University of Florida/Eml4507/s13.team4ever.R7
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Problem R7.1a: Verify the dim of the matrices (fead.f08.mtgs.[37-41] pg. 2) edit
On our honor, we did this assignment on our own.
Given: The desired dimensions for k and d with index of 6 edit
Find: edit
1.Use an index of 6 to verify the dimensions of k and d edit
Obtain:
With supporting:
From:
Solution: Verify the dimensions of k and d edit
The constructed 6x6 matrix shown below represents
The following matrix represents the element stiffness matrix in local coordinates.
Using the above two matricies and the transpose of T, you can construct the following equation:
Multiplying by will yield the required
Problem R7.1b: Solve 2 element frame system (fead.f08.mtgs.[37-41] pg. 3) edit
On our honor, we did this assignment on our own.
Given: Information on the two element truss system edit
Assume a square cross section. Also use same data from fea.f08.mtgs.p5-4.
Element length: (1) =4
Element length: (2) =2
Young's modulus: (1) =3
Young's modulus: (2) =5
Cross section area: (1) =1
Cross section area: (2) =2
Inclination angle: (1) = 30 deg
Inclination angle: (2) = -45 deg
Find: edit
1. Plot Undeformed Shape edit
2. Plot Deformed Shape 2-bar Truss edit
3. Plot Deformed Shape 2-bar Frame edit
Solution: edit
1. Plot Undeformed Shape edit
Undeformed Shape.
View of element with single force member view.
2. Plot Deformed Shape 2-bar Truss edit
Plotting the deformed shape 2-bar truss for the given problem from p5-2 of the fead08 lecture notes. The deformation can be seen the truss using the forces given from p5-2.
3. Plot Deformed Shape 2-bar Frame edit
Plotting the deformed shape 2-bar frame for the given problem from p5-2 of the fead08 lecture notes. The constraints are preserved during the analysis in this mode to keep the frame.
ref edit
for i=1:2:5
x_1((i+1)/2)=P(1,((i+1)/2))+V(i,5);
end
for i=2:2:6
y_1((i/2))=P(2,(i/2))+V(i,1);
end
%Required deformation plotting
hold on
for i=1:2
l1=N(1,i);
l2=N(2,i);
x1=[x_1(l1),x_1(l2)];
y1=[y_1(l1),y_1(l2)];
axis([-1 5 -1 5])
plot(x1,y1,'b')
title(['Constrained Shape'])
hold on
end
%Required deformation plotting
for i=1:2:5
x_o((i+1)/2)=P(1,((i+1)/2));
end
for i=2:2:6
y_o((i/2))=P(2,(i/2));
end
for i=1:2
l1=N(1,i);
l2=N(2,i);
x1=[x_o(l1),x_o(l2)];
y1=[y_o(l1),y_o(l2)];
axis([-1 5 -1 5])
plot(x1,y1,'-.or')
hold on
end
end
Problem R7.2 edit
On my honor, I have neither given nor recieved unauthorized aid in doing this assignment.
Description edit
We are to resolve problem 5.7. We are to solve for motion of the truss using modal superposition using the three lowest eigenvalues.
Solution edit
The following code was used for problem 5.7 to obtain the K and M matrices and to obtain the eigenvalues and the eigenvectors:
function R7P2a
p = 2;
E = 5;
A = 0.5;
L = 1;
Ld = 1*sqrt(2);
%degree of freedom for each element
dof = zeros(10,5);
dof(1,:) = [1 1 2 3 4];
dof(2,:) = [2 1 2 5 6];
dof(3,:) = [3 3 4 5 6];
dof(4,:) = [4 5 6 9 10];
dof(5,:) = [5 5 6 7 8];
dof(6,:) = [6 3 4 9 10];
dof(7,:) = [7 3 4 7 8];
dof(8,:) = [8 7 8 9 10];
dof(9,:) = [9 9 10 11 12];
dof(10,:) = [10 7 8 11 12];
%position of each node
pos = zeros(2,6);
pos(:,1) = [0;0];
pos(:,2) = [L;0];
pos(:,3) = [L;L];
pos(:,4) = [2*L;0];
pos(:,5) = [2*L;L];
pos(:,6) = [3*L;0];
%Connections
conn = zeros(2,10);
conn(:,1) = [1;2];
conn(:,2) = [1;3];
conn(:,3) = [2;3];
conn(:,4) = [3;5];
conn(:,5) = [3;4];
conn(:,6) = [2;5];
conn(:,7) = [2;4];
conn(:,8) = [4;5];
conn(:,9) = [5;6];
conn(:,10) = [4;6];
%seperates into x and y coordinates
x = zeros(10,2);
y = zeros(10,2);
for i = 1:10
x(i,:) = [pos(1,conn(1,i)),pos(1,conn(2,i))];
y(i,:) = [pos(2,conn(1,i)),pos(2,conn(2,i))];
end
%Set up K and M matrix
K = zeros(12);
M = zeros(12);
for i = 1:10
xm = x(i,2)-x(i,1);
ym = y(i,2)-y(i,1);
L = sqrt(xm^2+ym^2);
l = xm/L;
m = ym/L;
k = E*A/L*[l^2 l*m -l^2 -l*m;l*m m^2 -l*m -m^2;-l^2 -l*m l^2 l*m;-l*m -m^2 l*m m^2;];
m = L*A*p;
m = [m/2 0 0 0;0 m/2 0 0;0 0 m/2 0;0 0 0 m/2];
Dof = dof(i,2:5);
K(Dof,Dof) = K(Dof,Dof)+k;
M(Dof,Dof) = M(Dof,Dof)+m;
end
bc = [1;2;12];
K
M
F = [0;0;0;0;0;0;0;-5;0;0;0;0];
[L,X] = eigen(K,M,bc)
From this, we obtain M and K matrices which are:
K =
Columns 1 through 9
3.3839 0.8839 -2.5000 0 -0.8839 -0.8839 0 0 0 0.8839 0.8839 0 0 -0.8839 -0.8839 0 0 0 -2.5000 0 5.8839 0.8839 0 0 -2.5000 0 -0.8839 0 0 0.8839 3.3839 0 -2.5000 0 0 -0.8839 -0.8839 -0.8839 0 0 4.2678 0 -0.8839 0.8839 -2.5000 -0.8839 -0.8839 0 -2.5000 0 4.2678 0.8839 -0.8839 0 0 0 -2.5000 0 -0.8839 0.8839 5.8839 -0.8839 0 0 0 0 0 0.8839 -0.8839 -0.8839 3.3839 0 0 0 -0.8839 -0.8839 -2.5000 0 0 0 4.2678 0 0 -0.8839 -0.8839 0 0 0 -2.5000 0 0 0 0 0 0 0 -2.5000 0 -0.8839 0 0 0 0 0 0 0 0 0.8839
Columns 10 through 12
0 0 0 0 0 0 -0.8839 0 0 -0.8839 0 0 0 0 0 0 0 0 0 -2.5000 0 -2.5000 0 0 0 -0.8839 0.8839 4.2678 0.8839 -0.8839 0.8839 3.3839 -0.8839 -0.8839 -0.8839 0.8839
M =
Columns 1 through 9
1.2071 0 0 0 0 0 0 0 0 0 1.2071 0 0 0 0 0 0 0 0 0 2.2071 0 0 0 0 0 0 0 0 0 2.2071 0 0 0 0 0 0 0 0 0 2.4142 0 0 0 0 0 0 0 0 0 2.4142 0 0 0 0 0 0 0 0 0 2.2071 0 0 0 0 0 0 0 0 0 2.2071 0 0 0 0 0 0 0 0 0 2.4142 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Columns 10 through 12
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2.4142 0 0 0 1.2071 0 0 0 1.2071
This will also output the lowest eigenpairs:
Then, with these, we will find what the modal equations are according to the equation:
This will yield 3 unique differential equations. We solve for the complete solution to these differential equation using the boundary conditions from:
and
With these solutions for z, we can then find and plot the actual displacements from modal superposition using:
Problem R7.2 edit
On my honor, I have neither given nor recieved unauthorized aid in doing this assignment.
Description edit
We are to resolve problem 5.7. We are to solve for motion of the truss using modal superposition using the three lowest eigenvalues.
Solution edit
The following code was used for problem 5.7 to obtain the K and M matrices and to obtain the eigenvalues and the eigenvectors:
function R7P2a
p = 2;
E = 5;
A = 0.5;
L = 1;
Ld = 1*sqrt(2);
%degree of freedom for each element
dof = zeros(10,5);
dof(1,:) = [1 1 2 3 4];
dof(2,:) = [2 1 2 5 6];
dof(3,:) = [3 3 4 5 6];
dof(4,:) = [4 5 6 9 10];
dof(5,:) = [5 5 6 7 8];
dof(6,:) = [6 3 4 9 10];
dof(7,:) = [7 3 4 7 8];
dof(8,:) = [8 7 8 9 10];
dof(9,:) = [9 9 10 11 12];
dof(10,:) = [10 7 8 11 12];
%position of each node
pos = zeros(2,6);
pos(:,1) = [0;0];
pos(:,2) = [L;0];
pos(:,3) = [L;L];
pos(:,4) = [2*L;0];
pos(:,5) = [2*L;L];
pos(:,6) = [3*L;0];
%Connections
conn = zeros(2,10);
conn(:,1) = [1;2];
conn(:,2) = [1;3];
conn(:,3) = [2;3];
conn(:,4) = [3;5];
conn(:,5) = [3;4];
conn(:,6) = [2;5];
conn(:,7) = [2;4];
conn(:,8) = [4;5];
conn(:,9) = [5;6];
conn(:,10) = [4;6];
%seperates into x and y coordinates
x = zeros(10,2);
y = zeros(10,2);
for i = 1:10
x(i,:) = [pos(1,conn(1,i)),pos(1,conn(2,i))];
y(i,:) = [pos(2,conn(1,i)),pos(2,conn(2,i))];
end
%Set up K and M matrix
K = zeros(12);
M = zeros(12);
for i = 1:10
xm = x(i,2)-x(i,1);
ym = y(i,2)-y(i,1);
L = sqrt(xm^2+ym^2);
l = xm/L;
m = ym/L;
k = E*A/L*[l^2 l*m -l^2 -l*m;l*m m^2 -l*m -m^2;-l^2 -l*m l^2 l*m;-l*m -m^2 l*m m^2;];
m = L*A*p;
m = [m/2 0 0 0;0 m/2 0 0;0 0 m/2 0;0 0 0 m/2];
Dof = dof(i,2:5);
K(Dof,Dof) = K(Dof,Dof)+k;
M(Dof,Dof) = M(Dof,Dof)+m;
end
bc = [1;2;12];
K
M
F = [0;0;0;0;0;0;0;-5;0;0;0;0];
[L,X] = eigen(K,M,bc)
From this, we obtain M and K matrices which are:
K =
Columns 1 through 9
3.3839 0.8839 -2.5000 0 -0.8839 -0.8839 0 0 0 0.8839 0.8839 0 0 -0.8839 -0.8839 0 0 0 -2.5000 0 5.8839 0.8839 0 0 -2.5000 0 -0.8839 0 0 0.8839 3.3839 0 -2.5000 0 0 -0.8839 -0.8839 -0.8839 0 0 4.2678 0 -0.8839 0.8839 -2.5000 -0.8839 -0.8839 0 -2.5000 0 4.2678 0.8839 -0.8839 0 0 0 -2.5000 0 -0.8839 0.8839 5.8839 -0.8839 0 0 0 0 0 0.8839 -0.8839 -0.8839 3.3839 0 0 0 -0.8839 -0.8839 -2.5000 0 0 0 4.2678 0 0 -0.8839 -0.8839 0 0 0 -2.5000 0 0 0 0 0 0 0 -2.5000 0 -0.8839 0 0 0 0 0 0 0 0 0.8839
Columns 10 through 12
0 0 0 0 0 0 -0.8839 0 0 -0.8839 0 0 0 0 0 0 0 0 0 -2.5000 0 -2.5000 0 0 0 -0.8839 0.8839 4.2678 0.8839 -0.8839 0.8839 3.3839 -0.8839 -0.8839 -0.8839 0.8839
M =
Columns 1 through 9
1.2071 0 0 0 0 0 0 0 0 0 1.2071 0 0 0 0 0 0 0 0 0 2.2071 0 0 0 0 0 0 0 0 0 2.2071 0 0 0 0 0 0 0 0 0 2.4142 0 0 0 0 0 0 0 0 0 2.4142 0 0 0 0 0 0 0 0 0 2.2071 0 0 0 0 0 0 0 0 0 2.2071 0 0 0 0 0 0 0 0 0 2.4142 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Columns 10 through 12
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2.4142 0 0 0 1.2071 0 0 0 1.2071
This will also output the lowest eigenpairs:
Then, with these, we will find what the modal equations are according to the equation:
This will yield 3 unique differential equations. We solve for the complete solution to these differential equation using the boundary conditions from:
and
With these solutions for z, we can then find and plot the actual displacements from modal superposition using:
Table of Assignments R7 edit
Problem Assignments R5 | ||
---|---|---|
Problem # | Solved&Typed by | Reviewed by |
1a | Vernon Babich, Chad Colocar | All |
1b | Vernon Babich, Tyler Wulterkens | All |
2 | David Bonner, Chad Colocar | All |