Reconstruction of blade tip-timing signals based on the MUSIC algorithm
Introduction
Blade vibration measurement is important to ensure safe operation for rotating machinery, such as compressors, turbines, and so on [1], [2]. As a non-contact measurement method, the BTT is a standard test technology for advanced aero-engine manufacturing companies, such as Rolls-Royce, MTU, etc [3], [4], [5]. Compared with strain gauges, the BTT has the advantages of low equipment cost, easy installation, and monitoring all blade vibrations [6]. It has been playing an important role in compressor surge monitoring [7], fatigue cracks [8], and foreign object damage (FOD) [9].
Since the BTT signals are acquired by under-sampling, it is very important to avoid spectrum aliasing in reconstruction. Currently, the reconstruction methods mainly include curve fitting and spectrum analysis. The curve fitting methods are usually used to analyze the blade vibration during the sweep of rotating speed. These methods are the single degree of freedom (SDOF) curve fit [10], [11], the two-parameter plot method (2PP) [12], [13], the autoregressive (AR) method [14], [15], the circumferential Fourier fit (CFF) method [16], [17], and the non-OPR fit method [18]. Since the blade operation time at cruising conditions is the longest, the blade vibration under constant speed attracts more attention. The spectrum analysis methods are the most suitable for vibration analysis in this situation. Spectrum analysis methods mainly include traveling wave analysis (TW) [19], the “5 + 2”method [20], the minimum variance spectrum estimation (MVSE) [21], the non-uniform discrete Fourier transform (NUDFT) [22], the cross-spectrum estimation (CSE) [23], the compressed sensing method (CS) [24], [25], [26], and so on. Prior knowledge of blades is usually needed in the spectrum analysis. The CS method is the most admired method to reconstruct the BTT signals. However, under-sampling can still lead to spectrum aliasing if the probe layout is not suitable [26].
As BTT technology is gradually applied to the airborne measurement, it is extremely urgent to find a high-performance reconstruction method. The new method should rely on less prior knowledge, with little computation time and high accuracy. As a multi-frequency identification method, the MUSIC method has received more and more attention in the field of BTT signal analysis [26], [27], [28]. Chen Xuefeng is the first to introduce the MUSIC method into the BTT signal reconstruction [26]. It showed that the traditional MUSIC method [29] has the potential to overcome frequency aliasing. However, the limitation of the number of probes resulted in some problems, such as low identification accuracy and less number of identifiable frequencies. Yang Zhibo improved the traditional MUSIC method through reusing sampled data (The MUSIC method improved by Yang is named Yang’s method in the following text) [27]. Although the weakness of the traditional MUSIC is overcome by Yang’s method, the scale of the snapshot matrix is increased. The bigger size of the snapshot matrix will result in more computation time for the autocorrelation matrix calculation and eigenvalues decomposition. Subsequently, a subspace dimension reduced MUSIC method (SDR-MUSIC) was proposed to reduce the running time of Yang’s method [28]. Although the SDR-MUSIC reduces the frequency traversal time, it still has to rely on the expanded snapshot matrix to overcome spectrum aliasing. It is difficult to further improve the computational efficiency if the scale of the snapshot matrix is not reduced. In this paper, reconstruction conditions were proposed based on the MUSIC algorithm. The vibration frequency can be identified accurately by the traditional MUSIC if the reconstruction conditions are met. It need not expand the snapshot matrix or other modifications. Compared with Yang’s method, the computational complexity is greatly reduced. This is more conducive to real-time processing. The feasibility of the reconstruction conditions was verified through simulations and aero-engine experiments. Under the reconstruction conditions, the frequency identification accuracy of the traditional MUSIC method is consistent with Yang's method. Since the blade vibration amplitude cannot be directly extracted from the MUSIC pseudo spectrum, an amplitude identification method was proposed based on the discrete Fourier transformation (DFT) and the remainder theorem [30]. It does not depend on the process of frequency identification but only on frequency results. It is a supplement to the spectrum methods that can only obtain vibration frequency but not amplitude, for example, the ESPRIT method [31], Yang’s method [27], and the SDR-MUSIC method [28].
The outline of this paper is organized as follows: In Section 2, a mathematical model of BTT signals based on the MUSIC algorithm was established. In Section 3, reconstruction conditions were proposed to let the traditional MUSIC method overcome the spectrum aliasing. Besides, an amplitude identification method based on the DFT and remainder theorem was derived. In Section 4, the performance of the traditional MUSIC method under the reconstruction conditions was discussed by simulations. And the error of the amplitude identification was analyzed. In Section 5, the feasibility of the reconstruction conditions and amplitude identification method was verified through the commercial turbofan engine test. In Section 6, the advantages and disadvantages of the reconstruction conditions were discussed. Besides, the ways to improve the accuracy of the amplitude estimation were pointed out. The main conclusions were summarized in Section 7. The expanded snapshot matrix of Yang's method was introduced in Appendix A and the computational complexity was analyzed in Appendix B.
Section snippets
Blade tip-timing measurement
Assuming that there are G probes installed on the casing, and the probe installation angle is βg (g = 0, 1, … G − 1). Based on the principle of BTT measurement [32], the BTT sampling period is T = 1/fr, where fr is the rotating speed (Unit: Hz). In each sampling period T, the sampling time of the probe g relative to the reference probe (in the absence of instructions, probe 0 is used as the reference probe) is
Then the blade vibration displacement yg measured by the probe g can be given
Traditional MUSIC algorithm
The MUSIC is a multi-frequency identification method based on the orthogonality of the signal and noise subspace [29], [35], [36]. The autocorrelation matrix Ryy of the snapshot matrix Y can be expressed aswhere E represents the expectation operator, H represents the conjugate transpose, I is the G × G order unit matrix, and Rs is the autocorrelation matrix of the matrix S. The matrix Rs is given as
Let the eigenvalue decomposition of Ryy bewhere Σ = diag (λ1
Simulation parameter setting
The simulations were carried out according to Eqs. (1)–(5). Assuming there are 6 probes to monitor the blade vibration. The probe circumferential positions were shown in Table 1. Let the simulated rotating speed be 14000 rpm (fr = 233 Hz). Two probe groups were selected to verify the feasibility of the reconstruction conditions. The probe groups including 5 probes (P1 ~ P5) and 4 probes (P1 ~ P4), respectively. For each probe group, the position of the first probe was set as the reference.
Experiments
In the commercial turbofan test, the aero-engine ran at the normalized rotating speeds of 91%, 96%, 99%, 101%, 102%, and 104%, respectively. The rotating speed normalized reference is the economic cruise rotating speed. The 1st stage of the high-pressure compressor contains 25 blades. In this test, the blade excitation force mainly comes from airflow induce. Six fiber probes were arranged on the 1st stage. The fiber probe circumferential position is the same as that in Table 1. The axial
Discussion
The reconstruction conditions consider blade vibration characteristics, under-sampling frequency, and probe circumferential position. It reduces the data redundancy by optimizing the sampling model. The reconstruction conditions enable the traditional MUSIC method to overcome spectrum aliasing without any modification. Compared with Yang's method or SDR-MUISC, since the snapshot matrix maintained its original size, the minimum computational complexity of the MUSIC method is achieved (Please
Conclusion
The reconstruction conditions based on the MUSIC algorithm were proposed in this paper. It makes the traditional MUSIC method can complete frequency identification without any modification. Under the reconstruction conditions, the identification accuracy of the traditional MUSIC method can reach that of Yang’s method. In a sense, the reconstruction conditions can replace the expanded snapshot matrix. The computational complexity is greatly reduced. That is more conducive to real-time
CRediT authorship contribution statement
Zhibo Liu: Methodology, Software, Formal analysis, Writing - original draft, Writing - review & editing. Fajie Duan: Conceptualization, Supervision, Funding acquisition. Guangyue Niu: Investigation. Dechao Ye: Data curation. Junnan Feng: Resources. Zhonghai Cheng: Visualization. Xiao Fu: Funding acquisition. Jiajia Jiang: Funding acquisition. Jing Zhu: Resources. Meiru Liu: Resources.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This work was supported in part by National Natural Science Foundations of China (51775377), National key research and development plan project (2020YFB2010800), National Science and Technology Major Project (2017-V-0009), National Natural Science Foundations of China (61971307,61905175), the Fok Ying Tung education foundation (171055), China Postdoctoral Science Foundation (2020 M680878), Guangdong Province key research and development plan project (2020B0404030001), Tianjin Science and
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