VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY INSTITUTE OF MECHANICS o0o TRẦN THANH HẢI CRACK IDENTIFICATION FOR BEAM BY USING THE VIBRATION METHOD Specialized in Mechanics of Solids Code: 62 44 21 01 SUMMARY OF PhD THESIS HANOI – 2012 The thesis has been completed at The Institute of Mechanics, Vietnam Academy of Science and Technology Supervisors: Professor, Dr.Sc. Nguyen Tien Khiem. Assoc. Prof. Dr. Nguyen Viet Cuong Reviewer 1: Professor, Dr.Sc. Dao Huy Bich Reviewer 2: Prof. Dr. Hoang Xuan Luong Reviewer 3: Assoc. Prof. Dr. Tran van Lien Thesis is defended at The Institute of Mechanics, VAST on , Date Month Year 2012 Hardcopy of the thesis can be found at 1. Vietnam National Library 2. Library of Institute of Mechanics 1 INTRODUCTION Since early prognosis of cracks in structures is most important to keep away from accident, a lot of efforts have been focused to study cracked structures. This leads to many papers of the topic published in journals on structural engineering; fracture mechanics; vibrations and control etc. Content of the intensive study consists mainly of two problems. The first one is known as the forward problem that investigates structural behaviour with given position and size of cracks. The remaider acknowleged as the inverse problem relates to identify the potential cracks from measured behaviour of structure under consideration. The latter problem has recently received more interest from practical point of view. There are two approaches to the inverse problem, now could be called generally structural damage detection problem or structural health monitoring. The first one is based on “symptoms” of damage occurrence that are directly extracted from the measured data by using different methods of signal processing. This approach is termed by direct technique. The other one makes use of not only the measured data but also an artificial model of damaged structure for damage detection. The damage parameters are identified from fitting the simulated by model behaviour to the measured one. The latter approach is called the model-based methods. The advantage of the model-based methods in comparison with the symptom-based methods is its ability to encourage the useful knowledge acquired from study of the forward problem. The structural behaviour used as diagnostic signal for damage detection problem is frequently the vibrational characteristics or dynamic response of structure. This is because of the dynamical properties of structure contain more information on the structure condition than the static ones. The methods that use the vibrational characteristics for damage detection in structure are called the vibration-based method. 2 Using the model-based methods for structural damage detection faces the difficulty associated with measurement and modeling errors and incompleteness of measured data. The latter issues may make the problem of structural damage detection be ill- posed. The common way to overcome the disturbs is (a) refining the model of damaged structure with aim to decrease modeling error in compution of the structure behaviour; (b) applying the modern mathematical methods that could reduce the effect of measurement error and allow to obtain consistent solution of the ill-posed inverse problem; (c) exposing more information on the structure condition from limited amount of measured data. The purpose of present thesis is to develop an efficient procedure for crack identification in beam from measured mode shape and response to moving load using the well known Tikhonov regularization method and wavelet transform. Outline of thesis is as follows Chapter 1. Overview on vibration-based method for structural damage detection problem. Chapter 2. Theoretical background for regularization method, wavelet transform and model of cracked beam. Chapter 3. Mode shape-based procedure for multiple crack detection by using the Tikhonov’s regularization method. Chapter 4. Reponse of cracked beam to moving load and crack detection by the wavelet analysis of moving vehicle vibration. Concluding remarks and Referenses. Chapter 1. OVERVIEW Structural damage is understood as a change in either physical or geometrical properies of structure in comparison with a baseline configuration of intact structure. Damage is often described by its position and degree. The structural damage detection problem was firstly investigated by Adams etc [1] for a bar with single damage modeled by an axial spring at a position in bar. The authors have obtained an equation allowing one to determine the damage position from 3 measured pair of natural frequencies. Latter in [28] Liang and his coworkers extended the result for the case of beam by establishing general form of frequency equation of beam with single crack represented by a torsional spring. Morassi [39] proposed a first order approximate frequency equation for cracked beam with variable stiffness. Narkis [40] has calculated analytical solution of the problem for crack localization from measured two frequencies of simply supported beam. Nguyen Tien Khiem và Dao Nhu Mai [41] investigated in detail change in natural frequencies versus crack position and depth for beam with different cases of boundary conditions. Salawu [52] presented an overview on the frequency- based damage detection for structure. The problem gets to be more complicated when number of damages increased. Ostachowicz và Krawczuk [43] constructed frequency equation for beam with double crack in the form of 12×12 determinant. Shifrin và Ruotolo [55], by using the delta function for representation of change in stiffness at crack position obtained the frequency equation for beam with arbitrary number n of cracks in the form of determinant of order n+4. Khiem và Lien [21] used the transfer matrix method for deriving the frequency equation of beam with n cracks of the form of 4x4 determinants. Zhang and coworkers [61] have engaged the equation given by Khiem và Lien [21] for multi-crack detection from measured natural frequencies. Following Liang [28], Patil and Maiti [45] have obtained perturbation equation for natural frequencies of multiple cracked beam based on the energy conception. Recently, Lee [24] developed the sensitivity method for crack detection from natural freuencies by using the finite element formulation. Fernández-Sáez etc [14] introduced Rayleigh quotient to represent explicitly the first natural frequency of beam with single crack in term crack position and size. This generalized Rayleigh quotient gives good approximation only for the fundamental frequency so that it is inadequate for solution of the crack detection. Generally, the frequency-based method of crack detection is limited by that very small number of frequencies could be available and cracks at different positions may produce identical change in a frequency. Therefore, solution of the damage detection problem by 4 using frequencies is often non-unique. In such the case in order to have unique solution one has to engage other features of structure that could be axtracted from measurements. The mode shape of structure was early used for structural damage detection by Rizos et al [51], Yuen [60], West [65]…. At the first time, the mode shape has been taken in use for calculating different damage indices such as Modal Assurance Criteria (MAC) that is unable to be used for damage localization in structure. Then, Kim and his coworkers [20] have developed PMAC hay COMAC for the problem but they exposed to be insensitive to damage. Despite that Ho and Ewins [16], Parloo et al [46] proposed different damage indices calculated from given mode shape or its derivatives, Pandey etc [44] demonstrated that the mode shape is less sensitive to damage than mode shape curvature. Based on the idea, Ratcliffe [49], Wahab and De Roeck [63] have developed different procedures for damage localization from mode shape and mode shape curvature. Through studying vibration of multiple cracked beam, Li [27] has derived a recurrent relationship between mode shape if the beam in both sides of a crack. However, this is not a closed form solution for the mode shape so that it cannot be used for crack detection from measured mode shape. By using the step function Caddemi và Caliò [4] obtained closed form solution for mode shape of beam with arbitrary number of cracks that is an explicit expression of the mode shape through crack parameters. The closed form solution of mode shape has not been taken in use for multiple crack detection because it contains Dirac delta function. Although the mode shape of damaged structure could provide more useful information on the damage circumstance of a structure, measurement of mode shape is more difficult. The change in mode shape due to damage is usually more difficult to be monitored than its shift caused by measurement error. Hence, solving the problem associated with measurement erroneous in using mode shape for structural damage detection presents a great interest. 5 Chapter 2. THEORETICAL BACKGROUND 2.1. Tikhonov’s regularization method 2.1.1. Conception of inverse problem The essential content of inverse problem is to determine the “cause” with known “consequent”, Tarantola [56]. This problem was investigated early in Mechanics as determining the force applied to a body from given its movement trajectory. Recently, a novel formulation of the inverse problem has come from the practical demands: “establishing model of an existing object from its observed current behavior”. This is very complicated problem that is research subject in different fields of science and engineering and so far to be completely solved. The problem, called system identification, is also the substance of the condition monitoring of existing structures. The crucial attributes of the inverse problem are strong sensitivity of solution to either modeling or measurement inaccuracy and having non-unique solution because of incompleteness of given data. One of the most effective methods in solving the problem is proposed by A.N. Tikhonov that is called the regularization method and briefly described below. 2.1.2. Regularization method A lot of problems in science and engineering leads to solve the equation ,bAx = (2.1) where A is an arbitrary matrix (might be nonsquare or singular), b is a vector that is determined only as an approximation to unknown exact one .b Tikhonov và Arsenin [57], [58] suggested that solution of equation (2.1) can be found from solution of the mean square problem },{minarg 2 0 2 )xL(xbAxx −+−= α x RLS (2.2) with 0 x,, L α being regularization factor, regularization operator and a prior information about solution respectively. Leaving outside the equivalence between equations (2.1) and (2,2) one is going to consider equation (2,2). 6 Theorema. For 0f α solution of equation (2.2) can be found uniquely from the equation .)( 0 LxLbAxLLAA TTTT α+=+ α (2.3) Obviously, when 0→ α solution of equation (2.3) becomes the conventional mean square solution .)(xx 1 bAAA TT LSRLS − =→ The regularization factor α is chosen as solution of the equation , RLS δα =− bAx )( (2.4) where δ - noise level of vector b. In turn, equation (2.3) can be solved by using the Singular Value Decomposition (SVD) of matrix A T VΣUA = (2.5) where VU, are square matrices of order m and n respectively and n T m T IVV,IUU == , matrix ).,min(},, ,{)( 1 nmqdiagnm q = = × σ σ Σ Hence, solution of equation (2.3) with 0x = 0 is , ˆ 1 2 k r k k T kk v bu x ∑ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + = = σα σ (2.6) 2.2. Wavelet transform The Fourie transform was the most powerful tool in signal analysis in frequency domain. However, it cannot be used for analysis of non-stationary processes when frequency is dependent upon time. This gap can be fulfilled by using the newly developed wavelet transform that is briefly described below. 2.2.1. Difinition of wavelet transform The continuous wavelet transform is defined as ,)()(),( , ∫ = +∞ ∞− dtttfbaW ba ψ (2.7) where ),()/1()( * , a bt at ba − = ψψ (2.8) a is a real number acknowledged as scale or dilation factor, b is transition, () t ψ is called mother wavelet function and ( ) t * ψ is the 7 complex conjugate of ( ) t ψ . For every value of a and b, W(a,b) is determined as wavelet coefficient of the given signal f(t). Inverse wavelet transform is ∫∫ = +∞ ∞− +∞ ∞− − 2 , 1 ),()( a da dbbaWCtf bag ψ (2.9) where . )( ˆ 2 2 ∞< ∫ = ∞+ ∞− ξ ξ ξψ π dC g (2.10) From mathematical point of view, wavelet is convolution of the signal and wavelet function, allowing compressing a signal. 2.2.2. Application of wavelet Wavelet is widely used in signal processing, especially in detecting descontineous of a signal. It can be utilized also for detecting similarity of signals, filtering or compressing signals. Recently, wavelet has been used for local damage detection in structures through wavelet analysis of response of structure. 2.3. A model of multiple cracked beam 2.3.1. A crack model It was approved in Fracture Mechanics that a crack occurred at a section of beam member introduces local flexibility could be calculated by the formulae ),()/6(/ sFbEIhMc I π φ = Δ = (2.11) where h, b are high and wide of the beam’s rectacular cross section, EI is the bending stiffness, s = a/h is relative crack depth and )(sF I is an experimental function . In such the case, crack can be represented by a torsional spring of the stiffness )).(6/(/1 shFbEIcK I π = = (2.12) This idea constitutes the most simple and efficient model of crack in beam member. 2.3.2. Model of beam with crack The rotational spring model of crack allows one to represent a crack at a section in beam in a form of compatibility conditions at crack that should be satisfied by the displacements and forces of beam 8 in both sides of crack. The study of cracked beam with the crack model can be carried out by dividing the beam into sub-beam bordered by crack position and beam ends, Rizos và cộng sự [51]. This approach enables to use the governed equations without any change for solving the problems of cracked structures. 2.3.3. Finite element model of beam with crack Qian et al [48], have shown that stiffness matrix of a cracked beam can be expressed as . Tee c TCTK 1 ~ − = (2.13) Where T e l ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − −− = 1010 011 T and the flexibility matrix )1()0( ~ ijij e ij CCC += with . 2 23 2 23 )0( ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = EI l EI l EI l EI l ee ee C (2.14) and ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ + = 11 2 1 2 2 1 2 )1( 2 2 2 nRRnL RnLmR RnL e e e C , (2.15) .,, ' 36 , ' 36 0 2 2 0 2 1 24 ∫ = ∫ === a II a I daaFRdaaFR bhE m bhE n ππ l e – the length of element, a is crack depth. Summary of Chapter 2 In this Chapter, the fundamental of the Tikhonov’s relarization and Wavelet transform methods has been presented. The regularization method allows obtaining unique solution of the standard inverse problem with noisy measured data. The wavelet transform is an helpful tool in detecting small change in response of structure due to damage. Additionally, the continuous and finite element models of cracked beam are briefly described. [...]... Journal of Mechanics, Vol.32, No.4, 2010, pp 222-233 3 Nguyễn Tiến Khiêm, Trần Thanh Hải “Lời giải chính xác của bài toán dao động của dầm đàn hồi có nhiều vết nứt và ứng dụng” Hội nghị khoa học kỷ niệm 35 năm Viện Khoa học và công nghệ Việt nam 1975-2010 ISBN: 978-604-913-009-0, Hà Nội, 2010 4 Khoa Viet Nguyen, Hai Thanh Tran and Dung Dinh Nguyen “Infuence of the road surface unevenness on the multi-crack... complicated structures such as frames LIST OF THE AUTHOR’S PUBLICATIONS 1 Khoa Viet Nguyen, Hai Thanh Tran “Multi-cracks detection of a beam-like structure based on the on-vehicle vibration signal and wavelet analysis” Journal of Sound and Vibration, Vol 329, 2010, pp 4455-4465 2 Nguyen Viet Khoa and Tran Thanh Hai “Wavelet based technique for multi-crack detection of a beam like structure using the vibration... of a beam like structure subjected to moving vehicle” Proceedings of the International Conference on Engineering Mechanics and Automation – ICEMA 2010 Ha Noi, July 1-2, 2010 5 Nguyen Tien Khiem, Tran Thanh Hai “Rayleigh’s quotients for multiple cracked beam and application” Vietnam Journal of Mechanics Vol.33, No.1, 2011, pp 1-12 6 N.T.Khiem, T.T.Hai and L.K Toan “A novel formulation and solution to . VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY INSTITUTE OF MECHANICS o0o TRẦN THANH HẢI CRACK IDENTIFICATION FOR BEAM BY USING THE VIBRATION METHOD Specialized. Dr.Sc. Nguyen Tien Khiem. Assoc. Prof. Dr. Nguyen Viet Cuong Reviewer 1: Professor, Dr.Sc. Dao Huy Bich Reviewer 2: Prof. Dr. Hoang Xuan Luong Reviewer 3: Assoc. Prof. Dr. Tran van. crack localization from measured two frequencies of simply supported beam. Nguyen Tien Khiem và Dao Nhu Mai [41] investigated in detail change in natural frequencies versus crack position and