Problem 5.1
On my honor, I have neither given nor received unauthorized aid in doing this assignment.
Solve by hand the gen. eigenvalue problem for the spring-mass-damper system on p.53-13, using the data for the masses in (2) p.53-13b, and the data for the stiffness coefficients in (4) p.53-13b.
First plug given values into the stiffness matrix:
Next plug stiffness matrix in equation below:
![{\displaystyle [K-\gamma _{I}]x=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9dd50004a26c154db77dec7e19170ea01ee66a03)
Take the determinant:
![{\displaystyle det{\begin{Bmatrix}30-\gamma &-20\\-20&35-\gamma \end{Bmatrix}}=[(30-\gamma )(35-\gamma )]-(-20)(-20)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dfe9b7dfc520c4a3e3ef2196e033952d79f7ad82)
Simplifying:
Using the Quadratic formula to solve:
Final gamma values:
To find the eigenvectors, first we:
![{\displaystyle [K-\gamma _{I}]x=[{\begin{bmatrix}30&-20\\-20&35\\\end{bmatrix}}-\gamma *{\begin{bmatrix}1&0\\0&1\\\end{bmatrix}}{\begin{bmatrix}]x_{1}\\x_{2}\\\end{bmatrix}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8e6250b7b1c15771a4dea0a38cf4e5ee9976bad)
To solve, lets set 
equal to 1:
![{\displaystyle [K-\gamma _{I}]x=[{\begin{bmatrix}30&-20\\-20&35\\\end{bmatrix}}-52.66*{\begin{bmatrix}1&0\\0&1\\\end{bmatrix}}]{\begin{bmatrix}1\\x_{2}\\\end{bmatrix}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88bef8d04e92c5ffc5b742595df9375840ef662e)
![{\displaystyle [K-\gamma _{I}]x=[{\begin{bmatrix}30&-20\\-20&35\\\end{bmatrix}}+{\begin{bmatrix}-52.66&0\\0&-52.66\\\end{bmatrix}}]{\begin{bmatrix}1\\x_{2}\\\end{bmatrix}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/519f4040b8966636bf815b5cd55c9199ea093803)
![{\displaystyle [K-\gamma _{I}]x=[{\begin{bmatrix}30-52.66&-20\\-20&35-52.66\\\end{bmatrix}}{\begin{bmatrix}1\\x_{2}\\\end{bmatrix}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4c4639ec67eb97f02f9ce2ee09fd789868dca39)
![{\displaystyle [K-\gamma _{I}]x=[{\begin{bmatrix}-22.66&-20\\-20&-17.66\\\end{bmatrix}}{\begin{bmatrix}1\\x_{2}\\\end{bmatrix}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5231b9a23042a2f4f953e65a2dbf3c43ebc2733f)
Multiplying out the matrices, we obtain:
Using the same process we find :
![{\displaystyle [K-\gamma _{I}]x=[{\begin{bmatrix}30&-20\\-20&35\\\end{bmatrix}}-52.66*{\begin{bmatrix}1&0\\0&1\\\end{bmatrix}}]{\begin{bmatrix}x_{1}\\1\\\end{bmatrix}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6c2985493b34ed083af2d94efe539310745e72f)
![{\displaystyle [K-\gamma _{I}]x=[{\begin{bmatrix}-22.66&-20\\-20&-17.66\\\end{bmatrix}}{\begin{bmatrix}x_{1}\\1\\\end{bmatrix}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec454254356d0a2fc4c29657024bc0a2b551cece)
Multiplying out the matrices, we obtain:
