ACTA AERONAUTICAET ASTRONAUTICA SINICA >
Review on non⁃contact dynamic stress measurement methods of rotating blades
Received date: 2023-02-01
Revised date: 2023-03-01
Accepted date: 2023-06-02
Online published: 2023-06-21
Supported by
National Key Research and Development Program of China(2020YFB2010803);Key Project of Major Research Programs of the National Natural Science Foundation of China(92160203)
Blade is an important energy conversion component of aero-engines and gas turbines. The fatigue fracture problem of blades seriously affects the operation safety of the unit， so it is particularly important to monitor its health status. Real-time dynamic stress monitoring and load spectrum construction for blades can help to predict the remaining life of blades， warn the initiation of blade cracks， and realize the health management for blades. Since the non-contact dynamic stress measurement method was proposed， it has shown great potential in improving the safe operation capability of aero-engines， gas turbines and other equipment. This paper reviews the basic principle of non-contact dynamic stress measurement and the main research achievements in recent years. Besides， it summarizes the key methods and technologies in dynamic stress inversion， including the accurate identification method of tip vibration displacement， the determination method of stress amplitude ratio， multi-mode dynamic stress calculation method， etc. Furthermore， it analyzes the error sources of dynamic stress inversion and two commonly used error calibration tools for non-contact measurement of dynamic stress， and presents an outlook on the key research directions.
Weimin WANG , Dongfang HU . Review on non⁃contact dynamic stress measurement methods of rotating blades[J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2023 , 44(22) : 28516 -028516 . DOI: 10.7527/S1000-6893.2023.28516
| 1 | NIEBERDING W C， POLLACK J L. Optical detection of blade flutter［C］∥ Proceedings of the ASME 1977 International Gas Turbine Conference and Products Show. Philadelphia： International Gas Turbine Institute， 1977： V001T01A064. |
| 2 | KURKOV A， DICUS J.Synthesis of blade flutter vibratory patterns using stationary transducers［J］.Molecular Imaging， 1978， 11（1）：67-76. |
| 3 | MCCARTY P E， THOMPSON J W. A noninterference technique for measurement of turbine engine compressor blade stress［J］. Journal of Aircraft， 1982， 19（1）： 65-70. |
| 4 | MCCARTY P E， THOMPSON J W. Performance evaluation of a prototype noninterference technique for measurement of turbine engine com-pressor blade stress［C］∥ 16th Joint Propulsion Conference. Reston： AIAA； 1980： 1141. |
| 5 | RUSSHARD P. Development of a blade tip timing based engine health monitoring system［D］. Manchester： The University of Manchester， 2010：33-34. |
| 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）： 161-166. |
| 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）： 161-166 （in Chinese）. | |
| 7 | WANG W M， HU D F， LI Q H， et al. An improved non-contact dynamic stress measurement method for turbomachinery rotating blades based on fundamental mistuning model［J］. Mechanical Systems and Signal Processing， 2020， 144： 106851. |
| 8 | 朱昱达， 乔百杰， 符顺国， 等. 基于响应传递比的转子叶片动应变重构［J］. 航空动力学报， 2021，36（8）：1690-1701. |
| ZHU Y D， QIAO B J， FU S G. Dynamic strain reconstruction of rotating blade based on response transmissibility［J］. Journal of Aerospace Power， 2021， 36（8）：1690‐1701 （in Chinese）. | |
| 9 | CHEN Z S， SHENG H， XIA Y M， et al. A comprehensive review on blade tip timing-based health monitoring： status and future［J］. Mechanical Systems and Signal Processing， 2021， 149： 107330. |
| 10 | BATTIATO G. Vibrations prediction and measurement of multi-stage bladed disks with non-linear behavior due to friction contacts［D］. Turin：Politecnico di Torino， 2017. |
| 11 | JOUSSELIN O. Development of blade tip timing techniques in turbo machinery［D］. Manchester： The University of Manchester， 2013：25-27. |
| 12 | KHARYTON V. Faults detection in blades of an aviation engine in operation［D］. Lyon ：Ecole Centrale de Lyon， 2009： 2-3. |
| 13 | PICKERING T M. Methods for validation of a turbomachinery rotor blade tip timing system［D］. Blacksburg：Virginia Polytechnic Institute and State University， 2014： 6-8. |
| 14 | OSBURN N G. Implementation of a two probe tip-timing technique to determine compressor blade vibrations［D］. Stockholm： Royal Institute of Technology， 2000： 11-16. |
| 15 | MOHAMED M E E. Towards reliable and efficient calibration of blade tip timing measurements against finite element model predictions［D］. Manchester： The University of Manchester， 2019：28-35. |
| 16 | LEE G M. Prediction of strain responses from the measurements of displacement responses［J］. Mechanical Systems and Signal Processing， 2007， 21（2）： 1143-1152. |
| 17 | GEVINSKI J R， PEDERIVA R. Prediction of dynamic strain using modal parameters［J］. Journal of the Brazilian Society of Mechanical Sciences and Engineering， 2016， 38（1）： 49-57. |
| 18 | ZHANG X J， WANG Y R， JIANG X H， et al. Blade vibration stress determination method based on blade tip timing simulator and finite element method［J］. Journal of Engineering for Gas Turbines and Power， 2020， 142（3）： 031001. |
| 19 | 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（4）： 497-506. |
| 20 | IWATA T， BABA Y， SAEKI K， et al. Experimental evaluation of mechanical reliability of the impeller blade for large integrally geared compressors［C］∥ 46th Turbomachinery & 33rd Pump Symposia. Houston： Texas A&M Engineering Experiment Station， 2017： 520-529. |
| 21 | 李刚. 基于叶尖定时的叶片动应力非接触测量方法研究［D］. 天津： 天津大学， 2016： 25-41. |
| LI G. Research for No-contact measurement method of blade dynamic stress based on tip-timing［D］. Tianjin： Tianjin University， 2016： 25-41 （in Chinese）. | |
| 22 | AO C Y， QIAO B J， CHEN L， et al. Blade dynamic strain non-intrusive measurement using L1/2-norm regularization and transmissibility［J］. Measurement， 2022， 190： 110677. |
| 23 | RANTALAINEN T T， MIKKOLA A M， BJ?RK T J. Sub-modeling approach for obtaining structural stress histories during dynamic analysis［J］. Mechanical Sciences， 2013， 4（1）： 21-31. |
| 24 | ROY R， GHERLONE M， SURACE C， et al. Full-field strain reconstruction using uniaxial strain measurements： application to damage detection［J］. Applied Sciences， 2021， 11（4）： 1681. |
| 25 | WU R， ZHANG D S， YU Q F， et al. Health monitoring of wind turbine blades in operation using three-dimensional digital image correlation［J］. Mechanical Systems and Signal Processing， 2019， 130： 470-483. |
| 26 | 李馨， 赵加清， 张征明， 等. 基于数字图像相关的自适应应变场计算［J］. 光学学报， 2021， 41（23）： 125-132. |
| LI X， ZHAO J Q， ZHANG Z M， et al. Self-adaptive strain field calculation based on digital image correlation［J］. Acta Optica Sinica， 2021， 41（23）： 125-132 （in Chinese）. | |
| 27 | GWASHAVANHU B， OBERHOLSTER A J， HEYNS P S. Rotating blade vibration analysis using photogrammetry and tracking laser Doppler vibrometry［J］. Mechanical Systems and Signal Processing， 2016， 76-77： 174-186. |
| 28 | CASAVOLA C， PAPPALETTERA G. Strain field analysis in electronic components by ESPI： bad thermal contact and damage evaluation［J］. Journal of Nondestructive Evaluation， 2018， 37（1）： 11. |
| 29 | YANG Y C， JUNG H K， DORN C， et al. Estimation of full-field dynamic strains from digital video measurements of output-only beam structures by video motion processing and modal superposition［J］. Structural Control and Health Monitoring， 2019， 26（10）： e2408. |
| 30 | JANG H L， HAN D H， HWANG M Y， et al. Displacement， strain and failure estimation for multi-material structure using the displacement-strain transformation matrix［J］. Materials， 2020， 13（1）： 190. |
| 31 | LAXALDE D， THOUVEREZ F. Complex non-linear modal analysis for mechanical systems： application to turbomachinery bladings with friction interfaces［J］. Journal of Sound and Vibration， 2009， 322（4-5）： 1009-1025. |
| 32 | KRACK M. Nonlinear modal analysis of nonconservative systems： extension of the periodic motion concept［J］. Computers & Structures， 2015， 154： 59-71. |
| 33 | FERHATOGLU E， ZUCCA S. On the non-uniqueness of friction forces and the systematic computation of dynamic response boundaries for turbine bladed disks with contacts［J］. Mechanical Systems and Signal Processing， 2021， 160： 107917. |
| 34 | KHARYTON V， BLADH R. Using tiptiming and strain gauge data for the estimation of consumed life in a compressor blisk subjected to stall-induced loading［C］∥ Proceedings of ASME Turbo Expo 2014. Düsseldorf： ASME， 2014： 27251. |
| 35 | MOHAMED M， BONELLO P， RUSSHARD P. A novel method for the determination of the change in blade tip timing probe sensing position due to steady movements［J］. Mechanical Systems and Signal Processing， 2019， 126： 686-710. |
| 36 | MOHAMED M， BONELLO P， RUSSHARD P. Determination of simultaneous steady-state movements using blade tip timing data［J］.Journal of Vibration and Acoustics： Transactions of the ASME，2020，142（1）：011017. |
| 37 | ZHANG X L， WANG W M， CHEN K， et al. Five dimensional movement measurement method for rotating blade based on blade tip timing measuring point position tracking［J］. Mechanical Systems and Signal Processing， 2021， 161： 107898. |
| 38 | GUO H N， YANG Y M， GUAN F J， et al. A torsional vibration measurement method for rotating blades based on blade tip timing［J］. Shock and Vibration， 2021，2021：9950269. |
| 39 | ZHANG X J， WANG Y R， JIANG X H， et al. Parameter identification and sensor configuration in tip timing for asynchronous vibration of a deformed blade with finite element method simulated data verification［J］. Journal of Vibration and Acoustics， 2020， 142（2）： 021010. |
| 40 | 徐芝纶. 弹性力学简明教程［M］. 4版. 北京： 高等教育出版， 2013： 16-18. |
| XU Z L. Elastic mechanics concise course［M］. 4th ed. Beijing： Higher Education Press， 2013： 16-18 （in Chinese）. | |
| 41 | KHAN R， KAMILI O M， NAGABHUSHANA S T， et al. Correlation between FEA and testing for HCF-safety assessment of an aero engine turbine rotor blade［C］∥ ASME Gas Turbine India Conference. Bangalore： ASME， 2013： 3697. |
| 42 | MOHAMED M E， BONELLO P， RUSSHARD P. Uncertainties in the calibration process of blade tip timing data against finite element model predictions［C］/∥ 12th International Conference on Vibrations in Rotating Machinery. London： Institute of Mechanical Engineers， 2020. |
| 43 | MOHAMED M E， BONELLO P， RUSSHARD P， et al. Experimental validation of FEM-computed stress to tip deflection ratios of aero-engine compressor blade vibration modes and quantification of associated uncertainties［J］. Mechanical Systems and Signal Processing， 2022， 178： 109257. |
| 44 | KASZYNSKI A A， BROWN J M. Accurate blade tip timing limits through geometry mistuning modeling［C］∥ Proceedings of ASME Turbo Expo 2015. Montréal： ASME， 2015：43192. |
| 45 | GUO H N， YANG Y M， GUAN F J， et al. A blade dynamic strain inversion method for rotating blades based on blade tip timing［J］. Journal of Physics： Conference Series， 2021， 2029（1）： 012073. |
| 46 | DIMITRIADIS G， CARRINGTON I B， WRIGHT J R， et al. Blade-tip timing measurement of synchronous vibrations of rotating bladed assemblies［J］. Mechanical Systems and Signal Processing， 2002， 16（4）： 599-622. |
| 47 | LI J， AYE-ADDO N， KIELB R， et al. Mistuned higher-order mode forced response of an embedded compressor rotor：Part II—mistuned forced response prediction［J］. Journal of Turbomachinery， 2018， 140（3）： 031006. |
| 48 | BORNASSI S， FIRRONE C M， BERRUTI T M. Vibration parameters estimation by blade tip-timing in mistuned bladed disks in presence of close resonances［J］. Applied Sciences， 2020， 10（17）： 5930. |
| 49 | 李琳， 郭雅妮. 发动机静子尾流激励对叶盘结构高阶共振可激性的研究［J］. 航空动力学报， 2017， 32（4）： 780-790. |
| LI L， GUO Y N. Research on excitability of the wake of stator in aero-engine to high order resonance of bladed-disk［J］. Journal of Aerospace Power， 2017， 32（4）： 780-790 （in Chinese）. | |
| 50 | 王建军， 卿立伟， 李其汉. 旋转叶片频率转向与振型转换特性［J］. 航空动力学报， 2007， 22（1）： 8-11. |
| WANG J J， QING L W， LI Q H. Frequency veering and mode shape interaction for rotating blades［J］. Journal of Aerospace Power， 2007， 22（1）： 8-11 （in Chinese）. | |
| 51 | KELLEY N， OSGOOD R， BIALASIEWICZ J， et al. Using time-frequency and wavelet analysis to assess turbulence/rotor interactions［C］∥ 2000 ASME Wind Energy Symposium. Reston： AIAA， 2000： 36. |
| 52 | 刘海， 娄金伟， 胡伟， 等. 基于叶尖定时原理的整体叶盘振动测试与分析［J］. 航空发动机， 2021， 47（6）： 74-79. |
| LIU H， LOU J W， HU W， et al. Vibration test and analysis of blisk based on tip-timing theory［J］. Aeroengine， 2021， 47（6）： 74-79 （in Chinese）. | |
| 53 | GROSS J， KRACK M. Multi-wave vibration caused by flutter instability and nonlinear tip-shroud friction［J］. Journal of Engineering for Gas Turbines and Power， 2020， 142（2）： 021013. |
| 54 | DAY I J. Stall， surge， and 75 years of research［J］. Journal of Turbomachinery， 2016， 138（1）： 011001. |
| 55 | KAMARAJ A， YADAV V， VENKATARAMAN S. Estimating uncertainty bounds for blade vibratory stresses under multi-mode excitation［C］∥ 54th AIAA Aerospace Sciences Meeting. Reston： AIAA， 2016： 1887. |
| 56 | NI K， WANG X Q， MIGNOLET M. Blade stress estimation during multiple vibratory modes［C］∥ 54th AIAA/ASME/ASCE/AHS/ASC Structures， Structural Dynamics， and Materials Conference. Reston： AIAA， 2013： 1772. |
| 57 | SALHI B， LARDIES J， BERTHILLIER M. Identification of modal parameters and aeroelastic coefficients in bladed disk assemblies［J］. Mechanical Systems and Signal Processing， 2009， 23（6）： 1894-1908. |
| 58 | CHEN Z S， YANG Y M， GUO B， et al. Blade damage prognosis based on kernel principal component analysis and grey model using subsampled tip-timing signals［J］. Proceedings of the Institution of Mechanical Engineers， Part C： Journal of Mechanical Engineering Science， 2014， 228（17）： 3178-3185. |
| 59 | CHEN Z S， YANG Y M， XIE Y， et al. Non-contact crack detection of high-speed blades based on principal component analysis and Euclidian angles using optical-fiber sensors［J］. Sensors and Actuators A： Physical， 2013， 201： 66-72. |
| 60 | CHEN Z S， HE J， ZHAN C. Undersampled blade tip-timing vibration reconstruction under rotating speed fluctuation： uniform and nonuniform sensor configurations［J］. Shock and Vibration， 2019， 2019： 8103216. |
| 61 | CHEN Z S， LIU J H， ZHAN C， et al. Reconstructed order analysis-based vibration monitoring under variable rotation speed by using multiple blade tip-timing sensors［J］. Sensors， 2018， 18（10）： 3235. |
| 62 | WANG P， KARG D， FAN Z Y， et al. Non-contact identification of rotating blade vibration［J］. Mechanical Engineering Journal， 2015， 2（3）： 15-25. |
| 63 | HU Z， LIN J， CHEN Z S， et al. A non-uniformly under-sampled blade tip-timing signal reconstruction method for blade vibration monitoring［J］. Sensors， 2015， 15（2）： 2419-2437. |
| 64 | XU J H， QIAO B J， YANG Z B， et al. Optimal placement of blade tip timing sensors considering multi-mode vibration using evolutionary algorithms［C］∥ 2020 International Conference on Sensing， Measurement & Data Analytics in the era of Artificial Intelligence （ICSMD）. Piscataway： IEEE Press， 2020： 367-372. |
| 65 | CHEN S Y， YANG Y M， HU H F， et al. Blind interpolation for multi-frequency blade tip timing signals［J］. Mechanical Systems and Signal Processing， 2022， 172： 108946. |
| 66 | AO C Y， QIAO B J， LIU M R， et al. Non-contact full-field dynamic strain reconstruction of rotating blades under multi-mode vibration［J］. Mechanical Systems and Signal Processing， 2023， 186： 109840. |
| 67 | LIN J， HU Z， CHEN Z S， et al. Sparse reconstruction of blade tip-timing signals for multi-mode blade vibration monitoring［J］. Mechanical Systems and Signal Processing， 2016， 81： 250-258. |
| 68 | PAN M H， YANG Y M， GUAN F J， 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. |
| 69 | BOUCHAIN A， PICHERAL J， LAHALLE E， et al. Blade vibration study by spectral analysis of tip-timing signals with OMP algorithm［J］. Mechanical Systems and Signal Processing， 2019， 130： 108-121. |
| 70 | XU J H， QIAO B J， LIU J J， et al. Sparse reconstruction for blade tip timing signal using generalized minimax-concave penalty［J］. Mechanical Systems and Signal Processing， 2021， 161： 107961. |
| 71 | CHEN Z S， SHENG H， XIA Y M. Multi-coset angular sampling-based compressed sensing of blade tip-timing vibration signals under variable speeds［J］. Chinese Journal of Aeronautics， 2021， 34（9）： 83-93. |
| 72 | BIAN Z F， HU H F， YANG Y M， et al. A spectrum reconstruction method for blade vibration measurement based on probe waveform analysis［J］. Measurement， 2022， 199： 111425. |
| 73 | 户东方. 涡轮叶片动应力场重构关键技术研究［D］. 北京： 北京化工大学， 2020： 52-67. |
| HU D F. Research on key technology of dynamic stress field reconstruction for turbine blade［D］. Beijing： Beijing University of Chemical Technology， 2020： 52-67 （in Chinese）. | |
| 74 | BATTIATO G， FIRRONE C M， BERRUTI T M. Forced response of rotating bladed disks： Blade Tip-Timing measurements［J］. Mechanical Systems and Signal Processing， 2017， 85： 912-926. |
| 75 | RUSSHARD P. Blade tip timing （BTT） uncertainties［C］∥ Proceedings of the 12th International Aivela Conference on Vibration Measurements by Laser & Noncontact Techniques： Advances & Applications. Ancona： AIP Publishing LLC， 2016：020003. |
| 76 | PROCHáZKA P. Methods and measuring systems for calibration of non-contact vibrodiagnostics systems［C］∥ 2017 IEEE International Instrumentation and Measurement Technology Conference （I2MTC）. Piscataway： IEEE Press， 2017： 1-6. |
| 77 | DIAMOND D H， HEYNS P S， OBERHOLSTER A J. Improved Blade Tip Timing measurements during transient conditions using a State Space Model［J］. Mechanical Systems and Signal Processing， 2019， 122： 555-579. |
| 78 | FAN Z F， LI H K， DONG J N， et al. An improved multiple per revolution-based blade tip timing method and its applications on large-scale compressor blades［J］. Mechanical Systems and Signal Processing， 2022， 167： 108538. |
| 79 | REN S Q， XIANG X R， ZHAO W， et al. An error correction blade tip-timing method to improve the measured accuracy of blade vibration displacement during unstable rotation speed［J］. Mechanical Systems and Signal Processing， 2022， 162： 108030. |
| 80 | WANG W M， CHEN K， ZHANG X L， et al. A novel method to improve the precision of BTT under rapid speed fluctuation conditions［J］. Mechanical Systems and Signal Processing， 2022， 177： 109203. |
| 81 | HOCKADAY B D. Quantifying optical tip-timing probe error with laboratory apparatus［C］∥ Proceedings of the ASME Turbo Expo： Turbine Technical Conference and Exposition. Vancouver：ASME，2011： 437-444. |
| 82 | DIAMOND D H， HEYNS P S， OBERHOLSTER A J. Constant speed tip deflection determination using the instantaneous phase of blade tip timing data［J］. Mechanical Systems and Signal Processing， 2021， 150（7）：107151. |
| 83 | JAMIA N， FRISWELL M I， EL-BORGI S， et al. Modelling and experimental validation of active and passive eddy current sensors for blade tip timing［J］. Sensors and Actuators A： Physical， 2019， 285： 98-110. |
| 84 | FRISWELL M I， MOTTERSHEAD J E. Finite element model updating in structural dynamics［M］. Dordrecht： Kluwer Academic Publishers， 1995. |
| 85 | HU D F， WANG W M， ZHANG X L， et al. On-line real-time mistuning identification and model calibration method for rotating blisks based on blade tip timing （BTT）［J］. Mechanical Systems and Signal Processing， 2021， 147： 107074. |
| 86 | KASZYNSKI A A， BECK J A， BROWN J M. Uncertainties of an automated optical 3D geometry measurement， modeling， and analysis process for mistuned integrally bladed rotor reverse engineering［J］. Journal of Engineering for Gas Turbines and Power， 2013， 135（10）： 102504. |
| 87 | GILLAUGH D， JANCZEWSKI T， KASZYNSKI A， et al. Forced response variation of a compressor utilizing blade tip timing， strain gages， and as-manufactured finite element models［J］. Journal of Engineering for Gas Turbines and Power： Transactions of the ASME， 2020， 143（11）： 111023. |
| 88 | 刘豪. 与航空发动机涡轮叶片一体化集成的薄膜应变计研究［D］. 成都： 电子科技大学， 2019： 17-21. |
| LIU H. Research on thin film strain gauge integrated with aeroengine turbine blade［D］. Chengdu： University of Electronic Science and Technology of China， 2019： 17-21 （in Chinese）. | |
| 89 | ARPIN-PONT J， GAGNON M， TAHAN S A， et al. Strain gauge measurement uncertainties on hydraulic turbine runner blade［J］. IOP Conference Series： Earth and Environmental Science， 2012， 15（6）： 062042. |
| 90 | TEUFEL P， POLAND J. Robust and optimal strain gauge positions for blade vibration testing［C］∥ 2015 SIMULIA Community Conference. Berlin： SIMULIA Community， 2015. |
| 91 | KLAUKE T， KU¨HHORN A， BEIROW B， et al. Blade vibration phenomena of HPC bliscs considering manufacturing effects and strain gauge application［C］∥Proceedings of ASME Turbo Expo ： Power for Land， Sea， and Air. Berlin： ASME， 2008： 403-413. |
| 92 | ZHAO X W， LI H K， YANG S H， et al. Blade vibration measurement and numerical analysis of a mistuned industrial impeller in a single-stage centrifugal compressor［J］. Journal of Sound and Vibration， 2021， 501： 116068. |
| 93 | WEI D T， LI H K， CHEN Y G， et al. Development of blade tip timing signal simulator based on a novel model reduction method of bladed disks［J］. Journal of Sound and Vibration， 2022， 534： 117053. |
| 94 | MOHAMED M E， BONELLO P， RUSSHARD P. An experimentally validated modal model simulator for the assessment of different Blade Tip Timing algorithms［J］. Mechanical Systems and Signal Processing， 2020， 136： 106484. |
| 95 | FIGASCHEWSKY F， HANSCHKE B， KüHHORN A. Efficient generation of engine representative tip timing data based on a reduced order model for bladed rotors［J］. Journal of Engineering for Gas Turbines and Power， 2019， 141（1）： 012503. |
| 96 | CHIACHíO M， MEGíA M， CHIACHíO J， et al. Structural digital twin framework： formulation and technology integration［J］. Automation in Construction， 2022， 140： 104333. |
| 97 | 陶飞， 张辰源， 张贺， 等. 未来装备探索：数字孪生装备［J］. 计算机集成制造系统， 2022， 28（1）： 1-16. |
| TAO F， ZHANG C Y， ZHANG H， et al. Future equipment exploration： digital twin equipment［J］. Computer Integrated Manufacturing Systems， 2022， 28（1）： 1-16 （in Chinese）. | |
| 98 | 李杰林， 刘晋飞， 鞠传海， 等. 可靠性数字孪生概念内涵和技术框架研究［J］. 测控技术， 2021， 40（2）： 18-24. |
| LI J L， LIU J F， JU C H， et al. Research on conceptual connotation and technical framework of reliability-digital twin［J］. Measurement & Control Technology， 2021， 40（2）： 18-24 （in Chinese）. | |
| 99 | 沈如松， 徐焕翔， 矫永康， 等. 数字孪生及其在飞行器领域应用［J］. 指挥与控制学报， 2021， 7（3）： 241-248. |
| SHEN R S， XU H X， JIAO Y K， et al. Digital twin and its application in aircraft field［J］. Journal of Command and Control， 2021， 7（3）： 241-248 （in Chinese）. | |
| 100 | BECK J A， BROWN J M， GILLAUGH D L， et al. Integrally bladed rotor mistuning identification and model updating using geometric mistuning models［J］. Journal of Engineering for Gas Turbines and Power， 2021， 143（12）： 121012. |
| 101 | MAYWALD T， BACKHAUS T， SCHRAPE S， et al. Geometric model update of blisks and its experimental validation for a wide frequency［C］∥ ASME Turbo Expo ： Turbomachinery Technical Conference and Exposition， 2017： 63446. |
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