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General Solution Model for Non-CNC Machining Special Rotary Milling Cutter
1 Introduction
The special rotary milling cutter is an essential tool for machining complex free-form surfaces, commonly used in aerospace components and metal molds. It can be integrated with CNC machine tools or machining centers to achieve high-precision and efficient cutting. As CNC machining technology advances and the complexity of workpieces increases, the demand for these specialized rotary cutters is growing rapidly. Currently, most of these cutters are manufactured using multi-axis CNC grinding machines, which are expensive—often costing over one million dollars—and thus significantly increasing production costs. If conventional, non-CNC grinding machines could be utilized to produce such tools, manufacturing costs could be drastically reduced. This paper presents a general design model based on non-CNC machining techniques suitable for large-scale production of various types of special rotary milling cutters. It also explores the implementation of axial and radial feed motions. The author has successfully fabricated a ball-end spiral milling cutter using a non-CNC machining approach, and this article serves as a broader application of that previous work.
2 General Mathematical Model of the Cutting Edge Curve
The working profile of a special rotary milling cutter is a rotational surface, and the spiral cutting edge curve on this surface (as shown in Figure 1) is defined as an oblique line at a fixed angle relative to the surface. This curve can be described mathematically by the following general equation:
r = {x, y, z} = {f(u)cosv, f(u)sinv, g(u)} (1)
Here, u and v are parameter variables; f(u) represents the radius of rotation in the plane of revolution, where f(u) ≥ 0; and v is the angle between f(u) and the positive x-axis. To determine the oblique line on the surface, we first compute the first fundamental form. From equation (1), we have:
Ru = {f(u)cosv, f(u)sinv, g(u)}
Rv = {-f(u)sinv, f(u)cosv, 0}
From these, the coefficients of the first fundamental form are:
E = |Ru|² = f² + g²
F = Ru · Rv = 0
G = |Rv|² = f²
The tangent vector along the oblique line is dr, and the warp cut vector through that point is also dr. Since dv = 0, we have:
Dr = ru du | rv dv = ru du
Given that the oblique line makes a fixed angle j with the warp, the cosine of the angle between dr and dr is given by:
cos²j = (dr · dr)² / (|dr|² |dr|²) = (E du²) / (E du² + G dv²)
Solving this yields:
E sin²j du² = G cos²j dv² → dv = tanj (E/G) du = tanj [(f² + g²)/f] du (2)
Integrating both sides gives:
v = tanj ∫ [ (f² + g²)/f ] du + C (3)
Where C is determined by the initial conditions of the specific problem, ensuring continuity between different sections of the cutting edge. Substituting equation (3) into equation (1) results in a general mathematical model of the edge curve with a fixed angle j relative to the warp.
3 Realization of Relative Feed Motion
During non-CNC machining of a special rotary milling cutter, the tool only performs uniform rotational motion. The formation of the spiral groove is achieved by the axial feed of the grinding wheel while it rotates at a constant speed. Additionally, because the radii of different circular sections vary, the grinding wheel is typically designed for the maximum radius. Therefore, radial feed motion is necessary to accommodate varying radii. In non-CNC machining, the relative feed motion refers to the combination of axial and radial movements of the grinding wheel relative to the milling cutter.
Axial Feed Motion
This paper employs an end cam mechanism to achieve axial feed during non-CNC machining. The cam mechanism can generate precise movement trajectories based on a given function. The displacement function of the cam is derived from the edge curve equation on the rotary surface. Let the angular velocity of the rotating milling cutter be ω = dv/dt (where v is the angular parameter in the edge curve equation (1)), and the angular velocity of the cam during machining is also ω. By inverting the relationship v = v(u) from the edge curve equation under initial conditions, we obtain u = u(v). Substituting this into the axial component z = g(u) = g[u(v)] gives the cam displacement function. Because the cutting edge curve is designed for smooth continuity between surfaces, the resulting cam displacement function is also smooth and continuous.
Radial Feed Motion
The radial feed motion is implemented using a cam mechanism on the grinding machine. The feed amount varies depending on the radius of the rotary surface. When the radius of the milling cutter decreases from R to 0, the feed amount also decreases from r to 0, preventing overcutting and leaving a residual surface that can be easily compensated by the grinding wheel. The radial feed s at a point with gyration radius f(u) is calculated as:
s = r - (r/R)f(u) = r - (r/R)(x² + y²) (4)
By substituting the coordinates of any point on the rotary surface into equation (4), the radial feed at that point can be determined. Due to the smooth transition between surfaces and the continuity of the edge curve, the resulting profile is also smooth and continuous.
4 Solution Example
The following table provides examples of the cutting edge curve and relative motion equations for two types of special rotary milling cutters: the ball-end circular milling cutter and the angular conical milling cutter. Each example includes details of the surface geometry and the corresponding motion parameters.
5 Conclusion
While many studies have explored the spiral grooves of special rotary cutters and their grinding wheel designs, this paper focuses on a novel manufacturing method using non-CNC machining. This approach simplifies the process and reduces costs, making it ideal for mass production of special rotary cutters. However, it is not well-suited for experimental development or small-batch production. Overall, this method offers a promising solution for the manufacturing industry.