1. Introduction
Since the Industrial Revolution, energy consumption has increased rapidly with economic development, population increases, and social improvements. The resulting overconsumption of traditional fossil fuel energy sources has led to the continuous deterioration of the environment. In particular, global warming caused by greenhouse gases leads to severe challenges to the sustainable development of human society [
1]. Therefore, to increase the energy supply, ensure energy security, protect the environment, and promote sustainable societal development, the development and utilization of renewable energy have drawn international attention [
2]. Wind power generation is a type of power generation method that is relatively mature, offering at-scale development and good prospects for renewable energy generation, which is highly valued worldwide [
3,
4].
To drive domestic consumption, economic growth, and societal stabilization and development, the Chinese government has increased its investment in new energy since 2009. As a renewable energy source, wind energy has promising prospects and has been strongly supported by the government. In 2018, the cumulative capacity of wind power installed in China accounted for a record high of 35.7% of the total global capacity, ranking first worldwide [
5]. Chinese government officials suggest that this capacity could increase to 581 GW by 2025. As the core component of wind power equipment, the cost of wind turbine blades accounts for 1/4 to 1/3 of the total price of the equipment. Summarizing the existing literature, studies on wind turbine blades have generally focused on blade design, modeling, and vibration characteristics [
6,
7,
8]. For instance, Zhang et al. [
9] investigated the influence of tuned mass dampers on the vibration control of monopile offshore wind turbines under wind–wave loadings. For the design of an aerodynamic thrust-matched blade model, an innovative methodology was proposed by Ma et al. [
10] based on a genetic algorithm. With the continuous development of wind power equipment, the size of wind turbine blades is becoming increasingly large. Due to the complex and harsh working environment and long-term alternating load, blade shedding, cracking, wear, and other damage can easily occur. Therefore, the health monitoring of wind turbine blades is highly important for the development of wind power generation technology.
Many methodologies have been proposed for wind turbine blade damage detection based on vibration signals, including empirical mode decomposition [
11], support vector machine [
12], deep learning [
13,
14], artificial neural networks [
15], the Bayesian framework [
16], and so on. Nevertheless, there is a lack of research on damage detection in wind turbine blades through modal localization theory. Hodges [
17] first introduced Anderson’s localization concept from the field of solid-state physics to the field of structural dynamics, following which many studies on the mode localization of structures were conducted. Pierre and Dowell [
18] simulated the detuning of a two-span bridge by changing the constraint position and confirmed the existence of modal localization. ** one point in a thin plate. In recent years, there has been an increasing amount of research on mode localization. Shaat et al. [
22] discovered that surface roughness may affect the propagation of vibration energy in a microbeam, resulting in mode localization. Ying et al. [
23] explored the dynamic characteristics of quasi-periodic multiple-supported beam structures with local weak coupling by using analytical and numerical methods. Rabenimanana et al. [
24] presented a sensor using the mode localization phenomenon to detect a mass perturbation. Chen et al. [
25] investigated the mode localization behaviors of two-span beams through theoretical and experimental methods. Morozov et al. [
26] revealed the principal difference in the characteristics of the mode localization phenomenon for a microelectromechanical accelerometer model. Lyu et al. [
27] improved the parameter sensitivity of a ΔE-effect magnetic sensor using the mode localization effect. Although there is a great deal of research on the modal localization theory, there is a lack of studies on damage to wind turbine blades based on this theory.
With the increasing scale of wind turbines, the lengthening of wind turbine blades results in a growth in flexibility. The coupling between the hub and the wind turbine blades is becoming weaker. Generally, a large-scale wind turbine blade structure has 120-degree rotational symmetry and thus is a typical weakly coupled cyclically symmetric structure. In the design stage of wind turbine blades, they are generally modeled in conceptual states. However, due to manufacturing or construction errors, material defects, structural damage, and other factors, there is always a certain deviation between the actual blades and the conceptual blades; this deviation is denoted as detuning. This small detuning is not easily detected. Because of detuning, the cyclic symmetric property of the blades is destroyed, and thus, the blades become detuned structures. Generally, modal localization may lead to more severe damage to the wind turbine blades. Therefore, this paper investigates whether modal localization occurs under small detuning of wind turbine blades.
2. The Mechanism of Modal Localization
Matrix perturbation theory is a powerful tool for studying the modal localization of structures [
28,
29,
30,
31]. The basic concept of this method is to approximately express the eigenvalues and eigenvectors of structures after disturbance by utilizing the eigenvalues and eigenvectors before disturbance. The specific derivation process is described in the literature [
32], and a brief explanation is given here. For a discrete structure without dam**, the characteristic equation of vibration is
where
and
are the stiffness matrix and mass matrix, respectively;
is the
ith-order eigenvalue; and
is the
ith-order mass normalized modal vector. The subscript “0′’ represents the structural state without any detuning.
When physical parameters are slightly disturbed, the first-order perturbations of the stiffness matrix and mass matrix can be expressed as
where
is a variation in the stiffness matrix and
is a variation in the mass matrix. The subscript “1” represents the first-order perturbation, and
is a small constant. Similarly, the first-order perturbations of the modal parameters can be expressed as
where
and
are the
ith-order eigenvalue and eigenvector after disturbance, respectively. The expansion theorem is introduced
where
n denotes the degree of freedom (DOF), and
denotes the first-order perturbation expansion coefficient of the
i-order mode corresponding to the
s-order mode and is a definite constant. For structures with closely spaced modes,
is large, and
is not a small vector compared with
; thus, the modal vectors with similar frequencies change remarkably, resulting in the mode localization phenomenon. This finding illustrates that perturbation theory qualitatively analyses the mechanism of modal localization. To analyze the variation trend of modal vectors when the parameters of a structure with closely spaced modes have a small variety, the mechanism of mode localization is described quantitatively using the degree of detuning, the intensive degree of modes and the mode assurance criterion based on perturbation theory.
According to Equations (4) and (5), the detuning modal vector can be expressed as
where
is the linear superposition coefficient and
is a small vector compared to
. When the absolute value of
is not a small constant, the vector
can be neglected in
. When the value of
is close to
, the vector
cannot be neglected compared to
. The detuning eigenvector
is linearly superposed by a few harmonic eigenvectors, not all of which are for a structure with closely spaced modes. If the orders of the harmonic eigenvectors can be determined, then we can obtain the detuning eigenvector by linear superposition of these harmonic eigenvectors. Therefore, the proposed method can not only greatly reduce the workload but also satisfy the actual precision requirements. To determine the order of magnitude for scaling the system, the concepts of the degree of detuning [
33] and the degree of modal density [
34] are introduced as follows:
The calculation process of the linear superposition scale of harmonic modal vectors in detuning vectors is given as follows:
(1) Calculate the degree of detuning:
(2) Calculate the intensive degree of modes:
(3) Determine whether or is satisfied in the same order as . If one of the two conditions is satisfied, the i-order harmonic eigenvector component in the j-order detuning eigenvector cannot be neglected; otherwise it can be neglected.
(4) The orders that satisfy the conditions in (3) are the linear superposition ranges of the harmonic eigenvectors.
For the sake of confirming the linear superposition accuracy of the detuning modal vectors, the model assurance criterion [
35] is introduced as Equation (12):
assumption
where
is the
d-order detuning modal vector;
,
, and
are the
i-order,
j-order, and
k-order harmonic eigenvectors, respectively; and signifying
,
and
are the same as in Equation (8). The summation of the model assurance criterion (
MAC) can be expressed as
The values of of the detuning modal vector are calculated with respect to the harmonic vectors, which are in the linear superposition range; the closer is to 1, the higher the precision.
4. Modeling and Modal Analysis
In this paper, the NACA63-415 airfoil is used for blade modeling. The length of each blade model is 49 m, and the material density and elastic modulus are 1950 kg/m
3 and 2.8
10
10 N/m
2, respectively. The shell 181 element was used, and each node has 6 DOF. The corresponding blade model is displayed in
Figure 4. The mode analysis was conducted employing ANASYS software (R15.0). It assumes isotropic properties for all materials and the blade structure in a state of elastic deformation throughout the analysis. The model employed in this study is a simplified representation that utilized the same airfoil shapes to establish the modal and does not account for the coupling between different materials.
Table 3 presents the frequencies of the wind turbine blades, which are obtained via finite element analysis. For this perfectly periodic structure, four coupled modes with the same frequency exist: the 2nd mode and the 3rd mode, the 5th and 6th modes, the 8th and the 9th modes, as well as the 11th and the 12th modes. Moreover, the frequencies of certain neighboring modes exhibit proximity, which can easily result in the occurrence of modal localization phenomena.
Figure 5 shows that (1) the amplitudes of the first-order harmonic mode shapes of the three blades are very similar; (2) the amplitudes of the second-order mode shapes are shown for blade 1, blade 3 and blade 2, from large to small, while the amplitudes of blade 2 and blade 3 are close to one another in the third-order mode shape.
5. Simulation of Wind Turbine Blade Detuning
To study the influence of different detuning types on the modal localization of wind turbine blades, three different detuning types were simulated: mass detuning, stiffness detuning, and geometry detuning. For comparing the sensitivity of wind turbine blades to different detuning types, the detuning variance values were all set to 0.002. ANSYS software (R15.0) was used to conduct the numerical simulation and the establishment of the blades model was the same as that in the previous section.
5.1. Mass Detuning
In blade manufacturing, due to the occurrence of material defects and small discrepancies in blade size, mass detuning occurs in wind turbine blade operation. In addition, due to the complex and harsh operating environment of wind turbine sets, blades will be delaminated by lightning or coated with ice so that the mass distribution of the blades will be uneven. Therefore, mass detuning is used to simulate the influence of uneven mass distribution on blades.
In this section, the material density of the elements was varied to simulate the mass detuning of blades, and the detuning variance was 0.002 with a mean of 0.002. By comparing
Figure 5 and
Figure 6, it can be seen that after mass detuning, the vibration amplitude of the first-order mode of blade 2 significantly decreases, and the amplitude changes of blades 1 and 3 are relatively small, indicating a fairly obvious localization in the structure. The displacement of the second-order mode mainly occurred in blades 1 and 3, and the amplitudes are quite close, while the displacement of the tuned structure mode presents a stepwise pattern across the three blades, making it relatively difficult to distinguish the degree of modal localization before and after detuning. The displacement of the third-order mode was mainly concentrated on blade 2, revealing a more distinct localization phenomenon in the structure.
5.2. Stiffness Detuning
Under long-term and alternating loading, the bending moment of the blade roots is fairly large, so the roots are vulnerable to damage during service, leading to a decrease in stiffness.
In this section, the elastic modulus is varied to simulate the stiffness detuning of blades, and the detuning variance is 0.002 with a mean of 0.002. Comparing
Figure 5 and
Figure 7, it can be seen that after stiffness detuning, the displacement of the first-order mode is mainly concentrated on blade 2, the displacement of the second-order mode is concentrated on the third blade, and the displacement of the third-order mode is mainly concentrated on blade 1. When subjected to stiffness detuning, the first three orders of detuned modes show different degrees of modal localization.
5.3. Geometry Detuning
Blade element theory is a common method for designing wind turbine blade profiles and is conducive to improving the conversion rate of wind energy. The central concept of this theory is to decompose a wind turbine blade into tiny segments in the spreading direction. The angle of each blade element is different along the radial direction in one blade. In practice, it is difficult to ensure that the installation angles of the blade elements at the corresponding positions of the three blades are the same; thus, the geometry is simulated by changing the installation angles of the blade elements, and the detuning variance is also set to 0.002. Comparing
Figure 5 and
Figure 8, it can be seen that after geometric detuning, the displacement of the first-order mode shape mainly occurred on blade 2, with the displacement amplitudes of blades 1 and 3 being quite similar. The modal localization in the structure has already appeared. The displacement of the second-order mode shape is concentrated on blade 1, with smaller displacement amplitudes for blades 2 and 3. The displacement of the third-order mode shape is mainly concentrated on blade 3, with the smallest displacement amplitude on blade 2. This case analysis demonstrates that under geometric detuning, the first three orders of detuned mode shapes all exhibit different degrees of modal localization.
7. Conclusions
The mechanism of mode localization is quantitatively described using the degree of detuning and the intensiveness of the mode and mode assurance criterion based on perturbation theory, and the applicability of the proposed method is verified by a case study simulation. The turbine blade is analyzed via numerical simulation, and three different detuning forms are simulated by changing the density elastic modulus and installation angles of the blade elements. Moreover, the improved localization factor is proposed to quantitatively analyze the degree of modal localization. By changing the magnitude of detuning, the degree to which the location of the first third-order modal shapes is positively correlated with the detuning degree is determined, and the compositions of the harmonic modal shapes in the first third-order detuning shapes in the three detuning cases are studied. Based on the results, conclusion are drawn as follows.
(1) The wind turbine blades exhibit dense modal distribution, which can easily result in the occurrence of modal localization phenomenon;
(2) A small detuning of mass, stiffness, and geometry will have a great influence on the modal shapes of wind turbine blades;
(3) The degree of modal localization increases with increasing degree of detuning, but the increase in the modal localization gradually slows;
(4) Different order modal shapes have different sensitivities to different detuning types and detuning degrees;
(5) In detuning modal shapes, a sparser distribution of harmonic modal components corresponds to a smaller degree of modal localization, while a closer alignment of each component’s proportion results in a greater degree of modal localization.