Rayleigh’s quotient for multiple cracked beam and application - Nguyen Tien Khiem

Vietnam Journal of Mechanics, VAST, Vol. 33, No. 1 (2011), pp. 1 – 12 RAYLEIGH’S QUOTIENT FOR MULTIPLE CRACKED BEAM AND APPLICATION Nguyen Tien Khiem, Tran Thanh Hai Institute of Mechanics, VAST Abstract. Rayleigh’s quotient for Euler-Bernoulli multiple cracked beam with different boundary conditions has been derived from the governed equation of free vibration. An appropriate choosing of approximate shape function in terms of mode shape of uncracked beam and specific functions satisfying conditions at cracks and boundaries leads to an explicit expression of natural frequencies through crack parameters that can simplify not only the analysis of natural frequencies of cracked beam but also the crack detection problem. Numerical analysis of natural frequencies of the cracked beam by using the ob- tained expression in comparison with the well known methods such as the characteristic equation and finite element method shows their good agreement. The analytical expres- sion of natural frequencies applied to the crack detection problem allows the result of detection to be improved. Key words: Natural frequencies, multiple cracked beam, Rayleigh’s quotient. 1. INTRODUCTION Recently, a significant effort of researchers as well as engineers has been devoted to solve forward and inverse problems in vibration of cracked beam. The most important tool for solving the problems is so-called the characteristic equation relating implicitly natural frequencies with crack parameters. It has been shown in several studies, for instance [1, 2] that for a beam with single crack of extent γ at position xc the characteristic equation has the form F0(ω) + γF1(ω, xc) = 0 (1) where F0 and F1 are functions given by boundary conditions of the beam. If the crack extent γ is small, from the equation an explicit expression of natural frequencies through crack parameters can be derived as ∆ωk = ωk − ωk0 = γg(ωk0, xc) (2) with ωk, ωk0 are frequencies of cracked and uncracked beam respectively. This is an ap- proximation available only for small extent of single crack [3, 4]. J. Fernandez-Saez, L Rubio and C. Navarro [5] have obtained a formulae for fundamental frequency of sim- ply supported beam using Rayleigh’s method without assumption on smallness of crack depth. Comparison of the obtained analytical result [5] with numerical one shows that both the results are almost identical for the crack depth reached 75% beam thickness. 2 Nguyen Tien Khiem, Tran Thanh Hai The characteristic equation (1) has been developed for multiple cracked Bernoulli-Euler beam in [6, 7] by using the transfer matrix method and obtained explicitly in an ana- lytical form in [8]. The author of reference [9] has developed the finite element model of multiple cracked beam and established the characteristic equation accordingly to the FEM. The approximate equations similar to the equation (2) for multiple cracked beam was also established in [2, 7] assuming that the mode shapes of cracked beam remain the same as those of uncracked one. An important idea arisen from [5] is to obtain an explicit expression of natural frequencies of higher modes with other boundary conditions and for multiple cracked beam based on the choosing mode shapes of cracked beam. This idea has been developed in [10] to obtain higher frequencies (but not in analytical form) for no uniform beam using Galerkin’s method. This paper is devoted to derive an analytical expression for any natural frequency of multiple cracked beam with arbitrary boundary conditions using Rayleigh method. Theoretical section is devoted to establish general form of Rayleigh’s quotient for multiple cracked beam, that is applied in subsequent section to obtain analytical expressions of natural frequencies of the cracked beam through crack pa- rameters for different cases of the classical boundary conditions. Numerical investigation is presented in the last section for comparison in order to validate the theory. 2. THEORY Consider an uniform with the material and geometrical constants: Young’s modulus E, mass density ρ, length L, cross section area and moment of inertia F , I and with given arbitrarily boundary conditions at the ends x = 0 and x = L. It is well known that for such the beam, the eigenparameters consisting of nat- ural frequencies and corresponding mode shapes { ω2k, φk(x), k = 1, 2, .... } satisfying the equation φ (IV ) k (x)− λ 4 kφk(x) = 0, λ 4 k = ρFω2k EI , k = 1, 2, ... (3) Suppose, furthermore, that the beam has been cracked at the positions (e1, . . . , eN) and the cracks are modeled as rotational springs of stiffness K1, . . . , KN respectively. Dividing the beam into N segments (xj−1, xj), j = 1, ..., N , each of those contains a crack (ej, Kj) so that xj−1 ≺ ej ≺ xj , j = 1, ..., N and x0 = 0, xN = L. Considering equation (1) in j-th beam segment (xj−1, xj) and denoting the mode shape for this segment by φkj(x), one will have φ (IV ) kj (x)− λ 4 kφkj(x) = 0, x ∈ (xj−1, xj). (4) Multiplying both sides of the equations by φkj(x), then taking integration along the interval (xj−1, xj) yields ω2k = (EI/ρF ) N∑ j=1 akj/ N∑ j=1 bkj, k = 1, 2, 3, .... (5) Rayleigh’s quotient for multiple cracked beam and application 3 where akj = xj∫ xj−1 φ (IV ) kj (x)φkj(x)dx, bkj = xj∫ xj−1 φ2kj(x)dx. (6) If functions φkj, φ ′ kj, φ ′′ kj, φ ′′′ kj are continuous in the interval (a, b), one has b∫ a φ (IV ) kj (x)φkj(x)dx = b∫ a φ”2kj(x)dx+Bkj(b)− Bkj(a) (7) where Bkj(x) = φ ′′′ kj(x)φkj(x)− φ ′′ kj(x)φ ′ kj(x). (8) Applying equations (7), (8) to two segments (xj−1, x¯j), (x¯j, xj) for arbitrary x¯j ∈ (xj−1, xj) leads the first integral in (6) to akj = xj∫ xj−1 φ ′′2 kj (x)dx+ [ Bkj(x¯ − j )−Bkj(x¯ + j ) ] + [ Bkj(x − j )−Bkj(x + j−1) ] and, in consequence, N∑ j=1 akj = N∑ j=1 xj∫ xj−1 φ ′′2 kj (x)dx+ N−1∑ j=1 [Bkj(xj)−Bk,j+1(xj)]+ + N∑ j=1 [ Bkj(x¯ − j )−Bkj(x¯ + j ) ] + [BkN (xN)− Bk1(x0)] . (9) Because the beam is uniform (continuous) at the joints xj, j = 1, ..., N − 1, the functions φkj(x) satisfy conditions φk,j−1(x − j ) = φkj(x + j ); φ ′ k,j−1(x − j ) = φ ′ kj(x + j ); φ ′′ k,j−1(x − j ) = φ ′′ kj(x + j ); φ ′′′ k,j−1(x − j ) = φ ′′′ kj(x + j ) (10) and following conditions at the crack position ej: φkj(e − j ) = φkj(e + j ) = φkj(ej); EIφ ′′′ kj(e − j ) = EIφ ′′′ kj(e + j ) = EIφ ′′′ kj(ej); Kj [ φ′kj(e + j )− φ ′ kj(e − j ) ] = EIφ′′kj(e + j ) = EIφ ′′ kj(e − j ) = EIφ ′′ kj(ej). (11) It’s not difficult to verify that the conditions (8) lead to N−1∑ j=1 [Bkj(xj)−Bk,j+1(xj)] = 0 and due to conditions (11) one has N∑ j=1 [ B ( kje − j )−B ( kje + j ) ] = N∑ j=1 [ γjφ” 2 kj(ej) ] , γj = EI/Kj. 4 Nguyen Tien Khiem, Tran Thanh Hai Therefore, the equation (9) can be rewritten as N∑ j=1 akj = N∑ j=1   xj∫ xj−1 φ ′′2 kj (x)dx+ γjφ ′′2 kj (xj)  + [BkN (L)− Bk1(0)] , and the equation (5) gets the form ω2k = EI ρF N∑ j=1 [ xj∫ xj−1 φ ′′2 kj (x)dx+ γjφ ′′2 kj (xj) ] + [BkN (L)−Bk1(0)] N∑ j=1 [ xj∫ xj−1 φ2kj(x)dx ] . (12) This is Rayleigh’s quotient generalized for multiple cracked uniform beam with arbitrary boundary conditions, some particular cases of which have been given in [10]. The equations (12), in fact, represent a relationship between unknown natural frequencies and corre- sponding mode shapes that allows to calculate natural frequencies by given mode shapes. However, as it is well known, finding an exact mode shape is strictly related to finding the natural frequency. An exact expression of the mode shape for multiple cracked beam has been obtained in [7], but it contains unknown natural frequency so that substituting the exact mode shape into equation (12) leads the latter into an identity. The major idea of the well known Rayleigh’s method is to choose a trial shape function satisfying only boundary conditions instead of the unknown mode shape for calculating natural frequencies. Al- though such calculated natural frequencies are approximate, it may be closely approached to exact frequencies by a proper choosing the trial shape function. Furthermore, in the case of cracked beam, the calculated from Rayleigh’s quotient natural frequencies give an explicit expression of crack parameters that is useful not only for analysis of natural fre- quencies but also to detect crack positions and depth. Below, a combination of mode shape for uncracked beam with specially chosen functions satisfying both the conditions (11) at cracks and arbitrary boundary conditions is developed to obtain an analytical expression of any natural frequencies through crack parameters. Based on most importance for cracked beam that only its slope is discontinuous at the crack position, the mode shape functions are assumed to be chosen in the form φkj(x) = φk0(x) + φ c kj(x), (13) where φk0(x) are mode shapes of uncracked beam, continuous together with its derivatives φ′kj0(x), φ ′′ kj0(x), φ ′′′ kj0(x) in whole the beam (0, L) and linear functions φckj(x) = Ckjx+Djk + { 0, xj−1 ≤ x ≺ ej γjφ ′′ k0(xj)S(x− ej), ej ≺ x ≤ xj (14) with constants Ckj , Dkj and function S(x) = [sinhλx+ sinλx] /2λ, satisfying conditions: S(0) = S ′′(0) = S ′′′(0) = 0, S ′(0) = 1. Substituting equations (13), (14) into equations (10), (11) yields Ck,j+1xj +Dk,j+1 = Ckjxj +Dkj + γjφ ′′ k0(xj)S(xj − ej), Rayleigh’s quotient for multiple cracked beam and application 5 Ck,j+1 = Ckj + γjφ ′′ kj(ej)S ′(xj − ej), j = 1, 2, ..., N, that in consequence lead to the expressions Ckj = Ck1 + j−1∑ i=1 γiφ ′′ k0(ei)S ′(xi − ei), Dkj = Dk1 + j−1∑ i=1 γiφ ′′ k0(ei)S¯(xi − ei), S¯(xi − ei) = [S(xi − ei)− xiS ′(xi − ei)] , ∀j = 1, ..., N. (15) with two arbitrary constants Ck1, Dk1 which would be determined from boundary condi- tions. To calculate frequencies by equation (12) with the chosen mode shapes (13)-(15), at first, let’s consider the last term in numerator of equation (12) that can be expressed as [BkN (xN )− Bk1(x0)] = [BkN (L)−Bk1(0)] = B 0 k +B c k − N∑ j=1 γjβkjφ” 2 k0(ej), (16) where B0k = φ ′′′ k0(L)φk0(L)− φ ′′ k0(L)φ ′ k0(L)− φ ′′′ k0(0)φk0(0) + φ ′′ k0(0)φ ′ k0(0), (17) Bck = [ φ′′′k0(L)L− φ ′′ k0(L) + φ ′′ k0(0) ] Ck0 + [ φ′′′k0(L)− φ ′′′ k0(0) ] Dk0, (18) βkj = [ φ′′′k0(L)(ej − L) + φ ′′ k0(L) ] /φ′′k0(ej). (19) The equations (17) show that B0k is derived from given boundary conditions at the beam ends separately for uncracked beam, the determined by equations (18), (19) term Bck and non-dimensional coefficient βkj express a combined effect of boundary conditions and cracks on natural frequencies. In some particular cases shown below the terms B0k , B c k, βkj can be annulled by choosing the constants Ck0, Dk0 accordingly to given boundary conditions of the beam. Thus, the complete numerator of of the equation (10) can be written in general form N∑ j=1   xj∫ xj−1 φ ′′2 kj (x)dx+ γjφ ′′2 kj (xj)   = L∫ 0 φ ′′2 k0(x)dx+ N∑ j=1 γj(1− βkj)φ ′′2 kj (xj)+B 0 k +B c k. (20) Now let’s go to calculate the denominator of equation (10) using the mode shapes (14), (15). First, it can be expressed as N∑ j=1   xj∫ xj−1 φ2kj(x)dx   = N∑ j=1   xj∫ xj−1 φˆ2k0(x)dx+Ψkj   = L∫ 0 φˆ2k0(x)dx+Ψ1 +Ψ2, where Ψ1 = 2EI ρFω2k0  Bck + N∑ j=1 γj(1− βkj)φ ′′2 k0(ej)   , (21) Ψ2 = A c k + N∑ j=1 γ2j fkjφ ′′2 k0(ej) + N∑ j=2 j−1∑ i=1 γjγihjiφ ′′ k0(ei)φ ′′ k0(ej) (22) 6 Nguyen Tien Khiem, Tran Thanh Hai with Bck, βkj given by (18), (19) and Ack = C 2 k0L 3/3 +Ck0Dk0L 2 +D2k0L, (23) hji = (L− ej) [ (L− ei) 2 + (L− ei)(L− ej)− (ej − ei) 2 ] /3, (24) fkj = (L− ej) 3 3 [ 1 + (2L− ej)Ck0 + 3Dk0 (L− ej)γjφ ′′ k0(ej) ] . (25) Finally, generalized Rayleigh’s quotient (10) for multiple cracked beam with mode shapes chosen in the form (14), (15) can be written in the form ω2k = EI ρF L∫ 0 φ ′′2 k0(x)dx+ N∑ j=1 [ γj(1− βkj)φ ′′2 k0(ej) ] +B0k +B c k L∫ 0 φ2k0(x)dx+Ψ1 [ γjφ 2 k0(ej) ] +Ψ2 [ γjφ ′′2 k0(ej) ] , (26) that is an analytical expression of natural frequency of cracked beam through crack pa- rameters for arbitrary boundary conditions and mode shape of uncracked beam. Note that Ψ1, Ψ2 are linear and bilinear forms of crack extent vector (γ1, ..., γN) with coefficients being functions of crack positions (e1, ..., eN). For uncracked beam, equation (26) leads to the well known Rayleigh’s quotient ω2k0 = EI ρF   L∫ 0 φ ′′2 k0(x)dx+B 0 k   / L∫ 0 φ2k0(x)dx. (27) At the end of this theoretical section, it has to emphasize that all equations ob- tained above are free from any assumption on the boundary conditions at the beam ends. Therefore, it provides a tool for investigation of either cracked or intact beam with differ- ent cases of boundary supports. In the next section, some cases of the classical boundary conditions are considered in more details. 3. APPLICATION Before application of the developed theory to some particular cases, let’s consider the classical boundary conditions applied for mode shape chosen in the form of (15) rewritten as φk(x) = φk0(x) + φˆk(x) with φˆk(x) = φˆkj(x), for x ∈ [xj−1, xj], j = 1, ..., N ; φˆkj(x) = Ckjx+Dkj + { γj(ej − x)φ ′′ k0(ej) for x ∈ [xj−1, ej); 0 for x ∈ (ej, xj], that result in φ′′k(x) = φ ′′ k0(x); φ ′′′ k (x) = φ ′′′ k0(x), ∀x ∈ [0, L] and φk(0) = φk0(0) +Dk0; φ ′ k(0) = φk0(0) +Ck0; φ ′ k(L) = φ ′ k0(L) + Ck0 + N∑ j=1 γjφ ′′ k0(ej); Rayleigh’s quotient for multiple cracked beam and application 7 φk(L) = φk0(L) + Ck0L+Dk0 + N∑ j=1 γj(L− ej)φ ′′ k0(ej). The following classical boundary conditions for a beam are well known: - Simple supports: φk(0) = φ ′′ k(0) = φk(L) = φ ′′ k(L) = 0; - Cantilever: φk(0) = φ ′ k(0) = φ ′′′ k (L) = φ ′′ k(L) = 0; - Free ends: φ′′′k (0) = φ ′′ k(0) = φ ′′′ k (L) = φ ′′ k(L) = 0; - Clamped ends: φk(0) = φ ′ k(0) = φk(L) = φ ′ k(L) = 0. If the mode shape of uncracked beam φk0(x) satisfying the given above classical boundary conditions has been used, one has two constants Ck0, Dk0 to be chosen from the conditions - Simple supports: φk(0) = Dk0 = 0, φk(L) = Ck0L+Dk0 + N∑ j=1 γj(L− ej)φ ′′ k0(ej) = 0 - Cantilever: φk(0) = Dk0 = 0, φ ′ k(0) = Ck0 = 0 - Free ends: φ′′′k (0) = φ ′′′ k0(0) = 0; φ ′′ k(0) = φ ′′ k0(0) = 0; φ ′′′ k (L) = φ ′′′ k0(L) = 0; φ ′′ k(L) = φ ′′ k0(L) = 0. - Clamped ends: φk(0) = Dk0 = 0, φk(L) = Ck0L+Dk0 + N∑ j=1 γj(L− ej)φ ′′ k0(ej) = 0; φ′k(0) = Ck0 = 0; φ ′ k(L) = Ck0 + N∑ j=1 γjφ ′′ k0(ej) = 0. From the latter equations it can be seen that in the case of simple supports and cantilever the constants Ck0, Dk0 will be chosen uniquely. For the free-free ends beam, the constants Ck0, Dk0 may be chosen arbitrary. For clamped end beam, the choosing of two constants is inadequate to satisfy four conditions φk(0) = φ ′ k(0) = φk(L) = φ ′ k(L) = 0. Because of this uncertainty that requires a separated study, only the simply supported and cantilevered beams are considered in this work. 3.1. Simply supported beam For simple supports at the beam ends, the boundary conditions would be satisfied if it is chosen φk0(x) = sin(kpix/L), k = 1, 2, ... and Dk0 = 0, Ck0 = N∑ j=1 ( ej L − 1)γjφ ′′ ki0(ej), 8 Nguyen Tien Khiem, Tran Thanh Hai i.e. Ckj = N∑ j=1 ej L γjφ ′′ ki0(ej) + N∑ i=j+1 γiφ ′′ k0(ei);Dkj = − j∑ i=1 eiγiφ ′′ k0(ei), j = 1, ..., N. (28) Substituting (28) into (26) and taking account of (27) yield ω2k = ω2k0 [ 1 + 2 N∑ j=1 γ¯j sin 2 kpie¯j ] [ 1 + 4 N∑ j=1 γ¯j sin 2 kpie¯j + 2(kpi)4 3 N∑ i,j=1 qij γ¯i sinkpie¯i · γ¯j sinkpie¯j ] , (29) where ωk0 = (kpi/L) 2 √ EI/ρF and γ¯j = γj/L; e¯j = ej/L; qij = 0, i  j ; qjj = e¯ 2 j (1− e¯j) 2; qij = e¯i(1− e¯j) [ 1− e2i − (1− e¯j) 2 ] , i ≺ j. If the crack extents are small, i.e. γj = εηj, in the first order with respect to the small parameter ε, the equation (29) is simplified to ω2k = ω 2 k0  1− 2 N∑ j=1 γ¯j sin 2 kpie¯j   . (30) In a particularity, for a beam with single crack, equations (29), (30) become ω2k = ω2k0 [ 1 + 2γ¯ sin2 kpie¯ ] 1 + 4γ¯ sin2 kpix¯c + 2γ¯2 [(e¯2(1− e¯)2)/3] (kpi)4 sin 2 kpie¯ , (31) ωˆ2k = ω 2 k0 [ 1− 2γ¯ sin2 kpie¯ ] . (32) Formulas similar to the equation (32) have been given in [2, 3, 4] and a particular case of the equation (31) when k = 1, i. e. for the fundamental frequency, was obtained by J. Fernandez-Saez, L Rubio and C. Navarro [5]. 3.2. Cantilevered beam For a cantilever beam, choosing Ck0 = Dk0 = 0, i. e. Ckj = j∑ i=1 γiφ ′′ ki0(ei); Dkj = − j∑ i=1 eiγiφ ′′ ki0(ei), j = 1, ..., N. (33) and mode shape φk0(x) = Ak [cosh(λk0x/L)− cos(λk0x/L)]− [sinh(λk0x/L)− sin(λk0x/L)] ; ω2k0 = λ 4 k0EI/ρFL 4; Ak = [sinhλk0 + sinλk0] / [cos λk0 + coshλk0] . λ10 = 1.8751, λ20 = 4.6941, λ30 = 7.8747, λ40 = 10.9955, λ50 = 14.1372, . . . (34) Rayleigh’s quotient for multiple cracked beam and application 9 satisfying the boundary conditions φk(0) = φ ′ k(0) = φ ′′′ k (L) = φ ′′ k(L) = 0, results in the equations ω2k = ω2k0 [ 1 + N∑ j=1 γ¯jΦ¯ 2 k(e¯j) ] 1 + 2 N∑ j=1 γ¯jΦ¯ 2 k(e¯j) + (λ 4 k0/3) [ N∑ i,j=1 qjiγ¯j γ¯iΦ¯k(e¯j)Φ¯k(e¯i) ] (35) with the notations ωk0 = λ 2 k0 √ EI/ρF and Φ¯k(x) = [cosh(λk0x) + cos(λk0x)]−A −1 k [sinh(λk0x) + sin(λk0x)] ; (36) γ¯j = γj/L, e¯j = ej/L; qij = 0, i  j, qjj = (1− e¯j) 3, qij = (1− e¯j) [ (1− e¯i) 2 + (1− e¯i)(1− e¯j)− (e¯j − e¯i) 2 ] , i ≺ j. (37) The first order approximation with respect to small crack depth of the equation (35) is ω2k = ω 2 k0  1− N∑ j=1 γ¯jΦ¯ 2 k(ej)   . (38) For a cantilever beam with a single crack, equations (35) and (37) are reduced to ω¯k = ωk ωk0 = [ [ 1 + γ¯Φ¯2k(e¯) ] 1 + 2γ¯ [ 1 + λ4k0γ¯(1− e¯) 3/6 ] Φ¯2k(e¯) ]1/2 (39) and ω˜k = [ 1− γ¯Φ¯2k(e¯) ]1/2 . (40) In the case of two cracks, corresponding equations are ω¯k = [ [ 1 + γ¯1Φ¯ 2 k(e¯1) + γ¯2Φ¯ 2 k(e¯2) ] 1 + 2 [ γ¯1Φ¯2k(e¯1) + γ¯2Φ¯ 2 k(e¯2) ] +Ψ(γ¯1, γ¯2, e¯1, e¯2) ]1/2 , (41) where Ψ = λ4k0 3 [ γ¯21(1− e¯1) 3Φ¯2k(e¯1) + γ¯ 2 2(1− e¯2) 3Φ¯2k(e¯2) + γ¯1γ¯2g(e¯1, e¯2)Φ¯k(e¯1)Φ¯k(e¯2) ] with g(e¯1, e¯2) = (1− e¯2) [ (1− e¯1) 2 + (1− e¯1)(1− e¯2)− (e¯2 − e¯1) 2 ] and ω˜k = [ 1− γ¯1Φ¯ 2 k(e¯1)− γ¯2Φ¯ 2 k(e¯2) ]1/2 . (42) Thus, an explicit expression of natural frequencies through crack positions and ex- tents for multiple cracked beam has been obtained in a form convenient for frequency analysis as well as multi-crack detection of beam. To validate applicability of the obtained above analytical formulas to practice, numerical calculation will be carried out below and results will be compared with those obtained by using the characteristic equation and experimental ones. 10 Nguyen Tien Khiem, Tran Thanh Hai 4. NUMERICAL INVESTIGATION AND COMPARISON Numerical investigation carried out in this section for illustration and validating the theory is devoted to a cantilevered beam with a single and double cracks. In all the cases, the crack extent is calculated as function of crack depth a by the formulae γ¯ = EI LK = (5.346h/L)I(δ), (43) with δ = a/h ( h is beam thickness) and I(δ) = 1.8624δ2− 3.95δ3+ 16.375δ4− 37.226δ5+ 76.81δ6− −126.9δ7 + 172δ8 − 143.97δ9+ 66.56δ10. Beam with single crack The first frequency ratio calculated by using equation (39) and those obtained from the characteristic equation [6, 7, 8] as functions of crack position with different relative crack depth δ = 0.1-0.6 (i. e. crack depth of 10-60% beam thickness) are shown in Fig. 1. Graphics in the figure show that the equation (39) is almost equivalent to the characteristic equation in calculating the fundamental frequency of cracked beam. For higher frequencies the equation (40) gives better results than the equation (39) in comparison with solution of the characteristic equation. Fig. 1. Comparison of first frequency ratio computed by equation (39) with that obtained from the characteristic Equation for crack depth ratios from 0.1 to 0.6 Fig. 2 shows the second frequency ratio computed from equation (40) and charac- teristic equation for relative crack depth running from 0.1 to 0.4. It can be seen from the figure that difference between frequencies calculated from equation (40) and characteristic equation has been visible only when crack depth reaches 40% beam thickness. Beam with double cracks First, three natural frequencies ratios (cracked to uncracked) computed from equa- tions (41) and (42) are compared with the experimental results obtained in [12] for a cantilever with two cracks e¯1 = 0.3175, γ1 = 0.2, e¯2 = 0.6812, γ2 = 0.3and shown in Table 1. From the table it can be seen that equation (41) results in frequencies really more close to the experimental ones than the equation (42) and error of both the equations for all Rayleigh’s quotient for multiple cracked beam and application 11 Fig. 2. Comparison of second frequency ratio computed by Eq. (40) with solution of Characteristics Eq. for crack depth ratios from 0.1 to 0.4 three frequencies does not exceed 1%. This fact confirms validity of the formulas derived above. Table 1. Comparison of three frequencies ratios with experimental results [11] Frequencies Eq. (42) Eq. (41) Exp. [12] Err. of (41)(%) Err. of (42)(%) First 0.993866 0.993938 0.994581 0.064637 0.07188 Second 0.98574 0.982846 0.981361 -0.15133 -0.44621 Third 0.971835 0.9696 0.964265 -0.5533 -0.78511 After comparing frequencies computed by FEM with the experimental results ob- tained by the authors of J. Lee [11] has suggested in [12] a correction of the formulae (43) to get numerical results more close to the measured one. Of course, crack model has a considerable effect on numerical analysis of cracked structures, but disagreement be- tween predicted and measured data can arose also from the inaccurate model parameters such as material constants (E, ρ, ν) or geometrical one (A, I, b, h) and from the non-ideal boundary conditions in experiments. Therefore, only crack model correction is insufficient to agree prediction with experiments. In this study, numerical results are in good agree- ments with experiments with no crack model correction because they have been obtained in non-dimensional form that are free from inaccuracy of the model parameters. 5. CONCLUSION The main result of this study is the Rayleigh’s quotient developed for multiple cracked beam in an analytical expression relating directly the frequencies to crack param- eters, which can be used to calculate natural frequencies of all modes with different cases of boundary conditions. The improvement has been made based on the special choosing the mode shape of cracked beam consisting of mode shape of uncracked beam and linear functions satisfying the continuity conditions at cracks. In the case of multiple cracked 12 Nguyen Tien Khiem, Tran Thanh Hai beam with simple supports, general formulae has been derived for calculating natural fre- quencies that contains as a particularity the formulae obtained by J. Fernandez-Saez, L Rubio and C. Navarro [5] for fundamental frequency. Asymptotic approximations of the obtained herein formulas are similar to those given by former authors as R.Y. Liang and his coworkers [2], A. Morassi [3] and Y. Narkis [4]. The corresponding formulas have been obtained also for a cantilever beam and are applied to numerical investigation of natural frequencies in comparison with solutions of the characteristic equation given in [6, 7, 8]. Further study should be carried out in developing formulas for a nonuniform beam with different boundary conditions and using the direct relationship between frequencies and crack parameters to multi-crack detection for beam. ACKNOWLEDGEMENT The authors have a great pleasure to thank the NAFOSTED of Vietnam for financial support in completing this paper. REFERENCES [1] R. Y. Liang, J. Hu and F. Choy, Theoretical Study of Crack-Induced Eigenfrequency Changes on Beam Structures, Journal of Engineering Mechanics, 118(2) (1992) 384-396. [2] R. Y. Liang, J. Hu and F. Choy, Quantitative NDE Technique for Assessing Damages in Beam Structures, Journal of Engineering Mechanics, 118(7) (1992), 1468-1487. [3] A. Morassi, Crack-Induced Changes in Eigenparameters of Beam Structures, Journal of En- gineering Mechanics, 119(9) (1993), 1798-1803. [4] Y. Narkis, Identification of crack location in vibrating simply supported beams, Journal of Sound and Vibration, 172 (1994), 549-558. [5] J. Fernandez-Saez, L Rubio and C. Navarro, Approximate calculation of the fundamental frequency for bending vibrations of cracked beam, Journal of Sound and Vibration, 225(2) (1999), 345-352. [6] N.T. Khiem and T.V. Lien, A simplified method for frequency analysis of multiple cracked beam, Journal of Sound and Vibrations, 245(1) (2001), 737-751. [7] D.P. Patil and S.K. Maiti, Detection of multiple cracks using frequency measurements, Engi- neering Fracture Mechanics, 70(12) (2003), 1553-1572. [8] S. Caddemi and I. Caliò, Exact closed-form solution for the vibration modes of the Euler- Bernoulli beam with multiple open cracks, Journal of Sound and Vibration, 327(3-5) (2009), 473-489. [9] J. Lee, Identification of multiple cracks in a beam using natural frequencies, Journal of Sound and Vibration, 320 (2009), 482-490. [10] D.Y. Zheng and S.C. Fan, Natural frequencies of a non-uniform beam with multiple cracks via modified Fourier series, Journal of Sound and Vibration, 242(4) (2001), 701-717. [11] R. Ruotolo and C. Surace, Damage assessing of multiple cracked beam: numerical results and experimental validation, Journal of Sound and Vibration, 206(4) (1997), 567-588. [12] J. Lee, Identification of multiple cracks in a beam using vibration amplitudes, Journal of Sound and Vibration, 326 (2009), 205-212. Received June 13, 2010