
Section 7.2: Response to Random Loads
7.4. Consider the two degree-of-freedom system shown in Figure 7.1 where F
1
= 0 and, in addition to
F
2
, random force F
3
is applied to the second mass. The force spectrum is given by
[S
F F
(ω)] =
S
F
2
F
2
(ω) S
F
2
F
3
(ω)
S
F
2
F
3
(ω) S
F
3
F
3
(ω)
.
Derive an expression for the matrix response spectrum [S
XX
(ω)] in terms of the force spectrum, the
modal matrix, and the matrix frequency response function [H (iω)].
Solution: The transfer function H
ij
(ω) relates the i
th
response to force applied at the j
th
coordinate. The
transfer function matrix is given by
(
¯
H (ω)
)
=
H
11
H
12
H
12
H
21
H
22
H
22
.
Since, F
1
= 0, the spectral densities are all equal ...