Civil Engineering Reference
In-Depth Information
2
22
40
i
t
t
Δ−
≤
⇒
Δ≤
(9.178)
ω
i
where (obviously),
is the largest eigen-frequency expected to play any significant
role in the response behaviour of the system.
In any case
ω
(
)
Δ
12
R
should not be chosen larger than about
ω
where
ω
is
R
max
max
the largest frequency contained in the load.
Numeric integration methods
The numeric integration methods are based on the assumption that higher order
quantities may be obtained from the situation at the onset of the time step and integration
of the response development between time steps, i.e. that
t
Δ
⎫
()
∫
d
r
=+
r
r
ττ
⎪
⎪
⎬
⎪
k
+
1
k
0
0
t
where
≤≤Δ
(9.179)
Δ
t
()
∫
r
=+
r
r
ττ
d
⎪
⎭
k
1
k
+
0
As illustrated in Fig. 9.10 the approximation is that the acceleration variation within the
time step is either assumed constant and equal to its initial value, it is assumed equal to
the average acceleration over the time step or it is assumed linear across the time step. If
constant initial acceleration is assumed (see Fig. 9.10.a), then
τ
()
()
∫
d
r
τ
=
r
⇒
r
τ
=+
r
r
τ
=+⋅
r
r
τ
(9.180)
k
k
k
k
k
0
and thus
t
Δ
⎫
∫
d
t
r
=+
r
r
τ
=+Δ⋅
r
r
⎪
⎪
⎬
k
+
1
k
k
k
k
0
(9.181)
Δ
t
2
t
Δ
⎪
(
)
∫
d
t
r
=+
r
r
+⋅
τ
r
τ
=+Δ⋅
r
r
+
r
⎪
⎭
k
1
k
k
k
k
k
k
+
2
0
If the concept of a constant average acceleration is adopted (see Fig. 9.10.b), then
