Geology Reference
In-Depth Information
−
1
z
= −
ω
B
Cz
(1.97)
If we now define the objective function as
S
C
m
=2
, for the sensitivities of this function
we can write
c
c
which after substituting in the first equation of
(1.95), gives
∂
∂
K
∂
∂
∂
M
z
z
T
2
−
Ω
−
ˆ
ˆ
S
S
x
x
2
−
1
Kz
C
=
0
,
K B
= +
ω
CB C
(1.98)
∂
∂
c
x
e
e
m
=
sign( )
c
∂
∂
K
M
C
z
∂
∂
e
z
T
2
z
2
Ω
z
T
−
Ω
+
C
C
C
S
x
∂
x
x
Using Equation (1.98), we can now write
e
e
e
(1.93)
ˆ
ˆ
∂
∂
K
z
∂
∂
z
∂
∂
K
z
∂
∂
z
0
ˆ
ˆ
T
T
= −
K
C
⇒
z
= −
z K
C
=
Ignoring damping effects, we have
C
= 0,
z
S
= 0, and
z
C
=
u
C
, and thus the above equation
reduces to Equation (1.73).
The vectors
z
C
and
z
S
in Equation (1.93) can
be obtained by solving Equation (1.80). Given
the damping matrix
C
, the term ∂
C
/∂
x
e
is also
calculable. For example, assuming the Reyliegh
damping formulation of Equation (1.2), we have
∂
C
C
C
C
x
x
x
x
e
e
e
e
(1.99)
Differentiating Equation (1.98), we get
K
∂
∂
=
∂
∂
B
∂
∂
C
B C
2
−
1
+
2
ω
−
x
x
x
e
e
e
∂
∂
B
B C
∂
∂
ω
e M e K e
. Having
the sensitivities, one can solve the optimization
problem using a suitable solution algorithm.
C
/
∂ = ∂
x
a
M
/
∂ + ∂
x
a
K
/
∂
x
ω
2
CB
−
1
−
1
+
2
ω
CB C
−
1
x
x
e
e
(1.100)
Pre- and post-multiplying this equation by
z
C
and using Equation (1.97), we obtain
8.2. Free Vibration with Damping
Using a similar approach followed in the previous
section, in case of free vibration, Eqs. (1.79) and
(1.80) need to be changed to
ˆ
∂
∂
K
z
∂
∂
B
z
∂
∂
C
z
T
T
z
=
z
−
2
ω
z
−
C
C
C
C
C
S
x
x
x
e
e
e
∂
∂
B
z
∂
∂
ω
x
e
z
T
−
2
z Cz
T
=
0
(
)
Mz Cz Kz
0,
z
z
i
z
e
ω
,
(1.94)
i
t
+
+
=
=
+
S
S
C
S
x
C
S
e
(1.101)
Bz
−
ω
ω
Cz
=
0
0
Differentiating Equation (1.96), we have
C
S
(1.95)
Bz
+
Cz
=
S
C
∂
∂
B
=
∂
∂
K
∂
∂
M
∂
∂
ω
2
−
ω
−
2
ω
M
x
x
x
x
respectively where
ω
is a natural frequency and
e
e
e
e
(1.102)
= −
ω
2
B K
M
(1.96)
Substituting this equation in Equation (1.101)
and rearranging yields
From the second equation in (1.95), for
z
S
we
can write
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