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they fell within a small distance from each other.
A more generalized mean eigenvalue objective
function has been considered by Ma et al. (1995).
For example considering the harmonic mean of
the first two eigenvalues, we can define the fol-
lowing objective function
all of its components being zero except for its
i
-th
component which is 1.
Due to this perturbation, the stiffness and mass
matrices will change to
∂
∂
K
K x
(
+
ε
∆
x
)
= +
K
ε
+
o
( )
ε
,
i
x
i
(1.36)
−
(
)
1
∗
−
−
λ
=
λ
1
+
λ
1
(1.33)
∂
∂
M
2
M x
(
+
ε
∆
x
)
= +
M
ε
+
o
( )
ε
1
2
i
x
i
Using this objective function in example 2
results in a smooth increase for the first two ei-
genvalues. Figure 6 shows the final solution and
the evolution history of the first three eigenvalues
considering the objective function defined in
Equation (1.33) in example 2.
and the eigenvalues and the eigenvectors will
change to
λ
(
x
+
ε
∆
x
)
= +
λ
εµ
+
o
( ),
ε
j
=
1
,
…
,
m
j
i
j
(1.37)
5.2. Sensitivity Analysis of Multiple
Eigenvalues
ϕ
(
x
+
ε
∆
x
)
=
ϕ
+
ε
ν
+
o
( ),
ε
j
=
1
,
…
,
m
j
i
j
j
(1.38)
It is also possible to solve the multimodal eigen-
frequency optimization problems directly. Using
a perturbation technique, Bratus and Seyranian
(1983) calculated the sensitivities of multiple
eigenvalues. Here we follow Seyranian et al.
(1994) and Lund (1994) to present the sensitivity
analysis of multiple eigenfrequencies.
Assume an
m
-fold multiple eigenvalue
where
μ
j
and
ν
j
are the unknown sensitivities
of the multiple eigenvalues and eigenvectors
respectively.
o
(
ε
) indicates higher order terms.
Note that
μ
j
≡
μ
j
(
x
, ∆
x
i
) and
ν
j
≡
ν
j
(
x
, ∆
x
i
), i.e. the
sensitivities depend on ∆
x
i
.
Using the perturbed values of Eqs. (1.36),
(1.37), and (1.38) in the main eigenvalue problem,
after ignoring the higher terms, one obtains
λ
=
λ
,
j
=
1
,
…
,
m
(1.34)
∂
∂
K
∂
∂
M
(
)
=
j
−
λ
ϕ
+
K M
−
λ
ν
µ
M
ϕ
j
j
j
j
x
x
i
i
Due to multiplicity, any linear combination
of the corresponding eigenvectors will satisfy
the main eigenvalue problem
Kφ
j
=
λ
j
Mφ
j
. Now
assume the following linear combination
(1.39)
Premultiplying Equation (1.39) by
ϕ
T
=
1
, the second term in the left-hand
side will cancel out and one obtains the following
m
equations
,
s
,
,
m
m
∑
α
,
j
1
,
…
,
m
ϕ
=
ϕ
=
(1.35)
j
jk
k
k
=
1
∂
∂
K
∂
∂
M
T
T
ϕ
−
λ
ϕ
=
µ
ϕ
M
ϕ
,
s
=
1
,
…
,
m
s
j
j
s
j
x
x
where the coefficients
α
jk
are unknown. If we
apply a perturbation
ε
to the
i
i-th optimization
variable the vector of design variables changes
to
x
+
ε
∆
x
i
where ∆
x
i
denotes a vector of size
N
,
i
i
(1.40)
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