Civil Engineering Reference
In-Depth Information
then the following modal dynamic equilibrium condition is obtained
(
)
(
)
Mη CC η KK η R
+−
+−
=
(9.143)
ae
ae
Due to the orthogonal properties of the mode shapes all the off diagonal terms in
M
,
C
and
K
are zeros. Thus
⎡⎤
⎫
M
=
diag
M
T
⎫
M
C
K
=
φ Mφ
φ Cφ
φ Kφ
⎣
⎪
⎪
⎡⎤
i
i
i
i
⎪
⎪
T
diag C
C
=
where
=
(9.144)
⎬
⎪
⎬
⎣⎦
⎪
⎡⎤
i
i
i
i
T
=
K
=
diag
K
⎪
⎭
⎪
i
i
i
⎣⎦⎭
i
However, it is readily seen that
K
may more conveniently be determined from Eq.
9.140. Pre-multiplying
⎡
2
i
⎤
T
i
φ
, and thus it is seen that
KMφ 0
by
−
ω
=
⎣
⎦
i
(9.145)
2
K
M
=
ω
i
i
i
Furthermore, it is common practice to introduce
N
ζ
modal damping ratios
, each
mod
i
associated with the corresponding modal critical damping
2
M
ω
, and thus
ii
CM
ω ζ
=
2
(9.146)
i
i
i
i
It is seen from Eqs. 9.52 - 9.55 that
c
and
k
are not symmetric, and thus, nor are
ae
ae
0
0
c
k
the corresponding element properties
and
defined in Eq. 9.63. Therefore, the
ae
n
ae
n
C
K
aerodynamic damping and stiffness matrices
and
are non-symmetric, and the
ae
ae
C
and
K
above. Thus
orthogonal mode shapes will not nullify off-diagonal terms in
ae
ae
⎡
⎤
%$
⎢
⎥
T
C
=
C
where
C
=
φ C φ
(9.147)
⎢
⎥
ae
ae
ij
ae
i
ae
j
ij
⎢
⎥
$%
⎢
⎥
⎣
⎦
and
⎡
⎤
%$
⎢
⎥
T
K
=
K
where
K
=
φ K φ
(9.148)
⎢
⎥
ae
ae
ij
ae
i
ae
j
ij
⎢
⎥
$%
⎢
⎥
⎣
⎦
