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General Solution Model for Non-CNC Machining Special Rotary Milling Cutter
1. Introduction
The special rotary milling cutter is a crucial tool used in machining complex free-form surfaces, such as those found in aerospace components and metal molds. It is typically used with CNC machine tools or machining centers to achieve high-precision and efficient cutting. As CNC machining technology advances and the complexity of machined parts increases, the demand for these specialized rotary tools continues to grow. However, the production of such tools often relies on expensive multi-axis CNC grinding machines, which are costly to purchase (often imported at around one million dollars) and thus significantly increase manufacturing costs. If it were possible to use conventional, non-CNC grinding machines to produce these tools, the overall cost could be drastically reduced. This paper presents a general design model that enables the large-scale production of various types of special rotary milling cutters using non-CNC machining techniques. It also explores the implementation of axial and radial feed motions during the process. The author has successfully fabricated a ball-end spiral milling cutter using a non-CNC program, and this article serves as an extension and 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 on this surface is defined as an oblique line with a fixed angle relative to the surface. The equation of this surface can be expressed as:
r = {x, y, z} = {f(u)cosv, f(u)sinv, g(u)} (1)
Here, u and v are parameters; f(u) represents the radius of rotation at a given point, and v is the angular position relative to the x-axis. To determine the oblique line on the surface, we first compute the first fundamental form. From equation (1), we derive the tangent vectors:
Ru = {f(u)cosv, f(u)sinv, g(u)}
Rv = {-f(u)sinv, f(u)cosv, 0}
From these, we calculate the coefficients of the first fundamental form:
E = |Ru|² = f² + g²
F = Ru · Rv = 0
G = |Rv|² = f²
The tangent vector along the oblique line is dr, and the angle between dr and the warp direction is constant. Using the cosine formula, we find:
cos²j = (dr · dr)² / (|dr|² |dr|²) = E du² / (E du² + G dv²)
Solving this leads to:
sin²j du² = cos²j dv² * (G/E) → dv = tanj * (E/G) du = tanj * [(f² + g²)/f] du (2)
Integrating this gives the parametric expression for v:
v = tanj ∫ [ (f² + g²)/f ] du + C (3)
This equation allows us to define the spiral cutting edge curve with a fixed angle j. Substituting equation (3) into equation (1) yields a complete mathematical model for the edge curve.
3. Realization of Relative Feed Motion
In non-CNC machining of special rotary milling cutters, the tool only rotates uniformly, while the formation of the spiral groove is achieved through the axial feed of the grinding wheel. Additionally, since different points on the rotating surface have varying radii, the grinding wheel must adjust its radial position accordingly. Therefore, the relative feed motion in non-CNC machining involves both axial and radial movements of the grinding wheel relative to the tool.
Axial Feed Motion
To implement the axial feed motion, a cam mechanism is used. The cam displacement function is derived from the edge curve equation. Assuming the rotational speed of the tool is ω = dv/dt, and the cam rotates at the same speed, the displacement function is calculated by inverting the relationship v = v(u) from the edge curve equation. This function ensures smooth and continuous motion across different sections of the surface.
Radial Feed Motion
For radial feed, the cam mechanism adjusts the grinding wheel’s position based on the radius of the surface. The feed amount at any point is determined by the formula:
s = r - (r/R)f(u) = r - (r/R)(x² + y²) (4)
This ensures accurate grinding without overcutting, especially when transitioning between different surface sections.
4. Solution Example
The following table provides examples of the cutting edge curve equations and relative motion functions for two types of special rotary milling cutters: the ball-end circular mill and the conical mill.
5. Conclusion
Although many studies have addressed the design of spiral grooves and grinding wheels for special rotary cutters, this paper introduces a novel non-CNC machining method. This approach simplifies the manufacturing process and reduces costs, making it ideal for mass production. However, it is less suitable for experimental development or small-batch production due to its reliance on pre-defined profiles. Overall, the method offers a promising solution for industrial applications.