Summary of the thesis of engineering and engineering engineering Mechanical: Modal analysis and Crack detection in stepped beams

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.

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 date…month…201 Hardcopy of the thesis be found at: - Library of Graduate University of Science and Technology - Vietnam National Library INTRODUCTION 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 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 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, Chapters and Conclusion Chapter describes an overview on the subject literature; Chapter – development of TMM; Chapter – the Rayleigh method and Chapter presents the experimental study CHAPTER 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: u ( x, z, t )  u0 ( x, t )  zw0 ( x, t )   ( z) ( x, t ); w( x, z, t )  w0 ( 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)  ( z )  - for Euler-Bernoulli beam theory (the classical beam theory) (b)  ( z )  z - for Timoshenko beam theory or the first order shear beam theory (c)  ( z)  z 1  z / 3h2  - the second shear beaa theory (d)  ( z)  z e{2( z / h) } - 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 ( z)  b  (t  b ) 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: g ( z )   ( z  h 2) / h  - the power law material n b) E-FGM: E( z)  Et e (12 z / h) ,   0.5ln( Et / Eb ) - the exponent law material c) S-FGM: g1 ( z)   0.5 1  z / h  ,  z  h / n d) g2 ( z)  1  z / h  / , h /  z  - Sigmoid law material n In this thesis only the FGM of power law is investigated 1.1.2 Crack model in homogeneous beams ' M ' e-0 M e+0 a e 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 EI 6 (1   )hI c (a / h) Kc  where EI is bending stiffness, h is heigh of beam and function I c ( z)  z (0.6272  0.17248z  5.92134z 10.7054z  31.5685z  67.47 z5  139.123z 146.682z  92.3552z8 ), Hence, compatibility conditions at the cracked sections are w(x,t) x  x e w(x,t) x  x e0 M (e)  w( x, t )  c Kc x w(e  0, t )  w(e  0, t ),  w(e  0, t )  w(e  0, t )  w(e  0, t )  w(e  0, t )  ,  x x x3 x3 For Timoshenko beam the conditions take the form w(e  0, t )  w(e  0, t );  x (e  0, t )   x (e  0, t )   x (e, t );  (e  0, t )   (e  0, t )   c x (e) ; wx (e  0, t )  wx (e  0, t )   c x (e, t ) 1.1.3 Modeling crack in FGM beam Crack in FGM beam can be modeled by a spring of stiffness calculated as 72 (1  ) F ( ) d , h2 E ( h) F ( )  1.910  2.752  4.782  146.776  770.75  a/h K  1/ C; C    1947.83  2409.17  1177.98 , E / E  0.2; F ( )  1.150  1.662  21.667  192.451  909.375   2124.31  2395.83  1031.75 , E / E  1.0; F ( )  0.650  0.859  12.511  72.627  267.91   535.236  545.139  211.706 , E / E  5.0 1.2 Vibration of cracked beams 1.2.1 Homogeneous beams Consider an Euler-Bernoulli beam with n cracks at positions  e1  e2   en  L and depth a j , j  1, 2, , n Free vibration of the beam is described by equation d 4 ( x) / dx4   4 ( x)  0,    F / EI in every beam segment (e j 1 , e j ), i  1, , n  1, e0  0, en 1  L , general solution of which is  j ( x)  Aj cosh  x  Bj sinh  x  C j cos  x  Dj sin  x, x (ej 1, ej ) Substituting the solution into conditions at the crack positions  j (ej )   j 1(ej ), j(ej )   j1(ej , j(ej )   j1(ej ),  j(ej )   j1(ej )   j j(ej ), j 1,2, , n one obtains 4n equations for 4(n+1) unknowns C  {A1 , B1 , C1 , D1 , , An 1 , Bn 1 , Cn 1 , Dn 1}T Therefore, combining the equations with boundary conditions allows one to get closed system of equations [D(, e1 , , en ,  , ,  n )].C  for determining the unknown constants Hence, frequency equation can be obtained as det[D(, e1, , en , 1, , n )]  0, that could be solved to give roots k , k  1, 2,3, from which natural frequencies are calculated as k  k2 EI /  F , k  1, 2,3, For Timoshenko beam, equations of free vibration are   W( x)   G(W )  ;   I ( x)  EI ( x)   GA(W )  , that would be solved together with conditions at cracks W(e j  0)  W(e j  0)  W(e j ) ; (e j  0)  (e j  0)  (e j ); (e j  0)  (e j  0)   j (e j ) ; W (e j  0)  W (e j  0)  (e j ) Similarly, putting general solution Wj ( x)  Aj cosh k1 x  B j sinh k1 x  C j cos k2 x  D j sin k2 x;  j ( x)  r1 Aj sinh k1 x  r1B j cosh k1 x  r2C j sin k2 x  r2 D j cos k2 x, r1  (    Gk12 ) /  Gk1 ; r2  (    Gk22 ) /  Gk2 ; k1  ( b2  4c  b) / 2, k2  ( b2  4c  b) / b   (1   ); c   (   );   / E;   E /  G;  F / I in beam segment (e j 1 , e j ) into condittions at cracks and boundaries, frequency equation is obtained also in the form det D(, e1, , en , 1, ,  n )  for determining natural frequencies k , k  1, 2,3, 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 I11u  A11u  I12  ; I11w  A33 (w  )  ; I12u  I 22  A22   A33 (w  )  0; with coefficients A11 , A22 , A33 , I11 , I12 , I 22 calculated from the material constants Eb , Et , b , t , n,  , Beside, from condition of neutral plane, actual position of the axis measured from the midplane is determined as h0  [n(r 1)h] / [2(n  2)(n  r )], r  Et / Eb Seeking solution of the equations of motion given above in the form e e e u( x, t)  U ( x)eit ; w(x,t)=W( x)eit ; (x,t)=( x)eit , one has got the equations ( I11U  A11U )   I12  ;  I11W  A33 (W  )  ; ( I22 A22)   I12U  A33 (W  )  0, that in turn give rise general solution z ( x,)  G0 ( x,)C , where z ( x,)  {U ( x,), ( x,),W ( x,)}T , C = {C1 , ,C6}T and 1ek x  ek x  3ek x  G ( x,  )   e k x ek x ek x  1ek x  ek x 3 ek x  1 3 2 1e k x  k1 x e  1e k x  e k x e  k2 x  3e k x   ; k x   3 e  e   e k x  k3 x  j   I12 / ( I11  k 2j A11 );  j  k j A33 / ( I11  k 2j A33 ), j  1, 2,3 In case, if the beam is cracked at position e the solution gets to be zc ( x)  Φc ( x).C, Φc ( x)  G0 ( x,)  K( x  e)G0 (e,) G ( x) : x  0; G  ( x) : x  0; K ( x)   c K ( x)   c :x0 :x0 0 0 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: {E j ,  j , Aj , I j , L j }, j  1, 2, , n , It is well known that general solution of free vibration problem in every beam segment is expressed in the form j(x)  Aj coshj x Bj sinhj xCj cosj x Dj sinj x, x(0,Lj), with  j   j ( )  (  j Aj / E j I j )1/4 Introducing the state vector Vj  { j (x), j (x),M j (x),Q j (x)} , M j ( x)  E j I j j ( x); Q j ( x)  E j I j j ( x) we would have got the expression Vj ( x)  H j ( x)C j ; C j  {Aj ,B j ,C j ,D j }T and H j ( x) is a matrix function acknowledged as shape function matrix From the continuity conditions at joints of the beam segments Vj ( L j )  Vj 1 (0) one gets V( j  1)  Hj 11 (0).H j ( L j ).V( j)  Tj , j 1.V( j) or V(n)  Tn, n 1.Tn 1, n 1 T21 V(1)  T.V(1) , with T being called transfer matrix of the beam Applying boundary conditions for the latter connection allows one to get B0 {V1 (0)}  0; B1{Vn (1)}  or B().V(1)=0 Consequently, frequency equation is obtained as det B() =0 This is content of the so-called Transfer Matrix Method that is appropriate for modal analysis of stepped beams 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 11 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 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 12 CHAPTER 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 EulerBernoulli beam element is  ( x)  C1 L1 ( x)  C2 L2 ( x)  C3 L3 ( x)  C4 L4 ( x) , where n Lk ( x)  L0k ( x)   kj K ( x  e j ), k  1, 2,3, ; j 1 L01 ( x)  (cosh x  cos x) / 2; L02 ( x)  (sinh x  sin  x) / 2; L03 ( x)  (cosh x  cos x) / 2; L04 ( x)  (sinh x  sin  x) / 2; j 1 kj   j  L0k (e j )   ki S (e j  ei )  , k  1, 2,3, i 1   2.1.2 The transfer matrix Using the solution for mode shape, transfer matrix for the beam with cracks is conducted in the form T()  Tn,n 1.Tn 1,n 1 T21 ; T(j) = H j (L j ).Hj1 (0) ; L j ( x) L j ( x) L j ( x)   L j1 ( x )  L ( x)   L ( x ) L ( x ) Lj ( x)  j1 j2 j3 H j ( x)    E j I j Lj1 ( x) E j I j Lj ( x) E j I j Lj ( x) E j I j Lj ( x)     E j I j Lj1 ( x) E j I j Lj ( x) E j I j Lj ( x) E j I j L j ( x)  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) 13 Fig 2.1 Two models of stepped beam used in numerical analysis L1=L2=L3=1m, b1=b2=b3=0.1 h1=h3=0.15;h2=0.1 1 a /h = 10% 0.99 10% a /h = 20% 20% a /h = 30% a /h = 10% 10% 10% 0.995 30% 20% 20% a /h = 30% 30% 0.985 Ty so tan so thu nhat Ty so tan so thu nhat 30% 40% 0.97 20% 30% 0.99 a /h = 40% 0.98 10% a /h = 20% 40% 40% 40% 0.98 a /h = 40% 0.975 0.97 0.96 L1=L2=L3=1m,b1=b2=b3=0.1m;h1=h3=0.1m,h2=0.15m 0.965 0.95 0.94 0.5 0.96 B c thu hai Bac thu nhat 1.5 2.5 0.955 Vi tri vet n t h1=h3=0.15;h2=0.1 L1=L2=L3=1m, b1=b2=b3=0.1 10% 10% 0.995 10% 20% 20% 1.5 Vi tri vet n t 2.5 10% 10% 20% 20% 20% 30% 30% 1 0.995 B c thu hai Bac thu nhat 0.5 0.99 20% 30% 20% 30% 0.99 40% 0.98 0.985 40% a /h = 30% Ty so tan so thu ba Ty so tan so thu hai 0.985 a /h = 30% 0.975 40% a /h = 40% 0.97 40% a /h = 40% 40% 0.98 a /h = 40% a /h = 40% a /h = 30% 0.975 0.97 0.965 0.965 0.96 40% B c thu hai Bac thu nhat 0.96 0.955 0.95 0.5 1.5 Vi tri vet n t 2.5 Bac thu nhat B c thu hai 0.5 1.5 Vi tri vet n t 2.5 10% 20% 40% 30% 0.995 10% 10% 10% 20% 20% 20% 0.99 0.985 10% 0.99 20% 30% 20% 30% 0.985 30% 40% 30% Ty so tan so thu ba 10% 0.995 Ty so tan so thu ba 0.955 0.98 0.975 40% 0.98 a /h = 40% a /h = 40% a /h = 30% 0.975 0.97 0.97 a /h = 40% 0.965 a /h = 40% 0.965 L1=L2=L3=1m, b1=b2=b3=0.1 h1=h3=0.15;h2=0.1 40% 0.96 0.955 0.96 Bac thu nhat 0.5 Bac thu nhat B c thu hai 1.5 Vi tri vet n t 2.5 0.955 0.5 B c thu hai 1.5 Vi tri vet n t 2.5 Fig 2.2 Effect of crack position and depth on three lowest natural frequencies of B1S (right) and B2S (left) Notice: Observing graphs given in Fig 2.2 allows the following remerks to be made: (1) Likely to the uniform beams, there exist on stepped beam positions crack occurred at which does 14 not change some natural frequencies; (2) Change in natural frequencies due to crack undergoes a jump for crack passing across step; (3) Increasing depth leads to reduced natural frequencies, but reduction of natural frequencies is different for cracks at different steps 2.2 Stepped Timoshenko beam with multiple cracks 2.2.1 General solution  Wc ( x)   W0 ( x)   K w ( x  xc )      0 ( xc )    Gc ( x)C  c ( x )    ( x )   K ( x  xc )  W0 ( x, )  C1 cosh k1 x  C2 sinh k1 x  C3 cos k2 x  C4 sin k2 x ; 0 ( x, )  r1C1 sinh k1 x  r1C2 cosh k1 x  r2C3 sin k2 x  r2C4 cosk2 x Gc ( x)  G0 ( x, )   Kc ( x  xc ) 2.2.2 Transfer matrix T = T(m)T(m-1) …T(1), T( j )  H j ( L j ).Hj (0) , j = 1, …, m   a1 b1  K ( x) K c ( x)   w  K ( x)   a1 b1  a2 a2 b2  b2   G0 j ( x)   j K cj ( x  xc )   G ( x)  H j  x   c     PGc ( x)  ( j )  PG0 j ( x)   j PK cj ( x  xc )  2.2.3 Numerical results Natural frequencies of two-step cantilever Timoshenko beam with single crack are computed versus varying crack position and shown in Fig 2.6 and 2.7 The constants of the stepped are L=0.5m;E=210Gpa;   7860kg / m3 ;b=12mm; h1=20mm; h2=16 mm 15 Note that variation of natural frequencies caused by crack at the step also has a jump; crack near the clamped end is more affecting on natural frequencies than that located near the free end Twostepped cantilever beam 1.05 1/01 0.95 0.9 0.85 0.8 ah = 0.1 ah = 0.3 ah = 0.5 0.75 0.7 0.1 0.2 0.3 0.4 0.5 x c /L 0.6 0.7 0.8 0.9 Fig 2.6 First natural frequency ratio of cantilever stepped beam versus crack position Twostepped cantilever beam 1.05 2/02 0.95 0.9 0.85 ah = 0.1 ah = 0.3 ah = 0.5 0.8 0.1 0.2 0.3 0.4 0.5 x c /L 0.6 0.7 0.8 0.9 Fig 2.7 Second natural frequency ratio of cantilever stepped beam versus crack position 16 2.3 Stepped FGM beams with cracks 2.3.1 General solution z( x)  Φc ( x).C, z  {U , ,W }T ; Φc ( x)  G0 ( x, )  K( x  e)G0 (e, ) ; G ( x) : x  0; G  ( x) : x  0; K ( x)   c K ( x)   c  : x  0;  : x  0; 2.3.2 Transfer matrix T = T(m)T(m-1) …T(1), T( j )  H j ( L j ).Hj (0)  A11 x x     Φc ( x)  H j ( x)     x Φc ( x) j A22  x  A33 ;   A33  x  2.3.3 Numerical results Clamped Beam, L1=L2=L3=1;a/h=5,10,20,30,40% Clamped Beam, L1=L2=L3=1;a/h=5,10,20,30,40% 1.01 S2 S2 S0 S1 S2 S0 S2 0.96 S2 S2 S1 S1 S2 S0 S0 0.98 S2 S0 S0 S1 0.97 S1 S0 S0 S2 S2 0.96 S1 S1 S2 0.99 S1 S1 S0 Ty so tan so thu nhat S2 S1 S1 S0 S1 Ty so tan so thu hai S1 0.98 0.94 S0: h1=h2=h3=0.1 S1: h1=h3=0.1;h2=0.2 S1 S1 0.95 S0: h1=h2=h3=0.1 S1: h1=h3=0.1;h2=0.2 S2: h1=h3=0.1;h2=0.05 S2: h1=h3=0.1;h2=0.05 0.92 0.94 0.93 0.9 0.5 1.5 2.5 0.5 1.5 Vi tri vet nut 2.5 3 Clamped Beam, L1=L2=L3=1;a/h=5,10,20,30,40% Vi tri vet nut 1.01 0.99 S2 Ty so tan so thu ba 0.98 S2 S2 S1 S1 S0 0.97 S0 S2 S2 S1 S2 S0 S0 S0 0.96 S1 S0 S0 S1 0.95 0.94 S0: h1=h2=h3=0.1 0.93 S1 S1: h1=h3=0.1;h2=0.2 S2: h1=h3=0.1;h2=0.05 0.92 0.91 0.5 1.5 2.5 Vi tri vet nut Fig 2.9 Normalized natural freqiencies of stepped FGM with clamped ends in dependence on crack position and depth (a/h) 17 Notice: Numerical analysis has shown that material gradient and thickness variation of beam are significantly affecting the crack-induced change in natural frequencies For instance, abrupt change in beam thickness produces a jump in variation of natural frequencies versus crack position that is continuous for uniform beam Concluding remarks for Chapter Results obtained in this Chapter are as follow:  TMM has been developed for modal analysis of stepped Euler-Bernoulli; Timoshenko and FGM beams with arbitrary number of transverse open cracks;  The developed TMM allows studying effect of change in cross section area on natural frequencies in combination with cracks and material parameters of beams CHAPTER RAYLEIGH METHOD IN ANALYSIS AND CRACK DETECTION FOR STEPPED BEAMS 3.1 Rayleigh formula m xj j 1 x j 1 2  S j {  kj ( x)dx   jkj (e j )} k2  m xj j 1 x j 1  m j   ( x)dx kj 3.2 Natural frequencies m xj j 1 x j 1 m 2  S j  [kj ( x)] dx   S j  j [kj (e j )] k2  m xj j 1 m m j 1 j 1 2   m j [kj ( x)] dx    j S j k [kj (e j )]   m j  j ( , e) j 1 x j 1 18 Dam goi tua hai dau Dam goi tua hai dau 1 0.95 0.95 0.9 2/ 20 1/ 10 0.9 0.85 0.85 0.8 a1/h1=0.1 a2/h2=0.2 a3/h3=0.3 a4/h4=0.4 a5/h5=0.5 0.75 0.7 0.1 0.2 0.3 0.4 0.5 e 0.6 0.7 0.8 0.9 a1/h1=0.1 a2/h2=0.2 a3/h3=0.3 a4/h4=0.4 a5/h5=0.5 0.8 0.75 0.1 0.2 0.3 0.4 0.5 e 0.6 0.7 0.8 0.9 Fig 3.3 Effect of crack position on natural frequencies of twostep beam with simply supported ends 3.3 Crack detection Using the established Rayleigh quotient, diagnostic equations are derived in the form [A  B(γ)]γ  b, A  [akj  kj02 (e j )S j ; k  1, , n; j  1, , m]; b  {bk  (ks0  k2 ), k  1, , n}; m B( γ)  [bkj   R(ei , e j ) iki0 (ei )kj0  (e j ); k  1, , n; j  1, , m] i And iteration process proposed to solvethe diagnostic equations is [A(i)]γ (i )  b, where A(i)  A  B(γ (i 1) ); i  1, 2,3, ; γ (0)  Because of incompleteness and errorneousness of measured data the Tikhonov’s regularization technique is applied [AT (i)A(i)   I ] {γ (i ) }  b, That gives rise to results of crack detection presented in Table 3.5 in various cases of crack scenarios 19 Table 3.5 Result of crack detection in cantilever beam Crack scenarios Actual Single crack Detected Actual Double cracks Detected Actual Detected Actual Triple cracks Detected Actual Detected Crack positions 0.06 0.06 0.06 0.15 0.06 0.15 (0.26) 0.06 0.15 0.24 0.06 0.15 0.24 0.06 0.15 0.24 0.06 0.15 0.24 0.06 0.15 0.24 0.06 0.15 0.24 0.2 0.1944 0.2 0.1901 0.4 0.3906 0.6 0.6045 Crack depth 0.2 0.2041 0.2 0.1985 0.2 0.1980 0.4 0.4009 0.6 0.6038 (0.1081) 0.2 0.1917 0.4 0.3964 0.6 0.6060 Concluding remarks for Chapter In this Chapter:  Rayleigh quotient has been established, first, for multiple cracked beam in general, then, that is applied calculating natural frequencies of stepped beam with multiple cracks  Numerical results showed that natural frequencies of stepped beam can be more easily calculated by Rayleigh qoutient in comparison with the TMM  The Rayleigh quotient is applied also for deriving an efficient peocedure to detect multiple cracks in stepped beam by natural frequencies 20 CHAPTER EXPERIMENTAL STUDY OF CRACKED STEPPED BEAMS z y x Fig 4.6 Experimental model of clamped beam Excitation point Measurement point Model and diagram of measuring point and excitation point for stepped cantilever beam Measured data are given in Table 4.1 and 4.2 and illustrated in Fig 4.12 21 2.5 1.5 0.5 -0.5 0% 3.5 2.5 1.5 0.5 -0.5 0% 10% 20% 30% 10% 20% 30% 40% 40% 50% 50% 30 25 20 15 10 -5 0% 10% 20% 30% 40% 50% Fig 4.12 Comparison of measured natural frequencies with computed in various crack depth for stepped beam with clamped ends Crack at position e = 0.45m (Solid line – theory; Dash line - experimental) 22 Table Natural frequencies of clamped stepped beam with two cracks, first crack is at e1=0.2m of depth from 0%-40% and the second crack at e2=0.45m with depth 40% Natural frequencies (Hz) a1/h1 (%) f1 f2 f3 f4 10 20 30 40 73.25 73.17 72.87 72.61 72.46 142.9 142.9 142.9 142.9 142.9 295.8 295.3 294.4 293.5 291.6 517.56 516.75 514.56 511.31 506.3 Table Natural frequencies of stepped cantilever beam with single crack 0.6m and depth varying from 0% to 50% Natural frequencies (Hz) a/h (%) 12.5 30 50 f1 f2 f3 f4 13.56 13.56 13.51 13.44 54.69 54.69 54.31 53.13 139.4 139.4 139.3 138.6 290.94 290.88 289.44 285.69 Concluding remarks for Chapter In this Chapter:  Two models of stepped cracked beam have been fabricated and examined in experimental study: one is clamped in both ends and the other is cantilever  Natural frequencies of the beam were measured and compared to the computed by TMM and Rayleigh qoutient 23 CONCLUSION The classical transfer matrix method has been expanded to calculate natural frequencies of Timoshenko and FGM stepped beams with arbitrary number of cracks The Rayleigh qoutient is expanded to calculate natural frequencies of multiple cracked stepped beam that provides an explicit expression for natural frequencies through crack parameters A procedure is proposed to detect multiple cracks in stepped beam by measurements of natural frequencies An experimental study is completed to validate the theoretical development proposed in this thesis MAJOR CONTRIBUTIONS OF THE THESIS The TMM is first developed for modal analysis of Timoshenko and FGM stepped beams; The Rayleigh qoutient is first established for stepped beam with arbitrary number of cracks that is simple tool for calculating natural frequencies of the beam instead of solving complicate frequency equation in TMM; The developed herein procedure for multiple crack detection in stepped beam is an important contribution of the thesis Cracked multistep beam is thoroughly studied in experimentation that enables not only to validate the theoretical development but also provide input data for crack detection in stepped beams 24 LIST OF THE AUTHOR’S PUBLICATIONS Nguyen Tien Khiem, Duong The Hung, Vu Thi An Ninh, Multiple crack identification in stepped beam by measurements of natural frequencies, Vietnam Journal of Mechanics, 2014, 36(2), 119-132 Nguyen Tien Khiem, Vu Thi An Ninh, An application of Rayleigh quotient for multiple crack identification in beam, Tuyển tập cơng trình Hội nghị Cơ học kỹ thuật tồn quốc Kỷ niệm 35 năm Viện Cơ học 10/4/2014, Tập 1, 99-105 Nguyen Tien Khiem, Tran Thanh Hai, Vu Thi An Ninh, Free vibration of cracked multistep Timoshenko beam, Proceedings of the 2nd National Conference on Mechanical Engineering and Automation, Oct 7-8, 2016, Hanoi University of Science and Technology, 2016, 392-396 Nguyen Tien Khiem, Lê Khanh Toan, Ha Thanh Ngoc, Vu Thi An Ninh, Experimental study of cracked multistep beam, Proceedings of the 2nd National Conference on Mechanical Engineering and Automation, Oct 7-8, 2016, Hanoi University of Science and Technology, 2016, 397-400 Vu Thi An Ninh, Luu Quynh Huong, Tran Thanh Hai, Nguyen Tien Khiem, The transfer matrix method for modal analysis of cracked multistep beam, Journal of Science and Technology, 2017, 55(5), 598-611 N.T Khiem, T.V Lien, V.T.A Ninh (2017), Natural frequencies of stepped functionally graded beam with multiple cracks, Iranian Journal of Science and Technology – The Transactions in Mechanical Engineering (Accepted 3/2017) 7.N.T Khiem, T H Tran, V.T.A Ninh (2017), A closed-form solution to the problem of crack identification for multistepcantilever beam based on Rayleight quotient, International Journal of Solids and Structures (Submitted July 2017) ... methods for solving the problems of various types of stepped beams is really demanded Objective of the thesis The aim of this thesis is to study crack- induced change in natural frequencies and to develop... for detecting cracks in stepped beams by measurement of natural frequencies The research contents of the thesis (1) Developing the Transfer Matrix Method (TMM) for modal analysis of stepped Euler... 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 –