11.1.2 NC machining

2-D Milling : 2-D milling refers to the contouring capability of a machine tool limited to the -plane. By moving along the and axes simultaneously, while keeping constant, a complete 360 degrees contouring capability can be achieved.

-D Milling : -D milling has a capability between 2-D and 3-D milling. In -D milling, the cutting tool can follow any arbitrary curve in the -plane, but can only move stepwise in the -direction. This -D milling is also referred as pocket machining.

3-D Milling : 3-D milling refers to a cutting tool moving simultaneously along the , and axes, but not capable of performing tool rotation with respect to the workpiece.

5-D Milling : A rotation around two of the axes , and is added to , and translations, hence the tool orientation can vary. The 5-D milling is suitable for large production runs, because the two additional rotations reduce the required setups significantly [287].

The cutter motion for machining a part consists of roughing, semi-roughing and finishing, and should be considered separately, as illustrated in Fig. 11.7 [232]. For each process, an appropriate tool size and tool path needs to be determined.

Rough machining : It should be as simple as possible and preferably consist of a linear type motion only to minimize machining time. In other words, the cutter path should be as short as possible and the depth of cut and feedrate should be as large as possible.

Semi-rough machining : After rough machining, the shoulders left on the part should be removed.

Finishing machining : The cutter should follow the profile during these operations and the deviations of the cutter from the profile should always be maintained within a designated tolerance.

Figure 11.7: (a) Pocket machining with flat-end mill in roughing, (b) semi-roughing with large ball-end mill (adapted from [232])

More than 80% of all mechanical parts which are manufactured by milling machines can be cut by NC pocket machining [157]. This is based on the facts that most mechanical parts consist of faces parallel or normal to a single plane, and that free-form objects are usually produced from a raw stock by -D roughing and 3-D or 5-D finishing. When a cylindrical end-mill cutter is used in -D pocket machining, tool paths are generated by offsetting at a distance equal to the radius of the cutter from the boundary curve. When the cutter is located on the side of the curve where the center of curvature lies, the cutter radius must be smaller than the smallest radius of curvature of the boundary curve of the part to be machined to avoid local overcut (gouging). Gouging is one of the most critical problems in NC pocket machining. To avoid gouging, we need to determine the distribution of the curvatures along the boundary curve to select an appropriate cutter size.

Figure 11.1 (a) shows the tool path of a cylindrical cutter pocket machining a region where the center of curvature of the parabolic boundary curve lies. The parabola has the maximum curvature at with curvature value . Thus if the radius of the cylindrical cutter exceeds , there will be a region of gouging as depicted in Fig. 11.10 (a), where the cutter has a radius . Also the offset with has one self-intersection and two cusps. Points on the segment of the offset bounded by the self-intersecting points on the offset have distance less than the nominal offset distance 0.8 from the generator and this fact causes gouging. Therefore, if we trim off the region of the offset bounded by the two parameter values associated with the self-intersection, the cutter will not overcut the part but will leave an undercut region as shown in Fig. 11.10 (b). The undercut region must be revisited with the smaller size cutter. Each point on the trimmed offset curve is at least distance from every point on the progenitor [102]. Therefore computing the self-intersection points of the offset of a progenitor curve is important.

Most of the tool path generation algorithms for -D pocket machining based on offset contouring, first approximate the input curve with a combination of straight lines and circular arc segments, since traditional CNC interpolators accommodate only such elements and also the offsets of those elements are also straight lines and circular arc segments. Then the approximated boundary curves are offset. The difficult part is to identify and remove all the loops arising from self-intersections. There are two different approaches to remove such loops, namely the Voronoi diagram method [157] and pairwise intersection method [153]. Persson's early work [307] is one of the first to study spiral pocket machining using Voronoi diagrams. A book by Held [157] reviews all the related work until roughly 1991 and introduces an algorithm for the determination of tool paths for spiral and zig-zag milling, and the optimization of tool paths. Held's spiral algorithm, based on an extension of Persson's method [307] provides a general approach for fully automated pocket machining.

In the pairwise intersection method, computing the self-intersections of the offsets reduces to computing the intersections of a straight line to a straight line, a circle to a circle or a straight line to a circle. A brute force approach takes time for computation where is the number of segments plus the number of reflex vertices. Reflex vertices have an interior angle larger than . Hansen and Arbab [153] showed that careful elimination of non-intersecting segments reduces the computation time complexity to for a given shape .

Rohmfeld [349] developed an algorithm to generate tool paths for arbitrary simple piecewise smooth generator curves. The redundant global loops are removed by interval operations on the parameter space of the generator curves using the invariance of Gauss-Bonnet values between the generator and the so-called IGB (Invariant Gauss-Bonnet)-offset, which is equivalent to the rolling ball offset.

When a ball-end mill cutter is used in 3-D machining, the cutter will not gouge the design surface as long as the center of the ball-end mill moves on the trimmed offset surface, where loops arising from self-intersections are removed, and with the offset distance equal to the radius of the cutter (see Fig. 11.1 (b)). A detailed literature review on this topic is given in [183,78]. Among many methods, Sakuta et al. [361], Kuragano et al. [216], Kuragano [215], Kim and Kim [197], Lartigue et al. [223] employ the offset surface-plane intersection method. Sakuta et al. [361] approximate an offset surface by offsetting a quadrilateral mesh of points ignoring small gaps, while Kuragano et al. [216] and Kuragano [215] generate a polygonal offset surface by connecting the offset points, where points along the normal of the free-form surface are offset by the radius of the ball-end mill, to the desired accuracy. When there is a self-intersection in the polygonal offset surface, the portion bounded by the self-intersection lines is trimmed off. Then the approximated (trimmed) offset surface is intersected with parallel planes, which are called tool driving planes, at a regular interval resulting in a series of intersection lines (see Fig. 5.2). The interval between two successive parallel planes is called pick feed. The intersection curves of the approximated polygonal offset surface with these parallel planes generate the required tool paths.