1 Introduction
Special rotary milling cutters are essential tools used in the machining of complex free-form surfaces, such as those found in aerospace components and metal molds. These tools are typically used with CNC machine tools or machining centers to achieve high-quality and efficient machining. With the advancement of numerical control (NC) technology and the increasing complexity of workpieces, the application and consumption of special rotary surface tools have been on the rise. Currently, these tools are mainly manufactured using multi-axis linkage CNC grinders, which are expensive—importing one can cost over a million US dollars—making the manufacturing process costly. If it were possible to use standard tool grinders for non-NC machining of these special rotary cutters, the production costs could be significantly reduced. This paper presents a universal design model based on non-NC machining programs, suitable for large-scale production of various types of special rotary milling cutters. It also explores the realization methods of axial and radial relative feed motions. The author has successfully manufactured ball-end spiral milling cutters using non-NC machining solutions, and this article represents an extension and application of the previous research findings.
2 General Mathematical Model of Edge Curve
The working profile of a special rotating milling cutter is a revolving surface. The spiral cutting edge curve on this surface (see Figure 1) is defined as an inclined line that maintains a fixed angle with the meridian on the surface. The mathematical equation for this surface can be expressed as:
r = {x, y, z} = {f(u)cosv, f(u)sinv, g(u)} (1)
Figure 1: Spiral edge curve on a revolving plane
In this formula, u and v are parameter variables; f(u) represents the radius of the surface at any point, where f(u) ≥ 0; and v is the angle between the positive x-axis and f(u). To determine the slope line on the revolving surface, we first calculate the first fundamental form. From equation (1), we get:
ru = {f’(u)cosv, f’(u)sinv, g’(u)}
rv = {-f(u)sinv, f(u)cosv, 0}
Thus, the coefficients of the first fundamental form are:
E = ru · ru = f’² + g’²
F = ru · rv = 0
G = rv · rv = f²
It is noted that the tangent vector of the oblique line is dr, and the tangent vector of the meridian passing through the same point is also dr. Considering that for the meridian, dv = 0, we have dr = ru du + rv dv. Since the oblique line forms a fixed angle j with the meridian, the angle between dr and dr is constant. Using the dot product, we find:
cos²j = (dr · dr)² / (|dr|² |dr|²) = E du² / (E du² + G dv²)
After simplifying, we obtain:
E sin²j du² = G cos²j dv²
Solving for dv gives:
dv = tanj √(E/G) du = tanj [(f’² + g’²)¹â„² / f] du (2)
Integrating both sides yields:
v = tanj ∫ [√(f’² + g’²) / f] du + C (3)
Where C is determined by initial conditions, ensuring continuity of the cutting edge curve across different turning surfaces. Substituting equation (3) into equation (1) gives a general mathematical model of the blade curve with a fixed angle j relative to the meridian.
3 Realization of Relative Feed Motion
When a special rotary milling cutter is machined without NC, the tool rotates at a constant speed, while the grinding wheel simultaneously performs the rotation to form the spiral groove. Additionally, due to varying radii of the rounded sections on the revolving surface, and since the grinding wheel is typically designed for the maximum radius, radial feed motion is necessary to accommodate different groove sizes. In non-NC machining of special rotary cutters, the relative feed motion involves both axial and radial movements of the grinding wheel relative to the tool.
Axial Feed Motion
This paper uses a face cam mechanism to realize the non-NC axial feed motion of the special rotary milling cutter. The cam mechanism can generate a specific displacement function. The displacement function is derived from the edge curve equation of the rotating surface. Assuming the angular velocity of the tool is w = dv/dt, and the angular velocity of the cam is w', then setting w' = a'w (a' > 0), we can derive the general displacement function of the cam. By solving the inverse function of the edge curve equation under initial conditions, and substituting it into the axial component of the surface equation, we obtain the cam’s displacement function. Because the cutting edge curve is smooth and continuous at the junction of two rotating surfaces, the cam displacement function is also smooth and continuous at those points.
Radial Feed Motion
For non-NC machining of special rotary milling cutters, the radial feed motion is achieved using a master mechanism on a tool grinder. The radial feed rate varies with the radius of the rotating surface. As the radius decreases from R to 0, the feed rate also decreases, preventing overcutting and allowing the remaining surface to be easily compensated by the grinding wheel. Assuming the feed radius at a point f(u) on the surface is s, the radial feed can be calculated using the formula:
s = r - (r/R)f(u) = r - (r/R)(x² + y²)¹â„² (4)
By substituting the coordinates of any point on the surface into equation (4), the corresponding radial feed value can be obtained. Due to the smooth connection between rotating surfaces and the continuous nature of the cutting edge curve, the radial feed curve is also smooth and continuous at the intersection of the two surfaces.
4 Examples of Solving
The table below presents examples of solving the cutting edge curves and relative kinematic equations for two types of special rotary milling cutters: ball-end circular cutters and conical cutters. The parameters in the table are illustrated in Figures 2 and 3. The q value in the table represents the angle of the cam displacement curve between different rotating surfaces.
Table: Solving Example
Ball-end circular arc milling cutter vs. corner round conical milling cutter
Ball-end partial arc, rotating surface angle, circular surface, conical surface
Figure 2: Ball end circular cutter
Figure 3: Tapered circular cone cutter
5 Conclusions
The design method for spiral grooves and grinding wheels for special rotary milling cutters has been widely discussed in literature. This paper introduces a manufacturing method for special rotary milling cutters using non-NC machining programs, offering a simple and low-cost alternative, especially suitable for mass production. While this approach has good market potential, it is not ideal for experimental development or single-piece production.
DC200V-DC500V 5.5KW New Energy Power Source
Dc200V-Dc500V 5.5Kw New Energy Power Source,5.5Kw Dc200V New Energy Power Source,5.5Kw Dc500V New Energy Power Source,5.5Kw Dc New Energy Power Source
Huai'an Sur Hydraulic Technology Co., Ltd , https://www.surpowerunit.com