Calculation Using Taguchi Method For Tensile Strength

Calculation Using Taguchi Method for Tensile Strength S/N Ratio Temp T (℃) 210 210 210 230 230 230 250 250 250

Table 3.1 S/N Ratio for Tensile Strength Pressure Speed Yield Strength, yi (Mpa) P (bar) S (RPM) Average,ӯi 1 2 3 100 200 27.57 32.70 30.50 19.50 125 300 33.40 28.20 32.30 31.30 150 400 31.50 33.60 35.40 33.50 100 300 36.00 30.80 35.40 34.07 125 400 32.00 27.80 34.50 31.43 150 200 31.80 30.00 27.50 29.77 100 400 14.00 23.00 25.00 20.67 125 200 22.00 22.70 19.60 21.43 150 300 20.80 26.80 22.00 23.20 Total 254.20 253.40 251.20

Calculation Steps for S/N ratio: Formula: ŋi = -10 log [ 1/n i=1n(1/yi²)] where ŋi = S/N ratio for ith experiment yi= Observed value for ith experiment n = Total number of observed value for ith experiment ŋ1 = -10 log [1/3 (1/32.702+1/30.50² +1/19.50²)] = 28.11 ŋ2 = -10 log [1/3 (1/33.402+1/28.202+1/32.302) ] = 29.84 ŋ3 = -10 log [1/3 (1/31.502+1/33.602+1/35.402) ] = 30.47 ŋ4 = -10 log [1/3 (1/36.002+1/30.802+1/35.402) ] = 30.58 ŋ5 = -10 log [1/3 (1/32.002+1/27.802+1/34.502) ] = 29.84 ŋ6 = -10 log [1/3 (1/31.802+1/30.002+1/27.502) ] = 29.43 ŋ7 = -10 log [1/3 (1/14.002+1/23.002+1/25.002) ] = 25.43 ŋ8 = -10 log [1/3 (1/22.002+1/22.702+1/19.602) ] = 26.57 ŋ9 = -10 log [1/3 (1/20.802+1/26.802+1/22.002) ] = 27.16

Average S/N Ratio

Level 1 2

Table 3.2 Average S/N Ratio for Each Parameter and Level S/N Ratio Temp,T Pressure,P Speed,S 29.47 28.04 28.03 29.95 28.75 29.19

S/N Ratio 28.11 29.84 30.47 30.58 29.84 29.43 25.43 26.57 27.16

3 Difference Rank

26.39 3.56 1

29.02 0.98 3

28.58 1.16 2

Calculation Steps for Average S/N Ratio: 1. For parameter of Temperature, T

Formula: SNTj = Total signal-to-noise ratio, ŋi for parameter of temperature at level j 3 Where j = 1, 2, 3 SNT1= (28.11+29.84+30.47) /3 = 29.47 SNT2= (30.58+29.84+29.43) /3 = 29.95 SNT3=(25.43+26.57+27.16) /3 = 26.39 2. For parameter of Pressure, P

Formula: SNPj = Total signal-to-noise ratio, ŋi for parameter of pressure at level j 3 Where j = 1, 2, 3 SNP1 = (28.11+30.58+25.43) /3 = 28.04 SNP2 = (29.84+29.84+26.57) /3 = 28.75 SNP3 = (30.47+29.43+27.16) /3 = 29.02

3. For parameter of Speed, S Formula: SNSj = Total signal-to-noise ratio, ŋi for parameter of speed at level j 3 Where j = 1, 2, 3 SNS1 = (28.11+29.43+26.57)/3 = 28.03 SNS2 = (29.84+30.58+27.16)/3 = 29.19

SNS3 = (30.47+29.84+25.43)/3 = 28.58 Calculation Steps for S/N Ratio Difference: Difference S/N ratio = Max S/N ratio – Min S/N ratio Difference T = 29.95-26.39 = 3.56 Difference P =29.02-28.04 = 0.98 Difference S = 29.19-28.03 = 1.16

Mean Tensile Strength Table 3.3 Mean Tensile Strength for Each parameter and Level Mean Tensile Strength (MPa) Level Temp,T Pressure,P Speed,S 1 30.79 27.43 26.26 2 31.76 28.06 29.52 3 21.77 28.82 28.53 Difference Rank

9.99 1

1.39 3

3.27 2

Calculation Steps for Mean Modulus: 1. For parameter of Temperature, T

Formula: Mean Tensile Strength at level j = Total ӯi for parameter of temperature at level j 3 Where j = 1, 2, 3 Mean Tensile Strength at level 1 = (27.57+31.30+33.50) /3 = 30.79 Mean Tensile Strength at level 2 = (34.07+31.43+29.77) /3 = 31.76 Mean Tensile Strength at level 3 = (20.67+21.43+23.20) /3 = 21.77 2. For parameter of Pressure, P

Formula: Mean Tensile Strength at level j = Total ӯi for parameter of pressure at level j 3 Where j = 1, 2, 3 Mean Tensile Strength at level 1 = (27.57+34.07+20.67)/3 = 27.43 Mean Tensile Strength at level 2 = (31.30+31.43+21.43) /3 = 28.06 Mean Tensile Strength at level 3 = (33.50+29.77+23.20) /3 = 28.82

3. For parameter of speed, S

Formula: Mean Tensile Strength at level j = Total ӯi for parameter of speed at level j 3 Where j = 1, 2, 3 Mean Tensile Strength at level 1 = (27.57+29.77+21.43) /3 = 26.26 Mean Tensile Strength at level 2 = (31.30+34.07+23.20) /3 = 29.52 Mean Tensile Strength at level 3 = (33.50+31.43+20.67) /3 = 28.53 Calculation Steps for Mean Result Difference: Difference observed values = Max observed values – Min observed values Difference T = 31.76-21.77= 9.99 Difference P =28.82-27.43 = 1.39 Difference S = 29.52-26.26 = 3.27

Predicted Optimum Performance for Tensile Strength Table 3.4 Optimum Parameter Level and Results Temperature(

Pressure(bar)

Speed(RPM)

2

3

2

30.79

28.82

29.52

℃) Optimum Level Average Tensile Strength (MPa)

Predicted Optimum Performance: 32.93 MPa Calculation Steps for Predicted Optimum performance: Formula: Yopt = T/N + (Popt –T/N) + ( Topt – T/N) +( Sopt – T/N) Where Yopt = Predicted optimum performance T = Total of observed values for all the experiment N = Total number of observed values Popt , Topt , Sopt = Average yield strength resulted from pressure, temperature and speed at optimum level

*Average hardness for each optimum parameter level is got from table 3.3 T/N = (254.20+253.40+251.20) /27 = 758.8/27 = 28.10 Yopt = 28.1+ (30.79–28.1) + (28.82–28.1) + (29.52-28.1) = 32.93 MPa

Anova (Tensile Strength) Table 3.5 Calculation results from Anova

Parameter Temperature,T Pressure,P Speed,S Error Total

Degree of Freedom, f

Factor Sum of Squares, SS

Pure Sum of Squares, SS'

Contribution, ρ (%)

2 2 2 20 26

507.34 8.80 50.60 337.22 903.96

437.62 -24.92 16.88 438.36 　

48.41 2.76 1.87 48.49 100.00

Calcalation Steps for Degree of Freedom (DOF) 1.

DOF of a parameter = number of level of the parameter – 1 DOF for temperature = 3 – 1 = 2 DOF for pressure = 3 – 1 = 2 DOF for speed = 3 – 1 = 2

2. Total DOFs = Total number of observed values (yi) – 1

= 27 – 1 = 26 3. DOF of error, fe = Total DOFs – Total of all parameter DOFs fe = 26– (2 + 2 + 2) = 20 Calculation Steps for Factor Sum of Squares, SS Formula: SSp = P12/NP1 + P22/NP2 + P32/NP3 – C.F Where P1 = Total of yi that includes level 1 of parameter P NP1 = Total number of yi in which level 1 of parameter P is present C.F = correction factor = T²/N T= Total of observed values for all the experiment = 758.8 N= Total number of observed values C.F = 758.8²/ 27 = 21325

Table 3.6 Total of yi for each parameter and level Total yi

T1

T2

T3

277.10

285.80

195.00

P1

P2

P3

246.90

252.50

259.40

S1

S2

S3

236.30

265.70

256.80

1. SST = (277.10²/9 + 285.80²/9 + 195.00²/9) –21325 = 507.34 2. SSP = (246.90²/9+ 252.50²/9+259.40²/9) –21325 = 8.80 3. SSS = (236.30²/9+ 265.70²/9+ 256.80²/9) –21325 = 50.60

Calculation Steps for Total Sum of Squares, SST Formula: SST =i=1Nyi2 – C.F Where yi= Observed values C.F = correction factor N= Total number of observed values SST = (32.7² +30.5²+……+22²) - 21325= 903.96 Calculation Steps for Sum of Squares of error, SSe Formula: SSe = SST – SStemp – SSpressure– SSspeed Where SST = Total sum of squares of the variance SSpressure = Total sum of squares due to parameter of pressure SStemp = Total sum of squares due to parameter of temperature SSspeed = Total sum of squares due to parameter of speed SSe = 903.96 - 507.34 - 8.80 - 50.60 = 337.22

Calculation steps for Pure Sum of Squares SS’P = SSP – ( SSe/ fe * fP ) Where SSP = Sum of squares of factor P

SSe = Sum of squares of error fP = Degree of freedom for parameter P fe = Degree of freedom of error 1. SS’T = 507.34 – (337.2/20 * 2) = 437.62 2. SS’P = 8.80 – (337.2/20 * 2) = -24.92 3. SS’S = 50.60 – (337.2/20 * 2) = 16.88

SS’e = SSe + ( SSe/ fe * fT ) Where fT = Total degree of freedom of all the parameters SS’e = 337.2 + [337.2/20 * (2 + 2 + 2) ] = 438.36

Calculation Steps for Percent of Contribution, ρ Formula: ρ = [SS’P / SST ] x 100 % Where SST = Total sum of squares of the variance SS’P = Pure sum of squares for factor P 1. Ρ Temperature = [437.62/903.96] * 100 = 48.41 % 2. ΡPressure = [-24.92/903.96] * 100 = 2.76% 3. ΡSpeed

= [16.88/903.96] * 100 = 1.87 %

4. Ρerror

= [438.36/903.96] * 100 = 48.49 %