Chapter 5 Robust Performance Tailoring with Tuning - SSL - MIT

the eigenvalue equation, the derivatives of the frequencies are obtained by differenti-
ating Equation 2.5:
∂ωj
∂x
1
= φ
2ωj
T j
�
−ω 2 ∂M
j
∂x
�
∂K
+ φj
∂x
(2.21)
Obtaining the derivatives of the eigenvectors, or mode shapes, is slightly more in-
volved. Using Nelson’s method [98], it is assumed that the j th eigenvector derivative
is written as a linear combination of the j th eigenvector and a linearly independent
vector, ψj:
∂φj
∂x = ψj + ajφj
The scalar aj is obtained through derivation of the mass normalization equation:
aj = −φ T j
�
Mψj + 1
2
∂M
∂x φj
�
The vector ψj is found by solving the following matrix equation:
where Kj and bj are defined as:
bj = Kjψj
(2.22)
(2.23)
(2.24)
Kj = −ω 2 j M + K (2.25)
bj = ∂ω2 j
∂x Mφj
�
− −ω 2 �
∂M ∂K
j + φj
(2.26)
∂x ∂x
The matrix Kj is singular with rank of n−1 if the eigenvalues are distinct. This issue is
addressed by arbitrarily removing one element from bj as well as the corresponding row
and columns from Kj. The corresponding element in ψj is set to zero. Equations 2.21
and 2.22 provide expressions for the derivatives of the modal quantities with respect
to the design variables as a function of the mass and stiffness matrix derivatives, ∂M
∂x
and ∂K.
These final quantities will differ depending on how the design variables enter
∂x
the stiffness and mass matrices. These gradients are derived in the following section
for the specific design variables chosen for the SCI model PT optimization.
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