As a critical component for transmitting motion and power between intersecting axes, spiral bevel gears are widely used in automotive, aerospace, and other high-precision industries due to their smooth transmission and high load-bearing capacity. The theoretical research and manufacturing level of spiral bevel gears are important indicators of a country’s mechanical industry prowess. With the rapid development of the machinery manufacturing sector and increasingly fierce international competition, the requirements for gear accuracy and strength have become more stringent. The manufacturing process of spiral bevel gears is a key factor affecting their performance and has become a focal point of research worldwide.
Traditional spiral bevel gear machining relies on cradle-type machines, where the cradle simulates an imaginary gear, and the workpiece rotates relative to this imaginary gear to generate the tooth surface through a process akin to meshing. However, cradle-type machines have long transmission chains, leading to significant errors and low tooth surface accuracy. In contrast, full CNC milling offers high efficiency and precision, making it a superior alternative. This paper explores the conversion of traditional cradle-type machining motions to five-axis full CNC milling for spiral bevel gears, establishes a mathematical model for the cutting edge sweep trajectory, and performs finite element analysis to validate the meshing performance.
The complex tooth surface of spiral bevel gears necessitates specialized machining techniques. In full CNC milling, the five servo axes (X, Y, Z, A, B) of the CNC machine simulate the motions of the cradle, workpiece rotation, and other adjustments. The coordinate transformation from the cradle-type machine to the CNC system is derived as follows. Let $S_o\{o_o; x_o, y_o, z_o\}$ represent the machine coordinate system, $S_t\{o_t; x_t, y_t, z_t\}$ the tool coordinate system, and $S_p\{o_p; x_p, y_p, z_p\}$ the workpiece coordinate system. The transformation matrix from $S_o$ to $S_t$ is given by:
$$ M_{ot} = \begin{bmatrix}
\cos q & -\sin q & 0 & S \\
\sin q & \cos q & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix} $$
where $q$ is the cradle angle and $S$ is the tool radial distance. The transformation from $S_t$ to $S_b$ (bed coordinate system) involves the tool inclination angle $i$:
$$ M_{tb} = \begin{bmatrix}
\cos i & 0 & \sin i & 0 \\
0 & 1 & 0 & 0 \\
-\sin i & 0 & \cos i & 0 \\
0 & 0 & 0 & 1
\end{bmatrix} $$
The workpiece installation parameters, such as the vertical offset $E_m$, horizontal offset $X_p$, and bed slide position $X_b$, are incorporated through additional transformation matrices. The overall transformation from the tool to the workpiece coordinate system enables the derivation of the five-axis CNC motions. The tool center position $R_o$ in the machine coordinate system is:
$$ R_o = M_{ot} M_{tc} M_{tb} R_b $$
where $R_b = [0, 0, 0, 1]^T$ is the tool center in the tool coordinate system. The corresponding position in the workpiece coordinate system $R_p$ is:
$$ R_p = M_{pq} M_p M_o R_o $$
The CNC machine’s rotational axes A and B are derived from the unit orientation vector of the tool. For the CNC machine, the tool orientation vector $N_m = [0, 0, 1, 0]^T$ in the machine coordinate system must align with the transformed vector from the cradle system. Solving for A and B yields:
$$ A = \arctan\left(\frac{-N_y}{N_z}\right), \quad B = \arctan\left(\frac{-N_x}{\sqrt{N_y^2 + N_z^2}}\right) $$
The linear axes X, Y, Z are then computed from the tool center position in the CNC coordinate system. This transformation allows the complex generating motion of the cradle-type machine to be replicated by the five-axis CNC system.
The cutting edge sweep surface, or blade sweep trajectory, is modeled based on the tool geometry and the generating motion. The tool profile in the $x_t o_t z_t$ plane is defined by parameters such as the nominal radius, blade pressure angles, and edge length. Let $l$ be the edge length and $\theta$ the rotation angle around the $z$-axis. The tool surface equation is:
$$ \begin{cases}
x = x(l, \theta) \\
y = y(l, \theta) \\
z = z(l, \theta)
\end{cases} $$
By discretizing the generating motion into $n$ steps, the sweep surface equation becomes:
$$ \begin{cases}
x\{n\} = x(l, \theta) \\
y\{n\} = y(l, \theta) \\
z\{n\} = z(l, \theta)
\end{cases} $$
where $n$, $l$, and $\theta$ are parameters of the sweep surface. This model enables precise simulation of the cutting process.
To validate the method, a pair of spiral bevel gears is analyzed. The gear parameters are summarized in Table 1.
| Parameter | Pinion | Gear |
|---|---|---|
| Number of Teeth | 11 | 43 |
| Shaft Angle (°) | 90 | 90 |
| Module (mm) | 11 | 11 |
| Outer Cone Distance (mm) | 146.683 | 244.116 |
| Spiral Direction | Right-hand | Left-hand |
The cutter parameters for the gear and pinion are provided in Table 2.
| Parameter | Gear Cutter | Pinion Cutter |
|---|---|---|
| Nominal Radius (mm) | 152.400 | 152.400 |
| Blade Pressure Angle (Inner) | 21°16’56” | 20°31’57” |
| Blade Pressure Angle (Outer) | 18°43’4″ | 19°28’33” |
The machine setting parameters for the gear are listed in Table 3.
| Parameter | Value |
|---|---|
| Tool Radial Distance (mm) | 170.892 |
| Basic Cradle Angle (°) | 46.92922 |
| Workpiece Installation Angle | 73°24’57” |
| Vertical Offset (mm) | 0 |
| Roll Ratio | 1.03142 |
Using these parameters, the five-axis CNC motions are computed and converted into G-code. The machining process is simulated in VERICUT, and the resulting gear tooth surface is generated through Boolean operations between the tool sweep surface and the gear blank. The discrete tool positions are arrayed to form the complete tooth surface, as shown in the simulation results.
The tooth surface model is then used for finite element analysis. The gear pair is assembled with a shaft angle of 90°, and the pinion is set as the driving component. The material for both gears is 20CrMnTi, with a density of $7.8 \times 10^3 \, \text{kg/m}^3$, elastic modulus of $207 \, \text{GPa}$, and Poisson’s ratio of 0.25. The mesh is generated in HyperMesh using hexahedral elements for high accuracy and deformation resistance. A single tooth model is meshed and arrayed to form the full gear model, as illustrated in the grid generation process.
The boundary conditions include fixed supports at the gear hubs and coupling constraints at the inner surfaces. The contact between the gear and pinion is defined with a friction coefficient of 0.1. The analysis consists of two steps: a load application step and a rotational acceleration step. Loads of 200 N (light), 500 N (medium), and 700 N (heavy) are applied to the gear to study the contact behavior under different conditions.
The finite element results show that the contact pattern covers multiple tooth pairs, with one pair in full contact and others partially engaged. The contact area shape remains consistent across different loads, indicating stable meshing. The equivalent total strain and stress distributions are analyzed. The maximum strain values for 200 N, 500 N, and 700 N loads are $2.4495 \times 10^{-2}$, $2.4515 \times 10^{-2}$, and $2.4516 \times 10^{-2}$, respectively, demonstrating minimal variation with load. The stress analysis reveals that the maximum contact stress occurs at the tooth tips during engagement and disengagement. The stress-time curve for the 200 N load case shows a periodic pattern, with peaks at the start and end of contact. The maximum stresses for the three load cases are $4.70 \times 10^9 \, \text{Pa}$, $4.72 \times 10^9 \, \text{Pa}$, and $4.74 \times 10^9 \, \text{Pa}$, respectively, confirming that the gear pair meets the meshing requirements under various loads.
In conclusion, the full CNC milling method for spiral bevel gears effectively replaces traditional cradle-type machining by converting complex generating motions into five-axis CNC movements. The mathematical model of the cutting edge sweep trajectory enables accurate tooth surface generation, and the finite element analysis validates the meshing performance. The results indicate that the spiral bevel gear pair produced by this method exhibits stable contact patterns, low deformation sensitivity to load changes, and satisfactory stress distribution. This approach provides a foundation for further research on tooth surface error analysis, modification, and machine parameter optimization. The spiral bevel gear full CNC milling technique proves feasible and advantageous for high-precision applications.
The transformation from cradle-type to CNC machining involves detailed coordinate transformations and parameter adjustments. The tool sweep surface model is derived from the tool geometry and discretized generating motion. The Boolean operations between the tool positions and the gear blank yield a precise tooth surface model. The finite element model, constructed with hexahedral elements and appropriate boundary conditions, simulates the meshing behavior under different loads. The analysis of contact patterns, strain, and stress confirms the robustness of the spiral bevel gear design and manufacturing process. Future work could focus on optimizing the cutter path and incorporating tooth surface modifications to enhance performance. The spiral bevel gear full CNC milling method represents a significant advancement in gear manufacturing technology, offering improved accuracy and efficiency for industrial applications.
The mathematical formulation of the coordinate transformations ensures the accurate replication of cradle-type motions on a five-axis CNC machine. The derivation of the rotational axes A and B and the linear axes X, Y, Z from the cradle parameters is crucial for this conversion. The tool sweep surface model, parameterized by the edge length and rotation angle, captures the essence of the cutting process. The discretization of the generating motion into multiple steps allows for the generation of a smooth tooth surface through envelope formation. The finite element analysis, conducted with realistic material properties and loading conditions, provides insights into the meshing behavior of the spiral bevel gear pair. The results demonstrate that the gear pair maintains proper contact and stress levels across a range of loads, validating the proposed modeling and analysis approach. The spiral bevel gear full CNC milling method is thus a viable and efficient solution for modern gear production.
In summary, this study presents a comprehensive methodology for modeling and analyzing spiral bevel gears manufactured through full CNC milling. The conversion from cradle-type to CNC machining, the mathematical modeling of the tool sweep surface, and the finite element-based meshing analysis collectively ensure the reliability and performance of the gear pair. The spiral bevel gear full CNC milling technique offers a promising direction for advancing gear manufacturing capabilities, with potential applications in various high-precision industries. The continued refinement of this method will contribute to the development of more efficient and durable spiral bevel gears for future mechanical systems.