Problem 5.1

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


## Given

${\displaystyle m_{1}=3}$

${\displaystyle m_{2}=2}$

${\displaystyle k_{1}=10}$

${\displaystyle k_{2}=20}$

${\displaystyle k_{3}=15}$

## Find

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.

## Solution

First plug given values into the stiffness matrix:

${\displaystyle K={\begin{bmatrix}a_{1}&0&0&-a_{1}&0&0\\0&12a_{2}&6La_{2}&0&-12a_{2}&6La_{2}\\0&6La_{2}&4L_{2}^{2}&0&-6a_{2}&2La_{2}^{2}\\-a_{1}&0&0&a_{1}&0&0\\0&-12a_{2}&-6La_{2}&0&12a_{2}&-6a_{2}\\0&6La_{2}&4L_{2}^{2}&0&-6a_{2}&4La_{2}^{2}\\\end{bmatrix}}}$

${\displaystyle K={\begin{bmatrix}(10+20)&-20\\-20&(20+15)\\\end{bmatrix}}}$