APPLICATION OF WAVELET ANALYSIS IN DAMAGE DETECTION AND .

MAGDALENA RUCKAKRZYSZTOF WILDEAPPLICATIONOF WAVELET ANALYSISIN DAMAGE DETECTIONAND LOCALIZATIONWYDAWNICTWOPOLITECHNIKI GDAŃSKIEJ

MAGDALENA RUCKAKRZYSZTOF WILDEAPPLICATIONOF WAVELET ANALYSISIN DAMAGE DETECTIONAND LOCALIZATIONGDAŃSK 2007

PRZEWODNICZĄCY KOMITETU REDAKCYJNEGOWYDAWNICTWA POLITECHNIKI GDAŃSKIEJRomuald SzymkiewiczREDAKTOR PUBLIKACJI NAUKOWYCHJanusz T. CieślińskiRECENZENCIPaweł KłosowskiZbigniew ZembatyPROJEKT OKŁADKIKatarzyna OlszonowiczWydano za zgodąRektora Politechniki GdańskiejWydawnictwa PG można nabywać w Księgarni PG (Gmach Główny, I piętro)i zamawiać faksem, pocztą elektroniczną lub listownie pod adresem:Wydawnictwo Politechniki Gdańskiejul. G. Narutowicza 11/12, 80 952 Gdańsk, tel./fax 058 347 16 18e-mail: wydaw@pg.gda.pl, www.pg.gda.pl/WydawnictwoPG Copyright by Wydawnictwo Politechniki GdańskiejGdańsk 2007Utwór nie może być powielany i rozpowszechniany, w jakiejkolwiek formiei w jakikolwiek sposób, bez pisemnej zgody wydawcyISBN 978–83–7348–192–3

CONTENTSLIST OF SYMBOLS AND ABBREVIATIONS .51. INTRODUCTION . 91.1. Damage detection in civil engineering structures . 91.2. Wavelet transform application in damage detection . 111.3. Aim and scope of study . 132. WAVELET THEORY .2.1. Introduction to wavelet analysis .2.2. One-dimensional wavelet transform .2.2.1. Continuous wavelet transform .2.2.1.1. Vanishing moments .2.2.1.2. Detection of singularities .2.2.2. Discrete wavelet transform .2.2.2.1. Orthogonal wavelet transform .2.2.2.2. Biorthogonal wavelet transform .2.2.3. Examples of wavelets .2.3. Two-dimensional wavelet transform .2.3.1. Continuous wavelet transform .2.3.2. Discrete wavelet transform .2.3.2.1. Orthogonal wavelet transform .2.3.2.2. Biorthogonal wavelet transform .2.3.3. Examples of wavelets .2.4. Summary .14141717181819192122282829293031313. WAVELET ANALYSIS IN DAMAGE DETECTION .3.1. Input signals .3.1.1. Experimental procedure for deflection lines determination .3.1.2. Experimental procedure for mode shapes determination .3.2. Wavelet selection for damage detection .3.2.1. One-dimensional wavelets .3.2.2. Two-dimensional wavelets .3.3. Discrete and continuous wavelet transform in damage detection .3.3.1. One-dimensional wavelet transform .3.3.2. Two-dimensional wavelet transform .3.4. Boundary effects .3.4.1. One-dimensional wavelet transform .3.4.2. Two-dimensional wavelet transform .3.5. Summary and conclusions .34343435404048494950505051524. DAMAGE DETECTION ON EXPERIMENTAL EXAMPLES .4.1. Beam – static deflection lines .4.1.1. Experimental investigations of beam deflection lines .4.1.2. Numerical simulations .4.1.3. Results of wavelet analysis .5353535657

4Contents4.2. Beam – mode shapes .4.2.1. Experimental investigations of beam mode shapes .4.2.2. Numerical simulations .4.2.3. Results of wavelet analysis .4.3. Plate – mode shapes .4.3.1. Experimental investigations of plate mode shapes .4.3.2. Numerical simulations .4.3.3. Results of wavelet analysis .4.4. Cylindrical shell – mode shapes .4.4.1. Experimental investigations of cylindrical shell mode shapes .4.4.2. Numerical simulations .4.4.3. Results of wavelet analysis .4.5. Summary and conclusions .686871737777798084848888935. DAMAGE DETECTION SYSTEMS BASED ON NEURAL NETWORKS .5.1. Fundamentals .5.2. Damage assessment using neural networks .5.3. Backpropagation neural network .5.4. Neural network defect detection system .5.4.1. Architecture .5.4.2. Training .5.4.3. Results of testing on experimental beam deflection lines .5.4.4. Results of testing on experimental beam mode shapes .5.4.5. Results of testing on experimental plate mode shapes .5.4.6. Results of testing on experimental shell mode shapes .5.5. Summary and conclusions .949494959898981021041051061076. FINAL REMARKS . 1086.1. General remarks . 1086.2. Original elements of the study . 109ACKNOWLEDGEMENTS . 110REFERENCES . 111SUMMARY IN ENGLISH . 117SUMMARY IN POLISH . 117

LIST OF SYMBOLS AND ABBREVIATIONSSymbolsaaj[k]aj[k,m]Af(u,v,s)A( )bBB1BrB(s)CCdj[k]d ij [k,m]D1, D2EEffafcf (t )f (x )f(t)f(x,y)FF( )ggu, (t)GFFGFXGXFGXXhHH1HrH( )H1( ), H2( )H(s)H( )– depth of a defect– one-dimensional discrete approximation– two-dimensional discrete approximation– angle of the wavelet transform vector– accelerance– bias of neural network– width– distance from a support to a defect along width– width of a defect– system matrix– damping matrix– modal damping matrix– one-dimensional discrete wavelet coefficient– two-dimensional discrete wavelet coefficient– diameters– Young’s modulus (chapter 4)– error minimized by training algorithm (chapter 5)– frequency– pseudo-frequency of wavelet transform– centre frequency of a wavelet– one-dimensional time signal– one-dimensional space signal– force vector– two-dimensional space signal– value of concentrated static load– Fourier transform of one-dimensional signal– high-pass filter– window function– autospectrum of the force– cross spectrum between the response and the force– cross spectrum between the force and the response– autospectrum of the response– low-pass filter– height– distance from a support to a defect along height– height of a defect– frequency response function– estimators of frequency response function– transfer function matrix– frequency response function matrix (receptance)

6iKKKLL1L2LrL2 (R )L2 (R 2 )mmsemseregmswMMf(u,v,s)MMnndnetopPP Vj fP Wj fq(t)RR(s)RssSf(u, )ttktrTuu0uFuUVjwWf(u,s)Wf(u,v,s)List of symbols and abbreviations– imaginary unit– number of neurons in output layer– stiffness matrix– modal stiffness matrix– length– distance from a support to a defect along length– distance from the free end of a beam to the load– length of a defect– Hilbert space of measurable, square-integrable one-dimensional functions– Hilbert space of measurable, square-integrable two-dimensional functions– number of modes– mean sum of squares of the network errors– mean squared error with regularization performance function– mean of the sum of squares of the network weights and biases– number of neurons in hidden layer– modulus wavelet transform of two-dimensional signal– mass matrix– modal mass matrix– number of vanishing moments– number degree of freedom– net function– output of neural network– pole– number of patterns– orthogonal projection of function f(x) on space Vj– orthogonal projection of function f(x) on space Wj– modal coordinate vector– number of neurons in input layer– residue matrix– real numbers– scale parameter– Laplace variable (section 3.1.2)– windowed Fourier transform of one-dimensional signal– time– scaling factor– threshold value– sampling period– translation parameter (position)– beam displacement under dead load– beam displacement under concentrated static load– eigen vector– modal matrix– approximations space– weight of neural network– continuous wavelet transform of one-dimensional signal– continuous wavelet transform of two-dimensional signal

List of symbols and abbreviationsWjxx (t )x(t)X( )yzZ (x) (x,y) t (x) j,k(x) ( ) (x) j,k(x) u,s(x) (x,y) (x,y) (x,y) d n – details space– space coordinate– displacement signal– displacement vector– Fourier transform of displacement signal– space coordinate– input of neural network– integers– performance ratio– learning rate– one-dimensional smoothing function– two-dimensional smoothing function– Poisson ratio– damping ratio– mass density– time interval– frequency interval– scaling function– family of discrete scaling function– Fourier transform of one-dimensional mother wavelet function– one-dimensional mother wavelet function– family of discrete wavelets– family of wavelets– two-dimensional horizontal wavelet function– two-dimensional vertical wavelet function– two-dimensional diagonal wavelet function– circular frequency– circular damped frequency– circular natural frequency– spectral HMWFTWT– artificial neural network– cyan, magenta, yellow, black space of colours– continuous wavelet transform– discrete wavelet transform– finite element method– frequency response function– Fourier transform– Global Positioning System– non-destructive testing– red, green, blue space of colours– structural health monitoring– windowed Fourier transform– wavelet transform7

Chapter 1INTRODUCTIONTo, że matematycy znajdują szczęście w przestrzeniachBanacha, mogłoby być rzeczą dość zrozumiałą, ale dlaczego przestrzenie te pojawiają się tak często, ilekroć chcemy rozszyfrowaćstrukturę rzeczywistego świata? Czy świat jest zbudowany wedługrecept na szczęście matematyków?The fact that mathematicians find happiness in Banachspaces could be quite understandable, but why these spaces appear every time we want to decipher the structure of the realworld? Is the world constructed in accordance to formulas formathematicians’ happiness?Michał HellerSzczęście w przestrzeniach Banacha, 1997Happiness in Banach spaces, 19971.1. Damage detection in civil engineering structuresAll structures raised by humans have a limited lifespan. They wear out and undergoself-destruction in the course of time. Fatigue, corrosion, dynamic phenomena, overloading and environmental conditions can cause their degradation. In recent years,structural damage detection and health monitoring have emerged as the subject of intensiveinvestigation due to their practical importance. For structures like offshore platforms, dams,transmission towers, bridges, aircraft, etc. (Fig. 1.1) early detection of damage is essentialsince propagation of defects might lead to a catastrophic failure. Accurate detection ofdamage is also necessary in structural strengthening or reconstruction.A damage detection system can have four levels of the defect identification accuracyproposed by Rytter in 1993 [99]:— level 1: the presence of damage,— level 2: the geometric location of damage,— level 3: the quantification of the severity of damage,— level 4: the prediction of the remaining service life of the structure.The most common method of a non-destructive assessment of the structure integrity isa routine visual inspection, mandatory for important structures. For example, bridges haveto be regularly checked by experienced engineers. Damage detection can be facilitated bynon-destructive testing (NDT) based on radiography [37, 103], acoustic emission [77, 91],ultrasonic testing [128], magnetic fields methods [54, 120], eddy current methods [35], etc.Although such diagnostic methods can be effectively applied to damage detection in a fewknown a priori areas in a structure, however, they are impractical for a search of potentialdamage through all engineering object. Additionally, the mentioned NDT methods do notallow an on-line inspection but they are done at periodic maintenance check.