In modern industrial applications, the demand for large straight bevel gears has grown significantly, particularly in sectors such as power generation, shipbuilding, and mining machinery. The split structure of these gears addresses challenges in manufacturing and assembly, but it introduces complexities in machining processes. Traditional methods often rely on specialized tools like form cutters or ball-end mills, which can be costly and inefficient. This study explores the use of standard end mills for side milling split straight bevel gears, offering a cost-effective and efficient alternative. We focus on developing a precise cutter position solution to enable high-quality gear production on general-purpose CNC machines. The approach involves mathematical modeling of the gear tooth surface, tool path planning, and validation through simulation and experiments. By leveraging the versatility of end mills, we aim to reduce manufacturing costs and enhance the accessibility of large straight bevel gear production.
The tooth surface of a straight bevel gear can be represented as a ruled surface, which is generated by the motion of a straight line along two directrices—the gear’s large and small ends. This representation simplifies the modeling process and facilitates tool path calculation. The general form of a ruled surface is given by:
$$ P(u,v) = (1-v)E(u) + vF(u), $$
where \( E(u) \) and \( F(u) \) are vector functions describing the directrices. For a straight bevel gear, the tooth surface \( S(r,\phi) \) is expressed as:
$$ S(r,\phi) = (1-r)W(\phi) + rQ(\phi), $$
with \( W(\phi) \) representing the small-end tooth profile and \( Q(\phi) \) the large-end tooth profile, where \( r \) ranges from 0 to 1. This formulation allows for efficient computation of surface points and normals, which are critical for determining cutter positions during side milling.
To machine the split straight bevel gear, we establish a coordinate system that accounts for the gear’s geometry and the machine tool’s kinematics. The workpiece coordinate system \( S_O(O_O – X_o, Y_o, Z_o) \) is fixed to the gear blank, with its origin at the cone apex \( O \) of the base cone and the Z-axis passing through the center of the cone base. The tool coordinate system \( S_c(O_c – i_c, j_c, k_c) \) is attached to the cutter, with the origin at the tool center and axes aligned with the tool’s orientation. This setup enables the calculation of tool positions and orientations relative to the gear tooth surface. For split gears, the gear center is offset from the machine’s rotary table center, necessitating unique cutter positions for each tooth slot.
The side milling process involves moving the end mill along the tooth surface in a series of passes. We adopt a Z-shaped tool path, where the cutter starts at the large end, moves to the small end along the cone generatrix, and then returns in a reciprocating manner. This path minimizes idle time and ensures consistent surface finish. The cutter’s position is defined by the tool center coordinates \( M \) and the tool axis vector \( P \), which must remain perpendicular to the surface normal vector \( N \) at each point. The tool center is offset from the surface by the tool radius \( R_d \) in the direction of the unit normal vector.
To compute the tool center coordinates, we first determine the surface normal vector. The partial derivatives of the surface equation with respect to \( r \) and \( \phi \) yield the tangent vectors:
$$ S_r = \frac{\partial S(r,\phi)}{\partial r} = Q(\phi) – W(\phi), $$
$$ S_\phi = \frac{\partial S(r,\phi)}{\partial \phi} = (1-r)W'(\phi) + rQ'(\phi). $$
The normal vector \( F_s \) is then obtained from the cross product:
$$ F_s = S_r \times S_\phi = [f_x, f_y, f_z], $$
where:
$$ f_x = S_{ry} S_{\phi z} – S_{rz} S_{\phi y}, $$
$$ f_y = S_{rz} S_{\phi x} – S_{rx} S_{\phi z}, $$
$$ f_z = S_{rx} S_{\phi y} – S_{ry} S_{\phi x}. $$
The unit normal vector \( e \) is:
$$ e = \frac{F_s}{|F_s|} = [a_x, a_y, a_z]. $$
Given a surface point \( (X, Y, Z) \), the tool center coordinates \( (x_o, y_o, z_o) \) are calculated as:
$$ x_o = R \cdot [\cos(\phi \sin \delta_b) \sin \delta_b \cos \phi + \sin(\phi \sin \delta_b) \sin \phi] \cdot a_x \cdot R_d, $$
$$ y_o = R \cdot [\cos(\phi \sin \delta_b) \sin \delta_b \sin \phi – \sin(\phi \sin \delta_b) \cos \phi] \cdot a_x \cdot R_d, $$
$$ z_o = R \cdot \cos(\phi \sin \delta_b) \cos \delta_b \cdot a_x \cdot R_d. $$
Here, \( R \) is the gear’s reference radius, and \( \delta_b \) is the base cone angle. The tool axis vector \( P = (p_x, p_y, p_z) \) is derived by rotating the initial axis vector \( S_r \) by 90° around the Y-axis using the transformation matrix \( M_{oc} \):
$$ M_{oc} = \begin{bmatrix} \cos b & 0 & \sin b \\ 0 & 1 & 0 \\ -\sin b & 0 & \cos b \end{bmatrix}, $$
where \( b \) is the rotation angle. The resulting tool axis vector components are:
$$ p_x = R \cdot [\cos(\phi \sin \delta_b) \sin \delta_b \cos \phi + \sin(\phi \sin \delta_b) \sin \phi] [\cos b – \sin b], $$
$$ p_y = R \cdot [\cos(\phi \sin \delta_b) \sin \delta_b \sin \phi – \sin(\phi \sin \delta_b) \cos \phi], $$
$$ p_z = R \cdot \cos(\phi \sin \delta_b) \cos \delta_b [\sin b + \cos b]. $$
These equations provide explicit functions for the cutter position and orientation, enabling precise CNC programming for the split straight bevel gear.
To validate the cutter position solution, we conducted a simulation using VERICUT software. The gear parameters are summarized in Table 1, which includes key dimensions and design values for the split straight bevel gear.
| Parameter | Symbol | Value |
|---|---|---|
| Module | \( m_n \) | 20 mm |
| Number of Teeth | \( Z \) | 72 |
| Pressure Angle | \( \alpha_n \) | 20° |
| Face Width | \( B \) | 100 mm |
| Addendum Modification Coefficient | \( x_n \) | 0 |
| Pitch Diameter at Large End | \( D \) | 1440 mm |
We modeled a five-axis CNC machine in VERICUT, incorporating the calculated tool paths. The rough machining phase used a form cutter to remove bulk material, while the finishing phase employed a \( \phi 10 \, \text{mm} \) end mill for side milling. The simulation results showed minimal undercut and overcut errors, with most of the tooth surface matching the theoretical model. The Z-shaped tool path effectively covered the entire surface, and the cutter positions ensured accurate material removal. This simulation confirmed the feasibility of the proposed method for machining split straight bevel gears.
Following the simulation, we performed actual machining on a DMU100 five-axis CNC machine. The gear blank was pre-processed with rough cutting to leave a 0.1 mm allowance for finishing. The finishing operations used the derived cutter positions to achieve the final tooth surface. After machining, we measured the gear using a Hexagon coordinate measuring machine (CMM) to assess dimensional accuracy. The measurement data were compared against theoretical values to compute deviations in tooth profile, pitch, and runout.
The results of the gear measurement are summarized in Table 2, which lists the key deviation parameters and their values. All measured deviations fall within acceptable limits according to industry standards, such as GB/T 10095.1, confirming the accuracy of the machined split straight bevel gear.
| Deviation Parameter | Symbol | Value (μm) |
|---|---|---|
| Tooth Profile Total Deviation | \( F_f \) | 5.6 |
| Tooth Profile Form Deviation | \( f_f \) | 2.6 |
| Single Pitch Deviation | \( f_{pt} \) | 25.3 |
| Total Cumulative Pitch Deviation | \( F_p \) | 21.8 |
| Radial Runout Deviation | \( F_r \) | 38.3 |
The mathematical models for cutter position and tool axis vector are derived from the geometry of the straight bevel gear tooth surface. The surface normal vector plays a crucial role in determining the tool’s orientation. For a given point on the surface, the normal vector ensures that the cutter maintains contact without gouging. The unit normal vector \( e \) is used to offset the tool center by the radius \( R_d \), as shown in the equations for \( x_o \), \( y_o \), and \( z_o \). The tool axis vector \( P \) is aligned to be perpendicular to the surface normal, which is achieved through the rotation matrix \( M_{oc} \). This alignment is essential for effective side milling of the straight bevel gear tooth surface.
In the context of split straight bevel gears, the offset of the gear center from the machine table introduces additional complexity. Each tooth slot requires unique cutter positions, which are computed based on the local surface geometry. The Z-shaped tool path ensures that the cutter moves efficiently along the tooth surface, reducing machining time and improving surface finish. The reciprocating motion allows for continuous cutting, minimizing tool retractions and air cuts. This path is particularly suitable for straight bevel gears, as it follows the natural curvature of the tooth surface.
The simulation in VERICUT provided a virtual environment to test the tool paths and identify potential issues. The model included the machine kinematics, tool geometry, and gear blank. The automatic comparison feature in VERICUT highlighted areas with overcut or undercut errors, which were within the set tolerance of 0.05 mm. The simulation demonstrated that the cutter positions derived from the mathematical model accurately produce the desired tooth surface for the split straight bevel gear.
During actual machining, the CNC program generated from the cutter positions was executed on the DMU100 machine. The use of a standard end mill for finishing proved effective, as it offered flexibility and cost savings compared to specialized tools. The measured deviations from the CMM data confirm that the gear meets the required accuracy standards. The tooth profile deviations are minimal, indicating that the surface normals and tool orientations were correctly computed. The pitch and runout deviations are within the specified limits, ensuring proper meshing and operation of the straight bevel gear in practical applications.
In conclusion, the developed method for cutter position solution in side milling of split straight bevel gears is both feasible and accurate. The mathematical models for tool center coordinates and axis vectors enable precise CNC programming, while the Z-shaped tool path optimizes machining efficiency. The simulation and experimental results validate the approach, showing that standard end mills can be used to produce high-quality straight bevel gears on general-purpose machines. This method reduces manufacturing costs and expands the accessibility of large gear production, making it a valuable contribution to the field of gear manufacturing.
Further research could explore the application of this method to other types of bevel gears, such as spiral or hypoid gears, and investigate the effects of different tool paths on surface finish and tool wear. Additionally, real-time adjustment of cutter positions based on in-process measurements could enhance accuracy and compensate for machining errors. The integration of advanced materials and coatings for end mills may also improve performance and longevity in machining straight bevel gears.