In modern heavy machinery industries, such as power generation, shipbuilding, and mining, large straight bevel gears play a critical role as core transmission components. The increasing demand for these gears has highlighted challenges related to their large dimensions, high dependency on specialized equipment, and manufacturing difficulties. To address issues like transportation and machining constraints, the split straight bevel gear design has emerged as a practical solution. This paper explores the numerical control (NC) machining motion solution for split straight bevel gears using a universal five-axis CNC machine. By leveraging a cylindrical end mill for flank milling, we derive explicit functions for linear and rotary motion trajectories, validate the approach through simulation and practical experiments, and demonstrate the feasibility of machining large straight bevel gears on standard-sized equipment.
The manufacturing of straight bevel gears typically involves specialized tools like form cutters or ball-end mills, which can be inefficient or require large-scale machinery. In contrast, cylindrical end mills offer advantages in cost, versatility, and efficiency. For split straight bevel gears, which are segmented to facilitate handling and machining, NC machining requires precise motion planning to achieve accurate tooth profiles. This study focuses on solving the cutter location data and transforming it into machine tool trajectories through post-processing. The key aspects include coordinate system transformations, derivation of motion equations, and verification via simulation and physical machining.
To begin, we establish a series of coordinate systems to model the machining process for the split straight bevel gear. The workpiece coordinate system \( S_o (O_o – X_o Y_o Z_o) \) represents the gear model derived from the tooth surface equations. Transition coordinate systems \( S_i (O_i – X_i Y_i Z_i) \) and \( S_j (O_j – X_j Y_j Z_j) \) are introduced to align the gear model with the machine axes. The installation coordinate system \( S_w (O_w – X_w Y_w Z_w) \) accounts for the offset between the split gear’s installation center and the theoretical rotation center. Finally, the machine coordinate system \( S_m (O_m – X_m Y_m Z_m) \) defines the linear and rotary motions (X, Y, Z, A, B) of the CNC machine. The transformation matrices between these systems are as follows:
The transformation from \( S_o \) to \( S_i \) is given by:
$$ M_{io} = \begin{bmatrix}
\cos a & -\sin a & 0 & 0 \\
\sin a & \cos a & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix} $$
where \( a \) is the rotation angle about the Z-axis. The transformation from \( S_i \) to \( S_j \) is:
$$ M_{ji} = \begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & \cos c & -\sin c & 0 \\
0 & \sin c & \cos c & 0 \\
0 & 0 & 0 & 1
\end{bmatrix} $$
where \( c \) is the rotation angle about the X-axis. The transformation from \( S_j \) to \( S_w \) includes translations:
$$ M_{wj} = \begin{bmatrix}
1 & 0 & 0 & a_0 \\
0 & 1 & 0 & b_0 \\
0 & 0 & 1 & c_0 \\
0 & 0 & 0 & 1
\end{bmatrix} $$
and from \( S_w \) to \( S_m \):
$$ M_{mw} = \begin{bmatrix}
1 & 0 & 0 & a_1 \\
0 & 1 & 0 & b_1 \\
0 & 0 & 1 & c_1 \\
0 & 0 & 0 & 1
\end{bmatrix} $$
These transformations are crucial for converting cutter location data from the workpiece coordinate system to the machine coordinate system. For a cutter center point \( [X’_o, Y’_o, Z’_o] \) in \( S_o \), the corresponding point in \( S_m \) is computed as:
$$ [X’_m, Y’_m, Z’_m] = M_{mw} M_{wj} M_{ji} M_{io} [X’_o, Y’_o, Z’_o] $$
Simplifying this yields the linear motion equations:
$$ X’_m = X’_o \cos a – Z’_o \sin a + a_0 + a_1 $$
$$ Y’_m = X’_o \sin a \cos c + Y’_o \cos a \cos c – Z’_o \cos c + b_0 + b_1 $$
$$ Z’_m = X’_o \sin a \sin c + Y’_o \cos a \sin c + Z’_o \cos c + c_0 + c_1 $$
Similarly, the cutter axis vector \( \mathbf{p} = [p_X, p_Y, p_Z] \) in \( S_o \) is transformed to \( \mathbf{p}_m = [p_{mX}, p_{mY}, p_{mZ}] \) in \( S_m \) by applying rotational transformations:
$$ \mathbf{p}_m = M_{mw} M_{wj} \mathbf{p} $$
which expands to:
$$ p_{mX} = p_X \cos a – p_Z \sin a $$
$$ p_{mY} = p_X \sin a \cos c + p_Y \cos a \cos c – p_Z \cos c $$
$$ p_{mZ} = p_X \sin a \sin c + p_Y \cos a \sin c + p_Z \cos c $$
For rotary motion, the XYZAB-type five-axis machine is considered, where the tool orientation is defined by swing angles A and B. The swing angle A is calculated based on the Y-component of the cutter axis vector:
$$ A = \begin{cases}
360^\circ – \arctan \left( \frac{p_Y}{\sqrt{p_X^2 + p_Z^2}} \right) & \text{if } p_Y > 0 \\
0 & \text{if } p_Y = 0 \\
-\arctan \left( \frac{p_Y}{\sqrt{p_X^2 + p_Z^2}} \right) & \text{if } p_Y < 0
\end{cases} $$
The swing angle B depends on the signs of \( p_X \) and \( p_Z \):
$$ B = \begin{cases}
\arctan \left( \frac{p_X}{p_Z} \right) & \text{if } p_X \geq 0, p_Z \geq 0 \\
180^\circ – \arctan \left( \frac{p_X}{p_Z} \right) & \text{if } p_X \geq 0, p_Z \leq 0 \\
180^\circ + \arctan \left( \frac{p_X}{p_Z} \right) & \text{if } p_X \leq 0, p_Z \leq 0 \\
360^\circ – \arctan \left( \frac{p_X}{p_Z} \right) & \text{if } p_X \leq 0, p_Z \geq 0
\end{cases} $$
with \( \arctan(p_X / p_Z) = 90^\circ \) when \( p_Z = 0 \). These equations enable the generation of NC code for machining the split straight bevel gear.
To validate the motion solution, a case study was conducted on a split straight bevel gear with the parameters listed in Table 1. The gear undergoes rough machining using a form cutter for slotting, followed by finish machining with a cylindrical end mill for flank milling. The machining process is illustrated below, showing the setup for the split straight bevel gear.
Table 1: Basic parameters of the straight bevel gear used in the study.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| Module \( m_n \) (mm) | 20 | Face width \( b \) (mm) | 100 |
| Number of teeth \( z \) | 72 | Profile shift coefficient \( x_n \) | 0 |
| Pressure angle \( \alpha_n \) (°) | 20 | Pitch diameter at large end \( D \) (mm) | 1440 |
Simulation was performed in VERICUT software to verify the tool paths. A machine model matching the XYZAB configuration was built, and the NC program derived from the motion equations was loaded. The simulation results, as shown in Table 2, indicate that the tooth profile accuracy meets the required standards, with minor undercuts and overcuts within acceptable tolerances of 0.05 mm. The automatic comparison function in VERICUT confirmed that the machined surface closely matches the theoretical model, demonstrating the correctness of the motion trajectories.
Table 2: Simulation results for tooth profile deviations.
| Deviation Type | Value (μm) | Tolerance (μm) |
|---|---|---|
| Tooth-to-tooth composite deviation \( f_{pt} \) | 25.3 | 30 |
| Total composite deviation \( F_p \) | 21.8 | 25 |
| Radial runout \( F_r \) | 38.3 | 40 |
| Profile total deviation \( F_f \) | 5.6 | 10 |
| Profile form deviation \( f_f \) | 2.6 | 5 |
Practical machining was carried out on a DMU100 five-axis CNC machine. The split straight bevel gear was first rough-machined with a form cutter to create tooth slots, and then finish-machined using a 10 mm diameter cylindrical end mill for flank milling. The actual machining process confirmed the feasibility of the approach, with the gear being successfully produced. Post-machining measurement using a coordinate measuring machine revealed that the gear parameters comply with GB/T 10095.1-2022 standards, achieving grade 7 accuracy. This further validates the motion solution for the split straight bevel gear.
In conclusion, the numerical control machining motion solution for split straight bevel gears using a cylindrical end mill has been successfully developed and verified. The coordinate transformation matrices and motion equations provide a robust framework for generating accurate NC code. Simulation and practical experiments demonstrate that this method enables the machining of large straight bevel gears on standard five-axis CNC machines, reducing costs and improving accessibility. Future work could focus on optimizing tool paths for higher efficiency and extending the approach to other gear types. The repeated emphasis on straight bevel gear throughout this study underscores its significance in advancing manufacturing technologies for heavy machinery.
The derivation of the motion equations involves solving for the cutter location based on the gear geometry. For a straight bevel gear, the tooth surface can be represented parametrically. Let \( u \) and \( \theta \) be parameters defining the surface. The surface point \( \mathbf{r}(u, \theta) \) in the workpiece coordinate system is given by:
$$ \mathbf{r}(u, \theta) = \begin{bmatrix} x(u, \theta) \\ y(u, \theta) \\ z(u, \theta) \end{bmatrix} $$
For flank milling with a cylindrical cutter, the cutter axis must align with the surface normal at each point to minimize errors. The surface normal vector \( \mathbf{n}(u, \theta) \) is computed as:
$$ \mathbf{n}(u, \theta) = \frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial \theta} $$
The cutter location data, including the cutter center point \( \mathbf{C} \) and axis vector \( \mathbf{A} \), are then derived by offsetting the surface point along the normal direction by the cutter radius \( R \):
$$ \mathbf{C} = \mathbf{r}(u, \theta) + R \cdot \frac{\mathbf{n}(u, \theta)}{\|\mathbf{n}(u, \theta)\|} $$
$$ \mathbf{A} = \frac{\mathbf{n}(u, \theta)}{\|\mathbf{n}(u, \theta)\|} $$
These values are used in the coordinate transformations to obtain the machine motions. The overall process for solving the motion for the split straight bevel gear is summarized in Table 3.
Table 3: Steps in NC motion solution for split straight bevel gear.
| Step | Description | Key Equations |
|---|---|---|
| 1 | Define gear parameters and tooth surface | \( \mathbf{r}(u, \theta) \), \( \mathbf{n}(u, \theta) \) |
| 2 | Compute cutter location data | \( \mathbf{C} = \mathbf{r} + R \cdot \mathbf{n}/\|\mathbf{n}\| \), \( \mathbf{A} = \mathbf{n}/\|\mathbf{n}\| \) |
| 3 | Transform to machine coordinates | \( [X’_m, Y’_m, Z’_m] = M_{mw} M_{wj} M_{ji} M_{io} [X’_o, Y’_o, Z’_o] \) |
| 4 | Solve for rotary motions | \( A = f(p_Y, p_X, p_Z) \), \( B = g(p_X, p_Z) \) |
| 5 | Generate NC code | Based on linear and rotary trajectories |
The advantages of using a cylindrical end mill for straight bevel gear machining include reduced tooling costs and increased flexibility. However, challenges such as tool wear and dynamic errors must be considered. The motion solution accounts for these by incorporating precise coordinate transformations and optimizing tool paths. For instance, the transformation matrices ensure that the split straight bevel gear is correctly positioned relative to the machine axes, compensating for the offset in the installation center.
In the simulation phase, the VERICUT model was configured with the same kinematic chain as the physical machine. The NC program was executed, and the material removal process was visualized. The results showed that the tooth profile deviations were within acceptable limits, as detailed in Table 2. The automatic comparison highlighted areas with minor undercuts, primarily due to the rough machining stage, but the overall accuracy sufficed for industrial applications. This confirms that the motion solution for the straight bevel gear is effective.
During actual machining, the split straight bevel gear was mounted on the CNC machine using custom fixtures to maintain stability. The finish milling with the cylindrical end mill produced a smooth tooth surface, and subsequent measurements validated the geometric accuracy. The success of this practical implementation demonstrates that the proposed method can be adopted for manufacturing large straight bevel gears without requiring specialized equipment, thus lowering production barriers.
In summary, this study presents a comprehensive approach to NC machining of split straight bevel gears. The derivation of motion equations through coordinate transformations and post-processing enables accurate tool path generation. The validation via simulation and physical machining proves the method’s reliability. Future research could explore real-time error compensation and adaptive control to further enhance precision. The focus on straight bevel gear throughout this work highlights its importance in advancing gear manufacturing technologies.