546
15 Random Vibrations
Fig. 15.9 White noise
Example 15.6 Determine the response spectral density and mean square of response
of a SDF system having mass m, damping constant c and spring stiffness k due to a
white noise of magnitude S 0 .
If the spectral density of a random process remains constant at all ranges of the
frequency, then it is referred to as white noise (Fig. 15.9). From Eq. (3.136), we get
|H ( f )|
2
=
1
(k − 4π 2 m f 2 ) 2 + 4π 2 f 2 c 2
From Eq. (15.50), we get
S xx ( f ) =
1
(k − 4π 2 m f 2 ) 2 + 4π 2 f 2 c 2 ,
S F F ( f ) =
S 0
(k − 4π 2 m f 2 ) 2 + 4π 2 f 2 c 2
The mean square of the response is given by
x
2
=
∞
−∞
S xx ( f )d f = 2
∞
0
S xx ( f )d f
= 2
∞
0
S 0
(k − 4π 2 m f 2 ) 2 + 4π 2 f 2 c 2 d f
15.9 Random Response of MDF Systems
Complex structures have multiple degrees of freedom. The random response analysis of these structures can be carried out in the frequency domain. There are two
approaches to the problem.