Despite the wide application of the blade tip timing as a non-contact measurement technique, its acquired displacement signal is undersampled in many situations. Compressed sensing is an effective means to reconstruct the undersampled signal. However, the reconstruction process introduces the regularization penalty to achieve sparsity, reducing the amplitude accuracy of the reconstructed signal at the same time, and affecting the accurate identification of blade vibration amplitude, which is highly significant in reconstructing the dynamic stress of the blade. This paper combines the design matrix of the blade vibration equation and the compressed sensing dictionary to identify parameters of the blade vibration equation without prior information. First, a compressed sensing dictionary is constructed according to the form of design matrix elements and the maximum vibration frequency of interests. Second, atoms are extracted from the dictionary according to the indices of non-zero elements in the sparse representation of the blade tip timing signal, and parameters of the vibration equation are calculated. Third, Blade Tip Timing (BTT) simulation results demonstrate that the proposed method can accurately identify parameters of the vibration equation with both single-mode and multi-mode vibrations. Finally, experimental results of rotor blades show that the relative error of the identified natural frequency between using strain gage and using blade tip timing with compressed sensing is only 0.14%; the transmissibility of blade tip displacement to certain points strain of the four blades is compared and the maximum percentage of deviation from the mean is only 2.15%.
[1] 李克安,林左鸣,杨胜群,等. 航空发动机转子叶片振动方程及其频率计算[J]. 航空学报, 2013, 34(12):2733-2739. LI K A, LIN Z M, YANG S Q, et al. Vibration equation and frequency computation of an aero-engine rotor blade[J]. 2013, 34(12):2733-2739(in Chinese).
[2] ZHAO F, JING X, YANG M, et al. Experimental study of rotor blades vibration and noise in multistage high pressure compressor and their relevance[J]. Chinese Journal of Aeronautics, 2020, 33(3):870-878.
[3] KNAPPETT D, GARCIA J. Blade tip timing and strain gauge correlation on compressor blades[J]. Proceedings of the Institution of Mechanical Engineers Part G-Journal of Aerospace Engineering, 2008, 222(G4):497-506.
[4] 刘美茹,朱靖,梁恩波,等. 基于叶尖定时的航空发动机压气机叶片振动测量[J]. 航空动力学报, 2019,34(9):1895-1904. LIU M R, ZHU J, LIANG E B, et al. Vibration measurement on compressor rotor blades of aero-engine based on tip-timing[J]. Journal of Aerospace Power, 2019,34(9):1895-1904(in Chinese).
[5] WANG W, ZHANG X, HU D, et al. A novel none once per revolution blade tip timing based blade vibration parameters identification method[J]. Chinese Journal of Aeronautics, 2020, 33(7):1953-1968.
[6] 段发阶,李刚,叶德超,等. 基于叶尖定时的叶片动应力反演方法[J]. 纳米技术与精密工程, 2016, 14(3):7-12. DUAN F J, LI G, YE D C, et al. Method for inversing dynamic stress of blade based on tip-timing[J]. Nanotechnology and Precision Engineering, 2016, 14(3):7-12(in Chinese).
[7] HU Z, LIN J, CHEN Z, et al. A non-uniformly under-sampled blade tip-timing signal reconstruction method for blade vibration monitoring[J]. Sensors, 2015, 15(2):2419-2437.
[8] HEATH S, IMREGUN M. An improved single-parameter tip-timing method for turbomachinery blade vibration measurements using optical laser probes[J]. International Journal of Mechanical Sciences, 1996, 38(10):1047-1058.
[9] 欧阳涛,段发阶,闫明,等. 双参数法辨识叶片同步振动的研究[J]. 传感器与微系统, 2010, 29(3):42-45. OU Y T, DUAN F J, YAN M, et al. Research on identifying synchronous blade vibration using two-parameter-plot method[J]. Transducer and Microsystem Technologies, 2010, 29(3):42-45(in Chinese).
[10] 王维民,任三群,陈立芳,等. 涡轮机叶片同步振动参数辨识方法研究(建模仿真)[J]. 振动与冲击, 2017, 36(17):120-126. WANG W M, REN S Q, CHEN L F, et al. Synchronous vibration parametric identification method for blades of a turbine[J]. Journal of Vibration and Shock, 2017, 36(17):120-126(in Chinese).
[11] 王维民,任三群,陈立芳,等. 涡轮机叶片同步振动参数辨识方法研究(实验研究)[J]. 振动与冲击, 2017, 36(17):127-133. WANG W M, REN S Q, CHEN L F, et al. Tests for synchronous vibration parametric identification method of a turbine's blades[J]. Journal of Vibration and Shock, 2017, 36(17):127-133(in Chinese).
[12] Di MAIO D, EWINS D J. Experimental measurements of out-of-plane vibrations of a simple blisk design using blade tip timing and scanning LDV measurement methods[J]. Mechanical Systems and Signal Processing, 2012, 28(SI):517-527.
[13] JOUNG K, KANG S, PAENG K, et al. Analysis of vibration of the turbine blades using non-intrusive stress measurement system[C]//ASME 2006 Power Conference, 2006, 42053:391-397.
[14] CHEN S, DONOHO D L, SAUNDERS M A. Atomic decomposition by basis pursuit[J]. Siam Review, 2001, 43(1):129-159.
[15] PAN M, YANG Y, GUAN F, et al. Sparse representation based frequency detection and uncertainty reduction in blade tip timing measurement for multi-mode blade vibration monitoring[J]. Sensors, 2017, 17(8):1745.
[16] 吴淑明,胡海峰,赵志斌,等. 增强稀疏分解及其在叶片振动参数识别中的应用[J]. 机械工程学报, 2019, 55(19):19-27. WU S M, HU H F, ZHAO Z B, et al. Enhancing sparse decomposition based blade vibration parameter identification[J]. Journal of Mechanical Engineering, 2019, 55(19):19-27(in Chinese).
[17] WU S, RUSSHARD P, YAN R, et al. An adaptive online blade health monitoring method:from raw data to parameters identification[J]. IEEE Transactions on Instrumentation and Measurement, 2020, 69(5):2581-2592.
[18] DIAMOND D H, HEYNS P S. A novel method for the design of proximity sensor configuration for rotor blade tip timing[J]. Journal of Vibration and Acoustics-Transactions of the Asme, 2018, 140(6):061003.
[19] QIAO B, ZHANG X, WANG C, et al. Sparse regularization for force identification using dictionaries[J]. Journal of Sound and Vibration, 2016, 368:71-86.
[20] BOYD, VANDENBERGHE, FAYBUSOVICH. Convex optimization[M]. Cambridge:Cambridge University Press, 2004:291-345.
[21] CANDES E J, TAO T. Decoding by linear programming[J]. IEEE Transactions On Information Theory, 2005, 51(12):4203-4215.
[22] QIAO B, ZHANG X, GAO J, et al. Sparse deconvolution for the large-scale ill-posed inverse problem of impact force reconstruction[J]. Mechanical Systems and Signal Processing, 2017, 83:93-115.
[23] 敖春燕,乔百杰,刘美茹,等. 基于非接触式测量的旋转叶片动应变重构方法[J]. 航空动力学报, 2020(3):569-580. AO C Y, QIAO B J, LIU M R, et al. Dynamic strain reconstructing method of rotating blades based on no-contact measurement[J]. Journal of Aerospace Power, 2020(3):569-580(in Chinese).
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