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In-Depth Information
One procedure that is often effective is the central difference method in which the velocity
and acceleration are expressed as
1
1
V
n
V
n
=
t
[
V
n
+
1
−
V
n
−
1
]
,
=
t
2
[
V
n
+
1
−
2
V
n
+
V
n
−
1
]
(10.98)
2
where subscripts
n
−
1
,n
, and
n
+
1 represent the time
(
n
−
1
)
t, n
t
, and
(
n
+
1
)
t,
respectively, with
n
....
The displacement at time
=
0
,
1
,
(
n
+
1
)
t
is obtained by considering Eq. (10.97) at time
n
t
,
i.e.,
M V
n
+
C V
n
+
KV
n
=
P
n
(10.99)
V
n
and
V
n
of Eq. (10.98) into Eq. (10.99), giving
Substitute the relations for
M
V
n
+
1
=
K
V
n
−
M
V
n
−
1
C
2
M
C
t
2
+
P
n
−
−
t
2
−
2
t
t
2
2
t
or
KV
n
+
1
=
P
n
+
1
(10.100)
with
M
C
K
=
t
2
+
2
t
K
t
2
V
n
M
V
n
−
1
2
M
C
P
n
+
1
=
P
n
−
−
−
t
2
−
2
t
The constant matrix
K
is called an
effective stiffness matrix
, and
P
n
+
1
is the
effective load
at
time step
n
.
Since
M
and
C
are diagonal matrices, it is not necessary to solve a system of simultaneous
equations. Note that
K
is not a function of
K
.
In these relations, the equation for
V
n
+
1
involves
V
n
and
V
n
−
1
+
1. Equation (10.100) is a set of linear equations and can be used to find
V
n
+
1
.
Therefore, in order to
calculate the solution at the first time step
t
, a special starting procedure must be used.
For example, use the second relationship of Eq. (10.98) with
n
=
0 to derive an expression
V
0
.
=
(
−
)/
for
V
−
1
With
V
1
V
0
t,
this leads to
+
t
2
2
t
V
0
V
0
=
−
V
−
1
V
0
(10.101)
V
0
can be obtained directly from Eq. (10.97) with
V
0
and
V
0
known. Table 10.2
where
summarizes the procedure for computer implementation.
Stability and Accuracy
Consider a single-DOF sys
te
m so that
K
,
C
,
M
,
and
P
reduce to
k, c, m,
and
p
. Rearrange
Eq. (10.100), and set
c
=
0
, p
=
0
.
=
X
n
+
1
AX
n
(10.102)
where
V
n
+
1
V
n
V
n
V
n
−
1
X
n
+
1
=
X
n
=
and
2
2
t
2
−
ω
−
1
2
A
=
ω
=
k
/
m
(10.103)
1
0
Matrix
A
is called the
amplification matrix
. Stability and accuracy of an integration algo-
rithm depend on the eigenvalues of this amplification matrix, and in order to have a stable
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