In the field of heavy-duty vehicle transmissions, cycloid bevel gears have become the preferred choice for axle gears due to their high efficiency in continuous indexing gear cutting and superior meshing performance. However, the high cost of advanced six-axis CNC gear cutting machines and the underutilization of their capabilities when processing large gears in a four-axis mode present significant economic challenges for manufacturers. To address this, developing economical four-axis CNC gear milling machines specifically for large gear gear cutting is crucial. This study focuses on simulating the gear cutting process for cycloid bevel gears using four-axis generating methods to reduce trial-and-error costs and guide machine tool development. Based on VERICUT software, I establish a comprehensive gear cutting simulation model that accurately replicates the actual machining process, including tool geometry, machine kinematics, and numerical control programming. The goal is to validate the correctness of the simulation approach and provide a reliable reference for practical gear cutting and domestic machine tool innovation.
The core of this research lies in creating a virtual environment that mirrors real-world gear cutting operations. I begin by analyzing the structure and motion of a four-axis linkage gear milling machine. This machine typically consists of three linear axes (X, Y, Z) and two rotational axes (C, A). The X and Y axes control the horizontal and vertical movement of the cutter head, simulating the cradle rotation during generation. The Z axis controls the depth of cut. The C axis drives the cutter head rotation, while the A axis controls the workpiece rotation. For large gear generating gear cutting, the workpiece spindle A must synchronize with the cutter head spindle C for continuous indexing and also coordinate with the linear axes X and Y to fulfill the generating motion, resulting in a complex four-axis linkage. The gear cutting process involves two main steps: rough slotting via forming and finish generating. Understanding these motions is essential for accurate simulation.
To simulate this process in VERICUT, I construct a motion-equivalent machine model. Since VERICUT does not have a built-in four-axis gear milling machine, I utilize its component tree functionality to build a simplified model that replicates the kinematic chain of the actual machine. The model includes bases, slides, spindles, and fixtures corresponding to the X, Y, Z, C, and A axes. The workpiece, a gear blank, is modeled in UG software based on geometric parameters and imported into VERICUT. The machine model and blank setup form the foundation for the gear cutting simulation.
A critical component in accurate gear cutting simulation is the cutter head model. The Gleason TRIAC cutter head used for cycloid bevel gear cutting has a complex structure with grouped teeth, each consisting of an inside and an outside blade. The cutting edges are not on the cone generatrix; instead, their projections on the cutter head end plane are tangent to an offset circle from the center. The blade radius is measured at the blade nodal plane. I develop a mathematical model to describe this geometry. Let’s define the coordinate systems: $S_e(X_e, Y_e, Z_e)$ is fixed to the cutter head, rotating with it. $S_{e1}(X_{e1}, Y_{e1}, Z_{e1})$ is attached to the cutting edge of an inside blade, with origin at the inside blade nodal point $IP$. The cutting edge lies in a plane $H$ perpendicular to the nodal plane. For a point $G_I$ on the inside cutting edge, parameterized by $u$ (positive upward) and blade profile angle $\alpha$, the cutting edge vector in $S_{e1}$ is:
$$ \mathbf{r}_{e1} = \begin{bmatrix} u \sin\alpha & 0 & u \cos\alpha & 1 \end{bmatrix}^T $$
Transforming to the cutter head coordinate system $S_e$:
$$ \mathbf{r}_e = \mathbf{M}_{e1e} \mathbf{r}_{e1} $$
where $\mathbf{M}_{e1e}$ is the coordinate transformation matrix. Similar equations apply to outside blades with their respective parameters. Based on this mathematical model, I create a parametric 3D cutter head model in UG software, incorporating key design parameters such as nominal radius, blade profile angles, blade group number, and offset. This model closely matches the physical cutter head, ensuring realistic material removal in simulation.
The next step is motion analysis and NC programming. The relationship between the cutter head, cradle, and workpiece during generating gear cutting is governed by:
$$ \omega_A = \omega_g \left( \frac{z_0}{z_p} + \frac{z_0}{z} \right) + \omega_C $$
where $\omega_A$ is workpiece angular velocity, $\omega_g$ is cradle angular velocity, $\omega_C$ is cutter head angular velocity, $z_0$ is the number of blade groups, $z_p$ is the generating gear number of teeth, and $z$ is the workpiece number of teeth. In practice, the continuous indexing motion (cutter head and workpiece) is achieved using an electronic gearbox function for high-speed synchronization, while the generating motion (workpiece and cradle) is realized through linear interpolation where workpiece angular increments are coordinated with X and Y axis positions. I develop NC program code based on this analysis. The code includes commands for positioning, electronic gearbox activation, spindle rotation, rough cutting feed, and generating interpolation. For example, the generating segment uses G-code to move X, Y, and A axes simultaneously according to calculated increments derived from cradle angle steps. This program drives the virtual gear cutting process in VERICUT.
To verify the simulation results, I compute the theoretical tooth surface using a generating mathematical model. For a left-hand gear, I establish coordinate systems representing the cutter head, generating gear (cradle), and workpiece. The cutter head coordinate system $S_e$ rotates by $\phi_t$. The generating gear coordinate system $S_p$ rotates relative to the machine by $\phi_p = \phi_{p0} + \phi_g$, where $\phi_{p0}$ is from the pure rolling of the cutter head (related to $\phi_t$ by ratio $R_a$) and $\phi_g$ is the cradle angle for generation. The workpiece coordinate system $S_w$ rotates by $\phi_w = R_b \phi_g$, with $R_b$ as the gear cutting ratio. The tooth surface equation in the workpiece coordinate system is:
$$ \mathbf{r}_w(u, \phi_t, \phi_g) = \mathbf{M}_{wn} \mathbf{M}_{nm} \mathbf{M}_{mp}(\phi_g) \mathbf{M}_{pe0} \mathbf{M}_{e0e}(\phi_t) \mathbf{r}_e(u) $$
where $\mathbf{M}$ matrices denote coordinate transformations. The meshing equation must also be satisfied:
$$ f(u, \phi_t, \phi_g) = \left( \frac{\partial \mathbf{r}_w}{\partial u} \times \frac{\partial \mathbf{r}_w}{\partial \phi_t} \right) \cdot \frac{\partial \mathbf{r}_w}{\partial \phi_g} = 0 $$
Since the tooth surface is complex, I use a numerical method to solve it. I discretize the tooth surface into a grid with $m$ rows along the length and $n$ columns along the height. For each grid point $P_{ij}$ with coordinates $(X_{Lij}, R_{Lij})$ in the axial section, the corresponding point $M$ on the spatial tooth surface has coordinates $(x_M, y_M, z_M)$. The relationship is:
$$
\begin{cases}
R_{Lij} = \sqrt{y_M^2 + z_M^2} \\
X_{Lij} = x_M
\end{cases}
$$
Combining this with the meshing equation forms a nonlinear system for parameters $u$, $\phi_t$, $\phi_g$, which I solve using the Newton-Raphson method to obtain the theoretical tooth surface points.
For the gear cutting simulation example, I consider a gear pair with parameters shown in Table 1. The large left-hand gear is the target for simulation. Its gear cutting parameters are listed in Table 2.
| Parameter | Large Gear | Pinion |
|---|---|---|
| Number of Teeth | 28 | 17 |
| Module (mm) | 10.357 | |
| Pressure Angle (°) | 22.5 | |
| Midpoint Spiral Angle (°) | 35 | |
| Face Width (mm) | 50 | 50 |
| Outer Pitch Diameter (mm) | 290 | 176.07 |
| Whole Depth (mm) | 16.28 | 16.28 |
| Addendum (mm) | 7.24 | 7.24 |
| Pitch Cone Angle (°) | 58.73 | 31.27 |
| Parameter | Value |
|---|---|
| Cutter Head Radius (mm) | 125 |
| Number of Blade Groups | 13 |
| Blade Profile Angle (°) | 22.5 |
| Workpiece Installation Angle (°) | 58.74 |
| Radial Cutter Position (mm) | 168.5911 |
| Angular Cutter Position (°) | 46 |
| Horizontal Workpiece Position (mm) | 0 |
| Vertical Workpiece Position (mm) | 0 |
| Machine Center to Back (mm) | 0 |
| Gear Cutting Ratio | 1.169881 |
I configure the VERICUT simulation with the machine model, cutter head model, gear blank, and NC program. The gear cutting simulation process includes rough slotting and finish generating. After simulation, I extract the machined tooth surface data and compare it with the theoretical tooth surface calculated from the numerical model. The deviation is within 5 μm, confirming the accuracy of the simulation. This validates that the VERICUT gear cutting simulation model correctly replicates the four-axis generating gear cutting process.
Furthermore, the simulation allows for interference checking of non-cutting blade edges. In continuous indexing gear cutting, only the cutting edges engage the workpiece; the non-cutting edges must avoid interference with the already cut tooth surfaces. My initial cutter head design, with non-cutting edge profile angles of 22.5° for both inside and outside blades and a blade top width of 3 mm, shows no interference. However, if I increase the non-cutting edge angles to 24° for the outside blade and 23° for the inside blade, the simulation reveals interference at the toe of the tooth slot with an overcut of 0.2 mm. This demonstrates the utility of simulation in optimizing tool design for safe gear cutting.
Following the successful simulation, I conduct an actual gear cutting experiment on a developed four-axis machine using the same parameters. The gear cutting process proceeds smoothly with proper indexing and no interference. The machined gear is measured using a gear inspection system. The tooth surface error measurements show maximum deviations less than 6 μm, meeting precision requirements. This experimental result further corroborates the correctness of the VERICUT gear cutting simulation approach.
In conclusion, this research establishes a comprehensive method for simulating four-axis generating gear cutting of cycloid bevel gears using VERICUT software. I develop a motion-equivalent machine model, a parametric cutter head model based on mathematical analysis, and NC programs derived from kinematic relationships. The simulation accurately produces tooth surfaces that match theoretical calculations, and it serves as an effective tool for validating machine kinematics, NC code, and cutter head design prior to physical gear cutting. The method reduces development costs and risks, providing valuable guidance for the design of economical four-axis CNC gear milling machines and the optimization of gear cutting processes. Future work could extend this simulation framework to include cutting force analysis and thermal effects for more comprehensive gear cutting optimization.