-
The full nonlinear crack detection problem in uniformvibrating rods
Lourdes Rubio
Department of Mechanical Engineering, University Carlos III of Madrid, Avda. de la
Universidad 30, 28911 Leganes, Madrid, Spain. E-mail: [email protected]
Jose Fernandez-Saez
Department of Continuum Mechanics and Structural Analysis, University Carlos III of
Madrid, Avda. de la Universidad 30, 28911 Leganes, Madrid, Spain. E-mail:
Antonino Morassi
Corresponding author. Universita degli Studi di Udine, Dipartimento di Ingegneria Civile eArchitettura, via Cotonificio 114, 33100 Udine, Italy. Tel.: +39 0432 558739; fax: +39
0432 558700. E-mail: [email protected]
Abstract
The basic problem in structural diagnostics via dynamic methods consists in
determining the position and severity of a single open crack in a beam from the
knowledge of a pair of resonant or antiresonant frequencies. A well-established
theory is available for longitudinally vibrating uniform beams when the severity
of the crack is small. In this paper we fill the gap present in the literature
by showing that the results of the linearized theory for slight damage can be
extended to a crack with any level of severity.
Keywords: Damage identification, cracks, natural frequencies, antiresonant
frequencies, inverse problems, rods
1. Introduction
Dynamic testing is commonly used as a diagnostic tool to detect damage that
has occurred in a mechanical system during service. The final goal is to predict
Preprint submitted to Journal of Sound and Vibration August 22, 2014
-
location and level of severeness of the degradation from the measurement of the
changes induced by the damage on the vibrational behavior of the system.5
Within the large class of diagnostic problems arising in structural mechanics,
the crack detection problem in vibrating beams by frequency data has received
a lot of attention in the scientific community in last two-three decades [1]. The
reasons for this interest are various. Firstly, the beam model describes the be-
havior of structural members that play an important role in many civil and10
mechanical engineering applications. Secondly, the problem of identifying a
crack in a beam is the basic diagnostic problem and, therefore, it represents
an important benchmark to test the effectiveness of damage identification tech-
niques. In addition, concerning the type of input data, in most applications
researchers have used natural frequencies or antiresonant frequencies as effec-15
tive damage indicator. Frequencies can be measured more easily than can mode
shapes, and are usually less affected by experimental and modelling errors.
Among the various models that have been proposed in literature to describe
(open) cracks in beams, localized flexibility models enable one for simple and
effective representation of the behavior of damaged elements [2]. The results20
of extensive series of vibration tests carried out on steel beams with a single
and multiple cracks confirm that lower natural frequencies are predicted by
localized flexibility models with accuracy comparable to that of the classical
Euler-Bernoulli model for a beam without defects, see, for example, [3].
In this paper we shall mainly concerned with the inverse problem of identi-25
fying a single open crack in a longitudinally vibrating beam by frequency data.
The crack is modelled by inserting a translational linearly elastic spring at the
damage cross-section. On assuming that the undamaged configuration is com-
pletely known, the inverse problem consists in determining the location s of
the crack, and its magnitude or severity. This latter parameter is expressed in30
terms of the stiffness K of the spring simulating the crack, and the undamaged
configuration is obtained by taking the limit as the stiffness K tends to infinity
or, equivalently, as the flexibility 1K tends to zero.
When the crack is small, namely when the cracked rod is a perturbation of
2
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the undamaged rod, a well-established theory for the inverse diagnostic problem
is available, see Gladwell ([4], Chapter 15). The cornerstone property concerns
with the possibility to express the change (2n) of the (square of) nth natural
frequency n produced by a small single crack as a product of the flexibility1K
and the square of the axial force N(U)n (s) evaluated on the undamaged config-
uration, for the relevant mode shape, at the cracked cross section [5], [6], [7]:
(2n) = (N
(U)n (s))2
K. (1)
Then, in the case of small crack, the ratios of the change in different natural
frequencies depend on the damage location s only, not on the severity K. Hearn35
and Testa [1], Liang et al. [8], Rubio [9], among others, have used this property
for damage localization in beam-like structures. See also Lakshmanan et al.
[10] for identification of localized damage in rods from the iso-eigenvalue change
contours constructed between pairs of different frequencies.
Concerning the rigorous, i.e., mathematically proved, identification of a small40
crack in an axially vibrating rod, worth of mention is the result obtained by
Narkis [11]. Narkis proved that a single crack in a uniform free-free rod can
be uniquely localized (up to a symmetric position) by using the first two nat-
ural frequencies. Working on a linearized version of the frequency equation,
Narkis obtained a closed-form solution for the crack location s. Using relation45
(1), Morassi [12] extended Narkiss result to rods with single small crack under
different set of end conditions and for different pairs of natural frequencies, pro-
viding closed-form expressions also for the damage severity K. Later on, Dilena
and Morassi [13] proved that the measurement of the first natural frequency and
the first antiresonance of the driving point frequency response function evaluated50
at one end of a free-free uniform rod allows to uniquely determine the position
of the crack and its severity, by means of closed-form expressions. Extensions
to cracked rods with dissipation [14], cracked pipes [15] and multi-cracked rods
and beams [16], [17] are also available.
All the above mentioned results hold in the case of small damage. Therefore,55
3
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an important question is left open: Can the above results be extended to the
case of not necessarily small damage? More specifically, the following question
can be posed:
(Q) Do the Narkiss result [11] and the Dilena-Morassis result [13] continue
to hold even for a large crack?60
The a priori hypothesis of small crack is often considered to be a no very
restrictive limitation, since in several practical situations it is of interest to be
able to identify the damage as soon as it arises in a structure. However, there
are several motivation in support of the opportunity to extend the theory to not
necessarily small cracks. Firstly, it is not easy to determine rigorously when a65
crack can be considered small. The smallness of a crack is typically established
on the basis of the crack-induced changes on the lower natural frequencies.
However, this criterion is difficult to apply, since the vibration modes have
different sensitivity to damage according to the position of the crack along the
beam axis. The introduction of an average frequency shift does not simplify70
the analysis, since it should be clarified how many data must be included in the
calculation and how the threshold value corresponding to small damage should
be selected. Secondly, the linearized theories by Narkis and Morassi show some
limitation when the damage is located near a point of vanishing sensitivity for
a vibration mode. In [12] it was shown that this aspect also prejudices the75
reliability in assessment of the damage severity (see Table 3 in [12]). Analogous
indeterminacy was recently encountered in identifying multiple small cracks in
a longitudinally vibrating beam by frequency measurements [17]. Near a zero-
sensitivity point, the first order effect of the damage on the frequencies vanishes.
Therefore, it is expected that the indeterminacy can be removed by considering80
the full nonlinear inverse problem instead of its linearized version. Finally, it is
of course desirable to have a unifying general theory of the diagnostic problem
capable to include damages ranging from small to large severity.
The waiver to the linearized theory implies strong consequences and, in
particular, the inability to obtain an explicit relationship like (1) expressing85
4
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the change of a natural frequency in terms of the damage parameters and the
undamaged configuration. In the full nonlinear crack detection problem, the
relationship between damage parameters (s,K) and natural frequencies is im-
plicitly contained in the frequency equation of the cracked rod. Diagnostic
methods based on the analysis of the frequency equation have been already in-90
vestigated in literature. The approach generally consists in considering a pair
of frequency data (typically, the first two natural frequencies) and then solving
numerically the nonlinear system formed by the frequency equation written for
the two selected frequencies in terms of the damage parameters s and K. We
refer, among others, to the pioneering research developed by Adams et al. [5],95
the studies by Springer et al. [18] and Lin and Chang [19] on longitudinally
vibrating rods, and the contributions by Nandwana and Maiti [20], Cerri and
Vestroni [21], Vestroni and Capecchi [22] on cracked beams in bending vibra-
tion. The above results seem to suggest that the answer to question (Q) is
positive. However, at the best of our knowledge, a rigorous proof of this general100
property is not available, as the conclusions of the cited papers are drawn either
on the basis of numerical analysis of specific cases or on the study of particular
experimental situations.
In this paper we definitively prove that the answer to question (Q) is positive,
that is the results by Narkis [11] and Dilena-Morassi [13] continue to hold even105
for not necessarily small cracks. The corresponding analysis is developed in
Section 3 and in Section 4, respectively. Proofs are based on a careful analysis
of the solutions of the system of two frequency equations written for the pair
of frequency input data and on the use of classical results of the variational
theory of eigenvalues for longitudinally vibrating rods. A series of applications110
to measurements on cracked steel rods are presented in Section 5.
2. Formulation of the diagnostic problem and some frequency bound
Let us consider a straight thin rod under longitudinal vibration and with
free-free end conditions (F-F). We assume that the rod is uniform and has a
5
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single crack at the cross-section of abscissa xd, with 0 < xd < L, where L is the115
length of the rod. The crack is assumed to remain open during vibration and is
modelled as a massless longitudinal linearly elastic spring with stiffness K, see,
for example, [2] and [23]. The value of K can be determined in terms of the
geometry of the cracked cross-section and of the material properties of the rod.
The free undamped longitudinal vibrations of the rod with radian frequency120
and spatial amplitude w = w(x) are governed by the following eigenvalue
problem written in dimensionless form
w + 2w = 0, in (0, s) (s, 1),w(s) = w(s+),
w(s+) w(s) = Fw(s),w(0) = w(1) = 0,
(2)
(3)
(4)
(5)
where s = xdL , s (0, 1), and 2 = L2
E 2; E is the (constant) Youngs modulus
of the material; is the (uniform) volume mass density of the material; F = EALK ,1250 < F < +, is the normalized flexibility associated to the crack, where A is thearea of the transversal cross-section of the rod. The eigenvalue problem (2)(5)
has an infinite sequence of eigenvalues {n}n=0, with 0 = 0 < 1 < 2 < ... andlimnn = +. (Note that here and in the sequel we consider as eigenvaluethe positive square root of 2.) In correspondence of each eigenvalue n, there130
exists a no trivial solution wn = wn(x) of Eqs. (2)(5), e.g., the eigenfunction
associated to the eigenvalue n.
The eigenvalues {n}n=0 are the roots of the frequency equation associatedto the cracked rod
P(; s,F) (sinF sin(s) sin(1 s)) = 0, (6)
where P = P(; s,F) is the corresponding characteristic polynomial. Therefore,the eigenvalues n = n(s,F) are regular functions of the damage parameterss and F .135
The eigenvalue problem for the undamaged rod can be found by taking
K or, equivalently, F 0+ in Eqs. (2)(5). The corresponding frequency
6
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equation is
P(U)() sin = 0 (7)
and the normalized eigenpairs {(U)n , w(U)n (x)}n=0 have the explicit expressions
(U)n = n, w(U)n (x) =
2 cos(nx), n = 0, 1, 2, ... (8)
By Eq. (6) and Eq. (7), it is easy to see that the 0th eigenvalue is insensitive
to damage, e.g., 0 = (U)0 = 0, since the corresponding vibration mode is a
longitudinal rigid motion.
As it is well-known, closed-form expressions for the eigenvalues of the dam-
aged rod are in general not available. Nevertheless, useful upper and lower
bounds on n, n 1, can be provided by the Variational Theory of eigenvalues[24]. Let us add the constraint
l(w) w(s+) w(s) = 0 (9)
to the no trivial solutions of the eigenvalue problem (2)(5) for the cracked rod.
The above constraint can be interpreted as a real-valued continuous functional
acting on the functional spaceH1(0, s)H1(s, 1) of the admissible configurationsof the cracked rod. Here, H(a, b), < a < b < , is the Hilbert space ofthe functions f : (a, b) R such that b
af2 < and b
a(f )2 < , where f is
the weak derivative of f in the sense of distributions. Clearly, the eigenvalues
of the problem (2)(5) under the constraint (9) coincide with the eigenvalues
of the undamaged rod. The addition of the constraint (9) restricts the space
of the admissible configurations reachable by the cracked rod, and then, by a
Monotonicity theorem, the eigenvalues cannot decrease, namely
n (U)n , for every n 1. (10)
Moreover, by the Minimax Eigenvalue Principle [24], the nth eigenvalue of the
system with the constraint (9) cannot be bigger than the (n + 1)th eigenvalue
of the unconstrained system, namely
(U)n n+1, for every n 1. (11)
7
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Then, by Eq. (10) and Eq. (11) we deduce the following upper and lower bound
on the eigenvalues of the cracked rod
(U)n1 n (U)n , for every n 2. (12)
In this paper we are mainly concerned with the determination of the position
s and the severity F of the crack by the knowledge of the two eigenvalues 1 and1402. To solve the inverse problem, we form a two-equations system by writing
the frequency equation (6) for = 1 and = 2. More precisely, with the aim
of reproducing the inverse problem arising in practical applications, we assume
that specific values of the two eigenvalues have been given, say 1 = 1, 2 = 2,
and we investigate on the solutions (s,F) of the nonlinear system of equations145
P(1; s,F) = 0,P(2; s,F) = 0.
(13)
(14)
Actually, since both 1 and 2 are strictly positive, Eqs. (13)(14) can be
replaced by
sin1 = F1 sin(1s) sin(1(1 s)),sin2 = F2 sin(2s) sin(2(1 s)).
(15)
(16)
We note that, by the symmetry of the undamaged rod, the two damage con-
figurations (s,F) and (1 s,F) are indistinguishable with our choice of data.Then, without loss of generality, we can assume
0 < s 12. (17)
In Section 4 we shall show how to select appropriately the spectral data in order150
to remove the a priori assumption (17).
We conclude this section with the following simple result.
Proposition 2.1. Under the above notation, we have:
i) If 1 = (U)1 , then there is no crack.
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ii) If 2 = (U)2 , then either there is no crack or there is a crack located at155
s = 12 . In this latter case, the severity of the crack is indeterminate.
Proof. Case i). Let 1 = (U)1 . Then, by Eq. (8) and Eq. (15) we have
F sin(s) sin((1 s)) = 0. But, by Eq. (17), we have sin(s) sin((1 s)) > 0and, then, F must vanish.
Case ii). Let 2 = (U)2 . Then, by Eq. (8) and Eq. (16) we have160
F sin(2s) sin(2(1 s)) = 0, which implies, by Eq. (17), either s = 12 (and Fremains indeterminate) or F = 0 (i.e., there is no damage).
By taking Proposition 2.1 into account and recalling that eigenvalues are
simple, in the sequel we shall assume strict inequalities in Eq. (12), namely our
data 1, 2 satisfy
0 < 1 < , (18)
< 2 < 2. (19)
An additional inequality on 2 will be introduced later on.
3. Crack identification by the first two natural frequencies
Main result of this section is the following theorem.165
Theorem 3.1. Let s (0, 12) and 0 < F < . Let the value 1, 2 of the firstand second eigenvalues be given and satisfying Eq. (18) and Eq. (19). There
exists a unique pair of damage parameters (s,F) solution to Eqs. (15) and (16).
The rest of the section is devoted to the proof of Theorem 3.1. To simplify
the notation, we omit the overline in denoting the eigenvalue data and we simply170
indicate by 1, 2 the given (e.g., experimental) value of the first and second
eigenvalue.
The proof consists of several steps. In the first step, the dependence on Fin Eqs. (15) and (16) is removed, and we write a single equation on the crack
9
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position s only:
2 sin1 sin(2s) sin(2(1 s)) = 1 sin2 sin(1s) sin(1(1 s)). (20)
This fact was already observed in the literature, even for cracked beams in
bending vibration, see, for example, Cerri and Vestroni (2000). The relationship
analogous to Eq. (20) for the linearized problem is the well-known property -175
mentioned in the Introduction - that, in the case of a small crack, the ratio of
the change in two different natural frequencies depends on the damage location
only, not on the severity of the damage.
By comparing the sign of the factors appearing in Eq. (20), we deduce an
additional bound on 2. By Eqs. (17)(19) we have180
1 (0, ) sin1 (0, 1],1s (0, 2 ) sin(1s) (0, 1),1(1 s) (0, ) sin(1(1 s)) (0, 1],2 (, 2) sin2 [1, 0),2s (0, ) sin(2s) (0, 1],2(1 s) (2 , 2) sin(2(1 s)) (0, 1) if 2(1 s) (2 , ),2(1 s) (2 , 2) sin(2(1 s)) [1, 0) if 2(1 s) (, 2).
(21)
(22)
(23)
(24)
(25)
(26)
(27)
By inserting Eqs. (21)(27) in Eq. (20) we deduce that sin(2(1 s)) < 0, thatis
1 s < 2 0,h(0) = (22 21) sin1 sin2 < 0,h( 12 ) = 0.
(32)
(33)
(34)
(35)
Then, since h = h(s) is a regular function, there exists sd (0, 12 ) such thath(sd) = 0, and the existence part of Theorem 3.1 is proved.185
To conclude the proof we need to show that such a point sd is unique. To
prove uniqueness, it is enough to prove that the following Claim is true:
Claim: h vanishes in exactly one point inside(0,
1
2
). (36)
In fact, if Eq. (36) is satisfied at the point sh, sh (0, 12 ), then h(sh) < 0. (Ifh(sh) 0, then h must vanish in more than one point in (0, 12 ), a contradiction.)Therefore, it is enough to apply the Intermediate Value Theorem for continuous
functions to conclude that, since {h(sh) < 0, h( 12 ) > 0, h(s) > 0 in (sh, 12 )},there exists exactly one zero of h inside (sh,
12 ). Observing that h(s) < 0 for190
s (0, sh), we have the thesis.Therefore, it remains to prove Claim (36). The zeros of h(s) = 0 are the
roots of the equation
f(s) = g(s), s (0,
1
2
), (37)
where
f(s) = 22 sin1 sin(2(1 2s)), (38)
g(s) = 21 sin2 sin(1(1 2s)). (39)
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We analyze separately the functions f and g. Concerning the function g, by
Eqs. (29), (24) and since 1(1 2s) (0, ), we have
g(0) < 0, g
(1
2
)= 0, g(s) < 0 in
(0,
1
2
). (40)
The first derivative of g is
g(s) = 231 sin2 cos(1(1 2s)) (41)
and then g may have at most one zero in(0, 12
), say sg, given by
sg =1
2
41. (42)
We distinguish two cases:
Case i) (small damage) If 2 < 1, then there exists a unique sg (0, 12 )such that g(sg) = 0, and g(s) < 0 in (0, sg), g(s) > 0 in (sg, 1).
Case ii) (large damage) If 1 0 in (0, 12 ) and g = g(s) is195
monotonically increasing in (0, 12 ).
The two cases i) and ii) are sketched in Figure 1. Note that g(12
)= 231 sin2 >
0.
Let us consider the function f . Since 0 < 2(1 2s) < 2 for s (0, 12
), the
function f = f(s) vanishes exactly at the point sf (0, 12
)given by
sf =1
2
22. (43)
Note that, since < 2, the point sf always belong to(0, 12
). Moreover, by
direct computation we have
f(0) < 0, f(0) g(0) = (22 21) sin1 sin2 < 0, f(1
2
)= 0. (44)
Now we are in position to analyze the set of solutions to Eq. (37). Depending
on the behavior of the function g we distinguish two cases.200
The simplest is Case ii), corresponding to 0 < 1 sg 2 < 21,
(45)
The first derivative of f is
f (s) = 232 sin1 cos(2(1 2s)). (46)
A direct calculation shows that f vanishes at s2 = 12 42 (which alwaysbelongs to
(0, 12
)) and that may vanish at s1 =
12 342 if 2 > 32 . Obviously,210
s2 sf = sf s1 = 42 . Note also that f (12
)= 232 sin1 < 0. The
qualitative behavior of f is shown in Figure 3 and it is compared to that of the
function g. One can conclude that in all the cases shown in Figure 3 there exists
only one solution to the equation f(s) = g(s) in(0, 12
).
The proof of the Claim (36) is complete.215
Finally, once the position sd of the crack has been found, the flexibility Fcan be determined by Eq. (15), and the proof of Theorem 3.1 is complete.
4. Crack identification by resonant and antiresonant frequency data
Theorem 3.1 states that the knowledge of the first two natural frequencies
of the problem (2)(5) allows to uniquely identify a single crack in a free-free220
uniform rod, up to a symmetric position. In this section we shall prove that the
nonuniqueness of the solution due to the symmetry of the undamaged configura-
tion can be removed by using in connection resonant and antiresonant frequency
data. Precisely, in addition to the first natural frequency, we use the first an-
tiresonant frequency of the driving point frequency response function (FRF) of225
the rod H(, 0, 0), where is the frequency variable. Antiresonances are the
13
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zeros of the FRF H(, 0, 0) and coincide with the natural frequencies of the
rod when the longitudinal displacement at the cross-section x = 0 is hindered,
namely the antiresonances are the natural frequencies of the cantilever rod sup-
ported at x = 0. Under the same notation of Section 3, the corresponding230
eigenvalue problem in dimensionless variables is
w + 2w = 0, in (0, s) (s, 1),w(s) = w(s+),
w(s+) w(s) = Fw(s),w(0) = 0, w(1) = 0,
(47)
(48)
(49)
(50)
where s (0, 1) and 0 < F < . We denote by {n}n=1 the eigenvalues ofthe problem (47)(50), with 0 < 1 < 2 < ... and limn n = . (As inSection 2, hereinafter we consider as eigenvalues the positive roots of 2.) The235
eigenvalue problem for the undamaged rod is obtained by taking F 0+ inEqs. (47)(50).
The eigenvalues {n}n=1 are the roots of the frequency equation
Q(; s,F) (cosF cos(s) sin(1 s)) = 0. (51)
The equation analogous to Eq. (51) for the undamaged cantilever is
Q(U)() cos = 0 (52)
and the normalized eigenpairs {(U)n , v(U)n (x)}n=1 are
(U)n =
(1
2+ (n 1)
), v(U)n (x) =
2 sin
(1
2+ (n 1)
)x, n = 1, 2, ...
(53)
With a slight change of notation with respect to Section 2 and 3, in this section it
turns out to be convenient to enumerate the eigenvalues of the free-free cracked
and uncracked rod from n = 1, namely
0 = 1 < 2 < ..., {n}n=1, (54)
0 = (U)1 <
(U)2 < ..., {(U)n }n=1. (55)
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-
As in Eq. (10) we have
n (U)n , for every n 1. (56)
In order to compare the eigenvalues of the cantilever cracked rod with those of
the free-free cracked rod, we consider the effect of adding the constraint
m(w) w(0) = 0 (57)
on the admissible set of no trivial solutions of the eigenvalue problem (2)(5) of
the free-free cracked rod. Clearly, the eigenvalues of (2)(5) under the constraint
(57) coincide with the eigenvalues of the cantilever cracked rod {n}n=1 and,by the Minimax Eigenvalue Principle [24], we have
n n n+1, for every n 1. (58)
In particular, focussing the attention on 2 and 1, by Eqs. (18), (56) (for
n = 1) and Eq. (58) (for n = 2), we have
0 < 2 < , (59)
0 < 1 0,
h(1) = 0,
h(0) = (22 + 21) sin2 cos1 < 0,h(1) = (22 21) sin2 cos1 > 0.
(74)
(75)
(76)
(77)
The function h = h(s) is regular and then, by Eqs. (74)(77) and the Inter-
mediate Value Theorem, there exists a point sd (0, 1) such that h(sd) = 0(existence of a crack position). It remains to prove that such a point is unique.
To prove that h = h(s) has exactly one zero in (0, 1), it is enough to prove that
h = h(s) vanishes only at one point inside (0, 1). The zeros of h are the roots
of the equation
f(s) = g(s), (78)
where
f(s) = 21 sin2 cos(1(1 2s)), (79)
g(s) = 22 cos1 sin(2(1 2s)). (80)
We first analyze the function g = g(s). By definition we have g(1 s) = g(s),and g = g(s) is an odd function with respect to s = 12 , with g(0) = g(1) =22 cos1 sin2 < 0. Moreover, recalling Eq. (59), it easy to see that the only265zero of g belonging to (0, 1) is at s = 12 . The graph of g is sketched in Figure 4.
Since, by Eq. (60), 1(1 2s) (2 , 2 ), we have cos(1(1 2s)) > 0,
and the function f = f(s) is strictly positive in (0, 1). Moreover, by definition,
f(1 s) = f(s) and f is an even function with respect to s = 12 , with f(0) =f(1) = 21 sin2 cos1 > 0, f
(12
)= 21 sin2 > f(0). We also have
g(1) f(1) = (22 21) sin2 cos1 > 0, (81)
g(0) f(0) = (22 + 21) sin2 cos1 < 0. (82)
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The graph of f is sketched in Figure 5, together with the possible behavior of the
function g. Clearly, there exists only one point s in (0, 1) such that g(s) = f(s).
Therefore, the function h = h(s) vanishes in (0, 1) only at s = s and this
implies the uniqueness of the crack position. The severity coefficient F can be270determined by Eq. (63), and the proof of Theorem 4.1 is complete.
5. Applications
In Section 3 we have shown that the knowledge of the first two positive
natural frequencies allows to uniquely identify a single crack in a free-free uni-
form rod, up to a symmetric position. In Section 4 we have proved that the275
nonuniqueness of the solution due to the symmetry of the undamaged config-
uration can be removed by using the first positive natural frequency and the
first antiresonant frequency of the driving point frequency response function
H(, 0, 0) of the rod. The identification procedure has been checked on the ba-
sis of an exhaustive numerical investigation. The results show that the method280
allows for the exact determination of the actual solution of the diagnostic prob-
lem in absence of errors on the data, thus confirming numerically the proofs of
Theorem 3.1 and Theorem 4.1. This is true, in particular, for the identification
of a crack located near a point of the beam axis of vanishing sensitivity for a
vibration mode. Taking into account of the data used in this paper, namely285
the first two positive resonant frequencies and the first antiresonant frequency
of the free-free bar, the only case that falls in this class corresponds to a crack
positioned in the vicinity of the middle-point of the bar, so that the second
positive resonance frequency of the free-free bar is insensitive to damage.
In this section we check the ability of the identification methodology to deal290
with real cases in which experimental/modelling errors in the resonant/antiresonant
data are present. In particular, two different applications shall be considered in
the sequel. They correspond to the extreme cases in which either the analytical
model used for the interpretation of the measurements is extremely accurate
(Example 1) or rather large modelling errors are present in the data (Example295
18
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2).
The mechanical model of Example 1 is a free-free steel rod of length L =
2.925 m and square solid cross-section 22 22 mm, which has been damagedat the cross-section located at xd = 1.000 m from one end, see Figure 1(a) in
[12]. Three different damage configurations, D1, D2 and D3, were obtained300
by introducing a notch of increasing depth. Details of the experimental setup
and of the experimental modal analysis results can be found in [12]. Table 1
compares the experimental natural frequencies and their corresponding analyt-
ical estimates for the undamaged and damaged rod. For the definition of the
analytical model for the damaged rod, the theoretical value of the stiffness K,305
for each damage configuration, was obtained by assuming that the position xd
of the damage is known and by taking the measured value for the fundamen-
tal (positive) frequency of the damaged rod. Thus, the actual values of K for
the different damage configurations are the following: K = 3.09119 1010N/m(case D1), K = 7.84984109N/m (case D2), K = 4.37183108N/m (case D3).310The analytical model turns out to be extremely accurate for all the configura-
tions under investigation and the discrepancy between measured and analytical
values of the first two positive natural frequencies is lower than 0.04 per cent.
Frequency shifts induced by the damage are significantly larger than the mod-
elling errors, and are about 0.08, 0.32, 5 per cent of the undamaged value for315
damage configurations D1, D2 and D3, respectively.
Table 2 shows the estimated values of the position xNLd of the cracked sec-
tion and the spring stiffness KNL. The results of the linearized version of the
identification procedure proposed by Morassi [12] are also included in Table 2
for comparison.320
It is possible to observe that the estimated values obtained using the full
nonlinear solution agree well with the actual solution of the problem: the max-
imum difference is about 3% for the estimated position and less than 4% for
the spring stiffness. The discrepancies between identified and actual damage
parameters connected with this solution are exclusively due to the experimen-325
tal errors associated to frequency measurements and to modelling errors. De-
19
-
viations are typically bigger for less severe levels of damage. The maximum
discrepancy occurs for damage configuration D1. It is worthy to note that in
this case the linearized estimation procedure gives better results than the full
nonlinear method, whereas, as it was expected, the linearized solution deviates330
increasingly from the actual solution as the damage severity increases.
Example 2 is based on the experimental results presented in [13]. The me-
chanical model is a double T free-free steel rod of series HE100B. The length
is L = 2.747 m and the linear mass density is 20.4 kg/m. The damage was
obtained by saw cutting the rod at the cross-section at xd = 0.550 m far from335
the left end, see Figure 5(a) in [13]. Two different damage configurations, called
D4 and D5 in the following, were obtained by introducing a notch of increas-
ing depth. Details of the experimental setup and discussion on the results of
experimental modal analysis can be found in [13].
Table 3 compares the experimental values of the first two (positive) natural340
frequencies and the first antiresonance of H(, 0, 0) with their corresponding an-
alytical estimates, both for the undamaged and damaged rod. The actual value
of the spring stiffness K of the cracked rod was defined by assuming the position
xd of the damage as known and providing that, for each damage configuration,
the measured and the analytical fundamental (positive) natural frequency co-345
incide. Thus, the actual values of K for the different damage configurations
are the following: K = 3.507 1010N/m (case D4), K = 1.736 109N/m(case D5). Frequency shifts induced by damage in the first two resonances are
around 0.20.6 and 410 per cent for D4 and D5 configurations, respectively.
Antiresonance decreasing is about 7 per cent for configuration D5. As mod-350
elling errors are concerned, the analytical model turns out to be accurate for
natural frequencies, with maximum differences between experimental and the-
oretical values equal to 0.25 per cent. Modelling error for the first antiresonant
frequency is also around 1 per cent, with the exception of the configuration
D4 in which a discrepancy around 6 per cent was noticed. The source of this355
disagreement was not explained in [13], and it is expected that the important
error on first antiresonance will produce wrong estimates of the damage param-
20
-
eters for damage configuration D4. It should be also noticed that, although the
analytical model can be considered accurate in predicting natural frequencies,
percentage crack-induced changes in natural frequencies are comparable with360
the accuracy of the rod model for damage level D4.
Table 4 collects the estimated values of the position of the cracked section,
xNLd , and the spring stiffness, KNL, when the first two positives resonant fre-
quencies are used in the identification process. As expected, for configuration
D4, the inaccuracy of the data prejudices the reliability of the reconstruction.365
The results obtained by the linearized technique proposed in [12] are also in-
cluded for comparison in Table 4. As the small damage case (D4) is concerned,
the estimation obtained from the linearized procedure is better, whereas the
full nonlinear procedure gives an estimation closer to the actual solution as the
damage severity increases.370
Finally, Table 5 shows the results when the first positive resonant and the
first antiresonant frequency are used in the identification process. It can be
noted that the employment of the antiresonance, which is affected by large
modelling error in configuration D4, prevents obtaining accurate estimation of
the damage parameters. Conversely, the solution predicted by the theory is a375
satisfactory estimate of the actual solution of the damage problem for configu-
ration D5. In this case, in fact, modelling errors are small with respect to the
shifts induced by the crack and, as it was expected, the full nonlinear solution
gives a more accurate estimate of the damage parameters than the linearized
procedure.380
6. Conclusions
In this paper we have considered the problem of identifying a single open
crack in a uniform longitudinally vibrating rod by minimal frequency data. The
crack is modelled by a massless translational linearly elastic spring placed at
the damaged cross section. Two parameters, the position of the crack and its385
severity, are the unknowns of the inverse problem, and two spectral information
21
-
are considered as input data.
We have shown that the crack can be uniquely identified - up to a symmet-
rical position - by the knowledge of the first two positive natural frequencies of
the rod under free-free end conditions. Moreover, it was also shown that the390
symmetric position can be removed by replacing the second positive natural
frequency with the first antiresonant frequency of the driving point frequency
response function evaluated at one end of the rod.
The above results were known to hold in the case of small crack. The analy-
sis developed in this paper allows to extend the existing theory to a crack with395
any level of severity. The methodology used for the proof is completely different
from that adopted in the case of small damage. It is based on a careful analysis
of the solutions of the nonlinear system formed by the frequency equation writ-
ten for the pair of spectral input data, coupled with suitable upper and lower
bounds derived within the variational theory of eigenvalues. Numerical results400
are in agreement with the theory when exact analytical data are employed in
identification. The theory has been also checked on a series of dynamic tests on
cracked steel beams. Experiments show that if frequency data used in identifi-
cation are affected by relatively small errors with respect to the shifts induced
by the crack, then the proposed full nonlinear solution gives a more accurate405
estimate of the position and severity of the crack than the linearized procedure
based on the assumption of small damage.
The method proposed in this paper can be used, in principle, also to deal
with the corresponding diagnostic problem for a uniform beam in bending vi-
bration with a single open crack. In that case, the crack is represented by the410
insertion of a massless rotational spring at the damaged cross-section. A pre-
liminary analysis of the problem shows that the study of the solutions of the
system formed by the frequency equation written for, say, the first two natu-
ral frequencies is significantly more difficult with respect to the axial case. No
general results seem to be available yet on this challenging diagnostic problem.415
As a final remark, we point out that another aspect worth of investigation,
both from the theoretical and practical point of view, stands on the possibility
22
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of extending our analysis to axially vibrating rods with variable profile. Such an
extension seems to be not trivial at all, since, for the variable coefficient case, no
closed-form expression for the eigenpairs is available and the frequency equation420
is written in implicit form. It is likely that new ideas and mathematical tools
must be developed to deal with this class of problems.
Acknowledgement
The work of A. Morassi is supported by the University Carlos III of Madrid-
Banco de Santander Chairs of Excellence Programme for the 2013-2014 Aca-425
demic Year.
A. Morassi wishes to thank the colleagues of the University Carlos III of
Madrid, especially Professors L. Rubio and J. Fernandez-Saez, for the warm
hospitality at the Department of Engineering Mechanics.
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Table Captions
Table 1. Example 1: First two positive natural frequencies fn =n2 , n = 1, 2,
of the undamaged free-free rod and their values associated to the damaged cases495
Di, i = 1, 2, 3. Frequency values in Hz. Note: Modelling errors are reported
in brackets, = 100 (fModel fExp.)/fExp..
Table 2. Example 1: Damage parameter estimates xNLd , KNL obtained by the
first two natural frequencies fn =n2 , n = 1, 2. Actual solution: xd = 1.000
m; K = 3.09119 1010N/m (case D1), K = 7.84984 109N/m (case D2),500K = 4.37183 108N/m (case D3). xLd and KL are the corresponding estimatesobtained by the linearized procedure presented in [12].
Table 3. Example 2: First two positive natural frequencies fn =n2 , n =
2, 3, (note the ordering relation in Eq. (54)) and first antiresonant frequency
f1A =12 of the undamaged free-free rod, and their values associated to the505
damaged cases Di, i = 4, 5. Frequency values in Hz. Note: Modelling errors
are reported in brackets, = 100 (fModel fExp.)/fExp..
Table 4. Example 2: Damage parameter estimates xNLd , KNL obtained by
the first two positive natural frequencies fn =n2 , n = 2, 3 (note the ordering
relation in Eq. (54)). Actual solution: xd = 0.550 m; K = 3.507 1010N/m510(case D4), K = 1.736 109N/m (case D5). xLd and KL are the correspondingestimates obtained by the linearized procedure presented in [12].
Table 5. Example 2: Damage parameter estimates xNLd , KNL obtained by the
first positive natural frequency f2 =22 (note the ordering relation in Eq. (54))
and the first antiresonant frequency f1A =12 . Actual solution: xd = 0.550 m;515
K = 3.507 1010N/m (case D4), K = 1.736 109N/m (case D5). xLd and KLare the corresponding estimates obtained by the linearized procedure presented
in [13].
26
-
Figure Captions
Figure 1. The function g = g(s).520
Figure 2. Case ii): unique solution of f(s) = g(s).
Figure 3. Case i): unique solution of f(s) = g(s). Left: 32 < 2(< 2); right:
(
-
Table 1: Example 1: First two positive natural frequencies fn =n2
, n = 1, 2, of the undam-
aged free-free rod and their values associated to the damaged cases Di, i = 1, 2, 3. Frequency
values in Hz. Note: Modelling errors are reported in brackets, = 100 (fModel fExp.)/fExp..
Data Undamaged D1 D2 D3
Exper. Model Exper. Model Exper. Model Exper. Model
f1 882.25 882.25 881.5 881.5 879.3 879.3 831.0 831.0
(0.00) (0.00) (0.00) (0.00)
f2 1764.6 1764.5 1763.3 1763.1 1759.0 1759.2 1679.5 1680.1
(-0.006) (-0.011) (0.011) (0.036)
28
-
Table 2: Example 1: Damage parameter estimates xNLd , KNL obtained by the first two natural
frequencies fn =n2
, n = 1, 2. Actual solution: xd = 1.000 m; K = 3.09119 1010N/m (caseD1), K = 7.84984 109N/m (case D2), K = 4.37183 108N/m (case D3). xLd and KL arethe corresponding estimates obtained by the linearized procedure presented in [12].
Position [m] Spring stiffness [N/m]
Case xNLd xLd K
NL KL
D1 1.031 1.012 3.20720 1010 3.13548 1010
D2 0.992 0.989 7.78180 109 7.73130 109
D3 0.998 1.021 4.36188 108 4.77239 108
29
-
Table 3: Example 2: First two positive natural frequencies fn =n2
, n = 2, 3, (note the
ordering relation in Eq. (54)) and first antiresonant frequency f1A =12
of the undamaged
free-free rod, and their values associated to the damaged cases Di, i = 4, 5. Frequency values
in Hz. Note: Modelling errors are reported in brackets, = 100(fModelfExp.)/fExp..Data Undamaged D4 D5
Exper. Model Exper. Model Exper. Model
f2 941.1 941.1 939.3 939.3 901.8 901.8
(0.00) (0.00) (0.00)
f3 1879.1 1882.2 1868.3 1872.5 1693.3 1697.6
(0.16) (0.23) (0.25)
f1A 468.6 470.6 439.5 468.2 432.9 427.7
(0.41) (6.52) (-1.19)
30
-
Table 4: Example 2: Damage parameter estimates xNLd , KNL obtained by the first two
positive natural frequencies fn =n2
, n = 2, 3 (note the ordering relation in Eq. (54)).
Actual solution: xd = 0.550 m; K = 3.507 1010N/m (case D4), K = 1.736 109N/m(case D5). xLd and K
L are the corresponding estimates obtained by the linearized procedure
presented in [12].
Position [m] Spring stiffness [N/m]
Case xNLd xLd K
NL KL
D4 0.253 0.461 8.387 109 2.626 1010
D5 0.539 0.623 1.679 109 2.075 109
31
-
Table 5: Example 2: Damage parameter estimates xNLd , KNL obtained by the first positive
natural frequency f2 =22
(note the ordering relation in Eq. (54)) and the first antiresonant
frequency f1A =12
. Actual solution: xd = 0.550 m; K = 3.507 1010N/m (case D4),K = 1.736 109N/m (case D5). xLd and KL are the corresponding estimates obtained by thelinearized procedure presented in [13].
Position [m] Spring stiffness [N/m]
Case xNLd xLd K
NL KL
D4 0.143 0.157 2.760 109 3.301 109
D5 0.598 0.672 1.968 109 2.348 109
32
-
Figure 1: The function g = g(s).
33
-
Figure 2: Case ii): unique solution of f(s) = g(s).
34
- Figure 3: Case i): unique solution of f(s) = g(s). Left: 32 < 2(< 2); right: (
-
Figure 4: The function g = g(s) of Eq. (80).
36
-
Figure 5: The function f = f(s) of Eq. (79) and the intersection between f = f(s) and
g = g(s).
37
IntroductionFormulation of the diagnostic problem and some frequency boundCrack identification by the first two natural frequenciesCrack identification by resonant and antiresonant frequency dataApplicationsConclusions