User:EML4507.s13.team4ever.Tobin/R.4.2

Honor Pledge:
On my honor I have neither given nor received unauthorized aid in doing this problem.

Problem Description:
We are tasked with finding the eigenvector $$x_1$$ corresponding to the eigenvalue $$\lambda_1$$ for the same system as described in R4.2. We must also plot the mode shapes and compare with those from R4.2. Lastly we are tasked with creating an animation for each mode shape.

Given:
from Fead.s13.sec53b(4).djvu:

$$ K= \begin{bmatrix} 3 & -2\\ -2 & 5\\ \end{bmatrix} $$

$$ \gamma_1 = 4 - \sqrt{5} $$

$$ \gamma_2 = 4 + \sqrt{5} $$

and, from R4.1

$$ X_2 = \begin{bmatrix} -0.618 \\ 1 \\ \end{bmatrix} $$

Solution:
$$ \begin{bmatrix} K - \gamma_1*I\\ \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ \end{bmatrix} $$

So,

$$ \begin{bmatrix} -1 + \sqrt{5} & -2\\ -2 & 1 + \sqrt{5}\\ \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ \end{bmatrix} $$

Combining and reducing to row echelon form. $$ \begin{bmatrix} 1 & -0.618 & 0\\ 0 & 0 & 0\\ \end{bmatrix} $$

we may set $$ x_1 = 1 $$ and obtain:

$$ X_1 = \begin{bmatrix} 1 \\ 1.618 \\ \end{bmatrix} $$

Proving that they are orthogonal by ensuring the dot product between the two is zero: $$ X_1 $$ $$ \dot{} $$ $$ X_2 = (1)(-0.618)+ (1)(1.618) = 0$$ The eigenvalue corresponding to this eigenvector is the smallest possible and there this must be a fundamental mode. Mode shape plotted in picture below. Picture is adapted from Fead.s13.sec53b(4).djvu p.53-16.