User:EML4507.s13.team4ever.Bonner/Bonnerreport7p2

Problem R7.2
On my honor, I have neither given nor recieved unauthorized aid in doing this assignment.

Description
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
The following code was used for problem 5.7 to obtain the K and M matrices and to obtain the eigenvalues and the eigenvectors:

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:

$$ \gamma_{1}, \phi_{1} $$

$$ \gamma_{2}, \phi_{2} $$

$$ \gamma_{3}, \phi_{3} $$

Then, with these, we will find what the modal equations are according to the equation:

$$ z^{''} + \gamma_{i}z = \phi_{i}^{T} F(t) $$

This will yield 3 unique differential equations. We solve for the complete solution to these differential equation using the boundary conditions from:

$$ z_{i} (0) = \bar{\phi}_{i}^{T} M d(0) $$ and

$$ z_{i}^{'} (0) = \bar{\phi}_{i}^{T} M d^{'}(0) $$

With these solutions for z, we can then find and plot the actual displacements from modal superposition using:

$$ d(t) = \Sigma^{3}_{j=1} z_{j} \phi_{j} $$