The transfer matrix method for modal analysis of cracked multistep beam - Vu Thi An Ninh

Vietnam Journal of Science and Technology 55 (5) (2017) 598-611 DOI: 10.15625/2525-2518/55/5/9140 THE TRANSFER MATRIX METHOD FOR MODAL ANALYSIS OF CRACKED MULTISTEP BEAM Vu Thi An Ninh1, Luu Quynh Huong2, Tran Thanh Hai3, Nguyen Tien Khiem3, * 1University of Transport and Communications, 3 Cau Giay, Dong Da, Hanoi 2Thuy Loi University, Chua Boc, Dong Da, Hanoi 3Graduate University of Science and Technology, VAST, 18 Hoang Quoc Viet, Cau Giay, Hanoi *Email: ntkhiem@imech.vast.vn; khiemvch@gmail.com Received: 9 January 2017; Accepted for publication: 28 May 2017 ABSTRACT The present study addresses the modal analysis of multistep beam with arbitrary number of cracks by using the transfer matrix method and modal testing technique. First, there is conducted general solution of free vibration problem for uniform beam element with arbitrary number of cracks that allows one to simplify the transfer matrix for cracked multistep beam. The transferring beam state needs to undertake only at the steps of beam but not through crack positions. Such simplified the transfer matrix method makes straightforward to investigate effect of cracks mutually with cross-section step in beam on natural frequencies. It is revealed that step-down and step-up in beam could modify notably sensitivity of natural frequencies to crack so that the analysis provides useful indication for crack detection in multistep beam. The proposed theory was validated by an experimental case study. Keywords: stepped beam; cracked beam; modal analysis; transfer matrix method. 1. INTRODUCTION Stepped beam structures have found widespread application in engineering fields such as bridges, rotating machines, robotics and aerospace structures. In the engineering application, vibration of the structures is the problem of a great importance and it is studied in the enormous literature. Sato [1] 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 Transfer Matrix Method (TMM) combined with Finite Element Method (FEM) 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 equal or greater than 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. Latter, Jang and Bert [2, 3] used the conventional technique for calculating natural frequencies of two-step beam and shown that natural frequencies of the structure are dependent not only on the change in The transfer matrix method for modal analysis of cracked multistep beam 599 cross-section but also on the beam boundary conditions. Namely, stepping up (increasing height) loads to increasing natural frequencies for any boundary conditions except the clamped ends beam and stepping down (decreasing height) reduces the frequencies except the cantilevered beam. The findings are important to show the dynamic property of stepped beam and which method could be useful for vibration analysis of the beam. Other methods such as Adomian Decomposition Method (ADM) and Differential Quadrature Element Method (DEM) have been developed in [4] and [5], respectively, for free vibration analysis of multi-step beams. Cunha and Junior [6] investigated effect of elastic boundary supports on natural frequencies and mode shapes of multiple stepped beam. Kukla and Zamojska [7] studied effect of axial force on natural frequencies and longitudinal or torsional vibrations of stepped one-dimensional structures such as bars or shafts were studied in [8] by using Distributed Transfer Function Method (DTFM). Jaworski and Dowell [9] have compared different theoretical methods and beam theories used for free vibration analysis of multistep cantilever beam with experimental results. It was shown by the authors that there is disaggreement between theoretical and experimental results. Wattanasakulpong and Charoensuk [10] studied one-step beam made of functionally graded material. Vibration of stepped beam structures with cracks have been also intensively examined due to that cracks are potential to reduce the serviceability of a structure and in consequence may lead to a serious accidence if it is not early detected. To detect cracks in a structure its vibration analysis is crucially important. Nandwana and Maiti [11] have established frequency equation of an n-step Euler-Bernoulli beam with single crack in a form of 4(n+1) order determinant and used for crack detection by natural frequencies. Using TMM, Tsai and Wang [12] obtained frequency equation for cracked multistep Timoshenko beam in much simplified form of 4x4-dimention determinant that simplifies significantly computation of the beam’s natural frequencies. Maghsoodi et al. [13] have obtained an explicit expression of natural frequencies through the crack magnitudes for multistep Euler-Bernoulli beam that provide a system of linear algebraic equations for crack detection from natural frequencies. Li [14] was able to conduct a recurrent relationship between vibration modes of adjacent steps that is straightforward to obtain an explicit expression of frequency equation for multiple cracked and stepped beam. The TMM is completely developed and used for solving both the forward and inverse problems for multistep Euler-Bernoulli beam with arbitrary number of cracks by Attar in [15]. However, in the latter publication the transfer matrix is very complicated because it should be assembled not only at the steps of beam but also over the crack sections. This paper presents the TMM developed for modal analysis of cracked multistep beam based on an explicit expression for mode shape of multiple cracked uniform beam element. This enables to much simplify the transfer matrix of multiple cracked multistep beam compared to that was developed in [15] and it is validated by an experimental study. 2. GENERAL SHAPE FUNCTION FOR MULTIPLE CRACKED BEAM ELEMENT Consider a uniform beam element of length L; material density (ρ); elasticity (E) and shear (G) modulus; section area hbA ×= and moment of inertia 12/3bhI = . Assume furthermore that the beam is cracked at the positions Lee n <<<< ...0 1 and the cracks of depth naa ,...,1 are modeled by equivalent springs of stiffness nKK ,...,1 . The springs stiffness is calculated from the crack depths using formulas given in Appendix. For the beam element, free vibration is governed by equation Vu Thi An Ninh, Luu Quynh Huong, Tran Thanh Hai, Nguyen Tien Khiem 600 4 4 4 2 4( ) / ( ) 0, / ,d x dx x A EIϕ λ ϕ λ ρ ω− = = (2.1) that is solved under the conditions at the crack positions )0()0( −=+ jj ee φφ ; )0()0();()0()0( −′′′=+′′′′′=−′′=+′′ jjjjj eeeee φφφφφ ; ( 0) ( 0) ( ) ,j j j je e eϕ ϕ γ ϕ′ ′ ′′+ = − + (2.2) where njKEI jj ,...,2,1,/ ==γ called hereby magnitudes of the cracks. Introducing Krylov’s functions ,2/)sin(sinh)(;2/)cos(cosh)( ;2/)sin(sinh)(;2/)cos(cosh)( 0403 0201 xxxLxxxL xxxLxxxL λλλλ λλλλ +=−= −=+= (2.3) that are all continuous particular solutions of Eq. (2.1), we can prove that the functions ∑ = =−+= n j jkjkk kexKxLxL 1 0 ,4,3,2,1),()()( µ (2.4) where ( ) ( ) ( ) : 0; sinh sin( ) ( ) , 0,1,2,3, 20 : 0; p p S x x x xK x S x p x λ λ λ  ≥ + = = = < (2.5) p is derivative order and so-called damage parameters 4,3,2,1;,...,2,1, == knjkjµ are defined as ∑ − = =−′′+′′= 1 1 0 ,4,3,2,1),()([ j i ijkijkjkj keeSeL µγµ (2.6) are solutions of Eq. (2.1) satisfying also conditions (2.2). Since functions (2.3) and function )(xS defined in (2.5) are continuous solutions of Eq. (2.1), the functions (2.4) would satisfy also the equation except crack positions where they need to satisfy conditions (2.2). Indeed, since (0) (0) (0) 0; (0) 1,S S S S′′ ′′′ ′= = = = (2.7) one has got )0()()0()()0()0( 1 1 0 1 0 −=−+−=−++=+ ∑∑ − == jk j i ijkijk j i ijkijkjk eLeeSeLeeSeLeL µµ ; )0()()0()()0()0( 1 1 0 1 0 −′′=−′′+−′′=−′′++′′=+′′ ∑∑ − == jk j i ijkijk j i ijkijkjk eLeeSeLeeSeLeL µµ ; )0()()0()()0()0( 1 1 0 1 0 −′′′=−′′′+−′′′=−′′′++′′′=+′′′ ∑∑ − == jk j i ijkijk j i ijkijkjk eLeeSeLeeSeLeL µµ ; ).0()0(])()([)0( )()0()()0()0( 1 1 0 1 1 0 1 0 − ′′+−′=−′′+′′+−′= =+−′+−′=−′++′=+′ ∑ ∑∑ − = − == jkjjk j i ijkijkjjk kj j i ijkijk j i ijkijkjk eLeLeeSeLeL eeSeLeeSeLeL γµγ µµµ Thus, general solution of Eq. (2.1) satisfying conditions (2.2) can be found in the form )()()()()( 44332211 xLCxLCxLCxLCx +++=φ (2.8) The transfer matrix method for modal analysis of cracked multistep beam 601 where functions 4,3,2,1),( =kxLk are determined in (2.4)-(2.6) and 4321 ,,, CCCC are arbitrary constants would be found using boundary conditions for the beam. Using the expression (2.8) one is able to calculate displacement, slope, moment and shear force respectively as follows 44332211 )()()()()()( CxLCxLCxLCxLxxW +++=≡ φ ; 44332211 )()()()()()( CxLCxLCxLCxLxx ′+′+′+′=′≡Θ φ ; 1 1 2 2 3 3 4 4( ) ( ) ( ) ( ) ( ) ( ) ;M x EI x EIL x C EIL x C EIL x C EIL x Cϕ′′ ′′ ′′ ′′ ′′≡ = + + + (2.9) 443322111 )()()()()()( CxLEICxLEICxLEICxLEIxEIxQ ′′′+′′′+′′′+′′′=′′′≡ φ . that can be rewritten in the matrix form { ( )} [ ( )] { },x x= ⋅V H C (2.10) where vectors TxQxMxxWx )}(),(),(),({)}({ Θ=V , TCCCC },,,{}{ 4321=C and matrix             ′′′′′′′′′′′′ ′′′′′′′′ ′′′′ = )()()()( )()()()( )()()()( )()()()( )]([ 4321 4321 4321 4321 xLEIxLEIxLEIxLEI xLEIxLEIxLEIxLEI xLxLxLxL xLxLxLxL xH . (2.11) This representation (2.10) of beam state will be employed below to develop the transfer matrix method for multiple cracked stepped beam. 3. TRANSFER MATRIX FOR STEPPED BEAM WITH MULTIPLE CRACKS Let’s consider now a stepped beam composed of m uniform beam segments designated with subscript j, j=1,2,,m. Namely, material and geometry constants of j-th beam segment are denoted by jjjjj LhbE ,,,, ρ . Suppose that each of the beam segments contains a number ( jn ) of cracks represented by its position jji nie ,...,1, = and magnitude /ji j j jiE I Kγ = and a crack of magnitude /j j j jE I Kα = occurs at joint of (j+1)-th and j-th segments. Introduce state vector for j-th step as { }Tjjjjj xQxMxxWx )(),(),(),()( Θ=V defined in (2.9) symbolized with subscript j. Therefore, continuity conditions at step joints are 1 1 1 1(0) ( ); (0) ( ) ( ); (0) ( ); (0) ( )j j j j j j j j j j j j j j jW W L L M L M M L Q Q Lα+ + + += Θ = Θ + = = or 1(0) ( ) ( ), 1,2,..., ,j j j jL j mα+ = Γ =V V (3.1) where / 1 /j j j j jE I Kα γ= = and 1 0 0 0 0 1 0( ) 0 0 1 0 0 0 0 1 α α      Γ =       . (3.2) Vu Thi An Ninh, Luu Quynh Huong, Tran Thanh Hai, Nguyen Tien Khiem 602 On the other hand, using representation (2.10) the introduced state vector )(xjV could be now rewritten as ( ) [ ( )] ,j j jx x=V H C (3.3) where               ′′′′′′′′′′′′ ′′′′′′′′ ′′′′ = )()()()( )()()()( )()()()( )()()()( )]([ 4321 4321 4321 4321 xLIExLIExLIExLIE xLIExLIExLIExLIE xLxLxLxL xLxLxLxL x jjjjjjjjjjjj jjjjjjjjjjjj jjjj jjjj jH . (3.4) The shape functions )(xL jk defined in (2.4) where beam constants are replaced by those with subscript j. Specifically, the frequency parameter λ defined in (2.1) now is 4/12 )/( jjjjjj IEhbρωλ = . It is easily to show that expression (3.2) yields ( ) ( ) (0)j j jL j=V T V ; 1( ) ( ) (0)j j jj L −=T H H . (3.5) So, combining the relationship (3.5) with (3.1) for j =1, 2, , m one obtains finally 1( ) [ ]{ (0)};m mL =V T V (3.6) 1 1[ ] [ ( ) ( ) ( 1)... ( ) (1)]mm mα α−= Γ − ΓT T T T . (3.7) Usually, conventional boundary conditions are expressed by 0 1 L{ (0)} 0; { ( )} 0,m mL= =B V B V (3.8) where 0 , LB B are matrices of 2 4× dimension. For instance, if both ends of the beam are clamped the boundary matrices get the form 0 0 1 0 0 0 1 0 0 0 ; 0 1 0 0 1 0L Lα α     = =       B B . with 0 , Lα α being magnitudes of possible cracks at the end clamps. Consequently, 1[ ( )] (0) 0,ω =B V (3.9) where       = TB B B L 0)(ω . (3.10) Eq. (3.5) would have nontrivial solution with respect to )0(1V under the condition ( ) det[ ( )]) 0,D ω ω≡ =B (3.11) that is frequency equation desired for the stepped beam with cracks. For instance, if the left end of beam is clamped and the other one is free, i. e. the beam is cantilevered, the boundary conditions are 0)()()0()0( 11 ===Θ= mmmm LQLMW . Therefore, the frequency equation (3.10) gets to be The transfer matrix method for modal analysis of cracked multistep beam 603 33 34 33 44 43 34 43 44 ( ) det 0,CF T T D T T T T T T ω   ≡ = − =    (3.12) where 4,3,2,1,, =kiTik are elements of the total transfer matrix [T] defined in (3.4). Similarly, frequency equation of stepped FGM beam can be obtained as determinant of a 2x2 matrix for other cases of boundary conditions such as simple supports or clamped ends. Namely, for simply supported beam with 0)()()0()0( 11 ==== mmmm LMLWMW , frequency equation is 0det)( 14323412 3432 1412 =−=      ≡ TTTT TT TT DSS ω . (3.13) For beam with clamped ends where 0)()()0()0( 11 =Θ==Θ= mmmm LLWW , one has got 0det)( 14232413 2423 1413 =−=      ≡ TTTT TT TT DCC ω . (3.14) Solving the frequency equations gives rise natural frequencies ,...3,2,1, =kkω of the beam that in turn allow one to find corresponding solution of Eq. (3.8) as 11 )0( VV kD= with an arbitrary constant Dk and normalized solution 1V . Afterward, mode shape corresponding to natural frequency kω is determined for every beam step as follows kjjjjjjjjkjkjk CxLCxLCxLCxLDxWx ωω=+++== })()()()({)()( 44332211Φ ; })]{1()...2()1([)]0([ 11 VTTTHC −−= − jjjj , mj ,...,2,1= . (3.15) The arbitrary constant kD is determined by a chosen normalized condition, for example, 1)(max ),( =xjkjx Φ . Thus, the free vibration problem for stepped beam with multiple cracks is completely solved by the simplified transfer matrix method. 4. EXPERIMENTAL SETUP AND MODAL TESTING TECHNIQUE In this section experimental modal analysis is accomplished for the stepped beam with clamped ends as shown in Fig. 1. Geometry and material parameters of the beam models are given in Table 1. 3210 ; 7855 / ; 0.3E MPa kg mρ ν= = = . Crack is produced by saw cut with very small wide and different depth 0 %, 10 %, 20 %, 30 % and 40 % beam thickness at fixed positions on beam. Therefore, the saw-cut can be treated as an approximate model of open transverse crack described in [16]. Three scenarios of cracked beam are investigated: single crack at position 450 mm; double cracks at the positions 200 mm; 450 mm from the left end and triple cracks at positions 200 mm, 450 mm, 800 mm from the left end. In the first scenario, single crack of various depth (10 – 40 %) is examined. The second scenario is tested with various depth of crack at the first span and the crack at intermediate span Vu Thi An Ninh, Luu Quynh Huong, Tran Thanh Hai, Nguyen Tien Khiem 604 has fixed depth of 40 %. The last crack scenario is carried out for the beam with three cracks of equal depth 40 %. Figure 1. Model of stepped beam used in experimentation. Table 1. Geometrical dimensions and material properties of two-step (three span) beam. Geometrical parameters (mm) Beam spans 1st span 2nd span 3rd span Wide, b 20 20 20 Height, h 15.4 7.5 15.4 Length, L 315 400 315 Total length 1230 Material properties 3.0;/7855;210 3 === νρ mkgMPaE Figure 2. Measurement system PULSE 360. Figure 3. Experimental model with measurement points. 1 2 3 4 5 6 7 L1 L2 L3 L The transfer matrix method for modal analysis of cracked multistep beam 605 The PULSE B&K360 system, Fig. 2a, is employed for gathering and processing measured data. An impact hammer (Fig. 3) is used for generating an excitation at position A denoted by )(ωX in the frequency domain and an accelerometer has been employed for measuring response ( )(ωY ) at the position B on the beam. Hence, the signal processor installed in the measurement system provides Frequency Response Function (FRF) between the positions A, B calculated as ( )( ) ,( ) XY AB XX SH S ω ω ω = (4.1) where )(),( ωω XYXX SS are auto- and cross correlation functions respectively of the signals X and Y. Magnitude of the function (4.1) is shown for instance in Fig. 2b. Multiple measurement of FRF is performed for varying positions of excitation and response and all the measured data gathered should give rise the same modal parameters of testing structure. In the theory of structural vibration, it was shown that the FRF (4.1) can be expressed in term of natural vibration modes as ∑ = +− = n AB i BA H 1 22 ][ )()()( ℓ ℓℓ ℓℓ ωζωω φφ ω . (4.2) where ℓℓ ζω , are natural frequency and damping ratio respectively of mode ℓ and )(A ℓ φ - normalized th−ℓ mode shape measured at position A. Moreover, analysis of the function (4.2) in the frequency domain exhibited that in the case of small damping and sparse distribution of natural frequencies the frequency response function reaches its local maximums at resonant frequencies ...3,2,1,2/ˆ 22 =−= ℓ ℓℓℓ ζωω (4.3) The damping ratio is represented by sharpness of the resonant peak that determined by 2 1 ˆ ˆ( ) / 2 ,ζ ω ω ω= − ℓ ℓ ℓ ℓ (4.4) where 12 , ℓℓ ωω are two frequencies in both sides of ℓωˆ defined by 2/)ˆ()()( 11 ℓℓℓ ωωω ABABAB HHH == . (4.5) So that natural frequencies are determined from the measured data as ...3,2,1,2/ˆˆ 22 =+=∗ ℓ ℓℓℓ ζωω (4.6) and results are given in Table 2 in comparison with the numerically computed ones. 5. RESULTS AND DISCUSSION 5.1. Theoretical validation Note that in the case of uncracked beam the transfer matrix method proposed in section 3 leads to its classical version. This can be validated first by using the method for computing five lowest eigenvalues ( 24 / , 1,2,3, 4,5k kA EI kλ ρ ω= = ) of a uniform beam with clamped ends. Results of the computation compared to those obtained by the classical analytical method (see Table 2) show that the transfer matrix method is really an exact method equivalent to the Vu Thi An Ninh, Luu Quynh Huong, Tran Thanh Hai, Nguyen Tien Khiem 606 analytical one. Moreover, natural frequencies of an intact stepped beam calculated by the classical transfer matrix method are given in the first row of Table 3. The results compared to those obtained by FEM and measured demonstrate the fact that measured natural frequencies are more closed (almost identical for three lower frequencies) to the analytical ones than FEM results. This validates reliability of measured data. Table 2. Eigenvalues of uniform beam calculated by the TMM compared to analytical method [17]. Eigenvalues λ1 λ2 λ3 λ4 λ5 TMM 4.7300 7.8532 10.9956 14.1372 17.2788 Analytical 4.7300 7.8532 10.9956 14.1371 17.2787 5.2. Experimental validation Table 3. Comparison of calculated and measured natural frequencies of three-span stepped beam with clamped ends. Crack scenarios Natural frequencies (Hz) 1 2 3 4 5 Intact beam TMM 73.2781 144.5188 301.1640 529.0126 726.2999 Experiment 73.38 144.30 301.10 526.50 723.81 FEM 74.8296 146.6345 304.9121 530.7197 729.8934 Single crack TMM 72.4574 143.6455 294.4666 519.7713 721.3785 Experiment 72.31 143.70 294.40 517.56 714.88 Double cracks TMM 71.4113 143.0440 287.3652 491.1691 680.994 Experiment 71.63 143.70 290.90 493.06 689.81 Triple cracks TMM 70.9058 142.6157 285.9632 482.6887 671.1254 Experiment 71.06 142.90 287.50 480.06 674.69 Single crack at 450 mm; Double cracks at 200; 450 mm; Triple cracks at 200; 450; 800 mm from the left end of beam and all the cracks are of equal depth 40 %. Measurements of natural frequencies have been performed at 7 points (see Fig. 7) and measured data are processed accordingly to that procedure presented in section 4. In Table 3 there are depicted five lowest natural frequencies calculated and measured for three crack scenarios described in the last row of the table. The results show that discrepancy between calculated and measured natural frequencies is within 2 %. So the theoretical development proposed above in the sections 2 and 3 is thus experimentally validated and it can be surely used for analysis of crack effect on natural frequencies accomplished below. 5.3. Effect of crack position and depth The transfer matrix method for modal analysis of cracked multistep beam 607 For analysis of crack effect on natural frequencies of stepped beam two types of the beam are investigated. The beam of first type called down-stepped (B1S) is shown in Fig. 3 that was examined also in the experiment. The other one has intermediate span of thickness greater than that of the end spans and this type is called up-stepped beam (B2S). Both the types of stepped beam investigated below are clamped at the ends and have the following configurations: 1 2 3 1 2 3 1 3 2 1 2 3 1 2 3 1 3 2 B1S: 1 ; 0.1 ; 0.15 ; 0.10 ; B2S : 1 ; 0.1 ; 0.10 ; 0.15 . L L L m b b b m h h m h m L L L m b b b m h h m h m = = = = = = = = = = = = = = = = = = First, ratios of three lowest natural frequencies (cracked to intact) are computed for the beams with single crack of different depth from 10 % to 40 % and position running from the left to the right ends through the steps. Results are shown in Fig. 4 where the frequency ratios of beam B1 on the left and those of beam B2 on the right. Observing graphs presented in Fig. 4 allows one to make the following notations: (1) Likely to the uniform beam, there exist positions on the stepped beam crack occurred at which does not change a certain natural frequency. Such positions are called frequency node and they are given in Table 4 for uniform and stepped beams. Obviously, natural frequency nodes are located symmetrically about the beam middle for symmetric boundary conditions; (2) The frequency ratios undergo a jump when crack passing beam steps (this means discontinuity of frequency variation due to crack at beam steps). Expanse of the jumps is different for various modes and it is certainly dependent on height of the steps; (3) Natural frequencies, as well known, are monotonically decreasing with growing crack depth. Table 4. Frequency nodes of five lowest modes for uniform and stepped beams. Mode No Stepped beam B1 Uniform beam B0 Stepped beam B2 1 0.85-2.15 0.67-2.33 0.56-2.44 2 0.46-1.5 -2.54 0.4 - 1.5 - 2.6 0.38 - 1.5 - 2.62 3 0.3-1.16-1.84-2.7 0.28-1.07-1.93-2.72 0.27 - 0.94 -2.06 - 2.73 4 0.24-0.94-1.5-2.06-2.76 0.22-0.83-1.5-2.17-2.76 0.21- 0.76 - 1.5 - 2.24- 2.79 5 0.2-0.75-1.26-1.74-2.25- 2.8 0.18-0.68-1.23-1.77-2.32- 2.82 0.17-0.65-1.16-1.84-2.35- 2.83 Total beam length L = 3 m; Span length L1 = L2 = L3 = 1 m; Steps at 1 m and 2 m 5.1. Effect of beam steps and crack position In this subsection, aimed to study effect of steps (abrupt change in beam height), two uniform beams and two stepped beams (with changed both sizes of cross section) of the following geometry are investigated in addition to the stepped beams considered in the previous subsection. 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 BU1: 1 ; 0.1 ; 0.15 ; BU2 : 1 ; 0.1 ; 0.10 ; L L L m b b b m h h h m L L L m b b b m h h h m = = = = = = = = = = = = = = = = = = 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 B3S: 1 ; 0.15; 0.1; 0.15 ; 0.15; 0.1; 0.15 ; B4S : 1 ; 0.1; 0.15; 0.1 ; 0.1; 0.15; 0.10 L L L m b b b m h h h m L L L m b b b m h h h m = = = = = = = = = = = = = = = = = = Vu Thi An Ninh, Luu Quynh Huong, Tran Thanh Hai, Nguyen Tien Khiem 608 As the stepped beams B1S and B2S have uniform width, the beams B3S, B4S are stepped in both sizes of cross section (width and height). The first three frequency ratios computed in dependence on the crack position for the uniform and stepped beams are presented in Fig. 5. 0 0.5 1 1.5 2 2.5 3 0.94 0.95 0.96 0.97 0.98 0.99 1 Crack position Fi rs t f re qu en cy ra tio L1=L2=L3=1m, b1=b2=b3=0.1 h1=h3=0.15;h2=0.1 First Step Second Step a /h = 10% a /h = 20% a /h = 30% a /h = 40% 10% 20% 30% 40% 40% 30% 20% 10% 0 0.5 1 1.5 2 2.5 3 0.955 0.96 0.965 0.97 0.975 0.98 0.985 0.99 0.995 1 L1=L2=L3=1m,b1=b2=b3=0.1m;h1=h3=0.1m,h2=0.15m Crack position Fi rs t f re qu en cy ra tio First Step a /h = 30% a /h = 20% a /h = 10% 10% 20% 20% 10% 40% 40% a /h = 40% Second Step 30%30% 0 0.5 1 1.5 2 2.5 3 0.95 0.955 0.96 0.965 0.97 0.975 0.98 0.985 0.99 0.995 1 L1=L2=L3=1m, b1=b2=b3=0.1 h1=h3=0.15;h2=0.1 Se co n d fre qu en cy ra tio Crack position 10% a /h = 40% a /h = 30% 20% 30% 40%40% First Step Second Step a /h = 40% a /h = 30% 40% 10% 20% 30% 40% 20% 20% 0 0.5 1 1.5 2 2.5 3 0.96 0.965 0.97 0.975 0.98 0.985 0.99 0.995 1 1.005 L1=L2=L3=1m,b1=b2=b3=0.1m;h1=h3=0.1m,h2=0.15m Crack position Se co n d fr eq u en cy ra tio a /h = 40% a /h = 40% 40%40% 30% 10% 10% 20% 10% 20%20% a /h = 30% Second StepFirst Step 10% 30% 20% a /h = 30% 0 0.5 1 1.5 2 2.5 3 0.955 0.96 0.965 0.97 0.975 0.98 0.985 0.99 0.995 1 L1=L2=L3=1m, b1=b2=b3=0.1 h1=h3=0.15;h2=0.1 Th ird fre qu en cy ra tio Crack position Second StepFirst Step a /h = 40% a /h = 40% 30% 20% 10% 40% 40% 10% 20% 30%30% 20% 10% 0 0.5 1 1.5 2 2.5 3 0.955 0.96 0.965 0.97 0.975 0.98 0.985 0.99 0.995 1 L1=L2=L3=1m,b1=b2=b3=0.1m;h1=h3=0.1m,h2=0.15m Th ird fre qu en cy ra tio Crack position a /h = 30% a /h = 40%a /h = 40% 30% 10% 20% 10% 30% 20% 10% 20% First Step Second Step 40% 40% Figure 4. Effect of position and depth of crack on natural frequencies of beams B1 (left) and B2 (right). The transfer matrix method for modal analysis of cracked multistep beam 609 0 1 2 3 0.95 0.955 0.96 0.965 0.97 0.975 0.98 0.985 0.99 0.995 1 Crack position Fi rs t f re qu en cy ra tio B1S B1SB1S B2SB2S B3S Second StepFirst Step BU2 B2S BU1 B1S: b1=0.10;b2=0.10; b3=0.10m h1=0.15;h2=0.10; h3=0.15m B3S: b1=0.15;b2=0.10; b3=0.15m h1=0.15;h2=0.10; h3=0.15m B2S: b1=0.10;b2=0.10; b3=0.10m h1=0.10;h2=0.15; h3=0.10m B4S: b1=0.10;b2=0.15; b3=0.10m h1=0.10;h2=0.15; h3=0.10m B4SB4S B4S BU1: b1=0.10;b2=0.10; b3=0.10m h1=0.15;h2=0.15; h3=0.15m BU2: b1=0.10;b2=0.10; b3=0.10m h1=0.10;h2=0.10; h3=0.10m 0 1 2 3 0.95 0.955 0.96 0.965 0.97 0.975 0.98 0.985 0.99 0.995 1 Se co n d fr eq u en cy ra tio Crack position B2S B2S First Step Second Step BU2: b1=0.10;b2=0.10; b3=0.10m h1=0.10;h2=0.10; h3=0.10m BU1: b1=0.10;b2=0.10; b3=0.10m h1=0.15;h2=0.15; h3=0.15m B2S: b1=0.10;b2=0.10; b3=0.10m h1=0.10;h2=0.15; h3=0.10m B1S: b1=0.10;b2=0.10; b3=0.10m h1=0.15;h2=0.10; h3=0.15m B3S: b1=0.15;b2=0.10; b3=0.15m h1=0.15;h2=0.10; h3=0.15m B1S B1S B3S B4S B2SB2S B4S B3S B3S B3S B1SB1S B4S: b1=0.10;b2=0.15; b3=0.10m h1=0.10;h2=0.15; h3=0.10m BU1 BU2BU2 BU1 B4S B4S 0 1 2 3 0.955 0.96 0.965 0.97 0.975 0.98 0.985 0.99 0.995 1 Th ird fr eq ue nc y ra tio Crack position BU1 First Step Second Step B3S B1S BU2 BU2 BU1 B2S BU1 BU2 B2S B4S B2S B3S B3S BU1BU1 B4S B4S B4S B1S B4S: b1=0.10;b2=0.15; b3=0.10m h1=0.10;h2=0.15; h3=0.10m BU2: b1=0.10;b2=0.10; b3=0.10m h1=0.10;h2=0.10; h3=0.10m B2S: b1=0.10;b2=0.10; b3=0.10m h1=0.10;h2=0.15; h3=0.10m B1S BU1: b1=0.10;b2=0.10; b3=0.10m h1=0.15;h2=0.15; h3=0.15m B1S: b1=0.10;b2=0.10; b3=0.10m h1=0.15;h2=0.10; h3=0.15m B3S: b1=0.15;b2=0.10; b3=0.15m h1=0.15;h2=0.10; h3=0.15m Figure 5. Effect of beam thickness variaton (steps) and crack position on natural frequencies. Vu Thi An Ninh, Luu Quynh Huong, Tran Thanh Hai, Nguyen Tien Khiem 610 Comparing the ratios computed for beam BU1 and BU2 allows one to find that increasing thickness of uniform beam makes all its natural frequencies more sensitive to crack. This highlights the well-known fact that more stiff beam is more sensitive to crack. However, the increasing or decreasing thickness of only mid-span in stepped beam leads to diminish or magnify the second frequency sensitivity to crack occurred at the span. So, steps in beam thickness may increase or decrease natural frequency sensitivity to crack in dependence on where crack is located and which frequency is considered. Graphs given in Fig.5 show also that frequency nodes of stepped-down beam (B1) are thrusted to the beam middle and the nodes are pulled away from the middle for stepped-up beam (B2). Nevertheless, the steps do not shift the node located at the beam middle and uniformly increasing thickness of uniform beam does not change the frequency nodes. 6. CONCLUSION In the present paper a simplified version of the transfer matrix method has been developed for modal analysis of multiple cracked stepped beam based on an explicit expression of mode shape of multiple cracked uniform beam element. The simplification consists of that the beam state needs to be transferred only through steps of beam but not over the cracks as done in the earlier publications. An experimental modal analysis of cracked multistep beam has been carried out and comparison of computed and measured natural frequencies demonstrated a good agreement of the theory with experiment. Using the simplified TMM it was found that likely to the uniform beam there exist on beam positions crack appeared at which does not change a certain natural frequency. Such critical points on beam are called herein frequency nodes and it was shown that step-down shifts the nodes to the beam middle and step-up pulls them to the beam ends. Finally, the performed modal analysis shows significant influence of steps on the natural frequency sensitivity to cracks and this is a useful indication for crack detection in stepped beam by measurement of natural frequencies. Acknowledgement. This work was completed with financial support from NAFOSTED of Vietnam under Grant of number 107.01-2015.20. APPENDIX CALCULATION OF CRACK MAGNITUDE The so-called crack magnitude introduced above is calculated as [16] )/()1(6/ 020000 hahfKIE νpiγ −== ; (A.1) ).6.197556.401063.47 0351.332948.209736.95948.404533.16272.0()( 876 54322 0 zzz zzzzzzzf +−+ +−+−+−= The transfer matrix method for modal analysis of cracked multistep beam 611 REFERENCES 1. Sato H. - Free vibration of beams with abrupt changes of cross-section. Journal of Sound and Vibration 89 (1983) 59-64. 2. Jang S. 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