Wind Energy

56 Damage Detection on Structures of Offshore Wind Turbines 303
Table 56.1. Identified stiffness parameters by means of measured (simulated)
eigenfrequencies with an accuracy of three correct decimal places
Measured eigenfrequencies (Hz) Determined stiffness parameters (–)
0.407
2.782
7.618
� †
A
∆0 = det 11 A† 12
A †
21 A†
22
�
1.002
0.897
1.003
= A †
11A† 22 − A† 21A† 12 . (56.6)
It is shown in [3] that the development of determinants using Cramers Rule
and the application of the cofactor to (56.5) lead to a problem which is equivalent
to (56.1)
∆pfq − ∆qfp =0. (56.7)
With a non-singular linear combination of the determinantal maps one obtains
linear eigenvalue problems for the model parameters
fs = ∆ −1 ∆sf = λsf. (56.8)
These problems are well-posed. There is no incompleteness of information
because the number of required eigenfrequencies depends only on the number
of stiffness parameters.
Numerical Example: In the current state of method validation, measurement
data are not yet available. Therefore, artificial measurement data are
generated by applying scatter to the exact eigenfrequencies as resulting from
the analysis. It is supposed that the middle area of a cantilever beam model
with three elements (denoted by one to three beginning from the bottom) and
nine DOF has a stiffness degradation of 10%. In the first column of Table 56.1
the first three eigenfrequencies for the bending modes are shown with an accuracy
of three correct decimal places. The solution of the eigenvalue problems
determines the set of stiffness parameters as shown in the second column. A
stiffness parameter equal to one refers to an undamaged structural element.
The decrease of the second stiffness parameter points to a damage in this
region. The value of 0.897 indicates a stiffness reduction of 10.3% which is
quite close to the damage applied before. Furthermore, damage of the correct
structural element is reflected since the stiffness parameters of elements one
and three remain almost identical to one.
56.3 Validation of the Method
A validation of the method by means of a sensitivity analysis was carried out.
Therewith the influence of several parameters of the method on the accuracy
of the quantification was investigated. The parameters of the method