An Electromagnetic Load Identification Method Based on the Polynomial Structure Selection Technique

Mao, Wengui; Pei, Shixiong; Guo, Jie; Li, Jianhua; Wang, Buyao

doi:https://doi.org/10.1155/2024/1842508

Shock and Vibration

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Research Article | Open Access

Volume 2024 | Article ID 1842508 | https://doi.org/10.1155/2024/1842508

An Electromagnetic Load Identification Method Based on the Polynomial Structure Selection Technique

Wengui Mao,¹Shixiong Pei,¹Jie Guo,¹Jianhua Li,¹and Buyao Wang²

Academic Editor: Vasudevan Rajamohan

Received12 Jun 2023

Revised25 Dec 2023

Accepted08 Jan 2024

Published13 Jan 2024

Abstract

Electromagnetic loads can effectively monitor motor health and improve motor design. Considering the weak correlation of the modal shape and Chebyshev orthogonal polynomial in the space-time independent electromagnetic load identification method, a proposed method combining the polynomial structure selection technique together with limited measured displacement responses is presented, in which an error reduction ratio is used to pick out the significant mode shape matrix and the Chebyshev orthogonal polynomial. The time-history function of the electromagnetic load is reconstructed by combining the significant mode shape matrix and the identified concentrated load through modal transformation, and the corresponding spatial distribution function is fitted by the significant Chebyshev orthogonal polynomial. Eventually, a comparative numerical study considering the selection of significant components and measurement noise is carried out to prove the effectiveness of the presented method.

1. Introduction

Energy problems and the vibration and noise problems of wind turbines during operation have received widespread attention from scholars around the world [1–3]. Particularly, the radial electromagnetic load coming from the air-gap eccentricity variation pattern with time is the main factor of noise and vibration [4–6]. Its accurate identification has become a hot research topic for promoting the motor performance due to its assistance for vibration prediction, structural design, and health monitoring [7–9]. However, the electromagnetic load coming from the air-gap eccentricity occurring during motor operation due to disturbances from external excitations is distributed dynamic load, and its identification (or reconstruction) is the second inverse problem in structural dynamic analysis [10], which is the process of reconstructing the loads applied on the structure with the structural dynamic model and structural response information [11–14]. Different from the concentrated dynamic load, the representation of the electromagnetic load requires variables in both the time domain and the space domain. Consequently, the scale and difficulty of the calculation are greatly increased.

The subsistent investigates for distributed dynamic loads identification [15, 16] are mainly under the consideration of the multipoint concentrated dynamic load equivalented and basic functions fitted. The spatial distribution function may be fitted by linearly independent basis functions according to the generalized orthogonal domain theory [17]. Wang et al. [10] showed that the complex nonlinear spatial distribution function can also be effectively fitted by Chebyshev orthogonal polynomials. Jiang et al. [18] reconstructed the space-time coupling distributed load combining modal transformation together with orthogonal decomposition. Mao et al. [19] reconstructed the space-time independent electromagnetic load combining the equivalented concentrated dynamic load with basis function fitting. Zhang et al. [12] transformed distributed dynamic loads to equivalented concentrated loads acting on appropriate locations according to the strain modal theory. Li et al. [20] used the Tikhonov regularization scheme to identify the ratio of orthogonal polynomials and found that only enough measuring points are needed to meet the accuracy of distributed load identification. However, some shortcomings still existed in the methods abovementioned; the weak correlation of modal shape and basis function terms will cause the serious ill-posedness in time-history reconstruction and spatial distribution function fitting due to the determination of the modal shape and the basis function terms.

In this paper, a polynomial selection technique [20] is introduced to improve the ill-posedness problem, in which the contribution rate of each component is calculated by the error reduction ratio and the components with the small contribution rate are eliminated. The rest of this paper is organized as follows. Section 2 briefly introduces the electromagnetic load identification problem. In Section 3, the electromagnetic load reconstruction based on the polynomial selection technique is introduced in detail. In Section 4, a comparative numerical example is used to verify the effectiveness. In Section 5, some conclusions are briefly summarized.

2. Problem Statement

2.1. Electromagnetic Load

The authors in reference [21] show that the electromagnetic load presents the characteristics of space-time independence. Its spatial characteristic is mainly caused by the air-gap eccentricity (as shown in Figure 1), which is continuous distribution function changing with the spatial domain; its time characteristics are derived from the time-varying mechanical vibration frequency. Therefore, the electromagnetic load shown in the following equation is composed of space-time independent functions, namely, the time-history function and the spatial distribution function , independently.where and , respectively, denote the variables in the space-time domain.

2.2. The Electromagnetic Load Identification Method

The electromagnetic load can be transformed into a series of concentrated dynamic loads, which have the same time-history form at each action point. Therefore, the dynamic differential equation [19] of the wind turbine rotor under the electromagnetic load can be similar to the dynamic differential equation under the concentrated dynamic load, expressed as shown in the following equation:where , , and , respectively, denote the acceleration, velocity, and displacement vector of the wind turbine rotor, and , , and , respectively, denote the mass, stiffness, and damping matrix of the wind turbine rotor.

Performing modal transformation using the mode shape matrix and for equation (2), the kinetic equation (2) is thus transferred aswhere , , and , respectively, denote the modal mass, modal stiffness, and modal damping matrix of the wind turbine rotor. and , respectively, represent the modal displacement vector and the modal electromagnetic load in the modal space.

The modal electromagnetic load is shown in the following equation:where is the i-th order modal electromagnetic load; ; is the order of wind turbine rotor; and is a constant.

Equation (4) shows that there are same form and different amplitudes’ coefficient between and and the time-history can be replaced by an assumed-order mode electromagnetic load. Then, the electromagnetic load expression as shown in equation (1) can expressed as shown in the following equation:where and . Therefore, as long as the assumed-order mode electromagnetic load and the fitting spatial function are identified, the electromagnetic load can be reconstructed.

2.2.1. The Time-History Function Identification

Equation (5) shows that the time-history function can be replaced by the modal electromagnetic load. However, it is more difficult to reconstruct the modal load [22] than the concentrated dynamic load in the time domain based on the Green’s kernel function method because there must be enough modal orders and the modal response extracted difficultly in the solution process. In this paper, first, the equivalent concentrated dynamic load in the acting region is obtained by combining the conventional load identification method with the time-domain displacement response, as shown in equation (6) [23], and then the modal transformation is carried out to transform the concentrated dynamic load into the modal load .where k and n, respectively, refer to the number of the measured response node and the concentrated load node; represents the k-th measured response; represents the n-th concentrated dynamic load; and represents the Green’s function matrix.

2.2.2. Spatial Distribution Function Reconstruction

If the generalized Chebyshev polynomial basis function [24] is selected as the fitting function of , then equation (5) is rewritten as shown in the following equation:where and , respectively, denote the i-th generalized Chebyshev polynomial basis function and coefficient; and denotes the total number of terms of generalized Chebyshev polynomial basis function.

The generalized Chebyshev polynomial basic functions are discretized and equivalent to limited concentrated load nodes, then the electromagnetic load may be equivalent to a series of concentrated dynamic loads on corresponding concentrated nodes, which can be represented as shown in the following equation:

For the equivalent concentrated dynamic load and the modal electromagnetic load, according to the modal relationship , the relationship can be shown as follows:

Comparing equation (8) with equation (9),where represents the local mode shape matrix with rows and columns, which is extracted from the rows’ and columns’ mode shape matrix of the wind turbine rotor; represents the ratio between the modal load on j-th equivalent concentrated load node with ; represents the equivalent amplitude coefficients solved through the equivalent nodal load method [25]; and fitting is transformed into the solution of .

In summary, the main factors affecting the identification of the electromagnetic load are the selection of the local mode shape matrix and Chebyshev orthogonal polynomial terms. With the increase of modal natural frequency, in fact, not all orders have the same contribution to the equivalent concentrated load in equation (9), and the coefficients of basic functions in equation (10) is affected by the effective term of Chebyshev orthogonal polynomial which produces the equivalent amplitude coefficient . By selecting an effective mode shape matrix and eliminating the weakly correlated Chebyshev orthogonal polynomials, the ill-posedness in the identification process can be improved and the stability of the electromagnetic load identification can be improved.

3. Polynomial Selection Technique for Improving the Identification Method

Using the polynomial selection technique [20], significant components are effectively selected from all possible polynomial terms with the aid of the error reduction ratio (ERR) and Gram–Schmidt orthogonalization. First, the Gramm–Schmidt orthogonal change is performed on each item in each cycle, then the error reduction ratio of each item is calculated, and the item with the largest error ratio is selected. By selecting a polynomial in each loop, until the error reduction ratio is less than the threshold, the selection is stopped and the selected item is significant.

3.1. Significant Selection of the Mode Shape Matrix

For a wind turbine rotor with N freedom degrees, there are N-order modal vectors on each equivalent concentrated load action point. columns of shown in equation (9) is extracted from the -order modal vectors. The mode shape matrix associated with the equivalent concentrated load action point can be expressed as follows:

Through the error reduction ratio [20] of the mode shape matrix as equation (12), the columns of the most significant mode shape matrix will be picked out from all the N columns of based on the polynomial selection technique.where ; ; ; ; N is the total amount of modes order; and n is the total amount of the equivalent concentrated load point.

3.2. Significant Selection of the Chebyshev Orthogonal Polynomial

The equivalent amplitude coefficients shown in equation (9) are related to Chebyshev orthogonal polynomials equivalent to the concentrated load action point. The rows and columns equivalent amplitude coefficients in the equation (10) are expressed as follows:where ; ; ; ; and is the total amount of Chebyshev orthogonal polynomials’ terms.

Through the error reduction ratio of Chebyshev orthogonal polynomials reconstructing spatial distribution function as equation (14), the n columns’ significant Chebyshev orthogonal polynomial will be picked out from all the m-term Chebyshev orthogonal polynomials based on the polynomial selection technique.

3.3. The Improved Electromagnetic Load Identification

Through the abovementioned procedure, the most significant mode shape matrix and equivalent amplitude coefficients corresponding to significant Chebyshev orthogonal polynomials will be selected. Equation (10) can be rewritten as follows:

The regularization method [26, 27] is usually used to eliminate ill-posed problem in identifying the coefficients of Chebyshev orthogonal polynomials in equation (15) combined the known ratio together with the significant mode shape matrix and equivalent amplitude coefficients . To examine the identification accuracy, the correlation degree and the total error between the real and identified load are defined as shown in the following equation:

4. Procedure of Obtaining the Electromagnetic Load

Generally speaking, the implementation details of the improved electromagnetic load identification with the aid of polynomial selection technique is described in Figure 2, and the identification procedure can be described as follows: Step 1: with known , , and of the wind turbine rotor, specify the displacement measured node and get the corresponding displacement vector . Step 2: combine measured response together with the Green’s function matrix into the centralized dynamic load identification method to identify the equivalent dynamic load on the corresponding concentrated load node. Step 3: gain the significant mode shape matrix from the mode shape matrix shown in equation (11) using equation (12) based on the polynomial structure selection technique. Also, combine them with the equivalent dynamic load into the modal transformation to calculate the modal electromagnetic load and denote an assumed order modal load as the time-history function. Step 4: gain the corresponding coefficient regarding as the ratio . Combine them with equivalent amplitude coefficients shown in equation (13) using equation (14) into polynomial structure selection technique to gain the significant Chebyshev orthogonal polynomial. Step 5: compute the corresponding coefficient of the significant Chebyshev orthogonal polynomial through the regularization method using the obtained ratio with the significant mode shape matrix and equivalent amplitude coefficients . Obtain fitting spatial function . Step 6: reconstruct the electromagnetic load centering on and .

5. Numerical Examples

The numerical example (Table 1 and Figure 3) coming from the literature [19] is given to examine the proposed identification method. The electromagnetic loads acting vertically along the axis 335 mm–495 mm is selected as and . The six terms of Chebyshev orthogonal polynomials (, ) are discretely equivalented into six concentrated load nodes 12, 13, 14, 15, and 16 corresponding to the shaft length of 335 mm, 375 mm, 415 mm, 455 mm, and 495 mm. is applied to transform the interval into . The equivalent amplitude coefficients are shown in Table 2.

(a)

(b)

5.1. Identification Based on the Measured Displacement Response

The displacement responses in the time domain at five measuring points 1, 2, 5, 28, and 30 are obtained using assumed loads through the finite element method, as shown in Figure 4. The displacement responses are used as measured displacement responses . The equivalent concentrated dynamic load acting on the five corresponding concentrated load nodes 12, 13, 14, 15, and 16 have been identified, as shown in Figure 5. As shown in Figure 3, the rotor model is divided into 34 nodes with 136 order modes, and the 39th, 80th, 81th, 105th, and 117th order relative effective modal shape are picked out from the 136 order modes corresponding to five concentrated load nodes 12, 13, 14, 15, and 16 through the polynomial selection technique. Then, the significant mode shape matrix can be reconstructed based on the five-order effective modal shape mentioned above. Combine together with the equivalent concentrated dynamic load shown in Figure 5 into the modal transformation to calculate the 39th, 80th, 81th, 105th, and 117th effective modal electromagnetic load, as shown in Figure 6. The coefficient in the proposed method is different from the coefficient in literature [19] due to the different significant mode shape matrix corresponding to five concentrated load nodes 12, 13, 14, 15, and 16. The significant mode shape matrix in literature [19] is reconstructed coming from directly selecting the 12th, 13th, 14th, 15th, and 16th order relative modal shapes instead of the 39th, 80th, 81th, 105th, and 117th order relative effective modal shapes by the polynomial selection technique. Both methods can accurately reconstruct the time history, which shows that the selection of modal shape does not affect the time-history function identification.

The ratio between the modal loads on equivalent concentrated load nodes 12, 13, 14, 15, and 16 with a 39th order modal load is , , , , and , respectively (see the 4th row of Table 3). Combine these ratios together with equivalent amplitude coefficients shown in Table 2 into the polynomial structure selection technique to gain the significant Chebyshev orthogonal polynomial. The first and third of the six term Chebyshev orthogonal polynomial are picked out as the significant Chebyshev orthogonal polynomial. According to equation (15), the corresponding coefficient of the significant Chebyshev orthogonal polynomial through the regularization method is computed as , , , , , and (see the 4th row of Table 3). Compared with the identified and real spatial distribution function (see Figure 7), they have the same form and different coefficients . So, the spatial distribution function identification method is feasible. Furthermore, the 4th row of Table 4 shows that the correlation coefficient and the total relative error are 1 and 0.86, respectively, which is smaller than the 0.9979 and 4.87% from literature [19], which shows that the selection of the significant Chebyshev orthogonal polynomial is effective to the spatial characteristic identification.

We reconstructed the electromagnetic load centering on the present time-history function (see the green curve in Figure 6) and spatial distribution function (see the green curve in Figure 7). The identified and real electromagnetic load is shown in Figure 8, and the detailed results are shown in the 4th row of Table 4. Apparently, the error between them fluctuates between −0.04 N and 0.03 N, and its recognition error is smaller than the fluctuation between −0.2 N and 0.2 N, which shows that the identified electromagnetic load according to the proposed method is more accurate.

To further verify the electromagnetic load identification accuracy due to the selection of the significant mode shape matrix and the Chebyshev orthogonal polynomial, we compared with the following three types of programs: (I) only selecting the significant mode shape matrix; (II) only selecting the significant Chebyshev orthogonal polynomial; and (III) selecting both the significant mode shape matrix and the Chebyshev orthogonal polynomial (the proposed method). The identification errors of the three programs are shown in Figure 9, and the detailed results are shown in Tables 3 and 4 (see the 2nd, 3rd, and 4th row of Tables 3 and 4). Figure 9 shows that (III) has a smaller error compared with (I) and (II); the proposed method in the paper has a higher accuracy than the other programs where only the effective mode or Chebyshev polynomial are selected by polynomial selection techniques. It can find that selecting the significant mode shape matrix is important to the electromagnetic load identification accuracy. It directly affects the fitting accuracy of the spatial distribution function through the ratio which will cause ill-posed problem in identifying the coefficients .

(a)

(b)

(c)

5.2. Identification Based on the Measured Displacement Response with Noise

In the section, three different Gaussian noises of 3%, 5%, and 10% are used to check the robustness. Corresponding results are, respectively, shown in the 5th, 6th, and 7th row of Tables 3 and 4 and Figure 10. The results of 0.9998, 0.9998, and 0.9971 and 2.23%, 2.17%, and 8.01% (see Table 4) show that the improved method is robust when actual measurement noises are considered below 5%.

6. Conclusion

In the process of identifying the electromagnetic load, a polynomial selection technique for the significant mode shape matrix and the Chebyshev orthogonal polynomial is proposed in the present paper. A numerical example is studied and used to illustrate the feasibility and validity of the improved method. The following novelties and conclusions can be derived from this study:(1)This method demonstrates the similarity between the time-history function and the modal loads. The selection of the modal shape has a little effect on the time-history function identification.(2)The polynomial selection technique helped to pick out the significant mode shape matrix and the Chebyshev orthogonal polynomial, and the proposed method in the paper has a higher accuracy than the other programs where only the effective mode or Chebyshev polynomial are selected by polynomial selection techniques. Selecting the significant mode shape matrix does not affect the accuracy of time-history function identification but has a great influence on the accuracy of spatial distribution function reconstruction.(3)The identification results according to the measured displacement response with noise substantiate the effectiveness and robustness of the proposed method.

Data Availability

The data supporting the conclusion of the article are shown in the relevant figures and tables in the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (grant nos. 52375244 and 51775180), the Hunan Provincial Natural Science Foundation of China (grant nos. 2023JJ30192 and 2021JJ50101), the Scientific Research Foundation of Hunan Provincial Education Department (grant no. 21A0457), the Xiangtan Science and Technology Plan Key Project (grant no. GX-ZD20221008), and the Postgraduate Scientific Research Innovation Project of Hunan Province (grant no. CX20221298).

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Copyright © 2024 Wengui Mao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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