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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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