In the field of gear manufacturing, large modulus straight bevel gears present significant challenges due to their complex geometry and high load-bearing requirements. Traditional machining methods often rely on specialized equipment, which may not be readily available. This study explores an alternative approach using a finger-type milling cutter on a vertical milling machine. We focus on the principles of form machining for straight bevel gears, calculating key parameters, fitting tool paths, and analyzing tooth profile errors. Additionally, we propose a modification technique using modulus milling cutters to correct these errors, validated through VERICUT simulations and practical machining tests. The goal is to achieve high precision and improved meshing performance for straight bevel gears in applications demanding reliability and durability.
The straight bevel gear is widely used in heavy machinery, where its ability to transmit power between intersecting shafts is critical. However, machining large modulus straight bevel gears is often hindered by the lack of dedicated gear-cutting machines. We address this by employing a finger-type milling cutter, which approximates the spherical involute profile using the back-cone tooth shape. This method involves a single cutter designed for the mid-point tooth profile, with depth adjustments from the large end to the small end. Although this introduces tooth profile errors, they are symmetrically distributed and can be compensated through modification. Our work builds on existing research, such as studies incorporating meshing stiffness and transmission error into dynamic models, optimizing cutter profiles, and analyzing the impact of modification on noise and load distribution. By integrating geometric error analysis with practical modification, we aim to enhance the manufacturing process for straight bevel gears.
To begin, we consider a pair of straight bevel gears with specific parameters, as summarized in the table below. These parameters guide the machining and modification processes, ensuring accuracy in the straight bevel gear production.
| Parameter | Value |
|---|---|
| Number of Teeth (Large Gear) | 65 |
| Number of Teeth (Small Gear) | 22 |
| Module at Large End (mm) | 30 |
| Module at Small End (mm) | 22.131 |
| Face Width (mm) | 270 |
| Milling Cutter Module (mm) | 26.065 |
The machining process involves positioning the finger-type milling cutter perpendicular to the pitch cone generatrix of the straight bevel gear. The tooth profile on the back cone is controlled by the cutter’s form and the depth of cut. To ensure proper meshing, the tooth slot width at different cone distances must match the reference slot width on the equivalent gear’s pitch circle. This is achieved by calculating the depth variation, denoted as K, using the following equations for the tooth thickness and slot width on the equivalent gear.
For any radius on the equivalent gear, the tooth thickness is given by:
$$S_i = r_i \left[ \frac{S}{r} – 2(\text{inv} \alpha_i – \text{inv} \alpha) \right]$$
where \( r_i \) is the radius at an arbitrary point, \( S \) is the pitch circle tooth thickness, \( r \) is the pitch radius, \( \alpha_i = \arccos \left( \frac{r_b}{r_i} \right) \) is the pressure angle at radius \( r_i \), \( r_b \) is the base radius, and \( \alpha \) is the pressure angle at the pitch circle. The slot width at the pitch circle is:
$$e = \frac{\pi m}{2}$$
where \( m \) is the module. The slot width at any radius for the milling cutter is:
$$e_i = p_i – S_i$$
with \( p_i = \frac{2\pi m r_i}{z} \) being the tooth pitch at radius \( r_i \), and \( z \) the number of teeth. Setting \( e_i = e \) and solving using Newton’s iteration method yields the depth variation \( K = r_i – r \). The cutter is designed for the mid-face module, so the depth variation is zero at the mid-point. By computing \( r_i \) at different cone distances, we obtain continuous depth variations from the small to large end. Dividing the face width into eight segments, we calculate depth variations and fit a curve, as shown in the following table and derived formula.
| Face Width Segment (mm) | Depth Variation K (mm) |
|---|---|
| 0 | -6.0 |
| 33.75 | -4.5 |
| 67.5 | -3.0 |
| 101.25 | -1.5 |
| 135 | 0 |
| 168.75 | 1.5 |
| 202.5 | 3.0 |
| 236.25 | 4.5 |
| 270 | 6.0 |
The depth variation curve can be approximated by a polynomial function, such as:
$$K(B) = aB^2 + bB + c$$
where \( B \) is the face width position, and coefficients are determined through regression analysis. This approach ensures consistent tooth slot width across the straight bevel gear, maintaining proper meshing and transmission ratio.
However, using a single finger-type milling cutter inevitably introduces tooth profile errors. These errors arise because the cutter’s form is fixed at the mid-face profile, unable to adapt to varying cone distances. The error distribution is symmetrical, with excess material concentrated at the small end tooth tip and large end tooth root, while undercutting occurs at the small end root and large end tip. For a straight bevel gear pair, these errors can complement each other during meshing, but they still require correction for optimal performance. To quantify these errors, we analyze a cross-section of the equivalent gear, establishing a coordinate system with the origin at the gear center and the Y-axis along the cutter axis.
The reference tooth profile equation is:
$$\begin{cases}
X = r_i \sin(T + \theta_i) \\
Y = r_i \cos(T + \theta_i)
\end{cases}$$
where \( T \) is the tooth profile start angle, and \( \theta_i = \text{inv} \alpha_i \) is the roll angle. The milling cutter profile in the same coordinate system is:
$$\begin{cases}
X = r_o \sin(T’ + \theta_o) \\
Y = r_o \cos(T’ + \theta_o)
\end{cases}$$
with \( r_o = r_i + t + K \), \( t = e \tan \phi \) (where \( e \) is the distance from the mid-face to the section, and \( \phi \) is the base cone angle), \( K \) is the depth variation, \( T’ \) is the cutter profile start angle, and \( \theta_o = \text{inv} \alpha_o \), where \( \alpha_o \) is the pressure angle for the cutter base radius \( r’_b \). For a point \( (x_0, y_0) \) on the reference profile, the normal line equation is:
$$\begin{cases}
X = t \\
Y = y_1 – \frac{y_1 – y_0}{x_0} t
\end{cases}$$
where \( y_1 = \frac{r_b}{\cos(T + \tan \alpha)} \) and \( \cos \alpha = \frac{r_b}{\sqrt{x_0^2 + y_0^2}} \). Solving for the intersection \( (x’, y’) \) with the actual profile gives the error in the meshing direction:
$$E = \sqrt{(x’ – x_0)^2 + (y’ – y_0)^2}$$
This error calculation helps identify the magnitude and direction of deviations, guiding the modification process for the straight bevel gear.
Based on straight bevel gear modification theory, we aim to reduce edge contact and centralize the meshing area on the tooth surface. The small end tooth tip, being thicker than the theoretical profile, significantly affects meshing and requires modification. In contrast, the large end errors are smaller and complementary, so modification is unnecessary there. We use a modulus milling cutter with a profile matching the small end’s standard involute from the tooth tip to the pitch circle. The modification starts from the small end and progresses to the mid-face.
The slot width at any radius for the milling cutter on the equivalent gear is:
$$e = \frac{2\pi r_i}{z} – r_i \left[ \frac{S}{r} – 2(\text{inv} \alpha_i – \text{inv} \alpha) \right]$$
At different cone distances, the tooth tip slot width is:
$$e’ = \frac{2\pi r’_i}{z} – r’_i \left[ \frac{S’}{r’} – 2(\text{inv} \alpha’_i – \text{inv} \alpha’) \right]$$
where \( r’_i \) is the tip radius at various cone distances. Setting \( e = e’ \) and solving for \( r_i \) gives the depth variation \( K’ = r – r_i \) for modification, with the small end as the zero point. We compute multiple \( K’ \) values from the small end to mid-face and fit a tool path curve. This modification optimizes the tooth profile, reducing contact stress, minimizing the risk of surface pitting, and improving transmission stability and lifespan for the straight bevel gear.
To validate our approach, we conduct simulations in VERICUT, a CNC machining simulation software. We model the milling machine and workpiece, create a custom finger-type milling cutter based on mid-face involute parameters, and compile NC programs. The simulation involves importing STL format files, ensuring coordinate alignment, and defining the cutter profile using calculated involute data points. The custom cutter profile in VERICUT is designed to accurately represent the straight bevel gear machining process.
After loading the NC program and setting the machining zero point, we run the simulation and compare the resulting gear with a standard straight bevel gear model. The residual tooth profile before modification shows significant errors, particularly at the small end. Post-modification, the residuals are drastically reduced, confirming the effectiveness of our method. The following tables summarize the meshing direction errors before and after modification, demonstrating the improvement.
| From Tooth Tip | 0 | B/8 | B/4 | 3B/8 | B/2 |
|---|---|---|---|---|---|
| 0 | 2.158 | 1.588 | 1.051 | 0.523 | 0.003 |
| A/4 | 1.669 | 1.211 | 0.764 | 0.410 | 0.001 |
| A/2 | 1.117 | 0.804 | 0.520 | 0.246 | 0.004 |
| 3A/4 | 0.563 | 0.397 | 0.233 | 0.107 | 0.005 |
| A | 0.003 | 0.009 | 0.007 | 0.002 | 0.004 |
Note: A is the distance from tooth tip to pitch circle (mm), B is the face width (mm).
| From Tooth Tip | 0 | B/8 | B/4 | 3B/8 | B/2 |
|---|---|---|---|---|---|
| 0 | 0.002 | 0.003 | 0.002 | 0.004 | 0.001 |
| A/4 | 0.005 | 0.011 | 0.015 | 0.013 | 0.001 |
| A/2 | 0.001 | 0.009 | 0.018 | 0.022 | 0.003 |
| 3A/4 | 0.003 | 0.010 | 0.021 | 0.041 | 0.004 |
| A | 0.001 | 0.008 | 0.006 | 0.002 | 0.003 |
Practical machining tests are conducted on a vertical CNC milling machine using three-axis interpolation (Y, Z, and C axes). We measure the tooth profiles before and after modification with an optical coordinate measuring machine, comparing the data with theoretical profiles. The results show that modification reduces the maximum error from 0.563 mm to 0.041 mm in the meshing direction, achieving the desired precision. Meshing tests further reveal that stress concentration at the ends disappears post-modification, and the meshing area shifts toward the center of the tooth surface, enhancing the straight bevel gear’s performance.
In conclusion, our method for machining and modifying large modulus straight bevel gears using a finger-type milling cutter on a vertical milling machine proves effective. The tooth profile errors are small, symmetrically distributed, and complementary. Key considerations include designing the cutter tip larger than the standard tooth slot bottom to avoid interference at the small end root. Through simulation and experimentation, we demonstrate that modification centralizes the meshing area, reduces stress concentration, and improves overall performance. This approach offers a practical solution for manufacturing straight bevel gears where specialized equipment is unavailable, ensuring reliability in demanding applications.
The integration of geometric calculations, error analysis, and CNC simulation tools like VERICUT enables precise control over the machining process. Future work could explore dynamic modeling of straight bevel gears under load, further optimization of cutter profiles, and application to other gear types. Overall, this study contributes to advancing the manufacturing techniques for straight bevel gears, emphasizing the importance of modification in achieving high-quality gears.