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Probabilistic Models for Dynamical Systems, 2nd Edition
book

Probabilistic Models for Dynamical Systems, 2nd Edition

by Haym Benaroya, Seon Mi Han, Mark Nagurka
May 2013
Intermediate to advanced content levelIntermediate to advanced
764 pages
6h 43m
English
CRC Press
Content preview from Probabilistic Models for Dynamical Systems, 2nd Edition
Let
R
1
=
k
1
+ k
2
m
1
R
2
=
k
1
+ k
2
m
2
R
3
=
k
2
m
1
R
4
=
k
2
m
2
=
k
2
m
1
m
1
k
1
+ k
2
k
1
+ k
2
m
2
=
R
3
R
2
R
1
Then, Equations 3 to 5 become
R
1
= λ
3
R
1
+ R
2
= λ
1
+ λ
2
R
1
R
2
R
3
R
3
R
2
R
1
= λ
1
λ
2
R
1
= λ
3
.
We solve for R
1
, R
2
, and R
3
. We find
R
1
= λ
3
= 196π
2
R
2
= λ
1
+ λ
2
R
1
= λ
1
+ λ
2
λ
3
= 1504π
2
R
3
=
3
(R
1
R
2
λ
1
λ
2
)
R
1
R
2
=
3
λ
2
3
λ
1
λ
2
λ
3
λ
1
+ λ
2
λ
3
= 132.53π
2
R
4
=
R
3
R
2
R
1
=
3
λ
2
3
λ
1
λ
2
λ
3
λ
1
+ λ
2
λ
3
λ
1
+ λ
2
λ
3
λ
3
= 1016.7π
2
The spring constants are found by
k
2
=
m
1
R
3
+
1
R
4
=
10
1
196π
2
+
1
1016.7π
2
= 11572 N/m
k
1
=
m
1
R
1
+
1
R
2
m
1
R
3
+
1
R
4
= 5541.8 N/m
The mass is found by
m
1
=
k
2
R
3
=
m
R
3
1
R
3
+
1
R
4
= 8.8471 kg
m
2
=
k
2
R
4
=
m
R
4
1
R
3
+
1
R
4
= 1.1532 kg.
We can confirm our results by plugging in these values into the characteristic equations,
m
1
m
2
λ
2
(k
1
+ k
2
) (m
1
+ m
2
) λ + (k
1
+ k
2
)
2
k
2
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Publisher Resources

ISBN: 9781439849897