Abstract:
This paper delves into the principles of gear milling for spiral bevel gears and the numerical control (NC) machining process. By analyzing the structure of spiral bevel gear milling machines and their operational dynamics, a comprehensive machining coordinate system is established. Furthermore, the adjustment parameters and calculation methods for computer numerical control (CNC) spiral bevel gear milling machines are explored, leading to the development of an NC machining model. Practical examples are provided to illustrate the preparation of NC machining programs.
Keywords: spiral bevel gear; gear milling; NC machining; machining program
1. Introduction
Spiral bevel gear serve as fundamental components in mechanical transmissions, enabling motion transfer between intersecting axes. They are renowned for their high contact ratio, smooth transmission, low noise levels, and impressive load-bearing capacity. Consequently, they are extensively utilized in various machinery and equipment, particularly in high-precision, high-speed, and heavy-duty applications such as automobiles and airplanes. However, the manufacturing of spiral bevel gear presents a significant challenge in the gear-making industry, demanding precise machine adjustments. Therefore, research on NC machining technology for spiral bevel gear has become a focal point within the industry.
2. The Generating Principle of Spiral Bevel Gear Milling
The machine’s rocking table mechanism simulates an imaginary gear, and the rotating milling cutter disk installed on the rocking table forms a cutting surface, equivalent to a tooth surface of the imaginary gear. As the gear blank and the imaginary gear rotate around their respective axes at a specific gear ratio, the milling cutter disk cuts a tooth space on the gear blank. The rocking table then reverses to its initial position, while the workpiece box retreats along with the bed saddle. Simultaneously, the gear blank rotates through a certain indexing angle to enter the next cutting cycle. This process is repeated until the entire gear is completed. This method is known as the generating method, and the imaginary gear represented by the rocking table is termed the generating gear.
3. Structure and Mathematical Model of NC Spiral Bevel Gear Milling Machine
A simplified diagram of an NC spiral bevel gear milling machine. Based on the functions and motion laws of each machine component during actual processing, a machine coordinate system is established. The machine-centered total coordinate system Σm {Om Xm Ym Zm} has Om as the machine center, Zm passing through the rocking table center and pointing outside the rocking table, with the Xm-Ym plane representing the machine plane. This coordinate system is static. The rocking table Σc {Oc Xc Yc Zc} and the cutter disk Σk {Ok Xk Yk Zk} are dynamic coordinate systems. The origin Oc of the rocking table coordinate system coincides with the origin Om of the machine coordinate system. The origin Ok of the cutter disk coordinate system is located at the center of the cutter tip plane, which coincides with the machine XmOmYm plane. Σw {Ow Xw Yw Zw} is the coordinate system fixed to the workpiece, with Ow as the design intersection point of the workpiece gear.
During the generating process, the coordinate system Σc rotates with the rocking table around Zm by an angle φc, representing the current rotational angle of the generating gear (rocking table). The coordinate system Σw rotates with the workpiece around Zw by an angle φw, representing the current rotational angle of the workpiece gear relative to the generating gear angle φ1. The angle β denotes the angle between the line connecting the centers of the rocking table and the cutter disk and the horizontal line passing through the center of the rocking table. Sk represents the distance between the centers of the rocking table and the cutter disk. The angle δm indicates the installation angle of the workpiece box adjustment device. E represents the vertical wheel position, which is zero when using the imaginary flat-top gear principle to machine spiral bevel gear. Xb indicates the forward or backward movement of the cutter disk along the centerline of the rocking table relative to the standard position, directly affecting the cutting depth of the gear being cut. Xp represents the distance between the design intersection point of the workpiece gear and the machine center.
4. Parameter Calculation for NC Machining of Spiral Bevel Gear
4.1 Tool Position Parameter Calculation
When using the imaginary flat-top gear principle to machine spiral bevel gear, the nominal radius rk of the cutting tool disk must first be determined based on the geometric parameters of the gear being processed. Subsequently, the tool position, i.e., the position of the cutter disk center Ok in the machine coordinate system, is determined.
During the adjustment of the tool position on a spiral bevel gear milling machine, the vertical coordinate V and horizontal coordinate H of the cutter disk center Ok in the machine coordinate system XmOmYm plane are calculated as follows:
V = rk cos βm (1)
H = Lm – rk sin βm (2)
Where Lm is the midpoint cone distance, and βm is the midpoint helix angle. The radial tool position Sk and angular tool position q are then calculated as:
Sk = √(V² + H²) (3)
q = arctan(V/H) (4)
Before adjusting the tool position, the cutter disk center Ok coincides with the machine center Om. During adjustment, the eccentric wheel is rotated counterclockwise by an angle φ, causing the cutter disk fixed to the eccentric wheel to rotate to point B, making the distance between the cutter disk center Ok and the machine center Om equal to the radial tool position Sk. Then, the rocking table is rotated by an angle Q, making the angle between the line connecting the cutter disk center Ok and the machine center Om and the horizontal line passing through the machine center Om equal to the angular tool position q. The eccentric angle φ and rocking table angle Q are calculated as:
φ = 2arcsin(Sk / (2K)) (5)
Where K is the distance between the cutter disk center Ok and the eccentric wheel center Op.
Q = q ± φ/2 (6)
In equation (6), the “+” sign is used for left-hand gears, and the “-” sign is used for right-hand gears.
4.2 Motion Parameter Calculation for NC Spiral Bevel Gear Milling Machine
The calculation methods for the workpiece rotational displacement angle and program cycle times are briefly introduced below:
Zi = Z2 + Z’² / i = Zi / Z (7)
Where Z is the number of teeth on the gear being processed, Z’ is the number of teeth on the mating gear, and i is the rolling ratio.
360° / Z = a b (8)
Where a and b represent the angles corresponding to one tooth, with the calculated values rounded to three decimal places, with a as the rounded-up value and b as the rounded-down value.
R = Q · i (9)
Where R is the workpiece rotational displacement angle, and Q is the rocking table swing angle.
A = Q · i – a
B = Q · i – b (10)
Where A and B are the workpiece reverse rotational displacement angles.
Assuming that the A value is called x times and the B value is called y times, then:
x + y = Z
ax + by = 360 (11)
By solving the equations, the values of x and y are obtained, and the cycle times for the gear machining process are calculated by taking the common divisor c of x and y values. The A value is called x times, and the B value is called y times, ensuring that the angles a and b corresponding to the gear teeth are evenly distributed on the gear blank.
Using the above formulas, the machining parameters of the spiral bevel gear milling machine during the processing can be obtained.
5. Calculation Example and NC Programming
Combining the mathematical model of spiral bevel gears and the calculation method of machine machining parameters, we now take a gear with 39 teeth and a mating gear with 7 teeth as an example to perform parameter calculation and NC programming for spiral bevel gears.
5.1 Basic Parameters of Spiral Bevel Gear Pair
Table 1 lists the basic parameters of the spiral bevel gear pair, including the tooth number, modulus, helix angle, shaft angle, handedness, full-angle height, addendum height, dedendum height, pitch diameter, pitch cone angle, face cone angle, and root cone angle for both the large and small gears. These parameters are fundamental for gear design and manufacturing.
Table 1: Basic Parameters of Spiral Bevel Gear Pair
| Parameter | Large Gear | Small Gear |
|---|---|---|
| Tooth Number | 39 | 7 |
| Modulus (mm) | 8.03 | |
| Helix Angle | 35° | |
| Shaft Angle | 90° | |
| Handedness | Right | Left |
| Full-Angle Height (mm) | 12.69 | |
| Addendum Height (mm) | 11.04 | 1.65 |
| Dedendum Height (mm) | 3.53 | 12.63 |
| Pitch Diameter (mm) | 305.14 | 71.71 |
| Pitch Cone Angle | 78.24° | 11.13° |
| Face Cone Angle | 79.16° | 16.54° |
| Root Cone Angle | 72.32° | 10.23° |
5.2 Machine Adjustment Parameters for Spiral Bevel Gear Machining
Table 2 lists the adjustment parameters for spiral bevel gear machining, including radial tool position, angular tool position, eccentric angle, swing angle, horizontal wheel position, vertical wheel position, workpiece installation angle, rolling ratio, tool diameter, workpiece rotation displacement angle, and workpiece reverse rotation displacement angles A and B. These parameters are crucial for precise adjustment during the machining process.
Table 2: Machine Adjustment Parameters for Spiral Bevel Gear Machining
| Parameter | Value |
|---|---|
| Radial Tool Position (mm) | 124.97 |
| Angular Tool Position | 72.25° |
| Eccentric Angle | 73.39° |
| Swing Angle | 52.9° |
| Horizontal Wheel Position (mm) | 0 |
| Vertical Wheel Position (mm) | 0 |
| Workpiece Installation Angle | 72.25° |
| Rolling Ratio | 1.017 |
| Tool Diameter (mm) | 304.8 |
| Workpiece Rotation Displacement Angle | 50.84° |
| Workpiece Reverse Rotation Displacement Angle A | 40.527° |
| Workpiece Reverse Rotation Displacement Angle B | 40.526° |
Based on these parameters, we can calculate a = 9.473° and b = 9.474°. Then, using formula (11), we obtain x = 12 and y = 26. The loop count for the first-level subroutine is set to the greatest common divisor of x and y, which is 2. For the second-level subroutine, the loop is set to call value A once and value B twice, totaling 5 calls, followed by another call to value A once and value B three times, totaling 1 call. This ensures that the angles a and b corresponding to the gear teeth are evenly distributed on the gear blank, minimizing the distribution error between the teeth.
Based on the above parameters and calculations, the NC machining program for the hobbing method (generative method) of spiral bevel gear can be developed. The program consists of three parts: the main program, subroutines, and parameter calls. When changing the product, only the parameter values need to be adjusted, making the application very convenient.