I studied cycloid hypoid bevel gears under high-speed and heavy-load conditions because these gear pairs are widely used in rear-drive axle main reducers, where tooth-surface fatigue, wear, impact, vibration, and noise directly affect vehicle comfort and durability. The central problem I addressed is that heat treatment improves surface hardness, wear resistance, and fatigue resistance, but it also introduces distortion, residual stress, and tooth-surface deviation. For simple gears, quenching is often analyzed with a uniform cooling medium, and the main focus is thermal strain and microstructural strain. For cycloid hypoid bevel gears, however, the geometry is much more complex, so I had to include the effect of quenching-medium flow velocity on the heat-transfer coefficient at different gear surfaces. I therefore built a full numerical workflow that combined carburizing and quenching simulation, reverse reconstruction of the heat-treated gear, dynamic transmission-error analysis, and contact mechanics simulation. I also validated the numerical results with heat-treatment experiments and rolling-contact experiments.
I treated the cycloid hypoid bevel gears as a coupled multi-physics problem involving carbon diffusion, temperature evolution, phase transformation, stress, strain, and distortion. The material was 20CrMoH, a low-carbon alloy steel commonly used for gears because it offers stable performance, small thermal distortion, good carburizing behavior, and good manufacturability. I first defined the heat-treatment process, then calculated the thermo-physical properties with a specialized material simulation tool, then performed the carburizing and quenching simulation with a heat-treatment finite-element solver, and finally reconstructed the distorted tooth surface for dynamic contact analysis. The main gear parameters I used are listed below.
| Parameter | Large Gear | Small Gear | Symbol and Unit |
|---|---|---|---|
| Number of teeth | 47 | 11 | \(z_i\) |
| Reference normal module | 3.15216 | 3.15216 | \(m_n\) / mm |
| Reference spiral angle | 30.667 RH | 50 LH | \(\beta_m\) / deg |
| Shaft angle | 90 | 90 | \(\Sigma\) / deg |
| Face width | 32 | 36.69 | \(b_i\) / mm |
| Offset distance | 30 | 30 | \(E\) / mm |
| Pitch cone angle | 68.633 | 20.25 | \(\delta_i\) / deg |
| Reference pitch radius | 86.12 | 26.97 | \(R_m\) / mm |
| Addendum | 1.07 | 5.23 | \(h_a\) / mm |
| Dedendum | 6.23 | 2.07 | \(h_f\) / mm |
The chemical composition of 20CrMoH controlled the phase-transformation behavior and the final hardness distribution. I used the composition shown below as the basis for property calculation and heat-treatment simulation.
| 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 | Bal. |
I calculated the elastic and thermal properties of 20CrMoH with a material-property simulation tool linked to the heat-treatment solver. The temperature-dependent Poisson ratio and Young modulus are summarized below.
| Temperature / deg C | Poisson Ratio | Young Modulus / GPa |
|---|---|---|
| 0 | 0.290 | 210 |
| 300 | 0.301 | 193 |
| 600 | 0.319 | 152 |
| 900 | 0.345 | 117 |
The initial microstructure of 20CrMoH was mainly pearlite and ferrite. During heating, pearlite and ferrite began to transform into austenite at different critical temperatures. During quenching, austenite transformed into ferrite, pearlite, bainite, and martensite as the temperature decreased. The volume fraction of each phase depended on the local cooling rate, carbon content, and thermal history. The martensite fraction had the strongest influence on surface hardness, while the cooling rate also controlled thermal stress and distortion. Therefore, the quenching process had to be designed to obtain high surface hardness and a tough core while keeping distortion within tolerance. The transformation temperatures I used are listed below.
| Microstructural Constituent | Transformation Temperature / deg C |
|---|---|
| Pearlite | 739.4 |
| Bainite | 593.1 |
| Martensite start | 402.6 |
| Ferrite | 816.0 |
For the austenite formation during heating, I used a transformation law that relates the austenite fraction to the temperature interval between the lower and upper critical points:
$$
\beta_A = 1 – \exp\left[-\left(\frac{T-T_s}{T_e-T_s}\right)^4\right]
$$
where \(T\) is the local temperature, \(T_s\) is the lower critical temperature, \(T_e\) is the upper critical temperature, and \(\beta_A\) is the austenite volume fraction. For martensite formation during quenching, I used a temperature-dependent relationship:
$$
\beta_M = \frac{1-\beta_F-\beta_P-\beta_B}{1-\exp[-0.011(M_s-T)]}
$$
where \(M_s\) is the martensite start temperature and \(\beta_F\), \(\beta_P\), and \(\beta_B\) are the ferrite, pearlite, and bainite fractions. For the diffusion-controlled transformations, I used the Johnson-Mehl-Avrami-Kolmogorov form:
$$
\beta = 1 – \exp(-k t^n)
$$
To apply this in an incremental non-isothermal calculation, I used the Scheil additivity principle and wrote the update in the following form:
$$
\beta_i = \beta_{i-1} + (\beta_{ex} – \beta_{i-1})\exp(-k\Delta t)
$$
$$
t_i’ = \left(-\frac{\ln(1-X_i)}{k}\right)^{1/n}
$$
The carbon diffusion coefficient in austenite was calculated as a function of carbon content and temperature:
$$
D_c = 4.53\times10^{-7}\left[1 + (100C_A)^3\right]\left[1 – 2.22\times10^{-4}T\right]\exp\left(-\frac{17767}{T}\right)
$$
where \(C_A\) is the carbon content in austenite. The carbon flux during carburizing depended on the carbon concentration gradient, pressure, temperature, and solubility. I represented the flux in a generalized form as:
$$
J = -sD\left(\frac{\partial c}{\partial x} + \frac{c}{x} + \frac{\partial[\ln(T+273)]}{\partial x}K_s + \frac{\partial p}{\partial x}K_p\right)
$$
The heat-conduction equation included the latent heat released or absorbed by phase transformation:
$$
\rho C_p \dot{T} = \nabla \cdot (k_t \nabla T) + \Delta H_F\dot{\beta}_F + \Delta H_P\dot{\beta}_P + \Delta H_B\dot{\beta}_B + \Delta H_M\dot{\beta}_M
$$
The boundary heat flux was calculated from the surface heat-transfer coefficient and the difference between the external medium temperature and the gear surface temperature:
$$
q = h(T_e – T_s)
$$
The composite hardness was calculated from the volume fractions and individual phase hardness values:
$$
H = H_T\omega_T + H_P\omega_P + H_B\omega_B + H_A\omega_A + H_F\omega_F + H_M\omega_M
$$
The martensite hardness depended on the local carbon content, and I used the following polynomial form:
$$
H_M = 3626.58 – 463.45w_c + 24.55w_c^2 – 121.68w_c^3
$$
The hardness after tempering was estimated from the tempering temperature and time through a Hollomon-Jaffe-type parameter:
$$
H = H_1 + \frac{dH}{dT}\left[T\ln\left(\frac{t}{C}\right)\right]\left[1 – \frac{1}{I}\right]
$$
The total strain increment during quenching was decomposed into elastic, plastic, thermal, and transformation components:
$$
d\varepsilon = d\varepsilon_e + d\varepsilon_p + d\varepsilon_{th} + d\varepsilon_{tr}
$$
The flow stress and yield behavior were considered to depend on temperature, carbon content, phase state, and equivalent plastic strain:
$$
Y = Y(T,C,\varepsilon)
$$
I designed the heat-treatment cycle for 20CrMoH based on the gear geometry and the required case-core properties. The cycle included preheating, pre-carburizing, strong carburizing, diffusion, pre-cooling, oil quenching, and tempering. The carbon potential was controlled in each stage to obtain a gradual carbon profile and to avoid excessive carbide networks. The main stages are summarized below.
| Stage | Temperature / deg C | Carbon Potential / % | Time / s | Purpose |
|---|---|---|---|---|
| Preheating | 860-880 | — | 3600 | Prepare the gear for carburizing |
| Pre-carburizing | 900-920 | 1.00 | 10800 | Establish an initial carbon-rich layer |
| Strong carburizing | 920-930 | 1.15 | 9000 | Raise surface carbon content |
| Diffusion | 900-880 | 0.90 | 7200 | Smooth the carbon gradient |
| Hold | 870-850 | 0.90 | 600 | Stabilize the case carbon content |
| Oil quenching | 80-100 | — | 1200 | Form martensite and high hardness |
| Tempering | 170-190 | — | 3600 | Reduce residual stress and adjust hardness |
For the quenching boundary condition, I did not use a single constant heat-transfer coefficient. Instead, I divided the gear into inner ring, tooth surface, outer ring, and bottom surface, and I applied different heat-transfer coefficients according to the local quenching-oil flow velocity. This dynamic boundary condition made the simulation closer to the actual continuous furnace and oil-quench process used for cycloid hypoid bevel gears. The heat-transfer coefficient curves depended on the ISO-N32 quenching oil and the local flow velocity. The available curve functions are summarized below.
| Curve | Flow Velocity / m s^{-1} | Heat-Transfer Coefficient Function |
|---|---|---|
| 1 | 0 | \(h = 2284.35 + 4178.755\exp[-(T-526.88)/26163.28]\) |
| 2 | 0.200 | \(h = 2348 + 4687\exp[-(T-517.77)/22333.57]\) |
| 3 | 0.316 | Piecewise function with exponential and polynomial terms |
| 4 | 0.433 | Piecewise function with exponential and polynomial terms |
| 5 | 0.527 | \(h = 2720.3 + 5513\exp[-(T-549.31)/16287.96]\) |
| 6 | 0.600 | Piecewise function with exponential and polynomial terms |
I also estimated the average flow velocity at different gear surfaces. The inner ring, tooth surface, outer ring, and bottom surface experienced different local velocities, so I selected different heat-transfer curves for them.
| Medium Velocity / m s^{-1} | Inner Ring / m s^{-1} | Tooth Surface / m s^{-1} | Outer Ring / m s^{-1} | Bottom Surface / m s^{-1} |
|---|---|---|---|---|
| 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 |
| Velocity Range / m s^{-1} | Selected Curve |
|---|---|
| 0.5267-0.600 | Curve 6 |
| 0.433-0.5267 | Curve 5 |
| 0.3167-0.433 | Curve 4 |
| 0.200-0.3167 | Curve 3 |
| 0-0.200 | Curve 2 |
| 0-0.200 | Curve 1 |
After the heat-treatment simulation, I extracted the carbon distribution at each carburizing stage. The tooth tip reached the highest carbon content because of the sharp-corner effect. The tooth surface and bottom surface were similar, while the tooth root was lower because of its less exposed location. The tooth core had the lowest carbon content because carbon diffusion in the interior was slow. The evolution of carbon content is given below.
| Stage | Tooth Surface / % | Tooth Tip / % | Tooth Root / % | Bottom Surface / % | Tooth 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 | 0.2302 |
| Hold | 0.756 | 0.851 | 0.656 | 0.745 | 0.2306 |
The temperature field showed that the tooth tip cooled fastest, followed by the tooth surface, tooth root, bottom surface, and tooth core. The tooth core cooled mainly by internal heat conduction and did not contact the quenching oil directly, so it had the lowest cooling rate. The martensite transformation started at different times for different regions. The tooth tip began martensite transformation at about 1.3 s, the tooth surface at about 2.2 s, the tooth root at about 8 s, the bottom surface at about 10.1 s, and the tooth core at about 15.8 s. At around 700 deg C, the vapor film collapsed and the surfaces contacted the quenching oil more directly, giving the highest cooling rates. As latent heat was released and the heat-transfer coefficient changed, the cooling rate gradually decreased. After about 100 s, the maximum temperature difference among the tooth tip, tooth surface, tooth root, bottom surface, and tooth core was within about 50 deg C.
The microstructure distribution and hardness followed the cooling-rate distribution. The tooth tip had the highest martensite fraction, followed by the tooth surface, tooth root, bottom surface, and tooth core. The hardness was proportional to the martensite fraction because martensite provides high hardness and wear resistance. The predicted hardness and martensite fractions are listed below.
| Region | Tooth Tip | Tooth Surface | Tooth Root | Bottom Surface | Tooth Core |
|---|---|---|---|---|---|
| Hardness / HRC | 60.42 | 59.33 | 51.81 | 52.46 | 37.8 |
| Martensite Volume Fraction / % | 0.982 | 0.974 | 0.830 | 0.795 | 0.636 |
I also examined the stress and strain evolution. During the early quenching stage, the gear surface cooled quickly and contracted, so it was in tension, while the interior cooled slowly and constrained the surface contraction, so it was in compression. In the middle quenching stage, the surface cooling rate decreased and martensite began to form, causing volumetric expansion, while the interior cooling rate increased. As a result, the surface stress changed from tension to compression. After about 30 s, the stress state became relatively stable. The residual stress after quenching was compressive at the tooth surface and tensile in the tooth core. The main distortion regions were the root of the concave flank at the large end, the middle of the tooth tip, and the root of the convex flank at the small end. A crowned region formed near the middle of the tooth tip because of the combined effect of transformation stress and thermal stress. I set a strain threshold of 5 micrometers to identify this crowned region, and the measured length of the crowned zone was about 23.136 mm.
The heat-treatment quality was checked against the production requirements. The required surface hardness was 58-64 HRC, the required core hardness was 32-45 HRC, the required inner-hole ovality was less than 0.07 mm, and the required flatness of the inner and outer bottom surfaces was less than 0.07 mm. My simulation results are compared with these requirements below.
| Quantity | Simulated Value | Process Requirement | Result |
|---|---|---|---|
| Tooth-surface hardness / HRC | 59.33 | 58-64 | Satisfied |
| Tooth-core hardness / HRC | 37.8 | 32-45 | Satisfied |
| Inner-hole ovality / mm | 0.024 | <0.07 | Satisfied |
| Bottom-surface flatness / mm | 0.028 | <0.07 | Satisfied |
After the heat-treatment simulation, I exported the distorted point cloud of the cycloid hypoid bevel gears and used reverse-engineering software to process it. The exported point cloud could not be used directly for curve and surface fitting because it contained too many points and some local irregularities caused by distortion. I simplified, smoothed, visualized, cleaned, and segmented the point cloud before reconstructing the tooth surface. The initial point cloud contained 9730 points. After simplification with a distance tolerance of 1, the number was reduced to 6405 points, which was about 66 percent of the original count. This reduced memory use and improved processing speed without significantly reducing tooth-surface accuracy.
For smoothing, I used a Gaussian filter for the tooth surface because it preserves shape accuracy well, and I used a median filter for the gear body because it removes spikes effectively. I then segmented the point cloud into concave and convex tooth surfaces. The reconstructed curves were generated by intersecting the tooth-surface point cloud with transverse and longitudinal planes, and the surface was created from these curves. The reconstructed concave and convex surfaces were imported into a three-dimensional modeling environment to build a single tooth slot, and then the full gear model was generated by rotation, array, and Boolean subtraction. I repeated the process for the small gear and assembled the gear pair according to the theoretical mounting distance. I checked the assembly for interference and found no interference in the final reverse-built model.
To evaluate the reverse reconstruction accuracy, I compared the reconstructed tooth surfaces with the original heat-treated point cloud. The lateral deviation, negative-direction deviation, geometric deviation, and positive normal deviation were extracted. Since no specific national standard for reverse-engineered gears was available to me, I adopted the criterion that a deviation less than 0.5 percent of the gear diameter can be considered acceptable. The gear diameter was 203.82 mm, so the allowable deviation was about 1.0191 mm. The error results for the concave and convex surfaces are listed below.
| Surface | Error Type | Maximum | Average | Standard Deviation |
|---|---|---|---|---|
| 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 allowable 1.0191 mm, so I considered the reverse reconstruction to be acceptable and close to a first-class level for this application. The reconstructed cycloid hypoid bevel gears were then used for transmission-error simulation. I exported the assembled model to a multibody dynamics environment and defined contact forces with an impact function. The contact parameters I used are listed below.
| Parameter | Value |
|---|---|
| Force exponent | 1.5 |
| Damping coefficient | \(5\times10^4\) |
| Stiffness coefficient | \(3.16\times10^9\) |
| Static transition velocity | 0.0001 |
| Static friction coefficient | 0.08 |
| Dynamic transition velocity | 0.01 |
| Dynamic friction coefficient | 0.05 |
| Penetration depth | 0.0001 |
I studied four operating conditions: forward drive, reverse drive, neutral forward coast, and neutral reverse coast. For forward and reverse drive, the small gear was the driving gear and the large gear was the driven gear. For neutral forward and neutral reverse, the large gear was the driving gear and the small gear was the driven gear. The gear ratio was calculated from the tooth numbers:
$$
i = \frac{z_2}{z_1} = \frac{47}{11} = 4.2727
$$
The theoretical output speeds and torques were therefore obtained from the input values. The operating cases and the transmission-error results are summarized below.
| Condition | Driving Gear | Driven Gear | Input Speed / deg s^{-1} | Input Torque / N mm | Theoretical Output Speed / deg s^{-1} |
|---|---|---|---|---|---|
| Forward drive | Small gear, concave | Large gear, convex | 900 | 240 | 210.638 |
| Reverse drive | Small gear, convex | Large gear, concave | 900 | 240 | 210.638 |
| Neutral forward | Large gear, concave | Small gear, convex | 211 | 1025 | 901.545 |
| Neutral reverse | Large gear, convex | Small gear, concave | 211 | 1025 | 901.545 |
| Condition | Maximum Output Speed / deg s^{-1} | Average Output Speed / deg s^{-1} | Minimum Output Speed / deg s^{-1} | Relative Error / % |
|---|---|---|---|---|
| Forward drive | 247.9174 | 210.6575 | 183.0035 | 0.01 |
| Reverse drive | 236.7192 | 210.5223 | 188.3092 | 0.05 |
| Neutral forward | 1187.7295 | 900.3571 | 661.9666 | 0.13 |
| Neutral reverse | 1075.5680 | 901.1479 | 745.8160 | 0.04 |
The relative transmission error was calculated as:
$$
\varepsilon = \frac{\omega_{out,avg} – \omega_{out,theory}}{\omega_{out,theory}}\times100\%
$$
The forward and reverse drive cases produced relatively stable output speed. The neutral forward and neutral reverse cases showed much larger speed fluctuations. This indicated that using the small gear as the driving member gave better stability, while using the large gear as the driving member produced stronger vibration and impact. This agreed with the practical preference for small-gear driving in vehicle axles and indirectly supported the correctness of the reverse-built cycloid hypoid bevel gears.
For the dynamic contact analysis, I used a nonlinear finite-element formulation. The virtual work equation in an incremental Total Lagrange form was written as:
$$
\int_V \delta \varepsilon^T D \Delta \varepsilon\,dV + \int_V \delta \eta^T S\,dV = \int_V \delta u^T b\,dV + \int_A \delta u^T t\,dA – \int_V \delta \varepsilon^T S\,dV
$$
The nonlinear finite-element equilibrium equation was expressed as:
$$
(K_0 + K_u + K_\sigma)\Delta a = R_{n+1} – F_n
$$
where \(K_0\) is the small-displacement stiffness matrix, \(K_u\) is the large-displacement stiffness matrix, \(K_\sigma\) is the initial-stress stiffness matrix, \(\Delta a\) is the nodal displacement increment, \(R_{n+1}\) is the external load vector, and \(F_n\) is the internal force vector. For the contact interface, I used a hard contact pressure-overclosure relationship and a Coulomb friction model. The critical shear stress for frictional sliding was:
$$
\tau_{crit} = \min(\mu p, \tau_{max})
$$
The equivalent shear stress on the contact plane was:
$$
\tau_{eq} = \sqrt{\tau_1^2 + \tau_2^2}
$$
When the equivalent shear stress was below the critical value, the surfaces remained stuck; when it reached the critical value, sliding occurred. For anisotropic friction, the equivalent stress could be written as:
$$
\tau_{eq} = \sqrt{\left(\frac{\tau_1}{\mu_1}\right)^2 + \left(\frac{\tau_2}{\mu_2}\right)^2}\sqrt{\mu_1^2+\mu_2^2}
$$
I used a professional meshing tool to discretize the reverse-built cycloid hypoid bevel gears. Because the tooth surface is the main contact region, I refined the tooth-surface mesh and used a coarser mesh for the gear body. I generated one tooth, copied it by rotation, and then checked for gaps and duplicate nodes. The material properties assigned to the finite-element model are listed below.
| Property | Value |
|---|---|
| Elastic modulus | \(2.10\times10^{11}\) Pa |
| Density | \(7.84\times10^3\) kg m^{-3} |
| Poisson ratio | 0.278 |
The contact definition for the dynamic simulation is summarized below. I used a surface-to-surface contact formulation, an explicit dynamic solution, a penalty method, and a finite-sliding tracking scheme. The small-gear tooth surface was the master surface, and the large-gear tooth surface was the slave surface.
| Contact Setting | Selection |
|---|---|
| Master surface | Small-gear tooth surface |
| Slave surface | Large-gear tooth surface |
| Contact type | Surface-to-surface |
| Solution control | Explicit dynamic |
| Contact property | Dynamic friction |
| Contact method | Penalty function |
| Sliding formulation | Finite sliding |
I applied loads through reference points coupled to the inner rings of the two gears. For forward and reverse drive, the small gear received a rotational speed and the large gear received a torque. For neutral forward and neutral reverse, the large gear received a rotational speed and the small gear received a torque. The load cases are listed below.
| Condition | Driving Member | Driving Surface | Driven Member | Driven Surface | Speed / rad s^{-1} | Torque / N mm |
|---|---|---|---|---|---|---|
| Forward drive | Small gear | Concave | Large gear | Convex | 104.667 | 1025 |
| Reverse drive | Small gear | Convex | Large gear | Concave | 104.667 | 1025 |
| Neutral forward | Large gear | Concave | Small gear | Convex | 24.492 | 240 |
| Neutral reverse | Large gear | Convex | Small gear | Concave | 24.492 | 240 |
To avoid a sudden impact at the beginning of the explicit dynamic analysis, I applied the load through an amplitude curve. The load increased linearly during the first 1 ms and then remained constant. This allowed the cycloid hypoid bevel gears to enter contact smoothly and improved numerical stability.
For forward drive, four pairs of teeth participated in meshing at the same time, and the contact-stress locations were distributed in a stepped pattern. This confirmed the large overlap ratio and high load capacity of the cycloid hypoid bevel gears. The contact force fluctuated initially between 10 kN and 20 kN and then stabilized around 18 kN with a fluctuation band of about 15-20 kN. The contact area fluctuated mainly between 40 mm² and 80 mm². At the beginning, the contact area fluctuation was dense because the contact was not yet fully stable. As time increased, the fluctuation became more regular and periodic. I concluded that forward drive gave the best transmission stability among the four conditions.
| Forward Drive Quantity | Observed Range | Stabilized Range | Character |
|---|---|---|---|
| Contact force | 10-20 kN | 15-20 kN | Small and stable |
| Contact area | 40-80 mm² | 40-80 mm² | Periodic and small fluctuation |
| Number of meshing tooth pairs | 4 | 4 | Stepped contact |
For reverse drive, four pairs of teeth also participated in meshing, but the contact force and contact area showed interrupted pulses and obvious impact. The contact force fluctuated around 20 kN with a range of about 0-40 kN. The contact area initially ranged from 0 to 95 mm² and then stabilized between 10 mm² and 85 mm². The fluctuation range was larger than that of forward drive, so reverse drive produced more impact even though the load level was not as high as in neutral operation.
| Reverse Drive Quantity | Observed Range | Stabilized Range | Character |
|---|---|---|---|
| Contact force | 0-40 kN | Around 20 kN | Pulsed and impactive |
| Contact area | 0-95 mm² | 10-85 mm² | Large fluctuation |
| Number of meshing tooth pairs | 4 | 4 | Stepped contact |
For neutral forward, three pairs of teeth participated in meshing, and the contact force and contact area were much larger. The contact force initially fluctuated from 40 kN to 140 kN and then stabilized around 90 kN with a fluctuation band of about 80-100 kN. The contact area ranged from 60 mm² to 160 mm². The contact was unstable, and the fluctuation was more irregular than in forward drive. This condition produced severe impact and a high risk of gear damage.
| Neutral Forward Quantity | Observed Range | Stabilized Range | Character |
|---|---|---|---|
| Contact force | 40-140 kN | 80-100 kN | Large and unstable |
| Contact area | 60-160 mm² | 60-160 mm² | Wide fluctuation |
| Number of meshing tooth pairs | 3 | 3 | Stepped contact |
For neutral reverse, three pairs of teeth participated in meshing. The contact force and contact area were smaller than those in neutral forward but still larger than those in forward and reverse drive. The contact force fluctuated mainly from 20 kN to 70 kN, and the contact area ranged from 40 mm² to 120 mm². The contact was unstable and the fluctuation density was high. This condition was also unfavorable for the cycloid hypoid bevel gears.
| Neutral Reverse Quantity | Observed Range | Character |
|---|---|---|
| Contact force | 20-70 kN | Moderately large and unstable |
| Contact area | 40-120 mm² | Unstable and dense fluctuation |
| Number of meshing tooth pairs | 3 | Stepped contact |
By comparing the four conditions, I found that forward drive gave the smallest and most stable contact force, the smallest contact-area fluctuation, and the clearest periodicity. Reverse drive gave moderate force but larger fluctuation and impact. Neutral forward gave the largest contact force and contact area, with severe impact and unstable meshing. Neutral reverse gave slightly lower values than neutral forward but still had a wide fluctuation range. From the viewpoint of ride comfort and gear life, forward drive was clearly the best condition, while neutral operation should be avoided as much as possible. These results were consistent with the practical use of cycloid hypoid bevel gears in vehicle drive axles.
I validated the numerical model with heat-treatment experiments and contact-performance experiments. The carburizing, quenching, and tempering experiments were performed with a continuous heat-treatment furnace. The gears were loaded on fixtures and processed in batches. After heat treatment, I measured the hardness with a Rockwell hardness tester. The gear was sectioned perpendicular to the tooth surface, and the cut passed through the tooth root so that the surface and core hardness could be measured. The measured and simulated hardness values are compared below.
| Location | Simulated Hardness / HRC | Measured Hardness / HRC |
|---|---|---|
| Tooth surface | 59.33 | 59.2 |
| Tooth core | 36.9 | 37.8 |
The measured surface hardness was very close to the simulated value. The core hardness also agreed well. This indicated that the heat-treatment model had good accuracy. I also examined the microstructure with a field-emission scanning electron microscope. The tooth tip contained a small amount of fine granular carbide. The tooth surface contained fine needle-like martensite and some retained austenite. The tooth core contained martensite and a small amount of blocky ferrite. The non-martensitic layer depth was about 0.017 mm, and the internal oxidation depth was about 0.012 mm. Both values were below the maximum allowed depth of 0.02 mm. These results showed that the heat-treatment process produced acceptable microstructure and case-core properties.
| Microstructural Feature | Observation | Requirement |
|---|---|---|
| Tooth-tip carbide | Small amount of fine granular carbide | Acceptable |
| Tooth-surface martensite | Fine needle-like martensite with retained austenite | Acceptable |
| Tooth-core structure | Martensite with small blocky ferrite | Acceptable |
| Non-martensitic layer depth | 0.017 mm | <0.02 mm |
| Internal oxidation depth | 0.012 mm | <0.02 mm |
To analyze the tooth-surface error before and after heat treatment, I used a 45-point measurement method similar to production inspection. I measured the coordinates of 45 points on the concave and convex surfaces before and after heat treatment and plotted the difference surface. I also extracted corresponding points from the numerical model. The measured and simulated difference surfaces showed the same general trend. The largest errors appeared near the large end, small end, tooth root, and tooth tip. This agreement supported the correctness of the reverse-built cycloid hypoid bevel gears and the heat-treatment simulation.
For the contact-performance experiment, I used a bevel-gear rolling inspection machine. I set the driving shaft speed to 1000 r/min and the load to 1025 N·m. After running the gear pair, I observed the contact pattern on the concave and convex surfaces of the large gear. I also extracted the contact-stress points from the numerical simulation and fitted them into digital contact patterns. The measured and digital contact patterns had approximately the same position and similar inclination. The shape of the contact area showed a small difference. The concave contact zone was approximately in the middle third of the face width and was biased toward the tooth tip and the small end. The convex contact zone was also approximately in the middle third of the face width and was biased toward the tooth root and the small end. The concave contact pattern had a larger inclination than the convex pattern. In the numerical simulation, the concave contact zone occupied about 26-70 percent of the face width, with its centroid biased toward the tooth tip and small end. The convex contact zone occupied about 34.7-80 percent of the face width, with its centroid biased toward the tooth root and small end. These results were consistent with the experimental observations.
| Contact Case | Measured Pattern Position | Digital Pattern Position | Bias Direction | Inclination |
|---|---|---|---|---|
| Concave surface | Middle third of face width | 26-70% of face width | Tooth tip and small end | Larger |
| Convex surface | Middle third of face width | 34.7-80% of face width | Tooth root and small end | Smaller |
The difference between the measured and digital contact shapes came from two main sources. First, the reverse reconstruction of the cycloid hypoid bevel gears introduced some tooth-surface error. Second, the experimental gear pair had mounting errors, which changed the relative curvature at the meshing point and affected the contact shape. Even with these differences, the contact position, bias direction, and inclination trend were in good agreement. This gave me confidence that the integrated simulation method was reasonable for heat-treated cycloid hypoid bevel gears.
Overall, I reached several conclusions. The carbon content was highest at the tooth tip, followed by the tooth surface, bottom surface, tooth root, and tooth core. The cooling rate was highest at the tooth tip, followed by the tooth surface, tooth root, bottom surface, and tooth core. The martensite fraction and hardness followed the same order. The tooth surface experienced tension first and then compression, while the tooth core experienced compression first and then tension. After quenching, the tooth surface had compressive residual stress and the tooth core had tensile residual stress. The largest distortion appeared at the root of the concave flank at the large end, the middle of the tooth tip, and the root of the convex flank at the small end. A crowned zone formed near the middle of the tooth tip. The simulated surface hardness, core hardness, inner-hole ovality, and bottom-surface flatness all satisfied the production requirements.
The reverse reconstruction of the heat-treated cycloid hypoid bevel gears was accurate enough for dynamic analysis. The maximum lateral, negative-direction, geometric, and positive normal deviations of the concave and convex surfaces were below 0.5 percent of the gear diameter. The transmission-error simulation showed that the output speed fluctuated around the theoretical value under all four operating conditions and that the relative error remained within 0.13 percent. The small-gear driving conditions gave more stable transmission than the large-gear driving conditions. The dynamic contact simulation showed that forward drive gave the lowest contact force, the smallest contact-area fluctuation, and the most stable meshing. Neutral forward produced the largest contact force and contact area, the widest fluctuation, and the most severe impact. Neutral reverse was also unstable. These results were consistent with the practical observation that cycloid hypoid bevel gears operate best under normal forward drive and are more likely to be damaged under neutral or abusive conditions.
For future work, I would optimize the heat-treatment process to reduce the crowned zone near the tooth tip, because an excessively large crowned zone can cause edge contact, strong vibration, and impact loading. I would also seek a more direct coupling between the heat-treatment solver and the dynamic contact solver so that the reverse-reconstruction step can be reduced or eliminated. This would decrease the errors introduced by point-cloud processing and surface fitting. In addition, I would refine the mesh-generation strategy for the complex cycloid hypoid bevel gears and compare rigid-body dynamic results with flexible-body dynamic results. Flexible-body analysis would be especially important for predicting vibration, noise, and tooth-root stress more accurately. Finally, I would extend the study to include manufacturing errors, assembly errors, and lubrication effects, because these factors strongly influence the loaded contact behavior and fatigue life of cycloid hypoid bevel gears.
In summary, I established a complete numerical and experimental framework for cycloid hypoid bevel gears with heat-treatment distortion. The framework connected carburizing and quenching simulation, phase-transformation and hardness prediction, reverse reconstruction, transmission-error analysis, dynamic contact analysis, and experimental validation. The results showed that the heat-treatment process produced the desired hard surface and tough core, that the heat-treated geometry could be reconstructed with acceptable accuracy, and that forward drive gave the most stable meshing behavior. The numerical trends matched the experimental hardness, microstructure, difference surface, and contact pattern. I believe this framework can help designers and manufacturers of hypoid bevel gears evaluate heat-treatment distortion, improve tooth-surface quality, and predict meshing performance before costly trial production.