Ejemplo de Taguchi en Proceso de Perforado

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Journal of Materials Processing Technology 145 (2004) 84–92

Application of Taguchi method in the optimizationof end milling parameters

J.A. Ghani, I.A. Choudhury∗, H.H. Hassan Department of Engineering Design and Ma nufacture, Faculty of Engineering, Universiti Malaya, 50603 Kuala Lumpur, Malaysia

Received 17 April 2002; received in revised form 17 April 2002; accepted 25 June 2003

Abstract

This paperoutlines the Taguchi optimizationmethodology, whichis applied to optimize cuttingparametersin end millingwhen machining

hardened steel AISI H13 with TiN coated P10 carbide insert tool under semi-finishing and finishing conditions of high speed cutting. Themilling parameters evaluated are cutting speed, feed rate and depth of cut. An orthogonal array, signal-to-noise (S/N) ratio and Pareto

analysis of variance (ANOVA) are employed to analyze the effect of these milling parameters. The analysis of the result shows that the

optimal combination for low resultant cutting force and good surface finish are high cutting speed, low feed rate and low depth of cut.

Using Taguchi method for design of experiment (DOE), other significant effects such as the interaction among milling parameters are also

investigated. The study shows that the Taguchi method is suitable to solve the stated problem with minimum number of trials as compared

with a full factorial design.

© 2003 Elsevier B.V. All rights reserved.

Keywords: Taguchi method of DOE; High speed end milling; Hardened steel AISI H13; TiN coated carbide insert tool

1. Introduction

Robust design is an engineering methodology for obtain-ing product and process conditions, which are minimally

sensitive to the various causes of variation to produce

high-quality products with low development and manufac-

turing costs [1]. Taguchi’s parameter design is an important

tool for robust design. It offers a simple and systematic

approach to optimize design for performance, quality and

cost. Two major tools used in robust design are [1–3]:

• signal to noise ratio, which measures quality with empha-

sis on variation, and

• orthogonal arrays, which accommodate many design fac-

tors simultaneously.

When a critical quality characteristic deviates from the target

value, it causes a loss [2]. Continuously pursuing variability

reduction from the target value in critical quality character-

istics is the key to achieve high quality and reduce cost.

The successful applications of Taguchi methods by both

engineers and statisticians within British industry have lead

to the formation of UK Taguchi Club [4]. Taguchi’s ap-

proach is totally based on statistical design of experiments

[1], and this can economically satisfy the needs of problem

∗ Corresponding author.

solving and product/process design optimization [5]. By ap-

plying this technique one can significantly reduce the time

required for experimental investigation, as it is effective ininvestigating the effects of multiple factors on performance

as well as to study the influence of individual factors to de-

termine which factor has more influence, which less [1,5].

Some of the previous works that used the Taguchi method

as tool for design of experiment in various areas including

metal cutting are listed in Refs. [6–12].

This paper describes a case study on end milling parame-

ters at three levels each. The main objective is to find a com-

bination of milling parameters to achieve low cutting force

and surface roughness. Fig. 1 shows the overall set up of

the experiment, and Fig. 2 shows the geometry of two flutes

end milling cutter assembly and detailed insert dimension.

2. Taguchi method, design of experiment, and

experimental details

2.1. Taguchi method

Taguchi defines the quality of a product, in terms of the

loss imparted by the product to the society from the time

the product is shipped to the customer [13]. Some of these

losses are due to deviation of the product’s functional char-

acteristic from its desired target value, and these are called

0924-0136/$ – see front matter © 2003 Elsevier B.V. All rights reserved.

doi:10.1016/S0924-0136(03)00865-3



J.A. Ghani et al. / Journal of Materials Processing Technology 145 (2004) 84–92 85

Fig. 1. Schematic of the experimental set-up.

losses due to functional variation. The uncontrollable fac-

tors which cause the functional characteristics of a product

to deviate from their target values are called noise factors,

which can be classified as external factors (e.g. temperatures

and human errors), manufacturing imperfections (e.g. unit

to unit variation in product parameters) and product deteri-

oration. The overall aim of quality engineering is to make

products that are robust with respect to all noise factors.

The most important stage in the design of an experiment

lies in the selection of control factors. As many factors as

possible should be included, so that it would be possible toidentify non-significant variables at the earliest opportunity.

Taguchi creates a standard orthogonal array to accommo-

date this requirement. Depending on the number of factors,

Fig. 2. Geometries of two fluted end milling cutter and insert.

interactions and levels needed, the choice is left to the user

to select either the standard or column-merging method or

idle-column method etc.

Taguchi used the signal-to-noise (S/N) ratio as the quality

characteristic of choice [1,13]. S/N ratio is used as a measur-

able value instead of standard deviation due to the fact that

as the mean decreases, the standard deviation also decreases

and vice versa. In other words, the standard deviation cannot

be minimized first and the mean brought to the target.

Taguchi has empirically found that the two stage opti-

mization procedure involving S/N ratios indeed gives theparameter level combination, where the standard deviation

is minimum while keeping the mean on target [13]. This

implies that engineering systems behave in such a way that



86 J.A. Ghani et al. / Journal of Materials Processing Technology 145 (2004) 84–92

the manipulated production factors can be divided into three

categories:

1. Control factors, which affect process variability as mea-

sured by the S/N ratio.

2. Signal factors, which do not influence the S/N ratio or

process mean.

3. Factors, which do not affect the S/N ratio or processmean.

In practice, the target mean value may change during the

process development.

Two of the applications in which the concept of S/N ra-

tio is useful are the improvement of quality through vari-

ability reduction and the improvement of measurement. The

S/N ratio characteristics can be divided into three categories

when the characteristic is continuous:

nominal is the best characteristic : S/N = 10 logy

s2y

smaller the better characteristics :

S/N = −10 log1

n

y2

and

larger the better characteristics :

S/N = −10 log1

n

1

y2

where y is the average of observed data, s2y the variance of

y, n the number of observations, and y the observed data.

For each type of the characteristics, with the above S/Nratio transformation, the higher the S/N ratio the better is

the result.

2.2. Design of experiment

In this experiment with three factors at three levels each,

the fractional factorial design used is a standard L 27 (313)

orthogonal array [1]. This orthogonal array is chosen due to

its capability to check the interactions among factors. Each

row of the matrix represents one trial. However, the sequence

in which these trials are carried out is randomized. The three

levels of each factor are represented by a ‘0’ or a ‘1’ or a

‘2’ in the matrix.

The factors and levels are assigned as in Table 1 accord-

ing to semi-finishing and finishing conditions for the said

material when machining at high cutting speed.

Factors A, B, and C are arranged in columns 2, 5 and 6,

respectively, in the standard L27 (313) orthogonal array as

shown in Appendix A.

2.3. Experimental details

The machining trials were carried out on a Cincinnati Mi-

lacron Sabre 750 VMC in dry condition, as recommended

Table 1

Factors and levels used in the experiment (axial depth of cut is kept

constant at 3 mm)

Factor Level

0 1 2

A—speed (m/min) 224 280 355

B—feed (mm per tooth) 0.1 0.16 0.25C —radial depth of cut (mm) 0.3 0.5 0.8

Table 2

The chemical composition of workmaterial in percentage by weight

C 0.37

Si 0.9

Mn 0.46

P 0.014

S 0.02

Ni 0.11

Cr 5.34

Cu 0.4

Mo 1.25

V 1

by the tool supplier for the specific workmaterial. The insert

used was flat end mill TiN coated carbide. The detail geome-

try of two flutes end milling cutter assembly and detailed in-

sert dimension are shown in Fig. 2. The cutting forces in X , Y ,

and Z directions were measured online during the milling op-

eration using Kistler dynamometer model 9275 A. The sur-

face roughness was measured using surface roughness tester

model Mpi Mahr Perthometer. Table 2 shows the chemical

composition of workmaterial in percentage by weight.

3. Experimental results and data analysis

The objective of experiment is to optimize the milling pa-

rameters to get better (i.e. low value) surface roughness and

resultant force values, the smaller the better characteristics

are used. Table 3 shows the actual data for surface rough-

ness and resultant force along with their computed S/N

ratio. Whereas Tables 4 and 5 show the mean S/N ratio for

each levels of surface roughness and resultant force, respec-

tively. These data were then plotted as shown in Figs. 3 and

4, respectively.

3.1. Conceptual S/N ratio approach

Taguchi recommends analyzing the means and S/N ratio

using conceptual approach that involves graphing the effects

and visually identifying the factors that appear to be sig-

nificant, without using ANOVA, thus making the analysis

simple [3].

The average S/N ratios for smaller the better for surface

roughness and resultant force factors and significant inter-

action are shown in Figs. 3 and 4, respectively. Study of

Fig. 3 suggests that cutting speed (factor A) and interaction




Table 3

Experimental results for surface roughness, resultant force and their corresponding S/N ratio

Experimental

run

Factor Designation Measured parameters Calculated S/N ratio

A B C Surface roughness,

Ra (m)

Resultant

force (N)

S/N ratio for

Surface roughness

S/N ratio for

Resultant force

1 0 0 0 A0 B0C 0 0.207 469 1.368 −5.342

2 0 1 1 A0 B1C 1 0.169 1277 1.544 −6.2123 0 2 2 A0 B2C 2 0.513 2175 0.580 −6.675

4 1 0 0 A1 B0C 0 0.245 503 1.222 −5.403

5 1 1 1 A1 B1C 1 0.252 1167 1.197 −6.134

6 1 2 2 A1 B2C 2 0.27 2144 1.137 −6.662

7 2 0 0 A2 B0C 0 0.531 649 0.550 −5.624

8 2 1 1 A2 B1C 1 0.579 736 0.475 −5.734

9 2 2 2 A2 B2C 2 0.615 1831 0.422 −6.525

10 0 0 1 A0 B0C 1 0.231 1229 1.273 −6.179

11 0 1 2 A0 B1C 2 0.448 1746 0.697 −6.484

12 0 2 0 A0 B2C 0 0.418 1563 0.758 −6.388

13 1 0 1 A1 B0C 1 0.203 1125 1.385 −6.102

14 1 1 2 A1 B1C 2 0.671 1716 0.347 −6.469

15 1 2 0 A1 B2C 0 0.234 1446 1.262 −6.320

16 2 0 1 A2 B0C 1 0.263 1225 1.160 −6.176

17 2 1 2 A2 B1C 2 0.608 1334 0.432 −6.25018 2 2 0 A2 B2C 0 0.657 1208 0.365 −6.164

19 0 0 2 A0 B0C 2 1.045 1085 −0.038 −6.071

20 0 1 0 A0 B1C 0 0.756 949 0.243 −5.955

21 0 2 1 A0 B2C 1 0.872 1736 0.119 −6.479

22 1 0 2 A1 B0C 2 1.424 1203 −0.307 −6.161

23 1 1 0 A1 B1C 0 0.872 1179 0.119 −6.143

24 1 2 1 A1 B2C 1 0.888 1944 0.103 −6.577

25 2 0 2 A2 B0C 2 1.392 1761 −0.287 −6.492

26 2 1 0 A2 B1C 0 1.024 1270 −0.021 −6.208

27 2 2 1 A2 B2C 1 1.202 1826 −0.160 −6.523

between feed rate and depth of cut (interaction B × C) are

more significant. Feed rate (factor B) and depth of cut (fac-

tor C ) are insignificant. The highest cutting speed ( A2) ap-

pears to be the best choice to get low value of surface finish,

and thus making the process robust to the cutting speed in

particular. The feed rate (factor B) and depth of cut (factor

C ) are insignificant on the average S/N response. Since the

interaction B×C is significant, Park [1] has recommended

to use the two ways B × C table to select their levels as

calculated and tabulated in Appendix B(a). From the two

ways B × C table, the optimum combination for factor B

and factor C to get the best result is B0C 1 as explained in

[1]. Therefore, the optimal combination to get low value of

surface roughness is A2 B0C 1 within the tested range.

Table 4

Response table for average S/N ratio for surface roughness factors and

significant interaction

Symbol Cutting

parameter

Mean S/N ratio

Level 0 Level 1 Level 2 Maximum−

minimum

A Cutting speed 0.727 0.718 3.144 2.426

B Feed rate 0.703 0.559 0.605 0.144

C Depth of cut 0.652 0.788 0.331 0.457

B × C Interaction

B × C

0.944 0.853 −0.025 0.969

Study of Fig. 4 suggests that feed rate (factor B) and depth

of cut (factor C ) are more significant, followed by the inter-

action between feed rate and depth of cut (interaction B×C)

on average S/N response for cutting force. Cutting speed

(factor A) is insignificant as the slope gradient is very small.

The same procedure as in the surface roughness response is

used to find the best combination for interaction B × C as

calculated in Appendix B(b).

From the two ways B × C table, it is found that B0C 0is the best combination to lower the cutting force during

machining within the range of experiment. Since cutting

speed (factor A) is insignificant, it could be set at the highest

cutting speed to obtain high rate of metal removal or at the

lowest cutting speed to prolong the tool life depending on

Table 5

Response table for average S/N ratio for resultant force factors and

significant interaction

Symbol Cutting

parameter

Mean S/N ratio

Level 0 Level 1 Level 2 Maximum−

minimum

A Cutting speed −6.198 −6.219 −6.188 −0.0301

B Feed rate −5.950 −6.176 −6.475 −0.525

C Depth of cut −5.949 −6.235 −6.421 −0.472

B × C Interaction

B × C

−6.035 −6.281 −6.29 −0.255




Fig. 3. The smaller the better S/N graph for surface roughness.

application. However, analysis of Tables 6 and 7 suggests

choosing the highest cutting speed ( A2).

3.2. Pareto ANOVA: an alternative analysis

One of the methods to analyze data for process optimiza-

tion is the use of Pareto ANOVA [1]. Pareto ANOVA is a

simplified ANOVA method which uses Pareto principles. It

is a quick and easy method to analyze results of parameter

design. It does not require an ANOVA table and therefore

does not use F -tests.

Following are the Pareto ANOVA table for surface rough-

ness and resultant force analysis, respectively.

The Pareto ANOVA technique of analysis has been per-formed, which requires least knowledge about ANOVA

method and suitable for engineers and industrial practi-

tioners.

The use of S/N ratio for selecting the best levels com-

bination for surface roughness suggests that cutting speed

(factor A) and interaction B × C have strong effect on the

Fig. 4. The smaller the better S/N graph for resultant force.

surface roughness. From the result obtained, the best com-

bination to get low value of surface roughness is at level‘2’ of cutting speed, level ‘0’ of feed rate, and level ‘1’ of

depth of cut. Since the role of depth of cut is minimum in

obtaining good surface finish, it is indicated that in order to

achieve good surface finish, always use high cutting speed

and low feed rate.

On the other hand, the use of S/N ratio for selecting the

best levels combination for resultant force suggests that feed

(factor B), depth of cut (factor C ) and interaction B × C

have strong effect on the resultant force, whereas the effect

of cutting speed can be ignored. From the result, the best

combination to get low value of resultant cutting force is at

level ‘0’ of feed rate, level ‘0’ of depth of cut and level ‘2’ of cutting speed. Since the effect of cutting speed is negligible

in this case, it is suggested to cut at low feed rate and depth

of cut so that the force produced during the cutting process

is kept at minimum value. By increasing the cutting speed,

both the resultant cutting force and surface roughness values

are kept at minimum.




Table 6

Pareto ANOVA analysis for surface roughness

Factor and interaction

B × C A B C B × C A × B A × C A × B A × C

Sum at factor level

0 8.495 6.544 6.325 5.865 4.678 5.991 5.020 4.893 5.621

1 7.678 6.464 5.034 7.096 4.968 5.412 6.105 6.409 5.0072 −0.229 28.296 5.443 2.983 6.298 4.542 4.818 4.642 5.316

Sum of squares of difference (S ) 139.292 949.771 2.613 26.742 4.481 3.192 2.874 5.485 0.566

Contribution ratio (%) 12.272 83.679 0.230 2.356 0.395 0.281 0.253 0.483 0.050

Pareto diagram

83.679

12.272

2.356 0. 395 0. 48 3 0.281 0.253 0.23 0.05

A B × C C B × C A × B A × B B A × C A × C

Cumulative contribution 83.679 95.951 98.702 98.702 99.185 99.466 99.719 99.949 100

Check on significant interaction BC two-way table (Appendix B(a))

Optimum combination of

significant factor level

A2 B0C 1

Remarks The significant factors and interactions are chosen from the left-hand side in the above

Pareto diagram which cumulatively contribute about 90%

Estimate of error variance 0.026

Table 7

Pareto ANOVA analysis for resultant force

Factor and interaction

B × C A B C B × C A × B A × C A × B A × C

Sum at factor

0 −54.313 −55.785 −53.551 −53.548 −55.153 −55.344 −55.410 −55.551 −55.766

1 −56.534 −55.972 −55.589 −56.117 −55.676 −55.529 −56.004 −56.503 −56.159

2 −56.608 −55.696 −58.276 −57.789 −56.626 −56.580 −56.040 −55.400 −55.529

Sum of squares of difference (S ) 10.201 0.119 33.701 27.390 3.345 2.667 0.752 2.149 0.607

Contribution ratio (%) 12.605 0.147 41.642 33.844 4.13268 3.29564 0.929 2.655 0.749

Pareto diagram

41.642

33.844

12.605

4.133 3.296 2.6550.929 0.749 0.147

B C B × C B × C A × B A × B A × C A × C A

Cumulative contribution 41.642 75.486 88.091 92.224 95.519 98.174 99.103 99.852 100.000

Check on significant BC two-way table (Appendix B(b))

Optimum combination of

significant factor

B0C 0 A2

Remarks The significant factors and interactions are chosen from the left-hand side in the above

Pareto diagram which cumulatively contribute about 90%

Estimate of error variance 0.0198




4. Discussion

In the above case study two techniques of data analysis

have been used. Both techniques draw similar conclusions.

The cutting speed has found to be the most significant effect

to produce low value of average surface roughness ( Ra). The

explanation for the influence of cutting speed on surfacefinish is still not available. This could be explained in terms

of the velocity of chips that is faster at high cutting speed

than at low cutting speed. This leads to a shorter time for

the chips to be in contact with the newly formed surface

of workpiece and the tendency for the chips to wrap back

to the new face form is little as compared to low speed.

The condition of seizure and sublayer plastic flow occurred

at high speed and the term flow-zone is used to describe

secondary deformation in this range [16]. The time taken for

the chips at this flow-zone for high speed cutting is short as

compared to lower speed, as the velocity of chip is faster.

Further more, the chip formation process is influenced by

the shear length (ls) in the shear zone. The shear length (ls) isgiven as ls = t/Sin φ, where t is undeformed chip thickness,

and φ is the shear angle [17]. Philip [18] f ound the shear

angle (φ) is large at high cutting speeds, therefore the shear

length (ls) is small, as shown in Fig. 5. Consequently, the

chip will break away with less material deformation at the

immediate tool tip, which in turn preserved the machined

surface properties.

The surface roughness produced in milling operation de-

pends on feed rate [14], and the tool angular position de-

pends on the depth of cut and radius of the cutter [15].

Martelotti [14] describes the chip thickness model as fol-

lows: t = sSin b, where s and b represent feed per toothand tool angular position, respectively. Whereas the height

of the tooth mark is given by the following:

h =s2

8[R+ (sxN /π)]

Fig. 5. Shear length and shear angle in chip formation process.

where h is the height of tooth mark above point of lowest

level, mm; s the feed per tooth, mm; R the radius of cutter,

mm; N the number of teeth in cutter.

The height of tooth mark can be reduced by increasing

the radius of the cutter and by decreasing the feed per tooth

until the tooth mark becomes scarcely distinguishable, par-

ticularly at the lower feed rates.The use of S/N ratio for selecting the best levels of com-

bination for surface roughness ( Ra) value suggests the use of

low value of feed rate in order to obtain good finish. Smaller

angle of tool angular position is obtained at lower depth of

cut [15]. Therefore, it is preferable to set the depth of cut

to a low value. Therefore, one can say that the set values

for level ‘0’ and ‘1’ are both suitable to obtain good quality

of surface finish. From the result, the interaction of factor B

and factor C is more important than the effect of the individ-

ual factors. In other words, in order to get the best result it

requires experience to combine these two factors to achieve

a suitable combination of feed rate and depth of cut.

The S/N ratio suggests that cutting force depends on feedrate and depth of cut. Both the feed rate and depth of cut are

found to be at level ‘0’ for the best combination to obtain

low value of cutting force. The combination of feed rate and

depth of cut determines the undeformed chip section and

hence the amount of energy required to remove a specified

volume of material. The required force to form the chips is

dependent on the shear yield strength of the work material

under cutting conditions and on the area of the chip section

and the shear zone. The feed per tooth and the depth of

cut determine this area. The low value of cutting force is

desired to cut an unsupported beam or thin sections as well

as to preserve material properties against residual stress andchange in micro hardness at the subsurface.

5. Conclusions

From the analysis of result in end milling using conceptual

S/N ratio approach and Pareto ANOVA, the following can

be concluded from the present study:

1. Taguchi’s robust design method is suitable to analyze the

metal cutting problem as described in this paper.

2. Conceptual S/N ratio and Pareto ANOVA approaches for

data analysis draw similar conclusion.

3. In end milling, use of high cutting speed (355 m/min),

low feed rate (0.1 mm per tooth) and low depth of cut

(0.5 mm) are recommended to obtain better surface finish

for the specific test range.

4. Low feed rate (0.1 mm per tooth) and low depth of cut

(0.3 mm) lead to smaller value of resultant cutting force

the specific test range.

5. Generally, the use of high cutting speed, low feed rate

and low depth of cut leads to better surface finish and

low cutting force.




Appendix A

L27 (313) standard orthogonal array table with factors A, B, and C arranged in columns 2, 5 and 6, respectively. The interactions

among factors are indicated as in columns 1, 7, 8, 9, 11 and 12

Experimental run 1 2 3 4 5 6 7 8 9 10 11 12 13

1 0 0 0 0 0 0 0 0 0 0 0 0 02 0 0 0 0 1 1 1 1 1 1 1 1 1

3 0 0 0 0 2 2 2 2 2 2 2 2 2

4 0 1 1 1 0 0 0 1 1 1 2 2 2

5 0 1 1 1 1 1 1 2 2 2 0 0 0

6 0 1 1 1 2 2 2 0 0 0 1 1 1

7 0 2 2 2 0 0 0 2 2 2 1 1 1

8 0 2 2 2 1 1 1 0 0 0 2 2 2

9 0 2 2 2 2 2 2 1 1 1 0 0 0

10 1 0 1 2 0 1 2 0 1 2 0 1 2

11 1 0 1 2 1 2 0 1 2 1 1 2 0

12 1 0 1 2 2 0 1 2 0 0 2 0 1

13 1 1 2 0 0 1 2 1 2 0 2 0 114 1 1 2 0 1 2 0 2 0 1 0 1 2

15 1 1 2 0 2 0 1 0 1 2 1 2 0

16 1 2 0 1 0 1 2 2 0 1 1 2 0

17 1 2 0 1 1 2 0 0 1 2 2 0 1

18 1 2 0 1 2 0 1 1 2 0 0 1 2

19 2 0 2 0 0 2 1 0 2 1 0 2 1

20 2 0 2 0 1 0 2 1 0 2 1 0 2

21 2 0 2 0 2 1 0 2 1 0 2 1 0

22 2 1 0 1 0 2 1 1 0 2 2 1 0

23 2 1 0 1 1 0 2 2 1 0 0 2 1

24 2 1 0 1 2 1 0 0 2 1 1 0 2

25 2 2 1 2 0 2 1 2 1 0 1 0 2

26 2 2 1 2 1 0 2 0 2 1 2 1 0

27 2 2 1 2 2 1 0 1 0 2 0 2 1

B × C A B C B × C A × B A × C A × B A× C

Appendix B

(a) The calculated BC two-way table for surface roughness. From the BC two-way table, B0C 1 is found to be an optimal

condition.

B0 B1 B2 Total

C 0 1.368 + 1.222 + 0.550 = 3.14 0.243 + 0.119 − 0.021 = 0.341 0.758 + 1.262 + 0.365 = 2.385 5.866

C 1 1.273 + 1.385 + 1.160 = 3.818 1.544 + 1.197 + 0.475 = 3.216 0.119 + 0.103 − 0.160 = 0.206 7.24C 2 −0.038 − 0.287 − 0.307 = −0.632 0.432 + 0.347 + 0.697 = 1.476 0.580 + 1.137 + 0.422 = 2.139 2.983

Total 6.326 5.033 4.73 16.089

(b) The calculated BC two way table for resultant force. From the BC two-way table, B0C 0 is found to be an optimal condition.

B0 B1 B2 Total

C 0 −5.342 − 5.403 − 5.624 = −16.369 −5.955 − 6.143 − 6.208 = −18.306 −6.388 − 6.320 − 6.164 = −18.872 −53.547

C 1 −6.179 − 6.102 − 6.176 = −18.457 −6.212 − 6.134 − 5.734 = −18.08 −6.479 − 6.577 − 6.523 = −19.579 −56.116

C 2 −6.071 − 6.161 − 6.492 = −18.724 −6.484 − 6.469 − 6.250 = −19.203 −6.675 − 6.662 − 6.525 = −19.862 −57.789

Total −53.55 −55.589 −58.313 −167.452




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