User:Egm4507.s13.team4.colocar/FEA.s13.R5.2.cc

R5.2


Length of members 2-3 and 4-5 (truss height): $$ L_{23} = L_{45} = 1 $$

Length of members 1-2, 2-4, 4-6 (truss length): $$ L_{12} = L_{24} = L_{46} = 1 $$

Area of cross section: $$ A=1/2 $$

Young's modulus: $$ E=5 $$

Mass Density: $$ \rho=2 $$

Part One: Solve the generalized eigenvalue problem of the above truss. Display the results for the lowest 3 eigenpairs. Plot the mode shapes.

Part Two: Consider the same truss, but with 2 missing braces:

Solve the generalized eigenvalue for this truss, and plot the mode shapes.

Solution
On my honor, I have neither given nor received unauthorized aid in doing this problem.

function R5p2 % Part One E = 5; p = 2; %kg/m3a A = 0.5; %m2 L = 1;  %m h = p*A*L; v = h; d = (sqrt((L^2)+(L^2)))*A*p; m = [(h/2)+(d/2);(h/2)+(d/2); (h/2)+(h/2)+(d/2)+(v/2);(h/2)+(h/2)+(d/2)+(v/2); (h/2)+(d/2)+(d/2)+(v/2);(h/2)+(d/2)+(d/2)+(v/2); (h/2)+(h/2)+(d/2)+(v/2);(h/2)+(h/2)+(d/2)+(v/2); (h/2)+(d/2)+(d/2)+(v/2);(h/2)+(d/2)+(d/2)+(v/2); (h/2)+(d/2);(h/2)+(d/2)]; M = diag(m); Coord = [0 0;1 0;1 1;2 0;2 1;3 0]; Dof = [1 2;3 4;5 6;7 8;9 10;11 12]; Edof = [1 1 2 3 4;2 1 2 5 6;3 3 4 5 6;4 5 6 9 10;5 5 6 7 8; 6 3 4 9 10;7 3 4 7 8;8 7 8 9 10;9 9 10 11 12;10 7 8 11 12]; bc = [1;2;12]; [Ex,Ey] = coordxtr(Edof,Coord,Dof,2) K = zeros(12); F = zeros(12,1); ep = [E A]; for i=1:10 Ke = bar2e(Ex(i,:),Ey(i,:),ep); K = assem(Edof(i,:),K,Ke); end [L,X] = eigen(K,M,bc); eigval = L; eigvect = X; j1eig = eigval(1) j1eigvx = eigvect(:,1) j1eigvy = eigvect(:,2) j2eig = eigval(2) j2eigvx = eigvect(:,3) j2eigvy = eigvect(:,4) j3eig = eigval(3) j3eigvx = eigvect(:,5) j3eigvy = eigvect(:,6) plotpar = [1 4 0]; scale = .5; eldraw2(Ex,Ey); for j=1:3 clear plot ed = extract(Edof,X(:,j)); P = eldisp2(Ex,Ey,ed,plotpar,scale); W(j) = getframe; drawnow end

Three lowest eigenpairs

j1eig = 0.0890 j1eigvx = 0         0    -0.0938     0.2910    -0.1706     0.2804    -0.1679     0.2946    -0.1131     0.2756    -0.2330          0 j1eigvy = 0         0    -0.1713    -0.2076    -0.2639    -0.1197    -0.2637    -0.1637    -0.3443    -0.1391    -0.2757          0 j2eig = 0.2779 j2eigvx = 0         0    -0.1902     0.2970     0.1621     0.2185    -0.2188    -0.3370     0.1475    -0.2327    -0.0779          0 j2eigvy = 0         0     0.0382     0.1640    -0.2909     0.1947     0.2640    -0.0109    -0.1970    -0.2662     0.4295          0 j3eig = 0.5582 j3eigvx = 0         0     0.2733     0.2621    -0.0575    -0.1395     0.1216    -0.3929    -0.1489     0.2738    -0.1309          0 j3eigvy = 0         0     0.3282    -0.2744     0.1584     0.4169     0.0086    -0.0685    -0.1419    -0.0349    -0.2114          0 Mode Shape 1

Mode Shape 2

Mode Shape 3

% Part Two Coord = [0 0;1 0;1 1;2 0;2 1;3 0]; Dof = [1 2;3 4;5 6;7 8;9 10;11 12]; Edof = [1 1 2 3 4;2 1 2 5 6;3 3 4 5 6;4 5 6 9 10;5 9 10 11 12; 6 7 8 11 12;7 3 4 7 8;8 7 8 9 10]; [Ex,Ey] = coordxtr(Edof,Coord,Dof,2); K = zeros(12); bc = [1;2;12]; ep = [E A]; for i=1:8 Ke = bar2e(Ex(i,:),Ey(i,:),ep); K = assem(Edof(i,:),K,Ke); end [L,X] = eigen(K,M,bc); eigval = L; eigvect = X; j1eig = eigval(1) j1eigvx = eigvect(:,1) j1eigvy = eigvect(:,2) j2eig = eigval(2) j2eigvx = eigvect(:,3) j2eigvy = eigvect(:,4) j3eig = eigval(3) j3eigvx = eigvect(:,5) j3eigvy = eigvect(:,6) plotpar = [1 4 0]; scale = .5; eldraw2(Ex,Ey); for j=3 clear plot ed = extract(Edof,X(:,j)); P = eldisp2(Ex,Ey,ed,plotpar,scale); W(j) = getframe; drawnow end

Three lower eigen pairs

j1eig = 5.5511e-17 j1eigvx = 0         0     0.0000     0.2666    -0.2666     0.2666     0.0000    -0.2666    -0.2666    -0.2666     0.0000          0 j1eigvy = 0         0     0.0872    -0.2534     0.1188    -0.2333     0.1675    -0.3552     0.0680    -0.3270     0.2344          0 j2eig = 0.0902 j2eigvx = 0         0    -0.1544    -0.3189    -0.2063    -0.2326    -0.2669    -0.0106    -0.3004    -0.0077    -0.3073          0 j2eigvy = 0         0    -0.2636     0.1334     0.2701     0.0540    -0.3702    -0.2297     0.2087    -0.0929    -0.2564          0 j3eig = 0.3066 j3eigvx = 0         0    -0.1315    -0.2916    -0.3174     0.2561    -0.0160    -0.1685     0.3129     0.1479     0.1295          0 j3eigvy = 0         0    -0.3520    -0.0374    -0.0322     0.0346    -0.0265     0.3638     0.0364    -0.3364     0.3501          0

Mode Shape 1

Mode Shape 2

Mode Shape 3