University of Florida/Eml4507/s13.team4ever.Wulterkens.R4.4

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Problem 4.4 Eigenvalues and Eigenvectors for Mode Shape

Problem Statement[edit]

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]

Set up K matrix[edit]

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]

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]

[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]

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]

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]

R4.4.jpg