
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