Problem 5.1
On my honor, I have neither given nor received unauthorized aid in doing this assignment.
m
1
=
3
{\displaystyle m_{1}=3}
m
2
=
2
{\displaystyle m_{2}=2}
k
1
=
10
{\displaystyle k_{1}=10}
k
2
=
20
{\displaystyle k_{2}=20}
k
3
=
15
{\displaystyle k_{3}=15}
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:
K
=
[
a
1
0
0
−
a
1
0
0
0
12
a
2
6
L
a
2
0
−
12
a
2
6
L
a
2
0
6
L
a
2
4
L
2
2
0
−
6
a
2
2
L
a
2
2
−
a
1
0
0
a
1
0
0
0
−
12
a
2
−
6
L
a
2
0
12
a
2
−
6
a
2
0
6
L
a
2
4
L
2
2
0
−
6
a
2
4
L
a
2
2
]
{\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}}}
K
=
[
(
10
+
20
)
−
20
−
20
(
20
+
15
)
]
{\displaystyle K={\begin{bmatrix}(10+20)&-20\\-20&(20+15)\\\end{bmatrix}}}
