Calculation Using Taguchi Method For Hardness

Calculation Using Taguchi Method for Hardness S/N Ratio Table 3.1 S/N Ratio for Hardness Temp T (℃) 210 210 210 230 230 230 250 250 250

Pressur e P (bar) 100 125 150 100 125 150 100 125 150 Total

Speed

Hardness, yi (HRP)

S (RPM)

1

2

3

4

5

200 300 400 300 400 200 400 200 300

54.7 48.6 51.2 23.2 67.6 60.4 40.1 41.3 44.7 431. 8

57 51.8 54.5 29.8 47 52.7 37.8 39 45.9 415. 5

59.6 36.7 49 53.6 55.4 65.9 37.6 39.6 45.5 442. 9

37.3 57 51.9 59.1 66.2 65.9 37.5 40.2 50.1 465. 2

46.3 36.1 45.9 41.2 63.5 47.3 37.2 36.2 44.9 398. 6

Averag e,ӯi 50.98 46.04 50.5 41.38 59.94 58.44 38.04 39.26 46.22

S/N Ratio 33.75 32.82 34.02 30.75 35.31 35.11 31.60 31.85 33.27

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 [1/5 (1/54.72 + 1/57.0² + 1/59.6² + 1/37.3² + 1/46.3²) ]= 33.75 ŋ2 = -10 log [1/5 (1/48.62 + 1/51.8² + 1/36.7² + 1/57.0² + 1/36.1²) ]= 32.82 ŋ3 = -10 log [1/5 (1/51.22 + 1/54.5² + 1/49.0² + 1/51.9² + 1/45.9²) ]= 34.02 ŋ4 = -10 log [1/5 (1/23.22 + 1/29.8² + 1/53.6² + 1/59.1² + 1/41.2²) ]= 30.75 ŋ5 = -10 log [1/5 (1/67.62 + 1/47.0² + 1/55.4² + 1/66.2² + 1/63.5²) ]= 35.10 ŋ6 = -10 log [1/5 (1/60.42 + 1/52.7² + 1/65.9² + 1/65.9² + 1/47.3²) ]= 35.11 ŋ7 = -10 log [1/5 (1/40.12 + 1/37.8² + 1/37.6² + 1/37.5² + 1/37.2²) ]= 31.60 ŋ8 = -10 log [1/5 (1/41.32 + 1/39.0² + 1/39.6² + 1/40.2² + 1/36.2²) ]= 31.85 ŋ9 = -10 log [1/5 (1/44.72 + 1/45.9² + 1/45.5² + 1/50.1² + 1/44.9²) ]= 33.27

Average S/N Ratio

Level 1

Table 3.2 Average S/N Ratio for Each Parameter and Level S/N Ratio Temp,T Pressure,P Speed,S 33.53 32.03 33.57

2 3 Difference Rank

33.72 32.24 1.48 2

33.33 34.14 2.10 1

32.28 33.64 1.29 3

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: (33.75 + 33.82 + 34.02) /3 = 33.53 SNT2: (30.75 + 35.31 + 35.11) /3 = 33.72 SNT3: (31.60 + 31.85 + 33.27) /3 = 32.24 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 = (33.75 + 30.75 + 31.60) /3 = 32.03 SNP2 = (32.82+ 35.31 + 31.85) /3 = 33.33 SNP3 = (34.02 + 35.11 + 33.27) /3 = 34.14

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 = (33.75 + 35.11 + 31.85)/3 = 33.57

SNS2 = (32.82 + 30.75 + 33.27)/3 = 32.28 SNS3 = (34.02 + 35.31 + 31.60)/3 = 33.64 Calculation Steps for S/N Ratio Difference: Difference S/N ratio = Max S/N ratio – Min S/N ratio Difference T = 33.72 – 32.24 = 1.48 Difference P = 34.14 – 32.03 = 2.10 Difference S = 33.64 – 32.28 = 1.36

Mean Hardness

Level 1 2 3 Difference Rank

Table 3.3 Mean Hardness for Each parameter and Level Mean Hardness (MPa) Temp,T Pressure,P Speed,S 49.17 43.47 49.56 53.25 48.41 44.55 41.17 51.72 49.49 12.08 1

8.25 2

5.01 3

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

Formula: Mean Hardness at level j = Total ӯi for parameter of temperature at level j 3 Where j = 1, 2, 3 Mean Hardness at level 1 = (50.98 + 46.04 +50.50) /3 = 49.17 Mean Hardness at level 2 = (41.38 + 59.94 + 58.44) /3 = 53.25 Mean Hardness at level 3 = (38.04 + 39.26 + 46.22) /3 = 41.17

2. For parameter of Pressure, P

Formula: Mean Yield Strength at level j = Total ӯi for parameter of pressure at level j 3 Where j = 1, 2, 3 Mean Hardness at level 1 = (50.98 + 41.38 + 38.04) /3 = 43.47

Mean Hardness at level 2 = (46.04 + 59.94 + 39.26) /3 = 48.41 Mean Hardness at level 3 = (50.50 + 58.44 + 46.22) /3 = 51.72 3. For parameter of speed, S

Formula: Mean Yield Strength at level j = Total ӯi for parameter of speed at level j 3 Where j = 1, 2, 3 Mean Hardness at level 1 = (50.98 + 58.44 + 39.26) /3 = 49.56 Mean Hardness at level 2 = (46.04 + 41.38 + 46.22) /3 = 44.55 Mean Hardness at level 3 = (50.50 + 59.94 + 38.04) /3 = 49.49

Calculation Steps for Mean Result Difference: Difference observed values = Max observed values – Min observed values Difference P = 53.25 – 41.17 = 12.08 Difference T = 51.72 – 43.47 = 8.25 Difference S = 49.56 – 44.55 = 5.01

Predicted Optimum Performance for Hardness Table 3.4 Optimum Parameter Level and Results Temperature(℃)

Pressure(bar)

Speed(RPM)

Optimum Level 2 3 53.25 51.72 Average Hardness (MPa) Predicted Optimum Performance: 58.72HRP

3 49.49

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 = (431.8 + 415.5 + 442.9 + 465.2 + 398.6) / 45 = 2154/45 = 47.87 Yopt = 47.87 + (53.25 – 47.87 ) + (51.72 – 47.87) + (49.49 -47.87) = 58.72 HRP

Anova (Hardness)

Table 4.5 Calculation results from Anova Parameter Pressure,P Temperature,T Speed,S Error Total

Degree of Freedom, f 2 2 2 38 44

Factor Sum of Squares, SS 1132.864 517.605 248.037 2825.993 4724.500

Pure Sum of Squares, SS' 984.128 368.869 99.301 3272.203 　

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

= 45 – 1 = 44 3. DOF of error, fe = Total DOFs – Total of all parameter DOFs

fe = 44 – (2 + 2 + 2) = 38 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 = 2154 N= Total number of observed values C.F = 2154²/ 45 = 103104.8

Contribution, ρ (%) 20.83 7.81 2.10 69.26 100.00

Total yi T1

T2

T3

737.6

798.8

617.6

P1

P2

P3

652.0

726.2

775.8

S1

S2

S3

743.4

668.2

742.4

Table 4.6 Total of yi for each parameter and level 1. SST = (737.6²/15 + 798.8²/15 + 617.6²/15) – 103104.8 = 1132.864 2. SSP= (652.0²/15 + 726.2²/15 + 775.8²/15) – 103104.8 = 517.605 3. SSS = (743.4²/15 + 668.2²/15 + 742.4²/15) – 103104.8 = 248.037

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 = (54.7² + 57.0² + 59.6² + 37.3² + 46.3² +……+ 44.9²) - 103104.8 = 4724.5 Calculation Steps for Sum of Squares of error, SSe Formula: SSe = SST – SSpressure – SStemp – 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 = 4724.5 - 1132.864 - 517.605 - 248.037 = 2825.993

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. SST = 1132.864 – (2825.993/38 * 2) = 984.128 2. SSP = 517.605 – (2825.993/38 * 2) = 368.869 3. SSS = 248.037 – (2825.993/38 * 2) = 99.301

SS’e = SSe + ( SSe/ fe * fT ) Where fT = Total degree of freedom of all the parameters SS’e = 2825.993 + [2825.993/38 * (2 + 2 + 2) ] = 3272.203

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. ΡTemperature = [ 984.128/4724.5] * 100 = 20.83 % 2. ΡPressure = [ 368.869/4724.5] * 100 = 7.81 % 3. ΡSpeed

= [ 99.301/4724.5] * 100 = 2.10 %

4. Ρerror

= [ 3272.203/4724.5] * 100 = 69.26 %