LAMPIRAN
Contoh langkah untuk mendapatkan Plot Kurva Isomerit pada optimasi O
bjective
F unction
1 dengan variasi variabel L2 dan L3 :
1.
Langkah 1, menuliskan pada
M-file
Matlab seperti berikut:
function [objective3] = ConstructDiskObjectiveFunction3 (x) L1 =x(1);
L2 =x(1), L3 =x(2), L4 =3; L5 =3; L6 =3;
%L1, L2, L3, L4, L5, L6 satuannya dalam inches R2 = 6.0; %inches
R3 = 5; R4 = 3.5; R5 = 2; R6 = 1.0;
N = 100; % rpm
V = 1; % velocity, inch per sec nu = 0.3; % poisson ratio (v) rho = 0.284; % density
P2=0.0; % pressure at the outermost ring surface ,psi
P6=1001.0; % internal pressure at the bore, in this case P6=Pm P1=0; % external pressure at the periphery (P1 = P2)
%An
A2 = (((3+nu)*rho*10^4)/4)*((R2/R2)^2-(R4/R2)^2); A3 = (((3+nu)*rho*10^4)/4)*((R3/R2)^2-(R5/R2)^2); A4 = (((3+nu)*rho*10^4)/4)*((R4/R2)^2-(R6/R2)^2); A = [A2 A3 A4];
%Bn
B2 = (2*(R2/R3)^2)/((R2/R3)^2-1); B3 = (2*(R3/R4)^2)/((R3/R4)^2-1); B4 = (2*(R4/R5)^2)/((R4/R5)^2-1); B5 = (2*(R5/R6)^2)/((R5/R6)^2-1); B = [B2 B3 B4 B5];
%Cn
C2 = 2/((R3/R4)^2-1); C3 = 2/((R4/R5)^2-1); C4 = 2/((R5/R6)^2-1); C = [C2 C3 C4];
%Dn
D2 = (((1-nu)+(1+nu)*(R2/R3)^2)/((R2/R3)^2-1))+(L2/L3)*(((1+nu)+(1-nu)*(R3/R4)^2)/((R3/R4)^2-1));
D4 = (((1-nu)+(1+nu)*(R4/R5)^2)/((R4/R5)^2-1))+(L4/L5)*(((1+nu)+(1-nu)*(R5/R6)^2)/((R5/R6)^2-1));
D = [D2 D3 D4];
%Kn
K2 = A2/C2; K3 = A3/C3; K4 = A4/C4;
K = [K2 K3 K4];
%Un
U2 = D2/C2; U3 = D3/C3; U4 = D4/C4;
U = [U2 U3 U4];
%Qn
Q2 = (B2/C2)*(L1)/L2; Q3 = (B3/C3)*(L2)/L3; Q4 = (B4/C4)*(L3)/L4; Q = [Q2 Q3 Q4];
P3_g =(200-rand(1,2)*(200-100));%initial guess for P3, 2 random numbers between 100-200
for n=1:2,
P4_g=(K2*(V^2))-(Q2*P2)+U2*P3_g(1,:); %calculation for P6 P5_g=(K3*(V^2))-(Q3*P3_g(1,:))+(U3*P4_g); %using initial guess of P3
P6_g=(K4*(V^2))-(Q4*P4_g)+(U4*P5_g); %to obtain linear equation for interpolation
end
%interpolation, using actual value of P6 to obtain correct value of P3 P3 =
(((P6-P6_g(1,1))*(P3_g(1,2)-P3_g(1,1)))/(P6_g(1,2)-P6_g(1,1)))+P3_g(1,1);
%calculation of pressure for each interface P4 = (K2*V^2)-(Q2*P2)+(U2*P3);
P5 = K3*V^2-Q3*P3+U3*P4; P6 = K4*V^2-Q4*P4+U4*P5; P = [P2 P3 P4 P5 P6],
%radial stress calculation for each interface radial_stress3 = -((1+(L2/L3))*P3)/2;
radial_stress4 = -((1+(L3/L4))*P4)/2; radial_stress5 = -((1+(L4/L5))*P5)/2; radial_stress6 = -((1+(L5/L6))*P6)/2;
radial_stress = [radial_stress3 radial_stress4 radial_stress5 radial_stress6];
%En
E4 = 1/((R4/R5)^2-1); E5 = 1/((R5/R6)^2-1); E = [E2 E3 E4 E5];
F2 = (((((3+nu)*rho)*10^4)/4)*(R2/R2)^2)+(((1-nu) *rho*10^4)/4)*(R3/R2)^2;
F3 = (((((3+nu)*rho)*10^4)/4)*(R3/R2)^2)+(((1-nu)*rho*10^4)/4)*(R4/R2)^2;
F4 = (((((3+nu)*rho)*10^4)/4)*(R4/R2)^2)+(((1-nu)*rho*10^4)/4)*(R5/R2)^2;
F5 = (((((3+nu)*rho)*10^4)/4)*(R5/R2)^2)+(((1-nu)*rho*10^4)/4)*(R6/R2)^2;
F = [F2 F3 F4 F5];
%tangential stress (n)
tangential_stress3=-(B2*(L3/L2)*P2)+((E2+(B2/2)-nu/2*(L2/L3))*P3)+F2*V^2;
tangential_stress4=-(B3*(L4/L3)*P3)+((E3+(B3/2)-nu/2*(L3/L4))*P4)+F3*V^2;
tangential_stress5=-(B4*(L5/L4)*P4)+((E4+(B4/2)-nu/2*(L4/L5))*P5)+F4*V^2;
tangential_stress6=-(B5*(L6/L5)*P5)+((E5+(B5/2)-nu/2*(L5/L6))*P6)+F5*V^2;
tangential_stress = [abs(tangential_stress3) abs(tangential_stress4) abs(tangential_stress5) abs(tangential_stress6)],
%nilai tegangan tangensial maksimum
max_sigma_t = max(tangential_stress); %...(1)
%nilai tegangan tangensial minimum
min_sigma_t = min(tangential_stress); %...(2)
%objective3, max sigma(t) - min sigma(t) %substitusi dari persamaan (1) dan (2) objective3 = (max_sigma_t - min_sigma_t), end
Save As M-file di atas dengan nama: ConstructDiskObjectiveF unction3.
2.
Langkah 2, menuliskan
constraints
kosong pada
M-file
baru:
function [c, ceq] = constraintL (x) %Nonlinear inequality constraints c = [];
%Nonlinear equality constraints ceq = [];
Save As M-file di atas dengan nama: constraintL.
3.
Langkah 3, menuliskan batas atas, batas bawah, fungsi constraint (jika ada) dan
clc
clear all
x0 = [0.7 0.7] % Make a starting guess at solution f = ConstructDiskObjectiveFunction3 (x0)
%x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon)
options = optimset ('Display', 'iter', 'PlotFcns', @optimplotfval); [x,fval] = fmincon (@ConstructDiskObjectiveFunction3, x0, [], [], [], [], [0.6 0.6], [4 4], @constraintL, options)
Jalankan eksekusi optimasi tersebut, dan akan keluar nilai
objective function
minimum
serta nilai L2 dan L3 yang optimum. Catat senua data pada tiap iterasi, yakni data L2,
L3 dan nilai fungsinya (digunakan pada langkah 6 untuk menunjukkan jalannya
optimisasi dari tebakan awal hingga tercapai optimum).
4.
Langkah 4, menuliskan pada
M-file
baru sebagai berikut:
function [objective3] = ConstructDiskObjectiveFunction3bwt_data (x, y) L1 =x;
L2 =x; L3 =y; L4 =3; L5 =3; L6 =3;
R2 = 6.0; %inches R3 = 5;
R4 = 3.5; R5 = 2; R6 = 1.0;
N = 100; % rpm
V = 1; % velocity, inch per sec nu = 0.3; % poisson ratio (v) rho = 0.284; % density
P2=0.0; % pressure at the outermost ring surface ,psi
P6=1001.0; % internal pressure at the bore, in this case P6=Pm P1=0; % external pressure at the periphery (P1 = P2)
%An
A2 = (((3+nu)*rho*10^4)/4)*((R2/R2)^2-(R4/R2)^2); A3 = (((3+nu)*rho*10^4)/4)*((R3/R2)^2-(R5/R2)^2); A4 = (((3+nu)*rho*10^4)/4)*((R4/R2)^2-(R6/R2)^2); A = [A2 A3 A4];
%Bn
B2 = (2*(R2/R3)^2)/((R2/R3)^2-1); B3 = (2*(R3/R4)^2)/((R3/R4)^2-1); B4 = (2*(R4/R5)^2)/((R4/R5)^2-1); B5 = (2*(R5/R6)^2)/((R5/R6)^2-1); B = [B2 B3 B4 B5];
%Cn
C4 = 2/((R5/R6)^2-1); C = [C2 C3 C4];
%Dn
D2 = (((1-nu)+(1+nu)*(R2/R3)^2)/((R2/R3)^2-1))+(L2/L3)*(((1+nu)+(1-nu)*(R3/R4)^2)/((R3/R4)^2-1));
D3 = (((1-nu)+(1+nu)*(R3/R4)^2)/((R3/R4)^2-1))+(L3/L4)*(((1+nu)+(1-nu)*(R4/R5)^2)/((R4/R5)^2-1));
D4 = (((1-nu)+(1+nu)*(R4/R5)^2)/((R4/R5)^2-1))+(L4/L5)*(((1+nu)+(1-nu)*(R5/R6)^2)/((R5/R6)^2-1));
D = [D2 D3 D4];
%Kn
K2 = A2/C2; K3 = A3/C3; K4 = A4/C4;
K = [K2 K3 K4];
%Un
U2 = D2/C2; U3 = D3/C3; U4 = D4/C4;
U = [U2 U3 U4];
%Qn
Q2 = (B2/C2)*(L1)/L2; Q3 = (B3/C3)*(L2)/L3; Q4 = (B4/C4)*(L3)/L4; Q = [Q2 Q3 Q4];
P3_g =(200-rand(1,2)*(200-100));%initial guess for P3, 2 random numbers between 100-200
for n=1:2,
P4_g=(K2*(V^2))-(Q2*P2)+U2*P3_g(1,:); %calculation for P6 P5_g=(K3*(V^2))-(Q3*P3_g(1,:))+(U3*P4_g); %using initial guess of P3
P6_g=(K4*(V^2))-(Q4*P4_g)+(U4*P5_g); %to obtain linear equation for interpolation
end
%interpolation, using actual value of P6 to obtain correct value of P3 P3 =
(((P6-P6_g(1,1))*(P3_g(1,2)-P3_g(1,1)))/(P6_g(1,2)-P6_g(1,1)))+P3_g(1,1);
%calculation of pressure for each interface P4 = (K2*V^2)-(Q2*P2)+(U2*P3);
P5 = K3*V^2-Q3*P3+U3*P4; P6 = K4*V^2-Q4*P4+U4*P5; P = [P2 P3 P4 P5 P6];
%radial stress calculation for each interface radial_stress3 = -((1+(L2/L3))*P3)/2;
radial_stress5 = -((1+(L4/L5))*P5)/2; radial_stress6 = -((1+(L5/L6))*P6)/2;
radial_stress = [radial_stress3 radial_stress4 radial_stress5 radial_stress6];
%En
E2 = 1/((R2/R3)^2-1); E3 = 1/((R3/R4)^2-1); E4 = 1/((R4/R5)^2-1); E5 = 1/((R5/R6)^2-1); E = [E2 E3 E4 E5];
F2 = (((((3+nu)*rho)*10^4)/4)*(R2/R2)^2)+(((1-nu)*rho*10^4)/4)*(R3/R2)^2;
F3 = (((((3+nu)*rho)*10^4)/4)*(R3/R2)^2)+(((1-nu)*rho*10^4)/4)*(R4/R2)^2;
F4 = (((((3+nu)*rho)*10^4)/4)*(R4/R2)^2)+(((1-nu)*rho*10^4)/4)*(R5/R2)^2;
F5 = (((((3+nu)*rho)*10^4)/4)*(R5/R2)^2)+(((1-nu)*rho*10^4)/4)*(R6/R2)^2;
F = [F2 F3 F4 F5];
%tangential stress (n)
tangential_stress3=-(B2*(L3/L2)*P2)+((E2+(B2/2)-tangential_stress = [abs(tangential_stress3=-(B2*(L3/L2)*P2)+((E2+(B2/2)-tangential_stress3) abs(tangential_stress3=-(B2*(L3/L2)*P2)+((E2+(B2/2)-tangential_stress4) abs(tangential_stress5) abs(tangential_stress6)];
%nilai tegangan tangensial maksimum
max_sigma_t = max(tangential_stress); %...(1)
%nilai tegangan tangensial minimum
min_sigma_t = min(tangential_stress); %...(2)
%objective3, max sigma(t) - min sigma(t) %substitusi dari persamaan (1) dan (2) objective3 = (max_sigma_t - min_sigma_t);
end
Save As dengan nama file : ConstructDiskObjectiveFunction3bwt_data.
5.
Langkah 5, menuliskan pada
M-file
baru sebagai berikut:
clc
clear all
x=[0.25:0.25:5];% Make a starting guess at solution y=[0.25];
for n=1:20
[f(n)] = ConstructDiskObjectiveFunction3bwt_data (x(n),y);
Jalankan program di atas, kemudian lihat pada Workspace, klik kiri dua kali pada f.
Terdapat 20 data yang muncul di sana, copy pada excel. Kemudian ulangi langkah 5
dengan mengganti nilai y = 0.25 sampai 5. Lakukan langkah yang sama untuk
mengcopykan data ke excel, sehingga didapat data seperti berikut:
Gambar A Data objective function 1.
Gambar A tersebut hanya sebagian data saja (karena datanya terlalu panjang, sebagian
data tidak penulis munculkan).
6.
Langkah 6, menuliskan pada
M-file
baru sebagai berikut:
clc
clear all
x = [0.25:0.25:5; 0.25:0.25:5; 0.25:0.25:5; 0.25:0.25:5; 0.25:0.25:5; 0.25:0.25:5; 0.25:0.25:5; 0.25:0.25:5; 0.25:0.25:5; 0.25:0.25:5; 0.25:0.25:5; 0.25:0.25:5; 0.25:0.25:5; 0.25:0.25:5; 0.25:0.25:5; 0.25:0.25:5; 0.25:0.25:5; 0.25:0.25:5; 0.25:0.25:5; 0.25:0.25:5];
y = x';
z =[16539.31255 14276.91794 12235.45734 10384.07445 8697.403333 7154.39929 7434.678239 8010.465258 8542.764282 9036.32286 9495.221759 9922.988051 11134.42487 12357.54697 12539.61864 12226.61846 11931.7736 11983.21473 12260.69824 12523.37484 8483.034169 7496.999434 6569.940144 5696.718889 4872.778123 4124.489732 4484.716703 4826.200802 5150.367747 5458.502229
y. x 0.25 0.5 0,75 1 1,25 1,5 1,75 2 2,25
0,25 16539,31 14276,91794 12235,46 10384,07 8697,403 7154,39929 7434,678 8010,465 8542,764 0,5 8483,034 7496,999434 6569,94 5696,719 4872,778 4124,489732 4484,717 4826,201 5150,368 5 0,75 5723,873 5108,36504 4521,074 3960,103 3423,721 3036,563916 3272,567 3498,864 3716,041 3 1 4335,494 3893,429895 3468,37 3059,351 2665,482 2426,966645 2592,659 2752,537 2906,902 3 1,25 3502,865 3160,752198 2830,216 2510,679 2201,601 2025,104136 2146,11 2263,34 2376,969 2 1,5 2950,029 2672,695149 2403,855 2143,124 1890,14 1890,222622 1915,858 1940,764 1996,36 2 1,75 2557,668 2325,615182 2100,115 1892,408 1922,111 1951,010596 1979,137 2006,522 2033,194 2 2 2265,781 2067,096717 1910,636 1942,82 1974,177 2004,738025 2034,534 2063,592 2091,941 2 2,25 2040,906 1921,88339 1955,968 1989,216 2021,657 2053,320353 2084,234 2114,424 2143,915 2 2,5 1928,234 1963,799485 1998,525 2032,44 2065,573 2097,950897 2129,599 2160,541 2190,801 2 2,75 1967,776 2003,669683 2038,754 2073,057 2106,604 2139,419218 2171,527 2202,95 2233,709 2 3 2005,715 2041,734689 2076,976 2111,464 2145,222 2178,274046 2210,641 2242,345 2273,405 2 3,25 2042,18 2078,178721 2113,428 2147,953 2181,774 2214,912708 2247,39 2279,225 2310,438 2
3,5 2077,28 2113,1482 2148,297 2182,747 2216,519 2249,633088 2282,108 2313,963 2345,215 3,75 2111,106 2146,763143 2181,728 2216,021 2249,66 2282,665276 2315,053 2346,842 2378,047
5751.764935 6031.207148 6650.846687 7426.35978 8167.73849 8280.426845 8133.462958 7992.567091 7857.371031 7867.848963 5723.87297 5108.36504 4521.073931 3960.102799 3423.721092 3036.563916 3272.567275 3498.863958 3716.040744 3924.638048 4125.154418 4318.050506 4650.935495 5202.661761 5734.551399 6247.655319 6371.467325 6275.600288 6182.948808 6093.353769 4335.494207 3893.429895 3468.369626 3059.35082 2665.482205 2426.966645 2592.659171 2752.537434 2906.902198 3056.033832 3200.194015 3339.627262 3502.597816 3921.836408 4327.972812 4721.611788 5103.321444 5274.209214 5201.36296 5130.628338 3502.864886 3160.752198 2830.216218 2510.679107 2201.600854 2025.104136 2146.109627 2263.339808 2376.968649 2487.159589 2594.066327 2697.833534 2798.597505 3079.791269 3402.648517 3716.550397 4021.864417 4318.938245 4553.399048 4493.294387 2950.02924 2672.695149 2403.85469 2143.123558 1890.140289 1890.222622 1915.858034 1940.764172 1996.359669 2079.357831
2160.091502 2238.652109 2315.126224 2477.913158 2741.847865 2999.04129 3249.748449 3494.211658 3732.661307 3965.316594
2557.667845 2325.615182 2100.115062 1892.40758 1922.111425 1951.010596 1979.137353 2006.522257 2033.194274 2059.180885 2084.508174 2109.200918 2133.282669 2173.400103 2360.877829 2543.882655 2722.572769 2897.098984 3072.503077 3268.235712 2265.780862 2067.096717 1910.635898 1942.819754 1974.17664 2004.738025 2034.533802 2063.592384 2091.940796 2119.604758 2146.608762 2172.976146 2198.729161 2223.889032 2352.589017 2517.317767 2678.379178 2835.894369 2989.979187 3140.744483 2040.905898 1921.88339 1955.968131 1989.215808 2021.656882 2053.320353 2084.233849 2114.423704 2143.915033 2172.731802 2200.896893 2228.432166 2255.358512 2281.695909 2349.360697 2499.465612 2646.391642 2790.238713 2931.102602 3069.075159 1928.23444 1963.799485 1998.524939 2032.440187 2065.573256 2097.950897 2129.598654 2160.540932 2190.801061 2220.401353
2249.363157 2277.706911 2305.45219 2332.617752 2359.221577 2487.73272 2623.052634 2755.658549 2885.631303 3013.048556
1967.776042 2003.669683 2038.754367 2073.057138 2106.603847 2139.419218 2171.526908 2202.949566 2233.708883 2263.825645
2293.319778 2322.210392 2350.515825 2378.253677 2405.44085 2480.42287 2606.028006 2729.21162 2850.043068 2968.589085
2469.726285 2499.213825 2528.19407 2556.68 2584.684155 2612.218655 2639.295216 2665.925168 2756.15546 2845.316577
2175.24814 2210.318488 2244.747729 2278.553283 2311.751945 2344.359913 2376.392813 2407.865724 2438.793204 2469.189309 2499.067618 2528.441249 2557.322882 2585.724776 2613.658786 2641.136378 2668.168648 2694.766335 2747.178131 2832.30902 2205.699517 2240.420782 2274.524647 2308.027437 2340.944902 2373.292248 2405.084161 2436.334825 2467.057944 2497.266765 2526.974095 2556.192319 2584.933417 2613.208982 2641.030233 2668.408032 2695.352899 2721.875021 2747.984271 2821.634187 2235.149999 2269.497909 2303.250924 2336.424367 2369.033037 2401.091238 2432.612789 2463.611056 2494.09896 2524.089003 2553.593282 2582.623503 2611.191001 2639.306752 2666.981388 2694.225207 2721.048194 2747.460023 2773.470077 2812.900898 2263.652209 2297.609377 2330.993006 2363.817506 2396.096807
2427.844381 2459.073259 2489.79605 2520.024956 2549.771793 2579.048 2607.864659 2636.232505 2664.161944 2691.663059 2718.74563
2745.419138 2771.692783 2797.575489 2823.075916];
%batas bawah hold on;
m = [0.25; 5]; n = [0.6; 0.6]; plot (m, n,'-k');
hold on;
o = [0.6; 0.6]; p = [0.25; 5]; plot (o, p,'-k');
%batas atas hold on; q = [4; 4]; r = [0.25; 5]; plot (q, r,'-k');
hold on;
s = [5; 0.25]; t = [4; 4];
plot (s, t,'-k');
%iterasi optimasi hold on;
a = [0.7000;4.0000;3.5101;2.0550;1.3275;1.4775;1.4775;1.4922;1.4931; 1.4914;1.4357;1.4340;1.4275;1.4273;1.4272;1.4272;1.4272;1.4276;1.4282; 1.4282;1.4353];
b = [0.7000;4.0000;4.0000;2.3000;1.4500;1.3969;1.3969;1.3844;1.3829 ;1.3814;1.3688;1.3697;1.3682;1.3681;1.3681;1.3681;1.3681;1.3678;1.3673 ;1.3672;1.3612];
plot (a, b,':k');
hold on;
d = [0.7000;1.3612]; plot (c, d,'+k');
hold on;
e = [2.0000;1.6500;1.3875;1.4315;1.4321;1.4268;1.4207;1.4272;1.4277 ;1.4295;1.4297;1.4297;1.4297;1.4297;1.4297;1.4297;1.4297;1.4297];
f = [2.0000;1.6500;1.3875;1.3629;1.3835;1.3782;1.3721;1.3761;1.3744 ;1.3684;1.3676;1.3675;1.3675;1.3675;1.3675;1.3675;1.3675;1.3675]; plot (e, f,':r');
hold on;
g = [2.0000;1.4297]; h = [2.0000;1.3675]; plot (g, h,'+r');
hold on;
i = [1.0000;0.8000;0.7875;0.7846;0.7838;0.7838;0.7838;0.7838;0.7838; 1.154 ;1.3694;1.4234;1.4266;1.4274;1.4278;1.4279;1.4279;1.4280];
j = [3.5000;2.0500;1.9594;1.9381;1.9329;1.9326;1.9324;1.9324;1.9324; 1.5993;1.4064;1.3579;1.3550;1.3543;1.3539;1.3538;1.3538;1.3537]; plot (i, j,':g');
hold on;
k = [1.0000;1.4280]; l = [3.5000;1.3537]; plot (k, l,'+g');
