基于裂纹附加模态的梁裂纹损伤识别方法
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摘要
将梁中横向开裂纹等效为内部扭转弹簧,利用广义Delta函数和Heaviside函数,给出了具有任意条裂纹Euler-Bernoulli梁振动模态的统一显式解析表达式。在此基础上,引入裂纹附加模态的概念,并根据裂纹附加模态函数的构造特征,利用最小二乘拟合,建立了一种新的裂纹损伤参数识别方法。该方法计算简单,且仅需较少的测量点及测量点处某一模态的测量数据即可实现裂纹位置及深度的识别。最后,通过两个数值算例验证了裂纹损伤参数识别方法的适用性和可靠性,并考察了测量噪声对损伤识别的影响,结果表明裂纹位置识别精度高于裂纹等效弹簧刚度识别精度;前面裂纹识别结果影响后续裂纹的识别结果;随着测量噪声的增加,裂纹位置及裂纹等效弹簧刚度的识别误差增加,但仍在可接受的范围内,故该裂纹损伤识别方法在工程实际中具有一定的适用性。
Regarding the transverse open crack in a beam as an equivalent internal rotational spring,a unified explicit expression of the vibration mode of an Euler-Bernoulli beam with arbitrary number of cracks is obtained with the generalized Delta and Heaviside functions.On this basis,the concept of crack-induced additional vibration mode is proposed,and a novel method to identify the crack damage parameters is established with the constructive feature of the crack-induced additional vibration mode by using the least square fitting.The proposed method has the advantage of simple calculation and can identify the locations and equivalent rotational spring rigidities of the cracks using less mode measurement data.Finally,the validity and reliability of the proposed method for crack-damage identification are validated by two numerical examples,and the influence of the measurement noise on the identification results is examined.It is revealed that the identification precisions of the crack locations are higher than those of equivalent rotational spring rigidities of the cracks,and the identification result of present cracks has influence on the identification result of later ones.The identification errors of the crack location and the rigidity of the crack equivalent rotational spring increase with the increase of the measurement errors,but these errors are acceptable.Therefore,the proposed crack damage identification method can be applied in practical engineering.
Regarding the transverse open crack in a beam as an equivalent internal rotational spring,a unified explicit expression of the vibration mode of an Euler-Bernoulli beam with arbitrary number of cracks is obtained with the generalized Delta and Heaviside functions.On this basis,the concept of crack-induced additional vibration mode is proposed,and a novel method to identify the crack damage parameters is established with the constructive feature of the crack-induced additional vibration mode by using the least square fitting.The proposed method has the advantage of simple calculation and can identify the locations and equivalent rotational spring rigidities of the cracks using less mode measurement data.Finally,the validity and reliability of the proposed method for crack-damage identification are validated by two numerical examples,and the influence of the measurement noise on the identification results is examined.It is revealed that the identification precisions of the crack locations are higher than those of equivalent rotational spring rigidities of the cracks,and the identification result of present cracks has influence on the identification result of later ones.The identification errors of the crack location and the rigidity of the crack equivalent rotational spring increase with the increase of the measurement errors,but these errors are acceptable.Therefore,the proposed crack damage identification method can be applied in practical engineering.
引文
[1] Palmeri A,Cicirello A.Physically-based Dirac's delta functions in the static analysis of multi-cracked EulerBernoulli and Timoshenko beams[J].International Journal of Solids and Structures,2011,48(14-15):2184-2195.
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[3] Yan Y,Ren Q W,Xia N,et al.A close-form solution applied to the free vibration of the Euler-Bernoulli beam with edge cracks[J].Archive of Applied Mechanics,2016,86(9):1633-1646.
[4] Dimarogonas A D.Vibration of cracked structures:A state of the art review[J].Engineering Fracture Mechanics,1996,55(5):831-857.
[5] 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.
[6] Gentile A,Messina A.On the continuous wavelet transforms applied to discrete vibrational data for detecting open cracks in damaged beams[J].International Journal of Solids and Structures,2003,40(2):295-315.
[7] Kisa M.Vibration and stability of multi-cracked beams under compressive axial loading[J].International Journal of the Physical Sciences,2011,6(11):2681-2696.
[8] Fekrazadeh S,Khaji N.An analytical method for crack detection of Timoshenko beams with multiple open cracks using a test mass[J].European Journal of Environmental and Civil Engineering,2017,21(1):24-41.
[9] 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.
[10]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.
[11]FU C Y.The effect of switching cracks on the vibration of a continuous beam bridge subjected to moving vehicles[J].Journal of Sound and Vibration,2015,339(3):157-175.
[12]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.
[13]Jun O S,Eun H J,Earmme Y Y,et al.Modelling and vibration analysis of a simple rotor with breathing crack[J],Journal of Sound and Vibration,1992,155(2):273-290.
[14]胡家顺,冯新,周晶.呼吸裂纹梁非线性动力特性研究[J].振动与冲击,2009,28(1):76-80.Hu Jiashun,Feng Xin,Zhou Jing.Study on nonlinear dynamic response of a beam with a breathing crack[J].Journal of Vibration and Shock,2009,28(1):76-80.
[15]Bilello C.Theoretical and experimental investigation on damaged beams under moving systems[D].Italy:Universitàdegli Studi di Palermo,2001.
[16]Atluri S N.Computational Methods in the Mechanics of Fracture[M].New York:North-Holland,1986.
[17]Salawu O S.Detection of structural damage through changes in frequency:A review[J].Engineering Structures,1997,19(9):718-723.
[18]Yang X F,Swamidas A S J,Seshadri R.Crack identification in vibrating beams using the energy method[J].Journal of Sound and Vibration,2001,244(2):339-357.
[19]Kim J T,Ryu Y S,Cho H M,et al.Damage identification in beam-type structures:Frequency-based method vs.mode-shape-based method[J]. Engineering Structures,2003,25(1):57-67.
[20]Law S S,Lu Z R.Crack identification in beam from dynamic responses[J].Journal of Sound and Vibration,2005,285(4):967-987.
[21]王丹生,高智,杨海萍,等.基于特征正交分解的梁结构损伤识别[J].振动与冲击,2009,28(1):122-125.Wang Dansheng,Gao Zhi,Yang Haiping,et al.Damage identification for beam structures based on proper orthogonal decompostion[J].Journal of Vibration and Shock,2009,28(1):122-125.
[22]Labib A,Kennedy D,Featherston C.Free vibration analysis of beams and frames with multiple cracks for damage detection[J].Journal of Sound and Vibration,2014,333(20):4991-5003.
[23]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.
[24]Pandey A K,Biswas M,Samman M M.Damage detection from changes in curvature mode shapes[J].Journal of Sound and Vibration,1991,145(2):321-332.
[25]Douka E,Loutridis S,Trochidis A.Crack identification in beams using wavelet analysis[J].International Journal of Solids and Structures,2003,40(13):3557-3569.
[26]Chasalevris A C,Papadopoulos C A.Identification of multiple cracks in beams under bending[J].Mechanical Systems and Signal Processing,2006,20(7):1631-1673.
[27]胡家顺,冯新,李昕,等.裂纹梁振动分析和裂纹识别方法研究进展[J].振动与冲击,2007,26(11):146-152+189.Hu Jiashun,Feng Xin,Li Xin,et al.State-of-art of vibration analysis and crack indentification of cracked beams[J].Journal of Vibration and Shock,2007,26(11):146-152+189.
[28]Koo K Y,Lee J J,Yun C B,et al.Damage detection in beam-like structures using deflections obtained by modal flexibility matrices[J].Journal of Smart Structures and System,2008,4(5):605-628.
[29]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.Wang De-jiang,Yang Xiao.Crack non-destructive test in Timoshenko beams based on crack-indwced chordwise deflection[J].Engineering Mechanics,2016,33(12):186-195.
[30]Caddemi S,CaliòI.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.
[31]Yang X,Huang J,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.
[2] Caddemis S,CalióI.The influence of the axial force on the vibration of the Euler-Bernoulli beam with an arbitrary number of cracks[J].Archive of Applied Mechanics,2012,82(6):1-13.
[3] Yan Y,Ren Q W,Xia N,et al.A close-form solution applied to the free vibration of the Euler-Bernoulli beam with edge cracks[J].Archive of Applied Mechanics,2016,86(9):1633-1646.
[4] Dimarogonas A D.Vibration of cracked structures:A state of the art review[J].Engineering Fracture Mechanics,1996,55(5):831-857.
[5] 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.
[6] Gentile A,Messina A.On the continuous wavelet transforms applied to discrete vibrational data for detecting open cracks in damaged beams[J].International Journal of Solids and Structures,2003,40(2):295-315.
[7] Kisa M.Vibration and stability of multi-cracked beams under compressive axial loading[J].International Journal of the Physical Sciences,2011,6(11):2681-2696.
[8] Fekrazadeh S,Khaji N.An analytical method for crack detection of Timoshenko beams with multiple open cracks using a test mass[J].European Journal of Environmental and Civil Engineering,2017,21(1):24-41.
[9] 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.
[10]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.
[11]FU C Y.The effect of switching cracks on the vibration of a continuous beam bridge subjected to moving vehicles[J].Journal of Sound and Vibration,2015,339(3):157-175.
[12]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.
[13]Jun O S,Eun H J,Earmme Y Y,et al.Modelling and vibration analysis of a simple rotor with breathing crack[J],Journal of Sound and Vibration,1992,155(2):273-290.
[14]胡家顺,冯新,周晶.呼吸裂纹梁非线性动力特性研究[J].振动与冲击,2009,28(1):76-80.Hu Jiashun,Feng Xin,Zhou Jing.Study on nonlinear dynamic response of a beam with a breathing crack[J].Journal of Vibration and Shock,2009,28(1):76-80.
[15]Bilello C.Theoretical and experimental investigation on damaged beams under moving systems[D].Italy:Universitàdegli Studi di Palermo,2001.
[16]Atluri S N.Computational Methods in the Mechanics of Fracture[M].New York:North-Holland,1986.
[17]Salawu O S.Detection of structural damage through changes in frequency:A review[J].Engineering Structures,1997,19(9):718-723.
[18]Yang X F,Swamidas A S J,Seshadri R.Crack identification in vibrating beams using the energy method[J].Journal of Sound and Vibration,2001,244(2):339-357.
[19]Kim J T,Ryu Y S,Cho H M,et al.Damage identification in beam-type structures:Frequency-based method vs.mode-shape-based method[J]. Engineering Structures,2003,25(1):57-67.
[20]Law S S,Lu Z R.Crack identification in beam from dynamic responses[J].Journal of Sound and Vibration,2005,285(4):967-987.
[21]王丹生,高智,杨海萍,等.基于特征正交分解的梁结构损伤识别[J].振动与冲击,2009,28(1):122-125.Wang Dansheng,Gao Zhi,Yang Haiping,et al.Damage identification for beam structures based on proper orthogonal decompostion[J].Journal of Vibration and Shock,2009,28(1):122-125.
[22]Labib A,Kennedy D,Featherston C.Free vibration analysis of beams and frames with multiple cracks for damage detection[J].Journal of Sound and Vibration,2014,333(20):4991-5003.
[23]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.
[24]Pandey A K,Biswas M,Samman M M.Damage detection from changes in curvature mode shapes[J].Journal of Sound and Vibration,1991,145(2):321-332.
[25]Douka E,Loutridis S,Trochidis A.Crack identification in beams using wavelet analysis[J].International Journal of Solids and Structures,2003,40(13):3557-3569.
[26]Chasalevris A C,Papadopoulos C A.Identification of multiple cracks in beams under bending[J].Mechanical Systems and Signal Processing,2006,20(7):1631-1673.
[27]胡家顺,冯新,李昕,等.裂纹梁振动分析和裂纹识别方法研究进展[J].振动与冲击,2007,26(11):146-152+189.Hu Jiashun,Feng Xin,Li Xin,et al.State-of-art of vibration analysis and crack indentification of cracked beams[J].Journal of Vibration and Shock,2007,26(11):146-152+189.
[28]Koo K Y,Lee J J,Yun C B,et al.Damage detection in beam-like structures using deflections obtained by modal flexibility matrices[J].Journal of Smart Structures and System,2008,4(5):605-628.
[29]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.Wang De-jiang,Yang Xiao.Crack non-destructive test in Timoshenko beams based on crack-indwced chordwise deflection[J].Engineering Mechanics,2016,33(12):186-195.
[30]Caddemi S,CaliòI.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.
[31]Yang X,Huang J,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.