
The mean square response and the frequency limit is normalized such that
¯
R
XX
(0) = R
XX
(0)
k
2
S
0
ω
n
¯ω
c
= ω
c
/ω
n
.
Then,
¯
R
XX
(0) =
¯ω
c
−¯ω
c
1
(1 − r
2
)
2
+ (2ζr)
2
dr
=
1
2
A arctan
¯ω
c
B
−B arctan
¯ω
c
A
1
C
1
A
1
B
,
which can be plotted as a function of ¯ω
c
for various values of ζ. The figure below shows the normalized mean
squares obtained using the exact analytical expression in a solid line, and those obtained using numerical
integration using quad.m in Matlab as dots. The corresponding Matlab code is also shown. It took my
computer 0.011 s to plot the analytical expression whereas it took 0.68 s to numerically integrate and plot.
It is interesting to note that the mean