# The Grey Relational Analysis Biology Essay

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Vibration during end milling cause destructive effect, it produces poor surface finish, accelerates tool wear and reduces tool life. This paper presents the use of Taguchi method based grey relational analysis to optimize the machining parameters such as helix angle of cutting tool, cutting speed, feed rate, axial and radial depth of cut for reduced vibration amplitude in end milling operation. L25 Taguchi orthogonal design was employed for conducting the experiments. The experiments were conducted on aluminium Al 6063 by high speed steel end mill cutter and acceleration amplitude was measured using FFT analyzer. The online signals recorded for vibration amplitude picked by accelerometer at two positions, one fixed in the spindle (channel I) and other fixed in the work piece fixture (channel II). The signal-to-noise (S/N) ratio and analysis of variance (ANOVA) were employed to determine the optimum levels. Grey relational grade was used to optimize machining parameters by considering both responses at a time.

## Introduction

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Milling is basic metal cutting process in which the required shapes in metal components are obtained in removing unwanted materials. End milling operation is a type of peripheral milling process is quite complex. Many variables enter in such as tool geometry, tool materials, work piece materials, cutting conditions, etc., and cause the tool to react in a different way. The angle of entry and effective geometry of the tools may change during the machining as cutter tooth constantly changing position in relation to the work piece. It is the purpose to investigate and to reveal the effects of various variables to have better understanding of the end milling operation. Two major problem encountered with the end mill cutters related to rigidity are spring back and chatter. Spring back is caused by insufficient stiffness and excessive spring back of the cutter will results in a scratch mark during tool retraction. Chatter occurs during the feeding and retracting motions. The relative motion between cutting tool and work piece results in a vibration. The frequency of the vibration depends on the natural frequency of the machine tool. Chatter is a resonant vibration when the force acting on the tool - work piece system happens to vibrate same as of natural frequency of machine tool (1). Cutting is also not continuous in end milling, rather is periodically interrupted as cutting edges enter and leave the work piece. This leads to cyclic thermal and mechanical load which leads to fatigue failure.

Several investigations have been carried out to reveal the effect of various parameters in machining performance by vibration analysis. Klaus et al (2) proposed a simulation concept to predict for predicting regenerative work piece vibrations, which combines a finite element model for analyzing the dynamic behavior of the work piece. The author concluded that the dynamic behavior of work piece - tool system influences the quality of work piece surface. Sadettin et al (3) investigated the relationship between tool wear and vibration during end milling, reveals that the tool wear increase when the acceleration amplitude during machining increases. Lacerda and Lima (4) studied the cause of chatter vibrations between the cutter and the work piece, also suggested optimal selection of depth of cut and spindle rotation that result in maximum ship removal rate in milling. Julie and Joseph (5) demonstrated a tool condition monitoring approach in an end milling operation based on the vibration signal microcontroller-based data acquisition inbuilt with signal transducer. Ning et al (6) analyzed the effect of tool edge wear on the vibrations in high-speed turning using fast Fourier transform (FFT) and discrete wavelet transform technique. It is concluded that vibration amplitude increases with the increase in tool edge radius. Arnaud and Daniel (7) studied the problem of vibrations occurring during a machining operation and revealed that the vibrations may cause failures and defects to the process, like work piece surface alteration and rapid tool wear. Mann et al (8) implemented finite element analysis and semi-discretization methods to study the stability in milling process. The samples collected experimentally to access the stability of the system. Engin and Altintas (9) presented a generalized mathematical model of most helical end mills to predict the cutting forces, vibrations, dimensional surface finish and chatter stability lobes for an arbitrary end mill.

The dynamics of the spindle and cutter system determines the quality of the work parts (10). Henri and Gregoire (11) showed the influence of cutting edge defects on the stability of the machining and the raising of chatter phenomenon. Arizmendi (12) presented a model for predicting surface topography that includes the effects of tool vibration. Giuseppe and Nicolo (13) proposed an analytical - experimental model to predict chatter vibration phenomenon. Statistical prediction models for various performances (surface roughness, tool wear and cutting forces) to find the best levels of process parameters have been explored by many researchers. Hasan et al (14) employed Taguchi optimization method for low surface roughness in terms of process parameters such as feed, cutting speed, machining tolerance, axial and radial depth of cut. Eyup and Seref (15) applied Taguchi optimization method for low surface roughness in terms feed, cutting speed and depth of cut. Ghani et al (16) presented Taguchi optimization method, which is applied to optimize the cutting parameters such as speed, feed and depth of cut for low resultant cutting forces and good surface finish. They concluded that Taguchi method is suitable in solving the surface quality in machined surface. Grey relational analysis was introduced by Deng (17) used to measure the degree of relationship between various performance characteristic. This method can be used to solve interrelationship among different responses (18). Tosun (19) employed grey relational analysis to determine optimum parameters for multi performance characteristics in drilling operation. The generalized statistical predictive model by employing response surface methodology to determine acceleration amplitude has been devised in earlier work (20).

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End mill geometry is very complex and little work was pertaining to the influence of geometrical parameters in stability of milling process. The changes in geometrical parameter axial rake angle (helix angle) resulting in changes in machining performance particularly to the vibration amplitude have not been explored much. In this paper the aim is to analyze the influence of helix angle of end mill cutter in acceleration amplitude of vibration. Taguchi based grey relational analysis techniques has been employed to determine the optimum levels of process parameter such as helix angle of cutting tool, cutting speed, feed rate, axial and radial depth of cut for reduced vibration amplitude in end milling operation.

## Vibration amplitude and measurement

In End milling, the tool and work parts move relative to other with a frequency determined by the natural frequency of the machine tool. The constant pounding of frequency on the machine parts will result in increased tool wear and poor surface finish. There are several component; almost all the parts of the machine tool are responsible for the generation of vibration. The process parameters involved in the machining condition also contributes major magnitudes in vibration generation. Altering machine tool is not possible, but predicting the right cutting condition to reduce the vibration is possible by controlling the process parameter of the end milling. Altering the helix angle of the cutting tool will influence the metal cutting processes, especially respond much with vibration, which in turn controlling able to obtain machine surface with required finish and with reduced tool wear. The vibration amplitude is measured by using

twin-channel FFT analyzer (COCO 80), and the acceleration amplitudes are picked at two locations, one in the feed direction on the work piece holder and the other in the axial cutting direction in the spindle. The resulting vibration measurement in terms of displacement, velocity, and acceleration amplitude collected in the form of time waveforms and frequency spectrum. The vibration signals are collected in a time domain graph known as waveform graph. The time wave form is a time domain analysis which uses the history of the signal. The signal is stored in the analyzer, and any non-steady or transient impulses are noted. Discrete damages due to built-up edge formation, resonance condition can be identified from the wave form. The acceleration waveform indicates that pulse occurs periodically with a period of 2 s. The frequency spectrum is a plot of the amplitude of the vibration response versus frequency and can be derived by using digital fast Fourier analysis of the waveform. The peak level is the indication of maximum vibration generated in the milling, and the maximum acceleration amplitude of the milling is noted for our study. In milling, the dominant frequency components in the spectrum graph are around tooth passing frequency (ft) and their harmonics [21]. The vibration resulted by the interaction of the tool and work piece has characteristic frequency with the multiples of tooth passing frequency at 1Ã-, 2Ã-, 3Ã-, etc. The tooth passing frequency (ft) can be calculated from the following equation

Tooth passing frequency (ft) = (n X N)/ 60 (1)

where N=spindle speed (rpm) and n=number of teeth of end mill cutter.

## Grey based Taguchi method for optimization

Taguchi method is an experimental design technique used for engineering analyzes to optimize the levels of process parameters for the required performance characteristic [22, 23]. A large number of experiments have to be carried to study the characteristic influenced by number of parameters. This method reduces the magnitude of experiments by introducing a special design of orthogonal arrays to study the entire parameter space with minimum number of experiments. Thus it reduces time and cost of the experiment. Taguchi uses loss function to determine the performance characteristic deviating from the desired value. The loss function value is transformed into signal-to-noise ratio(S/N). The term signal represents the desirable (mean) values and the term noise represents the undesirable (standard deviation) values for the output characteristic. Three types of S/N ratio are available based on the output characteristic: lower is better, nominal is best (NB), or higher is better. In the present work the objective is to minimize vibration amplitude, hence the lower is better is adapted. The lower to better characteristic S/N ratio can be formulated as

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Examples of our workS/N ratio, Î·= -10 log i2) (2)

Where, n is the number of trial of the experiment and yi is the ith measured value in the trial. In addition to S/N ratio, a statistical technique analysis of variance (ANOVA) can be employed to determine the influence of the process parameters on the performance characteristic. Thus the optimum levels of process parameters can be estimated.

The controllable parameters in end milling operation such as geometrical nomenclature of the tool and cutting conditions found to influence vibration amplitude, which have a significant contribution in determining surface quality and tool life. In this study the influence of machining parameters such as helix angle of cutting tool, cutting speed, feed rate, axial and radial depth of cut (five levels for each parameter) in milling is considered and L25 (5 X 5) orthogonal array was employed for conducting the experiments. The machining parameters and their level are shown in the table 1. The range of the machining parameter is constrained by the limitations of the machine tools and by conducting trial runs.

Table 1. Parameters and levels in milling

## Parameter

## Units

## Factor levels

## 1

## 2

## 3

## 4

## 5

Helix angle (A)

degree ( 0 )

30

35

40

45

50

Cutting speed (B)

m/min

1.25

1.5

1.75

2

2.5

Feed rate (C)

mm/rev

0.02

0.03

0.04

0.05

0.06

Axial depth of cut (D)

mm

1.5

2

2.5

3

3.5

Radial depth of cut (D)

mm

1.5

2

2.5

3

3.5

Grey relational analysis is employed to optimize control parameters having multi-response through grey relational grade [24]. In grey relational analysis the first step is to normalize the S/N ratio calculated from the Taguchi method. This data preprocessing converts the original sequence to a set of comparable sequence. Depending upon performance characteristic different methods are adapted to normalize the raw data and linear normalization of the S/N ratio is performed. The normalized S/N ratio xij for the ith performance characteristic in the jth experiment can be expressed as

(3)

Where Î·ij is the jth experiment result in the ith trial, maxj Î·ij and minj Î·ij are the maximum and minimum value of Î·ij. Then the deviation sequence is calculated from the reference sequence of pre process data and the comparability sequence. The grey relation coefficient is calculated to express the relation between the ideal and normalized S/N ratio. Thus the grey relational coefficient Î³ij for the ith performance characteristic in the jth experiment is calculated using the following equation

(4)

Where (xio - xij) is the deviation sequence and Î¶ is the distinguishing coefficient. The value of Î¶ is chosen as 0.5. A weighting method is applied to integrate the grey relational coefficients of each experiment of different performance characteristic into grey relational grade. The overall evaluation of the multiple performance characteristics is based on the grey relational grade (Î´j),

Î´j = (1/m) iÎ³ij (5)

Where the wi is the weighting factor for the ith performance characteristic, m is the number of performance characteristic and Î´j is the grey relational grade for the jth experiment. The grey relational grade determines the relation between the reference sequence and comparability sequence.

The procedure of grey based Taguchi optimization method is outlined as,

Identifying the performance characteristics (Acceleration amplitude) and machining parameters (range and levels of the parameters)

Conducting experiments by setting appropriate orthogonal array (L25) and response of the performance characteristic is noted.

The mean value of vibration amplitude and S/N ratio is evaluated for the lower to better characteristic and the optimum levels of parameter are determined.

ANOVA is employed to determine the significant parameter that influences performance characteristics.

Normalize the experiment value of the vibration amplitude picked by two channels. Perform the grey relational generating and calculate the grey relational coefficient

Calculate grey relational grade by considering both acceleration amplitudes picked at two positions.

Analyze the experimental result and select the optimum levels of process parameter.

Verifying the optimal cutting parameters through confirmation experiment.

## Experimental set up

Machining vibration occurs due to relative movement between the tool and work piece in the machine tool. The resonant vibration occur when the force acting on the cutter cause to vibrate at a natural frequency of the machine tool. The minimum excitation may cause maximum amplitude which at constant pounding will increase tool wear and results in poor surface finish. The vibration amplitude is measured using twin channel FFT analyzer (COCO 80) and the magnitude of vibration are measured as acceleration amplitude and picked in two location of machine tool, one in the work holder in the feed direction and the other in the spindle in the cutting direction. The experiments were conducted on a HAAS vertical machining center with high speed end mill cutter and work piece as aluminum alloy (Al 6063) under dry condition. The dimension of the work piece specimen used was 32 mm X 32 mm in cross section and 40 mm in length. L25 orthogonal array was employed for conducting the experiments. The orthogonal array contains 25 rows and 5 columns. The experiments are conducted in a completely random manner in order to reduce experimental error. The vibration resulted by the interaction of the tool and work piece are measured as an acceleration amplitude in the feed direction on the work piece holder (channel I) and in the axial cutting direction in the spindle (channel II). The data are acquired in the FFT analyzers and are tabulated in the table (Table 2 & 3).

Table.2. Experimental result for acceleration amplitude for Channel I

## S.No

## Control Parameters

## Acceleration amplitude, mm/s2 for Channel I

## S/N ratio (Î·) , dB

## A

## B

## C

## D

## E

## Trial 1

## Trial 1

## Trial 1

## Average

1

1

1

1

1

1

6654.78

6796.23

6512.44

6654.48

-76.4636

2

1

2

2

2

2

5534.69

5647.03

5768.48

5650.07

-75.0423

3

1

3

3

3

3

5011.66

4797.04

4907.27

4905.32

-73.8147

4

1

4

4

4

4

3362.28

3435.50

3502.28

3433.35

-70.7156

5

1

5

5

5

5

3309.28

3158.49

3231.71

3233.16

-70.1941

6

2

1

2

3

4

3073.56

3151.13

3000.34

3075.00

-69.7587

7

2

2

3

4

5

1264.54

1311.45

1285.56

1287.18

-62.1938

8

2

3

4

5

1

2205.16

2253.05

2294.39

2250.86

-67.0481

9

2

4

5

1

2

1038.77

999.54

1017.43

1018.58

-60.161

10

2

5

1

2

3

4441.26

4529.23

4630.85

4533.78

-73.1305

11

3

1

3

5

2

628.80

609.38

618.17

618.78

-55.8315

12

3

2

4

1

3

1303.92

1335.54

1275.13

1304.86

-62.3129

13

3

3

5

2

4

687.93

675.13

699.55

687.54

-56.7469

14

3

4

1

3

5

2322.29

2363.91

2283.50

2323.23

-67.3227

15

3

5

2

4

1

842.91

861.70

842.91

849.17

-58.5804

16

4

1

4

2

5

898.02

867.60

886.40

884.00

-58.9299

17

4

2

5

3

1

1098.28

1119.91

1079.49

1099.22

-60.8227

18

4

3

1

4

2

680.66

693.45

705.07

693.06

-56.8163

19

4

4

2

5

3

876.29

897.91

863.50

879.23

-58.8832

20

4

5

3

1

4

811.60

824.40

842.02

826.01

-58.3407

21

5

1

5

4

3

2649.90

2559.49

2602.28

2603.89

-68.3133

22

5

2

1

5

4

801.97

787.35

774.55

787.96

-57.9309

23

5

3

2

1

5

2740.37

2847.79

2793.17

2793.78

-68.9249

24

5

4

3

2

1

911.21

883.79

896.59

897.20

-59.0584

25

5

5

4

3

2

1864.50

1879.12

1851.70

1865.10

-65.4142

Table.3. Experimental result for acceleration amplitude for Channel II

## S.No

## Control Parameters

## Acceleration amplitude, mm/s2 for Channel I

## S/N ratio (Î·) , dB

## A

## B

## C

## D

## E

## Trial 1

## Trial 1

## Trial 1

## Average

1

1

1

1

1

1

8960.11

8573.65

8770.88

8768.21

-78.8596

2

1

2

2

2

2

6779.55

6931.79

6651.32

6787.55

-76.6355

3

1

3

3

3

3

5426.96

5545.19

5667.42

5546.52

-74.8818

4

1

4

4

4

4

7067.78

6899.55

7220.01

7062.45

-76.9806

5

1

5

5

5

5

9139.10

9549.56

9347.33

9345.33

-79.4133

6

2

1

2

3

4

5874.05

5751.82

5643.59

5756.49

-75.2043

7

2

2

3

4

5

5560.37

5452.14

5682.61

5565.04

-74.9106

8

2

3

4

5

1

5964.53

5734.06

5842.29

5846.96

-75.3397

9

2

4

5

1

2

7063.33

7218.96

6915.10

7065.79

-76.9846

10

2

5

1

2

3

7703.30

7871.53

8057.16

7877.33

-77.929

11

3

1

3

5

2

5367.52

5483.15

5246.50

5365.72

-74.594

12

3

2

4

1

3

3888.99

3970.24

4057.89

3972.37

-71.9823

13

3

3

5

2

4

4613.02

4417.80

4509.81

4513.55

-73.0917

14

3

4

1

3

5

5627.17

5759.18

5882.39

5756.25

-75.2042

15

3

5

2

4

1

6171.22

6037.66

5891.67

6033.52

-75.613

16

4

1

4

2

5

5154.08

5383.63

5270.07

5269.26

-74.4364

17

4

2

5

3

1

3934.90

4010.67

4098.73

4014.77

-72.0744

18

4

3

1

4

2

3742.31

3578.48

3654.25

3658.35

-71.2672

19

4

4

2

5

3

4007.63

4091.69

3911.86

4003.73

-72.0507

20

4

5

3

1

4

6385.46

6674.89

6531.22

6530.52

-76.3004

21

5

1

5

4

3

5392.80

5268.03

5506.05

5388.96

-74.6315

22

5

2

1

5

4

5610.79

5477.55

5352.78

5480.37

-74.7778

23

5

3

2

1

5

5382.83

5258.06

5516.07

5385.65

-74.6264

24

5

4

3

2

1

4253.24

4119.99

3995.23

4122.82

-72.3067

25

5

5

4

3

2

5317.63

5079.62

5204.39

5200.55

-74.3225

## Results and Discussion

Analysis of S/N ratio

Table 2 and 3 shows the data was observed for three trails during experimentation. The average value of the acceleration amplitude picked at channel I and II are evaluated and noted. The S/N ratios are evaluated using the equation 2 by taking consideration that lower to better characteristic of acceleration amplitude and noted in Table 2 and 3. The vibration amplitude picked at the two channels for each parameter level is calculated by averaging the observed values when the parameter is maintained at that level. The mean acceleration amplitude (channel I and II) response table for each level of process parameters was created in the integrated manner. The acceleration amplitude (channel I and II) are given in the table 4 and 5. The effect table 4 and 5 indicates the mean of the response variable (acceleration amplitudes) means for each level of each control factor. The same procedure is applied for S/N ratio response for each level of process parameter and S/N ratio response table for acceleration amplitude (channel I and II) are given in the table 6 and 7. Table 6 and 7 indicates the mean of the S/N ratio for each level of control parameters.

From table 4 based on the mean value of the acceleration amplitude (Channel I) for each levels, the difference between the maximum and minimum values were calculated. The maximum difference will give the most significant parameters and rank for the significant parameters are depicted. From the table it is inferred that the optimal combination that yield reduced vibration amplitude which had been picked at the work piece holder (channel I) are A4 B4 C3 D5 E4. The rank of the significant parameters are rated as Helix angle (rank 1), feed rate (rank 2), axial depth of cut (rank 3), radial depth of cut (rank 4) and spindle speed (rank 5). From table 5 based on the mean value of the acceleration amplitude (Channel II) for each levels, the difference between the maximum and minimum values were calculated. The maximum difference will give the most significant parameters and rank for the significant parameters are depicted. From the table it is inferred that the optimal combination that yield reduced vibration amplitude which had been picked at the spindle (channel II) are A4 B3 C3 D3 E3. The ranks of the significant parameters are rated as Helix angle (rank 1), spindle speed (rank 2), axial depth of cut (rank 3), radial depth of cut (rank 4) and feed rate (rank 5). The effect of process parameters resulting from the optimization process are plotted in the fig 1 and fig 2.

Table.4. Mean response table for acceleration amplitude for Channel I

## levels

## A

## B

## C

## D

## E

1

4775.277

2767.233

2998.502

2519.543

2350.188

2

2433.083

2025.859

2649.451

2530.517

1969.119

3

1156.718

2266.112

## 1706.899*

2653.579

2845.418

4

## 876.3061*

## 1710.319*

1947.637

1773.331

## 1761.972*

5

1789.585

2261.446

1728.48

## 1553.999*

2104.272

## âˆ†

3898.97

1056.914

1291.604

1099.579

1083.446

## Rank

## 1

## 5

## 2

## 3

## 4

## *Optimum levels

Table.5. Mean response table for acceleration amplitude for Channel II

## levels

## A

## B

## C

## D

## E

1

7502.013

6109.729

6308.102

6344.512

5757.256

2

6422.322

5164.022

5593.387

5714.101

5615.594

3

5128.281

## 4990.205*

## 5426.125*

## 5254.913*

## 5357.782*

4

## 4695.324*

5602.206

5470.317

5541.662

5868.674

5

5115.671

6997.45

6065.68

6008.423

6264.305

## âˆ†

2806.688

2007.245

881.977

1089.598

906.5231

## Rank

## 1

## 2

## 5

## 3

## 4

## *Optimum levels

Table.6. S/N ratio response table for acceleration amplitude for Channel I

## levels

## A

## B

## C

## D

## E

1

-73.2461

-65.8594

-66.3328

-65.2406

-64.3947

2

-66.4584

-63.6605

-66.2379

-64.5816

-62.7531

3

-60.1589

-64.6702

## -61.7778

-67.4266

-67.2909

4

## -58.7586

## -63.2282

-64.8841

-63.3239

## -62.6985

5

-63.9284

-65.132

-63.2476

## -62.9776

-65.5131

## âˆ†

14.48749

2.631217

4.554988

4.449043

4.537863

## Rank

1

5

2

3

4

## *Optimum levels

Table.7. S/N ratio response table for acceleration amplitude for Channel II

## levels

## A

## B

## C

## D

## E

1

-77.3542

-75.5452

-75.6076

-75.7507

-74.8387

2

-76.0737

-74.0761

-74.826

-74.8799

-74.7607

3

-74.097

## -73.8414

## -74.5987

## -74.3264

## -74.2951

4

## -73.2258

-74.7054

-74.6123

-74.6806

-75.271

5

-74.133

-76.7156

-75.2391

-75.2351

-75.7182

## âˆ†

4.128345

2.874277

1.008877

1.42422

1.423102

## Rank

1

2

5

3

4

## *Optimum levels

Fig.1. Effect of process parameters on acceleration amplitude (Channel I & II)

Fig.2. Effect of process parameters on S/N ratio of acceleration amplitude (Channel I & II)

ANOVA

ANOVA is a statistically based, objective decision making tool was employed to examine the influence of process parameters on quality characteristics. It helps is testing the significance of all process parameters by comparing the mean square against an estimate of the experimental error at specific confidence levels. This is done by calculating the variability of the S/N ratios (sum of the squared deviations from the total mean S/N ratio) into contributions by each process parameter and error. The percentage contributions of variance are estimated by the following equations.

The total sum of the squared deviations (SST) from the total mean S/N ratio can be expressed as

SST = 2 (6)

Where 'n' is the number of experiment in the orthogonal array, is the S/N ratio of the ith experiment and is the total mean S/N ratio.

The percentage contribution of variance (Ï) can be calculated as follows

Ï = (SSD/ SST) (7)

Where SSD is the sum of the squares of deviation.

F-test is a statistical tool (the mean square error to residual) in ANOVA used to determine most significant process parameters that influence the quality characteristic. Higher the F-value will be most influential on the response quality characteristic. P-value demonstrate the significance level (significant or non significant) of the process parameter. Table 8 and 9 gives the results of ANOVA for acceleration amplitude (Channel I & II) respectively.

From table 8, it is observed that the most significant parameter that influence acceleration amplitude measured at the work piece (channel I) are of order of helix angle, A (68.58%); feed rate, C (9.68 %); axial depth of cut, D (7.19%); radial depth of cut, E (4.9%) and spindle speed, B (4.24%).

Table.8. ANOVA results for acceleration amplitude (Channel I)

## Parameters

## DF

## SS

## F

## P

## (Ï)%

## Sig

A

4

48476142

12.68

0.015

68.57549

1

B

4

2999104

0.78

0.59

4.242603

5

C

4

6842943

1.79

0.293

9.680187

2

D

4

5081271

1.33

0.395

7.188085

3

E

4

3466337

0.91

0.537

4.903562

4

Error

4

3824394

Total

28

70690191

From table 9, it is observed that the most significant parameter that influence acceleration amplitude that picked at the spindle (channel II) are of order of helix angle, A (53.52%); spindle speed, B (25.92%); axial depth of cut, D (6.98%); radial depth of cut, E (6.08%) and feed rate, C (4.42 %).

Table.9. ANOVA results for acceleration amplitude (Channel II)

## Parameters

## DF

## SS

## F

## P

## (Ï)%

## Sig

A

4

27101166

17.42

0.009

53.51502

1

B

4

13127279

8.44

0.031

25.92164

2

C

4

2239823

1.44

0.366

4.422842

5

D

4

3537247

2.27

0.223

6.984785

3

E

4

3080976

1.98

0.262

6.083815

4

Error

4

1555680

Total

28

50642171

Grey relational analysis

The optimized results of acceleration amplitude picked at two different positions using Taguchi method gives two different combinations. In order to investigate the optimization of machining parameters that takes the accountability of both acceleration amplitude (channel I & II), the analysis of multiple performance characteristics is required. Grey relational analysis is employed to determine the optimal machining parameters by considering acceleration amplitude picked at both positions. The S/N ratio calculated from Taguchi method were normalized by using equation 3 that converts the original sequence to a set of comparable sequence and listed in the table 10. The grey relational coefficient is calculated using equation 4 and corresponding combination grade and rank were manipulated and listed in the table. From the table 10, it is infer that out of the experimental run, exp no 18 is the optimized combination which will result in lesser acceleration amplitude. Their combination is A4 B3 C1 D4 E2.

Table.10. Calculated normalized, grey relational coefficient, grade and rank

## S.No

## Normalized

## Grey coeff

## Grade

## Rank

## Channel I

## Channel II

## Channel I

## Channel II

1.

0.0000

0.07447

0.3333

0.3507

0.342

25

2.

0.0689

0.37377

0.3494

0.444

0.3967

22

3.

0.1284

0.60977

0.3645

0.5617

0.4631

20

4.

0.2786

0.32733

0.4094

0.4264

0.4179

21

5.

0.3039

0.00000

0.418

0.3333

0.3757

24

6.

0.3250

0.56637

0.4255

0.5355

0.4805

19

7.

0.6916

0.60589

0.6185

0.5592

0.5889

12

8.

0.4564

0.54815

0.4791

0.5253

0.5022

18

9.

0.7901

0.32680

0.7044

0.4262

0.5653

14

10.

0.1616

0.19970

0.3736

0.3845

0.379

23

11.

1.0000

0.64850

1

0.5872

0.7936

7

12.

0.6858

0.99996

0.6141

0.9999

0.807

6

13.

0.9556

0.85066

0.9185

0.77

0.8442

3

14.

0.4430

0.56638

0.4731

0.5356

0.5043

17

15.

0.8667

0.51138

0.7896

0.5058

0.6477

10

16.

0.8498

0.66971

0.769

0.6022

0.6856

9

17.

0.7581

0.98756

0.6739

0.9757

0.8248

5

18.

0.9522

1.00000

0.9128

1

0.9564

1

19.

0.8521

0.99075

0.7717

0.9818

0.8768

2

20.

0.8784

0.41887

0.8043

0.4625

0.6334

11

21.

0.3950

0.64345

0.4525

0.5837

0.5181

15

22.

0.8982

0.62376

0.8309

0.5706

0.7007

8

23.

0.3654

0.64414

0.4407

0.5842

0.5124

16

24.

0.8436

0.95630

0.7617

0.9196

0.8407

4

25.

0.5355

0.68504

0.5184

0.6135

0.566

13

The effect of each machining parameters on the grey relational grade at different levels can be independent because the experimental design is orthogonal. From table 11, based on the mean value of the grey relation grade for each level, the difference between the maximum and minimum values was calculated. The maximum difference will give the most significant parameters and rank for the significant parameters are depicted. From the table it is inferred that the optimal combination that yield reduced vibration amplitude for multiple performance (both acceleration amplitude picked at both channels) are A4 B2 C3 D5 E2. The ranks of the significant parameters are rated as Helix angle (rank 1), spindle speed (rank 2), radial depth of cut (rank 3), feed rate (rank 4) and axial depth of cut (rank 5). The effect of process parameters resulting from the optimization process are plotted in the fig 3.

Table.11. Response table for grey relational grade

## levels

## A

## B

## C

## D

## E

1

0.3991

0.564

0.5765

0.572

0.6315

2

0.5032

## 0.6636*

0.5828

0.6292

## 0.6556*

3

0.7194

0.6557

## 0.6639*

0.5677

0.6088

4

## 0.7954*

0.641

0.5957

0.6258

0.6154

5

0.6276

0.5204

0.6256

## 0.6498*

0.5334

## âˆ†

0.3963

0.1432

0.0874

0.0821

0.1222

## Rank

1

2

4

5

3

## *Optimum levels

Fig.3. Effect of process parameters on grey relational grade

Table 12 gives the results of ANOVA for grey relational grade. From table 12, it is observed that the most significant parameter that influence acceleration amplitude are of order of helix angle, A (65.75 %); spindle speed, B (10.3 %); radial depth of cut, E (5.41 %); feed rate, C (3.47 %) and axial depth of cut, D (3.34 %).

Table.12. ANOVA results for grey relational grade

## Parameters

## DF

## SS

## F

## P

## (Ï)%

## Sig

A

4

0.51267

5.61

0.062

65.7472

1

B

4

0.08035

0.88

0.548

10.3045

2

C

4

0.02607

0.30

0.867

3.4793

4

D

4

0.02713

0.29

0.874

3.3433

5

E

4

0.04218

0.46

0.764

5.4094

3

Error

4

0.09136

Total

28

0.77976

Confirmation test

After evaluating the optimal parameter setting, the next step is to evaluate and verify the enhancement of enhancement of quality characteristics using the optimal parametric combination. From the experiments through Taguchi orthogonal array applying grey relational analysis the optimal combination identified as A4 B2 C3 D5 E2. An outcome of ANOVA indicates all the machining parameters are significantly contributing to the response. Hence all the parameters are included in predicting estimated grey relational grade. The estimated Grey relational grade using the optimal level of the design parameters can be calculated as:

Î´' = Î´m + Î´i - Î´m) (8)

Where Î´' is Grey relational grade for predicting the optimal machining parameters,

Î´i is the average Grey relational grade of the optimal level of machining parameters,

Î´m is the average Grey relational grade and

q is the number of machining parameters.

Grey relational grade for predicting optimal machining parameters can be computed as follows

Î´' = Î´m + Î´i - Î´m) = 0.9887

The confirmation is conducted by taking optimal combination of machining parameters A4 B2 C3 D5 E2. Comparison of the acceleration amplitude for the channel I & II between the initial parameters and optimal parameter combination is shown in the table 13. It is found that the optimal parameter combination reduces the acceleration amplitude in the grey relation grade for about 40.1%.

Table.13. Comparison between the initial and optimal parameters

## Initial parameters

## Optimal parameters from orthogonal array

## Optimal machining parameters

## Prediction

## Experiment

## Level

A2 B2 C3 D4 E5

A4 B3 C1 D4 E2

A4 B2 C3 D5 E2

A4 B2 C3 D5 E2

## Acceleration amplitude (Channel I)

1264.54 mm/s2

680.66 mm/s2

629.25 mm/s2

## Acceleration amplitude (Channel II)

5560.37 mm/s2

3658.35 mm/s2

3495.31 mm/s2

## Grey relational grade

0.5889

0.9564

0.9887

0.989

Improvement percentage of grey relational grade = 40.01%

## Conclusion

The vibration amplitude was measured as a performance measure in this study, which when increase indirectly produce poor surface finish and cause rapid tool wear. This study deals with the application grey based Taguchi approach to determine the optimal combination of machining parameters for reduced acceleration amplitude. By the experimental and analytical result, the conclusions that were drawn can be summarized into following points.

The effect of machining parameters on the acceleration picked at two prominent positions i.e., at work piece holder (channel I) and spindle (channel II) was evaluated using Taguchi method. The helix angle was the most significant parameter that influence acceleration amplitude measured at channel I and II. The optimal combinations of machining parameter for reduced amplitude were determined.

The grey relation analysis was done to determine optimal combination of machining parameters for multiple performance characteristics (i, e., considering both responses at a time). By grey based Taguchi technique reveals that the helix angle was the most significant parameter. The optimal combinations of machining parameter for reduced amplitude were determined and it found to be A4 B2 C3 D5 E2.

Confirmation experiments were done to evaluate the enhancement in the performance measure by using grey relation technique. The optimal combination parameter is compared with the initial parameter. Utilization of the optimal combination of machining parameter enhances a significant improvement of grey relation.