Spiral bevel gears transmit motion and power between intersecting shafts. Their high contact ratio, smooth running, and low noise make them common in automotive axle drives. Five-axis CNC gear milling machines are widely used because they offer strong flank correction capability and flexible adjustment of the tooth surface. That flexibility comes at a cost: the machine structure and control system are complex, and the machine price is high.
For non-offset spiral bevel gears with intersecting axes, a locally synthesized design with pre-controlled meshing performance can be manufactured by a generating method without cutter tilt and still achieve good meshing behavior. In that case, the fifth axis associated with root-cone-angle swing can be removed, reducing the machine to four controlled axes: X, Y, Z, and B. The mechanical generator settings are often expressed in cutting modes such as SGM, SFM, and SFT, and the conversion from a given mode to NC axis motion is mode-dependent. The four-axis formulation therefore has to be derived from the actual machine structure and the selected generating model, not from a universal axis transformation.
Machine Configuration and Generating Cycle
The four-axis arrangement uses two linear motions of the cutter head in X and Y, one linear motion of the workpiece box in Z, and rotation of the workpiece in B. The cutter head is driven independently by a variable-frequency motor through a single-stage gear transmission. It does not participate in the coordinated interpolation; its speed is set directly. The workpiece is driven by a servo motor through a worm gear reduction. The B axis performs both the generating rotation and the indexing rotation between tooth slots.
The generating cycle is organized as follows:
- Clamp the workpiece and enable coolant and chip removal.
- Return the machine to its reference position.
- Move Z to a safe position.
- Move X and Y to the centering reference point.
- Move X and Y to the generating start position while rotating B to its start angle.
- Feed Z to the tooth depth.
- Interpolate X, Y, and B through the generating sweep.
- Move X and Y to the generating end position and retract Z rapidly.
- Index the workpiece, return X and Y to the start position, and repeat the generating cycle until all tooth slots are cut.
In this architecture, the Z axis mainly establishes tooth depth. During the generating segment of the example described below, Z remains at a fixed position while X, Y, and B produce the relative motion between the cutter and the workpiece. The B axis therefore carries two functions: it is part of the generating motion and it also performs tooth indexing. This reduces hardware but demands accurate synchronization between B and the X/Y trajectory.
Kinematic Model and Motion Equivalence
The machine model uses a fixed frame \(S_m(X_m,Y_m,Z_m)\) connected to the machine, with \(X_mOY_m\) lying in the cutter tip plane, a cutter frame \(S_t\) rotating with the cutter head, and a workpiece frame \(S_w\) rotating with the B axis. The controlled cutter-center coordinates are \(C_X\), \(C_Y\), and \(C_Z\); the workpiece B-axis angle is \(C_B\); the blank mounting angle is \(C_\gamma\); and the axial wheel position correction is \(\Delta L\). The machine center \(O_0\) is defined as the cutter-center position when the blank mounting angle is \(90^\circ\) and the workpiece spindle and cutter spindle are coaxial. \(O_2\) is the design crossing point, and \(L\) is the distance from the workpiece-box rotation center to \(O_2\).
For the NC machine, the workpiece tooth surface can be written through the total coordinate transformation as
$$
\mathbf r_w=\mathbf M_C\,\mathbf r_b,\qquad
\mathbf M_C=\mathbf M_{wa}\mathbf M_{an}\mathbf M_{nm}\mathbf M_{mt}\mathbf M_{tb}
$$
where \(\mathbf M_C\) contains the machine geometry, the cutter phase, the B rotation, and the axial correction. For the mechanical generating model, the corresponding transformation is
$$
\mathbf r_w=\mathbf M_G\,\mathbf r_b,\qquad
\mathbf M_G=\mathbf M_{wa}\mathbf M_{an}\mathbf M_{nm}\mathbf M_{mp}\mathbf M_{pt}\mathbf M_{tb}
$$
Here the generating model is defined by the radial cutter position \(S_r\), angular cutter position \(q\), blank mounting angle \(\delta_M\), vertical wheel position \(E_m\), bed correction \(X_b\), horizontal wheel position correction \(X_g\), cradle angle \(\phi_g\), and workpiece angle \(\phi_w\). The generating relation is \(\phi_w=R_b\phi_g\), where \(R_b\) is the roll ratio. The angular cutter position is positive for a right-hand wheel and negative for a left-hand wheel; the vertical wheel position uses a hand-dependent sign convention.
The NC machine reproduces the intended tooth surface when the two total transformations are identical at every instant of the generating sweep:
$$
\mathbf M_C=\mathbf M_G
\quad\text{with}\quad
C_\gamma=\delta_M,\; C_A=\phi_c,\; \Delta L=X_g
$$
Solving the remaining terms gives the four controlled axis positions as functions of the cradle angle:
$$
\begin{aligned}
C_X &= X_g\cos\delta_M+g_{14}\cos\delta_M-g_{34}\sin\delta_M,\\
C_Y &= g_{24},\\
C_Z &= -X_g\sin\delta_M-g_{14}\sin\delta_M-g_{34}\cos\delta_M,\\
C_B &= \phi_w=R_b\phi_g.
\end{aligned}
$$
In these expressions, \(g_{14}\), \(g_{24}\), and \(g_{34}\) are the translation entries of the generating transformation \(\mathbf M_G\). The solution is evaluated at each cradle angle, so it produces a continuous X/Y/Z/B trajectory rather than a single fixed machine setup. The B coordinate follows the roll ratio, while X and Y compensate for the changing cutter-center position required by the generating motion.
Example Geometry and Axis Coordinates
The validation case is a right-hand large wheel of a 28-tooth/17-tooth spiral bevel gear pair. The main blank parameters are listed in Table 1. The large wheel has a large-end pitch diameter of 290 mm, a face width of 50 mm, and a whole tooth depth of 17.9487 mm. Its pitch cone angle is 58.7363°, its face cone angle is 62.0664°, and its root cone angle is 55.4062°.
| Parameter | Large wheel | Small wheel |
|---|---|---|
| Number of teeth | 28 | 17 |
| Module (mm) | 10.3572 | 10.3572 |
| Pressure angle (°) | 20 | 20 |
| Midpoint spiral angle (°) | 35 | 35 |
| Face width (mm) | 50 | 50 |
| Large-end pitch diameter (mm) | 290 | 176.07 |
| Whole tooth depth (mm) | 17.9487 | 17.9487 |
| Pitch cone angle (°) | 58.7363 | 31.2637 |
| Face cone angle (°) | 62.0664 | 34.5938 |
| Root cone angle (°) | 55.4062 | 27.9336 |
The cutting settings for the large wheel are given in Table 2. The concave and convex flanks use different cutter radii and profile angles, while the remaining generating settings are common. The blank mounting angle is 55.4062°, the radial cutter position is 131.7678 mm, the angular cutter position is 61.1561°, and the roll ratio is 1.167906. The horizontal wheel position, vertical wheel position, and bed correction are all zero in this example.
| Parameter | Concave flank | Convex flank |
|---|---|---|
| Cutter radius (mm) | 143.902 | 137.902 |
| Cutter tooth profile angle (°) | -18 | 22 |
| Blank mounting angle (°) | 55.4062 | 55.4062 |
| Radial cutter position (mm) | 131.7678 | 131.7678 |
| Angular cutter position (°) | 61.1561 | 61.1561 |
| Horizontal wheel position (mm) | 0 | 0 |
| Vertical wheel position (mm) | 0 | 0 |
| Bed position (mm) | 0 | 0 |
| Roll ratio | 1.167906 | 1.167906 |
For a cradle sweep from \(\phi_g=-30^\circ\) to \(30.9^\circ\), the computed trajectory gives \(C_Z=-71.325\) mm throughout the generating segment. The X coordinate varies from 111.7235 mm to \(-15.2586\) mm, the Y coordinate from 79.6475 mm to 136.4720 mm, and the B coordinate from 35.0262° to \(-36.0769°\) under the example’s axis-zero and sign convention. The large changes in X and Y show that the cutter center cannot be treated as a fixed offset; its path is part of the generating motion.
Simulation and Cutting Validation
The motion solution was checked in VERICUT by building an equivalent four-axis machine model, a cutter model from the cutter parameters, and a workpiece blank from the gear blank data. The computed X/Y/Z/B coordinates were imported into the CNC system of the virtual machine, and the generating process was simulated. The machined flank was compared with a theoretical solid model generated from the design data. In that simulation, the maximum deviation was less than 0.02 mm, indicating that the NC transformation reproduced the intended generating motion for the tested case.
A physical cutting trial was then performed on the large wheel using the same parameter set. The measured flank deviated from the theoretical surface, but the deviation remained within the tolerance adopted for the trial. This is expected because the real process includes cutter geometry errors, fixture error, machine geometric error, and alignment error. The physical trial therefore supports the practical use of the motion solution, but it does not isolate the kinematic transformation error from other error sources.
The simulated maximum deviation of less than 0.02 mm should be read as a case-specific result, not as a universal machine accuracy specification. It demonstrates consistency between the four-axis motion model and the theoretical generating model for the selected gear geometry and cutter settings. A production qualification would still require a larger sample of teeth, flanks, and machine conditions, together with a defined measurement uncertainty.
Implications and Limits
The four-axis generating method is technically meaningful for non-offset spiral bevel gears whose design does not require cutter tilt. By removing the root-cone-angle swing axis, the machine structure becomes simpler and less expensive, while the X/Y/B interpolation still reproduces the relative generating motion between the cutter and the workpiece. The Z axis establishes depth and can remain fixed during the generating sweep. The method is therefore an economic option for this class of spiral bevel gears.
The limitation is equally important. A four-axis machine does not provide the same flank-correction freedom as a five-axis machine. It is not a general replacement for five-axis generating when the design requires cutter tilt, hypoid offset, or aggressive local flank modification. The axis-position equations also depend on the selected cutting mode and sign conventions, so the conversion must be established for the specific machine and generating model. Under those constraints, the motion-equivalence approach provides a clear path from mechanical generator settings to NC trajectories, and the simulation and cutting trial support its use for the tested non-offset spiral bevel gear case.