Groove Milling

Synonyms

Slot milling; Slotting

Definition

Groove milling or slot milling is a machining operation where a milling cutter removes material on 180° of its circumference. The direction of feed is generally perpendicular to the rotational axis of the milling cutter. If the face of the tool has cutting edges along the whole radius (normally one cutting edge crosses the rotational axis of the cutter), the tool can be used for drilling operations and it is possible to machine pockets.

Theory and Application

Application

Groove milling is necessary for various applications. For a keyed joint, for example, it is necessary to produce keyseats, which are slots or pockets. Cooling fins may also be produced by groove milling.

In some cases the resulting dimensions and forms are very important; in other cases the resulting surface finish is more important. For economic reasons the results do not have to be as good as possible, but only as good as necessary. For achieving this, it is essential to know the machining result in advance, depending on the boundary conditions.

Real Shape of a Milled Slot

A slot ideally has a rectangular cross section with even surfaces. A milled slot will have surfaces which are not flat (Fig. 1) (Schröder 1974; Hann 1983; Gey 2002), and scallops will be visible from the single cutting edges of the tool.

Fig. 1

Shape of an ideal slot and a milled slot (exaggerated)

Figure 2 shows three different types of carbide cutters. The third one has chip-breaking grooves and is used for rough machining operations. The first and the second tools can be used for fine machining.

Fig. 2

Three carbide cutters: with three teeth, with two teeth, and with three teeth and chip-breaking grooves

Measurements of milled grooves using a coordinate measuring machine clearly show the form deviations of the slot flanks. Figure 3 shows the exaggerated flank forms of slots milled with a two-teeth cutter and with a three-teeth cutter.

Fig. 3

Left: milled with two-teeth cutter. Right: milled with three-teeth cutter

Looking at Figs. 2 and 4, it can be clearly seen that the tools do not have a rectangular outline. The slot flanks are machined only by short sections of the cutting edges at a time. In the case of a three-teeth cutter, these short sections are at different heights. This is the reason why a slot milled with a three-teeth cutter is not symmetric.

Fig. 4

Explanation for flank form deviations

In the places where the tool is in contact with the material, there is a reaction force acting on the tool. The force causes bending of the tool, and the bending causes a deviation of the flank surfaces (see Fig. 4).

Having a closer look at the slot flanks (Fig. 5), the scallops of the single teeth are clearly visible. These slots have been machined in AlCuMg1 F40 (Klobasa 2007).

Fig. 5

Scallops on the tooth flanks

The next chapters will deal with the mathematics behind these phenomena.

Trajectory of a Single Point of the Cutting Edge

The milling cutter has a rotational movement around its own axis and a translational movement in feed direction. The trajectory of each point of the tool can be described by a trochoid. A trochoid is defined in a way that a circle is rolling on a line; and each point that is fixed relatively to the circle will move on a trochoidal curve (Bronstein et al. 1995). Point A in Fig. 6 moves on its trochoid to point A′, and point B moves on its shifted trochoid to point B′.

Fig. 6

Cutting-edge points move on trochoidal paths

The formulas for describing a trochoid are

$$ {\displaystyle \begin{array}{l}\mathrm{x}\left(\upvarphi \right)=\mathrm{a}\cdot \upvarphi +\mathrm{r}\cdot \sin \left(\upvarphi \right)\\ {}\mathrm{y}\left(\upvarphi \right)=\mathrm{r}\cdot \cos \left(\upvarphi \right)\end{array}} $$

(1)

with:

• φ: angular position

• a: radius of the rolling circle

• r: radius of the tool

Now the open question is: How long is the radius of the circle, named a? The answer can be found in Fig. 6: The distance between the tool in any position and after one whole revolution is equal to the circumference of the rolling circle, which is 2 · π · a. On the other hand it is equal to the feed, which is the feed per tooth times the number of teeth (f_z · z). Combining these two formulas, a is given if the number of teeth of the cutter is known and if the feed per tooth is known:

$$ \mathrm{a}=\left({\mathrm{f}}_{\mathrm{z}}\cdot \mathrm{z}\right)/\left(2\cdot \pi \right) $$

(2)

Furthermore there is a relation between geometry and speeds: The quotient of the tool radius r and the rolling circle radius a is equal to the quotient of the cutting speed v_c and the feed velocity v_f (n is the rotational speed) (Tönshoff and Denkena 2004):

$$ \mathrm{r}/\mathrm{a}=\left(2\cdot \pi \cdot \mathrm{r}\cdot \mathrm{n}\right)/\left({\mathrm{f}}_{\mathrm{z}}\cdot \mathrm{z}\cdot \mathrm{n}\right)={\mathrm{v}}_{\mathrm{c}}/{\mathrm{v}}_{\mathrm{f}} $$

(3)

In Fig. 7 photographs of the ground of a milled slot and calculated trochoids are laid on top of each other and the coherence is visible.

Fig. 7

Comparison of a photograph of a milled slot ground and calculated trochoids

The material to be removed by one cutting edge is the area between two adjacent trochoidal trajectories. In Fig. 7 the trochoids rather look like circles, but if the cutting edge enters the material on one flank, it exits the material on the other flank at a different x-position (see Fig. 6).

For a better understanding of the shapes of the material to be removed, Fig. 8 shows the dependency of this shape on the feed per tooth and on the number of cutting edges. The distance d is the difference in x-positions.

Fig. 8

Dependency of d on the number of cutting edges and on the feed per tooth

Geometrical Description of the Flank Shape (Without Cutting Forces)

The examination of the trochoids at the flank sides helps in understanding the surface finish of the slot flanks (Fig. 9). The maximum scallop height is different for both sides, the upcut milling side and the synchronous milling side. The formulas for the trochoids are known. To be able to calculate the scallop height, the y values of the points P1, P2, P3, and P4 must be known. Then the maximum scallop height on the upcut milling side is

Fig. 9

Scallop height calculation

$$ {\mathrm{R}}_{\mathrm{Y}-\mathrm{upcut}}=\mathrm{y}\left(\mathrm{P}2\right)-\mathrm{y}\left(\mathrm{P}1\right) $$

(4)

and the maximum scallop height on the synchronous milling side is

$$ {\mathrm{R}}_{\mathrm{Y}-\mathrm{synchronous}}=\mathrm{y}\left(\mathrm{P}4\right)-\mathrm{y}\left(\mathrm{P}3\right) $$

(5)

The variables in the trochoid formula for y are y, φ, and r. The y value is wanted. The radius of the tool is given. The variable φ is known for P2 and for P3:

$$ \upvarphi \left(\mathrm{P}2\right)=0 $$

(6)

and

$$ \upvarphi \left(\mathrm{P}3\right)=\uppi $$

(7)

The φ values for P1 and P4 are not known but can be calculated using the trochoid formula for x. The rolling circle radius a can be calculated from the feed per tooth and the number of teeth. The tool radius r is known. The x value is zero for φ = 0 (at P2):

$$ \mathrm{x}\left(\mathrm{P}2\right)=0 $$

(8)

The x value for P1 is −f_z/2:

$$ \mathrm{x}\left(\mathrm{P}1\right)=-{\mathrm{f}}_{\mathrm{z}}/2 $$

(9)

The x value for P4 is x(P1) + d with d = (f_z · z)/ 2 (see Fig. 8):

$$ {\displaystyle \begin{array}{c}\mathrm{x}\left(\mathrm{P}4\right)=-{\mathrm{f}}_{\mathrm{z}}/2\kern0.5em +\left({\mathrm{f}}_{\mathrm{z}}\cdot \mathrm{z}\right)/2\\ {}={\mathrm{f}}_{\mathrm{z}}/2\cdot \left(\mathrm{z}-1\right)\end{array}} $$

(10)

With the known x values, φ can be calculated for P1 and P4 and hence the y values can be calculated.

To get an idea about what values will be achieved for realistic examples, in Fig. 10 two trochoids are shown where the feed and the tool diameter are to scale and slots have been milled with these values (Klobasa 2007).

Fig. 10

Values of machined trochoidal paths

Now the theoretic profile of the slot in a plane is known (without considering forces). As the milling cutter is rotating and moving forward, the points P1, P2, P3, and P4 are moving upward and forward. So the scallops are tilted by the angle α (Fig. 11).

Fig. 11

Evolution of angled scallops

The next task is to calculate the scallop angle α. Figure 12 leads to the formula which is needed to calculate alpha. On the left-hand side, there is the triangle with the two legs of the right-angled triangle (feed per tooth f_z and pitch h_z) and the wanted angle α. The height h_z can be calculated, as shown in Fig. 12 on the right-hand side, from the two legs of the triangle h_z · z and 2 · π · r and the helix angle λ of the tool. So the scallop angle depends on the feed per tooth, the number of teeth, the helix angle of the teeth, and the tool radius:

Fig. 12

Evolution of the formula to calculate the scallop angle alpha

$$ \upalpha =\arctan \left(\left({\mathrm{f}}_{\mathrm{z}}\cdot \mathrm{z}\cdot \tan \left(\uplambda \right)\right)/\left(2\cdot \uppi \cdot \mathrm{r}\right)\right) $$

(11)

Figure 13 helps in understanding the motion of the cutting edges.

Fig. 13

Rotating tool

Forces on the Tool

After the investigation of the pure geometric boundaries when milling a slot, it is necessary to analyze the forces on the tool, which lead to a tool deflection. The force depends on the amount of material which has to be removed by the teeth. Again the first analysis is being done in a plane (Fig. 14). The effective direction of motion of the tip of the cutting edge is tangential to the trochoid, and the effective cutting depth is perpendicular to the effective direction of motion. The effective cutting depth can be calculated for each point by finding the intersection point between the line along the effective cutting depth and the previous trochoid.

Fig. 14

Calculation of effective cutting depth

The extension of the 2D examination to the 3D milling cutter is shown in Fig. 15. Due to the helix angle, each point of the cutting edge is at a different rotational position on the tool and on a different height. So for an analytical calculation, the cutter has to be divided into height increments (e.g., Kline et al. 1982), and for each increment the effective cutting depth and thus the resulting force increment has to be calculated.

Fig. 15

Cutting depths at different locations on the tool

The forces can be calculated using the following formula, which is adapted to this milling process from the original (Altintas 2000):

$$ {\mathrm{F}}_{\mathrm{i}}=\mathrm{b}\cdot \left({\mathrm{K}}_{\mathrm{i}\mathrm{c}}\cdot \mathrm{h}\left(\upvarphi \right)+{\mathrm{K}}_{\mathrm{i}\mathrm{e}}\right) $$

(12)

• with i = e (effective), eN (normal to effective), a (axial)

• b = incremental height

• h = effective cutting depth

• K_ic and K_ie = constants, calculated from tests

• φ = rotational position of the cutter

The forces then have to be converted into an X- and Y-coordinate system.

The angle between the effective force direction and the tangent to the tool (cutting speed direction) is η, which is illustrated in Fig. 16. The lengths of the arrows in Fig. 16 representing the forces are not proportional to their values. They are drawn to clearly visualize the angle η. The formula to obtain η is

Fig. 16

Effective cutting force

$$ {\displaystyle \begin{array}{l}\upeta =\arctan \left(\sin \upvarphi /\left(\mathrm{r}/\mathrm{a}+\cos \upvarphi \right)\right)\\ {}\mathrm{or}\\ {}\upeta =\arctan \left(\sin \upvarphi /\left({\mathrm{v}}_{\mathrm{c}}/{\mathrm{v}}_{\mathrm{f}}+\cos \upvarphi \right)\right)\end{array}} $$

(13)

As shown in Fig. 17, v_c is tangential to the tool, v_f is in feed direction, and v_e is in the resulting effective direction.

Fig. 17

Cutting speed v_c, feed v_f, and effective cutting speed v_e

The resulting forces are shown exemplarily as measurements during slot milling using a two-teeth cutter (Fig. 18). On the left-hand side, the forces are plotted along the rotational position of the cutter. On the right-hand side, the forces are plotted in a 3D force system. There are several thin lines, which are bundled in two curves. The two dotted vertical lines in the left diagram correspond to the two arrows in the right diagram. The repeatability of the measured forces per revolution is high, but the two edges of the cutter produce slightly different forces. The reason may be the detailed micro-geometry of the cutting edges, which changes with time and wear and is never exactly the same on all cutting edges, or oscillations of the tool.

Fig. 18

Forces on milling cutter with two teeth

Bending of the Tool Due to Cutting Forces

After the examination of the occurring forces on the tool, the reaction of the tool has to be analyzed. The tool is not a simple cylinder, but a complex piece of metal with different moments of inertia on different heights and in different directions (Fig. 19).

Fig. 19

Neckings on different tools, depending on the rotational position

Simplifying, the area moment of inertia for the two-teeth cutter, which is displayed, e.g., in Fig. 16, can be calculated analytically by taking two half circles and using basic formulas as given in, e.g. (Beitz et al. 1995), using the parallel axis theorem and coordinate transformations for rotations.

The different bending lines of a two-teeth cutter are illustrated in Fig. 20.

Fig. 20

Different bending lines, depending on the rotational position

The deflections of the tool (bending and torsion) at the corresponding positions on the cutting edges can then be added to the theoretical slot geometry, and thus, it is possible to predict the slot surfaces. The whole procedure is explained in Klobasa (2007) in more detail.

In this method, possible vibrations are not considered.

Cross-References

• Cutting Fluid

• Cutting Force Modeling

• Cutting, Fundamentals