188
STATISTICAL RESPONSES FOR LINEAR STRUCTURES
real part of the largest eigenvalue, and successive matrix solutions are generated
by the following recursion formula:
Z„+1= Zn+V2"Zn(V2n)T
(7.98)
The quantités of équation (7.98) are defined as follows, in which I is the identity
matrix:
U = Q3I - F)-1
(7.99a)
V = U(/3I + F)
(7.99b)
W =2/3UBUt
(7.99c)
Zj = W + VWVT
(7.99d)
The génération of the successive terms in the sériés is aborted when the successive changes in partial sums becomes sufficiently small, or less than one percent.
If the System is not too large, then the following closed form solution to
équation (7.97), based on Laplacian transforms, can be used (Lin, 1967). That
is,
/»OO
Z=
exp(F • t)Bexp(FT ■ t)dt
(7.100a)
Jo
exp(F • t) = £-1(sI — F)-1
(7.100b)
exp(F ■ t) = L 1 (si - Ft) 1
(7.100c)
where C 1 is the inverse Laplace transform and s is the Laplace operator.
A Closed Form Solution
The solution to the covariance propagation équation for the single degree
of freedom System described by équations (7.70)-(7.81) was calculated using
the intégral solution of équations (7.100). One resuit is an expression for the
variance of the displacement in terms of the three structural constants (m. fci,ci)
and the three load excitation parameters (ô,w,<). That is
2an
Oj + a2
a3 + a4
+—7173 + 2,j7lT4 + 2e27273 ( 2e27374
(7 101)
ûi + a3
»! + a4
o2 + q3 1 Q3 + Q4
where