As mentioned in Chapter 2.3.1, the reduced uncut chip thickness h caused by a radial offsetΔR can be corrected by adjusting the tooth pitch p. For the sake of simplicity, a circular flute trajectory is assumed to calculate the necessary tooth pitch for an equally distributed load:
p = 2π Nt
1 – ΔR fz
(6.6)
From this equation it follows that a flute with a radial offset always has a smaller tooth pitch than the preceding flute with no offset (ΔR = 0µm). Conversely, this equation can be solved forΔR to determine the necessary radial offset to reach a balanced load for a given non-equal tooth pitch. This leads to an equal effective feed per tooth fj,u for all flutes (Eq. 4.7). In case of a four fluted end mill with a non-equal tooth pitch of p = [80◦, 100◦], a radial offset ofΔR = 13.3µm leads to a balanced load for all flutes. However, if the trochoidal trajectory of the flutes is considered, ΔR needs to be slightly reduced to reach a balanced load, as shown in Fig. 6.9.
For the sake of clarity, a helix angle of δ = 0◦ is chosen. The top left diagram shows the individual feed forces Ff,j of all j-flutes in case of no radial offset. The unequal tooth pitch leads to alternating undeformed chip thickness values and maximum values of Ff,j. In comparison, the top right diagram represents the forces in case of a radial offset ofΔR = 10µm. The gray lines correspond to the curves in the left diagram (ΔR = 0µm). Undeformed chip thicknesses and maximum values of Ff,j are almost equal for all flutes. However, entry and exit position of all flutes change due toΔR, which is represented by the markers. The detailed view A shows how the point of entry of flute z1 occurs earlier (square shaped marker). Simultaneously, the entry of flute z4 (ΔR = 10µm) is delayed (round shaped marker). The dotted trajectory shows the the position of flute z4 in case of no radial offset (z4∗). The exit situation is shown in the detailed view B. z4 is earlier out of cut (triangle shaped marker) while the immersion range of z1 increases (rhombus shaped marker). The position of the markers for all flutes are also included in the top right diagram. The shaded areas show the gaps which result from the shifted entry and exit position of the flutes. Accordingly, the course of the feed force changes (middle diagram).
However, the most noteworthy difference is shown in the lower diagrams. In case of the tool with unequal tooth pitches and no radial offset, two different time delays occur during one revolution (left diagram). As it can be seen, the radial offset leads to additional time delays. Nearφ ≈90◦, the earlier exit of z4 and delayed entry of z2 cause a gap. At this position, only the time delay of z1 is present. Same applies at φ ≈ 270◦. At the entry and exit position of z1 near φ ≈ 0◦ andφ ≈ 180◦, an additional time delay occurs (see detailed view C). Same applies for flute z3. At these positions the recessed flutes are not in cut. Therefore, the additional time delay corresponds to the tooth pitch p1,3= p3,1= 180◦ of the diametrical arranged flutes z1 and z3. The mentioned gaps combined with the additional time delays contribute to a disturbance of the regenerative effect. In analogy to end mills with unequal tooth pitches, this may induce additional
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radial depth of cut:
axial depth of cut:
spindle speed:
slot milling D
p d feed force Fftime delay θjfeed force Ff,j
without radial offset (∆R = 0 µm) with radial offset (∆R = 10 µm)
immersion angle f(t) immersion angle f(t)
change of entry and exit
change of time delay
fz
Fig. 6.9: Force and time delay variation in case of flutes withΔR > 0µm.
stable positions. By increasing the radial offset, the duration of these additional time delays can be expanded as shown in Fig. 6.10 a). The undeformed chip thickness area A2 of the recessed flute z2 decreases with increasingΔR. As a consequence, the flute z3 removes at its entry and exit position larger portions of material left behind by flute z1.
For the given tool from Fig. 6.10, the possible immersion rate of a tooth pitch pj,u in
de-IFW
radial depth of cut:
axial depth of cut:
process:
Fig. 6.10: Immersion conditions in dependance on the radial offset valueΔR.
pendance ofΔR can be determined by the following formula (Eq. 4.4):
rim,j,u =
As it can be seen, the immersion rate for the tooth pitch p3,1 increases with higher values for ΔR. However, this also results in a higher load for the flutes z3 and z1, as shown in Fig 6.10 b).
A balanced load based on the maximum value of the individual feed forces Ff,j is reached near ΔR = 10.7µm. It must be noted, however, that for the new tool concept it can be expected that the forces of the recessed flutes are higher due to induced damping forces of the chamfers. For ΔR = 110µm, the flutes with no offset reach their maximum value. At higher values the recessed flutes are in cut outside the range of the highest undeformed chip thickness of the flutes with no offset. Finally, atΔR = 133.5µm the recessed flutes have no engagement into the material.
The corresponding active tooth pitches are shown in the lower diagram. The immersion rate indicates how the radial offset affects the engagement length of a tooth pitch in comparison to the total engagement length in case of no radial offset. An increase ofΔR causes a decrease of the immersion rate of p2,1. Same applies for p3,2. However, the total immersion length for the regular flute z3does not decrease due to the increasing immersion range of p3,1. As already mentioned, this shift of the time delay makes a disturbance of the regenerative effect more likely.
The influence of the radial offsetΔR on process stability is to be determined with the aid of realistic and general input parameters. Therefore, an experimental modal analysis was carried out on a
IFW
© Gra/72797
hydraulic expansion chuck:
Schunk Tendo E tool dummy:
material: 90% WC, 10% Co density: 14.95 g/cm2 length: l2 = 92 mm diameter: D = 20 mm
machine tool:
Heller MCi16
triaxial accelerometer:
PCB T356A16
resolution: 0.001 m/s2 weight: 7.4 g
coherence
measured
fitted FRF of the tool in x-direction (feed forward)
magnitude |Hxx,w(w)| 10-6 coherence C(f)
z x y L = 51 mm
vf
frequency f
0 0.5 1 1.5 2 2.5 3 3.5 4 kHz 50
0.25 m/N
0.1 0.05 0.15
0
1
-0.4 0.6
0.2
Fig. 6.11: Experimental setup with dummy tool and obtained FRF.
Heller MCi16 machine tool. The experimental configuration is shown in Fig. 6.11. In order to obtain frequency responses which are independent of the tool geometry, e.g. the helix angle or the number of flutes, a dummy tool is used. The excitation of the dummy tool is performed at the uppermost exposed point with an impact hammer (PCB Piezotronics 086C03). The response is measured with a triaxial accelerometer. The experimental FRF in x-direction is shown in the lower part of Fig. 6.11. Furthermore, the spectral coherence is also given in the diagram to assure the causality between input (hammer impact) and output (tool deflection). The most dominant modes were selected for a fit of the experimental FRF, represented by the dashed line, using an adapted version of the presented PSO algorithm in Chapter 4.4. The resulting modal parameters are given in Table 6.5. It is assumed, for the sake of simplicity, that the compliance in x- and y-direction is similar. Thus, the modal parameters in y-direction have the same values. Furthermore, it is assumed that the workpiece has a high stiffness. Thus, in comparison to the dummy tool, its dynamical behavior is neglected. Stability charts for different values for the radial offsetΔR are shown in Fig. 6.12. For the calculation of the stability, the tool is modeled with equally distributed flutes and with a corner radius R = 0. Two diametrical arranged flutes have a radial offset with the same value. In order to underline the effect ofΔR on stability, a feed per tooth fzof 0.3 mm is selected. Thus, a noticeable influence ofΔR on time delays can be achieved. This is particularly necessary because, as it can be seen in Fig. 6.12, there is no noticeable differences in the stability limit between a reference tool (ΔR = 0µm, solid black line) and a tool with a low radial
Table 6.5: Modal parameters for the dummy tool in x-direction.
mode i f.,.i m.,.i k.,.i ζ.,.i
[–] [Hz] [kg] [N/µm] [–]
ft,xi 1 105.4 111.55 48.93 0.1349
2 468.86 3.74 32.48 0.0843
3 1218.46 0.82 47.92 0.0429
4 2364.13 0.28 62.41 0.0339
5 3357.20 0.34 151.64 0.0275
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© Gra/72844
axial depth of cut ap maximum forces max(F)
discretization:
tan. cutting force coeff.:
rad. cutting force coeff.:
tan. edge coeff.:
rad. edge coeff.:
tool diameter:
helix angle:
number of teeth:
radial depth of cut:
feed per tooth:
Fig. 6.12: Stability increase due to radial offsetΔR. However, absolute maximum values of process forces in feed and feed normal direction increase as well.
offset (ΔR = 10µm, dotted line). With increasing values for the radial offset, the gain in the stability limit also increases. ForΔR = fz, the recessed flutes are almost out of cut. Thus, the increase is stability is not assigned toΔR, but rather the reduced number of teeth contributing to the material removal. The resulting stability limit corresponds to the stability limit of a two fluted cutter. In this respect, these results are consistent with the knowledge that the asymptotic stability limit increases with decreasing tooth number Nt [Sil05, p. 1544 f.]. However, it should be noted that in this case, the actual tooth feed rate is doubled. This has an evident influence on the process forces, as shown in right diagram in Fig. 6.12. At the highest value forΔR, the maximum value for the feed force Ff is more than tripled. This means that it can be assumed that severe forced vibrations would occur in experimental investigations. Forced vibrations can be investigated with time-domain simulations, as it will be shown in the next section for tools with runout.