Civil Engineering Reference
In-Depth Information
routine
(
)
is a function of
r
,
r
and
r
at
t
t
tt
, while an implicit routine
contains some assumptions about the development of the motion in the time step
between
r
+Δ
=
k
. Obviously, the shorter time step, the easier it is to obtain a
good solution. Among the implicit methods there are two main classes of solution
strategies, i.e. the direct iteration procedures and the numeric integration methods. Some
of these are presented below.
t
and
t
+Δ
t
Fig. 9.9
Time domain
The second central difference method
Consider the situation at time step
t
,
t
and
t
. A Taylor series expansion of
r
−
1
+
1
k
+
1
and
r
is given by:
k
1
−
2
⎫
t
Δ
"
r
=+Δ⋅
r
t
r
+
⋅
r
+
⎪
⎪
⎬
k
1
k
k
k
+
2
(9.165)
2
t
Δ
⎪
"
r
=−Δ⋅
r
t
r
+
⋅
r
−
⎪
⎭
k
1
k
k
k
−
2
Thus, considering only the three first terms of the Taylor series expansion,
r
−
r
k
1
k
1
+
−
renders
1
2
(
)
r
≈⋅
r
−
r
(9.166)
k
k
1
k
1
+
−
t
Δ
