I have focused my research on the relationship between the transmission error of hypoid bevel gears and the noise, vibration, and harshness performance of the main reducer. Hypoid bevel gears are critical transmission components in the rear axle main reducer of rear-wheel-drive vehicles. They serve to reduce speed, increase torque, and change the direction of motion. The meshing performance of hypoid bevel gears is directly linked to the NVH behavior of the main reducer. Transmission error is one of the most important indices for evaluating the meshing quality of hypoid bevel gears. Lapping is the final modification process in the manufacturing of hypoid bevel gears and can effectively control and reduce transmission error. The transmission error values on the drive side and coast side after lapping determine the NVH performance of the main reducer under acceleration and coast conditions. My objective is to improve the NVH performance of the rear axle main reducer by optimizing and controlling the transmission error of hypoid bevel gears.
In this study, I established a vibration model for the meshing system of hypoid bevel gears, analyzed the transmission error curve and noise generation mechanism, and investigated the influence of gear manufacturing errors on transmission error. I used a Gleason 350GMM gear profile measuring instrument to detect changes in tooth profile errors before and after lapping. I also recorded transmission error values measured by a Gleason 600HTT rolling检查 machine. I then used A-weighted sound pressure level peak noise dB(A) of the main reducer as the evaluation index for NVH performance. Through vehicle road tests under 5th gear acceleration and 5th gear coast conditions, I explored the NVH level of the main reducer under different transmission error values. The experiments showed that, in the presence of installation errors and other factors, the transmission error after lapping is not always better when smaller. There is a range-based correspondence between the transmission error of hypoid bevel gears and the NVH performance of the main reducer.
To overcome the limitations of vehicle road tests, I used a rear axle NVH test bench combined with order analysis to simulate 5th gear acceleration and 5th gear coast conditions. I tested main reducers corresponding to hypoid bevel gears with a wide range of transmission error values. I fitted and analyzed the corresponding data of transmission error and peak noise. The analysis showed that when the drive-side transmission error of hypoid bevel gears is in the range of 8–12 μrad and the coast-side transmission error is in the range of 4–8 μrad, the corresponding main reducer peak noise under 5th gear acceleration is below 60 dB(A), and the peak noise under 5th gear coast is below 65 dB(A). I also optimized the lapping process parameters, including tooth flank clearance, spindle speed, and lapping cycle number, through a three-level, three-factor orthogonal experiment. I designed a comparative test for spindle speed and determined the best lapping parameters. The optimized parameters ensure that the transmission error of hypoid bevel gears after lapping falls within the desired ranges, and the vehicle road test confirms good NVH performance of the main reducer.
My research provides a reference for optimizing transmission error and improving the NVH performance of main reducers equipped with hypoid bevel gears.
1. Vibration Model and Noise Generation Mechanism of Hypoid Bevel Gears
Hypoid bevel gears are manufactured through a sequence of cutting, heat treatment, and lapping. The inherent errors from manufacturing cause dynamic excitation during meshing, which is the root cause of vibration and noise. Even without external excitation, hypoid bevel gears will vibrate due to internal excitation. I established a single-degree-of-freedom torsional vibration model to simulate the force condition of hypoid bevel gears during meshing. In this model, the meshing stiffness is represented by \(K\), the displacement excitation caused by tooth surface errors is represented by \(e\), and the viscous damping coefficient is represented by \(C\). The equation of motion for the hypoid bevel gear pair along the contact line can be written as:
$$
M\ddot{x} + C\dot{x} + K(t)\left(x – e(t)\right) = W
$$
where \(M\) is the equivalent mass of the hypoid bevel gears, \(x\) is the displacement along the tooth surface, \(C\) is the damping coefficient, \(e\) is the manufacturing error, \(K\) is the meshing stiffness, and \(W\) is the static normal load. The equivalent mass and load are defined as:
$$
M = \frac{J_1 J_2}{J_1 r_{b2}^2 + J_2 r_{b1}^2}, \quad
x = r_{b1}\theta_1 – r_{b2}\theta_2
$$
$$
W = \frac{T_1}{r_{b1}} = \frac{T_2}{r_{b2}}
$$
I separated the time-varying meshing stiffness into a constant part and a variable part, and the relative displacement into a static part and a dynamic part:
$$
K(t) = K + \Delta k, \quad x = x_s + \Delta x, \quad W = K x_s
$$
Substituting these into the equation of motion and neglecting higher-order terms, I obtained:
$$
M\Delta\ddot{x} + C\Delta\dot{x} + K\Delta x = -\Delta k x_s + \Delta k e + K e
$$
This equation shows that the main vibration excitations are manufacturing error excitation, time-varying stiffness excitation, and meshing impact excitation caused by the combination of the two. The transmission error of hypoid bevel gears after lapping is a component of manufacturing error, so it is closely related to the NVH performance of the main reducer.
Transmission error is defined as the difference between the actual rotation angle of the driven gear and the theoretical rotation angle calculated from the gear ratio. The expression is:
$$
TE = \left(\phi_2 – \phi_2^{(0)}\right) – \frac{z_1}{z_2}\left(\phi_1 – \phi_1^{(0)}\right)
$$
where \(\phi_1\) and \(\phi_2\) are the rotation angles of the driving and driven hypoid bevel gears, \(\phi_1^{(0)}\) and \(\phi_2^{(0)}\) are the initial angles, and \(z_1\) and \(z_2\) are the numbers of teeth. The transmission error curve is an important indicator of the meshing performance of hypoid bevel gears. In an ideal design, the transmission error curve should be a symmetric parabolic curve to ensure smooth transition of contact points. The theoretical transmission error is:
$$
\Delta\varphi_2 = m’_{21}\left(\varphi_1 – \varphi_1^{(0)}\right)^2
$$
where \(m’_{21}\) is the first derivative of the transmission ratio function. The intersection of adjacent curves is the transfer point, where the contact is transferred from one pair of teeth to the next. The velocity jump at this point determines the impact magnitude, which directly affects the NVH performance of hypoid bevel gears.
I used the single flank testing method with a Gleason 600HTT rolling检查 machine to measure the transmission error. The machine uses optical encoders to measure angular displacement errors. The phase difference between the input and output pulse sequences represents the transmission error. The measured values are output as SFT (single flank testing) values in μrad. I recorded the first-order transmission error values for both the drive side and coast side.
To analyze the influence of tooth profile errors on transmission error, I used a Gleason 350GMM coordinate measuring machine to detect the geometric errors before and after lapping. The changes in spiral angle, pressure angle, tooth profile, and cumulative pitch error are summarized in the following table.
| Parameter | Change after lapping (maximum) | Observation |
|---|---|---|
| Spiral angle error | 1.5 arcmin | Very small change |
| Pressure angle error | 4 arcmin | Small change |
| Tooth profile error | 4 μm | Small change |
| Cumulative pitch error | 11 μm | Significant reduction |
The data show that the cumulative pitch error changes most significantly after lapping, indicating that pitch error has a major influence on the transmission error of hypoid bevel gears. The pitch error along the normal direction is expressed as:
$$
\Delta t = t_2 – t_1
$$
where \(t_1\) and \(t_2\) are the pitches of two meshing teeth. A positive pitch deviation shifts the meshing point to one side, while a negative deviation shifts it to the other side. This breaks the normal meshing of hypoid bevel gears and causes uneven load distribution, leading to meshing impact and noise. Therefore, controlling pitch error is essential for reducing the transmission error of hypoid bevel gears.
2. Road Test of Main Reducer NVH under Different Transmission Errors
I used an LMS Test.Lab vibration and noise testing system for vehicle road tests. The system includes hardware front ends, microphones, speed sensors, and a data acquisition system. The microphones were placed at the rear seat and on the rear axle housing to collect noise signals. The LMS SCADAS system recorded the data. The output includes two sets of A-weighted sound pressure level signals: the overall level representing the interior noise, and the 12.52 order representing the meshing noise of the hypoid bevel gears. The 12.52 order is calculated from the gear ratio: the transmission ratio in 5th gear is 0.799, and the pinion has 10 teeth, so the meshing order is \(10 / 0.799 = 12.52\).
The A-weighted sound level is used because it closely matches the frequency response of the human ear. The sound pressure level is defined as:
$$
L_P = 10 \log_{10} \frac{P^2}{P_0^2}
$$
where \(P\) is the measured sound pressure and \(P_0 = 2 \times 10^{-5}\) Pa is the reference sound pressure. The A-weighted level is expressed in dB(A). For this main reducer, good NVH performance is defined as a peak noise below 60 dB(A) under 5th gear acceleration and below 65 dB(A) under 5th gear coast.
The road test conditions are shown in the following table.
| Main reducer condition | Engine speed range (r/min) |
|---|---|
| 5th gear acceleration | 2000–3600 |
| 5th gear coast | 3600–2000 |
I selected 10 sets of hypoid bevel gears from a batch of 360 sets. The contact patterns after lapping were all located in the middle-to-toe position and had good consistency. I also controlled the parameters of the reducer housing and differential housing to ensure that the transmission error was the only variable. The five key parameters of the housings were controlled as shown below.
| Control item | Differential housing | Reducer housing |
|---|---|---|
| Runout of driven gear mounting surface (mm) | 0.04 | — |
| Coaxiality of bearing position (mm) | 0.016 | — |
| Ear hole diameter (mm) | φ72 (-0.01, -0.03) | — |
| Offset (mm) | — | 25±0.02 |
| Perpendicularity of pinion and gear axes (mm) | — | 0.04 |
After improving the fixture structure and increasing the inspection frequency, the qualified rates of these parameters reached nearly 100%. This ensured that the road test could exclude interference from factors other than transmission error.
The road test results for the 10 sets of hypoid bevel gears are shown in the following table.
| Gear set | Drive side TE (μrad) | 5th gear acceleration peak noise (dB(A)) | Coast side TE (μrad) | 5th gear coast peak noise (dB(A)) |
|---|---|---|---|---|
| 1 | 17 | 68.67 | 15 | 68.49 |
| 2 | 10 | 60.30 | 7 | 63.09 |
| 3 | 7 | 71.39 | 5 | 70.87 |
| 4 | 22 | 72.62 | 14 | 71.43 |
| 5 | 16 | 69.54 | 10 | 67.55 |
| 6 | 13 | 60.34 | 8 | 65.46 |
| 7 | 15 | 67.51 | 16 | 69.52 |
| 8 | 17 | 66.87 | 17 | 70.84 |
| 9 | 20 | 70.61 | 15 | 67.45 |
| 10 | 11 | 61.36 | 13 | 68.91 |
The data show that gear set 2 has a drive-side transmission error of 10 μrad and a coast-side transmission error of 7 μrad, and its corresponding main reducer has a peak noise of 60.30 dB(A) under acceleration and 63.09 dB(A) under coast, indicating good NVH performance. Gear set 3 has a drive-side transmission error of 7 μrad and a coast-side transmission error of 5 μrad, which are lower than those of gear set 2, but its corresponding main reducer has a peak noise of 71.39 dB(A) under acceleration and 70.87 dB(A) under coast, indicating poor NVH performance. This confirms that the transmission error of hypoid bevel gears is not always better when smaller. There is a range-based correspondence between the transmission error and the NVH performance of the main reducer.
3. Bench Test and Order Analysis for Transmission Error Range Determination
To obtain sufficient experimental data, I used a rear axle NVH test bench. The bench includes a driving motor, an eddy current retarder for loading, a main reducer fixture, an anechoic chamber, sensors, and a temperature control system. The bench can simulate the 5th gear acceleration and coast conditions. The anechoic chamber has a background noise below 25 dB(A), ensuring accurate measurement.
I used order analysis to extract the noise signal of the hypoid bevel gears. The order analysis method is based on the angle domain. The rotation angle of the reference shaft can be expressed as a quadratic function of time:
$$
\theta(t) = a_0 + a_1 t + a_2 t^2
$$
For three pulse arrival times \(t_1, t_2, t_3\) with equal angular increments \(\Delta\Phi\), I have:
$$
\theta(t_1) = 0, \quad \theta(t_2) = \Delta\Phi, \quad \theta(t_3) = 2\Delta\Phi
$$
The time for a constant angle increment is:
$$
t = \frac{-a_1 + \sqrt{a_1^2 – 4a_2(a_0 – k\Delta\Phi)}}{2a_2}
$$
The relationship between meshing frequency \(f\), order \(c\), and reference shaft speed \(n_1\) is:
$$
c = \frac{60f}{n_1}
$$
For a gear pair, the meshing frequency is:
$$
f = \frac{z n_2}{60}
$$
Thus, the meshing order is:
$$
c = \frac{z n_2}{n_1}
$$
For the hypoid bevel gears in this study, the pinion has 10 teeth, and the transmission ratio in 5th gear is 0.799, so the meshing order is 12.52. The gear parameters are listed in the following table.
| Parameter | Pinion | Gear |
|---|---|---|
| Number of teeth | 10 | 43 |
| Transverse module | — | 4.186 |
| Pitch diameter (mm) | — | 180 |
| Midpoint spiral angle | 45°16′ | 27°3′ |
| Spiral direction | Left | Right |
| Addendum (mm) | — | 1.33 |
| Whole depth (mm) | — | 8.82 |
| Pitch deviation (μm) | ±0.018 | ±0.02 |
| Cumulative pitch error (μm) | 0.036 | 0.063 |
| Offset (mm) | 25 (downward) | |
| Mean pressure angle | 21°15′ | |
| Normal backlash (mm) | 0.06–0.18 | |
| Contact pattern along tooth length | 50%–70% | |
| Contact pattern along tooth height | 55%–75% | |
I verified the bench reliability by comparing road test and bench test results for three gear sets. The peak noise values and the speeds at which they occurred were very close. For example, one gear set had a drive-side peak noise of 66.07 dB(A) and a coast-side peak noise of 67.41 dB(A) on the bench, versus 68.67 dB(A) and 68.49 dB(A) on the road. The bench can accurately simulate the road conditions for hypoid bevel gears.
I tested 30 sets of hypoid bevel gears with different transmission error values. The drive-side transmission error ranged from 3 to 27 μrad, and the coast-side ranged from 3 to 17 μrad. The measured data are shown in the following table.
| Gear set | Drive side TE (μrad) | 5th gear acceleration peak noise (dB(A)) | Coast side TE (μrad) | 5th gear coast peak noise (dB(A)) |
|---|---|---|---|---|
| 1 | 17.05 | 68.67 | 15.54 | 68.49 |
| 2 | 10.67 | 59.30 | 7.67 | 63.09 |
| 3 | 18.32 | 68.39 | 16.21 | 71.87 |
| 4 | 22.56 | 69.62 | 14.44 | 70.43 |
| 5 | 16.06 | 69.54 | 10.02 | 67.55 |
| 6 | 13.99 | 60.34 | 8.98 | 65.46 |
| 7 | 15.27 | 64.51 | 16.90 | 69.52 |
| 8 | 17.55 | 66.87 | 17.57 | 70.84 |
| 9 | 20.08 | 69.61 | 15.00 | 67.45 |
| 10 | 8.23 | 55.36 | 13.00 | 66.47 |
| 11 | 11.67 | 56.80 | 3.87 | 65.89 |
| 12 | 19.19 | 63.29 | 11.34 | 67.27 |
| 13 | 26.63 | 70.54 | 5.56 | 63.88 |
| 14 | 9.17 | 56.73 | 12.64 | 68.44 |
| 15 | 10.88 | 58.34 | 10.54 | 65.89 |
| 16 | 12.44 | 59.16 | 4.88 | 64.78 |
| 17 | 7.32 | 56.87 | 6.36 | 64.01 |
| 18 | 6.58 | 65.45 | 8.34 | 65.08 |
| 19 | 21.22 | 71.19 | 3.15 | 68.34 |
| 20 | 5.31 | 63.28 | 7.88 | 63.18 |
| 21 | 14.69 | 63.45 | 4.45 | 64.19 |
| 22 | 13.20 | 62.08 | 12.14 | 66.78 |
| 23 | 8.34 | 58.61 | 13.67 | 67.46 |
| 24 | 10.13 | 59.22 | 5.66 | 63.07 |
| 25 | 11.05 | 58.18 | 4.25 | 62.77 |
| 26 | 22.51 | 68.73 | 6.17 | 63.39 |
| 27 | 23.03 | 65.41 | 7.15 | 64.14 |
| 28 | 24.89 | 66.07 | 9.57 | 66.92 |
| 29 | 25.17 | 70.66 | 10.89 | 67.22 |
| 30 | 7.87 | 60.18 | 12.05 | 68.55 |
I used a smoothing spline fit in MATLAB to analyze the relationship between transmission error and peak noise. For the drive side, the sum of squared errors (SSE) was 0.1634 with a smoothing parameter of 0.99990131, indicating a good fit. The fitted curve showed that when the drive-side transmission error is in the range of 8–12 μrad, the peak noise under 5th gear acceleration is below 60 dB(A). For the coast side, the SSE was 0.1566 with the same smoothing parameter. The fitted curve showed that when the coast-side transmission error is in the range of 4–8 μrad, the peak noise under 5th gear coast is below 65 dB(A). Therefore, the optimal ranges for the transmission error of hypoid bevel gears are 8–12 μrad on the drive side and 4–8 μrad on the coast side. These ranges ensure good NVH performance of the main reducer.
4. Lapping Parameter Optimization for Transmission Error Correction
Lapping is the final finishing process for hypoid bevel gears. It removes微小 amounts of metal from the tooth surface to improve surface finish and reduce transmission error. The lapping process involves the meshing of the pinion and gear with an abrasive slurry. The normal meshing force \(F\) is generated by the spindle torque \(T\):
$$
F = \frac{T}{R}
$$
where \(R\) is the radius. The normal force is decomposed into a tangential component \(F_t\) and a normal component \(F_1\):
$$
F_1 = F \cos\beta, \quad F_t = F \sin\beta
$$
Further decomposition gives the normal cutting force \(F_n\) and radial force \(F_r\):
$$
F_n = F_1 \cos\alpha = F \cos\beta \cos\alpha
$$
$$
F_r = F_1 \sin\alpha = F \cos\beta \sin\alpha
$$
where \(\alpha\) is the pressure angle and \(\beta\) is the spiral angle. The cutting depth \(a_p\) in lapping is related to the normal force by:
$$
a_p = \sqrt{ \frac{F_{n0}}{\pi \delta_s} 2 \tan^2 \varepsilon }
$$
where \(F_{n0}\) is the average normal force on each abrasive particle, \(\delta_s\) is the yield strength of the gear material, and \(\varepsilon\) is the half-cone angle of the abrasive particle. This equation shows that the cutting depth depends on the normal force, which is influenced by the spindle speed, loading torque, slurry flow, and tooth flank clearance.
I investigated three lapping parameters: tooth flank clearance, spindle speed, and lapping cycle number. The factors and levels are shown in the following table.
| Factor | Level 1 | Level 2 | Level 3 |
|---|---|---|---|
| A: Tooth flank clearance (mm) | 0.05 | 0.06 | 0.07 |
| B: Spindle speed (rpm) | 1200 | 1800 | 2400 |
| C: Lapping cycle number | 2 | 5 | 8 |
I designed a three-level, three-factor orthogonal experiment. The orthogonal array and measured transmission error values are shown below.
| Test | A | B | C | Drive side TE (μrad) | Coast side TE (μrad) |
|---|---|---|---|---|---|
| 1 | 0.05 | 1200 | 2 | 17 | 14 |
| 2 | 0.05 | 1800 | 5 | 16 | 12 |
| 3 | 0.05 | 2400 | 8 | 15 | 8 |
| 4 | 0.06 | 1200 | 5 | 15 | 16 |
| 5 | 0.06 | 1800 | 8 | 10 | 7 |
| 6 | 0.06 | 2400 | 2 | 18 | 13 |
| 7 | 0.07 | 1200 | 8 | 15 | 14 |
| 8 | 0.07 | 1800 | 2 | 10 | 5 |
| 9 | 0.07 | 2400 | 5 | 21 | 11 |
I performed a range analysis. For the drive side, the range values were \(R_A = 1.67\), \(R_B = 6\), and \(R_C = 4\), giving the order \(B > C > A\). The optimal combination was \(B_2C_3A_2\). For the coast side, the range values were \(R_A = 1.33\), \(R_B = 5\), and \(R_C = 3.33\), giving the same order \(B > C > A\). The optimal combination was \(B_2C_3A_3\). Considering both sides, I selected \(B_2C_3A_2\) as the final optimal scheme: tooth flank clearance 0.06 mm, spindle speed 1800 rpm, and lapping cycle number 5.
| Evaluation index | Factor order | Optimal scheme | Final scheme |
|---|---|---|---|
| Drive side TE | B > C > A | B2C3A2 | B2C3A2 |
| Coast side TE | B > C > A | B2C3A3 |
Because spindle speed had the largest influence, I conducted a further optimization experiment on spindle speed from 1500 rpm to 2400 rpm in steps of 100 rpm. The tooth flank clearance was fixed at 0.06 mm and the lapping cycle number at 5. The results are shown in the following table.
| Spindle speed (rpm) | Average drive side TE (μrad) | Average coast side TE (μrad) |
|---|---|---|
| 1500 | 19.0 | 14.4 |
| 1600 | 17.8 | 12.4 |
| 1700 | 17.0 | 11.8 |
| 1800 | 15.4 | 10.0 |
| 1900 | 12.6 | 9.4 |
| 2000 | 9.6 | 8.2 |
| 2100 | 11.8 | 9.0 |
| 2200 | 14.6 | 9.0 |
| 2300 | 15.0 | 10.8 |
| 2400 | 17.6 | 11.0 |
The results show that when the spindle speed is 2000 rpm, the average drive-side transmission error is 9.6 μrad and the average coast-side transmission error is 8.2 μrad, which are closest to the optimal ranges. Therefore, the final optimized lapping parameters are: tooth flank clearance 0.06 mm, spindle speed 2000 rpm, and lapping cycle number 5.
I used these optimized parameters to lap a new batch of hypoid bevel gears and selected 5 sets for vehicle road tests. The results are shown in the following table.
| Gear set | Drive side TE (μrad) | 5th gear acceleration peak noise (dB(A)) | Coast side TE (μrad) | 5th gear coast peak noise (dB(A)) |
|---|---|---|---|---|
| 1 | 10 | 59.63 | 5 | 63.03 |
| 2 | 12 | 60.20 | 6 | 66.30 |
| 3 | 11 | 61.38 | 6 | 64.75 |
| 4 | 9 | 62.31 | 8 | 65.53 |
| 5 | 10 | 60.01 | 7 | 63.43 |
Considering a test error of 2–3 dB(A), the peak noise under 5th gear acceleration is around 60 dB(A) with a variation of less than 3 dB(A), and the peak noise under 5th gear coast is below 65 dB(A) with a variation of less than 3.5 dB(A). This confirms that the optimized lapping parameters effectively control the transmission error of hypoid bevel gears within the desired ranges and ensure good NVH performance of the main reducer.
5. Conclusions
I studied the relationship between the transmission error of hypoid bevel gears and the NVH performance of the main reducer. I established a vibration model for hypoid bevel gears and analyzed the noise generation mechanism. I found that pitch error has a major influence on the transmission error of hypoid bevel gears. Through vehicle road tests, I demonstrated that the transmission error of hypoid bevel gears is not always better when smaller. There is a range-based correspondence between transmission error and main reducer NVH performance. Using a rear axle NVH test bench and order analysis, I obtained extensive data and fitted the relationship. The results show that when the drive-side transmission error of hypoid bevel gears is in the range of 8–12 μrad and the coast-side transmission error is in the range of 4–8 μrad, the main reducer peak noise under 5th gear acceleration is below 60 dB(A), and the peak noise under 5th gear coast is below 65 dB(A). I also optimized the lapping parameters. The optimal parameters are tooth flank clearance 0.06 mm, spindle speed 2000 rpm, and lapping cycle number 5. These parameters ensure that the transmission error of hypoid bevel gears after lapping falls within the desired ranges. Vehicle road tests confirmed that the main reducer NVH performance is good with these optimized hypoid bevel gears. My research provides a practical reference for controlling transmission error and improving the NVH performance of main reducers equipped with hypoid bevel gears.