174
STATISTICAL RESPONSES FOR LINEAR STRUCTURES
for which the load spectral density Spi(w) is known through équation (7.26), the
task now is to calculate the response spectral density S„(w) and the variance a?.
Because these results are so important in applications, ail of the assumptions and
mathematical details needed for this dérivation are included here. Because two
particular response functions for équation (7.27) are required eventually, these
are repeated for convenience. One is the harmonie response function given by
équation (5.62), or
//l.-'l = k^-mJ2 + jc-Liv + kl)-1
(7.28)
The other is the impulse response function given by équation (5.74), or
h(t) = —-—e~^ot sin üjdt
(7.29)
mwd
The Fourier Transform
A preliminary step is to relate H(w) to h(t) through the Fourier transform.
To do this, the steady-State solution to équation (7.27), as given by the convolution intégral in équation (5.76), is first written as
v = [ pi(r) h.(t — t)cIt
(7.30)
J —oo
Here the lower limit t = 0 was replaced by t = —oo since Pi(t) vanishes for
t < 0, leaving the value of the intégral unchanged. Further it is recalled that
h(t - t) is the response to a unit impulse at (t - r) = 0. For (t - r) < 0, the
response v is zéro because the unit impulse has not yet corne into existence.
Thus for t < t, h.(t — t) =0, and the upper limit t = t may be extended to
t = oo without changing the value of this intégral. That is
roo
v = / Pi(r) h(t — r)dr
(7.31)
J—oo
Now define a variable change: 0 = t - r, where dr = -d0. The lower limit
T — —00 now changes to 0 - oo, and the upper limit t — oo changes to
0 = —oo. It follows that
v = [
Pi(t - 0)h(0)(-d0)
(7.32)
Joo
Change the sign of the intégral and reverse the limits of intégration, or
v = [ Pi(t - 0)h(0)d0
(7.33)
Now rename the dummy variable of intégration, or let 0 = t, which gives
'
/ Pt(f - r)h(r)dr
(7.34)
J -oc