Page 435 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

422 Random Vibrations Chap. 13

The one-sided summation in the previous equation is complex and, hence,
the real part of the series must be stipulated for x(t) real. Because the real part of
a vector is one-half the sum of the vector and its conjugate [see Eq. (1.1-9)],
00 ^ 0 0
x(/) = Re i ; ^ E (Q e'"“'' +
M=1 n^\

By comparison with Eq. (1.2-6), we find
2 rT/ 2
= 2c„ = y i x{t)e
i J - r /2
= a„ - ib„ (13.2-9)
Example 13.2-1
Determine the mean square value of a record of random vibration x(t) containing
many discrete frequencies.
Solution: Because the record is periodic, we can represent it by the real part of the
Fourier series:
00
x(i) = ReEC„e'"“"'
1

1
where is a complex number, and C* is its complex conjugate. [See Eq. (13.2-9).]
Its mean square value is
___ 1 T 1
x^= lim ~ i ^ D dt
'
7--» ^ ■o =
= lim x; J ^ -I- 2C„C* + ^
_ j ^ \ ^ ^ -i2na)QT
<^1 °° 1 __
= E 2^nC: = E 2 ^ c f = E

n=l n =\

In this equation, for any t, is bounded between ± 1, and due to T -> oo in
the denominator, the first and last terms become zero. The middle term, however, is
independent of T. Thus, the mean square value of the periodic function is simply the
sum of the mean square value of each harmonic component present.

13.3 FREQUENCY RESPONSE FUNCTION

In any linear system, there is a direct linear relationship between the input and the
output. This relationship, which also holds for random functions, is represented by
the block diagram of Fig. 13.3-1.