Study and Quantitative Analysis of Mode Localization in Wind Turbine Blades

Abstract

The study of damage mechanisms for wind turbine blades is important. Generally, modal localization tends to accelerate structural damage. This is a new approach to studying these damage mechanisms for wind turbine blades through modal localization theory. Therefore, this paper investigates whether modal localization phenomena exist in wind turbine blades, as well as the impact of different forms of detuning on modal localization. Based on perturbation theory, a mechanism for mode localization is described quantitatively using the degree of detuning, the degree of mode density, and the mode assurance criterion. A finite element model for wind turbine blades was established using ANSYS software (R15.0), and three detuning cases were simulated by changing the density, elastic modulus, and installation angles of the blades. Moreover, an improved mode localization factor is proposed to quantitatively evaluate the degree of mode localization in wind turbine blades. The numerical results indicate that the degree of modal localization increases with an increasing degree of detuning, but the increase in modal localization gradually slows. Finally, the detuning modal shape composition, which includes harmonic components, is analyzed. The results show that the closer the composition of the detuning modes is, the stronger the degree of mode localization.

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. Xie et al. [19] studied vibration mode localization in randomly disordered cyclic periodic structures considering multiple substructure modes. Bendiksen [20] considered a weak-coupling antenna structure and found that mode localization occurs with detuning. Filoche et al. [21] discovered the strong localization of flexural waves induced by clamping 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 damping, the characteristic equation of vibration is

$$\left[K_0\right]\left\{u_{0i}\right\}={\lambda }_{0i}\left[M_0\right]\left\{u_{0i}\right\}$$

(1)

where $\left[K_0\right]$ and $\left[M_0\right]$ are the stiffness matrix and mass matrix, respectively; ${\lambda }_{0i}$ is the ith-order eigenvalue; and $\left\{u_{0i}\right\}$ 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

$$\left[K\right]=\left[K_0\right]+\epsilon \left[K_1\right],\left[M\right]=\left[M_0\right]+\epsilon \left[M_1\right]$$

(2)

where $\epsilon \left[K_1\right]$ is a variation in the stiffness matrix and $\epsilon \left[M_1\right]$ is a variation in the mass matrix. The subscript “1” represents the first-order perturbation, and $\epsilon$ is a small constant. Similarly, the first-order perturbations of the modal parameters can be expressed as

$${\lambda }_i={\lambda }_{0i}+\epsilon {\lambda }_{1i},\left\{u_i\right\}=\left\{u_{0i}\right\}+\epsilon \left\{u_{1i}\right\}$$

(3)

where ${\lambda }_i$ and $\left\{u_i\right\}$ are the ith-order eigenvalue and eigenvector after disturbance, respectively. The expansion theorem is introduced

$$\left\{u_{1i}\right\}={\sum }_{s=1}^n{C}_s^i\left\{u_{0s}\right\}$$

(4)

$$C_s^i=-\frac{1}{2}{\left\{u_{0i}\right\}}^T\left[M_1\right]\left\{u_{0i}\right\}(s=i)$$

(5)

$$\begin{gathered} C_s^i=\frac{\left({\left\{u_{0s}\right\}}^T\left[K_1\right]\left\{u_{0i}\right\}-{\lambda }_{0i}{\left\{u_{0s}\right\}}^T\left[M_1\right]\left\{u_{0i}\right\}\right)}{{\lambda }_{0i}-{\lambda }_{0s}} \\ \left(s\ne i\right) \end{gathered}$$

(6)

where n denotes the degree of freedom (DOF), and $C_s^i$ 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, $C_s^i(s\ne i)$ is large, and $\epsilon \left\{u_{1i}\right\}$ is not a small vector compared with $\left\{u_{0i}\right\}$; 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

$$\left\{u_i\right\}={\sum }_{s=1}^n{L}_s^i\left\{u_{0s}\right\}$$

(7)

$$L_s^i=\epsilon C_s^i(s\ne i)$$

(8)

$$L_s^i=\epsilon C_s^i+1 (s=i)$$

(9)

where $L_s^i$ is the linear superposition coefficient and $\epsilon C_s^i\left\{u_{0s}\right\}$ is a small vector compared to $\left\{u_i\right\}$. When the absolute value of ${\lambda }_i-{\lambda }_s(i\ne s)$ is not a small constant, the vector $\epsilon C_s^i\left\{u_{0s}\right\}$ can be neglected in $\left\{u_i\right\}$. When the value of ${\lambda }_i$ is close to ${\lambda }_s\left(s\ne i\right)$, the vector $\epsilon C_s^i\left\{u_{0s}\right\}$ cannot be neglected compared to $\left\{u_i\right\}$. The detuning eigenvector $\left\{u_i\right\}$ 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:

$$d_i=\frac{\left|{\lambda }_i-{\lambda }_i^{\prime }\right|}{{\lambda }_i}$$

(10)

$${\delta }_{ij}=\frac{\left|{\lambda }_i-{\lambda }_j\right|}{{\lambda }_i+{\lambda }_j}$$

(11)

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: $d_i$

(2) Calculate the intensive degree of modes: ${\delta }_{ij}$

(3) Determine whether ${\delta }_{ij}\le d_i$ or ${\delta }_{ij}$ is satisfied in the same order as $d_i$. 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):

$$MA{C}_{ij}=\frac{{\left|{\left(u_i\right)}^T{u}_j\right|}^2}{{\left(u_i\right)}^T\left(u_i\right){\left(u_j\right)}^T\left(u_j\right)}$$

(12)

assumption

$$u_d^{\prime }=L_i^d{u}_i+L_j^d{u}_j+L_k^d{u}_k+\cdots$$

(13)

where $u_d^{\prime }$ is the d-order detuning modal vector; $u_i$, $u_j$, and $u_k$ are the i-order, j-order, and k-order harmonic eigenvectors, respectively; and signifying $L_i^d$, $L_j^d$ and $L_k^d$ are the same as in Equation (8). The summation of the model assurance criterion (MAC) can be expressed as

$$\sum MAC\left(u_d^{\prime }\right)=MA{C}_{d^{\prime }i}+MA{C}_{d^{\prime }j}+MA{C}_{d^{\prime }k}+\cdots =1$$

(14)

The values of $\sum MAC$ of the detuning modal vector are calculated with respect to the harmonic vectors, which are in the linear superposition range; the closer $\sum MAC$ is to 1, the higher the precision.

3. Numerical Examples

3.1. Analysis Model

The spring-particle system shown in Figure 1 is used as the analysis model; this approach is simple cyclic symmetric structure. The mass of each particle is 100 kg, and the stiffness coefficients of each ring spring and each axial spring are 3.15 × 10⁵ N/m and 1.05 × 10⁷ N/m, respectively. The particle is allowed to move only in the plane, which signifies that the structure has 26 degrees of freedom. MATLAB (9.5 R2018b) was used to model the structure and for the detuning simulation and data calculation. The modal frequencies of the structure are presented in

Table 1. Frequencies of the 26-dof spring-particle system (Hz).

Mode	Frequency	Mode	Frequency	Mode	Frequency
1	35.467	10	40.095	19	81.511
2	36.737	11	40.095	20	81.545
3	36.737	12	40.344	21	81.545
4	37.471	13	57.484	22	81.577
5	37.471	14	57.484	23	81.577
6	38.450	15	81.477	24	81.608
7	38.450	16	81.486	25	138.969
8	39.407	17	81.486	26	138.969
9	39.407	18	81.511

. For this harmonic structure model, only 1st, 12th, 15th, and 24th modes are non-repeated modes, while every two frequencies of the remaining modes constitute a degenerate mode pair. In addition, there is minimal distinction between adjacent frequencies, making it prone to the occurrence of modal localization phenomenon.

3.2. Analysis of the Diagnostic Modality

The spring stiffness is subjected to random detuning with a mean value of 0 and a variance of $\sigma$ ($\sigma$ = 0.0001, 0.001, 0.01) by changing the elastic modulus of the springs.

Table 2. The degree of detuning and the density of the mode.

Mode	δij (i = 20; j = 11, 12, …26)	di (i = 20)
		σ = 0.0001	σ = 0.001	σ = 0.01
11	0.34076	0.000004	0.00020	0.00271
12	0.33802	0.000004	0.00020	0.00271
13	0.17306	0.000004	0.00020	0.00271
14	0.17306	0.000004	0.00020	0.00271
15	0.00041	0.000004	0.00020	0.00271
16	0.00035	0.000004	0.00020	0.00271
17	0.00035	0.000004	0.00020	0.00271
18	0.00020	0.000004	0.00020	0.00271
19	0.00020	0.000004	0.00020	0.00271
20	0.00000	0.000004	0.00020	0.00271
21	0.00000	0.000004	0.00020	0.00271
22	0.00020	0.000004	0.00020	0.00271
23	0.00020	0.000004	0.00020	0.00271
24	0.00093	0.000004	0.00020	0.00271
25	0.26041	0.000004	0.00020	0.00271
26	0.26041	0.000004	0.00020	0.00271

shows the degree of detuning and the intensity of the 20th-order detuning mode with the 11th~26th harmonic modes. The results show that (1) the 20th detuning mode is a linear combination of the 20th and 21st harmonic modes when $\sigma$ is 0.0001 and (2) the effective range of linear superposition is the 15th~24th harmonic modes when $\sigma$ is 0.001 or 0.01. Figure 2 displays the values of $MAC$ for the 20th detuning mode and the 15th~24th harmonic modes. These values are very close to each other when $\sigma$ = 0.01, which signifies that the proportions of harmonic modes in the 20th detuning mode are very close. The values of $\sum MAC$ are all greater than 0.99 (close to 1) when $\sigma$ is 0.0001, 0.001 and 0.01, signifying that the calculation precision is high and satisfies practical requirements.

In fact, each detuning modal vector can be obtained by linear superposition of all the harmonic vectors, and the approximate solution of the detuning vector can be obtained by linear superposition of parts of the harmonic ones. Figure 3 presents the approximate and exact solution of the 20th detuning modal vector. The results show that (1) there are minor differences between the approximate and exact solutions, meeting practical engineering requirements, and (2) the modal localization phenomenon already occurs when the value of $\sigma$ is 0.01.

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/m³ and 2.8 $\times$ 10¹⁰ N/m², 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. Frequencies of wind turbine blades (Hz).

Mode	Harmonic Frequencies	Mode	Harmonic Frequencies
1	0.16648	7	0.63440
2	0.16650	8	0.66146
3	0.16650	9	0.66146
4	0.42903	10	1.28278
5	0.43733	11	1.32847
6	0.43733	12	1.32847

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 $\sigma$ 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.

6. Quantitative Analysis of Wind Turbine Blades

6.1. Quantitative Analysis of the Model Location

When mode localization of a cycle symmetric structure occurs, the total energy of the mode shape is considered to be constant, and the energy is concentrated on a few DOFs. The proportion of the mode displacement’s absolute value in a certain DOF to the sum of the mode displacement’s absolute value reflects the aggregation degree of energy. Therefore, this paper defines an improved mode localization factor as the aggregation degree of energy, which can be expressed as

$$R_i=\frac{A_{i\max }^1}{A_{i\max }^0}\frac{(P_{i\max }^1-M_i^1)}{M_i^1}-\frac{(P_{i\max }^0-M_i^0)}{M_i^0}$$

(15)

$$M_i^0=\frac{A_{i\max }^0}{n}; M_i^1=\frac{A_{i\max }^1}{n}$$

(16)

where $P_{imax}^1$ is the maximum displacement of the i-th mode shape in the detuned structure and $P_{imax}^0$ is the maximum displacement of the i-th harmonic mode. $A_{imax}^1$ is the ratio of the maximum displacement of the i-th detuned mode to the sum of the absolute values of its mode displacements; $A_{imax}^0$ is the ratio of the maximum displacement of the i-th harmonic mode to the sum of the absolute values of its mode displacements; and n is the DOF number of the structure.

Using the wind turbine blade model from Section 3, we adjusted the detuning variance σ by changing the density, elastic modulus, and installation angles of blade elements to simulate the impact of different detuning levels on the modal localization of wind turbine blades. The degree of modal localization was quantitatively described by the improved local factor, as shown in Figure 9. The results indicate that (1) the increase in the modal localization degree becomes slower with increasing detuning degree; (2) the first-order mode shape is more sensitive to stiffness detuning than to the mass and geometry detuning; (3) the sensitivity of the second mode shape to mass detuning approaches that of geometry detuning and is less sensitive than to stiffness detuning; and (4) the sensitivity of the third mode shape to detuning decreases in the following order of parameters: mass > geometry > stiffness.

6.2. Quantitative Analysis of the Components of the Detuning Mode Shape

The components of the first-order detuning mode shape in the three detuning cases are calculated by using the previous method.

Table 4. The degree of mistuning and the intensive degree of mode of the wind turbine blades.

Mode	δij (i = 1; j = 1, 2, …6)	di (i = 1)
		Mass Detuning	Stiffness Detuning	Geometry Detuning
1	0.00000	0.00012	0.00006	0.00006
2	0.00006	0.00012	0.00006	0.00006
3	0.00006	0.00012	0.00006	0.00006
4	0.44090	0.00012	0.00006	0.00006
5	0.44860	0.00012	0.00006	0.00006
6	0.58430	0.00012	0.00006	0.00006

shows the values of the detuning degree and mode intensity degree. Notably, only the first three orders satisfy the condition that ${\delta }_{ij}\le d_i$ or the value of ${\delta }_{ij}$ is in the same order as $d_i$ in the three detuning cases. Therefore, the first-order detuning mode shape is linearly superposed by the first three orders of harmonics according to the previous criterion. The results show that the second-order and third-order detuning components are also linearly superposed by the first three order harmonic components for the three detuning types via the same method. The values of MAC and $\sum MAC$ for the first-order detuning mode shape with the first three-order harmonic shape of mass detuning are shown in

Table 5. MAC and ∑MAC of the first-order detuning mode shape of mass detuning.

MAC	Mass Detuning
$MA{C}_{(1\prime 1)}$	0.7050
$MA{C}_{(1\prime 2)}$	0.1800
$MA{C}_{(1\prime 3)}$	0.1132
${\displaystyle \sum MAC}$	0.9982

; note that the value of $\sum MAC$ is 0.9982, which is close to 1, indicating that the accuracy satisfies practical engineering requirements.

Figure 10 displays the linear superposition coefficient of the first third-order detuning mode shapes in three detuning cases, and the following conclusions are drawn: (1) the first-order harmonic mode shape accounts for the largest proportion of the first-order detuning mode shape in three detuning cases; (2) the second-order harmonic mode shape accounts for the largest proportion of the second-order detuning mode shape in mass and geometry detuning, and the main contributing factor of the second-order detuning mode shape is the third harmonic mode shape, followed by the first-order and second-order shapes in stiffness detuning; and (3) the compositions of the third detuning mode shape in mass and geometry detuning decrease according to third-order > second-order > first-order harmonic mode shapes, while the compositions of the third-order detuning in stiffness detuning decrease according to second-order > third-order > first-order.

To study the influence of the proportion of harmonic vectors on the detuning vectors, the coefficient of variation is introduced as

$$c_v=\frac{\sqrt{\sigma }}{\mu }$$

(17)

where $\sigma$ is variance, $\mu$ is the mean, and $c_v$ is the coefficient of variation.

Because the absolute value of the linear superposition coefficient can indicate the proportion of the harmonic vector component in the detuning coefficient, the coefficient of the absolute value of the linear superposition coefficient is calculated and shown in

Table 6. Values of the coefficient of variation.

Mode	cv
	Mass Detuning	Stiffness Detuning	Geometry Detuning
1	0.4854	0.4062	0.7300
2	0.6230	0.3579	0.7140
3	0.1522	0.8950	0.3331

. We can conclude that (1) the closer the component of the harmonic vectors is to the detuning vectors, the smaller is the coefficient of variation; and (2) the degree of mode localization decreases with increasing coefficient of variation in the same detuning order.

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.