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Assume that
c
=
0 and
p
=
0
,
and write Eq. (10.111) as
X
n
+
1
=
AX
n
(10.113)
where
V
n
+
1
V
n
V
n
−
1
V
n
V
n
−
1
V
n
−
2
X
n
+
1
=
X
n
=
and
/(
+
ω
2
t
2
)
−
/(
+
ω
2
t
2
)
/(
+
ω
2
t
2
)
5
2
4
2
1
2
ω
=
1
0
0
2
=
/
A
k
m
(10.114)
0
1
0
To study the stability criterion, follow the procedure outlined in Eqs. (10.104) and
(10.108). The same stability condition,
|
λ
i
|≤
1
,
can be obtained. It can be shown that
the three eigenvalues
λ
1
,
λ
2
,
and
λ
3
of
A
are always less than 1, and, hence, the stability
condition
1
,
2
,
3 is always satisfied. It is concluded that the Houbolt
method is always stable regardless of the size of the time step
|
λ
i
|≤
1 with
i
=
t
used for integration.
An integration scheme with this property, which is a characteristic of implicit integration
schemes, is said to be
unconditionally stable
. The time step
t
to be used for integration,
however, will be selected in order to achieve proper accuracy.
10.5.3
Newmark Method
In the
Newmark method
or
Newmark's
, are introduced
to indicate how much of the acceleration enters into the relations for velocity and dis-
placement at the end of the time interval. The relations that are adopted are
β
method
, two parameters,
γ
and
β
1
2
−
β
V
n
+
2
V
n
+
β(
2
V
n
+
1
V
n
+
1
=
V
n
+
t
(
t
)
t
)
(10.115a)
V
n
+
1
=
V
n
+
(
t
V
n
+
γ
t
V
n
+
1
1
−
γ)
(10.115b)
Equation (10.115a) can be used to express
V
n
+
1
in terms of
V
n
+
1
. Substitution of this
relation into Eq. (10.115b) gives
V
n
+
1
in terms of
V
n
+
1
. The recurrence relationship for
V
n
+
1
then can be obtained by substituting these two relations into the equations of motion
at time
(
n
+
1
)
t
.
The result is
KV
n
+
1
=
P
n
+
1
(10.116)
where the effective stiffness
K
and effective loading vector
P
n
+
1
are given by
1
β(
+
γ
β
1
K
=
K
+
2
M
t
C
t
)
M
1
β(
1
2
1
V
n
1
β
1
t
V
n
+
P
n
+
1
=
2
V
n
+
β
−
t
)
C
γ
β
γ
β
−
1
γ
β
−
2
V
n
1
t
V
n
+
+
t
V
n
+
+
P
n
+
1
(10.117)
2
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