In this study I investigate the residual stress and deformation produced during the cutting of an aviation spiral bevel gear large wheel after near-net forging and before heat treatment. I follow a process chain of near-net forging, gear machining, heat treatment, precision grinding, and surface strengthening. I concentrate on the gear machining stage because the cutting action changes the tooth surface state, introduces plastic deformation, and creates a coupled thermal-mechanical residual stress field. I use the forming method for the large wheel, and I model the process with three-dimensional geometry, forging simulation, single-tooth cutting simulation, and oblique cutting simulation. The purpose of the work is to determine how cutting speed, feed rate, and cutting depth affect tooth surface deformation and residual stress, to build predictive response surface models, and to optimize the gear machining parameters for low deformation and favorable residual stress.
I treat the spiral bevel gear large wheel as a complex surface component. Its tooth geometry, cutting kinematics, and material response make gear machining a strongly coupled process. The near-net forming step provides a shaped blank with internal residual stress, and the subsequent gear machining step removes material from the tooth surface. The initial residual stress from forging is not removed completely by cutting, so I include it in the residual stress simulation. I also simplify the real gear machining process into an equivalent oblique cutting model for residual stress analysis, while the global tooth deformation is studied through a single-tooth cutting model. This combination allows me to analyze both macroscopic deformation and surface-layer residual stress without losing the physical characteristics of gear machining.
| Process stage | Role in the study | Output considered |
|---|---|---|
| Near-net forging | Produces a near-final tooth form and initial residual stress | Forging residual stress field |
| Gear machining before heat treatment | Removes material and creates cutting deformation and residual stress | Tooth surface deformation and residual stress |
| Heat treatment | Not the main focus of this stage but influences final state | Downstream process reference |
| Precision grinding | Final geometry correction after heat treatment | Final accuracy and surface integrity |
| Surface strengthening | Improves fatigue and contact performance | Surface integrity and life |
Geometric modeling and gear machining principle. I use the geometric design relationships of spiral bevel gears to establish the large wheel model. The pitch diameter, pitch cone angle, outer cone distance, addendum, dedendum, face cone angle, and root cone angle are obtained from standard bevel gear equations. I use a three-dimensional CAD environment to import tooth surface point data and to construct the concave surface, convex surface, and tooth bottom. Then I trim one tooth slot and array it around the blank to obtain the complete large wheel. The main geometry equations are summarized below.
$$d = mz$$
$$\tan\delta_2 = \frac{z_2\sin\Sigma}{z_1+z_2\cos\Sigma}$$
$$R_e = \frac{d_2}{2\sin\delta_2}$$
$$L_m = L_a – \frac{b}{2}$$
$$h_a = (h_a^* \pm \xi)m$$
$$h_f = (h_f^* + c \mp \xi)m$$
$$\tan\gamma = \frac{h_f}{R_e}$$
$$D = d + 2h_a\cos\delta$$
$$S_d = \sqrt{H^2+V^2}$$
$$q_2 = \arctan\left(\frac{V}{H}\right)$$
$$\Delta x_B = L’\tan\Delta r” \approx \frac{L’\Delta r”}{3440}$$
$$W = W’ – (0.05\sim0.08)$$
$$N_0 = \frac{540\tan\beta_m}{Z_p\tan\alpha}\left(1-\frac{L_m}{r_d}\sin\beta_m\right)+0.5\sin\beta_m$$
For the large wheel of the aviation spiral bevel gear, the forming method is selected. In this method, the tooth profile of the generated gear corresponds closely to the cutting edge profile of the tool. The blank is fixed on the rotary table, the cutter head axis is set at a specific angle relative to the blank, and the cutter head feeds along its own axis. The inner and outer blades remove the tooth slot. After one slot is completed, the tool retracts, the blank indexes, and the next slot is machined. This is the core of the gear machining route used in this study.
| Parameter | Value |
|---|---|
| Module | 3 mm |
| Number of teeth | 44 |
| Hand | Left |
| Face width | 20.2 mm |
| Addendum coefficient | 0.85 |
| Pressure angle | 20 deg |
| Mean spiral angle | 35 deg |
| Shaft angle | 90 deg |
| Pitch diameter | 132 mm |
| Outer cone distance | 69.73 mm |
| Face cone angle | 73.39 deg |
| Mounting distance | 27 mm |
Tool generating surface and instantaneous gear machining parameters. The cutter head creates the tooth slot through the rotation of the inner and outer cutting edges. I define the tool generating surface in the cutter coordinate system. The cutting edge is represented as a conical surface, and the radius of the tool tip depends on the nominal cutter radius and the blade spacing. The face cone of the large wheel is also expressed in its own coordinate system. Coordinate transformation matrices are then used to move both the tool surface and the face cone into the machine coordinate system. The intersection of these two surfaces defines the instantaneous contact between the tool and the blank during gear machining.
$$r_t(\mu,\theta)=
\begin{bmatrix}
\pm r_c\sin\alpha_m\cos\theta\\
\pm r_c\sin\alpha_m\sin\theta\\
-r_c\cos\alpha_m+\mu
\end{bmatrix}$$
$$r_c = r_0 \pm 0.5W$$
$$r_a(p,\beta)=
\begin{bmatrix}
p\cos\delta_a\\
p\sin\delta_a\sin\beta\\
p\sin\delta_a\cos\beta\\
1
\end{bmatrix}$$
$$r_{tm}=M_{mt}r_t(\mu,\theta)$$
$$r_{am}=M_{mf}M_{f2}M_{2a}r_a(p,\beta)$$
$$L_M(\mu,\theta,p,\beta,z_j)=r_{tm}-r_{am}=0$$
For the machine coordinate system, the radial cutter position and angular cutter position are determined from the horizontal and vertical settings. The entry and exit angles are approximated from the face width, mean spiral angle, cutter radius, and rotation number. These values are used to identify the active cutting interval for a single blade during gear machining.
$$S_2 = \sqrt{H^2+V^2}$$
$$q_2 = \arctan\left(\frac{V}{H}\right)$$
$$\theta_{st}\approx \beta_m-\arcsin\left(\frac{B}{2kr_0\cos\beta_m}\right)$$
$$\theta_{ex}\approx \beta_m+\arcsin\left(\frac{B}{2kr_0\cos\beta_m}\right)$$
$$f_z=\frac{f_r}{Z_H}$$
The instantaneous undeformed cutting thickness and width are important for the cutting force and residual stress analysis. For the outer blade, the main cutting edge and the top edge have different thickness and width expressions. The same treatment is applied to the inner blade. These instantaneous values are used as the physical basis for the oblique cutting simplification and for selecting the cutting parameter ranges in the finite element simulations.
$$h_{ew}=2f_z\sin\alpha_e$$
$$b_{ew}=u_{ej}(\theta(t),z_j(t))$$
$$h_{iw}=2f_z\sin\alpha_i$$
$$b_{iw}=u_{ij}(\theta(t),z_j(t))$$
$$h_w=2f_z$$
$$b_w=S_b$$
$$S_b=(0.5\sim0.75)W$$
Material model and failure criteria for gear machining simulation. The large wheel material is AISI 4340 steel. This material has high strength, high toughness, and good hardenability, which makes it suitable for aviation transmission components. I use the Johnson-Cook constitutive model to describe the flow stress under large strain, high strain rate, and high temperature. I also use the Johnson-Cook damage criterion to describe material failure and chip separation during gear machining. The damage parameter accumulates with plastic strain, and failure occurs when the damage parameter reaches one. The fracture energy evolution is used to reduce mesh dependency after damage initiation.
| Property | Value |
|---|---|
| Density | 7.83×10-9 T/mm3 |
| Elastic modulus | 2×105 MPa |
| Poisson ratio | 0.29 |
| Yield strength | 792 MPa |
| Ultimate strength | 980 MPa |
| Thermal conductivity | 38 W/m/°C |
| Thermal expansion coefficient | 3.2×10-5 /°C |
| Specific heat | 4.77×108 kJ/T/°C |
| Melting point | 1519.85 °C |
$$\sigma = \left[A+B(\varepsilon^{pl})^n\right]\left[1+C\ln\left(\frac{\dot{\varepsilon}^{pl}}{\dot{\varepsilon}_0}\right)\right](1-\hat{T}^m)$$
$$\hat{T}=
\begin{cases}
0, & TT_m
\end{cases}$$
| Johnson-Cook parameter | Value |
|---|---|
| A | 792 MPa |
| B | 510 MPa |
| n | 0.26 |
| C | 0.014 |
| m | 1.03 |
$$\omega = \sum\left(\frac{\Delta\bar{\varepsilon}^{pl}}{\bar{\varepsilon}_f^{pl}}\right)$$
$$\bar{\varepsilon}_f^{pl}=\left[D_1+D_2\exp(-D_3\eta)\right]\left[1+D_4\ln\left(\frac{\dot{\bar{\varepsilon}}}{\dot{\varepsilon}_0}\right)\right]\left[1+D_5\hat{T}\right]$$
$$G_f=\int L\frac{\sigma_y}{\bar{\varepsilon}_0^{pl}}d\varepsilon^{pl}=\int_0^{u_f^{pl}}\sigma_y du^{pl}$$
| Failure parameter | Value |
|---|---|
| D1 | 0.05 |
| D2 | 3.44 |
| D3 | 2.12 |
| D4 | 0.002 |
| D5 | 0.61 |
Forging model and initial residual stress. To reflect the real gear machining condition after near-net forming, I build a forging model of the large wheel. I use a one-twelfth sector with symmetry boundaries to reduce computation time. The blank is treated as an elastoplastic body, while the upper and lower dies are treated as rigid bodies. I set the blank temperature, die temperature, friction coefficient, forging speed, and air cooling convection coefficient. After forging and cooling, I extract the residual stress along the tooth depth near the pitch circle. This stress field is then applied as an initial stress state in the subsequent gear machining residual stress simulation. In this way, the gear machining simulation includes the influence of the prior forging process.
| Forging simulation setting | Value |
|---|---|
| Blank material | AISI 4340 |
| Blank temperature | 900 °C |
| Die temperature | 200 °C |
| Friction coefficient | 0.25 |
| Upper die speed | 20 mm/s |
| Environment temperature | 25 °C |
| Air convection coefficient | 0.02 N/s/mm/°C |
| Symmetry sector | 1/12 of the wheel |
Single-tooth cutting model for deformation. Deformation during gear machining is macroscopic because the whole tooth can bend and displace under cutting load. I therefore build a single-tooth cutting model in DEFORM. The tool is rigid, and the tooth is elastoplastic. The bottom of the tooth is fixed, and the tool moves according to the selected cutting speed, feed rate, and cutting depth. The workpiece is meshed with tetrahedral elements. Friction and heat generation are enabled. After each simulation, I extract the total displacement at 45 points distributed over five lines along the tooth height and nine lines along the tooth width. The maximum value is taken as the tooth surface deformation for that simulation.
| Single-factor level | Cutting speed v (m/min) | Cutting depth a (mm) | Feed f (mm/r) |
|---|---|---|---|
| 1 | 80 | 1.0 | 0.2 |
| 2 | 100 | 1.5 | 0.3 |
| 3 | 120 | 2.0 | 0.4 |
| 4 | 140 | 2.5 | 0.5 |
The single-factor results show clear trends. When the feed rate increases from 0.2 mm/r to 0.5 mm/r, the tooth surface deformation increases. The reason is that the removed metal volume per unit time increases, the extrusion action on the tooth surface becomes stronger, and plastic deformation grows. When the cutting depth increases, the deformation also increases, but the change is smaller than that caused by feed rate. The cutting depth increases the total chip volume and friction heat, but the change in local contact stress per unit cutting edge length is limited. When the cutting speed increases, the deformation first increases and then decreases. At lower and moderate speeds, the cutting force and friction heat increase, which promotes plastic deformation. At higher speeds, thermal softening reduces the cutting force, and the chip leaves the cutting zone faster, which reduces the deformation. Therefore, the influence ranking for deformation is feed rate, then cutting speed, then cutting depth.
| Parameter | Trend of tooth surface deformation | Relative influence |
|---|---|---|
| Feed rate | Increases with feed rate | Strong |
| Cutting depth | Increases slightly with cutting depth | Moderate to weak |
| Cutting speed | First increases, then decreases | Strong |
Response surface analysis for cutting deformation. I use a Box-Behnken design to build a response surface model for tooth surface deformation. The three factors are cutting speed, cutting depth, and feed rate. The response is the maximum tooth surface deformation. I fit a second-order polynomial model and check its significance by analysis of variance. The model F value is much larger than the critical value, the P value is very small, and the adjusted coefficient of determination is high. The normal residual plot also shows that the residuals are close to a normal distribution. These results indicate that the deformation prediction model is reliable for the tested gear machining window.
$$\hat{Y}=\beta_0+\sum_{i=1}^{3}\beta_i x_i+\sum_{i=1}^{3}\sum_{j=i+1}^{3}\beta_{ij}x_i x_j+\sum_{i=1}^{3}\beta_{ii}x_i^2+\varepsilon_c$$
$$\hat{\delta}=f(v,a,f)$$
| Run | v (m/min) | a (mm) | f (mm/r) | Deformation (micrometers) |
|---|---|---|---|---|
| 1 | 140 | 1.00 | 0.35 | 20.181 |
| 2 | 110 | 2.50 | 0.50 | 31.370 |
| 3 | 110 | 2.50 | 0.20 | 21.600 |
| 4 | 80 | 1.75 | 0.50 | 29.001 |
| 5 | 110 | 1.75 | 0.35 | 22.600 |
| 6 | 80 | 1.00 | 0.35 | 19.595 |
| 7 | 140 | 2.50 | 0.35 | 18.500 |
| 8 | 80 | 1.75 | 0.20 | 19.050 |
| 9 | 80 | 2.50 | 0.35 | 22.819 |
| 10 | 110 | 1.75 | 0.35 | 22.401 |
| 11 | 110 | 1.00 | 0.20 | 20.229 |
| 12 | 110 | 1.00 | 0.50 | 29.900 |
| 13 | 140 | 1.75 | 0.50 | 27.065 |
| 14 | 110 | 1.75 | 0.35 | 22.500 |
| 15 | 140 | 1.75 | 0.20 | 17.201 |
| Source | DF | Sum of squares | Mean square | F value | P value |
|---|---|---|---|---|---|
| Model | 9 | 269.352 | 29.9279 | 602.84 | 0.0001 |
| Error | 5 | 0.248 | 0.0496 | ||
| Total | 14 | 269.600 |
The deformation optimization uses the minimum tooth surface deformation as the objective. The constraints are the tested ranges of cutting speed, cutting depth, and feed rate. The optimized result is a cutting depth of 2.5 mm, a feed rate of 0.2303 mm/r, and a cutting speed of 140 m/min. Under this combination, the predicted minimum tooth surface deformation is 16.6734 micrometers. This result provides a reference for reducing deformation in gear machining.
$$\min(\delta)=
\begin{cases}
80\le v\le 140\\
1.0\le a\le 2.5\\
0.2\le f\le 0.5
\end{cases}$$
| Optimization target | Cutting depth a (mm) | Feed f (mm/r) | Cutting speed v (m/min) | Predicted deformation (micrometers) |
|---|---|---|---|---|
| Minimum deformation | 2.5 | 0.2303 | 140 | 16.6734 |
Residual stress model for gear machining. The residual stress analysis focuses on the surface layer and the depth distribution. A full three-dimensional gear machining model would require extremely fine mesh near the cutting edge and would be inefficient. I therefore use an equivalent oblique cutting model. The cutting edge is inclined relative to the cutting velocity, which is consistent with the real forming method. I set the tool rake angle, relief angle, inclination angle, nose radius, friction coefficient, and thermal coupling. I also apply the forging-induced initial residual stress field before cutting. The workpiece is meshed with hexahedral elements, and the tool is meshed with tetrahedral elements. The simulation outputs the stress component normal to the cutting direction along the depth below the machined surface.
$$\varepsilon^{th}=\alpha(T-T_0)$$
$$\sigma_{ij}=C_{ijkl}(\varepsilon_{kl}-\varepsilon_{kl}^{th}-\varepsilon_{kl}^{pl})$$
$$\sigma_{res}=\sigma_{mech}+\sigma_{therm}$$
In the oblique cutting model, the cutting force is decomposed into tangential, feed, and radial components. Each component contains a shear term and a ploughing term. The shear flow stress is computed from the Johnson-Cook model. The normal stress on the shear plane and the ploughing stresses near the tool nose are also considered. These stress components explain why the machined surface shows tension while the subsurface shows compression after the tool passes. The thermal effect tends to produce tensile stress near the surface, while the mechanical effect tends to produce compressive stress below the surface. The final residual stress distribution is the result of the competition between these two effects during gear machining.
$$F_t=K_{tc}da_p t_c+K_{te}da_p$$
$$F_f=K_{fc}da_p t_c+K_{fe}da_p$$
$$F_r=K_{rc}da_p t_c+K_{re}da_p$$
$$\tau_s=\frac{1}{\sqrt{3}}\left[A+B\left(\frac{\varepsilon}{\sqrt{3}}\right)^n\right]\left[1+C\ln\left(\frac{\dot{\varepsilon}}{\dot{\varepsilon}_0}\right)\right]\left[1-\left(\frac{T-T_0}{T_m-T_0}\right)^m\right]$$
$$F_n=F_t\cos\phi_n\sin\beta+F_f\cos\phi_n+F_r\sin\phi_n\sin\beta$$
$$p_s=\frac{F_n}{fda_p}\cos\beta\sin\phi$$
$$L_{cut}=\frac{f}{\sin\phi_n\cos\phi_i}$$
$$p_e=\frac{P_{thrust}}{4da_p L_{plow}\cos\beta}$$
$$q_e=\frac{P_{cut}\mu}{da_p L_{plow}\cos\beta}$$
| Oblique cutting setting | Value |
|---|---|
| Tool rake angle | 15 deg |
| Tool relief angle | 6 deg |
| Inclination angle | 5 deg |
| Tool nose radius | 0.02 mm |
| Friction coefficient | 0.2 |
| Workpiece material | AISI 4340 |
| Tool material | Cemented carbide |
| Analysis type | Coupled thermal-mechanical |
Initial residual stress effect. I compare the residual stress distribution with and without the forging-induced initial stress. The two curves have the same general shape. The surface is tensile, and the stress quickly becomes compressive with increasing depth. The compressive stress reaches a peak and then decreases toward a stable value. When the initial residual stress is included, the maximum tensile stress at the surface is lower, and the maximum compressive stress is higher. In the deeper region, the curve without initial stress approaches zero, while the curve with initial stress remains compressive. This means that the prior forging state still influences the final residual stress after gear machining. The cutting process changes the near-surface stress state, but it does not completely erase the initial field.
| Condition | Surface stress | Maximum compressive stress | Deep region |
|---|---|---|---|
| Without initial stress | Tensile | Lower magnitude | Approaches zero |
| With initial stress | Tensile but lower | Higher magnitude | Remains compressive |
Single-factor residual stress results. I study the effects of cutting speed, feed rate, and cutting depth on the maximum tensile residual stress S1 and the maximum compressive residual stress S2. When the cutting speed increases, S1 increases and S2 decreases. Higher speed causes more severe friction and more heat, which increases the tensile stress near the surface. At the same time, thermal softening reduces the depth of mechanical loading, which lowers the compressive peak. When the feed rate increases, both S1 and S2 increase, and the stress-affected layer becomes deeper. A larger feed removes more material per revolution, raises the cutting resistance, increases plastic deformation, and pushes mechanical stress deeper into the tooth. When the cutting depth increases, the change in residual stress is relatively small. The active cutting edge length increases, but the local contact pressure and heat generation per unit edge length do not change greatly. Therefore, the influence ranking for residual stress is feed rate, then cutting speed, then cutting depth.
| Parameter | Effect on S1 | Effect on S2 | Effect on depth | Influence rank |
|---|---|---|---|---|
| Cutting speed | Increases | Decreases | Slight | Second |
| Feed rate | Increases | Increases | Increases | First |
| Cutting depth | Slight increase | First decreases then increases | Slight | Third |
Response surface analysis for residual stress. I use a Box-Behnken design for the residual stress response as well. The responses are the maximum tensile residual stress and the maximum compressive residual stress. I fit second-order models and evaluate them with analysis of variance. The F values are high, the P values are below the significance level, and the adjusted coefficients of determination are close to the ordinary coefficients of determination. The residual plots show that the models satisfy the normality assumption. I therefore use the models to optimize the cutting parameters for favorable residual stress in gear machining.
$$\hat{S}_1=g_1(v,a,f)$$
$$\hat{S}_2=g_2(v,a,f)$$
| Run | v (m/min) | a (mm) | f (mm/r) | S1 (MPa) | S2 (MPa) |
|---|---|---|---|---|---|
| 1 | 140 | 1.00 | 0.35 | 287.261 | -310.382 |
| 2 | 110 | 2.50 | 0.50 | 327.251 | -343.221 |
| 3 | 110 | 2.50 | 0.20 | 227.031 | -260.907 |
| 4 | 80 | 1.75 | 0.50 | 311.205 | -363.342 |
| 5 | 110 | 1.75 | 0.35 | 282.635 | -321.862 |
| 6 | 80 | 1.00 | 0.35 | 265.625 | -342.498 |
| 7 | 140 | 2.50 | 0.35 | 303.991 | -284.649 |
| 8 | 80 | 1.75 | 0.20 | 207.373 | -318.221 |
| 9 | 80 | 2.50 | 0.35 | 270.516 | -347.691 |
| 10 | 110 | 1.75 | 0.35 | 282.635 | -321.862 |
| 11 | 110 | 1.00 | 0.20 | 221.101 | -291.957 |
| 12 | 110 | 1.00 | 0.50 | 311.657 | -353.952 |
| 13 | 140 | 1.75 | 0.50 | 339.498 | -345.521 |
| 14 | 110 | 1.75 | 0.35 | 282.635 | -321.862 |
| 15 | 140 | 1.75 | 0.20 | 257.602 | -242.541 |
| Source for S1 | DF | Sum of squares | Mean square | F value | P value |
|---|---|---|---|---|---|
| Model | 9 | 20641.5 | 2293.5 | 159.95 | 0.0001 |
| Error | 5 | 71.7 | 23.9 | ||
| Total | 14 | 22713.2 |
| Source for S2 | DF | Sum of squares | Mean square | F value | P value |
|---|---|---|---|---|---|
| Model | 9 | 22376.1 | 2486.2 | 132.55 | 0.0001 |
| Error | 5 | 93.8 | 18.8 | ||
| Total | 14 | 22469.9 |
The optimization for residual stress seeks the minimum tensile residual stress and the maximum compressive residual stress. The constraints are the same as those used in the deformation study. The optimized parameters are a cutting depth of 2.5 mm, a feed rate of 0.2037 mm/r, and a cutting speed of 80 m/min. Under this condition, the predicted maximum tensile stress is 207.4106 MPa and the predicted maximum compressive stress is 322.2280 MPa. This combination is favorable because tensile stress at the surface is reduced and compressive stress in the subsurface is increased. Such a state is generally beneficial for fatigue resistance in gear machining and subsequent service.
$$\min(S_1),\ \max(S_2)=
\begin{cases}
80\le v\le 140\\
1.0\le a\le 2.5\\
0.2\le f\le 0.5
\end{cases}$$
| Optimization target | Cutting depth a (mm) | Feed f (mm/r) | Cutting speed v (m/min) | S1 (MPa) | S2 (MPa) |
|---|---|---|---|---|---|
| Minimum S1 and maximum S2 | 2.5 | 0.2037 | 80 | 207.4106 | 322.2280 |
Experimental validation of gear machining predictions. I use a CNC machine tool for the forming-method cutting experiment on the forged large wheel. The cutting parameters are set to the optimized values from the response surface models. After gear machining, I measure the tooth surface deformation on a gear measuring machine. I select five tooth slots and measure the deviation between the theoretical tooth surface and the actual tooth surface. The maximum deviation in each slot is taken as the deformation result. The experimental values are then compared with the predicted values. The maximum relative error is 8.39%, and the overall error remains below ten percent. This agreement indicates that the deformation simulation and the response surface model are effective for gear machining.
| Slot number | Experimental deformation (micrometers) | Predicted deformation (micrometers) | Relative error (%) |
|---|---|---|---|
| 1 | 17.7000 | 16.6734 | 5.80 |
| 2 | 18.2000 | 16.6734 | 8.39 |
| 3 | 17.3000 | 16.6734 | 3.62 |
| 4 | 15.8000 | 16.6734 | 5.53 |
| 5 | 17.4000 | 16.6734 | 4.18 |
For residual stress validation, I cut the machined wheel into single teeth using wire electrical discharge machining. The cutting position is kept away from the working tooth surface to reduce stress release caused by sectioning. I measure the residual stress on the concave working surface along the pitch line using an X-ray diffractometer. To obtain the depth distribution, I remove thin surface layers by electrolytic polishing and repeat the measurement. The experimental curves and simulation curves show the same pattern: tensile stress at the machined surface, a rapid transition to compressive stress, a compressive peak, and then a gradual decrease to a stable value deeper in the tooth. This consistency validates the oblique cutting residual stress model and the thermal-mechanical coupling treatment used in gear machining simulation.
| Group | Experimental S1 (MPa) | Predicted S1 (MPa) | Relative error for S1 (%) | Experimental S2 (MPa) | Predicted S2 (MPa) | Relative error for S2 (%) |
|---|---|---|---|---|---|---|
| 1 | 204.302 | 207.411 | 1.52 | 326.620 | 322.228 | 1.34 |
| 2 | 199.942 | 207.411 | 4.26 | 311.351 | 322.228 | 3.83 |
| 3 | 195.355 | 207.411 | 6.17 | 303.535 | 322.228 | 6.16 |
| 4 | 196.473 | 207.411 | 5.57 | 308.326 | 322.228 | 4.51 |
Discussion. The results show that gear machining parameters have a strong influence on both deformation and residual stress. Feed rate is the most influential factor for residual stress, while feed rate and cutting speed are the most influential factors for deformation. Cutting depth has a smaller effect within the tested range. The residual stress distribution is controlled by the competition between mechanical loading and thermal loading. Mechanical loading tends to create compressive stress below the surface, while thermal loading tends to create tensile stress near the surface. The final state depends on the cutting speed, feed rate, and the initial stress state from near-net forging. Because I include the initial residual stress in the model, the predicted deep-region stress is more realistic than a model that starts from a stress-free blank.
| Response | Most influential parameter | Second parameter | Least influential parameter | Optimized result |
|---|---|---|---|---|
| Deformation | Feed rate | Cutting speed | Cutting depth | 16.6734 micrometers |
| Tensile residual stress S1 | Feed rate | Cutting speed | Cutting depth | 207.4106 MPa |
| Compressive residual stress S2 | Feed rate | Cutting speed | Cutting depth | 322.2280 MPa |
Conclusions. I have studied the residual stress and deformation produced in the gear machining of an aviation spiral bevel gear large wheel after near-net forging and before heat treatment. I established a three-dimensional gear model, a forging model, a single-tooth cutting deformation model, and an equivalent oblique cutting residual stress model. I used single-factor simulations and Box-Behnken response surface designs to analyze the effects of cutting speed, cutting depth, and feed rate. The main conclusions are as follows.
First, tooth surface deformation increases with feed rate and cutting depth. With increasing cutting speed, the deformation first increases and then decreases. Feed rate and cutting speed have a stronger influence than cutting depth. The optimized parameters for minimum deformation are a cutting depth of 2.5 mm, a feed rate of 0.2303 mm/r, and a cutting speed of 140 m/min, giving a predicted minimum deformation of 16.6734 micrometers.
Second, the residual stress along the depth follows a characteristic curve. The machined surface is tensile. The stress quickly changes to compressive as depth increases. A compressive peak appears, and then the stress decreases toward a stable value. The maximum tensile stress increases with feed rate, cutting depth, and cutting speed. The maximum compressive stress increases with feed rate and decreases with cutting speed. The influence ranking is feed rate, then cutting speed, then cutting depth. The optimized parameters for minimum tensile stress and maximum compressive stress are a cutting depth of 2.5 mm, a feed rate of 0.2037 mm/r, and a cutting speed of 80 m/min. The predicted S1 is 207.4106 MPa, and the predicted S2 is 322.2280 MPa.
Third, the experimental validation supports the simulation and the predictive models. The maximum relative error for tooth surface deformation is 8.39%. The residual stress depth distributions from experiments and simulations follow the same trend, and the maximum relative error for the tensile and compressive stress peaks is 6.17%. These results show that the finite element model is feasible and that the response surface models are reliable for gear machining parameter selection and residual stress control.
Future work. I plan to add forging experiments to compare with the forging simulation and to better calibrate the initial residual stress field. I also intend to test additional optimization methods, such as neural network models and evolutionary algorithms, to improve the prediction of gear machining responses. In addition, I will study the influence of tool geometry, tool wear, and cooling conditions on residual stress and deformation. These extensions will provide a more complete understanding of gear machining quality control for aviation spiral bevel gears.
| Research item | Current finding | Planned extension |
|---|---|---|
| Initial residual stress | Forging simulation was applied before cutting | Add forging experiments for calibration |
| Optimization method | Response surface method was used | Use neural networks and evolutionary algorithms |
| Tool effect | Tool angles and nose radius were fixed | Study tool geometry and wear effects |
| Cooling condition | Thermal-mechanical coupling was included | Add different cooling and lubrication states |