Spectral method for vibration analysis of cracked beam subjected to moving load

conclusion. 3 Chapter 1 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 2 presents the methodology for spectral analysis of cracked beam subjected to moving force. Chapter 3 provides an exact solution in frequency domain of the moving load problem for intact beam and frequency response is thoroughly examined. Chapter 4 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 1. 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 )]([)( ),(),(),( 02 2 4 4 txxtP t txw F t txw F x txw EI           ; (1.1.1) (1.1.1) )]([)()()( tygmtkztzcmgtP   ; ],)([)(;)]()([)(;)()()()( 0000 ttxwtwtwtytztwmtkztzctzm   . In the latter equations ),( txw is the transverse displacement of beam, )(ty - vertical displacement of mass; )(0 tx is position of mass on the beam measured from the left end; )(t is delta Dirac function. From the given system the 4 following problems can be obtained for dynamic analysis of beam: 1. The moving force problem, when the force )(tP is known, for instance, }exp{)( 00  tPtP ; 2. The moving mass problem if )]([ 0 twgmP(t)  ; 3. The moving vehicle problem when Eq. (1.1) are solved for both the beam and vehicle. 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  w0 E, I, , F w(x,t) x x0 c v m v)( 0 tx k 5 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: 6 (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 7 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 2. METHODOLOGY 2.1. Frequency response Let’s consider vibration of an Euler-Bernoulli beam described by the equation ),( ),(),(),(),( 22 2 4 5 14 4 txp t txw t txw F tx txw x txw EI                        , where ),( txw is transverse deflection of the beam at section x; E, I, F, ρ, L - the beam’s material and geometry constants and 21, are damping coefficients. Under the Fourier transformation, the equation leads to ),(),( ),( 4 4 4   xQxW dx xWd  , ;/)( 21 24 EIiF   (2.1.1) ;),(),(; ),( ),(;),(),(         dtetxpxP EI xP xQdtetxwxW titi     )1/()/();1(/1 21212 2 1211   . The so-called frequency response ),( xW determined from Eq. (2.1.1) must satisfy boundary conditions. The frequency 8 response is complex function of frequency  , ),(),(),(  xiIxRxW ww  , the module of which ),(),(),(),( 22  xIxRxWxS www  , (2.1.2) is the frequency-amplitude characteristic of beam subjected to arbitrary load ),( txp . The function ),( xSw 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 0 is called deflection diagram of frequency 0 . Content of the frequency response method applied for moving load problem is first to solve Eq. (2.1.2) for a given moving load ),( txp . 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 )v()(),( txtPtxp   has the frequency-amplitude characteristic v/)/()v()(),( v/ EIevxPdtetxtPxQ xiti        (2.2.1) and general solution of Eq. (2.1.1) is represented as ),(),(),( 10  xxxW  (2.2.2) 0),(/),( 0 44 0 4   xdxxd   x dssQsxhx 0 1 ),()(),(  ; 32/)sin(sinh)(  xxxh  . Subsequently, solution (2.2.2) can be expressed in the form 3,2,1),,()()(),( 121  rxxDLxCLxW  (2.2.3) with )(),( 21 xLxL 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 9 )()()()( )(),()(),( )( 2 )( 1 )( 2 )( 1 )( 2 )( 1 )( 2 )( 1 1111 1111     pqqp qppq LLLL LL C    ; )()()()( )(),()(),( )( 2 )( 1 )( 2 )( 1 )( 1 )( 1 )( 1 )( 1 1111 1111     pqqp pqqp LLLL LL D    . (2.2.4) 2.3. Tikhonov regularization method A lot of problems in science and engineering leads to solve the equation ,bAx  (2.3.1) (2.3.1) where A is a matrix of arbitrary dimension and singularity and b is a vector that is known as an approximation of vector .b The conventional methods are inapplicable for such the system. A. N. Tikhonov proposed the so-called regularization method that suggests regularizing the Eq. (2.3.1) by 0LxLbAx)LLAA( TTTT α (2.3.2) with a prior solution 0x and regularizing matrix L and factor α . Finally, regularized solution is calculated by .vxv bux xˆ 1 0 1 2 0 k n rk kk r k k T kkk                    (2.3.3) Concluding remarks for Chapter 2 In this Chapter, the concept of frequency response of a structure subjected to a moving load is defined that provides basic instrument for developing the so-called frequency response method to spectral analysis of beam under moving force. Also, the Tikhonov’s regularization method is shortly described with the aim to use for solving the crack detection by measurement of frequency response to moving force. 10 Chapter 3. FREQUENCY RESPONSE OF BEAM SUBJECTED TO MOVING HARMONIC FORCE 3.1. Vibration of beam under constant moving force For convenience, the following dimensionless parameters are used: 1//  vcVv  - speed parameter (that is ratio of actual speed to the critical speed  /1cV ); ]2,0[/ 1   - frequency parameter (the ratio of frequency to the fundamental frequency of beam); /vv   is so- called driving frequency. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.5 1 1.5 2 2.5 Frequency/fundamental N o rm a li s e d a m p li tu d e v =0.40 0.50 0.04 0.20 0.25 0.30 0.10 Fig. 3.1. Response spectrum in dependence on the load speed Fig. 3.2. Eigenmode amplitude in dependence on load speed 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 dimensionless frequency M ids pa n de fle ct ion a m pli tu de Fig. 3.3. Response spectrum at the anti-resonant speeds 11 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Dimensionless frequency N or m al iz ed m id sp an d ef le ct io n Fig. 3.4. Response spectrum for harmonic load with 14.0  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 14.0  . 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. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Dimensionless frequency N or m al iz ed m id sp an d ef le ct io n Fig. 3.5. Response spectrum for harmonic load 14.0  at anti-resonant speeds. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 Load frequency factor S pe ed fa ct or k=1 k=2 k=3 k=4 k=5 k=6 k=10 k=15 k=20 k=30 Fig. 3.6. The map of anti-resonant speed in dependence of load frequency. 13 Concluding remark for Chapter 3 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 4 VIBRATION OF CRACKED BEAM SUBJECTED TO MOVING FORCE 4.1. Free vibration of cracked beam 14 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 nje j ,...,1,  as shown in Fig. 4.1. The crack is modeled by an equivalent spring of stiffness ),...,1(0 njK j  that is calculated from crack depth ),...,1( nja j  accordingly to the fracture mechanics theory. Free vibration of such the beam is described by the equation ),1,0(,0)()( 4)(  xxxIV  4 2 / EIFL   (4.1.1) everywhere in the beam except beam’s boundaries where the conditions must be satisfied 0)1()1(,0)0()0( )()( )()( 00  qpqp (4.1.2) and cracked sections where it is satisfied the condition ).0()0()0();0()0( );0()0();0()0(     jjjjjj jjjj eeeee eeee  (4.1.3) For the beam natural frequencies are seeking from the equation   ,0)I)()e,(B)γ(Γdet(),,( 0   Lef (4.1.4) L e1 ej aj a1 x y h b E, , F b h K0j 15  ,,...,)γ(Γ 1 ndiag  ],...,1,),,([),(B nkjeebbe kjjk   and mode shapes are determined as )(/),,(),,(,),,(),()( 0 1 kjkjk n j kjjkkk Lexexexxx     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 Freq.1 Freq.2 Freq.3 Freq.4 Freq. 5 Freq.6 Uncracked Eq.(4.1.4) 3.1416 3.9266 6.2832 7.0686 9.4248 10.2102 Ref.[36] π 3.9266 2π 7.0685 3π 10.2101 Span 1 Span 2 Eq. (4.1.1) uncracked 1.2 1.8 3.1056 3.9266 6.2395 7.0686 9.3911 10.2101 0.5 1.2 1.8 3.1056 3.7753 6.2395 7.0190 9.3911 9.7954 0.2 0.8 1.2 1.8 3.1056 3.7878 6.2395 6.6617 9.3911 9.5124 0.2 0.8 1.5 3.1157 3.7878 6.2832 6.6617 9.4270 9.5124 0.2 0.8 uncracked 3.1416 3.7878 6.2832 6.6617 9.4248 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 )]v[)( ),(),(),( 2 2 4 4 txtP t txw F t txw F x txw EI           (4.2.1) After Fourier transform the latter equation becomes ),(),( ),( 4 4 4   xQx dx xd  ; (4.2.2) General solution of Eq. (4.2.2) is   x dssQsxhxx 0 0 ),()(),(),(  , (4.2.3) ]sin)[sinh2/1()( 3 xxxh   ; 16 0),( ),( 0 4 4 0 4    x dx xd . (4.2.4) It was proved that free vibration of cracked is represented by    n k kk exKxLx 1 00 )(),(),(  (4.2.5)     1 1 0 ])(),([ j k kjkjjj eeSeL  . (4.2.6) So that after application of boundary condition for solution (4.2.3), (4.2.5) one obtains ),,,(),(),( 1 0  exxx k n k k   , (4.2.7) ),(),(),(),( 120100  xxLDxLCx  ; nkexKxLDxLCx kkkk ,...,1),(),(),(),( 21   . In the case of ti eePtP 0)(  one has e xivxi eQeEIPxQ     ˆ,v)/(),( v/ ˆ 0 /ˆ 0 ]v)/ˆ(/[)(),( 44v/ ˆ 0101     xieQxx ; xPxPxPxPx  sin)(cos)(sinh)(cosh)()( 432110  . 4.3. Influence of crack on frequency response of cracked beam For illustration, there is considered the beam of the following constants: m25 , 225.05.0 mhbF  , 3/7850,200 mkgMPaE   with various scenarios of cracks. Since the frequency response is a complex function, the following variations of the function are calculated 0( , ) ( , ) ( , )a cS x x x      , 0( , ) ( , ) ( , )m cS x x 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 ce Vf /v,/,/ 11   , where 1 - fundamental 17 frequency;  - load frequency; 1 /cV L  - critical speed of load. The frequency response variations are investigated in the frequency range from 0 to 12 , i. e. ]2,0[),2,0(  ef 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. 0.9 0.95 0.986 1 1.05 1.1 -1 -0.5 0 0.5 1 1.5 Dimensionless frequency S p e ct ru m d e vi a tio n v=0.1 v=0.2 v=0.3 v=1.0 v=0.5 v=0.4 (a) - Resonant frequency of load Fig. 4.2. Variation of response spectrum due to cracks for different load speed 0.9 0.92 0.94 0.96 0.986 1 1.02 1.04 1.06 1.08 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 S pe ct ru m d ev ia tio n Dimensionless frequency fe=0.4 &1.6 fe=0. & 2.0 fe=1.0fe=0.7 &1.3 fe=0.6 &1.4 fe=0.5 &1.5 fe=0.3 &1.7 fe=0.8 &1.2 fe=0.2 &1.8 fe=0.1 &1.9 (b) - speed=0.5 Fig. 4.3. Variation of response spectrum due to cracks for different load frequency 18 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.5 0 0.5 1 1.5 2 Dimensionless speed S pe ct ru m d ev ia tio n 1.2 0.7 0.6 1.4 fe=0 &2,0 omega=0.986 fe=1.0 0.9 1.1 0.8 1.3 Fig. 4.4. Variation of eigenmode amplitude versus load parameter 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Dimensionless frequency M ag ni tu de o f d ev ia tio n e=12 &13 e=9 &16 e=8 &17 e=7 &18 e=6&19 e=5 &20 e=10 &15 e=11&14 e=11& 14 e=11& 14 e=4 & 21 e=3 & 22 e=2 &23 e=1 &24 Fig 4.5. Variation of response spectrum vers crack parameter 0 5 10 L/2 15 20 25 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Span position F R d e vi a tio n m a g n itu d e e=14.5e=13.5 e=12.5 15.5 16.59.5 8.5 17.5 7.5 18.5 6.5 19.5 20.5 5.5 4.5 21.5 22.53.5 10.5 Fig. 4.6. Variation of vibration diagram vers. Crack position 19 0 5 10 L/2 15 20 25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Crack position FR d ev ia tio n m ag ni tu de v=0.1 v=0.2 v=0.3 v=0.4 v=0.5 v=1.0 Load frequency =omega1 Fig. 4.7. Variation of response spectrum vers. load speeds 0.9 0.95 1 1.05 1.1 1.15 0 0.2 0.4 0.6 0.8 1 1.2 Dimensionless frequency F R d e vi a tio n m a g n itu d e Number of cracks = 9 5 4 Number of cracks = 1 3 2 (b) - fe=1,v=0.5 8 7 6 Fig. 4.8. Variation of response spectrum vers. amount of cracks 0 5 10 15 20 25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Span location F R d e vi a tio n m a g n itu d e 9 cracks 7 cracks 6 cracks 8 cracks 5 cracks 4 cracks 3 cracks 2 cracks 1 cracks 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 nee ,...,1 ; (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 ),...,( 1 nγ 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 )(tP is measured at the positions )ˆ,...,ˆ( 1 mxx on beam. This implies that we have got the 21 data mjxf jj ,...,1),,ˆ()(   together with load given in the time domain )(tP . Using Eq. (4.2.7) one obtains )()(  bμA  , (4.4.1) where },...,1),()()({)( ];,...,1;,...,1),([)( 0 mjfb nkmj jjj jk     b A (4.4.2) },...,1;,...,1),,,()();,()({ 00 nkmjxx jkjkjj   e . 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 levels Actual crack depth Estimated crack depth, % (error, %) 1st crack 2nd crack 3rd crack 4th crack 5th crack 0% 5% 4.96 (0.80) 4.97 (0.60) 4.99 (0.20) 5.00 (0.00) 4.98 (0.40) 10% 9.92 (0.80) 9.94 (0.60) 9.98 (0.20) 10.00 (0.00) 9.96 (0.40) 15% 14.90 (0.66) 14.90 (0.66) 14.98 (0.13) 15.01(0.06) 14.94(0.40) 20% 19.87 (0.65) 19.92 (0.40) 19.97 (0.15) 20.00(0.00) 19.92(0.40) 30% 29.88 (0.40) 29.94 (0.50) 30.02 (0.15) 30.03(0.10) 29.91(0.30) 5% 5% 4.96 (0.80) 5.00 (0.00) 5.11 (2.50) 5.04 (0.80) 4.27 (14.60) 10% 9.94 (0.60) 10.01 (0.10) 10.26 (2.60) 10.05 (0.50) 8.49 (15.10) 15% 14.90 (0.66) 15.05 (0.33) 15.40 (2.60) 14.98(0.13) 12.83 (14.50) 20% 19.89 (0.55) 20.08 (0.40) 20.44 (2.20) 20.10 (0.50) 17.10 (14.50) 30% 29.69 (1.03) 30.31 (1.03) 30.64 (3.13) 30.11 (3.30) 26.03 (13.20) 10% 5% 4.99 (0.02) 5.09 (1.80) 5.19 (3.80) 4.89 (2.20) n/a 10% 9.99 (0.01) 10.15 (1.50) 10.31 (3.10) 9.78 (2.20) n/a 15% 15.16 (1.06) 15.30 (2.00) 15.40 (2.60) 14.68(2.13) n/a 20% 20.08 (0.40) 20.49 (2.45) 20.55 (2.75) 19.52(2.40) n/a 30% 30.45 (1.50) 30.37 (1.23) 30.47 (3.07) 29.45(3.33) n/a 15% 5% 5.09 (1.80) 5.09 (1.80) 5.19 (3.80) 4.78 (4.40) n/a 10% 10.15 (1.50) 10.21 (2.10) 10.31(3.10) 9.67 (3.30) n/a 15% 15.12 (0.80) 15.44 (2.93) 15.44 (2.93) 14.46 (3.60) n/a 20% 20.16 (0.80) 20.49 (2.45) 20.75 (3.75) 19.19(4.05) n/a 30% 30.54 (1.80) 30.91 (3.03) 30.82 (2.73) 28.93(3.56) n/a Actual crack positions 5m 10m 15m 20m 22.5m 22 0 5 10 15 20 25 0 1 2 3 4 5 6 7 8 9 x 10 -3 Scanning crack position C o rr e c te d c ra c k m a g n it u d e Corrected crack detection with f = 0.9*f1 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 t