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FIGURE 10.11
Linear interpolation of the acceleration for the Wilson
θ
method.
Equations (10.119) and (10.120) give at time
(
n
+
θ)
t
V
n
+
θ
=
V
n
+
θ
t
V
n
+
θ
+
V
n
)
(
(10.121)
2
2
t
2
V
n
+
θ
V
n
+
θ
+
2
V
n
)
V
n
+
θ
=
V
n
+
θ
t
(
(10.122)
6
t
, solve for
V
n
+
θ
and
V
n
+
θ
from Eqs. (10.121) and
(10.122) and substitute into the equation of motion of Eq. (10.97) at time
To obtain the solution at time
(
n
+
1
)
(
n
+
θ)
t
, i.e.,
C V
n
+
θ
+
M V
n
+
θ
+
KV
n
+
θ
=
P
n
+
θ
(10.123)
with
P
n
).
A linear load vector is employed here because the accelerations are assumed to vary
linearly. Equation (10.123) reduces to
P
n
+
θ
=
P
n
+
θ(
P
n
+
1
−
KV
n
+
θ
=
P
n
+
θ
(10.124)
where
6
M
(θ
3
θ
K
=
K
+
2
+
t
C
t
)
and
M
6
(θ
2
V
n
6
θ
t
V
n
+
P
n
+
θ
=
P
n
+
θ(
P
n
+
1
−
P
n
)
+
2
V
n
+
t
)
C
3
θ
V
n
2
V
n
+
θ
t
+
t
V
n
+
2
(
+
)
After
V
n
+
θ
is found, the solution at time
n
1
t
can be obtained from Eqs. (10.121) and
(10.122) by setting
1. The algorithm for this technique is outlined in Table 10.6.
The stabil
it
y of the Wilson
τ
=
θ
method can be investigated using the single-DOF case
with
c
0
,
and the magnification matrix
A
, which can be obtained from Eqs.
(10.119) and (10.120) with
=
0
, p
=
τ
=
1as
X
n
+
1
=
AX
n
(10.125)
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