are listed in Table 6.
| Parameter | Minimum | Maximum |
|---|---|---|
| Cutter radius (mm) | -20 | 20 |
| Tilt angle (deg) | -5 | 5 |
| Swivel angle (deg) | -15 | 15 |
| Radial cutter position (mm) | -10 | 10 |
| Angular cutter position (deg) | -10 | 10 |
| Installation angle (deg) | -5 | 5 |
| Vertical offset (mm) | -10 | 10 |
| Axial offset (mm) | -10 | 10 |
| Machine bed (mm) | -10 | 10 |
| Roll ratio | -0.5 | 0.5 |
I implemented the sensitivity matrix calculation in a software tool. The computed sensitivity matrix for the pinion convex surface is partially shown below:
$$
J = \begin{bmatrix}
0.07000 & 0.1614 & 0.0424 & 0.2438 & 0 & 0.1743 & 0.7065 & 1.8462 & 0.9801 & 2.038 \\
0.06557 & 0.1557 & 0.0453 & 0.2334 & 0.00001 & 0.1489 & 0.7079 & 1.7187 & 0.7241 & 2.478 \\
0.06059 & 0.1488 & 0.0487 & 0.2217 & 0 & 0.1198 & 0.7105 & 1.5782 & 0.4646 & 2.967 \\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
0.04398 & 0.1842 & 0.0423 & 0.2462 & 0.00001 & 0.1537 & 0.6307 & 0.1312 & 0.8087 & 2.613 \\
0.04865 & 0.2156 & 0.0310 & 0.2537 & 0 & 0.1852 & 0.6276 & 0.1586 & 1.3328 & 1.877
\end{bmatrix}
$$
Using this matrix, I solved for the machining parameter corrections. The modified pinion convex surface machining parameters are listed in Table 7. The maximum deviation between the actual pinion tooth surface after modification and the target tooth surface was less than \(2.359 \mu m\), which is well within the \(10 \mu m\) (0.01 mm) tolerance. This confirmed that the reverse adjustment was successful.
| Parameter | Modified value | Correction |
|---|---|---|
| Cutter radius (mm) | 117.73 | 9.655 |
| Tilt angle (deg) | 15.90 | 0.33 |
| Swivel angle (deg) | 330.14 | 9.82 |
| Radial cutter position (mm) | 107.5162 | 3.0888 |
| Angular cutter position (deg) | 84.77 | 2.52 |
| Installation angle (deg) | 355.98 | 0.02 |
| Vertical offset (mm) | 33.6815 | -2.18294 |
| Axial offset (mm) | 1.5049 | 0.03833 |
| Machine bed (mm) | 18.7449 | -4.88165 |
| Roll ratio | 4.6279 | -0.0654 |
I performed TCA analysis before and after modification. The original tooth surface had an internal diagonal contact pattern with a transmission error amplitude of \(30 \mu rad\). After modification, the contact pattern had a smaller internal diagonal trend, larger contact area, and the transmission error amplitude was reduced to \(14.8 \mu rad\). This met my modification objectives.
4. Influence of Loaded Meshing Performance on Drive Axle NVH
I then investigated how the loaded meshing performance of the hypoid bevel gear affects the drive axle NVH. I first calculated the mesh misalignment under actual working conditions using MASTA software. The misalignment is caused by the elastic deformation of shafts, bearings, and housing under load. The misalignment components are: pinion axial displacement \(\Delta XP\), gear axial displacement \(\Delta XW\), offset change \(\Delta E\), and shaft angle change \(\Delta\Sigma\). The misalignment values for the reverse surface under different torques are listed in Table 8.
| Torque (N·m) | \(\Delta XP\) (μm) | \(\Delta XW\) (μm) | \(\Delta E\) (μm) | \(\Delta\Sigma\) (μm) |
|---|---|---|---|---|
| -40 | -22.428 | 27.625 | 14.6873 | -0.0232 |
| -60 | -30.742 | 43.097 | 21.1243 | -0.0094 |
| -80 | -39.1367 | 58.9671 | 26.6865 | 0.01384 |
| -100 | -47.3308 | 74.3284 | 31.757 | 0.03851 |
I applied these misalignments to the finite element model and performed loaded contact analysis for the original and modified tooth surfaces. The load cases were -40 N·m, -60 N·m, -80 N·m, and -100 N·m. I compared the loaded contact area, maximum contact stress, loaded transmission error, and root bending stress.
The loaded contact area results showed that as the load increased, the contact area increased for both original and modified surfaces. However, under the same load, the modified surface had a smaller internal diagonal and a larger contact area than the original surface. The maximum contact stress results are summarized in Table 9. The modified surface had lower maximum contact stress than the original surface at all loads.
| Torque (N·m) | Original surface (MPa) | Modified surface (MPa) | Reduction (%) |
|---|---|---|---|
| -40 | 18.5 | 16.2 | 12.4 |
| -60 | 21.3 | 18.7 | 12.2 |
| -80 | 24.1 | 21.0 | 12.9 |
| -100 | 27.8 | 23.9 | 14.0 |
The loaded transmission error results are summarized in Table 10. The modified surface had significantly lower transmission error amplitude than the original surface at all loads. The transmission error amplitude decreased as the load increased for both surfaces.
| Torque (N·m) | Original surface (μrad) | Modified surface (μrad) | Reduction (%) |
|---|---|---|---|
| -40 | 28.5 | 18.2 | 36.1 |
| -60 | 25.3 | 15.6 | 38.3 |
| -80 | 22.1 | 13.1 | 40.7 |
| -100 | 19.4 | 11.2 | 42.3 |
The root bending stress results showed a similar trend. As the load increased, the root bending stress increased. Under the same load, the modified surface had lower root bending stress than the original surface. The reduction was approximately 10-15%.
I then performed NVH simulation in MASTA for both original and modified hypoid bevel gear pairs. The vibration and noise curves were obtained at the measurement point on the pinion outer bearing housing. The results showed that under all four loads, the modified surface had lower vibration and noise curves than the original surface. This indicated that the tooth surface modification improved the drive axle NVH performance. Furthermore, the variation trend of the loaded transmission error amplitude and contact stress was consistent with the variation trend of the vibration and noise curves. This means that reducing the loaded transmission error amplitude and contact stress can improve the drive axle NVH performance. This provides a clear direction for hypoid bevel gear tooth surface modification.
5. Experimental Validation
To validate the proposed method, I conducted grinding experiments, end-of-line (EOL) bench tests, and road tests. The original and modified hypoid bevel gears were manufactured by grinding. I used a closed-loop manufacturing process: the gear measuring center measured the tooth surface errors, and the errors were corrected to obtain a tooth surface consistent with the design. The measurement results showed that the tooth surface errors were very small, and the manufactured tooth surfaces were equivalent to the theoretical design surfaces.
I performed roll inspection to obtain the actual contact pattern. The roll inspection contact pattern and the TCA simulated contact pattern matched well in shape, position, and size. This validated the accuracy of the tooth surface design and the TCA method.
I installed the original and modified hypoid bevel gears on the drive axle EOL bench. The EOL bench simulates actual driving conditions by connecting drive motors to the drive shaft and two half shafts. Torque sensors measure the dynamic torque curve. The test condition was a deceleration stage with constant torque of -60 N·m and speed decreasing from 3950 r/min to 1480 r/min. The dynamic torque curve had a red standard line obtained from a benchmark drive axle that meets NVH requirements. The original surface exceeded the standard line between 2000 r/min and 2500 r/min, indicating that it did not meet the NVH requirement. The modified surface remained below the standard line, indicating that the NVH performance was improved.
I also conducted road tests on a flat, dry highway. The test equipment included a 24-channel LMS data acquisition system, a three-axis vibration sensor, and a microphone. The vibration sensor was attached to the pinion outer bearing housing of the drive axle. The microphone was placed near the driver’s right ear. The test was conducted in 5th gear during deceleration. The gear order was \(8 / 0.76 = 10.53\). The noise curves for the original and modified surfaces were measured. For the original surface, the minimum difference between the gear order noise curve and the total noise curve was 4.21 dB. For the modified surface, the minimum difference was 9.49 dB. The gear order noise curve after modification was significantly lower than that before modification. This confirmed that the tooth surface modification improved the drive axle NVH performance.
The experimental results were consistent with the simulation results. The EOL bench tests and road tests verified the effectiveness of the tooth surface modification method and the drive axle NVH simulation method. My research provides a theoretical reference for the NVH performance improvement of automobile drive axles and the tooth surface modification of hypoid bevel gears.
6. Conclusion
I established a numerical tooth surface calculation method for the hypoid bevel gear based on the HFT grinding process. I derived the tooth surface equations, solved the numerical tooth surface using the rotation projection principle, and built a 3D model in UG software.
I established a finite element loaded contact analysis method and an NVH simulation method. I used ABAQUS to simulate the loaded meshing performance and extracted the complete loaded contact area using a Python script. I used MASTA to build the drive axle model and perform NVH simulation.
I proposed a tooth surface mismatch pre-control and topology modification method. I constructed the Ease-off topology based on the complete conjugate principle, decomposed it into a second-order surface, and calculated the mismatch coefficients. I obtained the target tooth surface by modifying these coefficients. I established a sensitivity matrix and used SQP to reverse-adjust the machining parameters. The deviation between the actual tooth surface and the target tooth surface was less than \(2.359 \mu m\).
I investigated the influence of loaded meshing performance on drive axle NVH. The results showed that the modified hypoid bevel gear had smaller contact stress, lower root bending stress, and smaller loaded transmission error amplitude. The NVH curves were also lower after modification. The variation trend of the loaded transmission error amplitude and contact stress was consistent with the NVH curves. The EOL bench tests and road tests validated the simulation results and the modification method.
For future work, I plan to compare different optimization algorithms for the machining parameter reverse adjustment. I also plan to perform multiple rounds of tooth surface modification to further improve the hypoid bevel gear performance. In addition, I will extend the method to other types of gears and drive axle configurations.