Environmental Engineering Reference
In-Depth Information
and, finally,
h
i
u
i
¼
H
i
u
i
(dim H
i
¼
2n
m
Þ:
1
B
w
d
;
i
¼
s
d
;
i
I
A
ð
13
:
22
Þ
Equation
13.22
shows the relationship among the desired eigenvalue s
d,i
and
eigenvector w
d,i
, the matrix of the original, uncontrolled system A and the cor-
responding control forces u
i
, i.e., the control forces able to make the controlled
system vibrate according to desired modal shape, frequency and damping ratio.
Should the matrix H
i
be invertible, calculation of the control forces u
i
would be
straightforward. However, it is generally not. An approximate approach to solve
Eq.
13.22
for u
i
is to consider the pseudo-inverse matrix H
i
^ofH
i
. In this case, u
i
can be approximately evaluated as
h
i
w
d
;
i
:
1
H
i
u
i
¼
H
i
w
d
;
i
¼
H
i
H
i
ð
13
:
23
Þ
However, by using the approximation expressed by Eq.
13.23
, the actual
eigenvector w
CL,i
of the CL system will be similar, but not exactly equal to the
desired one w
d,i
:
w
CL
;
i
¼
H
i
u
i
w
d
;
i
:
ð
13
:
24
Þ
If the approximation of Eq.
13.23
is acceptable, by selecting the desired fre-
quency and damping ratio (through s
d,i
) and the shape (through w
CL,i
) of each
mode of vibration, it is possible to calculate the corresponding values of the
desired control forces u
i
and the resulting CL eigenvector w
CL,i
, to be collected in
the matrices U and W
CL
, respectively:
U
¼
u
1
½
ð
dim U
¼
m
2n
Þ
u
2
... u
2n
ð
13
:
25
Þ
W
CL
¼
w
CL
;
1
w
CL
;
2
... w
CL
;
2n
ð
dim W
CL
¼
2n
2n
Þ:
Recalling Eq.
13.17
, it is:
U
¼
G W
CL
ð
26
Þ
and, therefore, the gain matrix can be found as
G
¼
U
W
1
CL
:
ð
13
:
27
Þ
Once G is calculated through Eq.
13.27
, the corresponding control forces f
u
defined by Eq.
13.14
are able to approximately transform the original structure so
that it has the desired modal properties:
• frequency and damping ratio of each selected mode;
• modal shapes.
