Environmental Engineering Reference
In-Depth Information
and
T
z
¼½
d d
T
¼
d
0
d
0
d
r
ð
13
:
9
Þ
... d
r
...
is the system state. The 2n eigenvalues and eigenvectors of A fully describe the
free motion of the system shown in Fig.
13.13
when uncontrolled (f
di
= 0). In
particular, for the structural system considered, the complex eigenvalues s
i
come in
conjugate pairs which correspond to angular frequencies x
i
and modal damping
ratios n
i
[
19
] as follows (j is the complex unity):
q
1
n
i
s
i
;
ic
¼
f
i
x
i
j x
i
ð
13
:
10
Þ
or, conversely, as
x
i
¼
s
jj
n
i
¼
Real
ð
s
i
Þ
s
jj
:
ð
13
:
11
Þ
Each modal frequency and damping ratio can be drawn through Eq.
13.11
from
either the corresponding eigenvalue s
i
or from its complex conjugate s
i
,
c
. Analo-
gously, the 2n eigenvectors of A come in complex conjugate pairs that can be
collected in a 2 9 2n matrix W:
ð
13
:
12
Þ
Each column of the matrix W can be thought as made of 2 n-component vectors.
The first n-component vector is actually the complex modal shape, the second part
is the modal shape times the corresponding complex frequency. If damping can be
neglected, the following equation holds:
ð
13
:
13
Þ
otherwise w
i
is the complex modal shape corresponding to the real, undamped
counterpart u
i
, but explicitly considering the damping. W* here denotes the subset
of W directly comparable to U.
Let us assume that each control force f
di
, at a given instant of time, has a
specific value f
ui
, function of the system's state through a gain matrix G as follows:
T
¼
f
u
ðÞ¼
f
u1
ð
t
Þ
T
¼
G
z
ðÞ:
ð
13
:
14
Þ
f
d
ðÞ¼
f
d1
ð
t
Þ
½
... f
dm
ð
t
Þ
½
... f
um
ð
t
Þ
