Design of Experiments DOE DOE- 1
Design of Experiments DOE 1. (Design of Experiments) (Designed Experiments), 1-1 1-1, 1-1 Input ; Output ; Controllable input factors X1,X2,,Xp; Uncontrollable input factors Z1,Z2,,Zq DOE- 2
, ( ), (, ) X1,X2,,Xp, ; Z1,Z2,,Zq,, : (1) ; (2) (On target) (3) ( ) (4),,, ( ) (Robust),, :, (1) ; (2) ; (3) (KPC) DOE- 3
,, Design and analysis of experiments (Douglas C.Montgomery) 538, ( ) Factorial experiments, Fractional Factorial, and Taguchi Method, 2. (1),? (2)?,,,, ( ),, DOE- 4
,,,,,, :,? 2-1 2-1 DOE- 5
?, (T); (C); (O), T C O, ( ),,,,,, T C O : T 1525 F C 0.6% O 95 F???,,,,,. DOE- 6
3., (One factor at a time), T, 1450 F 1600 F, C O C=0.5%,O=70 F, 4 8 3-1,1600 F, 1450 5%, C=0.5%, O=70 F, C=0.7% O=120 F 1450 F 1600 F?, 8 ( 4 ),, ( ), 8, DOE- 7
3-1 T, C O (One factor at a time) DOE- 8
, 24,, ( ), Ronald A.Fisher, 20,, (Factorial Design) (, ), 8 8 4. Factorial Design, Fisher 4-1 ( ), ; T( ) C( ) O( ),, 8, 2 3,, ( ) DOE- 9
4-1 DOE- 10
4-1 (Main effect) 4-1-1 T 4-2, T 1450 F 1600 F, 4, (, C O) T 4 4-2,,T T=1600 F ( ) Th= 79+ 75+ 90+87 /4 4-1 T=1450 F ( ) Tl= 67+ 61+ 59+ 52 /4,T Tm= Th- Tl 79+75+90+87-67-61-59-52 /4 4-2 4-3, X Xm, Xh Xl Xm= Xh- Xl 4-4 DOE- 11
4-2 T 1450 F 1600 F, 4, (, C O) DOE- 12
4-1-2 C, C O 4-3. 4-3 C O Carbon effect, Oil temperature effect DOE- 13
: Cm= Ch- Cl= 61+ 52+ 87+ 75 /4-67+ 59+ 90+ 79 /4= = 68.75-73.75= -5.0 4-5 Om= Oh- Ol= 59+ 52+ 87+ 90 /4-67+ 61+ 75+ 79 /4= = 72-70.5= +1.5 4-6 4-2 (Interaction effects) :(1) O=70 F, T ;(2) O=120 F, T 4-4 4-5, O, T ( ), T, A B,A B,,B A, T O (1) O=120 F, T 90 59 87 52 /2= 31+ 35 /2= 33 4-7 (2) O=70 F, T 79 67 75 61 /2= 12+ 14 /2= 13 4-8 T O 33 13 /2= 10 4-9 DOE- 14
4-4 T O DOE- 15
4-5 T O T O 90 59 87 52 /2 79 67 75 61 /2 /2= 90+ 87+ 67+ 61-59- 52-79- 75 /4 4-10 T C C O 4-6 DOE- 16
4-6 T O T C C O DOE- 17
4-3 T O T O 90 59 87 52 /2 79 67 75 61 /2 /2 4-11 4-4, (1) C=0.7%, T O 87 52 /2 75 61 /2 17.5-7.0= 10.5 4-12 (2) C=0.5%, T O 90 59 /2 79 67 /2= 15.5-6.0= 9.5 4-13 T,O,C T O C= 10.5-9.5 /2= 0.5 4-14 T O C= 87 52 /2 75 61 /2 90 59 /2 79 67 /2 /2= 87+61+59+79-52-75-90-67 /4 4-15 DOE- 18
4 4 4-7,, + y1,y2,,y8 4-7 DOE- 19
(1) T Tm= Th- Tl y2+y4+y6+y8 /40- y1+y3+y5+y7 /4= = -y1+y2-y3+y4-y5+y6-y7+y8 /4= = - + - + - + - + /4 4-16 4-7, 4-7 T, Cm= - - + + - - + + /4 4-17 Om= - - - - + + + + /4 4-18 (2) ( 4-8):, T O, T O= 90+87+67+61-59-52-79-75 /4= = y6+y8+y1+y3-y5-y7-y2-y4 /4= = y1-y2+y3-y4-y5+y6-y7+y8 /4= = + - + - - + - + /4 4-19 DOE- 20
4-8,, T O ( ) T O, T C= + - - + + - - + /4 C O= + + - - - - + + /4 4-20 4-21 RUN Y T C O TC TO CO TCO Yi 1 Y1 2 Y2 3 Y3 4 Y4 5 Y5 6 Y6 7 Y7 8 Y8 4-8 T O C, T O C= 87+61+59+79-52-75-90-67 /4= = y8+y3+y5+y2-y7-y4-y6-y1 /4= = -y1+y2+y3-y4+y5-y6-y7+y8 /4= = - + + - + - - + /4 4-22 DOE- 21
4-8, T O C ( ) T O C 4-8, +, ya ya= + + + + + + + + /8 4-23 4-5 :? y= a0+ a1 XT+ a2 XC+ a3 XO+ a4 XT XC+ a5 XT XO+ a6 XC XO+ a7 XT XC XO 4-24, y( ) T,C,O, XT= T- T + + T - /2 / T + - T - /2 = = 2T- T + + T - / T + - T - 4-25 XC= C- C + + C - /2 / C + - C - /2 = = 2C- C + + C - / C + - C - 4-26 XO= O- O + + O - /2 / O + - O - /2 = = 2O- O + + O - / O + - O - 4-27 DOE- 22
y a0,a1,,a7? a0= ya a1= Tm/2 a2= Cm/2 a3= Om/2 a4= T C/2 a5= T O/2 a6= C O/2 a7= T O C/2, a1,,a7, T,C,O T C, T O, C O, T O C ( ) (-1) + (+1), 2, a1,,a7 2,, a0, ya 4-6 ( ) 4, 2 4 ( ) 4-9 5,2 5 ( ) 4-10,, DOE- 23
4-9 4, 2 4 ( ) DOE- 24
4-10 5, 2 5 ( ) DOE- 25
4-7 1., (24:8) 2 3., 4-8 4-8-1 1:, (Ni) (Mn) ( : :(1) N (2) M : Y(1b/in 2 ), N- 0 N + 3 M- 1 M + 2 DOE- 26
Mean N M NM 1 35 2 46 3 42 4 40 2 2, 2 2 =4, ( 4-1), ( ),,,, 4-11 DOE- 27
4-11 4-8-2 2:,, (Brainstorming), : (1) M (2) D (3) A( Ni ) ( )?,?, 2 DOE- 28
:(1) M (2) D (3) A :, M- low M + high D- low D + high A- 0.5% Ni A + 0.8% Ni 4-12.,,, ( ) ( 4-13) DOE- 29
4-12 DOE- 30
4-13.,, ( ) 4 14 DOE- 31
4-14 5. (Blocking) (Randomization),, 5-1 (Blocking) : 5-1 2 3, 2 3 =8, 4, 4, 4 DOE- 32
,, 5-1 2 3 5-2, 2, :,, T O C D 5 3, D, D, TCO D, DOE- 33
5-2 T O C 5 1 D 5 2 DOE- 34
RUN T C O TC TO CO TCO D Yi 1 1 Y1 2 2 Y2 3 2 Y3 4 1 Y4 5 2 Y5 6 1 Y6 7 1 Y7 8 2 Y8 5 3 (D,D TCO, TCO D. Fisher,,,,,, : 8, DOE- 35
5-2 (Randomization),,, : (1) 4, 1,2,3,4 (2) (3), 3, 3 6.,, (Blocking) (Randomization),,,,,,, DOE- 36
,? 6-1 6-1 DOE- 37
6-1 (a) DOE- 38
6-1 (b) DOE- 39
6-1 (c) DOE- 40
? (1) (2),,,,, ( ), 6-2 2 4, P : (1) 6 2,, 15, A B C D AB AC AD BC BD CD ABC ABD BCD CDA ABCD (2), i (3) P=100(i-0.5)/15,i (4) :, 6-3 DOE- 41
6-2 2 4 DOE- 42
(5) ( ) :,, ( 6-3), ( ),,,, : ya= 72.25 Am= -8.0 Bm= 24.0 Dm= -5.5 BD= 4.5 (b) : y= ya+ Am/2 XA+ Bm/2 XB+ Dm/2 XD+ BD/2 XB XD = 72.25+ -8.0/2 XA+ -24.0/2 XB+ -5.5/2 XD+ 4.5/2 XB XD = yr- y yr 6-4 6-5 DOE- 43
6-3 DOE- 44
6-4 DOE- 45
6-5 DOE- 46
7. 2, 2 k, k,2 2 2 k,, k,, k=3 2 3 =8 k=4 2 4 =16 k=5 2 5 =32 k=6 2 6 =64 k=7 2 7 =128,,,,,, (Dr.Taguchi), Taguchi 7-1 (Orthogonal arrays) Taguchi, : L8, L12, L16, L32,, L8, : DOE- 47
L Latin square 8. 7 1 7 2 7 3 L8, L12, L16 7 4, 1 2,. 7 1 L8 DOE- 48
7 2 L12 DOE- 49
7 3 L16 DOE- 50
7 4 7-2 7-2-1 L8 2 3 DOE- 51
7 1 2 3 A B C AB AC BC ABC Yi Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 7 1 1, c=a b=b a=c, 7 1 7 2 2 7 2 7 3 3 7 3 7 4 4 7 4 1 2 a=1 b=2 ab=3 c=4 ac=5 bc=6 abc=7 7 5 L8 DOE- 52
7 2 2 3 A B C AB AC BC ABC Yi c b a bc ac ab abc Yi Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 DOE- 53
7 3 2 3 a b ab c ac bc abc Yi Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 DOE- 54
7 4 2 3 a b ab c ac bc abc Yi Y8 Y7 Y6 Y5 Y4 Y3 Y2 Y1 DOE- 55
7 5 2 3 L8 a b ab c ac bc abc Yi 1 1 1 1 1 1 1 Y8 1 1 1 2 2 2 2 Y7 1 2 2 1 1 2 2 Y6 1 2 2 2 2 1 1 Y5 2 1 2 1 2 1 2 Y4 2 1 2 2 1 2 1 Y3 2 2 1 1 2 2 1 Y2 2 2 1 2 1 1 2 Y1,, L8 2 3 ( 2 3 ), 1 +, 2, ( ), ( ). 2 3 ( ), 2, + 1., 1 M1 1+1+1+1+2+2+2+2 /4 = /4 = = +y8+y7+y6+y5-y4-y3-y2-y1 /4 = = -y1-y2-y3-y4+y5+y6+y7+y8 /4= Cm 7-1 DOE- 56
4 M4 = +1+2+1+2+1+2+1+2 /4 = /4 = = +y8-y7+y6-y5+y4-y3+y2-y1 /4 = = -y1+y2-y3+y4-y5+y6-y7+y8 /4= Am 7-1, L8, a b ab c ac bc abc 1 2 3 4 5 6 7,.,,. L8 ( 7 5 (1) 3 (A B C), L8 2 3, L8 3, 2 3, L8 7?, 1 2 4 (2) 4 (A B C D), 3, 1,2,4 4,? 1,2,4,7 7, A,B,C,D 1,2,4,7, L8,D ABC, L8 7 D, ABC,,D ABC,, L8 D 7, ABC,D DOE- 57
7 5, ( 3,5,6),,,A,B,C (3) 5,6,7 4 1,2,4,7, 3,5,6,, 7 5 7-2-2 L16 L16, 2 4, 1 +, 2, ( ), ( ) 2 4, 2, + 1, L16, a b ab c ac bc abc d ad bd abd cd acd bcd abcd 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15, L16 4,5,6,7,8,9,10,11,12,13,14,15,, L16 ( 7 6): (1) 4 L16 2 4, L16 4, 2 4 4 1,2,4,8 DOE- 58
7 5 L8 DOE- 59
7 6 L16 DOE- 60
(2) 5 (A,B,C,D,E) 5 1,2,4,8,15 L16, 15, 7 6 : 7 DE+ABC DE 11 CE+ABD CE 13 BE+ACD BE 14 AE+BCD AE 15 E+ABCD E, E DE,CE,BE,AE, 1,2,3,4,5,6,8,9,10,12, (3) 6-8 1,3,5,7,9,11,13,15,, 7 6, (4) 9-15 DOE- 61
7-2-3 L12 L12,, 11, 7 2, 7 7 7 7 DOE- 62
7-3, Blocking, Randomization 7-4 7-4-1 1: 7 8 7 9, 7 A=Lifter/Bore Clearance / B=Head Gasket Orifice C=Oil level in Pan D=Bearing Clearances E=Oil Pump Gear Clearance F=Regulator(psi) ( / ) G=Oil pan volume 2 y 1 -gallery oil pressure y2-oil pressure in head DOE- 63
7 8(A) DOE- 64
A=Lifter/Bore Clearance / B=Head Gasket Orifice C=Oil level in Pan D=Bearing Clearances E=Oil Pump Gear Clearance F=Regulator(psi) ( / ) G=Oil pan volume Filter,MAIN BRGS,ROD BRGS CAM BRGS, y 1 -gallery oil pressure y2-oil pressure in head 7 8 B 7 10 L8, L8 7,, (, + 1, 2 ): (1) Y1( ) Am= 309+345+377+270-271-276-294-172 /4= = 309+345+377+270 )/4 - ( 271+276+294+172 /4= = 325.25 253.25 DOE- 65
A 1 325.25 A 2 253.25 Min. Max. 7 9 ( ) DOE- 66
Bm= 309+345-377-270+271+276-294-172 /4= = 309+345+271+276 )/4 - ( 377+270+294+172 /4= = 300.25 278.25 B 1 300.25 B 2 278.25 (2) Y2( ) Fm= 232-254-356+271+151-153-262+164 /4= = 232+271+151+164 )/4 - ( 254+356+153+262 /4= = 204.5 256.25 F 1 204.5 F 2 256.25 Gm= 232-254-356+271-151+153+262-164 /4= = 232+271+153+262 )/4 - ( 254+356+151+164 /4= = 229.5 231.25 G 1 229.5 G 2 231.25 7 11 DOE- 67
7 10 L8 DOE- 68
7 11 DOE- 69
? : (1) Y1 Y2, 7 4, A( / ),B( ),D( ),F( ) (2) ( A D),,, (3) ( F) 75psi 65 psi (4) DOE- 70
7-4-2 2: Taguchi Taguchi : Taguchi : (1) (Control factors)=,,, Design factor, Nominal value (2) (Noise factors)=,,,,, Taguchi, Robust Taguchi 7 12 Y ( ) Response =y=temperature of seal( ) A=Primary Ring width(mm) B=Water Pressure (psi) DOE- 71
C=Face Runout(inch) D=Mating Ring Finish(µ inch) E=Primary Ring Finish(µ inch) F=Mating Ring Flatness(µ inch) G=Primary ring Flatness(µ inch) H=Spring Load (lbf) 8,? L16, 7 7 L16 8 8 L16 7 6, 8 B 7 13 L16, : Am= 2+6+9.5+1.7+6.4+8.9+4.6+4-20.1-15.9-2.7-0.6-6.6-0.6-1.7-3.9)/8= E2= = 5.3875 6.5125 2+6+9.5+1.7-6.4-8.9-4.6-4+20.1+15.9+2.7+0.6-6.6-0.6-1.7-3.9 /8= Bm= = 7.3125 4.5875 2+6+9.5+1.7-6.4-8.9-4.6-4-20.1-15.9-2.7-0.6+6.6+0.6+1.7+3.9 /8= E4= = 4 7.9 2+6-9.5-1.7+6.4+8.9-4.6-4+20.1+15.9-2.7-0.6+6.6+0.6-1.7-3.9 /8= = 8.3125 3.5875 DOE- 72
7 12 DOE- 73
7 13 L16 7-14?,, 4 E4 7 6 E4= AC+BD+EG+FH AC EG FH E4 BD DOE- 74
B D, B, 7-15 B D B,, D=30 2, D=6(1)? 7-14 DOE- 75
7-15 B D DOE- 76
B D B D [ 1 1 y1+ 1 1 y2+ 1 2 y3+ 1 2 y4+ + 2 2 y5+ 2 2 y6+ 2 1 y7+ 2 1 y8+ + 2 2 y9+ 2 2 y10+ 2 1 y11+ 2 1 y12+ + 1 1 y13+ 1 1 y14+ 1 2 y15+ 1 2 y16] /8= [ 1 1 y1+ 1 1 y2+ 2 1 y7+ 2 1 y8+ + 2 1 y11+ 2 1 y12+ 1 1 y13+ 1 1 y14] /8+ +[ 1 2 y3+ 1 2 y4+ 2 2 y5+ 2 2 y6+ + 2 2 y9+ 2 2 y10 + 1 2 y15+ 1 2 y16] /8= {[ 1 1 y1+ 1 1 y2+ 1 1 y13+ 1 1 y14]/8+ +[ 2 1 y7+ 2 1 y8+ 2 1 y11+ 2 1 y12] /8}+ +{[ 1 2 y3+ 1 2 y4+ 1 2 y15+ 1 2 y16]/8+ +[ 2 2 y5+ 2 2 y6+ 2 2 y9+ 2 2 y10] /8} : (1) D=1, B=1, Ra= y1+y2+y13+y14 /4= 2+6+6.6+0.6 /4= 3.8 (2) D=1, B=2, Ra= y7+y8+y11+y12 /4= 4.6+4+2.7+0.6 /4= 2.975 (3) D=2 B=1, DOE- 77
Ra= y3+y4+y15+y16 /4= 9.5+1.7+1.7+3.9 /4= 4.2 (4) D=2 B=2, Ra= y5+y6+y9+y10 /4= 6.4+8.9+20.1+15.9 /4= 12.825 7-15, D=6 1,B, B, Taguchi, D=6 1? : (1) 8, 4, B,C,D,G BD (2), (3), B,, (4) ( B), D 1, (5) C,D,G,, SPC DOE- 78
7-4-3 2: Cadillac, 5 1 :,, 2 :,, 3 ( ) : :,, : (1) 4 ; (2) : A ( ); B ( ); C ( ); D ( ) DOE- 79
7-16. Cadillac 7-6 ( ) 2 1 (A) lb/in * (B) * (C) * (D) * * DOE- 80
4 : L8 7-7 L8 A B C D [in] 1 1 1 1 1 8 2 1 1 2 2 7 3 1 2 1 2 3 4 1 2 2 1 6 5 2 1 1 2 2 6 2 1 2 1 4 7 2 2 1 1 5 8 2 2 2 2 1, 4 3 4 7-8 Z 10 log S 2 7-8 Z Y [in] S 1 0.25 0.40 0.60 0.20 0.3625 0.1797 14.91 2 0.00 0.10-0.40 0.20-0.025 0.2630 11.60 3 0.75 1.80 1.10 1.10 1.1875 0.4404 7.12 4 0.00 0.00 0.75 0.60 0.3375 0.3944 8.08 5 1.60 1.00 1.10 1.40 1.275 0.2754 11.20 6 0.50 0.80 0.90 0.40 0.650 0.2380 12.47 7 0.75 0.75-0.60 0.60 0.375 0.6538 3.69 8 0.90 0.35 0.00 0.90 0.538 0.4423 7.09 DOE- 81
A. 7-9 L8 Y A B C D [in] 1 1 1 1 1 0.3625 8 2 1 1 2 2-0.025 7 3 1 2 1 2 1.1875 3 4 1 2 2 1 0.3375 6 5 2 1 1 2 1.275 2 6 2 1 2 1 0.65 4 7 2 2 1 1 0.375 5 8 2 2 2 2 0.538 1 Y 7-17 Y 7-10 I 1 2 3 4 5 6 7-0.550-0.313-0.244-0.044 0.081 0.194 0.425 AB+CD D A B BC+AD AC+BD C P=100(i-0.5)/7 7.14 21.43 35.71 50 64.29 78.57 92.86 DOE- 82
7-18 Y A -0.244 C 0.425 D 0.313 CD -0.55 y'=.5875626+(-0.244/2)xa+(0.425/2)xc+(-0.313/2)xd+ +(-0.55/2)XC XD 7-11 Y y y' y-y' y y' y-y' 0.3625 0.247 0.116 1.275 1.353-0.078-0.025 0.135-0.160 0.65 0.616 0.034 1.1875 1.109 0.079 0.375 0.491-0.116 0.3375 0.372-0.035 0.538 0.378 0.160 DOE- 83
7-12 Y 1 2 3 4 5 6 7 8-0.160-0.116-0.078-0.035 0.034 0.079 0.116 0.160 100*(i-0.5)/8 6.25 18.75 31.25 43.75 56.25 68.75 81.25 93.75 7-19 Y 7-20 Y DOE- 84
B. Z 7-13 L8 Z A B C D Z [db] 1 1 1 1 1 14.91 8 2 1 1 2 2 11.60 7 3 1 2 1 2 7.12 3 4 1 2 2 1 8.08 6 5 2 1 1 2 11.2 2 6 2 1 2 1 12.47 4 7 2 2 1 1 3.69 5 8 2 2 2 2 7.09 1 Z db Z db 7-21 Z 7-14 Z I 1 2 3 4 5 6 7-0.580-0.395 0.535 1.600 1.755 1.815 6.050 C AB+CD D+ABC BC+AD AC+BD A B P=100(i-0.5)/7 7.14 21.43 35.71 50 64.29 78.57 92.86 DOE- 85
7-22 Z Z'= 9.52+(6.05/2)XB B 6.05 7-15 Z Z Z' Z-Z' Z Z' Z-Z' 14.91 12.545 2.365 11.2 12.545-1.345 11.60 12.545-0.945 12.47 12.545-0.075 7.12 6.495 0.625 3.69 6.495-2.805 8.08 6.495 1.585 7.09 6.495 0.595 7-16 Z 1 2 3 4 5 6 7 8-2.805-1.345-0.945-0.075 0.595 0.625 1.585 2.365 100*(i-0.5)/ 8 6.25 18.75 31.25 43.75 56.25 68.75 81.25 93.75 DOE- 86
7-23 Z ( ) 6 10 target= 0 : A (1 ); B (1 ) C (2 ); D (2 ) y'=.5875626+(-0.244/2)xa+(0.425/2)xc+(-0.313/2)xd+ +(-0.55/2)XC XD Z'= 9.52+(6.05/2)XB 7-6 ( ) 2 1 (A) lb/in * (B) * (C) * (D) * * ( ) 100 25 7-24 DOE- 87
7-24 DOE- 88
8., 8-1 : (1),, (2) :, (Brainstorming), (Fishbone diagram),,,, L8,L16,L12, DOE- 89
(3) (4) (5) 8-2 : (1),, (2),, (3),, (4),, (5) DOE- 90
,,,,, 8-3,,,, : (1), (2),, (3),, DOE- 91
DOE DOE- 92
(DOE) 1 2 : 1 2, 3 4 (Brainstorming), (Fishbone diagram) 5, 6, 7,, L8,L16,L12, 8 3 ( ) 1 2 4 1 2 3 4 5 6 Robust design 5 DOE- 93
DOE- 94
DOE- 95
DOE- 96
DOE- 97
A A1 A B C; A2 A B C 1 - - - 9, 7 2 + - - 10 12 3 - + - 9 11 4 + + - 12 15 5 - - + 11 10 6 + - + 10 13 7 - + + 10 8 8 + + + 16 14 A3 : A= 3.375 B= 1.625 C= 0.875 AB= 1.375 AC= 0.125 BC= -0.625 ABC= 1.125 A4 A5 : A, B, AB A6 A7 B DOE- 98