Modal analysis and crack detection in stepped beams

Crack is a damage usually happened in structural members but dangerous for safety of structure if it is not early detected. However, cracks are often difficult to identify by visual inspection as they occurred at the unfeasible locations. Therefore, cracks could be indirectly detected from measured total dynamical characteristics of structures such as natural frequencies, mode shapes and frequency or time history response. In order to identify location and size of a crack in a structure the problem of analysis of the crack’s effect on the dynamic properties is of great importance. It could give also useful tool for crack localization and size evaluation. On the other hand, beams are frequently used as structural member in the practice of structural engineering. So, crack detection for beam-like structures gets to be an important problem. Crack detection problem of beam with uniform cross section is thoroughly studied, but vibration of cracked beam with varying cross section presents a difficult problem. It is because vibration of such the structure is described by differential equations with varying coefficients that are nowadays not generally solved. The beam with piecewise uniform beam, acknowledged as stepped beam is the simplest model of beam with varying cross section. Although, vibration analysis and crack detection for stepped beam have been studied in some publications, developing more efficient methods for solving the problems of various types of stepped beams is really demanded

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MINISTRY OF EDUTATION AND TRAINING VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY ----------------------------- VU THI AN NINH MODAL ANALYSIS AND CRACK DETECTION IN STEPPED BEAMS Specialization: Engineering Mechanics Code: 62 52 01 01 ABSTRACT OF DOCTOR THESIS IN MECHANICAL ENGINEERING AND ENGINEERING MECHANICS HANOI - 2018 The thesis has been completed at: Graduate University of Science and Technology - Vietnam Academy of Science and Technology Supervisions: 1: Prof.DrSc. Nguyen Tien Khiem 2: Dr. Tran Thanh Hai Reviewer 1: Prof.DrSc. Nguyen Van Khang Reviewer 2: Prof.Dr. Nguyen Manh Yen Reviewer 3: Assoc.Prof.Dr. Nguyen Đang To Thesis is defended at Graduate University of Science and Technology - Vietnam Academy of Science and Technology at, on datemonth201 Hardcopy of the thesis be found at: - Library of Graduate University of Science and Technology - Vietnam National Library 1 INTRODUCTION 1. Necessarily of the thesis Crack is a damage usually happened in structural members but dangerous for safety of structure if it is not early detected. However, cracks are often difficult to identify by visual inspection as they occurred at the unfeasible locations. Therefore, cracks could be indirectly detected from measured total dynamical characteristics of structures such as natural frequencies, mode shapes and frequency or time history response. In order to identify location and size of a crack in a structure the problem of analysis of the crack’s effect on the dynamic properties is of great importance. It could give also useful tool for crack localization and size evaluation. On the other hand, beams are frequently used as structural member in the practice of structural engineering. So, crack detection for beam-like structures gets to be an important problem. Crack detection problem of beam with uniform cross section is thoroughly studied, but vibration of cracked beam with varying cross section presents a difficult problem. It is because vibration of such the structure is described by differential equations with varying coefficients that are nowadays not generally solved. The beam with piecewise uniform beam, acknowledged as stepped beam is the simplest model of beam with varying cross section. Although, vibration analysis and crack detection for stepped beam have been studied in some publications, developing more efficient methods for solving the problems of various types of stepped beams is really demanded. 2 2. Objective of the thesis The aim of this thesis is to study crack-induced change in natural frequencies and to develop a procedure for detecting cracks in stepped beams by measurement of natural frequencies. 3. The research contents of the thesis (1) Developing the Transfer Matrix Method (TMM) for modal analysis of stepped Euler – Bernoulli, Timosheko and FGM beams with arbitrary number of cracks. (2) Expanding the Rayleigh formula for computing natural frequencies of stepped beams with multiple cracks. (3) Employing the extended Rayleigh formula for developing an algorithm to detect unknown number of cracks in stepped beams by natural frequencies. (4) Experimental study of cracked stepped beams to validate the developed theory. Thesis composes of Introduction, 4 Chapters and Conclusion. Chapter 1 describes an overview on the subject literature; Chapter 2 – development of TMM; Chapter 3 – the Rayleigh method and Chapter 4 presents the experimental study. CHAPTER 1. OVERVIEW ON THE MODELS, METHODS AND PUBLISHED RESULTS 1.1. Model of cracked beams 1.1.1. On the beam theories Consider a homogeneous beam with axial and flexural displacements ( , , )u x z t , ( , , )w x z t at cross section x. Based on some assumptions the displacements can be represented as: 3 0 0 0( , , ) ( , ) ( , ) ( ) ( , ); ( , , ) ( , ),u x z t u x t zw x t z x t w x z t w x t     where u0(x, t), w0(x, t) are the displacements at the neutral axis, (x,t) – shear slop, z is heigh from the neutral axis. Function (z), representing shear distribution can be chosen as follow: (a) ( ) 0z  - for Euler-Bernoulli beam theory (the classical beam theory). (b) ( )z z  - for Timoshenko beam theory or the first order shear beam theory. (c)  2 2( ) 1 4 / 3z z z h   - the second shear beaa theory. (d) 2{ 2( / ) }( ) e z hz z  - the exponent shear beam theory. Recently, one of the composites is produced and called Functionally Graded Material (FGM), mechanical properties of which are varying continuously along corrdinate z or x. Denoting elasticity modulus E, shear modulus G and material density , a model of the FGM is represented as ( ) ( ) ( )b t bz g z     where ,b t  stand for the characteristics (E, , G) at the bottom and top beam surfaces and function g(z) could be chosen in the following forms: a) P-FGM:  ( ) ( 2) / n g z z h h  - the power law material. b) E-FGM: (1 2 / )( ) , 0.5ln( / )z ht t bE z E e E E     - the exponent law material. c) S-FGM:  1( ) 1 0.5 1 2 / n g z z h   , 0 / 2z h  . d)  2 ( ) 1 2 / / 2 n g z z h  , / 2 0h z   - Sigmoid law material. In this thesis only the FGM of power law is investigated. 4 1.1.2. Crack model in homogeneous beams Fig 1.2. edge crack model. Consider a homogeneous beam as shown in Fig. 1.2 that contains a crack with depth a at position e. Based on the fracture mechanics, Chondros, Dimagrogonas and Yao have proved that the crack can be represented by a rotational spring of stiffness where EI is bending stiffness, h is heigh of beam and function Hence, compatibility conditions at the cracked sections are For Timoshenko beam the conditions take the form ( 0, ) ( 0, );w e t w e t   ( 0, ) ( 0, ) ( , );x x xe t e t e t        ( 0, ) ( 0, ) ( )c xe t e t e       ; w ( 0, ) w ( 0, ) ( , )x x c xe t e t e t       . M a e ' M e-0 ' e+0 2 . 6 (1 ) ( / ) c c EI hI a h K     2 2 3 4 5 6 7 8 ( ) (0.6272 0.17248 5.92134 10.7054 31.5685 67.47 139.123 146.682 92.3552 ), cI z z z z z z z z z z          2 2 0 0 2 2 3 3 2 2 3 3 w(x,t) w(x,t) ( ) w( , ) w( 0, ) w( 0, ), w( 0, ) w( 0, ) w( 0, ) w( 0, ) , . c x e x e c M e x t x x K x e t e t e t e t e t e t x x x x                                5 1.1.3. Modeling crack in FGM beam Crack in FGM beam can be modeled by a spring of stiffness calculated as 1.2. Vibration of cracked beams 1.2.1. Homogeneous beams Consider an Euler-Bernoulli beam with n cracks at positions 1 20 ... ne e e L     and depth , 1,2,..., .ja j n Free vibration of the beam is described by equation 4 4 4 4 2( ) / ( ) 0, / .d x dx x F EI        in every beam segment 1 0 1( , ), 1,..., 1, 0,j j ne e i n e e L     , general solution of which is 1( ) cosh sinh cos sin , ( , ).j j j j j j jx A x B x C x D x x e e          Substituting the solution into conditions at the crack positions 1 1 1( ) ( ), ( ) ( , ( ) ( ),j j j j j j j j j j j je e e e e e             1( ) ( ) ( ), 1,2,..., .j j j j j j je e e j n        one obtains 4n equations for 4(n+1) unknowns 1 1 1 1 1 1 1 1{ , , , ,..., , , , } T n n n nA B C D A B C D   C . Therefore, combining the equations with 4 boundary conditions allows one to get closed system of equations 1 1[ ( , ,..., , ,..., )]. 0n ne e   D C . 2 2/ 2 0 72 (1 ) ( ) 1/ ; , ( ) a h F K C C d h E h           2 3 4 5 6 7 2 1 ( ) 1.910 2.752 4.782 146.776 770.75 1947.83 2409.17 1177.98 , / 0.2; F E E                   2 3 4 5 6 7 2 1 ( ) 1.150 1.662 21.667 192.451 909.375 2124.31 2395.83 1031.75 , / 1.0; F E E                   2 3 4 5 6 7 2 1 ( ) 0.650 0.859 12.511 72.627 267.91 535.236 545.139 211.706 , / 5.0. F E E                   6 for determining the unknown constants. Hence, frequency equation can be obtained as 1 1det[ ( , ,..., , ,..., )] 0,n ne e   D that could be solved to give roots , 1,2,3,...k k  from which natural frequencies are calculated as 2 / , 1,2,3,....k k EI F k    For Timoshenko beam, equations of free vibration are 2 W( ) (W ) 0x G       ; 2 ( ) ( ) (W ) 0I x EI x GA         , that would be solved together with conditions at cracks W( 0) W( 0) W( )j j je e e    ; ( 0) ( 0) ( );j j je e e        ( 0) ( 0) ( )j j j je e e        ; ( 0) ( 0) ( ).j j jW e W e e       Similarly, putting general solution 1 1 2 2W ( ) cosh sinh cos sin ;j j j j jx A k x B k x C k x D k x    1 1 1 1 2 2 2 2( ) sinh cosh sin cos ,j j j j jx r A k x r B k x r C k x r D k x     2 2 2 2 1 1 1 2 2 2( ) / ; ( ) / ;r Gk Gk r Gk Gk         2 2 1 2( 4 ) / 2, ( 4 ) / 2k b c b k b c b      2(1 ); ( ); / ; / ; /b c E E G F I                . in beam segment 1( , )j je e into condittions at cracks and boundaries, frequency equation is obtained also in the form 1 1det ( , ,..., , ,..., ) 0n ne e     D for determining natural frequencies , 1,2,3,...k k  1.2.2. Vibration of FGM beams Based on the model of Timoshenko FGM beam and taking account for actual position of neutral plane equations of motion of the beam can be established in the form 7 11 11 12 0I u A u I    ; 11 33( ) 0I w A w     ; 12 22 22 33( ) 0;I u I A A w        with coefficients 11 22 33 11 12 22, , , , ,A A A I I I calculated from the material constants , , , , , ,...b t b tE E n   Beside, from condition of neutral plane, actual position of the axis measured from the midplane is determined as 0 [ ] / / .( 1) [2( 2)( )],e e e t bh n r h n n r r E E     Seeking solution of the equations of motion given above in the form ( , ) ( ) ; ( ,t)=W( ) ; ( ,t)= ( )i t i t i tu x t U x e w x x e x x e    , one has got the equations 2 2 11 11 12( ) 0I U A U I    ; 2 11 33( ) 0I W A W     ; 2 2 22 22 12 33( ) ( ) 0,I A I U A W        that in turn give rise general solution 0 0( , ) ( , )x x z G C , where 0 ,( , ) { ( , ), ( , ), ( , )}Tx U x x W x    z 1 6,...,C{C } TC = and 2 2 2 2 2 12 11 11 33 11 33/ ( ); / ( ), 1,2,3.j j j j jI I k A k A I k A j         In case, if the beam is cracked at position e the solution gets to be ( ) ( ). ,c cx xz Φ C 0 0( ) ( , ) ( ) ( , )c x x x e e   Φ G K G . 3 31 2 1 2 3 31 2 1 2 3 31 2 1 2 1 2 3 1 2 3 0 1 2 3 1 2 3 ( , ) ; k x k xk x k x k x k x k x k xk x k x k x k x k x k xk x k x k x k x e e e e e e x e e e e e e e e e e e e                                G ( ) : 0; ( ) : 0; ( ) ( ) 0 : 0 0 : 0 c cx x x x x x x x          G G K K 8 1.2.3. Conventional formulation of TMM In this section, an Euler-Bernoulli homogeneous beam composed of uniform beam elements with the material and geometry constants: { , , , , }, 1,2,...,j j j j jE A I L j n  , It is well known that general solution of free vibration problem in every beam segment is expressed in the form ( ) cosh sinh cos sin , (0, ),j j j j j j j j j jx A x B x C x D x x L         with 2 1/4( ) ( / )j j j j j jA E I      . Introducing the state vector { ( ), ( ),M ( ),Q ( )}j j j j jx x x x V , ( ) ( ); ( ) ( )j j j j j j j jM x E I x Q x E I x    we would have got the expression ( ) ( )j j jx xV H C ; { , , , } T j j j j jA B C DC and ( )j xH is a matrix function acknowledged as shape function matrix. From the continuity conditions at joints of the beam segments 1( ) (0)j j jL V V one gets 1 1 , 1( 1) (0). ( ). ( ) . ( )j j j j jj L j j     V H H V T V or , 1 1, 1 21( ) . ... (1) . (1)n n n nn    V T T T .V T V , with T being called transfer matrix of the beam. Applying boundary conditions for the latter connection allows one to get 0 1 1{ (0)} 0; { (1)} 0n B V B V . or ( ). (1)=0.B V Consequently, frequency equation is obtained as det ( ) =0.  B This is content of the so-called Transfer Matrix Method that is appropriate for modal analysis of stepped beams. 9 1.2.4. Rayleigh method For a standard beam flexural deflection in in vibration of frequency  is ( , ) ( )sinv x t x t  with function ( )x called mode shape of vibration. In that case, potential and kinetic energies are 2 2 0 (1/ 2)(sin ) ( ) ; L xxt EI x dx    2 2 2 0 (1/ 2)(cos ) ( ) . L T t A x dx     . Obviously, one of the energies reaches maximum when the other gets to be minimum equal to 0. So that, due to the energy conservation one gets 2 2 2 0 0 ( ) ( ) L L xxEI x dx A x dx      , from that frequency can be calculated as 2 2 2 0 0 ( ) / ( ) L L xxEI x dx A x dx      . This is classical form of Rayleigh formula or ratio that expresses relationship between exact mode shape and frequency of undamped free vibration. The Rayleigh formula in the exact form has no meaning for application to calculate frequency from mode shape because both the modal parameters, the frequency and mode shape, are usually found together. However, if we might select approximately a function for mode shape, then natural frequency could be easily calculated by using the Rayleigh formula. Off course, this is an approximation of the natural frequency and it converges to the exact frequency if the chosen mode shape gets to be approached to the exact one. Such calculating natural frequency from appropriately chosen shape function is acknowledged as Rayleigh method. The Rayleigh formula was expanded for multiple cracked Euler-Bernoulli beam by N.T. Khiem and T.T. 10 Hai and applied for calculating natural frequencies of just uniform Euler-Bernoulli beams. 1.3. Crack detection problem for beams Contents of the crack detection problem is to localize and evaluate severity of crack based on the measured data gathered from testing on the structure of interest. There are two appoaches to solve the problem: first approach is based only on measured data that are often response of the structure to a given load; the second one involves additionally a model of the structure with assumed cracks of unknown loacation and depth. The crucial tool for the first approach is the method used for signal processing such as, for example, the Fourier or wavelet transform. The second approach finds the way to connect the measured data with the structure model in form of diagnostic equations of unknown crack parameters. The advantage of the model-based approach to crack detection is that enables to apply the latest achievements in both theoretical and numerical development of the system identification theory. In this thesis, the model-based approach is applied and the crack parameters are determined from the equations connecting the measured and calculated natural frequencies. 1.4. Overview on vibration of stepped beams 1.4.1. Spepped beams without cracks Free vibration of stepped beams was studied by numerious authors such as Jang and Bert; Jaworski and Dowell, Cunha et al.; Kukla et al. and Yang, ... The most important obtained results demonstrate that natural frequencies of stepped beam are 11 significantly affected by abrupt change in cross section area of stepped beams and the natural frequency variation is dependent also on the boundary conditions. Sato studied an interesting problem that proposed to calculate natural frequency of beam with a groove in dependence on size of the groove. Using a model of stepped beam and the Transfer Matrix Method combined with Finite Element Method the author demonstrated that (a) fundamental frequency of the structure increases with growing thickness and reducing length of the mid-step; (b) the mid-step could be modeled by a beam element, therefore, the TMM is reliably applicable for the stepped beam if ratio of its length to the beam thickness (r=L2/h) is equals or greater 4.0. Comparing with experimental results the author concluded that error of the TMM may be up to 20% if the ratio is less than 0.2. 1.4.2. Cracked stepped beams Kukla studied a cracked onestep column with a crack at the step under compression loading. Zheng et al calculted fundamental frequency of cracked Euler-Bernoulli stepped beam by using the Rayleigh method. Li solved the problem of free vibration of stepped beam with multiple cracks and concentrated masses by using recurent connection between vibration mode of beam steps. The crack detection problem for stepped beams was first solved by Tsai and Wang, then, it was studied by Nandwana and Maiti based on the so-called contour method for identification of single crack in three-step beam. Zhang vet al. solved the problem for multistep beam using wavelet analysis and TMM. Besides, Maghsoodi et al have 12 proposed an explicit expression of natural frequencies of stepped beam through crack magnitudes based on the energy method and solved the problem of detecting cracks by measurements of natural frequencies. The classical TMM was completely developed by Attar for both the forward and inverse problem of multistep beam with arbitrary number of cracks. Neverthenless, the frequency equation used for solving the inverse problem is still very complicated so that cannot be usefully employed for the case of nember of cracks larger than 2. 1.5. Formulation of problem for the thesis Based on the overview there will be formulated subjects for the thesis as follow: (1) Further developing the TMM for modal analysis of stepped Euler – Bernoulli; Timoshenko and FGM beams; (2) Extending the Rayleigh formula for calculating natural frequencies of stepped beam with multiple cracks; (3) Using the established Rayleigh formula to propose an algorithm for multi-crack detection in stepped beam from natural frequencies; (4) Overall, carrying out an experimental study on cracked stepped beam to validate the developed theories. 13 CHAPTER 2. THE TRANSFER MATRIX METHOD FOR VIBRATION ANALYSIS OF STEPPED BEAMS WITH MULTIPLE CRACKS 2.1. Stepped Euler-Bernoulli beam with multiple cracks 2.1.1. General solution for uniform homogeneous Euler- Bernoulli beam element is 1 1 2 2 3 3 4 4( ) ( ) ( ) ( ) ( )x C L x C L x C L x C L x     , where 0 1 ( ) ( ) ( ), 1,2,3,4 n k k kj j j L x L x K x e k      ; 01 02( ) (cos os ) / 2; ( ) (sin sin ) / 2;L x h x c x L x h x x       03 04( ) (cos os ) / 2; ( ) (sin sin ) / 2;L x h x c x L x h x x       1 0 1 ( ) ( ) , 1,2,3,4 j kj j k j ki j i i L e S e e k             . 2.1.2. The transfer matrix Using the solution for mode shape, transfer matrix for the beam with cracks is conducted in the form , 1 1, 1 21( ) . ...n n n n   T T T T ; 1(j) = ( ). (0)j j jL  T H H ; 2.1.3. Numerical results For illustration, two types of stepped beam as shown in Fig. 2.1 are numerical examined herein. The first is denoted by B1S and the second – B2S. Three lowest natural frequencies of the beams with single crack are computed versus crack location (Fig. 2.2). 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) . ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) j j j j j j j j j j j j j j j j j j j j j j j j j j j j j
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