Resonant and antiresonant frequencies of multiple cracked bar

 Vietnam Journal of Mechanics, VAST, Vol.41, No. 2 (2019), pp. 157 – 170
 DOI: https://doi.org/10.15625/0866-7136/13092
 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
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,
 Resonant and antiresonant frequenci ... ................................................................
 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)
 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
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
 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
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%)
 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. 
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 
 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.
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