I performed a full-process numerical and experimental investigation of hypoid bevel gears under carburizing, quenching, reverse reconstruction, and dynamic meshing conditions. My objective was to understand how heat-treatment distortion changes the tooth surface, how that changed surface can be reconstructed with high fidelity, and how the reconstructed hypoid bevel gears behave under forward, reverse, neutral-forward, and neutral-reverse operating conditions. The work connected material-phase simulation, heat-treatment simulation, point-cloud processing, solid reconstruction, transmission-error prediction, and explicit dynamic contact analysis. I used 20CrMoH as the gear material because it is widely applied in automotive hypoid bevel gears and provides a favorable combination of core toughness and surface hardness after carburizing and quenching.
Geometry and design basis. The hypoid bevel gears I studied had a large wheel and a pinion with an offset and a right-angle shaft intersection. The tooth form was generated by a face-hobbing process, which produced a cycloidal lengthwise curvature. I used the theoretical parameters to build the initial solid model and then compared it with the heat-treated reverse model. The main geometric values are summarized in the following table.
| Parameter | Large wheel | Pinion | Unit |
|---|---|---|---|
| Number of teeth | 47 | 11 | — |
| Reference normal module | 3.15216 | 3.15216 | mm |
| Reference spiral angle | 30.667 RH | 50 LH | deg |
| Shaft angle | 90 | 90 | deg |
| Face width | 32 | 36.69 | mm |
| Offset | 30 | 30 | mm |
| Pitch cone angle | 68.633 | 20.25 | deg |
| Reference pitch radius | 86.12 | 26.97 | mm |
| Addendum | 1.07 | 5.23 | mm |
| Dedendum | 6.23 | 2.07 | mm |
Heat-treatment route. I designed the carburizing and quenching schedule according to the material response of 20CrMoH and the geometric complexity of the hypoid bevel gears. The process included preheating, pre-carburizing, strong carburizing, diffusion, holding, oil quenching, and tempering. The carbon potential and temperature were varied in stages so that the surface carbon concentration could reach a controlled level while avoiding excessive carbide networks and retained austenite. The schedule I used is given below.
| Stage | Temperature | Carbon potential | Time | Purpose |
|---|---|---|---|---|
| Preheating | 860–880 °C | — | 3600 s | Thermal stabilization |
| Pre-carburizing | 900–920 °C | 1.00% | 10800 s | Initial carbon enrichment |
| Strong carburizing | 920–930 °C | 1.15% | 9000 s | High surface carbon supply |
| Diffusion | 880–900 °C | 0.90% | 7200 s | Carbon profile smoothing |
| Holding | 850–870 °C | 0.90% | 600 s | Final austenite homogenization |
| Oil quenching | 80–100 °C oil | — | 1200 s | Martensitic hardening |
| Tempering | 170–190 °C | — | 3600 s | Residual stress relief |
Material property modeling. The thermophysical and transformation properties of 20CrMoH were not fully available in a form suitable for multi-field simulation. I therefore used a material property calculation approach to obtain phase-transition temperatures, thermal conductivity, thermal expansion, heat capacity, latent heat, Young’s modulus, and Poisson’s ratio. The chemical composition I adopted is listed below.
| Element | C | Mn | Ni | P | S | Mo | Cr | Si | Fe |
|---|---|---|---|---|---|---|---|---|---|
| Mass fraction (%) | 0.20 | 0.82 | 0.10 | 0.01 | 0.01 | 0.20 | 1.07 | 0.25 | Balance |
The temperature-dependent elastic properties used for the hypoid bevel gears are summarized in the next table.
| Temperature (°C) | Poisson’s ratio | Young’s modulus (GPa) |
|---|---|---|
| 0 | 0.290 | 210 |
| 300 | 0.301 | 193 |
| 600 | 0.319 | 152 |
| 900 | 0.345 | 117 |
Multi-field governing equations. I treated the heat treatment of hypoid bevel gears as a coupled thermal, metallurgical, diffusion, and mechanical problem. The temperature field was governed by heat conduction with latent heat release:
$$ \rho C_p(T)\frac{\partial T}{\partial t}=\nabla\cdot\left(k_t(T)\nabla T\right)+\Delta H_F\dot{\beta}_F+\Delta H_P\dot{\beta}_P+\Delta H_B\dot{\beta}_B+\Delta H_M\dot{\beta}_M $$
where ρ is density, Cp is specific heat, T is temperature, t is time, kt is thermal conductivity, ΔH represents latent heat, and β with subscripts denotes the volume fraction rate of each phase.
The austenite formation during heating was represented by a sigmoidal transformation law:
$$ \beta_A=1-\exp\left[-\left(\frac{T-T_s}{T_e-T_s}\right)^4\right] $$
where Ts is the lower critical temperature and Te is the upper transformation temperature. The transformation temperatures I obtained are given below.
| Phase or event | Temperature (°C) |
|---|---|
| Pearlite to austenite start | 739.4 |
| Ferrite to austenite start | 816.0 |
| Bainite transformation start | 593.1 |
| Martensite transformation start | 402.6 |
For diffusion-controlled transformations, I used the Johnson–Mehl–Avrami–Kolmogorov form:
$$ \beta=1-\exp(-kt^n) $$
and for martensite, I used a temperature-dependent fraction equation:
$$ \beta_M=\frac{1-\beta_F-\beta_P-\beta_B}{1+\exp\left[-0.011\left(M_s-T\right)\right]} $$
where Ms is the martensite start temperature. The carbon diffusion flux in the carburized layer was expressed as:
$$ J=-sD\left(\frac{\partial c}{\partial x}+\frac{c}{RT}\frac{\partial \ln a}{\partial x}\right) $$
where J is carbon flux, s is carbon solubility, D is diffusivity, c is carbon concentration, R is the gas constant, and a is activity. The composite hardness of the hypoid bevel gears was calculated from phase fractions:
$$ H=\omega_AH_A+\omega_FH_F+\omega_PH_P+\omega_BH_B+\omega_MH_M $$
where ω is volume fraction and H is the hardness of each phase. I also used a martensite hardness relation of polynomial form:
$$ H_M=a_0+a_1w_C+a_2w_C^2+a_3w_C^3 $$
where wC is carbon content in the martensite. These equations allowed me to predict both the surface hardness and the core hardness of the hypoid bevel gears after quenching.
Quenching boundary conditions. A uniform heat-transfer coefficient is not sufficient for complex hypoid bevel gears because the quenchant flow velocity changes over the tooth flank, root, tip, bore, and back face. I therefore used dynamic boundary conditions based on measured or correlated heat-transfer coefficient curves for ISO-N32 quenching oil. The heat-transfer coefficient was a function of surface temperature and local flow velocity:
$$ h=h(T,v) $$
The average flow velocities on different gear surfaces are summarized below.
| Medium velocity (m/s) | Bore (m/s) | Tooth flank (m/s) | Outer rim (m/s) | Back face (m/s) |
|---|---|---|---|---|
| 0.4 | 0.1245 | 0.0936 | 0.0607 | 0.0397 |
| 0.8 | 0.2481 | 0.1896 | 0.1694 | 0.1005 |
| 1.2 | 0.4537 | 0.3428 | 0.3189 | 0.1564 |
| 1.6 | 0.5662 | 0.4721 | 0.4813 | 0.1785 |
I mapped these velocities to heat-transfer coefficient curves and applied them separately to the inner ring, tooth surface, outer ring, and bottom surface. This dynamic quenching boundary condition made the simulation closer to industrial oil quenching of hypoid bevel gears.
| Velocity range (m/s) | Selected heat-transfer curve |
|---|---|
| 0.600–0.5267 | Curve 6 |
| 0.5267–0.433 | Curve 5 |
| 0.433–0.3167 | Curve 4 |
| 0.3167–0.200 | Curve 3 |
| 0.200–0 | Curve 2 and Curve 1 |
Carbon distribution results. I extracted carbon concentration after each carburizing stage. The tip always showed the highest carbon content because of the sharp-corner effect. The flank and bottom face were similar because their geometric curvature and gas access were comparable. The root had lower carbon content because it was partly shadowed and located near a concave transition. The core had the lowest carbon content because carbon diffusion from the surface to the center of hypoid bevel gears is slow. The stage-by-stage carbon results are listed below.
| Stage | Flank (%) | Tip (%) | Root (%) | Bottom (%) | Core (%) |
|---|---|---|---|---|---|
| Pre-carburizing | 0.703 | 0.761 | 0.595 | 0.702 | 0.225 |
| Strong carburizing | 0.853 | 0.955 | 0.781 | 0.824 | 0.227 |
| Diffusion | 0.751 | 0.863 | 0.660 | 0.751 | 0.230 |
| Holding | 0.756 | 0.851 | 0.656 | 0.745 | 0.231 |
The carbon profile reached its maximum during strong carburizing and then decreased during diffusion. After holding, the surface carbon concentration became relatively stable. For hypoid bevel gears, this stable surface layer is important because it controls wear resistance and contact fatigue resistance.
Temperature evolution. During quenching, the tip cooled fastest, followed by the flank, root, bottom, and core. The core cooled mainly by internal conduction because it was not directly exposed to the quenchant. The cooling rate decreased over time because of latent heat release and because the heat-transfer coefficient fell as the surface temperature dropped. The approximate cooling behavior is summarized below.
| Region | Cooling rank | Approximate martensite start time | Thermal behavior |
|---|---|---|---|
| Tip | 1 | 1.3 s | Fastest cooling and earliest transformation |
| Flank | 2 | 2.2 s | High cooling rate near the outer surface |
| Root | 3 | 8.0 s | Moderate cooling and delayed transformation |
| Bottom | 4 | 10.1 s | Lower cooling rate than the tooth region |
| Core | 5 | 15.8 s | Slowest cooling controlled by conduction |
After about 100 s of quenching, the maximum temperature difference among the measured regions was about 50 °C. This gradient is significant because it drives thermal stress and distortion in hypoid bevel gears.
Microstructure and hardness. The martensite fraction followed the cooling rate: the tip had the highest fraction, followed by the flank, root, bottom, and core. Because hardness is strongly correlated with martensite fraction, the same order appeared in the hardness results. The predicted hardness values and martensite fractions are shown below.
| Region | Martensite volume fraction | Predicted hardness (HRC) | Process requirement (HRC) |
|---|---|---|---|
| Tip | 0.982 | 60.42 | 58–64 |
| Flank | 0.974 | 59.33 | 58–64 |
| Root | 0.830 | 51.81 | — |
| Bottom | 0.795 | 52.46 | — |
| Core | 0.636 | 37.80 | 32–45 |
The surface layer of the hypoid bevel gears consisted mainly of high-carbon plate martensite with a small amount of retained austenite, which gave high hardness and wear resistance. The core consisted mainly of low-carbon lath martensite with some retained bainite and austenite, which provided toughness and fatigue resistance. This outward-hard and inward-tough structure is exactly what I wanted for hypoid bevel gears in automotive drive axles.
Stress and strain fields. I used a thermo-elastoplastic constitutive model with phase transformation strain. The total strain increment was expressed as:
$$ d\varepsilon=d\varepsilon_e+d\varepsilon_p+d\varepsilon_{th}+d\varepsilon_{tr} $$
where dεe is elastic strain, dεp is plastic strain, dεth is thermal strain, and dεtr is transformation strain. The flow stress depended on temperature, carbon content, and hardening state:
$$ Y=Y(T,C,H,\bar{\varepsilon}) $$
At the beginning of quenching, the surface cooled and contracted faster than the core, so the surface was in tension while the core was in compression. In the middle stage, martensite formation expanded the surface, and the core contraction began to dominate, so the surface stress changed from tension to compression. After about 20 s, the stress state became more stable. After about 30 s, the stress distribution in the hypoid bevel gears approached a steady residual state. The final surface residual stress was compressive, while the core residual stress was tensile. This distribution is favorable for contact fatigue resistance because compressive residual stress on the tooth surface can suppress crack initiation.
The strain field showed that the large-end concave root, the middle of the tooth tip, and the small-end convex root were the most distorted regions. In these locations, the geometry changed sharply and the cooling was nonuniform. The middle of the tooth tip formed a convex drum-shaped zone because the flank cooled rapidly and contracted while the interior resisted that contraction. I used a strain threshold of 5 μm to identify the drum-shaped region. The measured drum length was about 23.136 mm. If this drum region becomes too large, the hypoid bevel gears may experience edge contact, which can produce noise, impact, and early pitting. Therefore, heat-treatment distortion control is critical for hypoid bevel gears.
Heat-treatment quality check. I compared the simulated heat-treatment results with the industrial requirements for hypoid bevel gears. The surface hardness, core hardness, bore ovality, and back-face flatness all satisfied the specified limits.
| Item | Simulated value | Process requirement | Result |
|---|---|---|---|
| Surface hardness | 59.33 HRC | 58–64 HRC | Qualified |
| Core hardness | 37.80 HRC | 32–45 HRC | Qualified |
| Bore ovality | 0.024 mm | <0.07 mm | Qualified |
| Back-face flatness | 0.028 mm | <0.07 mm | Qualified |
Reverse reconstruction of distorted hypoid bevel gears. After heat treatment, the tooth surfaces of the hypoid bevel gears were no longer identical to the theoretical surfaces. To capture the distorted geometry, I exported the post-heat-treatment point cloud and reconstructed the tooth surfaces. The point cloud was simplified, smoothed, visualized, and segmented before surface fitting. Simplification reduced the number of points while preserving the essential curvature. Smoothing removed noise without flattening the contact-relevant features. Segmentation separated the concave and convex flanks so that each surface could be fitted independently.
| Step | Operation | Purpose |
|---|---|---|
| 1 | Point-cloud import | Read the post-heat-treatment geometry |
| 2 | Simplification | Reduce redundant points and improve processing speed |
| 3 | Smoothing | Remove noise and small irregularities |
| 4 | Visualization | Inspect the overall point-cloud shape |
| 5 | Segmentation | Separate concave and convex flanks |
| 6 | Curve fitting | Create cross-sectional curves on each flank |
| 7 | Surface reconstruction | Build continuous tooth surfaces |
| 8 | Solid reconstruction | Create the full hypoid bevel gear model |
I used a distance tolerance of 1 for simplification because it gave a good balance between point density and computational cost. The original point cloud contained 9730 points, and after simplification it contained 6405 points, a reduction of about 34%. For the tooth flank, I used Gaussian filtering because it preserves shape accuracy. For the gear body, I used median filtering because it suppresses spikes. After segmentation, I obtained separate point clouds for the concave and convex flanks. I then fitted sectional curves and reconstructed the surfaces. The reconstructed surfaces were imported into a CAD environment to build the solid hypoid bevel gears.
Tooth-surface error analysis. I evaluated the deviation between the reconstructed surfaces and the original point cloud. The maximum lateral deviation, negative-direction deviation, geometric deviation, and positive-normal deviation were compared with the gear diameter. For a gear diameter of 203.82 mm, the 0.5% threshold was 1.0191 mm. The error values for the concave and convex flanks are given below.
| Surface | Error type | Maximum (mm) | Mean (mm) | Standard deviation (mm) |
|---|---|---|---|---|
| Concave | Lateral deviation | 0.0023 | 0.0000 | 0.0001 |
| Concave | Negative-direction deviation | −0.0519 | −0.0293 | 0.0122 |
| Concave | Geometric deviation | 0.0519 | 0.0292 | 0.0123 |
| Concave | Positive-normal deviation | 0.0017 | 0.0017 | 0.0275 |
| Convex | Lateral deviation | 0.0000 | 0.0000 | 0.0000 |
| Convex | Negative-direction deviation | −0.0447 | −0.0146 | 0.0100 |
| Convex | Geometric deviation | 0.0447 | 0.0125 | 0.0091 |
| Convex | Positive-normal deviation | 0.0200 | 0.0091 | 0.0068 |
All maximum deviations were below the 0.5% diameter threshold, so I considered the reverse reconstruction to meet the required accuracy level for hypoid bevel gears. This was important because the subsequent dynamic contact simulation depended strongly on the reconstructed tooth surface geometry.
Transmission error of the reconstructed hypoid bevel gears. I assembled the reconstructed large wheel and pinion and analyzed the transmission error under four operating conditions: forward drive, reverse drive, neutral forward, and neutral reverse. In forward and reverse drive, the pinion was the driving member. In neutral forward and neutral reverse, the large wheel was the driving member. The theoretical transmission ratio was:
$$ i=\frac{z_2}{z_1}=\frac{47}{11}=4.2727 $$
I applied rotational speeds and torques according to typical automotive drive-axle conditions. The transmission-error results are summarized below.
| Condition | Driving member | Theoretical speed (deg/s) | Average speed (deg/s) | Relative error (%) |
|---|---|---|---|---|
| Forward | Pinion | 210.638 | 210.6575 | 0.01 |
| Reverse | Pinion | 210.638 | 210.5223 | 0.05 |
| Neutral forward | Large wheel | 901.545 | 900.3571 | 0.13 |
| Neutral reverse | Large wheel | 901.545 | 901.1479 | 0.04 |
In all four conditions, the angular velocity output fluctuated around the average value, and the average value was close to the theoretical value. The relative errors were below 0.13%. Forward drive gave the most stable transmission. Reverse drive also performed reasonably well but showed larger fluctuations. Neutral forward and neutral reverse showed much wider speed variation, which indicates unstable operation and stronger impact. These results are consistent with the practical preference for pinion-driven operation in automotive hypoid bevel gears.
Dynamic contact analysis. I used a finite-element contact model to study the meshing behavior of the reconstructed hypoid bevel gears. The gear mesh was created in a professional preprocessor, and the contact analysis was performed in an explicit dynamic solver. The contact interaction was defined as surface-to-surface contact with a finite sliding formulation. The normal behavior was hard contact, and the tangential behavior used a Coulomb friction model. The critical shear stress was:
$$ \tau_{crit}=\min(\mu p,\tau_{\max}) $$
and the equivalent shear stress was:
$$ \tau_{eq}=\sqrt{\tau_1^2+\tau_2^2} $$
where μ is the friction coefficient, p is contact pressure, τmax is the maximum shear strength, and τ1 and τ2 are shear components on the contact plane. The contact setup is given below.
| Contact parameter | Setting |
|---|---|
| Master surface | Pinion tooth surface |
| Slave surface | Large-wheel tooth surface |
| Contact type | Surface-to-surface |
| Solver control | Explicit dynamic |
| Friction model | Dynamic friction |
| Friction coefficient | 0.3 |
| Contact method | Penalty function |
| Sliding formulation | Finite sliding |
The load cases I used for the dynamic contact analysis are listed below.
| Condition | Driving wheel | Driven wheel | Driving surface | Driven surface | Speed (rad/s) | Torque (N·mm) |
|---|---|---|---|---|---|---|
| Forward | Pinion | Large wheel | Concave | Convex | 104.667 | 1025 |
| Reverse | Pinion | Large wheel | Convex | Concave | 104.667 | 1025 |
| Neutral forward | Large wheel | Pinion | Concave | Convex | 24.492 | 240 |
| Neutral reverse | Large wheel | Pinion | Convex | Concave | 24.492 | 240 |
To avoid a sudden impact at the beginning of loading, I used an amplitude curve that increased the load smoothly during the first 1 ms and then held it constant. The contact force and contact area were extracted from the simulation. The results showed clear periodic behavior for the hypoid bevel gears.
| Condition | Contact force range | Contact area range | Stability | Impact level |
|---|---|---|---|---|
| Forward | 15–20 kN | 40–80 mm² | High | Low |
| Reverse | 0–40 kN | 0–95 mm² | Moderate | Moderate |
| Neutral forward | 80–100 kN | 60–160 mm² | Low | High |
| Neutral reverse | 20–70 kN | 40–120 mm² | Low | High |
Forward drive produced the smallest and most stable contact force, and the contact area fluctuated periodically with a small range. This indicates stable meshing and low vibration for the hypoid bevel gears. Reverse drive had larger fluctuations and more visible impact pulses. Neutral forward produced the highest contact force and the widest contact-area fluctuation, which suggests severe impact and a high risk of tooth-surface damage. Neutral reverse was less severe than neutral forward but still much less stable than forward drive. These findings support the practical recommendation that hypoid bevel gears should operate mainly in the forward, pinion-driven condition.
Experimental validation. I validated the numerical model with carburizing and quenching experiments, hardness measurements, microstructure observations, coordinate measurements, and rolling contact tests. The measured surface hardness and core hardness were close to the simulated values. The microstructure showed fine carbides near the tip, fine needle martensite with retained austenite on the flank, and some blocky ferrite in the core. The non-martensitic layer depth and internal oxidation depth were both below the required limit.
| Item | Simulated value | Measured value | Requirement |
|---|---|---|---|
| Surface hardness | 59.33 HRC | 59.2 HRC | 58–64 HRC |
| Core hardness | 36.9 HRC | 37.8 HRC | 32–45 HRC |
| Non-martensitic layer depth | — | 0.017 mm | <0.02 mm |
| Internal oxidation depth | — | 0.012 mm | <0.02 mm |
I measured the tooth-surface deviations before and after heat treatment using a 45-point measuring scheme. The largest deviations occurred near the large end, small end, root, and tip. The numerical and measured difference surfaces showed the same general trend. This agreement supports the heat-treatment model and the reverse reconstruction of the hypoid bevel gears.
I also compared the digital contact pattern with the actual rolling-test contact pattern. The contact positions were approximately the same, and the inclination was similar. The contact shape differed slightly because the reconstructed surface had small fitting errors and because the experimental setup introduced assembly errors. Even so, the contact patterns confirmed that the reconstructed hypoid bevel gears captured the main meshing behavior.
| Comparison item | Digital result | Experimental result | Agreement |
|---|---|---|---|
| Concave contact position | Near the middle, biased toward tip and small end | Near the middle, biased toward tip and small end | High |
| Convex contact position | Near the middle, biased toward root and small end | Near the middle, biased toward root and small end | High |
| Inclination | Concave larger than convex | Concave larger than convex | High |
| Contact shape | Slightly different | Slightly different | Moderate |
Summary of my findings. I found that the carburizing and quenching process produced a carbon-rich surface and a low-carbon core in the hypoid bevel gears. The tip reached the highest carbon content and the core reached the lowest. The quenching rate decreased from the tip to the core, which produced a martensite fraction gradient and a hardness gradient. The final hardness distribution satisfied the requirement for automotive hypoid bevel gears.
I also found that heat-treatment distortion was concentrated at the large-end concave root, the tooth-tip middle, and the small-end convex root. The tooth-tip middle formed a drum-shaped region. This distortion can change the contact pattern and increase the risk of edge contact. Reverse reconstruction using point-cloud processing and surface fitting allowed me to capture the distorted geometry with maximum deviations below 0.5% of the gear diameter. The reconstructed hypoid bevel gears showed small transmission errors under all four operating conditions, with the largest relative error below 0.13%.
In the dynamic contact analysis, forward drive gave the most stable contact force and contact area. Neutral forward produced the largest contact force and the widest contact-area fluctuation, which indicates severe impact and a high risk of damage. Reverse drive and neutral reverse were intermediate. These trends are important for understanding the durability of hypoid bevel gears in automotive drive axles.
Finally, the experiments confirmed that the simulated hardness, microstructure, and distortion trends were consistent with measured results. The digital contact pattern and the actual contact pattern were located in similar regions. The remaining differences were caused mainly by reverse reconstruction error and assembly error. Overall, the full-process simulation approach I used provides a practical way to study heat-treatment distortion and meshing characteristics of hypoid bevel gears.
Future improvements. The heat-treatment simulation and dynamic contact analysis can be improved further. First, the drum-shaped distortion region should be reduced by optimizing the quenching schedule, the fixturing method, and the flow field around the hypoid bevel gears. Second, a direct interface between the heat-treatment solver and the dynamic solver would reduce the errors introduced by reverse reconstruction. Third, a more refined mesh in the contact zone would improve the accuracy of the contact-force and contact-area predictions. Fourth, a flexible-body dynamic analysis would provide a more realistic representation of impact and vibration than a rigid-body analysis. These improvements would make the numerical prediction of hypoid bevel gears even more reliable for automotive applications.