Environmental Engineering Reference
In-Depth Information
z
r
=
rth
modal damping ratio
m
r
=
rth
modal mass (kg)
n = number of modes;
y
ir
=
ith
component of mode shape r
The next step of the theoretical development is to form the cross-correlation function of
two responses (
x
ik
and
x
jk
) to a white noise input at a particular input point
k
. The cross-
correlation function Â
ijk
(
T
) can be defined as the expected value of the product of two re-
sponses evaluated at a time separation of
T
:
i jk
(
T
) =
E
[
x
(
t
+
T
)
x
(
t
)]
(11-52)
jk
ik
where
E
is the
expectation operator
.
Substituting Equation (11-51) into (11-52), recognizing that
f
k
(t)
is the only random vari-
able, allows the expectation operator to depend only on the forcing function. Using the defi-
nition of the
autocorrelation function
, and assuming for simplicity that the forcing function
is white noise, the expectation operation collapses to a scalar times a
Dirac delta function
.
The Dirac delta function then collapses one of the Duhamel integrations embedded in the
cross-correlation function. The resulting equation can be simplified via a change of vari-
able of integration, setting l =
t
-t. Using the definition of
g
from Equation (11-51) and the
trigonometric identity for the sine of a sum results in all the terms involving
T
being separated
from those involving l. This separation allows terms that depend on
T
to be factored out of
the remaining integral and out of one of the modal summations, with the following results:
é
ë
ù
cos w
d
T
ê
û
A
i jk
exp - z
r
w
n
T
n
r
=1
i jk
(
T
) =
(11-53)
+
B
i jk
sin w
d
T
exp - z
r
w
n
T
where
A
ijk
and B
ijk
are independent of
T
, are functions of only the modal parameters, and
completely contain the remaining modal summation, as shown in the following equations:
æ
æ
æ
ç
ç
è
A
i jk
sin w
d
l
-z
r
w
n
ç
ç
è
ç
ç
ç
n
s
=1
a
k
y
ir
y
kr
y
js
y
ks
×
¥
°
d
l.
l
×
sin w
d
l
ò
exp
=
í
í
B
i jk
-z
s
w
n
cos w
d
l
è
ç
m
w
d
m
y
d
r
s
(11-54)
Equation (11-53) is the key result of this derivation. Examining Equation (11-53), the
cross-correlation function is seen to be a sum of decaying sinusoids, with the same charac-
teristics as the impulse response function of the original system. Thus, cross-correlation
functions can be used as impulse response functions in time-domain modal parameter esti-
mation schemes. This is more clearly seen after further simplification using a new constant
multiplier (
Gj
r
), as follows:
y
ir
G
r
j
m
r
w
d
n
r
=1
exp - z
r
w
n
T
sin w
d
T
+ Q
r
R
(
T
) =
.
(11-55)
i j
James
et al
. [1995] present more definition of the intermediate steps in this derivation and the
results of analytical verification checks of the derivation.
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