STRUCTURAL RESPONSE STATISTICS: PART H
185
from the two governing differential équations (7.70) and (7.74). These and the
other matrices of équation (7.77) are as follows:
z
T
= [zi 22 Z3 Z4]T = v û E» ÉL
m m
■ T
(7.78)
Z\ — i) — *2
(7.79a)
Z? — V — —
- a>QZi + Z3
(7.79b)
, _P1 _
23 —
—
m
z4
(7.79c)
z4 = —2Çô>z4 — û2Z3 +
m
(7.79d)
F =
’ 0
1
-col -2 0
0
0
0
0
0 '
1
0
0
1
——2£û
(7.80)
1 -2 - 1/2
—a) a ' w
m
(7-81)
Fw= 0 0 0
In this case, the only nonzero term of the 4 by 4 matrix T is T4 4 which is
Covariance Propagation Equation: Dérivation
The next task is to cast the State variable form équation (7.77) in its statistical counterpart, the covariance propagation équation, also in State variable
form. It will be shown that solutions to this latter matrix équation yield statistical responses, which include the variance of displacement
for a single degree
of freedom structure. The dérivation that follows, based on the expositions of
Hedrick (1984) and Lin (1967), is general in that the results are applicable to
multi-degree of freedom linear structures also. However, zéro mean is assumer!
for both the State variable z and the white noise w, or
E^z] — E[w] - 0
(7-82)
Define the covariance propagation matrix Z for zéro mean. Let
Z(t) = Z = E[zz']
(7-83)