5.2 Time-Domain Simulation
6.1.1 Comparison between Average and Instantaneous Cutting Force Method . 50
Gradišek et al. investigated the identification of the specific process coefficients with the ACFM for different types of end mills [Gra04]. As a first step, they simulated process forces to evaluate their identification method. The used input data is shown in Fig. 6.1. The theoretical and identified coefficients for all investigated radial immersions ae (full, up- and down-milling: 50%, 25%, and 10%) agrees well. As usual for the ACFM, the authors used the circular path model for the simulation of the process forces and identification of the coefficients. If the process forces are simulated with the developed trochoidal path model (Chapter 4.2), only small differences to the circular path model are apparent, as shown in Fig. 6.1 a) for the feed normal force FfN. The average cutting forcesF¯
fN,circularandF¯
fN,trochoidaldiffer by 7 N only. In this case, the percentage difference
Δ¯FfN(fz) =
F¯
fN,circular(fz)
¯FfN,trochoidal(fz) – 1
·100% (6.1)
equals 4.3%. This is the highest difference for the investigated process parameters in down-milling, as shown in Fig. 6.1 b). With increasing immersion and decreasing feed per tooth the deviations get smaller. The dotted line with the circular markers in Fig. 6.1 b) represents the deviation for 10%-immersion for different feed per tooth values. This low error levels might be considered as not relevant, initially. However, this error levels matter for the coefficient identifi-cation. The identified specific process force coefficients are listed in Table 6.1. If the trochoidal path model is used to simulate the process force data and the circular path model for the identi-fication, the presumed low error levels have an impact. For full immersion milling, the deviations are small. However, in case of low immersion the deviation between the values of the theoretical
IFW
© Gra/72794
a) b)
feed normal force FfN
10%-down milling (fz = 0.2 mm)
feed per tooth fz immersion
radial depth of cut:
axial depth of cut:
process:
feed normal force FfN trochoidal
circular
Fig. 6.1: Comparison between circular and trochoidal flute movement (data from [Gra04]).
Table 6.1: Identified specific process force coefficients by the ACFM. Trochoidal path model was used for the simulated process forces.
immersion Ktc Krc Kac Kte Kre Kae
[N/mm2] [N/mm2] [N/mm2] [N/mm] [N/mm] [N/mm]
full 1, 872.2 503.0 1, 145.0 33.2 61.1 –4.8
50%-down 1, 917.8 498.2 1, 154.2 32.6 60.7 –5.0
50%-up 1, 849.4 548.8 1, 131.3 33.4 60.4 –4.5
25%-down 1, 974.5 522.6 1, 181.7 32.2 60.4 –5.3
25%-up 1, 878.7 569.9 1, 146.8 33.0 60.5 –4.7
10%-down 2, 089.8 577.1 1, 235.5 32.1 60.5 –5.5
10%-up 1, 964.3 621.1 1, 190.9 32.7 60.6 –5.0
and identified coefficients is significant. The identified radial cutting force coefficient Krcfor 10%-up-milling deviates from the actual theoretical value by 21%.
To reduce this systematical error, the developed PSO algorithm (Chapter 4.4) is used to identify the coefficients from the instantaneous cutting forces. Based on the mean value of the simulated forces for fz= 0.16 mm, the following lower and upper boundary values were chosen:
pi,k b=
Numerical heuristic optimization methods tend to fall into a local optimum. Thus, the algorithm was executed 500 times to find the best solution. The results are listed in Table 6.2. As the results
Table 6.2: Identified specific process force coefficients by PSO algorithm based on the ICFM.
Trochoidal path model was used for the simulated process forces (fz= 0.16 mm).
immersion Ktc Krc Kac Kte Kre Kae
[N/mm2] [N/mm2] [N/mm2] [N/mm] [N/mm] [N/mm]
full 1, 843.8 513.1 1, 118.7 33.9 60.8 –4.2
50%-down 1, 844.1 513.0 1, 118.8 33.9 60.8 –4.2
50%-up 1, 844.3 513.0 1, 118.7 33.9 60.8 –4.2
25%-down 1, 844.0 513.0 1, 118.7 33.9 60.8 –4.2
25%-up 1, 844.1 513.0 1, 118.7 33.9 60.8 –4.2
10%-down 1, 843.8 511.9 1, 118.7 33.9 60.8 –4.2
10%-up 1, 843.7 512.4 1, 118.7 33.9 60.8 –4.2
show, there is almost no deviation (highest error:–0.2% for Ktc at 10%-down-milling). Fig. 6.2 shows an example of the progress of the PSO algorithm for 500 executions.
IFW
© Gra/72795
a) b)
process:
feed per tooth: full immersion milling fz = 0.16 mm
radial depth of cut:
axial depth of cut:
swarm iterations Ns
progress of PSO algorithm normal distribution
global swarm fitness Fg,k
best fitness fitness range
worst fitness mean fitness
specific radial cutting coefficient Ktc variance s2
Fig. 6.2: Evaluation and analysis of the PSO results.
After 20 swarm iterations, the global swarm fitness Fg,kconverges to a constant value, as shown in Fig. 6.2 a). For this theoretical identification with no noise or dynamical caused deviations, the best swarm fitness Fg,k reaches a deviation value of almost 0 N. Fig. 6.2 b) shows the resulting value for Ktc if all 500 results are considered. The identified values of Ktc are normally distributed and the resulting mean value μ = 512.7 N/mm2 almost corresponds to the exact reference value. This proves the effectiveness of the applied PSO algorithm. In the following section, the identification analysis is extended by tool runout.
6.1.2 Influence of Tool Runout on Process Force Coefficient Identification
The amplitudes of the process forces for each flute can noticeably vary in case of tool runout. The ACFM does not consider tool runout. An implementation into the analytical expression for the
co-IFW
location angle λr
feed normal force FfN (ρr = 10 µm)
feed per tooth fz runout
location angle λr
a) b)
c) d)
radial depth of cut:
axial depth of cut:
process:
Fig. 6.3: Influence of tool runout on the process forces. The dotted lines and round markers have the same runout error for all four diagrams.
efficient identification would increase the complexity significantly. For the ICFM, the consideration of tool runout errors is possible [Gro15]. For the following investigations, the same data as in the previous section is used (Fig. 6.1). Fig. 6.3 shows the resulting deviations for the mean values of the simulated process forces with and without tool runout. For the upper diagrams a) and b), λr is kept constant andρris varied. Similar to Eq. 6.1, the percentage difference is calculated as follows (Fig. 6.3 a) and b)):
With increasingλr and decreasing fz the differences are very significant. In comparison to the variation ofρr, the variation of the runout location angleλr leads to a smaller influence on the
Table 6.3: Identification with the ACFM with trochoidal flute paths and runout (data from:
[Gra04]).
immersion Ktc Krc Kac Kte Kre Kae
[N/mm2] [N/mm2] [N/mm2] [N/mm] [N/mm] [N/mm]
full 1, 891.6 538.0 1, 139.9 31.4 57.7 –4.1
50%-down 2, 044.7 536.3 1, 149.1 21.8 54.8 –4.3
50%-up 1, 829.1 677.2 1, 131.0 34.3 48.1 –3.8
25%-down 2, 183.4 668.2 1, 171.5 19.6 48.4 –4.3
25%-up 1, 919.5 824.7 1, 145.0 29.7 43.4 –3.7
10%-down 2, 481.6 1, 047.3 1, 202.4 16.3 37.7 –3.7
10%-up 2, 200.1 1, 212.3 1, 172.8 22.0 35.3 –3.3
mean process forces. The resulting percentage error is calculated by (Fig. 6.3 c) and d)):
Δ¯Ff(λr, fz) =
¯F
f λr,ρr = 10µm, fz
¯Ff λr = 0◦,ρr = 0µm, fz – 1
·100% (6.4)
Forλr = 135◦, the force deviation for the investigated process parameters almost disappears.
In such a case, and assuming that most runout errors are not caused by errors of the tool geom-etry shape, the influence of the tool runout on the process forces could be theoretically reduced by changing the orientation between the tool and the tool holder. However, the deviation would increase in case of an end mill with lower helix angle valuesδj. High helix angles lead to a more evenly distributed tool runout error (Fig. 2.10), which reduced the peak force differences between the teeth. For the coefficient identification, a runout offset ofρr = 10µm and runout location angle λr = 30◦ is chosen for the process force simulation. The identified coefficients based on the ACFM are listed in Table 6.3. As the results show, the consideration of runout is more than rec-ommended. For example, without consideration of tool runout, Ktc for 10%-down-milling differs by 637.5 N
mm2.
For the ICFM based on the PSO algorithm, the coefficients are initially determined without con-sidering the runout error to demonstrate the influence of process force deviations. The boundary values from Eq. 6.2 are used. For 10%-down-milling, the best determined global swarm fitness value Fg,kis 67.8N with the following identified coefficients: Ktc = 1, 435.0 N
mm2, Krc = 597.2 N
mm2, Kac = 1043.6 N
mm2, Kte = 54.8 mmN , Kre = 53.7 mmN and Kac = –1.4 mmN . This indicates that the application of heuristic algorithms is susceptible for a wrong coefficient identification if the data of process forces contains errors. Very often experimental data can contain even more errors, caused by tool wear, runout, non-homogeneity of the workpiece, machine state variation, instru-mental error, and human error [Ahn11]. The reason why the result of the best global swarm fitness value Fg,kis misleading lies in the force curve shape of the various coefficients (see [Sel12b, p.
53]). This explains why the identified value for Ktc is too low and the value of Kte is too high, because both coefficients have similar force curve shapes and therefore have a similar influence on the overall process force curve shape. Same applies for Krc and Kre. Thus, to overcome this problem, the mean values for the coefficients are calculated based on the normal distributed 500 results, which yield the following coefficient values: Ktc = 1, 834.6 N
mm2, Krc = 512.9 N
mm2,
Table 6.4: Identification with the ICFM and PSO algorithm with trochoidal flute paths and runout (data from: [Gra04]).
immersion Ktc Krc Kac Kte Kre Kae λr ρr
[N/mm2] [N/mm2] [N/mm2] [N/mm] [N/mm] [N/mm] [◦] [mm]
full 1, 838.7 515.2 1, 115.9 34.4 60.6 –4.0 21.6 10.2
50%-down 1, 798.5 509.7 1, 100.6 37.3 60.9 –3.0 30.5 10.8
50%-up 1, 815.8 505.8 1, 100.0 36.3 61.1 –2.7 14.9 10.7
25%-down 1, 771.4 484.1 1, 078.1 37.3 62.4 –2.2 28.3 10.3
25%-up 1, 832.7 510.1 1, 111.5 34.6 60.9 –3.8 30.2 10.2
10%-down 1, 801.6 499.5 1, 095.2 35.6 61.4 –3.2 32.3 10.3
10%-up 1, 954.3 569.7 1, 187.3 29.4 58.9 –6.7 13.2 9.8
Kac = 1, 056.0 N
mm2, Kte = 30.0 mmN , Kre = 58.8 mmN , and Kac = –1.3 mmN . In case of tool runout, the undeformed chip thickness changes for varying valuesρrandλr. This requires a recalculation of the undeformed chip thickness. Because of the amount of iterations, computational time would increase noticeably. Instead, the following procedure is performed: First, as already described and carried out, the coefficients are estimated without the consideration of tool runout.
Next, the identified coefficients are used for the identification ofρrandλr. This step still remains time consuming. However, if the coefficients are kept constant and only the runout values are varied, the likelihood that the PSO algorithm falls into a local optimum can be reduced. With the identified runout values, the calibration of the coefficients is repeated. The results listed in Table 6.4 show the validity of this procedure. In the next section, this method is applied for the calibration of coefficients and tool runout with experimentally obtained process force data.