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In order to obtain the solution at time
(
n
+
1
)
t
, we employ the equations of motion
of Eq. (10.97) at time
t
. Substitute Eq. (10.110) into Eq. (10.97) and arrange all of
the known terms on the right-hand side. This leads to
2
M
(
(
n
+
1
)
K
V
n
+
1
=
5
M
(
V
n
−
4
M
(
V
n
−
1
11
C
6
3
C
3
C
2
2
+
t
+
P
n
+
1
+
2
+
2
+
t
)
t
)
t
t
)
t
M
(
V
n
−
2
C
+
2
+
(10.111)
t
)
3
t
or
KV
n
+
1
=
P
n
+
1
where
2
M
(
11
C
6
K
=
2
+
t
+
K
)
t
5
M
(
V
n
−
4
M
(
V
n
−
1
+
M
(
V
n
−
2
3
C
3
C
2
C
P
n
+
1
=
P
n
+
1
+
2
+
2
+
2
+
t
)
t
t
)
t
t
)
3
t
As with the central difference method, this formulation needs a special starting pro-
cedure. Houbolt used the formulas for the derivatives at the third point of the four
successive points along the cubic curve. The formulas give the following values for
V
−
1
and
V
−
2
:
2
V
0
−
V
−
1
=
(
t
)
V
1
+
2
V
0
(10.112)
t
V
0
+
2
V
0
−
V
−
2
=
6
6
(
t
)
8
V
1
+
9
V
0
The algorithm for implementation of this method is outlined in Table 10.4.
To examine the stability of the Houbolt method, st
u
dy the single DOF case used f
or
the central difference method, where
K
,
C
,
M
,
and
P
are replaced by
k, c, m,
and
p
.
TABLE 10.4
Procedure for Computer Implementation of the Houbolt Method
Initial Step
1.
Select
t
2.
Calculate the constants related to
t
t
2
t
2
c
1
=
2
/
c
2
=
11
/
6
t
3
=
5
/
c
4
=
3
/
t
c
5
=−
c
6
=−
c
4
/
c
7
=
c
1
/
c
8
=
c
4
/
2
c
1
2
2
9
Initialize
V
0
,
V
0
, and
V
0
3.
4.
Calculate
V
−
1
and
V
−
2
from Eq. (10.112)
Form the effective stiffness matrix
K
=
+
+
5.
c
1
M
c
2
C
K
Decompose (triangularize)
K
6.
=
LDL
T
At Each Step:
1.
Calculate
P
n
+
1
=
P
n
+
1
+
(
c
3
M
+
c
4
C
)
V
n
+
(
c
5
M
+
c
6
C
)
V
n
−
1
+
(
c
7
M
+
c
8
C
)
V
n
−
2
Solve
KV
n
+
1
=
2.
P
n
+
1
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