SPECTRAL METHOD FOR VIBRATION ANALYSIS OF CRACKED BEAM SUBJECTED TO MOVING LOAD

1 VIET NAM ACADEMY OF SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY -o0o - PHI THI HANG SPECTRAL METHOD FOR VIBRATION ANALYSIS OF CRACKED BEAM SUBJECTED TO MOVING LOAD Specialized in: Engineering Mechanics Code: 62 52 01 01 SUMMARY OF PhD THESIS Hanoi, 2016 INTRODUCTION Supervisors: Prof.DrSc Nguyen Tien Khiem Dr Phạm Xuan Khang Reviewer 1: Reviewer 2: Reviewer 3: Thesis is defended at Graduate university of Science and Technology _18 Hoang Quoc Viet _ Hanoi on , 2016 Hardcopy of the thesis be found at Vietnam National Library and Library of Graduate university of Science and Technology 1 Necessity of the theme Dynamic analysis of structure subjected to moving load is an important problem in the practice of engineering, especially, for the bridge and railway engineerings This problem was investigated very early, in the 19th Century However, it is studying at present by the following reasons: (1) moving load models need to be improved to describe more accurately the moving vehicle-structure interaction; (2) structures subjected to moving loads become more complicated so that a lot of new problems in dynamic analysis of such the structures has been posed; (3) more exact methods of dynamic analysis need also to be developed for solving the problems The most popular method used for dynamic analysis of a structure under moving load is the Bubnov-Galerkin method that is based on the eigenfunctions of the structure and therefore is called also superposition or modal method This method is difficult to apply for the structures eigenfunctions of which are unavailable In that case, the Finite Element Method (FEM), the most powerful technique for structure analysis, is employed Finite element model of a structure is conducted basically on the specific shape functions that are static solution of a finite element of the structure Therefore, high frequency dynamic response of a structure could be investigated by the finite element model with very large number of elements Recently, the dynamic stiffness method that uses the dynamic shape functions instead of the static one for constructing a frequency dependent matrix called dynamic stiffness matrix is developed Such development of the FEM enables to study dynamic response of arbitrary frequency for a structure as a distributed system This method called Dynamic Stiffness Method (DSM) is then formulated as a method used for dynamic analysis of structure in the frequency domain and termed Spectral Element Method (SEM) Objective of the thesis This thesis aimed to apply the SEM for dynamic analysis of cracked beam subjected moving harmonic force in the frequency domain Namely, the frequency response of a cracked beam subjected to moving harmonic force is obtained explicitly and examined in dependence upon the load and crack parameters This task is acknowledged herein spectral analysis of cracked beam subjected to moving load Subject of research Subject of this study is a multiple cracked beam-like structure under loading of a concentrated force moving with constant speed The Euler-Bernoulli theory of beam is used and crack is modeled by an equivalent spring of stiffness calculated from its depth accordingly to the fracture mechanics theory Methodology of research Method used in this study is mostly analytical method that is illustrated by numerical results obtained by MATLAB Thesis’s content Thesis consists of introduction, chapters and a conclusion Chapter describes an overview of the moving load problem and conventional methods used for solving the problem; the crack detection problem is also presented in this chapter Chapter presents the methodology for spectral analysis of cracked beam subjected to moving force Chapter provides an exact solution in frequency domain of the moving load problem for intact beam and frequency response is thoroughly examined Chapter studies cracked beam subjected to moving force and proposes a method for calculating natural frequencies of continuous multispan cracked beam A procedure for crack detection by using frequency response is developed Conclusion chapter summaries major results obtained in the thesis and some problems for further investigation Chapter OVERVIEW 1.1 The moving load problem Consider a beam subjected to the load produced by a moving mass as shown in Fig 1.1 Equations of motion for the system are EI  w( x, t ) w( x, t )  w( x, t )  F  F  P(t ) [ x  x0 (t )] ; (1.1.1) x t t P(t )  mg  cz(t )  kz (t )  m[ g  y(t )] ; 0 (t ); z(t )  [ y(t )  w0 (t )]; w0 (t )  w[ x0 (t ), t ] mz(t )  cz(t )  kz (t )  mw In the latter equations w( x, t ) is the transverse displacement of beam, y (t ) - vertical displacement of mass; x0 (t ) is position of mass on the beam measured from the left end;  (t ) is delta Dirac function From the given system the following problems can be obtained for dynamic analysis of beam: The moving force problem, when the force P(t ) is known, for instance, P(t )  P0 exp{t   } ; 0 (t )] ; The moving mass problem if P(t)  m[ g  w The moving vehicle problem when Eq (1.1) are solved for both the beam and vehicle x0 (t )  v m v c k E, I, , F x0 w0 w(x,t) x  Fig 1.1 Model of beam under moving load 1.2 Conventional methods for moving load problem a) The Bubnov-Galerkin method is based on an expansion of time domain response of a structure in a series of its eigenfunctions and, as result, a system of ordinary differential equations is obtained and solved by using the well-developed methods Most important results in the moving load problem have been obtained for simple beam-like structures by using the method However, this method is difficult to apply for complicate structures such as cracked ones, eigenfunctions of which are unavailable b) The finite element method is the most powerfull technique that may be applied for arbitrary complicate structures due to involved specific shape functions being static solution of a finite element Nevertheless, since the static shape functions have been used the finite element method is unable to apply for studying high frequency response of a structure c) The dynamic stiffness method gets to be advanced in comparison with the finite element method by that allows one to investigate dynamic response of arbitrary frequency This is due to frequency-dependent shape functions are employed instead of the static ones However, applying the dynamic stiffness method for the moving load problem leads the Gibb’s phenomena to appear when shear force is converted from the frequency domain to the time domain So, the frequency response obtained by the dynamic stiffness method should be analyzed directly rather in the frequency domain than in the time domain This leads to spectral analysis of frequency response of beam subjected to a moving load that is subject of the present thesis 1.3 Crack detection problem The problem of crack detection in structures has attracted a great attention of researchers and engineers because of its vital importance to safely employ a structure and avoid serious catastrophe might be caused from not early recognized cracked members The methods developed for solving the problem can be categorized as follows: (1) Frequency-based method means crack location and depth being predicted by using only measured natural frequencies (2) Mode shape-based method proposes to evaluate the crack parameters from measurements of mode shapes of structures under consideration (3) Time domain method is that uses time history response measured in-situ of a structure for its crack detection (4) Frequency response function method proposes to carry out the crack detection task based on the Frequency Response Function (FRF) measured by the dynamic testing technique Though all of the aforementioned methods are helpful in solving various specific problems of crack detection, they are all faced with either insensitivity of chosen diagnostic criterion to crack or noisy measured signal used for the crack detection Among the diagnostic indicators the frequency response function is most accurately measured by the dynamic testing method However, the FRF-based method is limited by the following facts First, measurement of FRF needs the testing load measured at a large number of positions on structure and, secondly, the presence of crack may be hidden by the interaction of vibration modes predominated in the measured FRF The shortcomings of the FRF-based method in crack detection may be avoided by using frequency response of a testing structure subjected to controlled moving load 1.4 Determination of thesis’s subject The short overview allows one to conclude that, firstly, the most efficient approach to the moving load problem is the dynamic stiffness method but it must be used directly for dynamic analysis of a structure in the frequency domain Secondly, the frequency response of a structure subjected to a well-controlled moving load provides a constructive signal for crack detection, especially, in beam-like structures So, subject of the present thesis is to further develop the frequency response method proposed by N.T Khiem et al to spectral analysis of cracked beam under moving force and to use that method for multi-crack detection from measured frequency response Chapter METHODOLOGY 2.1 Frequency response Let’s consider vibration of an Euler-Bernoulli beam described by the equation   w( x, t )   w( x, t )  w( x, t )  w( x, t )  EI      F  2  p( x, t )   4 x t  t   x  t , where w( x, t ) is transverse deflection of the beam at section x; E, I, F, ρ, L - the beam’s material and geometry constants and 1 , 2 are damping coefficients Under the Fourier transformation, the equation leads to d 4W ( x,  )  4W ( x,  )  Q( x,  ) , 4  F (  i ) / EI ; (2.1.1) dx  W ( x,  )   w( x, t )e it dt; Q( x,  )    P( x,  ) ; P( x,  )   p( x, t )e it dt; EI     1 / (1  12 );   (1   / ) /(1  12 ) The so-called frequency response W ( x,  ) determined from Eq (2.1.1) must satisfy boundary conditions The frequency response is complex function of , frequency W ( x, )  Rw ( x, )  iI w ( x, ) , the module of which S w ( x,  )  W ( x,  )  Rw2 ( x,  )  I w2 ( x,  ) , (2.1.2) is the frequency-amplitude characteristic of beam subjected to arbitrary load p( x, t ) The function S w ( x,  ) considered with respect to frequency  for fixed x is called herein response spectrum of beam at the section x The function (2.1.2) of variable x with fixed frequency  is called deflection diagram of frequency  Content of the frequency response method applied for moving load problem is first to solve Eq (2.1.2) for a given moving load p( x, t ) 2.2 Frequency response method in the moving load problem As well-known, load produced by a moving force P(t) expressed in the form p( x, t )  P(t ) ( x  vt ) has the frequency-amplitude characteristic  Q( x,  )   P(t ) ( x  vt )e  it dt  P( x / v)e  ix / v / EIv (2.2.1)  and general solution of Eq (2.1.1) is represented as W ( x, )  0 ( x, )  1 ( x, ) (2.2.2) d 0 ( x, ) / dx   0 ( x, )  4 x 1 ( x,  )   h( x  s)Q(s,  )ds ; h( x)  (sinh x  sin x) / 23 Subsequently, solution (2.2.2) can be expressed in the form W ( x, )  CL1 (x)  DL2 (x)  1 ( x, ), r  1,2,3 (2.2.3) with L1 ( x), L2 ( x) being the independent particular solutions of homogeneous equation (2.1.1) and satisfying boundary conditions at the left end of beam Therefore, constants C, D can be determined from the boundary conditions at the right end as 11 4.5 Normalized midspan deflection 3.5 2.5 1.5 0.5 0 0.2 0.4 0.6 0.8 Dimensionless frequency 1.2 1.4 1.6 Fig 3.4 Response spectrum for harmonic load with   0.41 Note: In case of constant moving load, two peaks of response spectrum reach at zero and fundamental frequency (see Fig 3.1) The maximum amplitude at zero frequency implies that moving load acts as a static load and this is happen when load speed is less than 1/10 critical speed The second peak attained at the fundamental frequency implies predomination of eigenmode of response and it is observed for speed greater than 1/3 critical one Fig 3.2 shows that there exist values of the load speed that may cancelate amplitude of eigenmode response This is approved by graphs given in Fig 3.3 that were plotted for so-called anti-resonant speeds 3.2 Frequency response to harmonic moving force Fig 3.4 shows response spectrum in the case of moving harmonic force of frequency   0.41 The peak attains at load frequency for load speed less than 0.1vc This means predomination of forced mode of response However, the peak 12 is rapidly reduced and completely disappears when load speed reaches 0.3vc For the speed exceeding 0.3vc it is observed only peak at fundamental frequency Similarly, we can find the anti-speeds for the moving harmonic load as shown in Fig 3.5 and 3.6 4.5 Normalized midspan deflection 3.5 2.5 1.5 0.5 0 0.2 0.4 0.6 0.8 Dimensionless frequency 1.2 1.4 Fig 3.5 Response spectrum for harmonic load   0.41 at anti-resonant speeds 0.34 0.32 k=1 k=2 k=3 k=4 k=5 k=6 k=10 k=15 k=20 k=30 0.3 0.28 0.26 0.24 Speed factor 0.22 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0.2 0.4 0.6 0.8 1.2 Load frequency factor 1.4 1.6 1.8 Fig 3.6 The map of anti-resonant speed in dependence of load frequency 13 Concluding remark for Chapter The obtained numerical results allow one to make the following concluding remarks for Chapter 3: (a) Response spectrum enable one to identify various vibration modes that are predominated in dependence on the load speed Namely, for the load speed less than 0.1vc response behaviors as vibration mode of load frequency and eigenmode of the response becomes governed if load speed exceeds 1/3vc (b) There exist speeds of load that may concelate the vibration mode of natural frequencies and such speeds are called anti-resonant ones Antiresonant speeds are elementarily calculated from given natural and load frequencies (c) Action of combined harmonic forces with different frequencies is also investigated Namely, the constant load is predominate for low speeds and for high speed the load with frequency more closed to the natural one has more effect on the response of beam The loads with frequencies symmetrical about the fundament frequency are equally affecting on the beam vibration Chapter VIBRATION OF CRACKED BEAM SUBJECTED TO MOVING FORCE 4.1 Free vibration of cracked beam 14 x e1 a1 E, , F aj h ej b L b y h K0j Fig 4.1 Model of cracked beam Suppose that a beam of elasticity modulus E, mass density ρ, length L, cross section area F and moment of inertia I is cracked at n positions e j , j  1, , n as shown in Fig 4.1 The crack is modeled by an equivalent spring of stiffness K0 j ( j  1, , n) that is calculated from crack depth a j ( j  1, , n) accordingly to the fracture mechanics theory Free vibration of such the beam is described by the equation  ( IV ) ( x)  4 ( x)  0, x  (0,1),   L4 F / EI (4.1.1) everywhere in the beam except beam’s boundaries where the conditions must be satisfied  ( p0 ) (0)   ( q0 ) (0)  0,  ( p) (1)   (q) (1)  (4.1.2) and cracked sections where it is satisfied the condition (e j  0)  (e j  0); (e j  0)  (e j  0); (e j  0)  (e j  0); (e j  0)  (e j  0)   j (e j  0) (4.1.3) For the beam natural frequencies are seeking from the equation (4.1.4) f (,  , e)  det(Γ(γ)B(, e)  L0 ( )I)  0, 15 Γ(γ)  diag , , n , B(, e)  [b jk  b( , e j , ek ) j, k  1, ,n] and mode shapes are determined as n  k ( x)  ( x, k )    ( x, k , e j ) kj ,  ( x, k , e j )   ( x, k , e j ) / L0 (k ) j 1 Illustrating example: For illustration, natural frequencies of two span continuous beam with cracks are calculated and presented in Table 4.1 Table 4.1 Natural frequencies of two-span cracked beam Cracking scenarios Uncracked Eq.(4.1.4) Ref.[36] Span Span uncracked 1.2 1.8 0.5 1.2 1.8 0.2 0.8 1.2 1.8 0.2 0.8 1.5 0.2 0.8 uncracked Freq.1 Freq.2 3.1416 3.9266 π 3.9266 3.1056 3.1056 3.1056 3.1157 3.1416 3.9266 3.7753 3.7878 3.7878 3.7878 Freq.3 Freq.4 6.2832 7.0686 2π 7.0685 Eq (4.1.1) 6.2395 7.0686 6.2395 7.0190 6.2395 6.6617 6.6617 6.2832 6.6617 6.2832 Freq 9.4248 3π Freq.6 10.2102 10.2101 9.3911 9.3911 9.3911 9.4270 9.4248 10.2101 9.7954 9.5124 9.5124 9.5124 4.2 The frequency response of cracked beam subjected to moving force In this section response of cracked beam subjected to a moving force is obtained Vibration of the beam in the time domain is described by equation EI  w( x, t ) w( x, t )  w( x, t )   F    F  P(t ) [ x  vt )] x t t After Fourier transform the latter equation becomes d 4 ( x,  )  4 ( x,  )  Q( x,  ) ; dx (4.2.1) (4.2.2) General solution of Eq (4.2.2) is x  ( x,  )  0 ( x,  )   h( x  s)Q(s,  )ds , h( x)  (1 / 2 )[sinh x  sin x] ; (4.2.3) 16 d 40 ( x,  ) dx  40 ( x,  )  (4.2.4) It was proved that free vibration of cracked is represented by n 0 ( x,  )  L0 ( x,  )    k K ( x  ek ) (4.2.5)  j   j [ L0 (e j ,  )    k S (e j  ek ) ] (4.2.6) jk11 k 1 So that after application of boundary condition for solution (4.2.3), (4.2.5) one obtains n  ( x,  )   ( x,  )    k  k ( x, e,  ,  ) , (4.2.7) k 1  ( x, )  C0 L1 ( x,  )  D0 L2 ( x,  )  1 ( x, ) ;  k ( x, )  Ck L1 ( x,  )  Dk L2 ( x,  )  K ( x  ek ), k  1, ,n In the case of P(t )  P0 e iet one has Q( x, )  ( P0 / EIv)e ix / v  Q0 e ix / v , ˆ    e ˆ ˆ 1( x,)  10(x)  Q0eiˆx / v /[4  (ˆ / v)4 ] ; 10 (x)  P1 () cosh x  P2 () sinh x  P3 () cos x  P4 () sin x 4.3 Influence of crack on frequency response of cracked beam For illustration, there is considered the beam of the following constants:   25m , F  b  h  0.5  0.25m2 , E  200MPa,   7850kg / m with various scenarios of cracks Since the frequency response is a complex function, the following variations of the function are calculated Sa ( x,  )  c ( x,  )  0 ( x, ) , Sm ( x,  )  c ( x, )  0 ( x, ) The former is called variation of response spectrum and the latter – variation of frequency response The lower index “c” denotes the frequency response of cracked beam and that with index “0” - that of uncracked one The dimensionless    / 1 , f e   / 1 ,   v / Vc , where 1 - fundamental 17 frequency;  - load frequency; Vc  1L /  - critical speed of load The frequency response variations are investigated in the frequency range from to 21 , i e   (0,2), f e  [0,2] centered at fundamental frequency The variations are computed versus both the beam span variable x and frequency as well Results of computation are presented in Figs 4.2-4.9 The computed frequency response variations provide useful instructions for crack detection by measurements of frequency response to moving load 1.5 (a) - Resonant f requency of load v=0.1 Spectrum deviation v=0.2 v=0.5 v=0.3 0.5 v=0.4 v=1.0 -0.5 -1 0.9 0.95 0.986 1.05 1.1 Dimensionless frequency Fig 4.2 Variation of response spectrum due to cracks for different load speed 0.3 0.25 fe=0.8 &1.2 fe=1.0 fe=0.7 &1.3 fe=0.6 &1.4 fe=0.5 &1.5 0.2 fe=0.4 &1.6 0.15 Spectrum deviation (b) - speed=0.5 fe=0.3 &1.7 fe=0.1 &1.9 fe=0.2 &1.8 0.1 0.05 fe=0 & 2.0 -0.05 -0.1 -0.15 -0.2 0.9 0.92 0.94 0.96 0.986 1.02 Dimensionless frequency 1.04 1.06 1.08 Fig 4.3 Variation of response spectrum due to cracks for different load frequency 18 omega=0.986 1.5 Spectrum deviation 0.9 fe=1.0 1.1 0.8 0.5 1.2 0.7 0.6 1.3 1.4 fe=0 &2,0 -0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Dimensionless speed Fig 4.4 Variation of eigenmode amplitude versus load parameter 0.9 e=12 &13 e=11&14 e=10 &15 0.8 e=11& 14 e=9 &16 0.7 e=11& 14 e=8 &17 Magnitude of deviation 0.6 e=7 &18 0.5 e=4 & 21 e=6&19 0.4 e=3 & 22 e=5 &20 0.3 e=2 &23 e=1 &24 0.2 0.1 0.92 0.94 0.96 0.98 1.02 1.04 1.06 1.08 Dimensionless frequency Fig 4.5 Variation of response spectrum vers crack parameter 0.2 e=13.5 e=14.5 e=12.5 0.18 10.5 0.16 15.5 9.5 16.5 8.5 17.5 FR deviation magnitude 0.14 0.12 7.5 18.5 0.1 6.5 19.5 0.08 5.5 0.06 4.5 0.04 3.5 20.5 21.5 22.5 0.02 0 10 L/2 Span position 15 20 25 Fig 4.6 Variation of vibration diagram vers Crack position 19 Load frequency =omega1 v=0.1 0.9 0.8 FR deviation magnitude 0.7 0.6 0.5 v=0.2 0.4 v=0.3 0.3 v=0.4 0.2 v=0.5 0.1 v=1.0 0 10 L/2 Crack position 15 20 25 Fig 4.7 Variation of response spectrum vers load speeds Number of cracks = (b) - fe=1,v=0.5 1.2 FR deviation magnitude 0.8 0.6 0.4 0.2 Number of cracks = 0.9 0.95 Dimensionless frequency 1.05 1.1 1.15 Fig 4.8 Variation of response spectrum vers amount of cracks cracks cracks 0.9 cracks 0.8 cracks FR deviation magnitude 0.7 0.6 cracks 0.5 cracks 0.4 0.3 cracks 0.2 cracks 0.1 cracks 0 10 Span location 15 20 25 Fig 4.9 Variation of vibration diagram vers amount of cracks 20 4.4 Crack detection in beam by measured frequency response The crack detection procedure proposed in this section consists of the following steps: (1) A grid of cracks of unknown depths is assumed at positions e1 , , en ; (2) A model of beam with the cracks is constructed so that an explicit expression for frequency response of that cracked beam subjected to a moving harmonic force could be conducted (3) Based on the established model and measured data of frequency response unknown crack magnitudes are evaluated; (4) Mapping the evaluated crack magnitudes versus assumed crack positions allows one to find out the apparent peaks positions of which result in detected crack locations (5) The crack magnitudes corresponding to the peaks are used for estimating crack depth using formulas given in fracture mechanics and the procedure of crack detection is thus completed The major task in the crack detection procedure is to evaluate crack magnitude vector γ  ( , ,  n ) from given model of cracked beam and measured frequency response Subsequently, the governing equations for crack magnitude estimations are given below Suppose that frequency response  ( x ,  ) of beam subjected to a moving harmonic force P(t ) is measured at the positions ( xˆ1 , , xˆ m ) on beam This implies that we have got the 21 data f j ( )   ( xˆ j ,  ), j  1, , m together with load given in the time domain P(t ) Using Eq (4.2.7) one obtains A()μ  b() , (4.4.1) A( )  [ jk ( ), j  1, ,m; k  1, ,n]; b( )  {b j ( )  f j ( )   j ( ), j  1, ,m} (4.4.2) where { j ()   ( x j , );  jk ()   k ( x j , e, ), j  1, ,m; k  1, ,n} Applying the Tikhonov regularization method for Eq (4.1.1) one is able to evaluation crack magnitudes that are shown in Figs 4.10-4.12 and listed in Table 4.2 Table 4.2 Results of crack detection in dependence on the measurement noise level Noise Actual levels crack depth 5% 10% 0% 15% 20% 30% 5% 10% 5% 15% 20% 30% 5% 10% 10% 15% 20% 30% 5% 10% 15% 15% 20% 30% Actual crack positions 1st crack 4.96 (0.80) 9.92 (0.80) 14.90 (0.66) 19.87 (0.65) 29.88 (0.40) 4.96 (0.80) 9.94 (0.60) 14.90 (0.66) 19.89 (0.55) 29.69 (1.03) 4.99 (0.02) 9.99 (0.01) 15.16 (1.06) 20.08 (0.40) 30.45 (1.50) 5.09 (1.80) 10.15 (1.50) 15.12 (0.80) 20.16 (0.80) 30.54 (1.80) 5m Estimated crack depth, % (error, %) 2nd crack 3rd crack 4th crack 4.97 (0.60) 4.99 (0.20) 5.00 (0.00) 9.94 (0.60) 9.98 (0.20) 10.00 (0.00) 14.90 (0.66) 14.98 (0.13) 15.01(0.06) 19.92 (0.40) 19.97 (0.15) 20.00(0.00) 29.94 (0.50) 30.02 (0.15) 30.03(0.10) 5.00 (0.00) 5.11 (2.50) 5.04 (0.80) 10.01 (0.10) 10.26 (2.60) 10.05 (0.50) 15.05 (0.33) 15.40 (2.60) 14.98(0.13) 20.08 (0.40) 20.44 (2.20) 20.10 (0.50) 30.31 (1.03) 30.64 (3.13) 30.11 (3.30) 5.09 (1.80) 5.19 (3.80) 4.89 (2.20) 10.15 (1.50) 10.31 (3.10) 9.78 (2.20) 15.30 (2.00) 15.40 (2.60) 14.68(2.13) 20.49 (2.45) 20.55 (2.75) 19.52(2.40) 30.37 (1.23) 30.47 (3.07) 29.45(3.33) 5.09 (1.80) 5.19 (3.80) 4.78 (4.40) 10.21 (2.10) 10.31(3.10) 9.67 (3.30) 15.44 (2.93) 15.44 (2.93) 14.46 (3.60) 20.49 (2.45) 20.75 (3.75) 19.19(4.05) 30.91 (3.03) 30.82 (2.73) 28.93(3.56) 10m 15m 20m 5th crack 4.98 (0.40) 9.96 (0.40) 14.94(0.40) 19.92(0.40) 29.91(0.30) 4.27 (14.60) 8.49 (15.10) 12.83 (14.50) 17.10 (14.50) 26.03 (13.20) n/a n/a n/a n/a n/a n/a n/a n/a n/a n/a 22.5m 22 -3 Corrected crack detection with f = 0.9*f1 x 10 Corrected crack magnitude 0 10 15 Scanning crack position 20 25 Fig 4.10 Results of crack detection for load frequency 0.9ω1 Fig 4.11 Results of crack detection for load speed 0.5Vc Fig 4.12 Results of crack detection in the case of resonance 23 Concluding remarks for Chapter In the present Chapter, general theory of vibration of cracked beam subjected to arbitrary moving force is presented A novel method for calculating natural frequencies of multispan continuous beam with arbitrary number of cracks is proposed as an illustrating example of the theory application Frequency response of cracked beam subject to moving harmonic force is thoroughly investigated versus load parameters such as speed, frequency and crack parameters A procedure is proposed for crack identification by measurements of frequency response to moving harmonic force and it is validated by a numerical example The obtained results demonstrate that the frequency response to moving harmonic force is an efficient indicator for detecting multiple cracks in beam GENERAL CONCLUSION The major results obtained in the thesis are as follow: Using the spectral method an explicit expression for frequency response of multiple cracked beam subjected to concentrated harmonic force moving with constant speed has been conducted Based on the exact solution for frequency response, various vibration modes are identified versus speed of the load Namely, for the speed less than 1/10 the critical speed the response is governed by the vibration mode of load frequency (forced mode) and for the speed exceeding 1/3 critical one the vibration mode of natural frequency (eigenmode) is predominated For the speed between 1/10 24 and 1/3 critical speed, vibration mode of the response is a combination of the forced and eigenmodes of vibration The vibration mode of driving frequency is insignificant in comparison with the forced and eigen vibration modes for every speed of the load A novel method is proposed to calculate natural frequencies of continuous multispan beam with cracks that ignores the calculating friction at the supports There have been analyzed the combined effect of load parameters such as speed, frequency and crack parameters (depth, location and quantity of cracks) on the frequency response of beam This is an important indication for identification of cracks from measured frequency response of beam to moving load An efficient procedure is developed for crack identification in beam by frequency response to moving harmonic force and validated by a numerical examination 25 LIST OF THE AUTHOR’S PUBLICATIONS Nguyen Tien Khiem, Phi Thi Hang (2014) Spectral analysis of multiple cracked beam subjected to moving load Vietnam Journal of Mechanics 35(4): 245-254 Nguyen Tien Khiem, Pham Xuan Khang, Phi Thi Hang (2014) A method for calculating natural frequencies of multispan beam with arbitrary number of cracks Mechanics Nationwide Conference The 35th anniversary of the Institute of Mechanics, Ha Noi, 09- 04- 2014, Hanoi 2014 Vol I, pp 87-92 P.T Hang and N.T Khiem (2015) Frequency response of multiple cracked beam to moving harmonic load Mechanics Nationwide Conference, Danang 2015 (accepted) Nguyen Tien Khiem, Phi Thi Hang, Le Khanh Toan (2015) Crack detection in pile by measurement of frequency response function Journal of Nondestructive Testing and Evaluation, Online First 15 Sept 2015 DOI:10.1080/10589759.2015.1081904 N.T Khiem and P.T Hang Frequency response of a beam-like structure to harmonic forces Vietnam Journal of Mechanics (Submitted 2015) Nguyen Tien Khiem, Phi Thi Hang Analysis and identification of multiple cracked beam subjected to moving harmonic load Journal of Vibration and Control (Submitted 2015)