6. Tool Path Strategies in Surface Generation
6.2 Tool Paths for Symmetric Macro Shapes
The diamond turn machining process normally involves the face turning operation just as in any typical turning process. In a normal turning process, the work surface spins about the spindle axis with the tool feed motion in a direction perpendicular to this axis. The tool feed motion path typically lies in a plane (XZ) containing the spindle axis (Z) (Figure 6.2).
Thus, a typical DTM is a 2-axis simultaneously controllable machine.
The tool feed motion is usually not synchronised with the spindle rotation;
the spindle either runs at a constant rotational speed or in a monotonically
(a) (b) (c)
FIGURE 6.1
Example of (a) a rotationally symmetric shape, in contrast to (b) a rotationally asymmetric surface – the otherwise planar surface has several convex lens protrusions placed off-centered.
(c) A part surface filled with tiny features that are tens of micrometers in size.
θ(t)
work
Tool feed path motion is in XZ plane
Y X
Z Tool
Tool motion path View along Y-axis Symmetric work
surface
X Z
FIGURE 6.2
Tool motion path for symmetric shapes involves spindle rotation to provide cutting motion while the tool moves in the XZ plane to generate the necessary geometry.
81 Tool Path Strategies in Surface Generation
increasing or decreasing speed to maintain constant tangential speed at the cutting tool tip. Because of this lack of synchronicity, machining features produced naturally tend to be rotationally symmetric (Figure 6.1) about the axis. This is the normal arrangement in a simple DTM setup, which also leads to production of shapes symmetrical about the axis. Such shapes can be spherical or aspherical (elliptic, parabolic, etc.), but symmetric about the axis.
Let us now consider on how material is removed to create the desired symmetrical surface. Consider first creating a flat planar face. The face turning operation removes an entire disc-shaped volume of material from the front face of the work-piece in the form of a slowly peeling spiral of material (Figure 6.3). The spiral can be better visualised in the follow- ing way: If you can imagine the spindle to be brought to a stand-still, but allow the tool to have both rotational cutting and feed motion, then you may see that the tool traverses on the face of the work material in the form of a spiral, starting from the outside and slowly converging toward the center (Figure 6.3). This material removal in a spiral fashion holds true even when the face is not flat but has a curvature (e.g. a sphere) – like spirally peeling the skin of an orange. This path can be mathematically quantified as follows:
Consider the DTM operation of facing a flat surface such as in Figure 6.3.
Assuming that the spindle revolving speed is N (rev/s), and the radial
X Y
Z Material removed
as spirals
Total volume of material to be
removed
Process can be viewed as the tool moving in a spiral path scratching the
stationary work-piece FIGURE 6.3
Schematic illustrating how face turning operation, the basic process in DTM, removes material in the form of a uniform thickness spiral to produce symmetrical shapes – in this case a planar surface. This form of material removal applies even when the face is not flat but has a curvature (e.g. spherical surface).
feed of the tool is f mm/rev, the spiral path of the cutting tool would be an Archimedes spiral of the form (in polar notation):
r r f
= ref −
2πθ; (6.1)
where rref denotes a faraway position that the tool moves radially inwards from. Hence, the x and y coordinates of the spiral path can be found as (ori- gin at the center on the spindle axis at a suitable point):
x r r f
= = ref−
cosθ cos
πθ θ
2 (6.2)
y r r f
= = ref −
sinθ sin
πθ θ
2 (6.3)
assuming that the tool starts from an outer point (defined using Rref) and spirals inwards. For the flat surface, z is zero or a constant.
For a spherical surface (of radius R) of the type shown in Figure 6.2, the path will be a spiral in three dimensions, with an additional varying z-coordinate given by (assuming that tool starts from a reference point, zref)
z z= ref ± R2−r2 (6.4)
It is clear that since the feature is rotationally asymmetrical, z is not a func- tion of θ. For a parabolic surface of revolution characterized by constant a, the following would be the z-coordinate variation with r, again independent of θ:
z z r
ref a
= ± 2
4 (6.5)
For practical path execution with a rotating spindle, and radial feed path, set θ = 0. Then r = x, and the controller determines the position x and then based on this, the z-coordinate for tool motion.
For non-analytical surfaces, the following approach can be undertaken. On the projected flat surface (XY plane), based on a feed rate in the x-direction, an Archimedes spiral path is adopted and a discrete set of points, (r, θ) in polar coordinates, identified. At these points, the z-coordinate is determined based on the given data or non-parametric relationship. Sufficiently close points (r, z) (θ is set to zero since the spindle is rotating) can then be used along with linear interpolation (or other interpolations, such as a spline-based, based on
83 Tool Path Strategies in Surface Generation
controller sophistication and encoder resolution) to fill in the gaps between the points for tool movement.
This spiral motion path concept is important to understand, since this in turn will make it easier to visualise how, in the discussion that follows in the next section, asymmetrical shapes are created by synchronising tool feed motion (in the Z-axis) with the spindle rotation (C-axis). Note that only fin- ishing path motions are described here. Rough cut motion paths and asso- ciated algorithms are techniques that can be adopted from standard CNC literature.