Environmental Engineering Reference
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Therefore, the structural dynamic equation-based geometric nonlinearities
method instead of Eq. 9.1 can be expressed with modal coordinates as
Ma þ C a þ K 0 þ K 1 ð a Þ
½
a ¼ Q
ð 9 : 6 Þ
where a is the modal coordinate. K 1 (a) can be expanded based on the Taylor
theorem as
K 1 ðÞ¼ K 1 ðÞþ 1
2! K G ðÞþ 1
3! K B ðÞ
ð 9 : 7 Þ
where K G (a) is a linear function of modal coordinate a and K B (a) is its quadratic
function. Based on the method proposed by Sharf, the total stiffness matrix can be
calculated iteratively.
In this chapter, a finite element analysis considering the effect of stress stiffening
is employed to simulate the dynamic behavior of rotational blades. The effect of
stress stiffening is taken into account by generating and then using an additional
stiffness matrix, hereinafter called the stress stiffness matrix. The stress stiffness
matrix is added to the regular stiffness matrix to determine the total stiffness.
9.2.2 Damage Detection Methodology: Principal Component
Analysis
A principal component analysis (PCA) is a method based on the idea of reduced
order whose purpose is to reduce the dimensionality of the data, while retaining as
much as possible the characteristics of the original dataset. PCA has been applied
in structural health monitoring [ 43 ], modal analysis [ 44 ], and the elimination of
environmental effects in damage detection [ 23 , 24 ]. In the present work, it will be
shown that this method is useful for eliminating the effects of rotation in the
vibration-based damage detection of wind turbine blades.
The variations in environmental conditions (such as the rotational speed and
temperature) are known to have considerable effects on vibrational features. Let us
denote the n-dimension vector x k as a set of vibration features identified at time t k ,
(k = 1…N)withN being the number of samplings. All of the x k are collected in a
matrix x 2< n N , where n represents the number of selected modes, while the
natural frequencies are chosen as the vibrational features. Then, PCA can com-
press the data by reducing the original dimension n to a lower dimension m:
Y ¼ TX
ð 9 : 8 Þ
where X is called the scores matrix and T 2< m n is the loading matrix; the
dimension m is considered the number of principal components that affect the
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