Page 436 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 13.3 Frequency Response Function 423

Input F{f) System Output /( /)
/?(/) Figure 13.3-1. Input-output rela­
tionship of a linear system.

In the time domain, the system behavior can be determined in terms of the
system impulse response h(t) used in the convolution integral of Eq. (4.2-1).

y(0 = f ‘x ( O h ( t - (13.3-1)
•'n
A much simpler relationship is available for the frequency domain in terms of the
frequency response function H((o), which we can define as the ratio of the output
to the input under steady-state conditions, with the input equal to a harmonic time
function of unit amplitude. The transient solution is thus excluded in this consider­
ation. In random vibrations, the initial conditions and the phase have little
meaning and are therefore ignored. We are mainly concerned with the average
energy, which we can associate with the mean square value.
Applying this definition to a single-DOF system,

mÿ -\- cy ky = x{t) (13.3-2)
let the input be x(t) = The steady-state output will then be y =
where H{(o) is a complex function. Substituting these into the differential equation
and canceling from each side, we obtain
+ ¿CO) k)H{co) = 1

The frequency response function is then
1
Hiw)
k —mo)^ -f- ico)

1 1
(13.3-3)
1 -
As mentioned in Chapter 3, we will absorb the factor 1/k in with the force. H(co)
is then a nondimensional function of co/co^ and the damping factor
The input-output relationship in terms of the frequency-response function
can be written as
y(t) =H(co)F,e‘ (13.3-4)
where is a harmonic function.
For the mean square response, we follow the procedure of Example 13.2-1
and write
y = \F^{He'^^ -h (13.3-5)