
6.9. Let the random force F (t) have the autocorrelation function,
R
F F
(τ) = A exp (−α |τ |)
cos βτ +
α
β
sin β |τ|
.
Find the force spectrum, response spectrum, and the mean-square response.
Solution: Using the Euler identity, we can write
cos βτ =
e
iβτ
+ e
−iβτ
2
sin β |τ| =
e
iβ|τ|
−e
−iβ|τ |
2i
.
Taking the Fourier transform of the autocorrelation function, we can write the spectral density as
S
F F
(ω) =
1
2π
∞
−∞
R
XX
(τ) e
−iωτ
dτ
=
A
2π
1
2
1
2i
α
β
∞
−∞
e
−α|τ|
e
−iωτ
e
iβτ
+ e
−iβτ
e
iβ|τ|
−e
−iβ|τ |
dτ
=
A
2π
1
2
1
2i
α
β
∞
−∞
(e
−α|τ|−i(ωτ −βτ −β|τ|)
−e
−α|τ|−i(ωτ −βτ+β|τ|)
+e
−α|τ|−i(ωτ +βτ−β|τ|)
−e
−α|τ |−i(ωτ +βτ+β|τ|)
)dτ
=
A
2π
1
2
1
2i
α
β
∞
0
(e
−ατ−i(ωτ−2βτ)
−e
−ατ−i(ωτ)
+e
−ατ−i(ωτ )
−e
−ατ−i(ωτ+2βτ)
)dτ