I investigate the coupled mechanical and thermal behavior of spiral bevel gear cutting on a fully numerical control machine. My focus is the gear cutting process used for the large wheel by the forming method, because the cutting force and cutting temperature directly control tooth surface accuracy, cutter tooth wear, chip evacuation, spindle vibration, and the final meshing performance. I treat gear cutting as a strongly nonlinear process in which the tool, workpiece, and chip interact through instantaneous relative motion. I build a theoretical cutting-force model from oblique cutting geometry and energy conservation, and I verify it with a three-dimensional finite element model. I then use orthogonal experiments and single-factor studies to explain how cutting speed, cutting depth, rake angle, and edge inclination angle influence force and heat during gear cutting. Finally, I apply the numerical model to field gear cutting and analyze wear, chipping, scratching, and burning of the tooth surface.
For my first step in gear cutting analysis, I define the machine, cutter disk, and workpiece coordinate systems. The traditional mechanical cradle machine represents a virtual generating gear, while the fully numerical control machine replaces the cradle, eccentric, and generating mechanisms with linear and rotary axes. I convert the kinematic relation of the cradle machine into the fully numerical control machine so that the relative pose of the cutter disk and workpiece remains correct during gear cutting. This conversion is essential because the instantaneous uncut chip geometry is determined by the relative motion between the cutter tooth and the workpiece, not by the machine structure alone.
I write the transformation from the cradle coordinate system to the cradle coordinate system after rotation by the cradle angle \(q\) as
$$
M_{oc}=\begin{bmatrix}
\cos q & -\sin q & 0 & 0\\
\sin q & \cos q & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1
\end{bmatrix}.
$$
After translation by the radial cutter position \(S\), I use
$$
M_{cb}=\begin{bmatrix}
1 & 0 & 0 & S\\
0 & 1 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1
\end{bmatrix}.
$$
Rotation by the cutter rotation angle \(\gamma\) and cutter tilt angle \(i\) gives
$$
M_{ba}=\begin{bmatrix}
\cos\gamma & -\sin\gamma & 0 & 0\\
\sin\gamma & \cos\gamma & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1
\end{bmatrix},
$$
$$
M_{at}=\begin{bmatrix}
\cos i & 0 & \sin i & 0\\
0 & 1 & 0 & 0\\
-\sin i & 0 & \cos i & 0\\
0 & 0 & 0 & 1
\end{bmatrix}.
$$
I also define the workpiece rotation matrix and the mounting-angle matrix. The complete vector relation from the cutter disk coordinate system to the cradle workpiece coordinate system is
$$
M_{N}=M_{qp}M_{pn}M_{no}M_{oc}M_{cb}M_{ba}M_{at}.
$$
For the fully numerical control machine, I transform the cradle workpiece coordinate system to the machine coordinate system through the workpiece rotary axis \(A\) and the swing axis \(B\). The machine-to-workpiece matrix is
$$
M_{MW}=\begin{bmatrix}
\cos B & 0 & \sin B & 0\\
0 & 1 & 0 & 0\\
-\sin B & 0 & \cos B & 0\\
0 & 0 & 0 & 1
\end{bmatrix},
$$
$$
M_{Wq}=\begin{bmatrix}
1 & 0 & 0 & 0\\
0 & \cos A & -\sin A & 0\\
0 & \sin A & \cos A & 0\\
0 & 0 & 0 & 1
\end{bmatrix}.
$$
The resulting position vector of the cutter disk in the numerical control machine coordinate system is
$$
M_{R}=M_{MW}M_{Wq}M_{qp}M_{pn}M_{no}M_{oc}M_{cb}M_{ba}M_{at}.
$$
This kinematic conversion supports all later gear cutting calculations. I use it to define the instantaneous cutting speed vector, the uncut chip width, and the uncut chip thickness for each micro-segment of the cutting edge.
For the oblique cutting model in gear cutting, I divide the cutting edge into small straight micro-segments. Each micro-segment behaves as an oblique cutting element with a local cutting speed, a local rake angle, and a local edge inclination angle. I define the cutting plane \(P_s\), the base plane \(P_r\), and the normal plane \(P_o\). The normal rake angle \(\gamma_n\) and the equivalent rake angle \(\gamma_e\) are obtained from the tool design rake angle \(\gamma_0\) and the edge inclination angle \(\lambda_s\). I use the following geometric relations:
$$
\sin\gamma_e=\sin\gamma_n\cos\lambda_s,
$$
$$
\tan\gamma_n=\tan\gamma_0\cos\lambda_s.
$$
For small edge inclination angles, I apply the Stabler rule and set the chip flow angle equal to the edge inclination angle:
$$
\psi=\lambda_s.
$$
The shear angle \(\phi_e\) in the equivalent section is calculated from the maximum shear stress theory as
$$
\phi_e=45^\circ-\beta_e+\gamma_e,
$$
where \(\beta_e\) is the friction angle. I estimate the friction angle from the empirical relation
$$
\beta_e=C_\beta\gamma_e^{x}a_{kc}^{y}V^{z},
$$
where \(a_{kc}\) is the uncut chip thickness and \(V\) is the cutting speed. This relation reflects the fact that friction at the tool-chip interface in gear cutting depends on rake geometry, chip thickness, and sliding velocity.
I summarize the solved angle parameters for representative tool geometries in the table below. These values are used directly in the energy-based force model.
| Edge inclination \(\lambda_s\) (deg) | Tool rake \(\gamma_0\) (deg) | Equivalent rake \(\gamma_e\) (deg) | Normal rake \(\gamma_n\) (deg) |
|---|---|---|---|
| 5 | 10 | 10.20 | 9.84 |
| 10 | 10 | 14.83 | 14.49 |
| 15 | 10 | 19.16 | 18.84 |
| 5 | 15 | 11.26 | 9.80 |
| 10 | 15 | 15.77 | 14.42 |
| 15 | 15 | 19.99 | 18.75 |
| 5 | 20 | 12.99 | 9.74 |
| 10 | 20 | 17.31 | 14.31 |
| 15 | 20 | 21.36 | 18.57 |
Using the energy method, I assume that shear energy on the shear plane and friction energy on the rake face dominate the total cutting power in gear cutting. I neglect secondary energy losses such as elastic deformation and surface creation. The total cutting power is
$$
P_m=P_s+P_f,
$$
where \(P_s\) is the shear power and \(P_f\) is the friction power. The shear force \(F_s\) is the product of shear stress and shear area:
$$
F_s=\tau_s A.
$$
The shear area for a micro-segment is
$$
A=\frac{a_{kc}a_{kw}}{\sin\phi_e},
$$
where \(a_{kc}\) is the cutting layer thickness and \(a_{kw}\) is the cutting edge width. The cutting edge width is related to the cutting layer width \(b\) by
$$
a_{kw}=\frac{b}{\cos\lambda_s}.
$$
The shear force then becomes
$$
F_s=\tau_s\frac{a_{kc}b}{\sin\phi_e\cos\lambda_s}.
$$
The cutting speed \(V_x\), shear speed \(V_s\), and chip flow speed \(V_{ch}\) form a velocity triangle in the equivalent section. I calculate the chip flow speed as
$$
V_{ch}=V_x\frac{\sin\phi_e}{\cos(\phi_e-\gamma_e)},
$$
and the shear speed as
$$
V_s=V_x\frac{\cos\gamma_e}{\cos(\phi_e-\gamma_e)}.
$$
The shear power is therefore
$$
P_s=F_sV_s
=\frac{\tau_s a_{kc}bV_x\cos\gamma_e}{\sin\phi_e\cos\lambda_s\cos(\phi_e-\gamma_e)}.
$$
For the friction power, I balance the internal force on the chip with the external force on the rake face. The friction force is
$$
F_f=F_s\frac{\sin\beta_e}{\sin\phi_e\cos(\phi_e+\beta_e-\gamma_e)}.
$$
The friction power is
$$
P_f=F_fV_{ch}
=\frac{\tau_s a_{kc}bV_x\sin\beta_e\sin\phi_e}{\sin\phi_e\cos\lambda_s\cos(\phi_e+\beta_e-\gamma_e)\cos(\phi_e-\gamma_e)}.
$$
Combining the two power terms gives the total cutting power for a micro-segment in gear cutting:
$$
P_m=\frac{\tau_s a_{kc}bV_x}{\sin\phi_e\cos\lambda_s\cos(\phi_e-\gamma_e)}
\left[
\cos\gamma_e+
\frac{\sin\beta_e\sin\phi_e}{\cos(\phi_e+\beta_e-\gamma_e)}
\right].
$$
Because the cutting power in the cutting speed direction equals the product of the cutting force component and the cutting speed, the \(x\)-direction force of the micro-segment is
$$
F_x=\frac{P_m}{V_x}.
$$
I then project the friction force and the normal support force onto the \(y\) and \(z\) directions. The \(y\)-direction force is
$$
F_y=
\frac{a_{kc}b\tau_s\left(\gamma_n-\psi\right)\cos\gamma_e\sin\beta_e}{\sin\phi_e\cos\lambda_s\cos(\phi_e+\beta_e-\gamma_e)},
$$
and the \(z\)-direction force is
$$
F_z=
\frac{a_{kc}b\tau_s\left[
-\frac{\psi}{\cos\lambda_s}
-\tan\lambda_s\gamma_e+\gamma_n\sin\beta_e
\right]}{\sin\phi_e\cos\lambda_s\cos(\phi_e+\beta_e-\gamma_e)}.
$$
These three components define the micro-segment cutting force vector. I integrate them along the main cutting edge and the top edge to obtain the force contribution of each tooth.
For the main cutting edge of the outer tooth, the uncut chip width, uncut chip thickness, and maximum uncut chip width are
$$
b_o=\frac{f_t}{\cos\alpha_o},
$$
$$
h_o=f_z\sin\alpha_o,
$$
$$
b_{o,\max}=\frac{H_g-f_z}{\cos\alpha_o},
$$
where \(\alpha_o\) is the outer pressure angle, \(f_z\) is the feed per tooth, \(f_t\) is the feed per revolution, and \(H_g\) is the full tooth height. For the inner tooth, the corresponding pressure angle is \(\alpha_i\). For the top edge, I use
$$
b_a=\frac{W}{2},
$$
$$
h_a=f_z,
$$
where \(W\) is the cutter spacing. The integrated force components are
$$
F_{ix}=\int_0^{b_{i,\max}}F_x\,db,\quad
F_{iy}=\int_0^{b_{i,\max}}F_y\,db,\quad
F_{iz}=\int_0^{b_{i,\max}}F_z\,db,
$$
$$
F_{ox}=\int_0^{b_{o,\max}}F_x\,db,\quad
F_{oy}=\int_0^{b_{o,\max}}F_y\,db,\quad
F_{oz}=\int_0^{b_{o,\max}}F_z\,db,
$$
$$
F_{ax}=\int_0^{b_{a,\max}}F_x\,db,\quad
F_{ay}=\int_0^{b_{a,\max}}F_y\,db,\quad
F_{az}=\int_0^{b_{a,\max}}F_z\,db.
$$
After transforming these components into the machine coordinate system, I write the total force of the outer tooth as
$$
FO=F_o+F_a,
$$
and the total force of the inner tooth as
$$
FI=F_i+F_a.
$$
This formulation allows me to predict the instantaneous force during gear cutting for different tool geometries and cutting parameters.
For the finite element model, I use an explicit dynamic temperature-displacement analysis because gear cutting involves large deformation, high strain rate, contact, chip separation, and heat generation. The material behavior of the workpiece is described by the Johnson-Cook constitutive model:
$$
\sigma=\left(A+B\varepsilon_p^n\right)
\left(1+C\ln\frac{\dot{\varepsilon}_p}{\dot{\varepsilon}_0}\right)
\left[1-\left(\frac{T-T_0}{T_m-T_0}\right)^m\right].
$$
Here \(A\) is the yield strength, \(B\) is the strain hardening coefficient, \(n\) is the strain hardening exponent, \(C\) is the strain-rate sensitivity, \(m\) is the thermal softening exponent, \(T_m\) is the melting temperature, and \(T_0\) is the reference temperature. I use the Johnson-Cook shear failure criterion to describe chip separation:
$$
\varepsilon_f^{pl}=
\left[d_1+d_2\exp\left(d_3\frac{p}{q}\right)\right]
\left[1+d_4\ln\left(\frac{\dot{\varepsilon}^{pl}}{\dot{\varepsilon}_0}\right)\right]
\left[1+d_5\hat{\theta}\right].
$$
For mass scaling, I control the stable time increment using
$$
\Delta T=\frac{L}{c}=\sqrt{\frac{\rho}{E}}L,
$$
where \(L\) is the smallest element size, \(\rho\) is density, and \(E\) is Young’s modulus. I use local mass scaling only for elements whose stable time increment is below a target value. This reduces solution time without significantly changing the physical inertia of the gear cutting process.
I define the contact between the cutter tooth and the workpiece as a surface-to-surface contact with a modified Coulomb friction law. In the sticking region, the friction stress is limited by the shear yield stress; in the sliding region, the friction stress is proportional to the normal pressure:
$$
\tau_f=
\begin{cases}
\mu\sigma_n, & \mu\sigma_n<\tau_s,\\
\tau_s, & \mu\sigma_n\ge\tau_s.
\end{cases}
$$
I use a friction coefficient of 0.2 for the workpiece and tool pair. The tool is modeled as cemented carbide, and the workpiece is modeled as 40Cr steel. The material parameters are listed below.
| Property | Tool (WC) | Workpiece (40Cr) |
|---|---|---|
| Density (kg/m\(^3\)) | 15630 | 7850 |
| Elastic modulus (MPa) | \(6.5\times 10^5\) | \(2.11\times 10^5\) |
| Poisson ratio | 0.25 | 0.30 |
| Thermal expansion (1/K) | \(5\times 10^{-6}\) | \(1.18\times 10^{-5}\) |
| Specific heat (J/(kg K)) | 470 | 450 |
| Thermal conductivity (W/(m K)) | 59 | 32.66 |
I build the cutter disk and workpiece blank in a three-dimensional modeling environment. The cutter tooth is simplified to the cutting portion, while the workpiece blank is trimmed to the local region that participates in gear cutting. I use hexahedral elements with an advancing front algorithm, and I use C3D8T thermally coupled brick elements with linear interpolation. The cutter tooth contains approximately 6000 elements, and the workpiece blank contains approximately 38220 elements. The bottom of the workpiece is fully constrained in all six degrees of freedom, and the rotational and feed motions are applied to the cutter disk reference point.
The main tool design parameters used in my gear cutting simulation are listed below.
| Parameter | Value |
|---|---|
| Nominal cutter radius | 152.4 mm |
| Cutter spacing | 4.188 mm |
| Inner pressure angle | 21.28 deg |
| Outer pressure angle | -18.72 deg |
| Edge inclination angle | 10 deg |
| Rake angle | 15 deg |
The large wheel parameters used for the gear cutting model are listed below.
| Parameter | Value |
|---|---|
| Number of teeth | 43 |
| Face width | 80 mm |
| Outer cone distance | 244.116 mm |
| Pitch angle | 75.65 deg |
| Root angle | 73.42 deg |
| Face angle | 76.57 deg |
| Hand | Right |
I compare the theoretical force model with the finite element simulation for both single-tooth gear cutting and double-tooth gear cutting. The comparison uses the mean force over each cutting pass. I first study the single-tooth case because it isolates the action of one cutting edge and reduces the computational cost. The single-tooth force comparison is shown in the table below.
| Cutting speed (m/min) | Cutting depth (mm) | Rake angle (deg) | Edge inclination (deg) | Theoretical force (N) | Simulated force (N) |
|---|---|---|---|---|---|
| 120 | 0.2 | 15 | 10 | 401.5 | 325.23 |
| 240 | 0.2 | 15 | 10 | 408.34 | 328.67 |
| 480 | 0.2 | 15 | 10 | 405.75 | 296.71 |
| 120 | 0.2 | 15 | 10 | 409.94 | 325.23 |
| 120 | 0.3 | 15 | 10 | 547.70 | 521.99 |
| 120 | 0.5 | 15 | 10 | 988.55 | 665.81 |
| 120 | 0.2 | 10 | 10 | 570.58 | 325.23 |
| 120 | 0.2 | 15 | 10 | 409.94 | 285.99 |
| 120 | 0.2 | 20 | 10 | 326.34 | 376.65 |
| 120 | 0.2 | 15 | 5 | 368.91 | 195.46 |
| 120 | 0.2 | 15 | 10 | 409.94 | 325.23 |
| 120 | 0.2 | 15 | 15 | 439.39 | 399.04 |
I find that the theoretical and simulated forces share the same general trends in gear cutting. The theoretical model is smoother because it is based on idealized assumptions, while the finite element model includes thermal softening, friction variation, and chip morphology. For increasing cutting speed, the theoretical force changes only slightly, but the simulated force first rises and then falls as the cutting temperature reduces the flow stress. For increasing cutting depth, both models show a clear increase in force. For increasing rake angle, the theoretical force decreases, while the simulated force first decreases and then increases because a very large rake angle weakens the cutting edge and changes chip flow. For increasing edge inclination, both models show an increasing trend, although the slopes differ.
I also compare the three force components for single-tooth gear cutting. The table below lists the theoretical and simulated \(x\), \(y\), and \(z\) components.
| Case | Theoretical \(F_x\) (N) | Theoretical \(F_y\) (N) | Theoretical \(F_z\) (N) | Simulated \(F_x\) (N) | Simulated \(F_y\) (N) | Simulated \(F_z\) (N) |
|---|---|---|---|---|---|---|
| 1 | 399.00 | 48.07 | 33.29 | 282.48 | 146.67 | 92.77 |
| 2 | 399.86 | 59.29 | 35.63 | 242.79 | 127.96 | 79.57 |
| 3 | 407.70 | 57.27 | 40.64 | 259.67 | 122.73 | 74.34 |
| 4 | 399.00 | 48.07 | 33.29 | 282.48 | 146.67 | 92.77 |
| 5 | 532.42 | 112.03 | 46.43 | 464.90 | 199.59 | 127.94 |
| 6 | 956.67 | 316.67 | 88.47 | 566.39 | 295.51 | 208.33 |
| 7 | 428.01 | 51.17 | 35.61 | 282.48 | 146.67 | 92.77 |
| 8 | 399.00 | 48.07 | 33.29 | 238.43 | 147.13 | 72.36 |
| 9 | 360.47 | 43.79 | 30.06 | 335.29 | 138.17 | 113.28 |
| 10 | 317.26 | 55.00 | 32.28 | 164.83 | 96.04 | 45.71 |
| 11 | 399.00 | 48.07 | 33.29 | 282.48 | 146.67 | 92.77 |
| 12 | 556.77 | 34.76 | 36.45 | 337.87 | 155.37 | 143.22 |
The \(x\)-component is the main cutting force in gear cutting. The \(y\)-component is the radial force, and the \(z\)-component is the axial force. The theoretical and simulated trends are consistent. The radial force is usually smaller than the main force, and the axial force is often closer to the theoretical prediction. The differences are caused by simplification in the analytical model, especially the neglect of tool wear, thermal softening, and dynamic contact effects.
For double-tooth gear cutting, the inner and outer teeth participate sequentially. The outer tooth cuts the convex side, and the inner tooth cuts the concave side. The force comparison is listed below.
| Cutting speed (m/min) | Cutting depth (mm) | Rake angle (deg) | Edge inclination (deg) | Theoretical force (N) | Simulated force (N) |
|---|---|---|---|---|---|
| 120 | 0.2 | 15 | 10 | 787.87 | 579.79 |
| 240 | 0.2 | 15 | 10 | 801.32 | 643.01 |
| 480 | 0.2 | 15 | 10 | 796.18 | 692.42 |
| 120 | 0.2 | 15 | 10 | 804.60 | 579.79 |
| 120 | 0.3 | 15 | 10 | 1061.03 | 1047.25 |
| 120 | 0.5 | 15 | 10 | 1881.60 | 1679.75 |
| 120 | 0.2 | 10 | 10 | 843.35 | 655.04 |
| 120 | 0.2 | 15 | 10 | 804.60 | 579.79 |
| 120 | 0.2 | 20 | 10 | 710.31 | 602.73 |
| 120 | 0.2 | 15 | 5 | 626.71 | 487.79 |
| 120 | 0.2 | 15 | 10 | 804.60 | 579.79 |
| 120 | 0.2 | 15 | 15 | 1095.33 | 610.13 |
The double-tooth force follows the same physics as the single-tooth case, but the force is distributed between two teeth. The outer tooth enters first, so it carries the initial load and generates the first segment of the chip. The inner tooth then engages, and the two teeth cut simultaneously for part of the cycle. This overlap reduces the load per tooth, improves chip evacuation, and stabilizes the gear cutting process. The three-component comparison for double-tooth gear cutting is given below.
| Case | Theoretical \(F_x\) (N) | Theoretical \(F_y\) (N) | Theoretical \(F_z\) (N) | Simulated \(F_x\) (N) | Simulated \(F_y\) (N) | Simulated \(F_z\) (N) |
|---|---|---|---|---|---|---|
| 1 | 786.02 | 52.82 | 65.38 | 539.15 | 157.71 | 143.55 |
| 2 | 786.06 | 56.36 | 69.98 | 597.99 | 183.05 | 104.30 |
| 3 | 801.59 | 65.82 | 79.82 | 663.91 | 160.06 | 114.28 |
| 4 | 786.02 | 52.82 | 65.38 | 539.15 | 157.71 | 143.55 |
| 5 | 1034.10 | 118.85 | 90.17 | 1013.80 | 217.14 | 147.65 |
| 6 | 1827.95 | 330.13 | 169.47 | 1586.78 | 476.20 | 277.37 |
| 7 | 841.39 | 56.36 | 69.93 | 611.10 | 152.32 | 179.95 |
| 8 | 786.02 | 52.82 | 65.38 | 539.15 | 157.71 | 143.55 |
| 9 | 708.63 | 47.91 | 59.04 | 569.28 | 148.90 | 130.54 |
| 10 | 623.70 | 62.95 | 63.40 | 441.01 | 139.51 | 155.43 |
| 11 | 786.02 | 52.82 | 65.38 | 539.15 | 157.71 | 143.55 |
| 12 | 1194.50 | 36.72 | 71.58 | 559.49 | 180.57 | 163.19 |
The validation results show that my energy-based model captures the main trends of gear cutting force. The model is not a replacement for finite element analysis, but it provides a fast prediction tool that can guide parameter selection before simulation or field trials.
To analyze the combined effect of multiple parameters, I design an orthogonal experiment with three factors and three levels. The factors are cutter speed \(A\), cutting depth \(B\), and rake angle \(C\). The level values are listed below.
| Level | Cutter speed (m/min) | Cutting depth (mm) | Rake angle (deg) |
|---|---|---|---|
| 1 | 120 | 0.2 | 10 |
| 2 | 240 | 0.3 | 15 |
| 3 | 480 | 0.5 | 20 |
The orthogonal array and the measured force and temperature responses are listed below. I record the maximum force, mean force, maximum temperature, and mean temperature for each gear cutting simulation.
| Run | Factor combination | Maximum force (N) | Mean force (N) | Maximum temperature (C) | Mean temperature (C) |
|---|---|---|---|---|---|
| 1 | A1B1C2 | 415.85 | 108.80 | 55.5 | 35.2 |
| 2 | A1B2C3 | 1782.98 | 501.41 | 132.4 | 75.6 |
| 3 | A1B3C1 | 2029.41 | 748.42 | 118.8 | 54.7 |
| 4 | A2B1C3 | 1973.53 | 320.30 | 126.5 | 69.9 |
| 5 | A2B2C1 | 1255.50 | 450.19 | 106.0 | 52.1 |
| 6 | A2B3C2 | 3316.74 | 608.89 | 120.2 | 55.4 |
| 7 | A3B1C1 | 1335.15 | 326.89 | 108.5 | 54.9 |
| 8 | A3B2C2 | 1677.03 | 464.74 | 150.0 | 78.5 |
| 9 | A3B3C3 | 3583.91 | 818.47 | 184.2 | 89.8 |
I use range analysis to rank the influence of each factor. For maximum and mean force, the range values show that cutting depth has the strongest effect, followed by rake angle, and then cutting speed. For maximum temperature, cutting speed has the strongest effect, followed by cutting depth, and then rake angle. For mean temperature, rake angle has the strongest effect, followed by cutting speed, and then cutting depth. The range analysis tables are given below.
| Response | Factor | Level 1 mean | Level 2 mean | Level 3 mean | Range |
|---|---|---|---|---|---|
| Maximum force | Cutting speed | 1409.41 | 2181.92 | 2198.70 | -789.28 |
| Maximum force | Cutting depth | 1241.51 | 1571.84 | 2976.69 | -1735.18 |
| Maximum force | Rake angle | 1540.02 | 1803.21 | 2446.81 | -906.79 |
| Mean force | Cutting speed | 452.64 | 459.79 | 536.70 | -84.06 |
| Mean force | Cutting depth | 251.76 | 472.11 | 725.26 | -473.50 |
| Mean force | Rake angle | 508.50 | 393.90 | 546.73 | -152.82 |
| Response | Factor | Level 1 mean | Level 2 mean | Level 3 mean | Range |
|---|---|---|---|---|---|
| Maximum temperature | Cutting speed | 102.23 | 117.57 | 147.57 | -45.33 |
| Maximum temperature | Cutting depth | 96.83 | 129.47 | 141.07 | -44.23 |
| Maximum temperature | Rake angle | 111.10 | 108.57 | 147.70 | -39.13 |
| Mean temperature | Cutting speed | 55.17 | 59.03 | 74.40 | -19.23 |
| Mean temperature | Cutting depth | 53.23 | 68.73 | 66.63 | -15.50 |
| Mean temperature | Rake angle | 53.90 | 56.37 | 78.33 | -24.43 |
From the orthogonal analysis, I obtain the preferred level combination for both force and temperature as a cutting speed of 120 m/min, a cutting depth of 0.2 mm, a rake angle of 15 deg, and an edge inclination angle of 5 deg. This combination keeps the uncut chip area small, avoids excessive friction, and maintains a stable chip flow in gear cutting.
I next perform single-factor studies for single-tooth gear cutting. The first factor is cutting speed. I keep cutting depth at 0.2 mm, rake angle at 15 deg, and edge inclination at 10 deg. The force results are shown below.
| Cutting speed (m/min) | Cutting depth (mm) | Rake angle (deg) | Edge inclination (deg) | Force (N) |
|---|---|---|---|---|
| 120 | 0.2 | 15 | 10 | 325.23 |
| 240 | 0.2 | 15 | 10 | 328.67 |
| 480 | 0.2 | 15 | 10 | 296.71 |
At low and medium speed, the force increases slightly with cutting speed. The chip has poor flowability, the contact length between the chip and rake face is large, and friction resistance increases. At high speed, the friction power raises the temperature, the chip bottom softens, and the flow stress decreases, so the force drops. This behavior is important in gear cutting because it shows that high speed can reduce force only when thermal softening dominates over strain-rate hardening.
For cutting depth, I keep cutting speed at 120 m/min, rake angle at 15 deg, and edge inclination at 10 deg. The force results are given below.
| Cutting depth (mm) | Cutting speed (m/min) | Rake angle (deg) | Edge inclination (deg) | Force (N) |
|---|---|---|---|---|
| 0.2 | 120 | 15 | 10 | 325.23 |
| 0.3 | 120 | 15 | 10 | 521.99 |
| 0.5 | 120 | 15 | 10 | 665.81 |
The force increases significantly with cutting depth. The contact length between the cutting edge and the workpiece increases, the chip becomes thicker, the removed area grows, and both friction and shear deformation increase. At the largest depth, the force increase is less steep because the chip flow becomes better and the contact length does not grow proportionally. Nevertheless, cutting depth remains the strongest force factor in gear cutting.
For rake angle, I keep cutting speed at 120 m/min, cutting depth at 0.2 mm, and edge inclination at 10 deg. The force results are shown below.
| Rake angle (deg) | Cutting speed (m/min) | Cutting depth (mm) | Edge inclination (deg) | Force (N) |
|---|---|---|---|---|
| 10 | 120 | 0.2 | 10 | 325.23 |
| 15 | 120 | 0.2 | 10 | 285.99 |
| 20 | 120 | 0.2 | 10 | 376.65 |
The force first decreases and then increases as the rake angle rises. At 15 deg, the cutting edge is sharp enough to reduce deformation, but still strong enough to avoid excessive deflection. When the rake angle becomes too large, the cutting edge loses strength, the chip flow direction changes, and the shear zone enlarges. This result is central to tool design for gear cutting because a moderate rake angle gives the best balance between force and edge durability.
For edge inclination angle, I keep cutting speed at 120 m/min, cutting depth at 0.2 mm, and rake angle at 15 deg. The force results are listed below.
| Edge inclination (deg) | Cutting speed (m/min) | Cutting depth (mm) | Rake angle (deg) | Force (N) |
|---|---|---|---|---|
| 5 | 120 | 0.2 | 15 | 195.46 |
| 10 | 120 | 0.2 | 15 | 325.23 |
| 15 | 120 | 0.2 | 15 | 399.04 |
The force increases almost linearly with edge inclination angle. As the edge inclination increases, the contact length between the cutting edge and the workpiece grows, the cutting area expands, and the chip becomes thicker. The friction and deformation resistance both increase. Therefore, a small edge inclination is preferred when the goal is to reduce force in gear cutting, although a small positive inclination can still help chip evacuation and cutting stability.
I then study the temperature behavior in single-tooth gear cutting. For cutting speed, I keep cutting depth at 0.2 mm, rake angle at 15 deg, and edge inclination at 10 deg. The temperature results are shown below.
| Cutting speed (m/min) | Cutting depth (mm) | Rake angle (deg) | Edge inclination (deg) | Temperature (C) |
|---|---|---|---|---|
| 120 | 0.2 | 15 | 10 | 159.9 |
| 240 | 0.2 | 15 | 10 | 173.3 |
| 480 | 0.2 | 15 | 10 | 185.7 |
The temperature rises with cutting speed. The cutting power increases, and the friction heat at the chip-tool interface accumulates because there is less time for heat diffusion. At high speed, the chip flow becomes stronger and carries away more heat, which slows the temperature rise. The high-temperature zone is concentrated near the rake face and the cutting edge, where the contact pressure and sliding velocity are highest.
For cutting depth, the temperature results are given below.
| Cutting depth (mm) | Cutting speed (m/min) | Rake angle (deg) | Edge inclination (deg) | Temperature (C) |
|---|---|---|---|---|
| 0.2 | 120 | 15 | 10 | 159.9 |
| 0.3 | 120 | 15 | 10 | 191.0 |
| 0.5 | 120 | 15 | 10 | 163.1 |
The temperature first increases and then decreases with cutting depth. A larger depth increases the contact area and friction heat, but it also produces a thicker chip that removes more heat. When the heat removal by the chip exceeds the additional heat generation, the measured temperature falls. This non-monotonic behavior is a key feature of gear cutting and shows why depth should not be selected only from a force viewpoint.
For rake angle, the temperature results are listed below.
| Rake angle (deg) | Cutting speed (m/min) | Cutting depth (mm) | Edge inclination (deg) | Temperature (C) |
|---|---|---|---|---|
| 10 | 120 | 0.2 | 10 | 153.7 |
| 15 | 120 | 0.2 | 10 | 159.9 |
| 20 | 120 | 0.2 | 10 | 196.9 |
The temperature increases with rake angle. A larger rake angle changes the chip flow direction and can reduce the contact length in the ideal case, but in the actual gear cutting process the chip may still adhere to the rake face and the tooth slot. Poor heat dissipation then raises the temperature. This result supports the choice of a moderate rake angle for thermal control.
For edge inclination angle, the temperature results are shown below.
| Edge inclination (deg) | Cutting speed (m/min) | Cutting depth (mm) | Rake angle (deg) | Temperature (C) |
|---|---|---|---|---|
| 5 | 120 | 0.2 | 15 | 119.26 |
| 10 | 120 | 0.2 | 15 | 159.90 |
| 15 | 120 | 0.2 | 15 | 181.56 |
The temperature increases with edge inclination. The contact length and contact area increase, friction intensifies, and the chip becomes thicker. The heat generated at the rake face accumulates, and the temperature rises. The temperature increase with edge inclination is more gradual than the force increase, but it is still significant for tool life in gear cutting.
I extend the analysis to double-tooth gear cutting because it is closer to the real process. The outer tooth enters first and cuts the convex flank. The inner tooth enters later and cuts the concave flank. During the overlapping period, both teeth cut at the same time. The force and temperature behavior in double-tooth gear cutting are summarized in the following tables.
| Cutting speed (m/min) | Cutting depth (mm) | Rake angle (deg) | Edge inclination (deg) | Maximum force (N) | Mean force (N) |
|---|---|---|---|---|---|
| 120 | 0.2 | 15 | 10 | 795.73 | 579.79 |
| 240 | 0.2 | 15 | 10 | 1527.08 | 643.01 |
| 480 | 0.2 | 15 | 10 | 1602.97 | 692.42 |
| Cutting depth (mm) | Cutting speed (m/min) | Rake angle (deg) | Edge inclination (deg) | Maximum force (N) | Mean force (N) |
|---|---|---|---|---|---|
| 0.2 | 120 | 15 | 10 | 795.73 | 579.79 |
| 0.3 | 120 | 15 | 10 | 2098.16 | 1047.25 |
| 0.5 | 120 | 15 | 10 | 3397.79 | 1679.75 |
| Rake angle (deg) | Cutting speed (m/min) | Cutting depth (mm) | Edge inclination (deg) | Maximum force (N) | Mean force (N) |
|---|---|---|---|---|---|
| 10 | 120 | 0.2 | 10 | 655.04 | 487.79 |
| 15 | 120 | 0.2 | 10 | 795.73 | 579.79 |
| 20 | 120 | 0.2 | 10 | 860.89 | 602.73 |
| Edge inclination (deg) | Cutting speed (m/min) | Cutting depth (mm) | Rake angle (deg) | Maximum force (N) | Mean force (N) |
|---|---|---|---|---|---|
| 5 | 120 | 0.2 | 15 | 727.11 | 487.97 |
| 10 | 120 | 0.2 | 15 | 795.73 | 579.79 |
| 15 | 120 | 0.2 | 15 | 938.42 | 610.13 |
In double-tooth gear cutting, the maximum force and mean force increase with cutting speed and cutting depth. The effect of cutting depth is especially strong. When the depth becomes large, the force fluctuation also becomes large, which indicates that the dynamic stability of gear cutting becomes worse. The rake angle has an optimum near 15 deg, where the force is moderate and the fluctuation is small. The edge inclination increases the force because it increases the contact length and the cutting area. The inner and outer teeth share the load, so the force per tooth is lower than in the single-tooth case, but the total force is higher when both teeth are engaged.
The temperature behavior in double-tooth gear cutting is summarized below.
| Cutting speed (m/min) | Cutting depth (mm) | Rake angle (deg) | Edge inclination (deg) | Outer maximum temperature (C) | Inner maximum temperature (C) |
|---|---|---|---|---|---|
| 120 | 0.2 | 15 | 10 | 1045 | 930 |
| 240 | 0.2 | 15 | 10 | 1125 | 980 |
| 480 | 0.2 | 15 | 10 | 1026 | 1010 |
| Cutting depth (mm) | Cutting speed (m/min) | Rake angle (deg) | Edge inclination (deg) | Outer maximum temperature (C) | Inner maximum temperature (C) |
|---|---|---|---|---|---|
| 0.2 | 120 | 15 | 10 | 1045 | 930 |
| 0.3 | 120 | 15 | 10 | 1120 | 1005 |
| 0.5 | 120 | 15 | 10 | 1180 | 1060 |
| Rake angle (deg) | Cutting speed (m/min) | Cutting depth (mm) | Edge inclination (deg) | Outer maximum temperature (C) | Inner maximum temperature (C) |
|---|---|---|---|---|---|
| 10 | 120 | 0.2 | 10 | 1033 | 880 |
| 15 | 120 | 0.2 | 10 | 1045 | 930 |
| 20 | 120 | 0.2 | 10 | 985 | 955 |
| Edge inclination (deg) | Cutting speed (m/min) | Cutting depth (mm) | Rake angle (deg) | Outer maximum temperature (C) | Inner maximum temperature (C) |
|---|---|---|---|---|---|
| 5 | 120 | 0.2 | 15 | 418.6 | 390.0 |
| 10 | 120 | 0.2 | 15 | 1045 | 930 |
| 15 | 120 | 0.2 | 15 | 1222 | 1100 |
The outer tooth temperature is generally higher than the inner tooth temperature because the outer tooth enters the gear cutting zone first. It receives the initial thermal load and has a longer accumulation time. The inner tooth enters later, when the chip evacuation path is already partially formed, so its thermal load is lower. The temperature difference between the two teeth decreases as the cutting speed increases because heat diffusion becomes more uniform.
For cutting depth, both outer and inner tooth temperatures increase because the contact area and friction heat grow. The chip becomes thicker and can carry away more heat, but the heat generation still dominates until the depth becomes very large. For rake angle, the outer tooth temperature first decreases and then increases, while the inner tooth temperature changes in the opposite direction in some ranges because the chip flow direction and contact state differ between the convex and concave surfaces. For edge inclination, both tooth temperatures first increase and then decrease because a larger inclination increases contact length but also produces a thicker chip that removes more heat.
I apply the numerical model to field gear cutting on a fully numerical control spiral bevel gear milling machine. The tool disk rotates at a nominal speed, the workpiece is clamped in a fixture, and the cutting zone is cooled by the machine cooling system. I observe tool wear, chipping, tooth surface scratching, and burning. The main observations are summarized below.
| Condition | Observation | Main cause |
|---|---|---|
| Nominal speed 118.8 m/min | Low wear, stable chip flow, acceptable surface | Balanced heat generation and force |
| Lower speed 98.8 m/min | Wear, chipping, scratches, burn marks | High friction, large deformation resistance, vibration |
| High speed | Localized heating and thermal wear | Friction heat concentration and thermal fatigue |
The finite element temperature field shows that the highest temperature in gear cutting appears at the rake face near the cutting edge, where the chip slides against the tool. The high-temperature zone expands as the cut proceeds. The top edge and the main edge both contribute to heat generation, but the rake face remains the critical region for wear. When the cutting speed is too low, the force becomes large and the tool edge experiences impact loading. This can cause chipping and fatigue cracks. When the cutting speed is too high, the heat becomes concentrated, the tool material softens, and oxidation wear or thermal cracking can occur.
I find that cutting speed is the most important factor controlling tool wear in gear cutting. A moderate cutting speed reduces both mechanical wear and thermal wear. It keeps the cutting force within a reasonable range, avoids excessive vibration, and allows the chip to carry away enough heat. The field results show that a speed near 118.8 m/min produces less heat and better surface quality, while a speed near 98.8 m/min causes severe wear and chipping. This supports the optimized parameters obtained from the orthogonal experiment: a cutting speed of 120 m/min, a cutting depth of 0.2 mm, a rake angle of 15 deg, and an edge inclination angle of 5 deg.
In my view, the main contribution of this work is the combination of an energy-based analytical force model with a three-dimensional thermal-mechanical finite element model for gear cutting. The analytical model provides fast predictions of the micro-segment force and the inner and outer tooth force under different tool geometries. The finite element model captures the chip formation, contact friction, heat generation, and temperature distribution in the tooth slot and on the rake face. The orthogonal experiment and single-factor studies reveal the influence ranking and the optimum levels. The application study connects the numerical results to real tool wear and surface defects in gear cutting.
I conclude that cutting depth should be kept small to control force, rake angle should be kept near 15 deg to balance edge sharpness and strength, edge inclination should be kept small to limit contact length, and cutting speed should be selected near the transition between mechanical wear and thermal wear. Under these conditions, gear cutting can achieve lower force, lower temperature, longer tool life, better surface integrity, and higher productivity. The proposed model can be used to guide machine adjustment, cutter disk design, and process parameter selection for spiral bevel gear cutting.
For future work, I plan to include the effect of the temperature field directly in the analytical force model, because thermal softening changes the flow stress during gear cutting. I also intend to study chip morphology, residual stress on the machined surface, and the dynamic response of the machine tool spindle under multi-tooth engagement. Different workpiece materials and cutter coatings can also be introduced to test the generality of the model. These extensions will make the gear cutting model more comprehensive and more useful for industrial application.