Resonant and antiresonant frequencies of multiple cracked bar

Nội dung text: Resonant and antiresonant frequencies of multiple cracked bar

1. Vietnam Journal of Mechanics, VAST, Vol.41, No. 2 (2019), pp. 157 – 170 DOI: RESONANT AND ANTIRESONANT FREQUENCIES OF MULTIPLE CRACKED BAR P. T. B. Lien1, N. T. Khiem2,∗ 1University of Transport and Communications, Hanoi, Vietnam 2Institute of Mechanics, VAST, Hanoi, Vietnam ∗E-mail: ntkhiem@imech.vast.vn Received: 12 September 2018 / Published online: 29 March 2019 Abstract. The natural frequencies or related resonant frequencies have been widely used for crack detection in structures by the vibration-based technique. However, antiresonant frequencies, the zeros of frequency response function, are less involved to use for the prob- lem because they have not been thoroughly studied. The present paper addresses analysis of antiresonant frequencies of multiple cracked bar in comparison with the resonant ones. First, exact characteristic equations for the resonant and antiresonant frequencies of bar with arbitrary number of cracks are conducted in a new form that is explicitly expressed in term of crack severities. Then, the conducted equations are employed for analysis of variation of resonant and antiresonant frequencies versus crack position and depth. Nu- merical results show that antiresonant frequencies are indeed useful indicators for crack detection in bar mutually with the resonant ones. Keywords: multi-cracked bar; longitudinal vibration; frequency equation; antiresonant fre- quency. 1. INTRODUCTION Natural frequencies of a structure are an important dynamical characteristic that is usually computed by solving the so-called characteristic or frequency equation of the structure. Establishing the frequency equation for a structure gets to be crucial for both the analysis and identification of the structure. Adams et al. [1] are the first authors who established exact frequency equation for bar with single crack adopted by the spring model. Narkis [2] and Morassi [3] first obtained closed form solution in locating a crack using frequency equation of longitudinal vibration. More comprehensive study on both the forward and inverse problems in free vibration of multiple cracked bar was accom- plished in References [4–10]. However, the study showed that unique solution of the crack detection cannot be found by using only natural frequencies. Some efforts have been made to solve the problem by encompassing other vibration characteristics such mode shapes [11–13] or frequency response function [14], but it was successful when an- tiresonant frequencies have been employed [15–17]. Nevertheless, using additionally the c 2019 Vietnam Academy of Science and Technology
2. 158 P. T. B. Lien, N. T. Khiem antiresonant frequencies for crack detection in bar enables to obtain unique solution of the crack detection problem only for free end bar. This may be caused from that the an- tiresonant frequencies of cracked bar with different boundary conditions have not been exhaustively investigated. The present paper is devoted to study systematically variation of antiresonant fre- quencies of bar versus crack parameters mutually with resonant frequencies. First, there is derived a new form of characteristic equations for both resonant and antiresonant fre- quencies of multiple cracked bars. Then, the established equations are used for investi- gating change in the frequencies caused by presence of cracks. Numerical results have been examined to illustration of the proposed herein theory. 2. GENERAL FREQUENCY EQUATION FOR MULTIPLE CRACKED BAR Let’s consider longitudinal vibration in a bar that is described by the equation [14] p Φ00(x) + λ2Φ(x) = 0, x ∈ (0, 1), λ = ωL ρ/E, (1) under general boundary conditions 0 0 α0Φ(0) + β0Φ (0) = 0, α1Φ(1) + β1Φ (1) = 0, (2) with the material, geometry and boundary constants E, ρ, L, α0, β0, α1, β1. Suppose that the bar is damaged to crack at arbitrary number n of positions ej: 0 ≤ e1 < < en ≤ 1. For cracks modeled by transitional spring of stiffness Kj, conditions at the crack positions are [18] 0 0 0 Φ (ej + 0) = Φ (ej − 0), Φ(ej + 0) = Φ(ej − 0) + γjΦ (ej), (3) 2 γj = EA/LKj = 2(1 − ν )(h/L)θ(aj/h), j = 1, , n, θ(z) = 0.9852z2 + 0.2381z3 − 1.0368z4 + 1.2055z5 + 0.5803z6 − 1.0368z7 + 0.7314z8. (4) It can be shown that any solution of equation (1) satisfying the first boundary condition in (2) at x = 0 and conditions (3) inside the bar is expressed in the form [14] Φ(x) = CL(x, λ), (5) where C is a constant and function n L(λx) = L0(λx) + ∑ µkK(x − ek), (6) k=1  0 for x < 0  0 for x < 0 K(x) = , K0(x) = , cos λx for x ≥ 0 −λ sin λx for x ≥ 0 L0(λx) = (α0 sin λx − λβ0 cos λx), " j−1 # 0 µj = γj L 0(λej) − λ ∑ µk sin λ(ej − ek) , j = 1, , n. (7) k=1 Substituting expression (5) into the second boundary condition in (2) at x = 1 yields 0 C[α1L(1, λ) + β1L (1, λ)] = 0,
3. Resonant and antiresonant frequencies of multiple cracked bar 159 that would have nontrivial solution with respect to constants C under the condition n D(λ) ≡ d0(λ) + ∑ H(1 − ej)µj = 0, (8) j=1 0 where d0(λ) = α1L0(λ) + β1L 0(λ); H(x) = α1 cos λx − λβ1 sin λx. The Eq. (8) is gen- eral form of frequency equation for multiple cracked bar that in combination with Eq. (7) enables to compute eigenvalues λ1, λ2, λ3, dependently on crack parameters. The obtained equation is implicit regarding crack magnitudes γ1, , γn, so that solving that equation with respect to the eigenvalues or natural frequencies needs to compute the so- called damage parameters µ1, , µn defined by Eqs. (7). It would be much simplified in solution of both the forward and inverse problems for cracked bar if an explicit expres- sion of the characteristic equation regarding the crack magnitudes γ1, , γn is available. Indeed, the recurrent relationships (7) can be rewritten as 0 µ1 = γ1L 0(λe1), 0 0 µ2 = γ2L 0(λe2) − λγ1γ2L 0(λe1) sin λ(e2 − e1), 0 0 0 µ3 = γ3L 0(λe3) − λγ1γ3L 0(λe1) sin λ(e3 − e1) − λγ2γ3L 0(λe2) sin λ(e3 − e2) 2 0 + λ γ1γ2γ3L 0(λe1) sin λ(e2 − e1) sin λ(e3 − e2), Substituting latter expressions into (8) one obtains n n j−1 D(λ) ≡ d0(λ) + ∑ γjd1(λ, ej) − λ ∑ ∑ d2(λ, ej, ek)γjγk j=1 j=2 k=1 n j−1 k−1 2 n−1 + λ ∑ ∑ ∑ d3(λ, ej, ek, er)γjγkγr + + (−λ) dn(λ, en, , e1)γ1γ2 γn (9) j=3 k=2 r=1 n = d ( ) + (− )k−1d ( e e e ) = 0 λ ∑ ∑ λ k λ, ik , ik−1 , , i1 γi1 γi2 γik 0, k=1 1≤i1<i2< <ik≤n where 0 d0(λ) = α1L0(λ) + β1L 0(λ), 0 d1(λ, ej) = H(1 − ej)L 0(λej), 0 d2(λ, ej, ek) = H(1 − ej) sin λ(ej − ek)L 0(λek), 0 d3(λ, ej, ek, er) = H(1 − ej) sin λ(ej − ek) sin λ(ek − er)L 0(λer), 0 dn(λ, en, , e1) = H(1 − en) sin λ(en − en−1) sin λ(en−1 − en−2) sin λ(e2 − e1)L 0(λe1). (10) The obtained equation (9) is desired explicit form of the characteristic equation that provides an efficient tool for solving not only the forward but also the inverse problem of multiple cracked bar. Note, another form of the characteristic equation for multiple
4. 160 P. T. B. Lien, N. T. Khiem cracked rod was exactly obtained by Shifrin in [7], but that equation is implicit with respect to the crack magnitudes, likely, the equation given by Khiem et al. in [14]. In the case of intact bar, Eq. (9) becomes 2 d0(λ) ≡ (α0α1 + λ β0β1) sin λ − λ(α1β0 − α0β1) cos λ = 0. (11) Next, for bar with single, double and triple cracks exact frequency equations get respectively the forms D1(λ) ≡ d0(λ) + γ1d1(λ, e1) = 0, (12) D2(λ) ≡ d0(λ) + γ1d1(λ, e1) + γ2d1(λ, e2) − λγ1γ2d2(λ, e2, e1) = 0, (13) D3(λ) ≡ d0(λ) + γ1d1(λ, e1) + γ2d1(λ, e2) + γ3d1(λ, e3) − λγ1γ2d2(λ, e2, e1), 2 − λγ1γ3d2(λ, e3, e1) − λγ2γ3d2(λ, e3, e2) + λ γ1γ2γ3d3(λ, e3, e2, e1) = 0. (14) Moreover, if the cracks are small so that asymptotic approximations of first, second and third order respectively for the frequency equation are n d0(λ) + ∑ γjd1(ej) = 0, (15) j=1 n n j−1 d0(λ) + ∑ γjd1(ej) − λ ∑ ∑ d2(λ, ej, ek)γjγk = 0, (16) j=1 j=2 k=1 n n j−1 n j−1 k−1 2 d0(λ) + ∑ γjd1(ej) − λ ∑ ∑ d2(λ, ej, ek)γjγk + λ ∑ ∑ ∑ d3(λ, ej, ek, er)γjγkγr = 0. j=1 j=2 k=1 j=3 k=2 r=1 (17) Finally, it has to note that general boundary conditions (2) include all the conven- tional end conditions and the elastic ones in dependence on the specific combinations of 0 0 parameters (α0, β0, α1, β1). Namely, for the case of free-free ends, (a) Φ (0) = Φ (1) = 0; fixed ends, (b) Φ(0) = Φ(1) = 0 and fixed-free ends, (c) Φ(0) = Φ0(1) = 0, the parame- ters get respectively (a) α0 = α1 = 0, β0 = β1 = 1, (b) α0 = α1 = 1, β0 = β1 = 0, (c) β0 = α1 = 0, α0 = β1 = 1. (18) So that, for the listed above boundary conditions one has 0 2 (a) Free-Free ends: L0(x) = −λ cos λx, L 0(x) = λ sin λx, H1(x) = −λ sin λx, and therefore 2 d0(λ) = λ sin λ, 3 d1(λ, e) = −λ sin λe sin λ(1 − e), (19) 3 d2(λ, e1, e2) = −λ sin λe1 sin λ(e2 − e1) sin λ(1 − e2),
5. Resonant and antiresonant frequencies of multiple cracked bar 161 0 (b) Fixed-Fixed ends: L0(x) = sin λx, L 0(x) = λ cos λx, H1(x) = cos λx and d0(λ) = sin λ, d1(λ, e) = cos λ(1 − e) sin λe, (20) d2(λ, e1, e2) = λ cos λ(1 − e2) sin λ(e2 − e1)cosλe1. 0 (c) Fixed-Free ends: L0(x) = sin λx, L 0(x) = λ cos λx, H1(x) = −λ sin λx and d0(λ) = λ cos λx, 2 d1(λ, e) = −λ sin λ(1 − e) cos λe, (21) 2 d2(λ, e1, e2) = −λ sin λ(1 − e2) sin λ(e2 − e1) cos λe1. If both the ends of bar are supported by translational springs of stiffness S0, S1, the pa- rameters (α0, β0, α1, β1) are defined as α0 = α1 = 1, β0 = −EA/S0, β1 = EA/S1, so that 0 2 L0(x) = sin λx + λβ0 cos λx, L0(x) = λ cos λx − λ β0 sin λx, H1(x) = cos λx − λβ1 sin λx, and 2 d0(λ) = λ(β0 + β1) cos λ + (1 − λ β0β1) sin λ, d1(λ, e) = λ[cos λ(1 − e) − λβ1 sin λ(1 − e)](cos λe − λβ0 sin λe), d2(λ, e1, e2) = λ[cos λ(1 − e2) − λβ1 sin λ(1 − e2)](cos λe1 − λβ0 sin λe1) sin λ(e2 − e1). (22) Moreover, in the case of small cracks first order asymptotic approximations of the frequency equation for the conventional (Free-Free; Fixed-Fixed; Fixed-Free) boundary conditions are n sin λ − λ ∑ γj sin λej sin λ(1 − ej) = 0, (23) j=1 n sin λ + λ ∑ γj cos λej cos λ(1 − ej) = 0, (24) j=1 n cos λ − λ ∑ γj cos λej sin λ(1 − ej) = 0. (25) j=1 Also, assuming λ = λ0 + ∆λ with λ0 being the frequency parameter of intact bar and small ∆λ, the latter equations yield all those obtained in earlier studies, for example, [8]. 3. ANTIRESONANT FREQUENCY EQUATION FOR MULTIPLE CRACKED BARS As well known in the vibration theory, resonant frequencies of a mechanical system are poles of the system’s frequency Response Function (FRF) while antiresonant ones are zeros of the FRF. In case of systems without damping, the resonant frequencies are identical to natural frequencies determined as roots of the frequency equations. As zeros of FRF, antiresonant frequencies of a multiple cracked bar are seeking as follows. General
6. 162 P. T. B. Lien, N. T. Khiem expression of FRF for multiple cracked bars has been obtained by Khiem et al. [12, 13] in the form  n   ( − )[ − + ( − )]   g 1 x0 α0 sin λx β0λ cos λx ∑ µjS x ej  L  j=1  FRF(ω, x0, x)= sin λ(x − x0) − n , λEF    d0(λ) + ∑ µj[α1 cos λ(1 − ej) − β1λ sin λ(1 − ej)]   j=1  (26) where g(x) = α1 sin λx + β1λ cos λx and function S(x) = {0 if x < 0; cos λx if x ≥ 0}. Letting x = x0 = 1, that implies the FRF determined with both the input and output applied at the right end of bar, Eq. (26) is simplified to n [α0 sin λ − β0λ cos λ + ∑ µj cos λ(1 − ej)] −β L j=1 FRF(ω, 1, 1) = 1 . (27) EF n d0(λ) + ∑ µj[α1 cos λ(1 − ej) − β1λ sin λ(1 − ej)] j=1 Therefore, antiresonant frequencies can be sought by solving the equation n ¯ ¯ ¯ ¯ α0 sin λ − β0λ cos λ + ∑ µj cos λ(1 − ej) = 0, j=1 with respect to λ¯ or n ¯ ¯ ¯ d0(λ) + ∑ H(1 − ej)µj = 0, (28) j=1 where d¯0(λ¯ ) = α0 sin λ¯ − β0λ¯ cos λ¯ ; H¯ (x) = cos λ¯ x. Eq. (28) has the same form as Eq. (8) where the functions d0(λ), H(x) are replaced by d¯0(λ), H¯ (x) and in both the equations the parameters µj are expressed by the same equations (7). Thus, equation for antiresonant frequencies (called herein antiresonant frequency equation) can be derived as n − d¯ ( ¯ ) + (− ¯ )k 1d¯ ( ¯ e e e ) = 0 λ ∑ ∑ λ k λ, i1 , i2 , , ik γi1 γi2 γik 0, (29) k=1 1≤i1<i2< <ik≤n where 0 d¯1(λ, ej) = H¯ (1 − ej)L 0(λ¯ ej), 0 d¯2(λ¯ , ej, ek) = H¯ (1 − ej) sin λ¯ (ej − ek)L 0(λ¯ ek), 0 d¯3(λ, ej, ek, er) = H¯ (1 − ej) sin λ¯ (ej − ek) sin λ¯ (ek − er)L 0(λ¯ er), 0 d¯n(λ¯ , en, , e1) = H¯ (1 − en) sin λ¯ (en − en−1) sin λ¯ (en−1 − en−2) sin λ¯ (e2 − e1)L 0(λ¯ e1). (30)
7. Resonant and antiresonant frequencies of multiple cracked bar 163 Similarly, the first, second and third order asymptotic approximations of the antiresonant frequency equation can be obtained respectively as n ¯ ¯ ¯ ¯ d0(λ) + ∑ γjd1(λ, ej) = 0, (31) j=1 n n j−1 ¯ ¯ ¯ ¯ ¯ ¯ ¯ d0(λ) + ∑ γjd1(λ, ej) − λ ∑ ∑ d2(λ, ej, ek)γjγk = 0, (32) j=1 j=2 k=1 n n j−1 n j−1 k−1 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ 2 ¯ ¯ d0(λ) + ∑ γjd1(λ, ej) − λ ∑ ∑ d2(λ, ej, ek)γjγk + λ ∑ ∑ ∑ d3(λ, ej, ek, er)γjγkγr = 0. j=1 j=2 k=1 j=3 k=2 r=1 (33) Since the FRF (26) is meaningless at the fixed ends of bar, the antiresonant frequency equations (29) and (31)–(33) are applied only for free end bar and cantilever bar. In the latter cases of boundary conditions, the first order approximate antiresonant frequency equation are n ¯ ¯ ¯ ¯ cos λ − λ ∑ γj cosλ(1 − ej) sin λej = 0, (34) j=1 n ¯ ¯ ¯ ¯ sin λ + λ ∑ γj cos λ(1 − ej) cos λej = 0. (35) j=1 Eq. (35) shows that antiresonant frequencies of fixed-free bar are resonant frequencies of fixed end bar (see Eq. (24), so they are the same for symmetric cracks. Nevertheless, antiresonant frequencies of free-free end bar, likely resonant frequencies of cantilever bar, have not the symmetric effect. The latter fact has been employed by Rubio et al. [17] to obtain unique solution in localization of single and double crack in free-free end rod from given resonant and antiresonant frequencies. However, as shown below, the result cannot be extended for other cases of boundary conditions, even if other pair of resonant and antiresonant frequencies are used. To validate the proposed theoretical development, antiresonant frequencies of the free end bar that was experimentally examined by the authors of Ref. [15] are computed and compared to the measured ones (see Tab.1). Obviously, calculated and measured antiresonant frequencies are excellently agreed (descrepancy between them is less than 1%. However, the descrepancy increases with severity of damage, esspecially, for higher frequencies. Note, deviation between calcu- lated and measured first antiresonant frequency is of the same order 7% for both the cases of damage severity D1 and D2. This is perhaps caused by inaccuracy of the crack model used for representing the saw cut in the experimentation. 4. CRACK-INDUCED CHANGE IN RESONANT AND ANTIRESONANT FREQUENCIES (NUMERICAL RESULTS) The problem of single crack detection in free end bar has been thoroughly studied by Morassi and his coworkers. However, it is necessary to note that the unique solution
8. 164 P. T. B. Lien, N. T. Khiem Table 1. Antiresonant frequencies of intact and cracked bar compared to the measured ones Intact bar Damage senario D1 Damage senario D2 Mode No Exp. Present Exp. Present Exp. Present [15] (deviation, %) [15] (deviation, %) [15] (deviation, %) 1 468.6 470.6 (0.42) 439.5 470.3 (7.0) 432.9 465.3 (7.48) 2 1411.7 1411.7 (0) 1409.3 1406.4 (0.2) 1365.6 1301.7 (4.67) 3 2328.4 2352.8 (1.05) 2337.0 2339.6 (0.1) 2324.4 2132.9 (8.23) 4 3265.8 3294.0 (0.86) - 3282.1 (-) 3102.5 3134.1 (1.01) 5 4216.6 4235.1 (0.43) - 4232.9 (-) 3722.1 4200.8 (12.86) 6 5145.1 5176.3 (0.67) - 5173.3 (-) 4866.6 5098.5 (4.76) Damage scenarios D1,D2 correspond to different depth (6 and 15 mm) of crack at position e = 0.55/2.747 in locating single crack was attained in [17] because only first resonant and antireso- nant frequencies have been used. The unique solution could not be obtained by using a pair of second or higher resonant and antiresonant frequencies. Obviously, the resonant and antiresonant frequencies used for obtaining unique solution have no critical point, crack occurred at which do not change the frequencies. Consequently, it can be expected that existence of the critical points for resonant and antiresonant frequencies destroys the uniqueness of solution in crack detection problem by using the frequencies. Therefore, knowing the critical points, that are called hereby nodes of resonant and antiresonant frequencies, is important in solving the crack detection problem. The above equations show that nodes of resonant frequencies can be sought from equation d1(λ0, x) = 0, where λ0 is solution of frequency equation in case of uncracked bar. Nodes of five lowest resonant frequencies are given in Tab.2 for the cases of classical boundary conditions. Table 2. Nodes of resonant frequencies for bar with classical boundary conditions Mode Fixed end bar Free-free end bar Fixed-free end bar No 1 1/2 not available not available 2 1/4 3/4 0.5 1/3 3 1/6 1/2 5/6 1/3 2/3 0.2 0.6 4 1/8 3/8 5/8 7/8 0.25 0.5 0.75 1/7 3/7 5/7 5 1/10 3/10 01/2 7/10 9/10 0.2 0.4 0.6 0.8 1/9 3/9 5/9 7/9 For finding nodes of antiresonant frequencies of free-free bar and fixed-free bar one has the following equations cos λ0(1 − x) sin λ0x = 0 and cos λ0(1 − x) cos λ0x = 0, respectively. Solutions of the equations for five modes are given in Tab.3. Evidently, node of resonant frequencies in fixed end bar exactly coincide with nodes of antiresonant frequencies in fixed-free end bar. All the calculated nodes of resonant and antiresonant
9. Resonant and antiresonant frequencies of multiple cracked bar 165 frequencies of the fixed-free (Fig.1) and free-free (Fig.2) bars can be observed in Figs.1–2 where there are shown ratios of the frequencies to those of intact bar. The ratios (resonant on the left and antiresonant – on the right) are plotted versus crack position (from 1 to 1) in different crack depth (10%–50%). Table 3. Nodes of antiresonant frequencies for bar with free ends and fixed-free ends Mode Fixed-free end bar Free-free bar No 1 1/2 not available 2 1/4 3/4 2/3 3 1/6 1/2 5/6 2/5 4/5 4 1/8 3/8 5/8 7/8 2/7 4/7 6/7 5 1/10 3/10 1/2 7/10 8/10 2/9 4/9 6/9 8/9
10. 166 P. T. B. Lien, N. T. Khiem Fig. 1. Variation of three lowest resonant (left) and antiresonant (right) frequencies of fixed-free bar versus crack position with different crack depth (10%–50%)
11. Resonant and antiresonant frequencies of multiple cracked bar 167 Fig. 2. Variation of three lowest resonant (left) and antiresonant (right) frequencies of free-free bar versus crack position with different crack depth (10%–50%) Observing graphics given in the Figures demonstrates that crack at free end of bar makes no effect on the resonant frequencies, while it would do significant change in antiresonant frequencies if the frequency response function is defined at this position. Likely to the resonant frequencies, antiresonant frequencies are all monotonically re- duced with increasing depth of crack except the nodes (Tab.3) where they are unaffected by the crack presence. The ratios of resonant and antiresonant frequencies computed for free-free end bar with two cracks are presented respectively in Figs.3–4. Obviously, symmetric cracks make the same effect on resonant frequencies of the bar, but this is not true for antireso- nant frequencies. Also, the larger number of cracks makes more reduction of antiresonant frequencies. Fig. 3. Variation of first and second resonant frequencies versus position of two cracks with equal depth 30% for free end bar.
12. 168 P. T. B. Lien, N. T. Khiem Fig. 3. VariationFig. 3. Variation of first of first and and second second resonant resonant frequencies frequencies versus position versus of positiontwo cracks with of two cracks with equalequal depth depth 30% 30% for free for end free bar. end bar Fig. 4. Variation of first and second antiresonant frequencies versus positions of two cracks with Fig. 4. Variation of first and secondequal antiresonant depth 30% for free frequencies end bar. versus positions of two cracks 5.Conclusions with equal depth 30% for free end bar Fig. 4In. Variation the present of work first andthere second has been antiresonant derived a novelfrequencies form of versus characteristic positions equation of two cracks for resonant with and antiresonant frequencies ofequal multiple depth cracked 30% for bar free that end is explicitlybar. expressed in terms of crack magnitudes. The conducted characteristic equations are general regarding boundary conditions and 5.Conclusions In the present work there has been derived a novel form of characteristic equation for resonant and antiresonant frequencies of multiple cracked bar that is explicitly expressed in terms of crack magnitudes. The conducted characteristic equations are general regarding boundary conditions and
13. Resonant and antiresonant frequencies of multiple cracked bar 169 5. CONCLUSIONS In the present work there has been derived a novel form of characteristic equation for resonant and antiresonant frequencies of multiple cracked bar that is explicitly expressed in terms of crack magnitudes. The conducted characteristic equations are general regard- ing boundary conditions and exact in comparison with the numerous approximate ones known in the literature. These characteristic equations provide a useful tool for develop- ing crack detection procedures in bar. The antiresonant frequencies of bar with single and double cracks have been exam- ined versus crack position and depth mutually with the resonant ones. The obtained results show that there exist also nodes for antiresonant frequencies but they are differ- ent from those of resonant ones. Furthermore, resonant frequencies are defined indepen- dently upon where frequency response is measured, while antiresonant frequencies are strongly dependent on the FRF’s measurement. Therefore, effect of crack position on an antiresonant frequency may be also different if the antiresonant frequency is extracted from different FRFs. The observed different properties of resonant and antiresonant fre- quencies may be helpful for detecting cracks in bar by using both of them. The question that is open in this study is how to determine antiresonant frequencies of cracked bar with fixed-fixed ends. This problem is easily solved for uncracked bar, but it is unsolved for a bar with a crack because the points selected for measurement of FRF may disregard effect of the crack on the FRF. The problem mentioned above is subject for further study of the authors. ACKNOWLEDGEMENT The first author is thankful to University of Transport and Communications for its financial support to complete this study under grant number T2019-CB-013. The second author is thankful to NAFOSTED of Vietnam for support under grant number 107.02-2015.34 in completing this work. REFERENCES [1] R. D. Adams, P. Cawley, C. J. Pye, and B. J. Stone. A vibration technique for non-destructively assessing the integrity of structures. Journal of Mechanical Engineering Science, 20, (2), (1978), pp. 93–100. [2] Y. Narkis. Identification of crack location in vibrating simply supported beams. Journal of Sound and Vibration, 172, (4), (1994), pp. 549–558. [3] A. Morassi. Identification of a crack in a rod based on changes in a pair of natural frequencies. Journal of Sound and Vibration, 242, (4), (2001), pp. 577–596. [4] M. Dilena and A. Morassi. Detecting cracks in a longitudinally vibrating beam with dis- sipative boundary conditions. Journal of Sound and Vibration, 267, (1), (2003), pp. 87–103. [5] R. Ruotolo and C. Surace. Natural frequencies of a bar with multiple cracks. Journal of Sound and Vibration, 272, (1-2), (2004), pp. 301–316.
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