基于裂纹附加模态Timoshenko梁的裂纹损伤识别
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摘要
将梁中横向裂纹等效为无质量扭转弹簧,并忽略其对梁剪切变形的影响,得到的具有任意裂纹数目Timoshenko梁自振模态的统一显示解析表达式.将裂纹梁的自振模态分为基本模态和裂纹附加模态,利用最小二乘拟合,建立了利用裂纹附加模态函数的梁裂纹损伤识别方法.通过数值模拟开展了简支单裂纹梁以及悬臂和固支双裂纹梁等的裂纹损伤识别,考察了测量误差对损伤识别的影响,数值结果表明本文所提出的裂纹损伤识别方法对裂纹位置的识别精度高于对裂纹损伤程度的识别精度;随着测量误差的增加,裂纹位置及裂纹损伤程度的识别误差增加,但仍在可接受的范围内,故该裂纹损伤识别方法在实际工程中具有一定的应用价值.
Regarding a transverse crack in a beam as a massless rotational spring and neglecting its influence on the shear deformation of beam, the unified explicit analytical expression of the vibration mode of Timoshenko beam with arbitrary number of cracks is obtained. The vibration modes of a cracked beam are then divided into the basic mode and the crack-induced additional mode, and a damage identification method of beam crack is established based on the crack-induced additional mode function through the least square fitting. Damage identifications of cracks of a simply-supported beam with single crack, cantilever and clamped-clamped beams with double cracks are carried out numerically, and the influence of measurement noise on the identification accuracy is examined. The numerical results show that, using the crack damage identification method presented in present paper, the identification accuracy of the crack position is higher than that of the crack damage degree. The identification errors of the positions and the damage degrees of the cracks increase with the increase of measurement errors, but are in acceptable range. Therefore, the proposed crack damage identification method is applicable for practical engineering.
Regarding a transverse crack in a beam as a massless rotational spring and neglecting its influence on the shear deformation of beam, the unified explicit analytical expression of the vibration mode of Timoshenko beam with arbitrary number of cracks is obtained. The vibration modes of a cracked beam are then divided into the basic mode and the crack-induced additional mode, and a damage identification method of beam crack is established based on the crack-induced additional mode function through the least square fitting. Damage identifications of cracks of a simply-supported beam with single crack, cantilever and clamped-clamped beams with double cracks are carried out numerically, and the influence of measurement noise on the identification accuracy is examined. The numerical results show that, using the crack damage identification method presented in present paper, the identification accuracy of the crack position is higher than that of the crack damage degree. The identification errors of the positions and the damage degrees of the cracks increase with the increase of measurement errors, but are in acceptable range. Therefore, the proposed crack damage identification method is applicable for practical engineering.
引文
[1]DIMAROGONAS A D.Vibration of cracked structures:a state of the art review[J].Engineering Fracture Mechanics,1996,55(5):831-857.
[2]CHALLAMEL N,XIANG Y.On the influence of the unilateral damage behaviour in the stability of cracked beam columns[J].Engineering Fracture Mechanics,2010,77(9):1467-1478.
[3]REZAEE M,HASSANNEJAD R.Free vibration analysis of simply supported beam with breathing crack using perturbation method[J].Acta Mechanica Solida Sinica,2010,23(5):459-470.
[4]JASSIM Z A,ALI N N,MUSTAPHA F,et al.A review on the vibration analysis for a damage occurrence of a cantilever beam[J].Engineering Failure Analysis,2013,31(7):442-461.
[5]YANG X,HUANG X,OUYANG Y.Bending of Timoshenko beam with effect of crack gap based on equivalent spring model[J].Applied Mathematics and Mechanics,2016,37(4):513-528.
[6]DOEBLING S W.Damage identification and health monitoring of structural and mechanical systems from changes in their vibration characteristics:a literature review[J].Shock and Vibration Digest,1996,30(11):2043-2049.
[7]ATTAR M.A transfer matrix method for free vibration analysis and crack identification of stepped beams with multiple edge cracks and different boundary conditions[J].International Journal of Mechanical Sciences,2012,57(1):19-33.
[8]汪德江,杨骁.基于裂纹诱导弦挠度的Timoshenko梁裂纹无损检测[J].工程力学,2016,33(12):186-195.
[9]杨骁,唐珂,黄瑾.基于裂纹诱导挠度的悬臂梁裂纹静力损伤识别[J].力学季刊,2017,38(2):318-326.
[10]RUOTOLO R,SURACE C.Natural frequencies of a bar with multiple cracks[J].Journal of Sound and Vibration,2004,272(1-2):301-316.
[11]ZHANG Z G,CHEN F,ZHANG Z Y,et al.Vibration analysis of non-uniform Timoshenko beams coupled with flexible attachments and multiple discontinuities[J].International Journal of Mechanical Sciences,2014,80(1):131-143.
[12]徐福后,张玉祥.含裂纹Timoshenko梁自由振动分析[J].船舶力学,2011,15(10):1166-1172.
[13]BUDA G,CADDEMI S.Identification of concentrated damages in Euler-Bernoulli beams under static loads[J].Journal of Engineering Mechanics,2007,133(8):942-956.
[14]CICIRELLO A,PALMERI A.Static analysis of Euler-Bernoulli beams with multiple unilateral cracks under combined axial and transverse loads[J].International Journal of Solids and Structures,2014,51(5):1020-1029.
[15]SWAMIDAS A S J,YANG X,SESHADRI R.Identification of cracking in beam structures using Timoshenko and Euler formulations[J].Journal of Engineering Mechanics,2004,130(11):1297-1308.
[16]KHIEM N T,LIEN T V.The dynamic stiffness matrix method in forced vibration analysis of multiple-cracked beam[J].Journal of Sound and Vibration,2002,254(3):541-555.
[17]LABIB A,KENNEDY D,FEATHERSTON C,et al.Free vibration analysis of beams and frames with multiple cracks for damage detection[J].Journal of Sound and Vibration,2014,333(20):4991-5003.
[18]CAWLEY P,ADAMS R D.The location of defects in structures from measurements of natural frequencies[J].Journal of Strain Analysis for Engineering Design,1979,14(2):49-57.
[19]KIM J T,STUBBS N.Crack detection in beam-type structures using frequency data[J].Journal of Sound and Vibration,2003,259(1):145-160.
[20]杜思义,殷学纲,陈淮.基于频率变化识别结构损伤的摄动有限元方法[J].工程力学,2007,24(4):66-70.
[21]DOEBLING S W,FARRAR C R,PRIME M B.A summary review of vibration-based damage identification methods[J].Shock and Vibration Digest,1998,30(2):91-105.
[22]RIZOS P F,ASPRAGATHOS N,DIMAROGONAS A D.Identification of crack location and magnitude in a cantilever beam from the vibration modes[J].Journal of Sound and Vibration,1990,138(3):381-388.
[23]GOUNARIS G D,PAPADOPOULOS C A.Analytical and experimental crack identification of beam structures in air or in fluid[J].Computers and Structures,1997,65(5):633-639.
[24]唐小兵,沈成武,付军.梁裂纹位置识别的模态能量法[J].武汉理工大学学报:交通科学与工程版,2001,25(3):241-243.
[25]唐天国,刘浩吾,陈春华,等.梁裂缝损伤检测的模态应变能法及试验研究[J].工程力学,2005,22(S1):39-45.
[26]孙国,顾元宪.连续梁结构损伤识别的改进柔度阵方法[J].工程力学,2003,20(4):50-54.
[27]CHEN W,ZHAO W G,YANG H Z,et al.Damage detection based on optimized incomplete mode shape and frequency[J].Acta Mechanica Solida Sinica,2015,28(1):74-82.
[28]FERNANDEZ-SAEZ J,RUBIO L,NAVARRO C.Approximate calculation of the fundamental frequency for bending vibrations of cracked beams[J].Journal of Sound and Vibration,1999,225(2):345-352.
[29]SHI Z Y,LAW S S,ZHANG L M.Damage localization by directly using incomplete mode shapes[J].Journal of Engineering Mechanics,2000,126(6):656-660.
[30]SEYED S K,ABDOLLAH B,GHOLAMREZA G A.Structural damage detection using incomplete modal data and incomplete static response[J].KSCE Journal of Civil Engineering,2013,17(1):216-223.
[31]BILELLO C.Theoretical and experimental investigation on damaged beams under moving systems[D].Italy:Universitàdegli Studi di Palermo,2001.
[32]PALMERI A,CICIRELLO A.Physically-based Dirac's delta functions in the static analysis of multi-cracked Euler-Bernoulli and Timoshenko beams[J].International Journal of Solids and Structures,2011,48(14-15):2184-2195.
[33]CADDEMI S,PALMERI A.Static analysis of Euler-Bernoulli beams with multiple unilateral cracks under combined axial and transverse loads[J].International Journal of Solids and Structures,2014,51(5):1020-1029.
[34]BRACEWELL R N.The Fourier transform and its applications[M].New York:McGraw-Hill,1986.
[35]HUANG T C.The effect of rotatory inertia and of shear deformation on the frequency and normal mode equations of uniform beams with simple end conditions[J].Journal of Applied Mechanics,1961,28(4):579-584.
[36]HAN S M,BENAROYA H,WEI T,et al.Dynamics of transversely vibrating beams using four engineering theories[J].Journal of Sound and Vibration,1999,225(5):935-988.
[37]CADDEMI S,CALIóI.Exact closed-form solution for the vibration modes of the Euler-Bernoulli beam with multiple open cracks[J].Journal of Sound and Vibration,2009,327(3-5):473-489.
[2]CHALLAMEL N,XIANG Y.On the influence of the unilateral damage behaviour in the stability of cracked beam columns[J].Engineering Fracture Mechanics,2010,77(9):1467-1478.
[3]REZAEE M,HASSANNEJAD R.Free vibration analysis of simply supported beam with breathing crack using perturbation method[J].Acta Mechanica Solida Sinica,2010,23(5):459-470.
[4]JASSIM Z A,ALI N N,MUSTAPHA F,et al.A review on the vibration analysis for a damage occurrence of a cantilever beam[J].Engineering Failure Analysis,2013,31(7):442-461.
[5]YANG X,HUANG X,OUYANG Y.Bending of Timoshenko beam with effect of crack gap based on equivalent spring model[J].Applied Mathematics and Mechanics,2016,37(4):513-528.
[6]DOEBLING S W.Damage identification and health monitoring of structural and mechanical systems from changes in their vibration characteristics:a literature review[J].Shock and Vibration Digest,1996,30(11):2043-2049.
[7]ATTAR M.A transfer matrix method for free vibration analysis and crack identification of stepped beams with multiple edge cracks and different boundary conditions[J].International Journal of Mechanical Sciences,2012,57(1):19-33.
[8]汪德江,杨骁.基于裂纹诱导弦挠度的Timoshenko梁裂纹无损检测[J].工程力学,2016,33(12):186-195.
[9]杨骁,唐珂,黄瑾.基于裂纹诱导挠度的悬臂梁裂纹静力损伤识别[J].力学季刊,2017,38(2):318-326.
[10]RUOTOLO R,SURACE C.Natural frequencies of a bar with multiple cracks[J].Journal of Sound and Vibration,2004,272(1-2):301-316.
[11]ZHANG Z G,CHEN F,ZHANG Z Y,et al.Vibration analysis of non-uniform Timoshenko beams coupled with flexible attachments and multiple discontinuities[J].International Journal of Mechanical Sciences,2014,80(1):131-143.
[12]徐福后,张玉祥.含裂纹Timoshenko梁自由振动分析[J].船舶力学,2011,15(10):1166-1172.
[13]BUDA G,CADDEMI S.Identification of concentrated damages in Euler-Bernoulli beams under static loads[J].Journal of Engineering Mechanics,2007,133(8):942-956.
[14]CICIRELLO A,PALMERI A.Static analysis of Euler-Bernoulli beams with multiple unilateral cracks under combined axial and transverse loads[J].International Journal of Solids and Structures,2014,51(5):1020-1029.
[15]SWAMIDAS A S J,YANG X,SESHADRI R.Identification of cracking in beam structures using Timoshenko and Euler formulations[J].Journal of Engineering Mechanics,2004,130(11):1297-1308.
[16]KHIEM N T,LIEN T V.The dynamic stiffness matrix method in forced vibration analysis of multiple-cracked beam[J].Journal of Sound and Vibration,2002,254(3):541-555.
[17]LABIB A,KENNEDY D,FEATHERSTON C,et al.Free vibration analysis of beams and frames with multiple cracks for damage detection[J].Journal of Sound and Vibration,2014,333(20):4991-5003.
[18]CAWLEY P,ADAMS R D.The location of defects in structures from measurements of natural frequencies[J].Journal of Strain Analysis for Engineering Design,1979,14(2):49-57.
[19]KIM J T,STUBBS N.Crack detection in beam-type structures using frequency data[J].Journal of Sound and Vibration,2003,259(1):145-160.
[20]杜思义,殷学纲,陈淮.基于频率变化识别结构损伤的摄动有限元方法[J].工程力学,2007,24(4):66-70.
[21]DOEBLING S W,FARRAR C R,PRIME M B.A summary review of vibration-based damage identification methods[J].Shock and Vibration Digest,1998,30(2):91-105.
[22]RIZOS P F,ASPRAGATHOS N,DIMAROGONAS A D.Identification of crack location and magnitude in a cantilever beam from the vibration modes[J].Journal of Sound and Vibration,1990,138(3):381-388.
[23]GOUNARIS G D,PAPADOPOULOS C A.Analytical and experimental crack identification of beam structures in air or in fluid[J].Computers and Structures,1997,65(5):633-639.
[24]唐小兵,沈成武,付军.梁裂纹位置识别的模态能量法[J].武汉理工大学学报:交通科学与工程版,2001,25(3):241-243.
[25]唐天国,刘浩吾,陈春华,等.梁裂缝损伤检测的模态应变能法及试验研究[J].工程力学,2005,22(S1):39-45.
[26]孙国,顾元宪.连续梁结构损伤识别的改进柔度阵方法[J].工程力学,2003,20(4):50-54.
[27]CHEN W,ZHAO W G,YANG H Z,et al.Damage detection based on optimized incomplete mode shape and frequency[J].Acta Mechanica Solida Sinica,2015,28(1):74-82.
[28]FERNANDEZ-SAEZ J,RUBIO L,NAVARRO C.Approximate calculation of the fundamental frequency for bending vibrations of cracked beams[J].Journal of Sound and Vibration,1999,225(2):345-352.
[29]SHI Z Y,LAW S S,ZHANG L M.Damage localization by directly using incomplete mode shapes[J].Journal of Engineering Mechanics,2000,126(6):656-660.
[30]SEYED S K,ABDOLLAH B,GHOLAMREZA G A.Structural damage detection using incomplete modal data and incomplete static response[J].KSCE Journal of Civil Engineering,2013,17(1):216-223.
[31]BILELLO C.Theoretical and experimental investigation on damaged beams under moving systems[D].Italy:Universitàdegli Studi di Palermo,2001.
[32]PALMERI A,CICIRELLO A.Physically-based Dirac's delta functions in the static analysis of multi-cracked Euler-Bernoulli and Timoshenko beams[J].International Journal of Solids and Structures,2011,48(14-15):2184-2195.
[33]CADDEMI S,PALMERI A.Static analysis of Euler-Bernoulli beams with multiple unilateral cracks under combined axial and transverse loads[J].International Journal of Solids and Structures,2014,51(5):1020-1029.
[34]BRACEWELL R N.The Fourier transform and its applications[M].New York:McGraw-Hill,1986.
[35]HUANG T C.The effect of rotatory inertia and of shear deformation on the frequency and normal mode equations of uniform beams with simple end conditions[J].Journal of Applied Mechanics,1961,28(4):579-584.
[36]HAN S M,BENAROYA H,WEI T,et al.Dynamics of transversely vibrating beams using four engineering theories[J].Journal of Sound and Vibration,1999,225(5):935-988.
[37]CADDEMI S,CALIóI.Exact closed-form solution for the vibration modes of the Euler-Bernoulli beam with multiple open cracks[J].Journal of Sound and Vibration,2009,327(3-5):473-489.