In my research, I study the manufacturing of hypoid bevel gears on a four-axis CNC gear cutting machine by using the modified roll method. Hypoid bevel gears are widely used in automotive drive axles, mining machinery, ship transmissions, and aerospace systems because they combine smooth motion, high load capacity, large transmission ratio, and compact layout. Compared with spiral bevel gears, hypoid bevel gears have an offset between the pinion axis and the gear axis, which lowers the center of gravity and improves strength and stability. However, the offset also makes the tooth surface geometry more complex, and the machining parameters must be designed carefully to obtain a favorable contact pattern and a controlled transmission error.
Traditional mechanical gear cutting machines have been the main equipment for producing hypoid bevel gears. These machines use a cradle, an eccentric drum, change gears, and sometimes a modified roll mechanism. Although they are mature, they require many manual adjustments, have long setup times, and depend heavily on the experience of the operator. The appearance of CNC gear cutting machines has changed this situation. A CNC machine has a simpler structure, higher stiffness, better repeatability, and greater flexibility. In particular, the four-axis CNC gear cutting machine is already common in many factories because it is affordable and reliable. In my work, I focus on how to use such a machine to cut hypoid bevel gears by the modified roll method, rather than limiting it to spiral bevel gears.
The main challenge is that the traditional cutting theory is based on mechanical machines with a cradle. On a four-axis CNC machine, the cradle rotation is replaced by the interpolation of linear axes. Therefore, the modified roll motion must be expressed as a numerical control polynomial, and the relative motion between the cutter and the workpiece must be transformed from the theoretical cutting coordinate system into the machine coordinate system. I developed a complete calculation procedure, a tooth contact analysis program, and a CNC machining program. I also performed a cutting experiment with a 6-tooth pinion and a 41-tooth gear. The contact pattern after lapping and rolling inspection was acceptable, which confirmed the feasibility of manufacturing hypoid bevel gears by the modified roll method on a four-axis CNC machine.
Local Synthesis and HFM Design Logic
I use local synthesis as the theoretical foundation. The gear is machined by the formate method, so its tooth surface is known from the cutter geometry. The pinion is machined by the modified roll method. The design goal is to ensure that the gear and pinion contact at a chosen reference point, that the contact ellipse has a prescribed length, that the contact path has a prescribed direction, and that the transmission error is a concave parabola with a controlled first derivative of the gear ratio. These conditions allow the hypoid bevel gears to have good meshing performance and reduced sensitivity to assembly errors.
The design procedure I follow can be summarized as follows:
- Determine the gear cutting parameters for the formate method.
- Calculate the principal curvatures and principal directions of the gear tooth surface at the reference point.
- Prescribe the contact ellipse length, the contact path direction, and the first derivative of the transmission ratio at the reference point.
- Use the point-contact equations to obtain the principal curvatures and principal directions of the pinion tooth surface.
- Use the line-contact equations between the pinion cutter and the pinion tooth surface to solve the pinion cutting parameters.
- Express the modified roll motion as a polynomial and convert the theoretical motion into CNC axis commands.
The gear cutting parameters for the formate method are not simply equal to the theoretical pressure angles. To keep the inner and outer cutters balanced, I introduce a small rotation of the cutter axis about a chosen axis. This correction changes the machine setting angle and the cutter position. The rotation angle is
$$ \Delta\alpha = \alpha – \alpha_{f1} $$
where \(\alpha\) is the mean pressure angle and \(\alpha_{f1}\) is the theoretical pressure angle of the gear convex side. The gear blank vectors before rotation can be written as
$$ \mathbf{MO}_{f2} = \begin{bmatrix} -R_{02}\cos\beta_{f2} \\ R_{02}\sin\beta_{f2} \\ h_{f2} \end{bmatrix} $$
$$ \mathbf{p}_{2} = \begin{bmatrix} \cos\delta_{f2}\cos\beta_{f2} \\ \cos\delta_{f2}\sin\beta_{f2} \\ \sin\delta_{f2} \end{bmatrix} $$
After the blank rotation, the vectors become
$$ \mathbf{MO}_{f2}’ = \begin{bmatrix} -R_{02}\cos\beta_{f2} \\ R_{02}\sin\beta_{f2}\cos\Delta\alpha – h_{f2}\sin\Delta\alpha \\ R_{02}\sin\beta_{f2}\sin\Delta\alpha + h_{f2}\cos\Delta\alpha \end{bmatrix} $$
$$ \mathbf{p}_{2}’ = \begin{bmatrix} \cos\delta_{f2}\cos\beta_{f2} \\ \cos\delta_{f2}\sin\beta_{f2}\cos\Delta\alpha – \sin\delta_{f2}\sin\Delta\alpha \\ \cos\delta_{f2}\sin\beta_{f2}\sin\Delta\alpha + \sin\delta_{f2}\cos\Delta\alpha \end{bmatrix} $$
The machine setting angle and the spiral angle at the reference point are related by
$$ \tan\beta_M = \frac{\tan\delta_{f2}\sin\Delta\alpha}{\cos\beta_{f2}} + \tan\beta_{f2}\cos\Delta\alpha $$
$$ \cos\delta_{M2} = \frac{\cos\delta_{f2}\cos\beta_{f2}}{\cos\beta_M} $$
The cutter center in the theoretical coordinate system is
$$ \mathbf{O}_c = \begin{bmatrix} 0 \\ r_0 \\ h_0 \end{bmatrix} $$
The vector from the gear root apex to the cutter center is
$$ \mathbf{O}_{f2}\mathbf{O}_c = \begin{bmatrix} -R_{02}\cos\beta_{f2} \\ r_0 – R_{02}\sin\beta_{f2}\cos\Delta\alpha – h_{f2}\sin\Delta\alpha \\ h_0 – R_{02}\sin\beta_{f2}\sin\Delta\alpha – h_{f2}\cos\Delta\alpha \end{bmatrix} $$
The vertical height correction and the horizontal and vertical cutter positions are
$$ \Delta h = h_{f2}(1-\cos\Delta\alpha) – R_{02}\sin\beta_{f2}\sin\Delta\alpha $$
$$ H = R_{02}\cos\beta_{f2} – \frac{\Delta h}{\tan\delta_{M2}} – \Delta r\cos\beta_M $$
$$ V = R_{02}\cos\beta_{f2}\sin\beta_M + \Delta r\sin\beta_M $$
$$ \Delta r = r_0 – R_{02}\sin\beta_{f2}\sin\Delta\alpha + h_{f2}\cos\Delta\alpha $$
Therefore, the radial cutter position and angular cutter position for the gear are
$$ S_2 = \sqrt{H^2 + V^2} $$
$$ q_2 = \tan^{-1}\left(\frac{V}{H}\right) $$
The machine setting angle is \(\delta_{M2}\), and the axial position of the gear blank is
$$ X_2 = Z_f – \frac{h_f}{\sin\delta_{M2}} $$
where \(Z_f\) is the distance from the gear root apex to the design crossing point. With these values, the gear formate cutting parameters are fully determined.
Mathematical Models
I established three main mathematical models: the gear cutting model, the pinion cutting model, and the meshing model. These models are necessary for deriving the tooth surface equations, the principal curvatures, the contact conditions, and the CNC axis motions.
Gear Cutting Model
In the gear cutting model, the gear is fixed during the cutting cycle because the formate method has no generating motion between the cradle and the workpiece. The cutter coordinate system is attached to the cutter head. The gear tooth surface is generated by the conical cutter surface. The cutter surface in its own coordinate system can be written as
$$ \mathbf{r}_e = \begin{bmatrix} -u_G\cos\alpha_2 \\ (r_c – u_G\sin\alpha_2)\sin\theta_G \\ (r_c – u_G\sin\alpha_2)\cos\theta_G \\ 1 \end{bmatrix} $$
where \(u_G\) and \(\theta_G\) are the surface parameters, \(\alpha_2\) is the cutter pressure angle, and \(r_c\) is the cutter tip radius. The unit normal to the cutter surface is
$$ \mathbf{n}_e = \begin{bmatrix} \sin\alpha_2 \\ -\cos\alpha_2\sin\theta_G \\ -\cos\alpha_2\cos\theta_G \end{bmatrix} $$
The gear tooth surface is obtained by transforming the cutter surface into the gear coordinate system:
$$ \mathbf{r}_2 = [M_{2m}][M_{mg}][M_{ge}]\mathbf{r}_e $$
$$ \mathbf{n}_2 = [L_{2m}][L_{mg}][L_{ge}]\mathbf{n}_e $$
The transformation matrices include the radial cutter position, the angular cutter position, the machine setting angle, and the blank offset. Because the formate method has no generating motion, the gear tooth surface is identical in shape to the cutter surface after the coordinate transform.
Pinion Cutting Model
The pinion is cut by the modified roll method. The cradle rotates while the pinion rotates, and the instantaneous ratio of the cradle angle to the pinion angle is not constant. The pinion cutter surface is a conical surface as well, but its coordinate system is different from that of the gear cutter. I write the pinion cutter surface as
$$ \mathbf{r}_f = \begin{bmatrix} (r_f – u_q\sin\alpha_1)\cos\theta_q \\ (r_f – u_q\sin\alpha_1)\sin\theta_q \\ -u_q\cos\alpha_1 \\ 1 \end{bmatrix} $$
The normal to the pinion cutter surface is
$$ \mathbf{n}_f = \begin{bmatrix} -\cos\alpha_1\cos\theta_q \\ -\cos\alpha_1\sin\theta_q \\ -\sin\alpha_1 \end{bmatrix} $$
The principal directions of the pinion cutter surface are
$$ \mathbf{e}_{If} = \begin{bmatrix} -\sin\theta_q \\ \cos\theta_q \\ 0 \end{bmatrix} $$
$$ \mathbf{e}_{IIf} = \begin{bmatrix} -\sin\alpha_1\cos\theta_q \\ -\sin\alpha_1\sin\theta_q \\ \cos\alpha_1 \end{bmatrix} $$
The corresponding principal curvatures are
$$ k_I^{(f)} = -\frac{\cos\alpha_1}{r_f + u_q\sin\alpha_1}, \qquad k_{II}^{(f)} = 0 $$
The relative position of the cutter and the pinion is controlled by the radial cutter position, the angular cutter position, the vertical offset, the horizontal offset, the sliding base, the machine root angle, and the roll ratio. These parameters are solved from the local synthesis conditions.
Meshing Model
The meshing model contains a fixed coordinate system, a pinion coordinate system, and a gear coordinate system. The pinion and gear axes are offset by the hypoid offset. The two tooth surfaces must satisfy the following conditions at the meshing point:
$$ \mathbf{r}_h^{(1)} = \mathbf{r}_h^{(2)} $$
$$ \mathbf{n}_h^{(1)} = \mathbf{n}_h^{(2)} $$
$$ \mathbf{n}_h^{(2)} \cdot \mathbf{v}_h^{(12)} = 0 $$
where the relative velocity is
$$ \mathbf{v}_h^{(12)} = \boldsymbol{\omega}_h^{(1)} \times \mathbf{r}_h^{(1)} – \boldsymbol{\omega}_h^{(2)} \times \mathbf{r}_h^{(2)} $$
These equations are used both for the design of the pinion cutting parameters and for tooth contact analysis of the hypoid bevel gears.
Gear Tooth Surface and Reference Point
I choose a reference point on the gear tooth surface, usually near the middle of the tooth. The reference point is defined by its longitudinal and radial positions:
$$ XL = (R_m + \Delta x)\cos\Gamma_2 – \Delta y\sin\Gamma_2 – G $$
$$ RL = (R_m + \Delta x)\sin\Gamma_2 + \Delta y\cos\Gamma_2 $$
The gear surface parameters at the reference point satisfy
$$ X_2(u_G,\theta_G) = XL $$
$$ Y_2^2(u_G,\theta_G) + Z_2^2(u_G,\theta_G) = RL^2 $$
Because these equations are nonlinear, I solve them iteratively. A good initial guess can be obtained from the middle of the tooth height and the middle spiral angle. Once \(u_G\) and \(\theta_G\) are obtained, the reference point coordinates and the normal vector are known.
The principal directions of the gear tooth surface are the same as those of the cutter surface because the formate method has no generating motion. They are
$$ \mathbf{e}_{se} = \begin{bmatrix} \sin\theta_G \\ \cos\theta_G \\ 0 \end{bmatrix} $$
$$ \mathbf{e}_{qe} = \begin{bmatrix} -\sin\alpha_2\cos\theta_G \\ -\sin\alpha_2\sin\theta_G \\ \cos\alpha_2 \end{bmatrix} $$
The principal curvatures of the gear tooth surface at the reference point are
$$ k_s = \frac{\cos\alpha_2}{r_c – u_G\sin\alpha_2}, \qquad k_q = 0 $$
After transforming these vectors into the meshing coordinate system, I obtain the gear principal directions and curvatures at the reference point. These values are the input for the local synthesis of the pinion.
Principal Curvatures and Directions of the Pinion
At the reference point, the gear and pinion are assumed to be in point contact. The gear surface data are already known. I prescribe three quantities for the pinion surface:
- The first derivative of the transmission ratio, denoted \(m_{21}’\). For a concave transmission error curve, \(m_{21}’\) should be negative.
- The contact path direction at the reference point, denoted by the angle \(\eta_2\).
- The semi-major axis of the contact ellipse, denoted by \(b\). Usually \(b\) is about 0.15 to 0.25 of the tooth width.
The point-contact equations are then solved in the fixed coordinate system. The gear surface and pinion surface must have the same position vector and the same unit normal at the contact point, and the relative velocity must be perpendicular to the common normal. After solving these equations, the principal directions and principal curvatures of the pinion at the reference point are obtained.
The solution procedure can be summarized in the following table.
| Step | Operation | Result |
|---|---|---|
| 1 | Transform the gear surface to the fixed frame | Gear position and normal at the reference point |
| 2 | Apply the meshing equation | Gear rotation angle at the reference point |
| 3 | Use the prescribed contact ellipse and contact path | Pinion principal directions in the fixed frame |
| 4 | Solve the point-contact curvature equations | Pinion principal curvatures |
| 5 | Transform to the pinion cutting frame | Input data for line-contact solution |
Pinion Machining Parameters
The pinion tooth surface and the pinion cutter surface are in line contact during generation. Therefore, the conditions for line contact must be satisfied. I write these conditions in terms of the coefficients \(b_{ij}\):
$$ b_{12}^2 = b_{11}b_{22} $$
$$ b_{11}b_{23} = b_{12}b_{13} $$
$$ b_{12}b_{33} = b_{13}b_{23} $$
The coefficients depend on the principal curvatures of the pinion and the cutter, the angle between their principal directions, the relative velocity, and the derivative of the transmission ratio. The main coefficient expressions are
$$ b_{11} = k_I^{(fM)}\cos^2\sigma – k_I^{(1M)}\cos^2\sigma – k_{II}^{(1M)}\sin^2\sigma $$
$$ b_{12} = b_{21} = 0.5(k_I^{(1M)} – k_{II}^{(1M)})\sin 2\sigma $$
$$ b_{22} = k_I^{(fM)}\sin^2\sigma – k_I^{(1M)}\sin^2\sigma – k_{II}^{(1M)}\cos^2\sigma $$
$$ b_{13} = b_{31} = (k_I^{(fM)} – k_I^{(1M)})\mathbf{w}_I^{(1M)} \cdot \mathbf{n}^{(fM)} \cdot \mathbf{e}_I^{(fM)} $$
$$ b_{23} = b_{32} = (k_{II}^{(fM)} – k_{II}^{(1M)})\mathbf{w}_{II}^{(1M)} \cdot \mathbf{n}^{(fM)} \cdot \mathbf{e}_{II}^{(fM)} $$
The third coefficient \(b_{33}\) contains the derivative of the roll ratio \(m_{1f}’\), the relative velocity, and the curvature terms. I solve the line-contact equations to obtain the pinion cutter radius, the roll ratio at the reference point, the vertical offset, the horizontal offset, the sliding base, the radial cutter position, and the angular cutter position.
The cutter radius is obtained from
$$ r_f = u_p + \frac{\cos\alpha_1}{k_I^{(fM)}}\sin\alpha_1 $$
The vertical offset and horizontal offset are
$$ E_1 = \frac{Y_f}{m_{1f}} + \frac{v_x^{(f)}}{m_{1f}} $$
$$ X_1 = \frac{v_Y^{(f)} – X_c m_{1f}}{m_{1f}\cos\gamma_1} $$
The sliding base is
$$ X_{B1} = (Z_F – X_1)\sin\gamma_1 $$
The radial and angular cutter positions are
$$ S_1 = \sqrt{H_1^2 + V_1^2} $$
$$ q_1 = \tan^{-1}\left(\frac{V_1}{H_1}\right) $$
where
$$ H_1 = X_c + X_1\cos\gamma_1 – r_f(u_q^*\sin\alpha_1 + \theta_q^*) $$
$$ V_1 = Y_c – E_1 – r_f(u_q^*\sin\alpha_1 + \theta_q^*) $$
These parameters define the pinion cutting setup on the mechanical cradle-type machine. I then convert them to the CNC machine by using the modified roll polynomial and the coordinate transformation.
Modified Roll Polynomial
The modified roll motion is the key to cutting hypoid bevel gears on a CNC machine. In a mechanical machine, the modified roll is produced by a cam or a special mechanism. On a CNC machine, the motion is produced by interpolation of the controlled axes. I express the pinion rotation as a fifth-order Taylor polynomial of the cradle rotation:
$$ \phi_1 = R_{ap}\phi_f – C\phi_f^2 – D\phi_f^3 – E\phi_f^4 – F\phi_f^5 $$
where \(R_{ap}\) is the roll ratio at the reference point, and \(C\), \(D\), \(E\), and \(F\) are modified roll coefficients. The first derivative at the reference point is
$$ \left.\frac{d\phi_1}{d\phi_f}\right|_{\phi_f=0} = R_{ap} $$
The second-order coefficient is related to the angular acceleration of the cradle:
$$ C = -\frac{1}{2R_{ap}}f”(0) $$
The third-order coefficient is
$$ D = -\frac{1}{6R_{ap}}f”'(0) $$
In practice, the higher-order coefficients are small, and the second- and third-order coefficients have the greatest influence on the tooth surface and the meshing performance. I therefore concentrate on the effects of \(C\) and \(D\) in my numerical study.
| Coefficient | Physical meaning | Main influence on hypoid bevel gears |
|---|---|---|
| \(R_{ap}\) | Roll ratio at the reference point | Tooth depth, pressure angle, and contact position |
| \(C\) | Second-order modified roll coefficient | Lengthwise curvature, transmission error amplitude, and contact pattern symmetry |
| \(D\) | Third-order modified roll coefficient | Tooth surface twist, pressure angle variation, and contact path curvature |
| \(E\) | Fourth-order coefficient | Small local correction of the tooth surface |
| \(F\) | Fifth-order coefficient | Small local correction of the tooth surface |
Tooth Contact Analysis and V/H Check
I developed a tooth contact analysis program for the gear formate method and the pinion modified roll method. The program simulates the rolling inspection of hypoid bevel gears on a computer. It calculates the contact point path, the contact ellipse, and the transmission error curve. The transmission error is defined as
$$ \delta(\phi_1) = (\phi_2 – \phi_2^0) – \frac{N_1}{N_2}(\phi_1 – \phi_1^0) $$
where \(\phi_1^0\) and \(\phi_2^0\) are the rotation angles at the reference point. A concave parabolic transmission error curve is preferred because it reduces the sensitivity to misalignment and improves the load distribution among the teeth.
I also simulate the V/H check. In a real rolling inspection, the gear and pinion are mounted at their theoretical positions, and then the offset and the pinion mounting distance are changed in small increments. The contact pattern moves along the tooth surface. The V/H check can be represented by the following equations:
$$ \mathbf{r}_h^{*(2)} = \mathbf{r}_h^{(2)} + \begin{bmatrix} 0 \\ 0 \\ V \end{bmatrix} $$
$$ \mathbf{r}_h^{*(1)} = \mathbf{r}_h^{(1)} + \begin{bmatrix} 0 \\ 0 \\ H \end{bmatrix} $$
$$ \mathbf{n}_h^{(2)} \cdot \mathbf{v}_h^{(12)} = 0 $$
The contact conditions are then solved for each V/H combination. The results show how the contact pattern moves when the hypoid bevel gears are misaligned. This is useful for determining the adjustment range in the axle assembly.
| Quantity | Meaning | Typical effect on contact pattern |
|---|---|---|
| \(V\) | Offset change | Moves contact along the tooth length; for right-hand gear and left-hand pinion, increasing V moves the convex side to the toe and the concave side to the heel |
| \(H\) | Pinion mounting distance change | Moves contact along the tooth height; increasing H moves the contact toward the top of the tooth |
| \(C\) | Second-order modified roll coefficient | Changes the lengthwise curvature and the transmission error amplitude |
| \(D\) | Third-order modified roll coefficient | Changes the twist and the pressure angle distribution |
Numerical Study of Modified Roll Coefficients
I performed a numerical study for a hypoid bevel gear pair with 6 pinion teeth and 37 gear teeth. The gear was cut by the formate method, and the pinion was cut by the modified roll method. The basic blank parameters are listed in Table 1. The gear cutting parameters are listed in Table 2. The initial pinion cutting parameters are listed in Table 3.
| Parameter | Pinion | Gear |
|---|---|---|
| Number of teeth | 6 | 37 |
| Module | 11.732 | 11.732 |
| Face width (mm) | 67.547 | 62 |
| Face angle (deg) | 14.816 | 78.937 |
| Pitch angle (deg) | 11.311 | 78.497 |
| Root angle (deg) | 10.878 | 74.934 |
| Offset (mm) | 35 | 35 |
| Spiral angle (deg) | 45 | 34.446 |
| Addendum (mm) | 12.956 | 1.601 |
| Whole depth (mm) | 16.791 | 16.791 |
| Gear cutting parameter | Value |
|---|---|
| Radial cutter position (mm) | 162.3350 |
| Angular cutter position (deg) | 48.3605 |
| Horizontal position (mm) | -2.2224 |
| Machine setting angle (deg) | 75.3413 |
| Pinion parameter | Concave side | Convex side |
|---|---|---|
| Cutter radius (mm) | 139.4384 | 160.9053 |
| Cutter profile angle (deg) | 21 | 25 |
| Machine root angle (deg) | 10.878 | 10.878 |
| Radial cutter position (mm) | 188.6833 | 145.4782 |
| Angular cutter position (deg) | -61.6111 | -59.0783 |
| Vertical offset (mm) | 61.3670 | 19.1900 |
| Horizontal position (mm) | 41.0634 | -11.7830 |
| Sliding base (mm) | -6.4456 | -1.5680 |
| Roll ratio | 7.46666 | 5.4720 |
| Second-order coefficient | 0.2635 | -0.1755 |
| Third-order coefficient | -0.13 | 0 |
For the concave side of the pinion, I increased the second-order coefficient from 0.2635 to 0.2835. The contact path remained nearly straight, and the transmission error curve remained nearly symmetric. However, the amplitude at the lower end of the transmission error curve increased from about 50 to about 100. The tooth surface deviation showed a U-shaped distribution along the tooth length, which means that the lengthwise curvature increased and the spiral angle decreased. Along the tooth height, the pressure angle changed from the toe to the heel, and the surface twist also changed.
| Second-order coefficient \(C\) | Contact path | Transmission error curve | Tooth surface deviation |
|---|---|---|---|
| 0.2635 | Nearly straight | Symmetric, lower end about 50 | Reference surface |
| 0.2835 | Nearly straight | Symmetric, lower end about 100 | U-shaped lengthwise deviation; pressure angle variation along tooth height |
For the convex side of the pinion, I increased the second-order coefficient from -0.1755 to -0.1555. The contact path was still nearly straight, and the transmission error curve was nearly symmetric. The intersection point changed from about 30 to about 10, and the lower amplitude changed from about 45 to about 15. The tooth surface deviation showed a U-shaped distribution along the tooth length, and the lengthwise curvature decreased. Near the top of the tooth, the deviation increased from toe to heel, while near the root, it decreased from toe to heel. This indicates a change in the twist of the tooth surface.
| Second-order coefficient \(C\) | Contact path | Transmission error curve | Tooth surface deviation |
|---|---|---|---|
| -0.1755 | Nearly straight | Symmetric, lower end about 45 | Reference surface |
| -0.1555 | Nearly straight | Symmetric, lower end about 15 | U-shaped lengthwise deviation; opposite height trends near top and root |
I also studied the third-order modified roll coefficient. For the concave side, I changed the third-order coefficient from -0.13 to -0.08. The contact path and the contact width did not change significantly, but the lower end of the transmission error curve increased from about 50 to about 70. The tooth surface deviation showed that the spiral angle decreased from toe to heel, and the pressure angle decreased from top to root. The height curvature also changed noticeably. For the convex side, I changed the third-order coefficient from 0 to 0.05. The influence on the convex and concave sides was similar. The deviation along the tooth length decreased from toe to heel, and the deviation along the tooth height decreased from top to root. These changes were nonlinear, which means that the third-order coefficient can be used to control the twist and the higher-order geometry of hypoid bevel gears.
| Third-order coefficient \(D\) | Side | Contact path | Transmission error | Main deviation trend |
|---|---|---|---|---|
| -0.13 | Concave | Nearly straight | Lower end about 50 | Reference surface |
| -0.08 | Concave | Nearly straight | Lower end about 70 | Spiral angle and pressure angle decrease from one end to the other |
| 0 | Convex | Nearly straight | Reference behavior | Reference surface |
| 0.05 | Convex | Nearly straight | Similar trend to concave side | Nonlinear lengthwise and heightwise deviation |
From this numerical study, I conclude that the second-order coefficient mainly controls the lengthwise curvature and the transmission error amplitude, while the third-order coefficient mainly controls the twist and the pressure angle distribution. By selecting these coefficients carefully, I can obtain a favorable contact pattern and a smooth transmission error curve for hypoid bevel gears.
Four-Axis CNC Machine Kinematics
The four-axis CNC gear cutting machine has four controlled axes: \(X\), \(Y\), \(Z\), and \(A\). The \(X\) axis moves horizontally, the \(Y\) axis moves vertically, the \(Z\) axis controls the cutting depth, and the \(A\) axis rotates the workpiece. The generating motion is produced by the interpolation of \(X\), \(Y\), and \(A\). The \(Z\) axis does not participate in the generating motion. This structure is simpler than a mechanical cradle-type machine because the cradle and the eccentric drum are eliminated.
The first task is to transform the theoretical cutting coordinate system into the machine coordinate system. The cutter center position in the theoretical frame is
$$ C_1 = \frac{H_1 + X_1}{\cos\delta_{f1}} $$
$$ D_1 = V_1 – E_1 $$
After considering the fixture and the workpiece mounting distance, the cutter center in the machine frame becomes
$$ C_0 = M + (H_1 + X_1 – L_1 – L_2 – L)\cos\delta_{f1} $$
$$ D_0 = N + V_1 – E_1 $$
where \(M\) and \(N\) are the machine constants, \(L_1\) is the fixture length, \(L_2\) is the gear mounting distance, and \(L\) is the distance between the mounting reference plane and the workpiece rotation center. I define the position where the cutter and workpiece have the same relative position as the theoretical reference point as the tool setting position. This position is set as the origin of the temporary workpiece coordinate system. This greatly simplifies the CNC program.
The cutting cycle does not start at the tool setting position. It starts at the beginning of the generating motion and ends at the end of the generating motion. For a generating angle \(\Theta\), the cutter center moves along a circular arc. The start and end positions of the cutter center in the temporary coordinate system are
$$ X_s = S_1\cos\left(q_1+\frac{\Theta}{2}\right) – S_1\cos q_1 $$
$$ Y_s = S_1\sin\left(q_1+\frac{\Theta}{2}\right) – S_1\sin q_1 $$
$$ X_e = S_1\cos\left(q_1-\frac{\Theta}{2}\right) – S_1\cos q_1 $$
$$ Y_e = S_1\sin\left(q_1-\frac{\Theta}{2}\right) – S_1\sin q_1 $$
During the generating motion, the instantaneous cutter center position is
$$ X_8 = S_1\cos(q_1+\theta) – S_1\cos q_1 $$
$$ Y_8 = S_1\sin(q_1+\theta) – S_1\sin q_1 $$
The workpiece rotation axis \(A\) follows the modified roll polynomial:
$$ A = A_0 + i_{01}\left[(\theta-\theta_0) – C(\theta-\theta_0)^2 – D(\theta-\theta_0)^3\right] $$
where \(A_0\) is the initial angle, \(i_{01}\) is the roll ratio, and \(\theta\) is the instantaneous cradle angle. The start and end values of \(A\) are
$$ A_s = A_0 + i_{01}\left[-\frac{\Theta}{2} – C\left(\frac{\Theta}{2}\right)^2 – D\left(\frac{\Theta}{2}\right)^3\right] $$
$$ A_e = A_0 + i_{01}\left[\frac{\Theta}{2} – C\left(\frac{\Theta}{2}\right)^2 – D\left(\frac{\Theta}{2}\right)^3\right] $$
The CNC program uses these expressions to calculate the coordinates of the \(X\), \(Y\), and \(A\) axes at each interpolation step. The \(Z\) axis is controlled separately for the cutting depth and the retract motion.
| Axis | Function | Expression |
|---|---|---|
| \(X\) | Horizontal generating motion | \(X_8 = S_1\cos(q_1+\theta) – S_1\cos q_1\) |
| \(Y\) | Vertical generating motion | \(Y_8 = S_1\sin(q_1+\theta) – S_1\sin q_1\) |
| \(A\) | Workpiece rotation with modified roll | \(A = A_0 + i_{01}[(\theta-\theta_0) – C(\theta-\theta_0)^2 – D(\theta-\theta_0)^3]\) |
| \(Z\) | Cutting depth and retract | Set by the depth of cut and the tooth height |
CNC Programming and Cutting Experiment
I wrote a CNC machining program for the four-axis machine. The program first moves the \(X\), \(Y\), and \(A\) axes to the tool setting position. The machine then pauses so that the operator can set the \(Z\) axis position. After the cycle start, the machine moves to the beginning of the generating motion, feeds the cutter to the required depth, and performs the interpolation. The interpolation step is \(0.125^\circ\) in the experiment. After one tooth is cut, the \(Z\) axis retracts, the workpiece indexes to the next tooth, and the cycle repeats until all teeth are cut.
The machining flow can be summarized as follows:
- Convert the theoretical cutting model into the machine model.
- Set the tool setting position as the origin of the temporary workpiece coordinate system.
- Move \(X\), \(Y\), and \(A\) to the tool setting position.
- Pause the program and set the \(Z\) axis position.
- Start the cycle and move to the beginning of the generating motion.
- Feed \(Z\) to the cutting depth.
- Interpolate \(X\), \(Y\), and \(A\) according to the modified roll polynomial.
- Retract \(Z\) after the generating motion is complete.
- Index the workpiece and repeat until all teeth are machined.
I performed the cutting experiment on a four-axis CNC gear cutting machine. The gear pair had 6 pinion teeth and 41 gear teeth. The gear was cut by the formate method, and the pinion was cut by the modified roll method. The initial pinion cutting parameters are listed in Table 4. After the first rolling inspection, the contact pattern on the convex side was too long and located near the root. I reduced the radial cutter position by 0.1 mm to move the contact pattern toward the toe, and I increased the roll ratio by 0.015 to move the contact pattern toward the top. The corrected contact pattern was better, but it was still too wide. I then increased the vertical offset by 6 mm, increased the radial cutter position by 4.7497 mm, increased the horizontal position by 0.95 mm, increased the roll ratio by 0.184, and reduced the cutter radius by 2.4 mm. After the second correction, the contact pattern was located in the middle of the tooth, with no top or root interference and no second-order or third-order contact defects. The rolling noise was acceptable.
| Pinion parameter | Initial concave side | Initial convex side | Final concave side | Final convex side |
|---|---|---|---|---|
| Cutter radius (mm) | 143.345 | 161.650 | 143.345 | 159.250 |
| Cutter profile angle (deg) | 20.133 | 24.500 | 20.133 | 24.500 |
| Machine root angle (deg) | 9.82 | 9.82 | 9.82 | 9.82 |
| Radial cutter position (mm) | 176.9484 | 148.2521 | 176.530 | 152.900 |
| Angular cutter position (deg) | -62.13335 | -63.0833 | -62.13335 | -63.0833 |
| Vertical offset (mm) | 49.86 | 19.31 | 49.86 | 25.31 |
| Horizontal position (mm) | 8.03 | -7.65 | 7.43 | -6.70 |
| Sliding base (mm) | -4.72 | -2.04 | -4.72 | -2.04 |
| Roll ratio | 7.703375 | 6.1785625 | 7.703375 | 6.377562 |
| Second-order coefficient | 0.209965 | -0.176999 | 0.209965 | -0.176999 |
| Third-order coefficient | 0 | 0.08 | 0 | 0.08 |
For the concave side, I reduced the radial cutter position by 0.42 mm and reduced the horizontal position by 0.60 mm. The final contact pattern was located in the middle of the tooth. Its length was about one third of the tooth width, and its height was about one half of the tooth height. There was no diagonal contact, and the rolling noise was normal. The final contact pattern satisfies the practical requirements for hypoid bevel gears.
| Gear cutting parameter | Value |
|---|---|
| Radial cutter position (mm) | 162.2794 |
| Angular cutter position (deg) | 50.8 |
| Horizontal position (mm) | -0.628 |
| Machine setting angle (deg) | 76.3333 |
| Cutter radius (mm) | 152.4 |
| Point width (mm) | 5.33 |
| Outer cutter profile angle (deg) | 22.5 |
| Inner cutter profile angle (deg) | 22.5 |
The experiment showed that the tooth surface finish was good, and the contact pattern was acceptable. The interpolation step of \(0.125^\circ\) was sufficiently small for the modified roll motion. The results confirm that the four-axis CNC gear cutting machine can be used to manufacture hypoid bevel gears by the modified roll method. This is important because it expands the capability of a widely used and affordable CNC machine and reduces the dependence on expensive five-axis or six-axis machines.
Summary and Outlook
In my research, I developed a complete method for manufacturing hypoid bevel gears by the modified roll method on a four-axis CNC gear cutting machine. I used local synthesis to design the gear and pinion tooth surfaces. I established the gear cutting model, the pinion cutting model, and the meshing model. I derived the gear principal curvatures and directions, solved the pinion principal curvatures and directions by point contact, and obtained the pinion cutting parameters by line contact. I expressed the modified roll motion as a fifth-order polynomial and studied the influence of the second-order and third-order coefficients on the contact pattern and the transmission error. I then transformed the theoretical cutting motion into the CNC machine coordinate system and derived the explicit expressions for the \(X\), \(Y\), \(A\), and \(Z\) axes. Finally, I performed a cutting experiment with a 6-tooth pinion and a 41-tooth gear. The rolling inspection showed a good contact pattern and normal noise, which verified the feasibility of the proposed method.
There are still several topics that I would like to study further. First, the tooth surface correction in my experiment was based on proportional adjustment, which is effective but not fully optimized for CNC machines. I would like to develop a correction method that directly uses the modified roll coefficients and the CNC axis motions to compensate for tooth surface deviations. Second, the calculation software was implemented in a scripting environment, and it does not yet have a user-friendly interface. I would like to integrate the calculation, tooth contact analysis, and CNC code generation into a single engineering tool. Third, I only performed one cutting experiment. More experiments with different gear sizes, offsets, and modified roll coefficients are needed to build a reliable database for industrial application. Fourth, the influence of machine errors, heat treatment distortion, and assembly errors on the final contact pattern of hypoid bevel gears should be included in future models. By addressing these issues, I believe that the modified roll method on four-axis CNC machines can become a practical and powerful solution for the production of high-quality hypoid bevel gears.