
8.14. A uniform fixed-fixed string is subjected to a weakly stationary random excitation F (x, t) that is
characterized by
E{F(x, t)} = 0
R
F F
(x
1
, x
2
; τ ) = σ
2
exp(−α |τ|)δ(x
1
−x
2
),
where δ (·) is the Dirac delta function. Compute the cross-correlation function and the cross-spectral
density of the response.
Solution: It is found that the response spectrum is given by
S
YY
(x
1
, x
2
; ω) =
∞
r=1
∞
k=1
Y
r
(x
1
)H
∗
r
(ω) S
Q
r
Q
k
(ω)H
k
(ω) Y
k
(x
2
),
where Y
r
(x) is the r
th
eigenfunction, H
r
(ω) is the transfer function between the generalized coordinate and
generalized force, given by
H
r
(ω) =
ω
2
r
−ω
2
+ i2ζ
r
ω
r
ω
−1
,
ω
r
is the r
th
natural frequency, ζ
r
is the r
th
damping ratio, S
Q
r
Q
k
(ω)