Civil Engineering Reference
In-Depth Information
()
()
tot
t
t
(9.162)
r
=+
r
r
A solution strategy may be pursued in the original finite element degrees of freedom
(developed in Chapter 9.4) or in modal coordinates (see chapter 9.8). If a solution in
original degrees of freedom is pursued, then the calculation of
()
t
r
will require the
solution of the dynamic equation given in Eq. 9.80 or 9.85, depending on the choice of
motion induced load description. It is seen that either of these equations may be written
on the general form
()
()
()
()
Mr
t
+
C
r
t
+
K
r
t
=
R
t
(9.163)
net
net
dyn
CCC
KKK
R
=−
⎪
=−
⎫
net
ae
if quasi-static properties are adopted
where
,
⎬
⎪
net
ae
()
()
dyn
t
t
=
R
⎭
(
)
CCC
KKK
=−
s
s
=
⎪
0
0
⎫
net
ae
(
)
=−
=
if indicial functions are adopted
.
⎬
⎪
net
ae
()
()
()
R
t
=
R
t
+ Δ
⎭
R
t
dyn
ae
If a solution in modal coordinates is pursued, then the calculation of
()
()
t
t
r Φη
=⋅
requires solution of the dynamic equation given in Eq. 9.143, i.e.
()
(
)
()
(
)
()
()
t
t
t
t
Mη
+−
CC η
+−
KK η R
=
(9.164)
ae
ae
where all quantities are defined in Eqs. 9.142, 9.144 - 9.148 and 9.155.
In any time domain solution it will be necessary to perform a stochastic simulation of
the stationary flow components
()
()
()
ut
,
vt
and
wt
contained in the buffeting load
R
()
()
vector
R
dyn
t
or
t
. Such a simulation procedure is presented in Appendix A.3.
There are a number of iteration procedures available for a time domain solution
strategy. Only a selected few are included below. In any case, as illustrated in Fig. 9.9 a
time domain solution will involve some discretisation of the load processes
()
dyn
t
R
or
R
()
t
at time step
t
(
k
1,2,,....,
N
), and a stepwise calculation of the corresponding
response (
r
or
η
). Based on the knowledge of the response at time step
t
and the
discretised values of the load, the task at hand is to calculate the response at time step
=
k
t
. Such a forward prediction routine is called explicit if it is based on the known
response history alone. It is called implicit if it contains assumptions about the response
situation or equilibrium condition in the unknown future of the system. I.e., in an explicit
+
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