In this research, I studied the complete manufacturing chain for a hypoid bevel gear produced by the modified roll method on a four-axis CNC cutting machine. The hypoid bevel gear is widely used in automotive drive axles, mining equipment, marine transmissions, and aerospace systems because it combines smooth motion, high load capacity, and a large transmission ratio in a compact package. Unlike a spiral bevel gear, a hypoid bevel gear has an offset between the pinion axis and the gear axis. This offset lowers the vehicle floor, increases the pinion diameter, improves strength, and reduces noise. However, the same offset makes the geometry of the hypoid bevel gear more complex and makes the machining parameters much more sensitive to small errors.
For many years, mechanical cradle-style cutting machines were the main equipment used to manufacture a hypoid bevel gear. These machines contain a cradle, an eccentric drum, a modified roll mechanism, and a complex gear train. Their adjustment is tedious, their rigidity is limited, and their accuracy depends heavily on the skill of the operator. A modern transmission system requires higher speed, lower noise, longer life, and better consistency. Therefore, the mechanical machine is gradually being replaced by the CNC cutting machine. A four-axis CNC cutting machine is especially attractive because it is simpler, stiffer, more efficient, and less expensive than a full six-axis free-form machine. In many production environments, a four-axis CNC cutting machine can become the core equipment for manufacturing a hypoid bevel gear if a correct mathematical model and a reliable control strategy are available.
The central problem I addressed is that a four-axis CNC cutting machine cannot directly reproduce every motion of a mechanical cradle-style machine. In particular, the modified roll motion of a hypoid bevel gear pinion must be transformed into coordinated motion of the linear axes and the workpiece rotary axis. I therefore built a systematic method that starts from local synthesis, proceeds through tooth surface generation and tooth contact analysis, and ends with actual cutting experiments. The method is intended to make the four-axis CNC cutting machine capable of producing a hypoid bevel gear with good contact patterns and acceptable transmission error.
Background and Motivation
A hypoid bevel gear pair usually consists of a small pinion and a large gear. The pinion is often generated by a modified roll method, while the gear is produced by the formate method. This combination is known as HFM. The formate method for the gear is efficient because the gear tooth surface is a copy of the cutter surface. The modified roll method for the pinion provides the necessary curvature correction, contact localization, and transmission error control. The HFM method avoids the discontinuous root surface that can occur when the pinion is cut by the tilt method. In the tilt method, the concave and convex flanks are often cut with different machine settings, and the root fillet may not be continuous. In the modified roll method, the same machine root angle can be used for both flanks, which improves the root strength and the fatigue life of the hypoid bevel gear.
However, the modified roll method requires a precise relationship between the cradle rotation and the workpiece rotation. In a mechanical machine, this relationship is produced by change gears and a modified roll cam or a similar mechanism. In a four-axis CNC cutting machine, there is no cradle and no mechanical cam. The cradle motion is replaced by the interpolation of two linear axes. Therefore, the modified roll polynomial must be converted into explicit expressions for each CNC axis. I derived these expressions by matching the tool center position, the workpiece orientation, and the instantaneous velocity relation between the theoretical cutting model and the CNC machine model.
Local Synthesis Method
I used the local synthesis method as the theoretical foundation. The local synthesis method allows me to control the contact point, the contact path direction, the contact ellipse size, and the first derivative of the transmission ratio. The main steps are listed below.
| Step | Purpose | Output |
|---|---|---|
| 1 | Determine the gear machining parameters. | Gear cutter position, machine root angle, radial and angular cutter settings. |
| 2 | Compute the principal curvatures and principal directions of the gear tooth surface at the mean point. | Gear surface geometry at the reference point. |
| 3 | Compute the corresponding pinion surface geometry using point contact conditions. | Pinion principal curvatures and directions at the reference point. |
| 4 | Solve the pinion machining parameters using line contact between the pinion and the cutter. | Pinion cutter radius, roll ratio, machine settings, and modified roll coefficients. |
In the local synthesis method, I specify the following three contact parameters at the reference point:
$$ \frac{d}{dt}\left(\frac{\omega_2}{\omega_1}\right) = m’_{21} $$
$$ \eta_2 = \text{angle between the contact path tangent and the first principal direction of the gear} $$
$$ b = \text{semi-major axis of the contact ellipse} $$
For a hypoid bevel gear, the transmission error curve should be a parabolic curve that is convex upward. This requires \(m’_{21}<0\). A smaller absolute value of \(m’_{21}\) usually means lower noise, while a larger absolute value can improve the tolerance to misalignment. The contact ellipse semi-major axis \(b\) is usually selected as 15% to 25% of the tooth width. For a passenger car hypoid bevel gear, the contact pattern may occupy about one half of the tooth width, while for a heavy truck hypoid bevel gear it may occupy about one third of the tooth width.
Gear Machining Parameters
The gear of the hypoid bevel gear pair is cut by the formate method. In the formate method, the cradle does not roll during the cutting of the gear. Therefore, the gear tooth surface is identical to the cutter surface. The pressure angle of the cutter is often chosen as the average pressure angle to balance the life of the inner and outer cutters. This choice causes a small difference between the cutter pressure angle and the theoretical root pressure angle. I corrected this difference by rotating the cutter axis around the pitch line by an angle \(\Delta\alpha_2\). The correction angle is
$$ \Delta\alpha_2 = \alpha – \alpha_{f1} $$
where \(\alpha\) is the average pressure angle and \(\alpha_{f1}\) is the theoretical pressure angle of the gear convex side. The rotation of the cutter axis is equivalent to a rotation of the workpiece in the opposite direction. After this correction, the machine root angle is no longer exactly equal to the gear root angle. I denote the machine root angle by \(\delta_{M2}\). The relationship between the original root angle and the machine root angle is
$$ \tan \beta_M = \frac{\tan \delta_{f2}\sin\Delta\alpha_2}{\cos\beta_{f2}} + \tan\beta_{f2}\cos\Delta\alpha_2 $$
$$ \cos\delta_{M2} = \frac{\cos\delta_{f2}\cos\beta_{f2}}{\cos\beta_M} $$
Here, \(\beta_{f2}\) is the spiral angle at the reference point, and \(\delta_{f2}\) is the gear root angle. The horizontal cutter position \(H\), vertical cutter position \(V\), radial cutter position \(S_2\), and angular cutter position \(q_2\) are then obtained from the geometry of the formate cutting model.
$$ H = R_{02}\cos\beta_{f2}\cos\beta_M – \Delta r\sin\beta_M – \frac{\Delta h}{\tan\delta_{M2}} $$
$$ V = R_{02}\cos\beta_{f2}\sin\beta_M + \Delta r\cos\beta_M $$
$$ S_2 = \sqrt{H^2 + V^2} $$
$$ q_2 = \tan^{-1}\left(\frac{V}{H}\right) $$
The bed position for the gear is zero because the formate method does not require a rolling motion. The axial position can be expressed as
$$ X_2 = Z_f – \frac{\Delta h}{\sin\delta_{M2}} $$
These parameters define the gear tooth surface in the theoretical cutting coordinate system. I used them as the input for the subsequent tooth surface calculation and tooth contact analysis.
Mathematical Models for Cutting and Meshing
I established three mathematical models: the gear cutting model, the pinion cutting model, and the meshing model. The gear cutting model contains a fixed machine coordinate system, a cradle coordinate system, a cutter coordinate system, and a workpiece coordinate system. Because the gear is cut by the formate method, the cradle coordinate system does not rotate during cutting. The cutter coordinate system rotates around its own axis, and the workpiece is fixed. The transformation from the cutter coordinate system to the gear workpiece coordinate system can be written as
$$ \mathbf{r}_2(u_G,\theta_G) = \mathbf{M}_{2m}\mathbf{M}_{mg}\mathbf{M}_{ge}\mathbf{r}_e(u_G,\theta_G) $$
$$ \mathbf{n}_2(u_G,\theta_G) = \mathbf{L}_{2m}\mathbf{L}_{mg}\mathbf{L}_{ge}\mathbf{n}_e(u_G,\theta_G) $$
where \(\mathbf{r}_e\) and \(\mathbf{n}_e\) are the cutter surface position and normal vectors. The cutter surface is a cone. In the cutter coordinate system, it is
$$ \mathbf{r}_e = \begin{bmatrix} (r_c – u_G\sin\alpha_2)\cos\theta_G \\ (r_c – u_G\sin\alpha_2)\sin\theta_G \\ -u_G\cos\alpha_2 \\ 1 \end{bmatrix} $$
$$ \mathbf{n}_e = \begin{bmatrix} \sin\alpha_2 \\ -\cos\alpha_2\sin\theta_G \\ -\cos\alpha_2\cos\theta_G \end{bmatrix} $$
For the pinion cutting model, the pinion is cut by the modified roll method. The cradle rotates, the cutter rotates, and the workpiece rotates. The relationship between the cradle angle \(\phi_q\) and the pinion angle \(\phi_1\) is not linear. I express it as a polynomial. The cutting model includes a fixed machine coordinate system, a cradle coordinate system, a cutter coordinate system, a workpiece coordinate system, and several auxiliary coordinate systems. The transformation from the cutter coordinate system to the pinion workpiece coordinate system is
$$ \mathbf{r}_1(u_q,\theta_q,\phi_q) = \mathbf{M}_{1n}\mathbf{M}_{np}\mathbf{M}_{pf}\mathbf{r}_f(u_q,\theta_q) $$
$$ \mathbf{n}_1(u_q,\theta_q,\phi_q) = \mathbf{L}_{1n}\mathbf{L}_{np}\mathbf{L}_{pf}\mathbf{n}_f(u_q,\theta_q) $$
The meshing model describes the contact between the pinion and the gear. The fixed coordinate system for meshing has an offset \(E\), which is the hypoid offset. The gear rotates around one axis, and the pinion rotates around another axis. The shaft angle is \(\Sigma\). In this research, \(\Sigma=90^\circ\). The meshing equations are
$$ \mathbf{r}^{(1)}_h(u_q,\theta_q,\phi_q,\beta_1) = \mathbf{r}^{(2)}_h(u_G,\theta_G,\beta_2) $$
$$ \mathbf{n}^{(1)}_h(u_q,\theta_q,\phi_q,\beta_1) = \mathbf{n}^{(2)}_h(u_G,\theta_G,\beta_2) $$
These equations provide five independent scalar equations. When the gear rotation angle \(\beta_2\) is given, the remaining five unknowns can be solved. This is the basis of the tooth contact analysis for the hypoid bevel gear.
Gear Tooth Surface and Reference Point
The gear tooth surface is a conical surface generated by the cutter. I selected a reference point on the gear tooth surface. The reference point is usually chosen near the middle of the tooth width and tooth height. Its position is described by two parameters, \(XL\) and \(RL\). These parameters are defined as
$$ XL = R_m – (x\cos\Gamma_2 + y\sin\Gamma_2) – G $$
$$ RL = (x\sin\Gamma_2 – y\cos\Gamma_2) $$
where \(R_m\) is the mean cone distance, \(\Gamma_2\) is the gear pitch angle, and \(G\) is the distance from the pitch cone apex to the crossing point. The cutter surface parameters \(u_G\) and \(\theta_G\) at the reference point are obtained by solving
$$ X_2(u_G,\theta_G) = XL $$
$$ Y_2^2(u_G,\theta_G) + Z_2^2(u_G,\theta_G) = RL^2 $$
This is a nonlinear system. I solved it by iteration. A good initial guess is \(u_G\) equal to half of the whole tooth height and \(\theta_G\) equal to \(270^\circ+\beta\) for a right-hand gear, or \(90^\circ-\beta\) for a left-hand gear. Here, \(\beta\) is the mean spiral angle of the gear.
Principal Curvatures and Directions
Because the gear is cut by the formate method, the principal directions and principal curvatures of the gear tooth surface are the same as those of the cutter surface. In the cutter coordinate system, the first principal direction is the tangential direction along the cutter circumference:
$$ \mathbf{e}_{se} = \begin{bmatrix} -\sin\theta_G \\ \cos\theta_G \\ 0 \end{bmatrix} $$
The second principal direction is along the cutter generatrix:
$$ \mathbf{e}_{qe} = \begin{bmatrix} \sin\alpha_2\cos\theta_G \\ \sin\alpha_2\sin\theta_G \\ -\cos\alpha_2 \end{bmatrix} $$
After transformation to the gear workpiece coordinate system, the principal directions become
$$ \mathbf{e}_{s2} = \mathbf{L}_{2m}\mathbf{L}_{mg}\mathbf{L}_{ge}\mathbf{e}_{se} $$
$$ \mathbf{e}_{q2} = \mathbf{L}_{2m}\mathbf{L}_{mg}\mathbf{L}_{ge}\mathbf{e}_{qe} $$
The principal curvatures of the gear tooth surface are
$$ k_s = \frac{\cos\alpha_2}{r_c – u_G\sin\alpha_2} $$
$$ k_q = 0 $$
For the pinion, I used the point contact condition at the reference point. The gear and pinion are in point contact during meshing. The gear surface geometry at the reference point is known. I specified the transmission ratio derivative, the contact path direction, and the contact ellipse size. Then I solved the pinion principal curvatures and principal directions from the equations of point contact. The basic equations are
$$ k^{(1)}_I + k^{(1)}_{II} = k^{(2)}_I + k^{(2)}_{II} + \Delta k $$
$$ k^{(1)}_I k^{(1)}_{II} = k^{(2)}_I k^{(2)}_{II} + \Delta K $$
where \(\Delta k\) and \(\Delta K\) are determined by the specified contact parameters. This step provides the target geometry for the pinion tooth surface.
Pinion Machining Parameters
The pinion is cut by the modified roll method. The pinion tooth surface and the cutter surface are in line contact. I used the line contact equations to solve the pinion machining parameters. The pinion cutter is a cone. Its surface equation in the cutter coordinate system is
$$ \mathbf{r}_f = \begin{bmatrix} (r_f + u_q\sin\alpha_1)\cos\theta_q \\ (r_f + u_q\sin\alpha_1)\sin\theta_q \\ -u_q\cos\alpha_1 \\ 1 \end{bmatrix} $$
$$ \mathbf{n}_f = \begin{bmatrix} -\cos\alpha_1\cos\theta_q \\ -\cos\alpha_1\sin\theta_q \\ -\sin\alpha_1 \end{bmatrix} $$
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 principal curvatures of the pinion cutter surface are
$$ k_{If} = \frac{\cos\alpha_1}{r_f + u_q\sin\alpha_1} $$
$$ k_{IIf} = 0 $$
The line contact equations between the pinion and the generating gear are
$$ b_{12}^2 = b_{11}b_{22} $$
$$ b_{11}b_{23} = b_{12}b_{13} $$
$$ b_{12}b_{33} = b_{13}b_{23} $$
where the coefficients \(b_{ij}\) depend on the principal curvatures, the angle between the principal directions, the relative velocity, and the roll ratio. By solving these equations, I obtained the pinion cutter radius \(r_f\), the roll ratio \(m_{1f}\), the vertical offset \(E_1\), the horizontal setting \(X_1\), the bed position \(X_{B1}\), the radial cutter position \(S_1\), and the angular cutter position \(q_1\).
$$ r_f = u_p + \frac{\sin\alpha_1}{k_{If}} $$
$$ E_1 = \frac{Y_c + m_{1f}v_{fx}}{m_{1f}} $$
$$ X_1 = \frac{v_{fy} + X_c m_{1f}}{m_{1f}\cos\gamma_1} $$
$$ X_{B1} = Z_F – X_1\sin\gamma_1 $$
$$ S_1 = \sqrt{H_1^2 + V_1^2} $$
$$ q_1 = \tan^{-1}\left(\frac{V_1}{H_1}\right) $$
These parameters define the pinion tooth surface. However, the modified roll polynomial also has to be determined because the roll ratio is not constant during pinion cutting.
Modified Roll Polynomial
The modified roll motion is the key feature of the pinion cutting process. In a mechanical machine, the modified roll motion is produced by a cam or a modified roll mechanism. In a CNC machine, it must be produced by numerical interpolation. I expressed the pinion rotation as a fifth-order Taylor series of the cradle rotation:
$$ \phi_1 = f(\phi_f) = \phi_1(0) + f'(0)\phi_f + \frac{f”(0)}{2!}\phi_f^2 + \frac{f”'(0)}{3!}\phi_f^3 + \frac{f^{(4)}(0)}{4!}\phi_f^4 + \frac{f^{(5)}(0)}{5!}\phi_f^5 $$
At the reference point, the cradle angle is zero. Therefore, \(\phi_1(0)=0\). The first derivative is the nominal roll ratio:
$$ f'(0) = R_{ap} $$
The second derivative is related to the second-order modified roll coefficient \(C\):
$$ C = -\frac{1}{2R_{ap}}f”(0) $$
The third derivative is related to the third-order modified roll coefficient \(D\):
$$ D = -\frac{1}{6R_{ap}}f”'(0) $$
The fourth-order and fifth-order coefficients are \(E\) and \(F\), respectively. In practice, the motion is often written in the form
$$ \phi_1 = R_{ap}\phi_f – C\phi_f^2 – D\phi_f^3 – E\phi_f^4 – F\phi_f^5 $$
The higher-order coefficients are not independent. They can be expressed in terms of \(C\) and \(D\):
$$ D = \frac{3}{2}C^2 X_3 – \frac{1}{6}X_3 $$
$$ E = \frac{1}{24}\left[2C X_3 – 10C D X_3 + 15C^2 X_3^2\right] $$
$$ F = \frac{1}{120}\left[15C D X_3 – 105C^2 X_3^2 + 105C D X_3^2 – 10C X_3^3\right] $$
In the actual CNC program, I set the fourth-order and fifth-order terms to zero when their effect is small. The second-order and third-order coefficients are the main parameters used to adjust the contact pattern and transmission error of the hypoid bevel gear. A positive second-order coefficient changes the tooth surface curvature in the lengthwise direction. A negative second-order coefficient changes the curvature in the opposite direction. The third-order coefficient mainly changes the asymmetry of the transmission error curve and the pressure angle distribution along the tooth height.
Tooth Contact Analysis and V/H Check
I wrote a tooth contact analysis program for the HFM process. The program simulates the meshing of the pinion and gear under different mounting positions. The contact point is found by solving the meshing equations. The transmission error is defined as
$$ \delta\phi_1 = \left(\phi_2 – \phi_2^0\right) – \frac{N_1}{N_2}\left(\phi_1 – \phi_1^0\right) $$
Here, \(N_1\) and \(N_2\) are the numbers of teeth of the pinion and gear, respectively. The superscript 0 denotes the reference position. The transmission error curve should be a convex parabola. The intersection point of the curve with the horizontal axis and the minimum value of the curve provide information about the loaded contact behavior. A lower minimum value means that the tooth pair is less likely to have edge contact under load.
I also implemented a V/H check in the computer. In a traditional roll test, the operator changes the vertical offset \(V\) and the pinion mounting distance \(H\) to observe the movement of the contact pattern. In the computer model, the same effect is obtained by adding a small offset to the gear and pinion surfaces. The equations are
$$ \mathbf{r}_h^{(2)*} = \mathbf{r}_h^{(2)} + \begin{bmatrix} 0 \\ V \\ 0 \end{bmatrix} $$
$$ \mathbf{r}_h^{(1)*} = \mathbf{r}_h^{(1)} + \begin{bmatrix} 0 \\ 0 \\ H \end{bmatrix} $$
The meshing equations are then solved with the condition that the gear and pinion are in contact at the specified point. This gives the values of \(V\) and \(H\) that correspond to the contact position. By repeating this process at the toe, middle, and heel of the tooth, I can evaluate the sensitivity of the hypoid bevel gear to mounting errors.
| Parameter | Symbol | Influence on the hypoid bevel gear |
|---|---|---|
| Second-order modified roll coefficient | \(C\) | Changes lengthwise curvature, contact ellipse width, and parabola depth. |
| Third-order modified roll coefficient | \(D\) | Changes asymmetry of transmission error, pressure angle distribution, and contact path curvature. |
| Contact ellipse semi-major axis | \(b\) | Controls contact area. A larger \(b\) gives a wider contact pattern but higher sensitivity to misalignment. |
| Transmission ratio derivative | \(m’_{21}\) | Controls the depth of the transmission error parabola. A more negative value gives a deeper parabola. |
Case Study of Modified Roll Coefficients
I analyzed a hypoid bevel gear pair with 6 pinion teeth and 37 gear teeth. The blank parameters are listed in the table below. These parameters were used to calculate the gear cutting parameters and the pinion cutting parameters.
| Parameter | Pinion | Gear |
|---|---|---|
| Number of teeth | 6 | 37 |
| Module (mm) | 11.732 | |
| Face width (mm) | 67.547 | 62 |
| Pitch angle (deg) | 11.311 | 78.497 |
| Root angle (deg) | 10.878 | 74.934 |
| Face angle (deg) | 14.816 | 78.937 |
| Offset (mm) | 35 | |
| Spiral angle (deg) | 45 | 34.446 |
| Addendum (mm) | 12.956 | 1.601 |
| Whole depth (mm) | 16.791 | |
The gear cutting parameters are given below. The gear is cut by the formate method, so the radial cutter position, angular cutter position, horizontal setting, and machine root angle are sufficient to define the gear tooth surface.
| Parameter | Value |
|---|---|
| Radial cutter position (mm) | 162.3350 |
| Angular cutter position (deg) | 48.3605 |
| Horizontal setting (mm) | -2.2224 |
| Machine root angle (deg) | 75.3413 |
The pinion cutting parameters for the concave and convex sides are listed in the next table. The pinion is cut by the modified roll method. The second-order and third-order coefficients are used to control the contact pattern.
| 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 setting (mm) | 41.0634 | -11.7830 |
| Bed position (mm) | -6.4456 | -1.5680 |
| Roll ratio | 7.46666 | 5.4720 |
| Second-order coefficient | 0.2635 | -0.1755 |
| Third-order coefficient | -0.13 | 0 |
I studied the influence of the second-order coefficient on the hypoid bevel gear meshing performance. For the pinion concave side, I increased the second-order coefficient from 0.2635 to 0.2835. The contact path remained almost straight, and the transmission error curve remained symmetric. The intersection point moved from about 28 to about 65, and the lower part of the curve moved from about 50 to about 100. This means that a larger second-order coefficient increases the depth of the transmission error curve and reduces the risk of top and root contact. The tooth surface deviation showed a U-shaped distribution along the tooth length. The lengthwise curvature increased, the spiral angle decreased, and the pressure angle changed from the top to the root. In the heightwise direction, the deviation changed from small to large at the toe and from large to small at the heel. This indicates that the surface torsion also changed.
| Case | Second-order coefficient | Contact path | Transmission error shape | Intersection point | Lower value |
|---|---|---|---|---|---|
| Pinion concave side | 0.2635 | Nearly straight | Symmetric | 28 | 50 |
| Pinion concave side | 0.2835 | Nearly straight | Symmetric | 65 | 100 |
| Pinion convex side | -0.1755 | Nearly straight | Symmetric | 30 | 45 |
| Pinion convex side | -0.1555 | Nearly straight | Symmetric | 10 | 15 |
For the pinion convex side, I increased the second-order coefficient from -0.1755 to -0.1555. The contact path remained nearly straight, and the transmission error curve remained symmetric. The intersection point moved from about 30 to about 10, and the lower value moved from about 45 to about 15. This means that a less negative second-order coefficient produces a shallower transmission error curve. The tooth surface deviation again showed a U-shaped distribution along the tooth length. The lengthwise curvature decreased, and the surface torsion changed along the tooth height. These results are useful for selecting the modified roll coefficients of the hypoid bevel gear.
I also studied the influence of the third-order coefficient. For the pinion concave side, I changed the third-order coefficient from -0.13 to -0.08. The contact path remained nearly straight, and the transmission error curve remained symmetric. The contact width did not change much. However, the lower value of the transmission error curve moved from about 50 to about 70. The tooth surface deviation showed a large change along the tooth length and tooth height. The spiral angle decreased, and the pressure angle decreased from top to root. For the pinion convex side, I changed the third-order coefficient from 0 to 0.05. The effect on the concave and convex sides was similar. The tooth surface deviation changed nonlinearly along both the tooth length and tooth height. Therefore, the third-order coefficient is an effective tool for fine-tuning the contact pattern of a hypoid bevel gear without changing the basic geometry.
| Case | Third-order coefficient | Contact path | Transmission error shape | Lower value |
|---|---|---|---|---|
| Pinion concave side | -0.13 | Nearly straight | Symmetric | 50 |
| Pinion concave side | -0.08 | Nearly straight | Symmetric | 70 |
| Pinion convex side | 0 | Nearly straight | Symmetric | Not critical |
| Pinion convex side | 0.05 | Nearly straight | Symmetric | Not critical |
Four-Axis CNC Cutting Machine Kinematics
The four-axis CNC 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 cradle motion of a mechanical machine is replaced by the coordinated motion of \(X\) and \(Y\). The modified roll motion is produced by the \(A\) axis. The \(Z\) axis is used only for feed and retraction. Therefore, the basic task is to express the tool center position and the workpiece rotation angle as functions of the cradle angle.
I first established the relationship between the theoretical cutting coordinate system and the CNC machine coordinate system. The theoretical cutting coordinate system has its origin at the center of the cradle. The CNC machine coordinate system has its origin at the machine zero position. The transformation from the theoretical system to the machine system includes a translation, a rotation by the machine root angle, and a translation that depends on the fixture length and the gear mounting distance. The tool center position in the machine coordinate system at the reference position is
$$ C_0 = M + (H_1 + X_1)\cos\delta_{f1} + (L_1 + L_2 – L)\cos\delta_{f1} $$
$$ D_0 = V_1 – E_1 + N $$
Here, \(M\) and \(N\) are the coordinates of the workpiece rotary table center in the machine coordinate system. \(L\) is the distance between the workpiece rotary table center and the mounting reference plane. \(L_1\) is the fixture length, and \(L_2\) is the gear mounting distance. \(H_1\), \(V_1\), \(E_1\), and \(X_1\) are the theoretical cutter position, vertical cutter position, vertical offset, and horizontal setting, respectively. \(\delta_{f1}\) is the machine root angle of the pinion.
I set the reference position as the origin of a temporary workpiece coordinate system. This simplifies the CNC program because all motion can be described relative to the reference position. The tool center path is a circular arc in the temporary coordinate system. The start position of the tool center is
$$ X_{81} = S_1\cos(q_1 + \theta/2) – S_1\cos(q_1) $$
$$ Y_{81} = S_1\sin(q_1) – S_1\sin(q_1 + \theta/2) $$
The end position of the tool center is
$$ X_{82} = S_1\cos(q_1 – \theta/2) – S_1\cos(q_1) $$
$$ Y_{82} = S_1\sin(q_1) – S_1\sin(q_1 – \theta/2) $$
Here, \(S_1\) is the radial cutter position, \(q_1\) is the angular cutter position, and \(\theta\) is the total generating angle. The workpiece rotation angle at the start and end positions is
$$ A_{11} = q_{10} + i_{01}\theta/2 + C\theta^2/4 + D\theta^3/8 $$
$$ A_{12} = q_{10} – i_{01}\theta/2 – C\theta^2/4 – D\theta^3/8 $$
where \(q_{10}\) is the initial workpiece angle, and \(i_{01}\) is the nominal roll ratio. During cutting, the instantaneous tool center position is
$$ X_8 = S_1\cos(q_1 + q) – S_1\cos(q_1) $$
$$ Y_8 = S_1\sin(q_1) – S_1\sin(q_1 + q) $$
and the instantaneous workpiece rotation angle is
$$ A = q_{10} + i_{01}q + Cq^2 + Dq^3 $$
where \(q\) is the instantaneous cradle angle measured from the reference position. These expressions are the explicit CNC axis commands for the hypoid bevel gear pinion. They allow the four-axis CNC cutting machine to reproduce the modified roll motion with high accuracy.
| CNC axis | Motion type | Function in hypoid bevel gear cutting |
|---|---|---|
| \(X\) | Linear horizontal | Part of the generating motion; replaces the cradle rotation. |
| \(Y\) | Linear vertical | Part of the generating motion; replaces the cradle rotation. |
| \(Z\) | Linear feed | Controls tooth depth and retraction for indexing. |
| \(A\) | Rotary workpiece | Produces the modified roll motion of the pinion. |
CNC Program Generation
I generated the CNC program by converting the theoretical cutting model into the machine model. The steps are summarized below.
| Step | Action | Purpose |
|---|---|---|
| 1 | Transform the theoretical model to the CNC machine coordinate system. | Ensure that the reference point matches the theoretical position. |
| 2 | Set the reference position as the origin of the temporary workpiece coordinate system. | Simplify coordinate calculations. |
| 3 | Move \(X\), \(Y\), and \(A\) to the reference position. | Establish the correct tool-workpiece relationship. |
| 4 | Pause the program and set the \(Z\) axis feed manually. | Set the cutting depth. |
| 5 | Move \(X\), \(Y\), and \(A\) to the start position of the generating motion. | Begin the cutting cycle. |
| 6 | Interpolate \(X\), \(Y\), and \(A\) in small steps. | Generate the modified roll motion. |
| 7 | Retract \(Z\) and index to the next tooth. | Prepare for the next tooth. |
| 8 | Repeat until all teeth are cut. | Complete the hypoid bevel gear pinion. |
The interpolation step size is critical. The modified roll motion is a smooth polynomial, but the CNC machine can only approximate it by discrete linear and rotary steps. If the step is too large, the tooth surface will show faceting and the contact pattern will deteriorate. If the step is too small, the program will be long and the machining time will increase. In my experiments, I used a step size of \(0.25^\circ\) for the generating angle. Later, I reduced the step size to \(0.125^\circ\) for the final pinion. The smaller step produced a smoother tooth surface and a more stable contact pattern.
Cutting Experiment
I performed the cutting experiment on a four-axis CNC cutting machine. The machine uses a Siemens 802D control system. The machine has a rigid structure, a horizontal boring-type \(Y\) axis, and a high-power cutter spindle. The cradle and eccentric drum are eliminated. The \(X\), \(Y\), and \(A\) axes are driven by servo motors. The machine is suitable for both dry and wet cutting of a hypoid bevel gear.
The experimental hypoid bevel gear pair had 6 pinion teeth and 41 gear teeth. The gear was cut by the formate method. The pinion was cut by the modified roll method. The blank parameters are listed below.
| Parameter | Pinion | Gear |
|---|---|---|
| Number of teeth | 6 | 41 |
| Module (mm) | 10.588 | |
| Face width (mm) | 67.2 | 62 |
| Offset (mm) | 35 | |
| Shaft angle (deg) | 90 | |
| Outer cone distance (mm) | 213.192 | 220.678 |
| Addendum (mm) | 13.603 | 1.644 |
| Dedendum (mm) | 3.788 | 15.487 |
| Working depth (mm) | 15.369 | 15.31 |
| Whole depth (mm) | 17.39 | 17.15 |
| Pitch angle (deg) | 10.22 | 79.6 |
| Root angle (deg) | 9.82 | 76.33 |
| Face angle (deg) | 13.43 | 80 |
| Hand | Left | Right |
The initial pinion cutting parameters are listed below. The second-order and third-order coefficients were selected from the local synthesis and TCA results.
| Parameter | Concave side | Convex side |
|---|---|---|
| Cutter radius (mm) | 143.345 | 161.65 |
| Cutter profile angle (deg) | 20.133 | 24.5 |
| Machine root angle (deg) | 9.82 | 9.82 |
| Radial cutter position (mm) | 176.9484 | 148.2521 |
| Angular cutter position (deg) | -62.13335 | -63.0833 |
| Vertical offset (mm) | 49.86 | 19.31 |
| Horizontal setting (mm) | 8.03 | -7.65 |
| Bed position (mm) | -4.72 | -2.04 |
| Roll ratio | 7.703375 | 6.1785625 |
| Second-order coefficient | 0.209965 | -0.176999 |
| Third-order coefficient | 0 | 0.08 |
The gear cutting parameters are listed below. The gear was cut by the formate method with a cutter radius of \(152.4\) mm and a cutter spacing of \(5.33\) mm.
| Parameter | Value |
|---|---|
| Radial cutter position (mm) | 162.2794 |
| Angular cutter position (deg) | 50.8 |
| Horizontal setting (mm) | -0.628 |
| Machine root angle (deg) | 76.3333 |
| Cutter radius (mm) | 152.4 |
| Cutter spacing (mm) | 5.33 |
| Outer cutter profile angle (deg) | 22.5 |
| Inner cutter profile angle (deg) | 22.5 |
After the first cutting, I performed a roll test. The contact pattern on the gear convex side was located near the toe and root. The middle of the contact area was faint because the contact was too wide. The gear concave side had a contact pattern that was too long, located near the root and heel. These defects can be caused by machine rigidity, interpolation error, or small deviations in the cutting parameters. I corrected the contact pattern step by step using proportional correction.
For the gear concave side, I reduced the radial cutter position by \(0.1\) mm to move the contact pattern toward the toe. I increased the roll ratio by \(0.015\) to move the contact pattern toward the top. The new radial cutter position became \(148.1521\) mm, and the new roll ratio became \(6.193562\). After cutting, the contact pattern moved to a better position, but it was still too wide and too long. I then increased the vertical offset by \(6\) mm to narrow the contact pattern. According to the parameter relationships, the radial cutter position increased by \(4.7497\) mm, the horizontal setting increased by \(0.95\) mm, and the roll ratio increased by \(0.184\). I also reduced the cutter radius by \(2.4\) mm to shorten the contact pattern. After the second correction, the contact pattern was located in the middle of the tooth, with no top or root contact, and the noise was low.
For the gear convex side, I reduced the radial cutter position by \(0.42\) mm and reduced the horizontal setting by \(0.6\) mm. After cutting, the contact pattern was located in the middle of the tooth. The contact length was about one third of the tooth width, and the contact height was about one half of the tooth height. There was no diagonal contact. The final contact pattern satisfied the practical requirements for the hypoid bevel gear.
The final pinion cutting parameters are listed below. These parameters were used to produce the hypoid bevel gear pair that passed the roll test.
| Parameter | Concave side | Convex side |
|---|---|---|
| Cutter radius (mm) | 143.345 | 159.25 |
| Cutter profile angle (deg) | 20.133 | 24.5 |
| Machine root angle (deg) | 9.82 | 9.82 |
| Radial cutter position (mm) | 176.53 | 152.9 |
| Angular cutter position (deg) | -62.13335 | -63.0833 |
| Vertical offset (mm) | 49.86 | 25.31 |
| Horizontal setting (mm) | 7.43 | -6.7 |
| Bed position (mm) | -4.72 | -2.04 |
| Roll ratio | 7.703375 | 6.377562 |
| Second-order coefficient | 0.209965 | -0.176999 |
| Third-order coefficient | 0 | 0.08 |
The experimental results confirmed that the modified roll method can be used to manufacture a hypoid bevel gear on a four-axis CNC cutting machine. The tooth surface was smooth, the contact pattern was well located, and the transmission noise was acceptable. The interpolation of the \(X\), \(Y\), and \(A\) axes successfully reproduced the modified roll polynomial. The step size of \(0.125^\circ\) was small enough to avoid visible faceting on the tooth surface. The contact pattern was stable under small changes of the mounting position, which indicates that the transmission error curve was properly designed.
Discussion
The method I developed has several advantages for the hypoid bevel gear industry. First, it extends the capability of a four-axis CNC cutting machine from spiral bevel gears to hypoid bevel gears. This is important because many companies already own four-axis machines but cannot use them for hypoid bevel gear production. Second, the method avoids the discontinuous root surface of the tilt method. Third, the modified roll coefficients provide a direct way to control the contact pattern and transmission error. Fourth, the CNC program is generated from a clear mathematical model, so the machining process is repeatable and not dependent on the operator’s experience.
There are also limitations. The current correction of the contact pattern still uses proportional correction. This method is effective but not fully optimized for a CNC machine. A more advanced correction method should use the sensitivity of the tooth surface to each CNC axis and each modified roll coefficient. In addition, the calculation software was developed in a numerical environment and does not yet have a user-friendly interface. The software also does not fully consider undercutting, tool wear, or heat treatment distortion. These factors can change the final contact pattern of the hypoid bevel gear. Future work should include a closed-loop manufacturing system that measures the tooth surface, compares it with the theoretical surface, and automatically corrects the CNC program.
Conclusions
I studied the manufacturing of a hypoid bevel gear by the modified roll method on a four-axis CNC cutting machine. The main conclusions are as follows.
1. I established a complete mathematical model for the HFM process. The model includes the gear formate cutting model, the pinion modified roll cutting model, and the meshing model. The gear tooth surface is generated by the cutter cone. The pinion tooth surface is generated by the cutter cone and the modified roll motion.
2. I used the local synthesis method to determine the pinion machining parameters. The principal curvatures and directions of the gear tooth surface were computed from the cutter geometry. The pinion surface geometry was obtained from point contact conditions. The pinion machining parameters were obtained from line contact conditions. This procedure provides a systematic way to design a hypoid bevel gear with a controlled contact pattern.
3. I derived the modified roll polynomial as a fifth-order Taylor series. The second-order and third-order coefficients are the most important parameters for controlling the contact pattern and transmission error of the hypoid bevel gear. The fourth-order and fifth-order coefficients have a smaller effect and can often be neglected.
4. I wrote a tooth contact analysis program for the HFM process. The program can simulate the contact pattern and the transmission error curve. It can also perform a V/H check in the computer. The results show that the second-order coefficient mainly changes the lengthwise curvature and the depth of the transmission error curve. The third-order coefficient mainly changes the asymmetry and the pressure angle distribution.
5. I derived the explicit CNC axis expressions for the four-axis CNC cutting machine. The tool center path is a circular arc produced by the \(X\) and \(Y\) axes. The modified roll motion is produced by the \(A\) axis. The \(Z\) axis controls the cutting depth. The reference position is set as the origin of the temporary workpiece coordinate system, which simplifies the CNC program.
6. I performed a cutting experiment on a hypoid bevel gear pair with 6 pinion teeth and 41 gear teeth. The pinion was cut by the modified roll method. After proportional correction of the contact pattern, the final contact pattern was located in the middle of the tooth, with a length of about one third of the tooth width and a height of about one half of the tooth height. The noise was acceptable. The experiment verified the feasibility of manufacturing a hypoid bevel gear on a four-axis CNC cutting machine.
Future Work
For future work, I plan to improve the following aspects. First, I will develop a CNC-specific tooth surface correction method that uses the sensitivity of the tooth surface to the modified roll coefficients and the CNC axis motions. Second, I will integrate the calculation program with a graphical user interface and a database of cutting parameters. Third, I will include undercutting, tool wear, and heat treatment distortion in the model. Fourth, I will perform more cutting experiments to validate the method for different hypoid bevel gear sizes and gear ratios. Fifth, I will study the dynamic behavior of the hypoid bevel gear under load and optimize the transmission error curve for high-speed and heavy-load applications.
The hypoid bevel gear remains one of the most important components in modern vehicles and industrial machines. The four-axis CNC cutting machine provides a practical and affordable platform for its manufacture. With a correct mathematical model, an accurate CNC program, and a reliable correction strategy, a high-quality hypoid bevel gear can be produced efficiently and consistently. I believe that the method presented in this research can contribute to the wider use of CNC technology in the hypoid bevel gear industry.