Wind Energy

302 J. Reetz
information about the current stiffness parameters by means of the given
model and the measured eigenfrequencies. Due to the formulation as multiparameter
eigenvalue problem the ill-posed problem is transformed into
common linear eigenvalue problems. The last step is the diagnostics. Thereby
the interpretation of parameters provides detection, quantification and localisation
of the damage. To simplify matters the equations hold for a number
of two parameters without loss of generality. The problem of identification of
the stiffness parameters by means of measured eigenfrequencies is
�
(−ˆω1M + a1K1 + a2K2) g1 = 0,
(56.1)
(−ˆω2M + a1K1 + a2K2) g2 = 0.
These eigenvalue equations comprise the known model matrices Ki and Mi,
the known measured eigenfrequencies ˆωi, the unknown eigenvectors gi and
the unknown stiffness parameters ai, which have to be identified. The same
elements in a generalised notation are given by
�
(λ0A10 + λ1A11 + λ2A12) g1 = B1g1 = 0 ,
(56.2)
(λ0A20 + λ1A21 + λ2A22) g2 = B2g2 = 0
with the stiffness parameters λi = λ0ai and the model matrices Aik. Every
row is posed in the space of physical coordinates. There is a dependence of
the required information on the number of degrees of freedom (DOF) of the
model and hence a lack of information. This problem is ill-posed. But with
the linear maps B one can define linear induced maps B † as follows:
B † : R 3 ⊗ R 3 → R 3 ⊗ R 3 , (56.3)
B †
1 :(g1 ⊗ g2) ↦→ B1g1 ⊗ g2,
B †
2 :(g1 ⊗ g2) ↦→ g1 ⊗ B2g2.
The maps B1 and B2 act on the element g1 and g2, respectively and the
identity acts on the other elements. The application of these induced linear
maps on the problem shows in (56.4) that if and only if the right-hand side
holds, the left-hand side also holds:
� B1g1 =0
B2g2 =0 ⇔
� B †
The condition written in model matrices is
2�
2�
1 (g1 ⊗ g2) =B †
1f =0,
B †
2 (g1 ⊗ g2) =B †
2f =0.
(56.4)
A
s=0
† rsλsf = A
s=0
† rsfs =0. (56.5)
This problem is posed in the space of tensor products. By extension of the
use of determinants for systems of linear equations one can apply the rules of
determinants on this linear induced operators. Then one obtains compositions
of maps (see (56.6)) called determinantal maps, i.e.