University of Florida/Eml4507/s13.team4ever.Wulterkens.R4.4
Problem 4.4 Eigenvalues and Eigenvectors for Mode Shape
Problem Statement
[edit | edit source]Assume the mass density to be 5,000 . Contruct the diagonal mass matrix and find the eigenpairs. With this, design a matlab and calfem program to determine the deformation shape.
Matlab Solution
[edit | edit source]Set up K matrix
[edit | edit source]k = 100000000000*0.0001/0.3;
K = zeros(28,28);
K(1,1)=k*1;K(3,3)=k*1;
K(3,3)=K(3,3)+k*1;K(5,5)=K(5,5)+k*1;
K(5,5)=K(5,5)+k*1;K(7,7)=K(7,7)+k*1;
K(7,7)=K(7,7)+k*1;K(9,9)=K(9,9)+k*1;
K(11,11)=K(11,11)+k*1;K(9,9)=K(9,9)+k*1;
K(11,11)=K(11,11)+k*1;K(13,13)=K(13,13)+k*1;
K(15,15)=K(15,15)+k*1;K(17,17)=K(17,17)+k*1;
K(19,19)=K(19,19)+k*1;K(17,17)=K(17,17)+k*1;
K(19,19)=K(19,19)+k*1;K(21,21)=K(21,21)+k*1;
K(23,23)=K(23,23)+k*1;K(21,21)=K(21,21)+k*1;
K(23,23)=K(23,23)+k*1;K(25,25)=K(25,25)+k*1;
K(27,27)=K(27,27)+k*1;K(25,25)=K(25,25)+k*1;
K(2,2)=K(2,2)+k*1;K(16,16)=K(16,16)+k*1;
K(4,4)=K(4,4)+k*1;K(18,18)=K(18,18)+k*1;
K(6,6)=K(6,6)+k*1;K(20,20)=K(20,20)+k*1;
K(8,8)=K(8,8)+k*1;K(22,22)=K(22,22)+k*1;
K(10,10)=K(10,10)+k*1;K(24,24)=K(24,24)+k*1;
K(12,12)=K(12,12)+k*1;K(26,26)=K(26,26)+k*1;
K(14,14)=K(14,14)+k*1;K(28,28)=K(28,28)+k*1;
l = cos(45)^2;
kl = 100000000000*0.0001/(0.3*sqrt(2));
for i = 0:5
j = 1+i*2;
K(j,j)=K(j,j)+l*kl;K(j,j+1)=K(j,j+1)+l*kl;K(j+1,j)=K(j+1,j)+l*kl;K(j+1,j+1)=K(j+1,j+1)+l*kl;
K(j+16,j)=K(j+16,j)+l*kl;K(j+16,j+1)=K(j+16,j+1)+l*kl;K(j+17,j)=K(j+17,j)+l*kl;K(j+17,j+1)=K(j+17,j+1)+l*kl;
K(j,j+16)=K(j,j+16)+l*kl;K(j,j+17)=K(j,j+17)+l*kl;K(j+1,j+16)=K(j+1,j+16)+l*kl;K(j+1,j+17)=K(j+1,j+17)+l*kl;
K(j+16,j+16)=K(j+16,j+16)+l*kl;K(j+16,j+17)=K(j+16,j+17)+l*kl;K(j+17,j+16)=K(j+17,j+16)+l*kl;K(j+17,j+17)=K(j+17,j+17)+l*kl;
end
Set up the M matrix
[edit | edit source]M = zeros(28,28);
M(i,i)=0.3*0.0001*5000;
for i = 1:28
M(i,i)=0.3*0.0001*5000;
end
for i = 1:12
M(i,i)=0.3*0.0001*5000*sqrt(2)+M(i,i);
M(i+16,i+16)=0.3*0.0001*5000*sqrt(2)+M(i+16,i+16);
end
for i = 3:12
M(i,i) = M(i,i)+0.3*0.0001*5000/2;
M(i+14,i+14) = M(i+14,i+14)+0.3*0.0001*5000/2;
end
b = zeros(28,1);
Determine the Eigenvalues and Eigenvectors
[edit | edit source][L,X] = eigen(K,M)
L =
1.0e+08 *
0.7625 0.7625 0.7625 0.7625 0.7984 0.7984 0.9205 0.9205 0.9578 0.9578 0.9578 0.9578 1.2259 1.2259 1.5251 1.5251 1.5251 1.5251 1.8406 1.8406 1.9251 1.9251 1.9251 1.9251 2.2222 2.2222 2.2222 2.2222
X =
Columns 1 through 5
0.0000 0.0000 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 -0.0000 -0.0000 -0.0000 0.0000 -0.0000 0.0000 0.0000 0.0000 -0.8946 -0.5809 0.0362 -0.0684 -0.0000 0.0000 0.0000 -0.0000 -0.0000 -0.0000 0.0759 -0.2460 -0.1249 1.0305 0.0000 0.0000 0.0000 0.0000 -0.0000 0.0000 0.5811 -0.8619 0.0773 -0.2392 -0.0000 -0.0000 0.0000 0.0000 0.0000 0.0000 0.0029 -0.0537 -1.0587 -0.1414 -0.0000 0.0000 -0.0000 -0.0000 -0.0000 0.0679 -0.0000 0.0000 0.0000 0.0000 -1.3744 -0.0000 -0.0000 0.0000 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 -0.0000 -0.0000 0.0000 -0.0000 0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 0.0000 0.0000 -0.0000 -0.0000 -0.0000 0.0000 0.0000 0.8946 0.5809 -0.0362 0.0684 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0759 0.2460 0.1249 -1.0305 -0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.5811 0.8619 -0.0773 0.2392 0.0000 0.0000 -0.0000 -0.0000 -0.0000 0.0000 -0.0029 0.0537 1.0587 0.1414 0.0000 0.0000 -0.0000 -0.0000 -0.0000 0.4876 0.0000 -0.0000 -0.0000 -0.0000 0.4876
Columns 6 through 10
0.4876 -1.1750 -0.0000 0.0000 -0.0000 0.4876 1.1750 -0.0000 0.0000 0.0000 -0.0000 0.0000 0.0000 -0.0348 -0.3197 0.0000 -0.0000 -0.0000 0.1011 0.9291 0.0000 0.0000 -0.0000 0.0386 0.0973 0.0000 -0.0000 0.0000 -0.1121 -0.2827 -0.0000 0.0000 0.0000 -0.3438 0.0468 0.0000 0.0000 -0.0000 0.9991 -0.1361 0.0000 0.0000 0.0000 -0.0144 -0.0850 -0.0000 -0.0000 -0.0000 0.0420 0.2471 -0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 -0.0000 -0.0000 -0.0000 0.0000 0.0000 -0.0000 0.0000 -0.0000 -0.0000 0.0000 0.0000 -0.0000 0.0000 0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 0.0679 -0.0000 0.0000 -0.0000 -0.0000 -1.3744 -0.0000 0.0000 -0.0000 -0.0000 0.0000 0.0000 0.0000 -0.0348 -0.3197 -0.0000 -0.0000 -0.0000 0.1011 0.9291 0.0000 0.0000 -0.0000 0.0386 0.0973 -0.0000 -0.0000 0.0000 -0.1121 -0.2827 0.0000 -0.0000 0.0000 -0.3438 0.0468 -0.0000 -0.0000 -0.0000 0.9991 -0.1361 0.0000 0.0000 0.0000 -0.0144 -0.0850 0.0000 -0.0000 -0.0000 0.0420 0.2471 -0.0000 -0.0000 1.1750 0.0000 0.0000 0.0000 -0.0000 -1.1750 -0.0000 -0.0000
Columns 11 through 15
-0.0000 -0.0000 -0.0000 -0.9256 0.0000 0.0000 0.0000 -0.0000 -0.9256 0.0000 -0.0818 -0.1048 -0.0000 -0.0000 0.8937 0.2379 0.3045 0.0000 -0.0000 -0.0000 0.0270 -0.3307 -0.0000 0.0000 -0.5386 -0.0786 0.9613 0.0000 -0.0000 -0.0000 -0.0028 -0.0266 -0.0000 0.0000 0.2345 0.0083 0.0772 0.0000 0.0000 -0.0000 0.3371 0.0009 -0.0000 0.0000 0.0000 -0.9798 -0.0025 0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.7827 -0.0000 0.0000 0.0000 0.0000 0.5054 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0000 -0.0000 0.0000 -0.0000 -0.0000 0.0000 0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 -0.0000 0.0000 0.0000 0.7827 0.0000 -0.0000 -0.0000 -0.0000 -0.5054 -0.0000 -0.0818 -0.1048 0.0000 -0.0000 -0.8937 0.2379 0.3045 0.0000 0.0000 -0.0000 0.0270 -0.3307 0.0000 -0.0000 0.5386 -0.0786 0.9613 -0.0000 -0.0000 -0.0000 -0.0028 -0.0266 0.0000 -0.0000 -0.2345 0.0083 0.0772 0.0000 0.0000 -0.0000 0.3371 0.0009 0.0000 0.0000 0.0000 -0.9798 -0.0025 0.0000 0.0000 0.0000 -0.0000 -0.0000 0.9256 0.0000 -0.0000 0.0000 0.0000 0.9256 -0.0000 0
Columns 16 through 20
0.0000 -0.0000 0.0000 -0.5350 -0.0000 0.0000 -0.0000 -0.0000 -0.5350 -0.0000 0.1161 0.0021 -0.5759 0.0000 0.0000 0.0000 -0.0000 -0.0000 0.0000 0.0000 0.5802 -0.0106 -0.7189 -0.0000 0.0000 0.0000 0.0000 -0.0000 -0.0000 0.0000 0.8903 -0.0322 0.5433 -0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 -0.0000 0.0324 1.0690 0.0104 0.0000 -0.0000 0.0000 0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 1.2924 -0.0000 -0.0000 -0.0000 -0.0000 0.3783 0.0000 0.0000 -0.0000 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 -0.0000 -0.0000 0.0000 0.0000 -0.0000 -0.0000 -0.0000 0.0000 0.0000 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 -0.0000 -1.2924 -0.0000 0.0000 -0.0000 -0.0000 -0.3783 -0.0000 -0.1161 -0.0021 0.5759 0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 0.0000 -0.5802 0.0106 0.7189 -0.0000 0.0000 0.0000 -0.0000 -0.0000 -0.0000 0.0000 -0.8903 0.0322 -0.5433 -0.0000 -0.0000 0.0000 -0.0000 0.0000 0.0000 0.0000 -0.0324 -1.0690 -0.0104 0.0000 0.0000 -0.0000 -0.0000 0.0000 -0.0000 -0.0000 0.0000 0.0000 0.0000 0.0000 0.5350 0.0000 0.0000 0.0000 -0.0000 0.5350
Columns 21 through 25
-0.0000 -0.0000 0.0000 0.0000 0.0000 -0.0000 -0.0000 0.0000 0.0000 0.0000 -0.8943 -0.4535 0.0330 0.1273 -0.0000 -0.3077 -0.1560 0.0113 0.0438 -0.0000 -0.3734 0.8333 -0.1855 0.3932 0.0000 -0.1285 0.2867 -0.0638 0.1353 0.0000 -0.2824 0.2924 -0.0745 -0.9230 0.0000 -0.0972 0.1006 -0.0256 -0.3176 0.0000 0.0614 -0.1931 -0.9908 0.0000 -0.0000 0.0211 -0.0664 -0.3409 0.0000 0.0000 0.0000 -0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 -0.0000 0.0000 -0.0000 0.0000 0.0000 -0.0000 -0.0000 -0.5675 0.0000 0.0000 -0.0000 0.0000 -0.3423 0.0000 0.0000 -0.0000 -0.0000 -2.4917 -0.0000 -0.0000 0.0000 0.0000 -0.1374 -0.0000 -0.0000 0.0000 0.0000 0.0000 -0.0000 -0.0000 0.0000 0.0000 -0.0000 -0.8943 -0.4535 0.0330 0.1273 0.0000 -0.3077 -0.1560 0.0113 0.0438 -0.0000 -0.3734 0.8333 -0.1855 0.3932 -0.0000 -0.1285 0.2867 -0.0638 0.1353 0.0000 -0.2824 0.2924 -0.0745 -0.9230 0.0000 -0.0972 0.1006 -0.0256 -0.3176 -0.0000 0.0614 -0.1931 -0.9908 0.0000 0.0000 0.0211 -0.0664 -0.3409 0.0000 0.0000 -0.0000 0.0000 0.0000 -0.0000 0.0000 0.0000 -0.0000 -0.0000 0 0
Columns 26 through 28
-0.0000 -0.0000 0.0000
-0.0000 0.0000 0.0000
-0.0000 -0.0000 -0.0000
-0.0000 -0.0000 0.0000
0.0000 0.0000 0.0000
-0.0000 -0.0000 0.0000
0.0000 -0.0000 0.0000
0.0000 -0.0000 0.0000
0.0000 0.0000 0.0000
-0.0000 -0.0000 -0.0000
0.0000 0.0000 0.0000
0.0000 0.0000 0.0000
-0.9357 2.3366 -0.0972
2.3783 0.8522 -0.4088
-0.1325 -0.6608 -0.0622
0.3428 0.2098 2.5468
0.0000 -0.0000 -0.0000
-0.0000 -0.0000 -0.0000
-0.0000 0.0000 0.0000
0.0000 -0.0000 -0.0000
-0.0000 -0.0000 -0.0000
-0.0000 -0.0000 0.0000
0.0000 -0.0000 0.0000
0.0000 0.0000 0.0000
-0.0000 -0.0000 -0.0000
-0.0000 -0.0000 -0.0000
-0.0000 -0.0000 -0.0000
0 0 0
Set up the Origional Shape
[edit | edit source]L=0.3;
for i=1:7
pos(:,i) = [(i-1)*L;0];
pos(:,i+7) = [(i-1)*L;L];
end
for i = 1:14,
x(i) = pos(1,i);
y(i) = pos(2,i);
end
for i = 1:6
conn(1,i)= i;
conn(2,i)= i+1;
end
for i = 7:12
conn(1,i) = i+1;
conn(2,i) = i+2;
end
for i = 13:19
conn(1,i) = i-12;
conn(2,i) = i-5;
end
for i = 20:25;
conn(1,i) = i-19;
conn(2,i) = i-11;
end
figure(1)
for i = 1:25
node_1 = conn(1,i);
node_2 = conn(2,i);
xx = [x(node_1),x(node_2)];
yy = [y(node_1),y(node_2)];
plot(xx,yy)
hold on
end
Add in the Eigenvectors
[edit | edit source]for i=1:14
j = i*2;
y(i) = y(i)+X(j,1);
end
for i=1:14
j = i*2-1;
x(i) = x(i)+X(j,1);
end
for i = 1:25
node_1 = conn(1,i);
node_2 = conn(2,i);
xx = [x(node_1),x(node_2)];
yy = [y(node_1),y(node_2)];
plot(xx,yy)
end
Final
[edit | edit source]