Page 461 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

448 Random Vibrations Chap. 13
For a harmonic excitation fit) = the preceding equation beeomes

x(t) = r e ‘^^'~^^h(r) dr
A)
= rh{T)e dr
f)
Beeause the steady-state output for the input yit) = e""' is x = H{co)e‘'^', the
frequeney-response function is

H{w)= rh{T)e ‘^^^ dr = r h{r)e •^'^^dr
f) J-yc
which is the FT of the impulse response funetion hit). The lower limit in the
preeeding integral has been ehanged from 0 to -^c because hit) = 0 for negative t.

13.8 FTs AND RESPONSE
In engineering design, we often need to know the relationship between different
points in the system. For example, how much of the roughness of a typical road is
transmitted through the suspension system to the body of an automobile? (Here
the term transfer function^ is often used for the frequency-response function.)
Furthermore, it is often not possible to introduce a harmonic excitation to the
input point of the system. It may be necessary to accept measurements xit) and
y (0 at two different points in the system for which the frequency response function
is desired. The frequency response function for these points can be obtained by
taking the FT of the input and output. The quantity Hico) is then available from
y{(o) FT of output
H{co) = (13.8-1)
X{oj) FT of input
where X{co) and Vico) are the FT of xit) and yit).
If we multiply and divide this equation by the complex conjugate A^*(o)), the
result is
Y{co)X^{co)
H(co) (13.8-2)
X(co)X^(co)
The denominator X{co)X"'{o)) is now a real quantity. The numerator is the cross
spectrum Y(oj)X*ico) between the input and the output and is a complex quantity.
The phase of Hico) is then found from the real and imaginary parts of the cross
spectrum, which is simply
=\Y{oj)X*(co)\/4>y - 4>, (13.8-3)

^Strictly speaking, the transfer function is the ratio of the Laplace transform of the output to the
Laplace transform of the input. In the frequency domain, however, the real part of s = a + ico is zero,
and the LT becomes the FT.