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: lrubio@ing.uc3m.es

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:

ppfer@ing.uc3m.es

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: antonino.morassi@uniud.it

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

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

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

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

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

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

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.

8

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

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)

11

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

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)

14

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)

17

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

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

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.

References430

[1] G. Hearn, R.B. Testa, Modal analysis for damage detection in structures.

Journal of Structural Engineering ASCE 117(10) (1991) 30423063.

[2] L.B. Freund, G. Herrmann, Dynamic fracture of a beam or plate in plane

bending. Journal of Applied Mechanics 76-APM-15 (1976) 112116.

[3] S. Caddemi, A. Morassi, Multi-cracked Euler-Bernoulli beams: mathemati-435

cal modelling and exact solutions. International Journal of Solids and Struc-

tures 50(6) (2013) 944956.

[4] G.M.L. Gladwell, Inverse problems in vibration, Kluwer Academic Publish-

ers, Dordrecht, The Netherlands, Second edition, 2004.

23

[5] R.D. Adams, P. Cawley, C.J. Pye, B.J. Stone, A vibration technique for440

non-destructively assessing the integrity of structures. Journal of Mechanical

Engineering Science 20(2) (1978) 93100.

[6] P. Gudmundson, The dynamic behaviour of slender structures with cross-

sectional cracks. Journal of the Mechanics and Physics of Solids 31 (1983)

329345.445

[7] A. Morassi, Crack-induced changes in eigenparameters of beam structures.

Journal of Engineering Mechanics ASCE 119(9) (1993) 17981803.

[8] R.Y. Liang, J. Hu, F. Choy, Quantitative NDE technique for assessing dam-

ages in beam structures. Journal of Engineering Mechanics ASCE 118(7)

(1992) 1468-1487.450

[9] L. Rubio L, An efficient method for crack identification in simply supported

Euler-Bernoulli beams. Journal of Vibration and Acoustics 131 (2009) Paper

051001.

[10] N. Lakshmanan, B.K. Raghuprasad, N. Gopalakrishnan, K. Sathishkumar,

S.G.N. Murthy, Detection of contiguous and distributed damage through455

contours of equal frequency change. Journal of Sound and Vibration 329

(2010) 13101331.

[11] Y. Narkis, Identification of crack location in vibrating simply supported

beams. Journal of Sound and Vibration 172(4) (1994) 549558.

[12] A. Morassi, Identification of a crack in a rod based on changes in a pair of460

natural frequencies. Journal of Sound and Vibration 242(4) (2001) 577596.

[13] M. Dilena, A. Morassi, The use of antiresonances for crack detection in

beams. Journal of Sound and Vibration 276(1-2) (2004) 195214.

[14] M. Dilena, A. Morassi, Detecting cracks in a longitudinally vibrating beam

with dissipative boundary conditions. Journal of Sound and Vibration 267465

(2003) 87103.

24

[15] M. Dilena, M. Fedele DellOste, A. Morassi, Detecting cracks in pipes filled

with fluid from changes in natural frequencies. Mechanical Systems and Sig-

nal Processing 25(8) (2011) 31863197.

[16] A. Morassi, M. Rollo, Identification of two cracks in a simply supported470

beam from minimal frequency measurements. Journal of Vibration and Con-

trol 7(5) (2001) 729740.

[17] L. Rubio, J. Fernandez-Saez, A. Morassi, Identification of two cracks in a

rod by minimal natural frequency and antiresonant frequency data. Submit-

ted, 2013.475

[18] W.T. Springer, K.L. Lawrence, T.J. Lawley, Damage assessment based on

the structural frequency-response function. Experimental Mechanics 28(1)

(1998) 34-37.

[19] H.-P. Lin, S.-C. Chang, Identification of crack location in a rod from mea-

surements of natural frequencies, Preprint, 2004.480

[20] B.P. Nandwana, S.K. Maiti, Detection of the location and size of a crack

in stepped cantilever beams based on measurements of natural frequencies.

Journal of Sound and Vibration 192(3) (1997) 324335.

[21] M.N. Cerri, F. Vestroni, Detection of damage in beams subjected to diffused

cracking. Journal of Sound and Vibration 234(2) (2000) 259276.485

[22] F. Vestroni, D. Capecchi, Damage detection in beam structures based

on frequency measurements. Journal of Engineering Mechanics ASCE 126

(2000) 761-768.

[23] E. Cabib, L. Freddi, A. Morassi, D. Percivale, Thin notched beams. Journal

of Elasticity 64(2/3) (2001) 157178.490

[24] R. Courant, D. Hilbert, Methods of Mathematical Physics (volume I), First

English Edition, Interscience Publishers, Inc., New York, 1966.

25

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