6.20 Reduction Methods for Dynamic Analysis
243
⎧
⎨
⎩
x 1
x 2
x 3
⎫
⎬
⎭
=
⎡
⎣
1 0
1
2
1
2
0 1
⎤
⎦
x 1
x 3
Therefore,
[M] red = [H ]
T
[M] [H ]
=
⎡
⎢
⎣
m +
mk
2
4k 2
k
2 m
4k 2
k
2 m
4k 2
m +
mk
2
4k 2
⎤
⎥
⎦
=
⎡
⎣
5m
4
m
4
m
4
5m
4
⎤
⎦
[K ] red = [H ]
T
[K ] [H ]
=
⎡
⎢
⎣
2k −
k
2
2k
−
k
2
2k
−
k
2
2k
2k −
k
2
2k
⎤
⎥
⎦
=
⎡
⎣
3k
2
−
k
2
−
k
2
3k
2
⎤
⎦
It is suggested that the master degrees of freedom should be chosen in the region
of high flexibility and slave degrees of freedom in regions of high stiffness [17, 18].
Reference [19] describes automatic procedure for this.
According to Ref. [17], retained (master) and internal (slave) d.o.f. can be selected
on the basis of ratio of diagonal terms of [K] and [M] matrices of Eq. (6.144). Those
d.o.f. which yield the largest values of the ratio K ii /M ii are selected as slave d.o.f.
Attempts have been made to reduce the error inherent in the static consideration
method applied to dynamic problems. A dynamic consideration method has been
proposed which is as follows [20].
To start the process, an approximate value is assigned to the first eigenvalue
p
2
1 , and then, the dynamic condensation is applied to the dynamic matrix [D 1 ] =
[K ] − p
2
1 [M], and then solving the reduced eigenvalue problem to determine the
first and second eigenvalues p
2
2 and p
2
3 . Next, the dynamic condensation is applied
to the dynamic matrix [D 2 ] = [K ] − p
2
2 [M] to reduce the problem, and the
second and third eigenvalues p
2
1 and p
2
2 are calculated. The procedure is repeated
with one exact eigenvalue and an approximation of the next eigenvalue calculated at
each step.
243
⎧
⎨
⎩
x 1
x 2
x 3
⎫
⎬
⎭
=
⎡
⎣
1 0
1
2
1
2
0 1
⎤
⎦
x 1
x 3
Therefore,
[M] red = [H ]
T
[M] [H ]
=
⎡
⎢
⎣
m +
mk
2
4k 2
k
2 m
4k 2
k
2 m
4k 2
m +
mk
2
4k 2
⎤
⎥
⎦
=
⎡
⎣
5m
4
m
4
m
4
5m
4
⎤
⎦
[K ] red = [H ]
T
[K ] [H ]
=
⎡
⎢
⎣
2k −
k
2
2k
−
k
2
2k
−
k
2
2k
2k −
k
2
2k
⎤
⎥
⎦
=
⎡
⎣
3k
2
−
k
2
−
k
2
3k
2
⎤
⎦
It is suggested that the master degrees of freedom should be chosen in the region
of high flexibility and slave degrees of freedom in regions of high stiffness [17, 18].
Reference [19] describes automatic procedure for this.
According to Ref. [17], retained (master) and internal (slave) d.o.f. can be selected
on the basis of ratio of diagonal terms of [K] and [M] matrices of Eq. (6.144). Those
d.o.f. which yield the largest values of the ratio K ii /M ii are selected as slave d.o.f.
Attempts have been made to reduce the error inherent in the static consideration
method applied to dynamic problems. A dynamic consideration method has been
proposed which is as follows [20].
To start the process, an approximate value is assigned to the first eigenvalue
p
2
1 , and then, the dynamic condensation is applied to the dynamic matrix [D 1 ] =
[K ] − p
2
1 [M], and then solving the reduced eigenvalue problem to determine the
first and second eigenvalues p
2
2 and p
2
3 . Next, the dynamic condensation is applied
to the dynamic matrix [D 2 ] = [K ] − p
2
2 [M] to reduce the problem, and the
second and third eigenvalues p
2
1 and p
2
2 are calculated. The procedure is repeated
with one exact eigenvalue and an approximation of the next eigenvalue calculated at
each step.