
The characteristic equation is given by
det
k
11
−mω
2
k
12
k
21
k
22
−mω
2
= 0.
For our problem, the natural frequencies are ω
1
= 19.1 rad/s and ω
2
= 35.7 rad/s. The modal matrix is
given by
[P ] =
1
√
2
1 1
1 −1
.
The force spectrum is given by
S
F F
(ω) =
1
2π
∞
−∞
R
F F
(τ) e
−iωτ
dτ
=
T
−T
1 −
|τ|
T
e
−iωτ
dτ.
Then, the force spectrum is
S
F F
(ω) =
4 sin
2
(ωT/2)
T ω
2
.
The next figure show the force spectrum.
S
F F
(ω) for T = 2.
The response spectrum is given by
[S
XX
(ω)] = [H
∗
(ω)] [S
F F
(ω)] [H (ω)]
T
,
where
[H (ω)] =
[K] −ω
2
[M]
−1
or, equivalently the response spectrum is given by
[S
XX
(ω)] = [P ] [H
∗
(ω)] [P ]
T
[S
F F
(ω)] [P ] [H(ω)] [P ]
T
,
where H
11
(ω) = 1/
ω
2
− ω
2
1
.
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