Identification of two cracks with different severity in beams and rods from minimal frequency data

Abstract

It has been known for a long time that the problem of identifying two small cracks in a simply supported beam from the first three natural frequencies can be analytically formulated and solved if the two cracks have equal severity. In this paper we extend this result to the case of cracks with different severity. Each crack is simulated by a rotational elastic spring and the inverse problem is solved in terms of the damage-induced changes in the first four natural frequencies. Closed-form expressions of the damage parameters in terms of the measured frequencies are obtained. The results can be extended to the identification of two cracks in a longitudinally vibrating beam based on a suitable set of natural frequency and antiresonant frequency data. Numerical simulations support the theory, and show that if accurate input data are available and the cracks are not too close, then damage identification leads to satisfactory results.

Keywords

Multiple cracks damage identification natural frequencies antiresonant frequencies beams rods

1. Introduction

This paper is focussed on the identification of two cracks in beams and rods from natural frequency and antiresonant frequency data.

Morassi and Rollo (2001) considered the problem of identifying two cracks in a simply supported uniform beam in bending vibration from minimal natural frequency data. Each crack was assumed to remain open during vibration and was modeled as a linearly elastic rotational spring located at the damaged cross-section. By considering the damaged beam as a perturbation of the undamaged one (e.g. under the assumption of small damage), the authors proved that the inverse problem can be formulated and solved in terms of the first three natural frequencies, provided that the two cracks have equal severity. The inverse problem turns out to be ill-posed; namely, even by leaving symmetrical positions aside, cracks in two sets of different locations can produce identical changes in first three natural frequencies. An analogous result holds for a free–free longitudinal rod with two cracks. It is worth noticing that closed-form expressions of the damage parameters in terms of the frequency shifts were obtained in Morassi and Rollo (2001). This fact is not trivial at all, since the resolving system of equations is highly nonlinear in the crack position.

The above result has been recently extended in the case of free–free uniform longitudinally vibrating rods with two small open cracks. Under the assumption of cracks of equal severity, it was shown in Rubio et al. (2013) that a suitable use of lower resonant and antiresonant frequency data allows us to exclude all the symmetrical crack locations occurring in the formulation by Morassi and Rollo (2001). Even in this case, closed-form expressions of the damage parameters in terms of the data were provided.

Both the papers Morassi and Rollo (2001) and Rubio et al. (2013) leave an important question unsolved, namely the identification of two cracks having different severity. The present paper addresses this problem.

In the case of a simply supported uniform beam in bending vibration with two small open cracks, we show in Section 2 that the inverse problem can be formulated and solved in closed form in terms of the first four natural frequencies of the beam. The strategy of the proof is different from that of the papers Morassi and Rollo (2001) and Rubio et al. (2013), and it strongly relies on the peculiar structure of the resolving system of four nonlinear equations obtained by linearizing the function expressing the dependence of the natural frequency on the severity of the damage in a neighborhood of the undamaged configuration. More precisely, by introducing two auxiliary variables defined both in terms of the position and the severity of each crack, it was possible to find the crack positions as solutions of a second-degree polynomial equation and, next, to recover information on damage severity. Since only natural frequencies are used as data and the undamaged configuration is symmetrical, the two cracks are uniquely determined up to symmetry with respect to the mid-span cross-section.

Concerning the indeterminacy due to the structural symmetry, in the second part of the paper (Section 3) we consider the identification of two small cracks of different severity in a free–free uniform rod under longitudinal vibration. We show that the use of the first two natural frequencies and the first two antiresonant frequencies of the driving point frequency response function (FRF) evaluated at one end of the rod allows us to exclude the spurious solutions due to the symmetry of the undamaged configuration. Even in this case, closed-form solutions are available for the damage parameters in terms of the data.

The last part of both Sections 2 and 3 collects the results of an exhaustive numerical analysis of the inverse problem for beams and rods with two cracks. In particular, the analysis of cases with neighboring or distant cracks having different severities is carefully developed. Numerical simulations support the theoretical results and show that if the cracks are sufficiently distant from each other and frequency data are affected by errors relatively small with respect to the frequency changes induced by the damage, then the proposed methodology leads to satisfactory indications.

The mathematical tool used to prove the above results strongly relies on the general property that in beam-like structures the change in a natural frequency produced by a small single crack may be represented as a product of two terms, of which one is proportional to the severity and the other depending solely on the location of the damage (Adams et al., 1978; Gudmundson, 1983; Morassi, 1993; Gladwell, 2004). An important consequence is the following: if a single crack is present, then the ratios of the change in different natural frequencies depend only on the damage location, not on its severity. Hearn and Testa (1991), Liang et al. (1992) and Rubio (2009), among others, have used this property for a damage localization analysis in beam-like structures. Narkis (1994) followed this approach for identifying a single crack in a uniform free–free rod under longitudinal vibration and proved that a single crack can be uniquely localized (up to symmetry) by using the first two natural frequencies. Morassi (2001) extended Narkis’s results to beams and rods with a single crack under different sets of end conditions and for different pairs of natural frequencies. He showed that the problem of determining the location of the crack from changes in two natural frequencies is generally ill-posed: if the system is symmetrical, then damage at any one of a set of symmetrical points will produce identical changes in natural frequencies. Even if the system is not symmetrical, damage at different locations can still produce identical changes in two natural frequencies. In Dilena and Morassi (2004) it was shown how the use of antiresonant frequency data coupled with natural frequency data can be useful to exclude the spurious symmetrical positions of the damage occurring in initially symmetrical beams.

In the case of multiple cracks, the damage-induced change on a natural frequency takes into account the global damage pattern; see equations (8) and (62) for bending and longitudinal vibrations, respectively. The general property mentioned above connecting ratios of different natural frequencies and damage locations obviously is no longer true. Therefore, the identification procedure becomes more involved with respect to the case in which the damage is restricted to one location, as emerges from the analysis shown in Sections 2 and 3.

Finally, it is appropriate to recall that other approaches have been followed in the literature to identify multiple cracks in beams. Without claim of completeness, and referring to Sekhar (2008) and Caddemi and Caliò (2014) for an updated overview of the topic, and to Caddemi and Caliò (2009) for a comprehensive presentation of the direct vibrational problem, here we restrict attention to methods in which only lower resonant/antiresonant frequency data are used in identification.

Assuming, as above, the linear concentrated flexibility model to describe cracks in rods and beams, one approach consists in considering (at least) as many natural frequencies as the unknowns of the problem (e.g. two unknowns for each crack, the position and the severity), and then numerically solving the system formed by the characteristic equation written for all the natural frequencies in terms of the damage parameters (Cerri and Vestroni, 2000; Vestroni and Capecchi, 2000; Attar, 2012; Mazanoglu and Sabuncu, 2012). Inverse transcendental eigenvalue problems for the identification of multiple open cracks in a longitudinally vibrating rod were considered by Singh (2009). The author noticed that the possible presence of spurious solutions (due to the symmetry of the undamaged configuration) found in solving the nonlinear system of characteristic equations can be avoided by carefully selecting the data and using simultaneously natural frequency and antiresonant frequency measurements. In general terms, this is a powerful class of methods, but has the drawback of requiring strong support on numerical simulation, with the consequence of making it difficult to find out general properties, such as, for example, the indication of optimal data to be used in order to reduce nonuniqueness effects in the inverse problem solution.

Another common approach to multi-cracked identification in beams consists in transforming the inverse problem into an optimization problem. In general terms, the damage parameters are determined such that the natural frequencies of the mechanical model are closest (in a least squares sense) to those found experimentally; see Ruotolo and Surace (1997), Cerri and Vestroni (2000), Vestroni and Capecchi (2000), Khiem and Lien (2004) and, for a linearized version suitable in the case of small cracks, Patil and Maiti (2003) and Rubio (2009). More recently, Khiem and Toan (2014) proposed a procedure for multiple-crack detection in beams by natural frequency measurements. Their method combines a second-order approximation of natural frequencies with respect to crack magnitudes via Rayleigh’s quotient, a suitable Tikhonov regularization method, and the so-called crack scanning method to estimate the number of cracks. This class of techniques allows us to consider systems of high complexity and beams with a large number of cracks. However, the approach generally requires subtle analysis in order to deal with the possible nonconvexity of the error function and, as a consequence, with the appearance of multiple local and global minima. In connection with this point, see the interesting analysis presented in Vestroni and Capecchi (2000) and the recent contribution by Greco and Pau (2012) on crack identification in a frame.

2. Inverse problem in a bending vibrating beam

2.1. Formulation of the problem

Let us consider a straight thin simply supported beam under bending vibration. We assume that the beam is uniform and has two cracks at two different cross-sections of abscissa s₁, s₂, with 0 < s_i < L for i = 1, 2, where L is the length of the beam. Assuming that cracks remain open during the vibration, every crack is represented by inserting a massless rotational linearly elastic spring with stiffness K_i, i = 1, 2, at the cracked cross-section. We refer, for example, to Freund and Herrmann (1976) for a justification of this localized flexibility model of a crack based on fracture mechanics arguments, and to Caddemi and Morassi (2013) for an alternative derivation. The magnitude of the spring stiffness depends on the geometry of the cracked cross-section and on the material properties of the beam. The free undamped bending vibrations of the cracked beam with radian frequency $\sqrt{λ}$ and spatial amplitude v = v(z) are governed by the following eigenvalue problem:

{EIv IV - λ γ Av = 0, z \in (0, s 1) \cup (s 1, s 2) \cup (s 2, L) (1) v (s_{i}^{-}) = v (s_{i}^{+}), v ″ (s_{i}^{-}) = v ″ (s_{i}^{+}), v ″' (s_{i}^{-}) = v ″' (s_{i}^{+}), i = 1, 2 (2) K i (v' (s_{i}^{+}) - v' (s_{i}^{-})) = EIv ″ (s i), i = 1, 2 (3) v (0) = v (L) = 0 (4) v ″ (0) = v ″ (L) = 0 (5)

where

ϕ (z_{0}^{-}) = lim z \to z_{0}^{-} ϕ (z)

ϕ (z_{0}^{+}) = lim z \to z_{0}^{+} ϕ (z)

denote the left and right limit value of the function ϕ = ϕ(z) at z₀. In the above equations, E is the (constant) Young’s modulus of the material; I and A are the moment of inertia and the area of the transversal cross-section, respectively; γ is the (uniform) volume mass density of the material. Under our assumptions, there exists an infinite sequence of eigenvalues

{λ n} n = 1 \infty

, with

0 < λ 1 < λ 2 < \dots

and

lim_{n \to \infty} λ n = \infty

, such that the problem (1) to (5) has no trivial solution v_n = v_n(z) (eigenfunction associated to the eigenvalue λ_n). The eigenvalue problem for the undamaged beam can be obtained as the limit in (1) to (5) as

K i \to \infty, i = 1, 2

. The mass-normalized eigenpairs of the undamaged beam

{λ_{n}^{(U)}, v_{n}^{(U)} (z)} n = 1 \infty

can be evaluated explicitly and are equal to

λ_{n}^{(U)} = \frac{EI}{γ A} (\frac{n π}{L}) 4, v_{n}^{(U)} (z) = \sqrt{\frac{2}{γ AL}} \sin (\frac{n π z}{L}), n = 1, 2, \dots

(6)

If cracks are small, namely, both K₁ and K₂ are large enough, then we may find the first-order damage-induced variation in the eigenvalues as shown, for example, in Morassi (1993), Loya et al. (2006) and Rubio (2009). By taking

λ n = λ_{n}^{(U)} + δ λ n

(7)

we find

δ λ n = - \frac{(M_{n}^{(U)} (s 1)) 2}{K 1} - \frac{(M_{n}^{(U)} (s 2)) 2}{K 2}

(8)

for every n ≥ 1, where the bending moment associated to the nth undamaged vibration mode is given by

M_{n}^{(U)} (z) = - EI \frac{d 2}{dz 2} (v_{n}^{(U)} (z))

(9)

Inserting the explicit expression (6) of the eigenpairs of the undamaged beam into equation (8), we obtain

C n = \frac{\sin 2 \frac{n π s 1}{L}}{K 1} + \frac{\sin 2 \frac{n π s 2}{L}}{K 2}, n \geq 1

(10)

where the nonnegative quantity

C n = - \frac{δ λ n}{\frac{2 EI}{L} λ_{n}^{(U)}}, n \geq 1

(11)

only depends on the undamaged beam and on the measured eigenvalue shift induced by the damage on the nth eigenvalue.

We are concerned with the problem of finding the position and severity of the two cracks, namely the four parameters {(s₁, K₁), (s₂, K₂)}, from a minimal set of natural frequency data. Before embarking on the calculations, it should be observed that, due to the symmetry of the undamaged beam, the damage configurations {(s₁, K₁), (s₂, K₂)}, {(L−s₁, K₁), (L−s₂, K₂)}, {(L−s₁, K₁), (s₂, K₂)}, {(s₁, K₁), (L−s₂, K₂)} are indistinguishable from natural frequency input data. In fact, all of them correspond to the same natural frequency shifts evaluated from the undamaged configuration. Therefore, taking into account this intrinsic lack of uniqueness of the problem, as in Morassi and Rollo (2001) we shall assume

0 < s 1 < s 2 \leq \frac{L}{2}

(12)

2.2. Identification of two cracks in a simply supported beam by frequency data

In this section we shall show how to select minimal frequency data in order to properly formulate the diagnostic problem and to find closed-form expressions of the damage parameters.

Since four parameters need to be determined, we consider as minimal data the changes in the first four natural frequencies. This choice has the merit of allowing an analytical treatment of the inverse problem. Moreover, it is also justified from the point of view of applications, since experiments show that the one-dimensional model (1) to (5) is able to reproduce the low-frequency behavior well, while it loses precision as the order of the natural frequency increases (Davini et al., 1995; Caddemi and Morassi, 2013). By writing equation (10) for the first four frequencies, we obtain the following system of four nonlinear equations in the unknowns (s₁, K₁) and (s₂, K₂):

{\frac{1}{K 1} \sin 2 \frac{π s 1}{L} + \frac{1}{K 2} \sin 2 \frac{π s 2}{L} = C 1 (13) \frac{1}{K 1} \sin 2 \frac{2 π s 1}{L} + \frac{1}{K 2} \sin 2 \frac{2 π s 2}{L} = C 2 (14) \frac{1}{K 1} \sin 2 \frac{3 π s 1}{L} + \frac{1}{K 2} \sin 2 \frac{3 π s 2}{L} = C 3 (15) \frac{1}{K 1} \sin 2 \frac{4 π s 1}{L} + \frac{1}{K 2} \sin 2 \frac{4 π s 2}{L} = C 4 (16)

where

C i > 0, i = 1, 2, 3, C 4 \geq 0

(17)

The particular case in which C₄ = 0 is straightforward. By equations (16) and (12), if C₄ vanishes, then

s 1 = \frac{L}{4}

and

s 2 = \frac{L}{2}

. Inserting these expressions into equations (13), (14), one easily gets

K 1 = \frac{1}{C 2}, K 2 = \frac{2}{2 C 1 - C 2}

(18)

In order to discuss the general case, let us introduce the following position variables:

x = x (s 1) = \cos \frac{2 π s 1}{L} \in [- 1, 1), y = y (s 2) = \cos \frac{2 π s 2}{L} \in [- 1, 1)

(19)

Note that for

s \in (0, L / 2]

the function

f (s) = \cos (\frac{2 π s}{L})

is a one-to-one correspondence between the interval (0, L/2] and the interval [−1, 1). Therefore, if we were able to find the two variables {x, y}, then we could determine uniquely the positions {s₁, s₂} of the two cracks.

By using standard trigonometric identities and recalling that

\sin 23 α = \frac{1}{2} - 2 \cos 32 α + \frac{3}{2} \cos 2 α

(20)

after a rearranging of the terms, the system (13) to (16) takes the form

{\frac{1 - x}{K 1} + \frac{1 - y}{K 2} = 2 C 1 (21) \frac{1 - x}{K 1} (1 + x) + \frac{1 - y}{K 2} (1 + y) = C 2 (22) \frac{1 - x}{K 1} (1 + 2 x) 2 + \frac{1 - y}{K 2} (1 + 2 y) 2 = 2 C 3 (23) \frac{1 - x}{K 1} (1 + x) x 2 + \frac{1 - y}{K 2} (1 + y) y 2 = \frac{C 4}{4} (24)

Let us introduce the two auxiliary variables

ξ 1 = ξ 1 (x, K 1) = \frac{1 - x}{K 1} \in (0, + \infty), ξ 2 = ξ 2 (y, K 2) = \frac{1 - y}{K 2} \in (0, + \infty)

(25)

Then, system (21) to (24) becomes

{ξ 1 + ξ 2 = 2 C 1 (26) ξ 1 (1 + x) + ξ 2 (1 + y) = C 2 (27) ξ 1 (1 + 2 x) 2 + ξ 2 (1 + 2 y) 2 = 2 C 3 (28) ξ 1 (1 + x) x 2 + ξ 2 (1 + y) y 2 = \frac{C 4}{4} (29)

The next step consists of expressing the auxiliary variables ξ₁, ξ₂ in terms of the position variables x, y by means of the equations (26) and (27). By (26) we have

ξ 2 = 2 C 1 - ξ 1

(30)

and then, by plugging this expression into (27), we get

ξ 1 (x - y) = C 2 - 2 C 1 (1 + y)

(31)

Under assumptions (12) we have

x \neq y

(32)

and then, by inverting (31) and (30), we find

ξ 1 = \frac{C 2 - 2 C 1 (1 + y)}{x - y}

(33)

ξ 2 = - \frac{C 2 - 2 C 1 (1 + x)}{x - y}

(34)

Now we use equations (28) and (29) to find the damage position variables x and y. It is exactly at this point that the peculiar structure of the nonlinear system (26) to (29) plays a crucial role. It is convenient to write equation (28) in the equivalent form

(ξ 1 + ξ 2) + 4 (ξ 1 x + ξ 2 y) + 4 (ξ 1 x 2 + ξ 2 y 2) = 2 C 3

(35)

The first two expressions appearing on the left-hand side of equation (35) can be evaluated in terms of the input data by means of equations (26) and (27), respectively. After a rearrangement of the terms, equation (35) takes the form

ξ 1 x 2 + ξ 2 y 2 = \frac{C 3}{2} + \frac{3 C 1}{2} - C 2

(36)

We proceed analogously with equation (29). By using, this time, equation (36) also, we have

ξ 1 x 3 + ξ 2 y 3 = \frac{C 4}{4} - \frac{C 3}{2} - \frac{3 C 1}{2} + C 2

(37)

and we end up with the reduced system

{ξ 1 x 2 + ξ 2 y 2 = A (38) ξ 1 x 3 + ξ 2 y 3 = B (39)

where

A = \frac{C 3}{2} + \frac{3 C 1}{2} - C 2, B = \frac{C 4}{4} - A

(40)

To conclude, we replace the expressions (33), (34) of ξ₁ and ξ₂, respectively, in (38), (39) and we elaborate. After straightforward calculation, the following system is obtained:

{(C 2 - 2 C 1) (x + y) - 2 C 1 xy = A (41) (C 2 - 2 C 1) (x 2 + y 2 + xy) - 2 C 1 xy (x + y) = B (42)

Under the position

S = x + y, P = xy

(43)

system (41) to (42) can be rewritten as a linear system in the variables S and P, namely

{(C 2 - 2 C 1) S - 2 C 1 P = A (44) AS - (C 2 - 2 C 1) P = B (45)

The determinant d of the two-by-two matrix of coefficients in (44) and (45) never vanishes. In fact, recalling (40) and the expressions (21) to (23) of the coefficients C_i, i = 1, 2, 3, we have

d = C 1 C 3 - (C 1 - C 2) 2 = \frac{(1 - x) (1 - y) (x - y) 2}{K 1 K 2}

(46)

which is clearly strictly positive since

x, y \in [- 1, 1), K i > 0

, i = 1, 2, and x ≠ y by (12). Therefore, the system (44) and (45) admits a unique solution and we have obtained closed-form expressions for S and P in terms of the data C_i, i = 1, 2, 3, 4.

Now, by the definition (43) of P and S, the variable position x turns out to be the root of the second-order polynomial equation

x 2 - Sx + P = 0

(47)

namely

x \mp = \frac{S \mp \sqrt{S 2 - 4 P}}{2}

(48)

where the notation x₋, x₊ corresponds to − sign and + sign on the right-hand side, respectively. Denote by y₋, y₊ the solution corresponding to x₋, x₊, respectively. To recover the damage severities we use the definition (25) of the auxiliary variables ξ_i, i = 1, 2. Let (x,y) be one of the two solutions (x₋,y₋), (x₊,y₊). By (33), (34) we obtain

K 1 = \frac{(1 - x) (x - y)}{C 2 - 2 C 1 (1 + y)}

(49)

K 2 = \frac{(1 - y) (y - x)}{C 2 - 2 C 1 (1 + x)}

(50)

and, in conclusion, the complete set of solutions is given by

{(s 1 -, K 1 -), (s 2 -, K 2 -)}, {(s 1 +, K 1 +), (s 2 +, K 2 +)}

(51)

where (K₁₋, K₂₋), (K₁₊, K₂₊) are evaluated by (49) with (x = x₋, y = y₋) and by (50) with (x = x₊, y = y₊), respectively. The crack positions

s i \mp

, i = 1, 2, are obtained by inverting the one-to-one function

\cos \frac{2 π s}{L} on (0, L / 2]

appearing in (19).

Finally, by noticing that y₋ = x₊, y₊ = x₋ and that K₁₊ = K₂₋, K₂₊ = K₁₋, it is easy to show that the two damage configurations (51) actually coincide. Therefore, we have shown that the knowledge of the first four natural frequencies allows us to uniquely determine the two cracks, up to symmetry with respect to the mid-span cross-section (see condition (12)).

Nonuniqueness of the solution is a typical pathology of inverse problems in vibration, and it is known that it can occur even when, as in the present case, the number of pieces of data is equal to the number of unknowns to be determined. A way to reduce, or even to remove, such an indeterminacy consists in introducing information coming from a spectrum associated to somehow different end conditions. This idea is developed in Section 3 with the aim of identifying two cracks having different severity in a free–free longitudinally vibrating rod.

2.3. Applications

The present section is devoted to outlining some applications of numerical character. The inverse problem is formulated in terms of pseudo-experimental data, that is, natural frequency values are obtained by solving the direct problem in undamaged conditions and for a given set of damage scenarios {(s₁, K₁), (s₂, K₂)}. The specimen is a uniform simply supported beam, of length L, with rectangular cross-section b × h, where $\frac{h}{L} = 0.1$ ; see Figure 1.

Figure 1.

Simply supported beam with two cracks.

A single open edge crack, with crack front parallel to the side b and depth a_i, is supposed to be present at the cross-section of abscissa s_i, i = 1, 2. The stiffness of the rotational spring simulating the damage at s_i, i = 1, 2, is

K i = \frac{EI}{L δ bi}

(52)

where the normalized flexibility δ_bi has the expression

δ bi = 2 \frac{h}{L} (\frac{α i}{1 - α i}) 2 (5.93 - 19.69 α i + 34.14 α_{i}^{2} - 35.84 α_{i}^{3} + 13.20 α_{i}^{4})

(53)

and

α i = \frac{a i}{h}

is the crack ratio; see, for example, Tada et al. (1985).

An exhaustive set of numerical simulations has been carried out for different locations of the cracks and various levels of damage. Three main different damage scenarios among several studied are presented and discussed in the sequel: they are illustrative of the main features of the inverse problem and of the identification technique. The first case, δ_b₁ = 0.00139 and δ_b₂ = 0.00531 (corresponding to α₁ = 0.05, α₂ = 0.10, respectively) is characterized by ‘very small–small’ damage (XS–S), that is, the values of the flexibility δ_bi are chosen such that the variations of the first four natural frequencies are up to 0.6% of the undamaged values. The second case involves ‘moderate–large’ damage (M–L), δ_b₁ = 0.0115157 and δ_b₂ = 0.0313333 (α₁ = 0.15, α₂ = 0.25), and it corresponds to variations of the same spectral data up to 3.5%. In the third case there are two equal ‘small–small’ cracks (S–S), δ_b₁ = δ_b₂ = 0.00531 (α₁ = α₂ = 0.10), corresponding to frequency variations of up to 0.9% of the referential values. Identification results are presented for three sets of damage locations, namely s₁ = L/4, s₂ = 7 L/20 (close cracks, denoted by C in what follows), s₁ = L/4, s₂ = 9 L/20 (distant cracks, case D1), and finally s₁ = L/4, s₂ = L/2 (distant cracks, with one centered crack, case Dc).

We start by considering the results obtained in the absence of errors on the data. The eigenvalues of the undamaged beam and their values associated with the cases of damage are shown in Table 1. The latter are obtained by solving in an exact way the eigenvalue problem (1) to (5) with the actual values of the damage parameters. The results of identification are summarized in Table 2. Note that the case Dc corresponds to C₄ = 0 and, therefore, it has been solved via equation (18).

Table 1.

First four dimensionless frequencies for the undamaged beam ( $f_{n}^{(U)} = L 2 \sqrt{\frac{γ A}{E 1}} \sqrt{λ_{n}^{(U)}}$ ) and their values f_n associated to the cases of damage (free-of-error data). C=close cracks; D = distant cracks; Dc = distant cracks, with one centered crack. Note: in brackets $Δ = 100 \times (f_{n}^{(U)} - f n) / f_{n}^{(U)}$ .

		Very small–small damage (XS–S)			Moderate–large damage (M–L)			Small–small damage (S–S)
f_n	Undam.	Case C	Case D1	Case Dc	Case C	Case D1	Case Dc	Case C	Case D1	Case Dc
f ₁	9.8696	9.8215	9.8121	9.8109	9.5793	9.5290	9.5225	9.8025	9.7932	9.7920
		(0.49)	(0.58)	(0.59)	(2.94)	(3.45)	(3.52)	(0.68)	(0.77)	(0.79)
f ₂	39.4784	39.2894	39.4041	39.4235	38.3364	38.9411	39.0348	39.1406	39.2531	39.2714
		(0.48)	(0.19)	(0.14)	(2.89)	(1.36)	(1.12)	(0.86)	(0.57)	(0.52)
f ₃	88.8264	88.7530	88.3949	88.3007	88.2517	86.2526	85.7819	88.5824	88.2223	88.1301
		(0.08)	(0.49)	(0.59)	(0.65)	(2.90)	(3.43)	(0.27)	(0.68)	(0.78)
f ₄	157.9140	157.1690	157.6300	157.9140	153.8560	156.3840	157.9140	157.1690	157.6300	157.9140
		(0.47)	(0.18)	(0.00)	(2.57)	(0.97)	(0.00)	(0.47)	(0.18)	(0.00)

Table 2.

Results of damage identification from frequency data (cases free of error). Determination of the damage severity $δ bl = \frac{E 1}{LK i}$ and the corresponding damage locations s_i, i = 1, 2. C = close cracks; D1 = distant cracks; Dc = distant cracks, with one centered crack. Percentage of error is indicated in brackets.

	XS–S Damage δ_b₁ = 0.00139 δ_b₂ = 0.00531			M–L Damage δ_b₁ = 0.0115157 δ_b₂ = 0.0313333			S–S Damage δ_b₁ = 0.00531 δ_b₂ = 0.00531
Estimated damage parameters	Case C	Case D1	Case Dc	Case C	Case D1	Case Dc	Case C	Case D1	Case Dc
	s₁/L = 0.25	s₁/L = 0.25	s₁/L = 0.25	s₁/L = 0.25	s₁/L = 0.25	s₁/L = 0.25	s₁/L = 0.25	s₁/L = 0.25	s₁/L = 0.25
	s₂/L = 0.35	s₂/L = 0.45	s₂/L = 0.50	s₂/L = 0.35	s₂/L = 0.45	s₂/L = 0.50	s₂/L = 0.35	s₂/L = 0.45	s₂/L = 0.50
δ_b1	0.00152	0.00138	0.00140	0.01507	0.01087	0.01120	0.00538	0.00519	0.00520
	(−9.35)	(0.36)	(−0.72)	(−30.82)	(5.64)	(2.78)	(−1.32)	(2.26)	(2.07)
δ_b2	0.00509	0.00524	0.00520	0.02448	0.02908	0.02900	0.00502	0.00524	0.00520
	(3.96)	(1.20)	(1.89)	(21.86)	(7.18)	(7.44)	(5.46)	(1.32)	(2.07)
s ₁	0.25659	0.25099	0.25000	0.27337	0.25223	0.25000	0.25286	0.24987	0.25000
	(−2.64)	(−0.40)	(0.00)	(−9.35)	(−0.89)	(0.00)	(−1.14)	(0.05)	(0.00)
s ₂	0.35236	0.45024	0.50000	0.36508	0.45120	0.50000	0.35463	0.45028	0.50000
	(−0.67)	(−0.05)	(0.00)	(−4.31)	(−0.27)	(0.00)	(−1.32)	(−0.06)	(0.00)

It is possible to observe that the solution predicted by the theory generally is a satisfactory estimate of the actual solution of the diagnostic problem. The discrepancies between identified and actual damage parameters are exclusively due to the perturbation assumption of small damage. Deviations are typically smaller for less severe damage, as it is expected because the inverse problem is linearized in a neighborhood of the undamaged beam. For the sake of completeness, we note that numerical simulations have not led to accurate results in the case of close cracks. The motivation of this discrepancy is connected with the reconstruction procedure illustrated in Section 2.2 and, particularly, with the inversion of the two-by-two linear system (44) and (45). By (46), the determinant of the corresponding two-by-two matrix vanishes with order $| x - y | 2$ as $x \to y$ or, equivalently, as $s 1 \to s 2$ . Therefore, the inversion of (44) and (45) is ill-posed for close cracks, and the effects of the assumption of small damage are amplified strongly.

In order to test the robustness of identification to possible errors, numerical simulation has been repeated in the presence of random noise on the data. In practical applications, one of the main sources of error is due to the inaccuracy of the analytical model that is used to interpret the experiments. Damage-induced changes are typically small and, therefore, it may happen that even small modeling errors can affect the outcome of identification. Frequency values ${\sqrt{λ n}} n = 1 4$ were perturbed as

(\sqrt{λ n}) pert = \sqrt{λ n} (1 + n τ)

(54)

where τ is a real random Gaussian variable with zero mean. Two normal distributions with standard deviations σ = 0.00033, σ = 0.00067 have been considered for two error levels, namely, Error Level 1 and Error Level 2, respectively. It is worth noticing that in (54) the magnitude of the error increases linearly with the vibration mode number n, for example, maximum errors on natural frequencies

\sqrt{λ n}

are 0.1 n% and 0.2 n%, n = 1, 2, 3, 4, for Error Levels 1 and 2, respectively. This assumption is motivated by the experimental observation that the mathematical model of a cracked beam loses accuracy as the mode order increases; see, for example, Davini et al. (1995) and Caddemi and Morassi (2013).

For each damage configuration, a Monte Carlo simulation on a population of 10,000 samples has been carried out. Table 3 shows the statistical properties of the results of identification. It can be seen that random errors on the data have different effects in the identification problem solution. As was expected, the cases concerning ‘very small–small’ damage are not well identified, as the crack-induced shifts in the eigenfrequencies are generally lower than the errors on the data. This situation is partially avoided in cases corresponding to ‘small–small’ damage, for which the diagnostic method gives, in general, good results. Different explanation is required for ‘moderate–large’ damage severities. In this case, identification results are also influenced by the fact that we are dealing with severity values for which the first-order perturbation of eigensolutions is applicable with less accuracy. In spite of this indeterminacy, the identified values of damage position are satisfactory enough.

Table 3.

Results of damage identification (cases with random error). C=close cracks; D1 = distant cracks; Dc = distant cracks, with one centered crack. XS–S = very small–small damage; M–L= moderate–large damage; S–S = small–small damage.

		Damage severity δ				Damage positions				Percentage of complex values
		δ ₋		δ ₊		s₋(δ)/L		s₊(δ)/L		Percentage of complex values
Case	Statistical property	Err. Lev. 1	Err. Lev. 2	Err. Lev. 1	Err. Lev. 2	Err. Lev. 1	Err. Lev. 2	Err. Lev. 1	Err. Lev. 2	Err. Lev. 1	Err. Lev. 2
C^XS⁻^S	Mean	0.00218	0.00195	0.00471	0.00501	0.22265	0.20109	0.36417	0.37050	40.09	54.01
	Std. Dev.	0.00521	0.00884	0.00307	0.00322	0.06579	0.06697	0.03089	0.00392
	Mean/Exact	1.57122	1.40288	0.88701	0.94539	0.89060	0.81636	1.04049	1.05857
D1^XS⁻^S	Mean	0.00170	0.00158	0.00520	0.00547	0.22355	0.19781	0.45185	0.44305	33.12	55.24
	Std. Dev.	0.00485	0.00665	0.00056	0.00132	0.05167	0.03640	0.01580	0.02321
	Mean/Exact	1.22302	1.13669	0.97928	1.03013	0.89420	0.79124	1.00418	0.98456
Dc^XS⁻^S	Mean	0.00145	0.00079	0.00540	0.00575	0.21685	0.19540	0.46972	0.45548	68.20	72.46
	Std. Dev.	0.00173	0.00774	0.00062	0.00127	0.04770	0.06544	0.01350	0.02008
	Mean/Exact	1.04165	0.56834	1.02260	1.08286	0.86740	0.78156	0.93944	0.91098
C^M⁻^L	Mean	0.01584	0.01651	0.02373	0.02313	0.27371	0.27025	0.36699	0.37079	0.00	4.74
	Std. Dev.	0.00385	0.00686	0.00398	0.00705	0.01507	0.02810	0.00955	0.02114
	Mean/Exact	1.37500	1.43316	0.76229	0.74301	1.09484	1.08100	1.04854	1.0594
D1^M⁻^L	Mean	0.00902	0.00927	0.03344	0.03340	0.16685	0.16561	0.42579	0.42597	0.00	0.00
	Std. Dev.	0.00073	0.00158	0.00046	0.00092	0.00902	0.01938	0.00197	0.00398
	Mean/Exact	0.78299	0.80469	1.07420	1.07292	0.66740	0.66244	0.94620	0.94662
Dc^M⁻^L	Mean	0.01093	0.01076	0.02909	0.02938	0.25179	0.24713	0.48609	0.48049	55.04	55.65
	Std. Dev.	0.00069	0.00136	0.00052	0.01010	0.00719	0.01348	0.00583	0.0083
	Mean/Exact	0.94878	0.93403	0.93447	0.94378	1.00716	0.98852	0.97218	0.96098
C^S⁻^S	Mean	0.00550	0.00566	0.00514	0.00519	0.24166	0.22941	0.36156	0.37102	11.16	38.28
	Std. Dev.	0.00620	0.00370	0.00212	0.00300	0.03828	0.05165	0.03342	0.04211
	Mean/Exact	1.03580	1.06591	0.96680	0.97740	0.96664	0.91764	1.03303	1.06006
D1^S⁻^S	Mean	0.00521	0.00514	0.00529	0.00554	0.24620	0.23403	0.45050	0.44245	14.89	37.49
	Std. Dev.	0.00073	0.00139	0.00060	0.00109	0.01701	0.03004	0.01619	0.02302
	Mean/Exact	0.98117	0.96798	0.99623	1.04520	0.98480	0.93612	1.00111	0.98324
Dc^S ⁻ ^S	Mean	0.00550	0.0050	0.0055	0.00571	0.24264	0.23316	0.46814	0.45574	56.34	60.88
	Std. Dev.	0.00068	0.00134	0.00051	0.00100	0.01491	0.02750	0.01348	0.01942
	Mean/Exact	0.94727	0.94162	1.03578	1.07533	0.97060	0.93264	0.93628	0.91148

For the sake of completeness, we notice that complex values of the damage parameters are obtained in a certain number of simulations: see the last two columns of Table 3. The highest percentage of complex values has been found in case Dc, that is, when the cracks are located at $s 1 = \frac{L}{4}$ and $s 2 = \frac{L}{2}$ . In fact, a detailed analysis of the data shows that equation (19) may lead to imaginary values in case Dc, since the value of the position variable y is outside of the interval [−1, 1). A possible explanation of this discrepancy is the following. When random errors are considered on the data, the first-order variation of the fourth natural frequency is not exactly equal to zero. Then, differently from the case of error-free data, the identification procedure illustrated at the beginning of Section 2.2 for C₄ = 0 cannot be applied, and the general identification algorithm has been adopted. Therefore, by (38) and (39), we easily get

ξ 1 x 2 (1 + x) + ξ 2 y 2 (1 + y) = \frac{C 4}{4}

(55)

If the random error is such that the sign of C₄ is negative, then clearly equation (55) does not have real solutions.

Finally, as expected from previous free-of-error simulations, the case of close cracks still remains the most problematic.

3. Inverse problem in a longitudinally vibrating rod

3.1. Formulation of the problem

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 two cracks at two different cross-sections of abscissa s₁, s₂, with 0 < s_i < L, i = 1, 2, where L is the length of the rod. Each crack is assumed to remain open during vibration and is modeled as a massless longitudinal linearly elastic spring with stiffness K_i, i = 1, 2. The value of K_i can be determined in terms of the geometry of the cracked cross-section and of the material properties of the beam (Freund and Herrmann, 1976). In this section we determine the first-order changes induced by damage both on natural frequencies and a specific set of antiresonances of the rod. The free undamped vibrations of the rod with radian frequency $\sqrt{λ}$ and spatial amplitude w = w(z) are governed by the following eigenvalue problem:

{EAw ″ + λ γ Aw = 0, z \in (0, s 1) \cup (s 1, s 2) \cup (s 2, L) (56) w' (s_{i}^{-}) = w' (s_{i}^{+}), i = 1, 2 (57) K i (w (s_{i}^{+}) - w (s_{i}^{-})) = EAw' (s i), i = 1, 2 (58) w' (0) = w' (L) = 0 (59)

where E is the (constant) Young’s modulus of the material, A is the area of the transversal cross-section, and γ is the (uniform) volume mass density of the material. It is well known that there exists an infinite sequence of eigenvalues

{λ_{n}^{F - F}} n = 0 \infty

, with

0 = λ_{0}^{F - F} < λ_{1}^{F - F} < λ_{2}^{F - F} < \dots

and

lim_{n \to \infty} λ_{n}^{F - F} = \infty

, such that the problem (56) to (59) has no trivial solution

w_{n}^{F - F} = w_{n}^{F - F} (z)

. The undamaged system corresponds to taking in (56)–(59) the limit as

K i \to \infty

, i = 1, 2. The mass-normalized eigenpairs of the undamaged rod

{λ_{n}^{F - F (U)}, w_{n}^{F - F (U)} (z)} n = 0 \infty

can be evaluated explicitly and are equal to

λ_{n}^{F - F (U)} = \frac{E}{γ} (\frac{n π}{L}) 2, w_{n}^{F - F (U)} (z) = \sqrt{\frac{2}{γ AL}} \cos (\frac{n π z}{L}), n = 0, 1, 2, \dots

(60)

As the cracks are small, namely both K₁ and K₂ are large enough, we can use a standard approach to find the first-order perturbation on the eigenvalues induced by the damage (Morassi, 1993; Loya et al., 2006). By taking

λ_{n}^{F - F} = λ_{n}^{F - F (U)} + δ λ_{n}^{F - F}

(61)

we find

δ λ_{n}^{F - F} = - \frac{(N_{n}^{(U)} (s 1)) 2}{K 1} - \frac{(N_{n}^{(U)} (s 2)) 2}{K 2}

(62)

for every n ≥ 0, where the axial force associated to the nth undamaged vibration mode is given by

N_{n}^{(U)} (z) = EA \frac{d}{dz} (w_{n}^{F - F (U)} (z))

(63)

The 0th vibration mode is a longitudinal rigid motion, hence the corresponding eigenvalue is insensitive to damage. Inserting the explicit expression (60) of the eigenpairs of the undamaged rod into equation (62) and elaborating, we have

C_{n}^{F - F} = \frac{\sin 2 \frac{n π s 1}{L}}{K 1} + \frac{\sin 2 \frac{n π s 2}{L}}{K 2}, n \geq 1

(64)

where the nonnegative quantity

C_{n}^{F - F} = - \frac{δ λ_{n}^{F - F}}{\frac{2 EA}{L} λ_{n}^{F - F (U)}}, n \geq 1

(65)

only depends on the undamaged rod properties and on the measured eigenvalue shift induced by the damage on the nth eigenvalue.

In addition to resonant frequencies, we consider the antiresonant frequencies of the driving point FRF of the rod $H F - F (ω, 0, 0)$ obtained by taking the excitation point and the measurement point coincident with the left cross-section z = 0. Here, ω is the frequency variable. Antiresonances are the zeros of the FRF $H F - F (ω, 0, 0)$ and coincide with the natural frequencies of the rod when the longitudinal displacement at the cross-section z = 0 is hindered, namely, antiresonances coincide with the natural frequencies of the supported–free (S–F) rod. Therefore, under the assumption of small cracks, the first-order variation of the (square of the) mth antiresonance with respect to damage can be evaluated as above (see equations (61), (62)).

The eigenvalues of the supported–free rod will be denoted by $λ_{m}^{S - F}$ in the sequel. The eigenpairs of the undamaged supported–free rod are given by

λ_{m}^{S - F (U)} = \frac{E}{γ} (\frac{(1 + 2 m) π}{2 L}) 2, w_{m}^{S - F (U)} (z) = \sqrt{\frac{2}{γ AL}} \sin \frac{(1 + 2 m) π z}{2 L}

(66)

m = 0, 1, 2, \dots

Let us introduce the quantities

C_{m}^{S - F}

analogous to those appearing in (65):

C_{m}^{S - F} = - \frac{δ λ_{m}^{S - F}}{\frac{2 EA}{L} λ_{m}^{S - F (U)}}, m \geq 0

(67)

By proceeding as exemplified above and with the same notation, we can write the following additional relationships between the measured quantities C_m^S^- ^F and the damage parameters {(s₁,K₁), (s₂,K₂)}:

C_{m}^{S - F} = \frac{\cos 2 \frac{(2 m + 1) π s 1}{2 L}}{K 1} + \frac{\cos 2 \frac{(2 m + 1) π s 2}{2 L}}{K 2}, m \geq 0

(68)

3.2. Identification of two cracks by resonant/antiresonant frequency data

In this section we are concerned with the problem of determining the two cracks, namely the four damage parameters {s₁, K₁), (s₂, K₂)}, from minimal spectral data. By the analogy between this inverse problem and the corresponding problem for the cracked beam in bending discussed in Section 2.2 (compare the expressions (10) and (64)), the use of the first four (positive) natural frequencies of the free–free rod clearly leads to nonunique solution, because of the symmetry of the undamaged configuration. In the sequel, we show how one can properly select resonant and antiresonant data in order to avoid the nonuniqueness of the solution and to find closed-form expressions for the damage parameters. We assume that frequency data consist of the first and second antiresonant frequencies of the driving point FRF $H F - F (ω, 0, 0)$ , namely $λ_{0}^{S - F}$ and $λ_{1}^{S - F}$ , and the second and third resonant frequencies of the rod, namely $λ_{1}^{F - F}$ and $λ_{2}^{F - F}$ (note that both the resonances and antiresonances are enumerated from 0, and that the lower resonant frequency vanishes). Then, by (64), (65), (67) and (68) we obtain the following system of four nonlinear equations in the unknowns (s₁, K₁) and (s₂, K₂):

{\frac{1}{K 1} \sin 2 \frac{π s 1}{2 L} + \frac{1}{K 2} \sin 2 \frac{π s 2}{2 L} - (\frac{1}{K 1} + \frac{1}{K 2}) = - C 1 (69) \frac{1}{K 1} \sin 2 \frac{π s 1}{L} + \frac{1}{K 2} \sin 2 \frac{π s 2}{L} = C 2 (70) \frac{1}{K 1} \sin 2 \frac{3 π s 1}{2 L} + \frac{1}{K 2} \sin 2 \frac{3 π s 2}{2 L} - (\frac{1}{K 1} + \frac{1}{K 2}) = - C 3 (71) \frac{1}{K 1} \sin 2 \frac{2 π s 1}{L} + \frac{1}{K 2} \sin 2 \frac{2 π s 2}{L} = C 4 (72)

where

C 1 = C_{0}^{(S - F)}, C 2 = C_{1}^{(F - F)}, C 3 = C_{1}^{(S - F)}, C 4 = C_{2}^{(F - F)}

(73)

and, since s₁ ≠ s₂,

C i > 0, i = 1, 2, 3, 4

(74)

The structure of system (69) to (72) and that of the corresponding system (13) to (16) for the cracked beam in bending are different. Nevertheless, we shall show that the introduction of a suitable pair of damage position variables and auxiliary variables allows us to adapt the solving procedure illustrated in Section 2 to the present case. For the sake of completeness, the main steps of the analysis are sketched in the sequel.

Let us define

x = x (s 1) = \cos \frac{π s 1}{L} \in (- 1, 1), y = y (s 2) = \cos \frac{π s 2}{L} \in (- 1, 1)

(75)

and

η 1 = η 1 (x, K 1) = \frac{1 + x}{K 1} \in (0, + \infty), η 2 = η 2 (y, K 2) = \frac{1 + y}{K 2} \in (0, + \infty)

(76)

Then, system (69) to (72) can be written in the equivalent form

{η 1 + η 2 = 2 C 1 (77) η 1 (1 - x) + η 2 (1 - y) = C 2 (78) η 1 (1 - 2 x) 2 + η 2 (1 - 2 y) 2 = 2 C 3 (79) η 1 (1 - x) x 2 + η 2 (1 - y) y 2 = \frac{C 4}{4} (80)

The structure of (77) to (80) is analogous to that of system (26) to (29), with the exception of the sign on the first-degree expressions involving the variables x and y. Despite this difference, by adapting the analysis of Section 2.2, one can show that the two quantities

S = x + y, P = xy

(81)

satisfy the two-by-two linear system

{(C 2 - 2 C 1) S - 2 C 1 P = A (82) A S - (C 2 - 2 C 1) P = B (83)

with

A = \frac{C 3}{2} + \frac{3 C 1}{2} - C 2, B = A - \frac{C 4}{4}

(84)

The determinant of the two-by-two matrix of the system (82) and (83) is always different from zero, for example,

d = C 1 C 3 - (C 1 - C 2) 2 = \frac{(1 + x) (1 + y) (x - y) 2}{K 1 K 2}

(85)

and

S

and

P

can be uniquely determined in terms of the data. This allows us to determine the variable x as

x \mp = \frac{S \mp \sqrt{S 2 - 4 P}}{2}

(86)

where the notation x₋, x₊ corresponds to sign − and sign + on the right-hand side. Let us denote by y₋, y₊ the solution corresponding to x₋, x₊, respectively. Let (x, y) be one of the two solutions (x₋, y₋), (x₊, y₊). By (76), the corresponding damage severities are

K 1 = \frac{(1 + x) (y - x)}{C 2 - 2 C 1 (1 - y)}

(87)

K 2 = \frac{(1 + y) (- y + x)}{C 2 - 2 C 1 (1 - x)}

(88)

and the complete set of solutions is given by

{(s 1 -, K 1 -), (s 2 -, K 2 -)}, {(s 1 +, K 1 +), (s 2 +, K 2 +)}

(89)

where (K₁₋, K₂₋) and (K₁₊, K₂₊) are evaluated by (87) with (x = x₋, y = y₋) and by (88) with (x = x₊, y = y₊), respectively. Here, the crack positions

s i \mp

are determined by inverting the (one-to-one) function

\cos \frac{π s}{L}

in the interval (0, L) . Finally, by noticing that y₋ = x₊, y₊ = x₋ and that K₁₊ = K₂₋, K₂₊ = K₁₋, one can easily prove that the two damage configurations (89) actually coincide.

In conclusion, we have shown that the two cracks {(s₁₋, K₁₋), (s₂₋, K₂₋)} can be uniquely determined by the knowledge of the first and second antiresonant frequencies of H(ω, 0, 0) and the second and third natural frequencies of the rod.

3.3. Applications

An exhaustive series of numerical simulations has been carried out for different locations of the cracks and various damage levels in a free–free rod, of length L, with rectangular cross-section b × h, where $\frac{h}{L} = 0.1$ : see Figure 2.

Figure 2.

Free–free rod with two cracks.

A pair of symmetric open edge cracks, each with crack front parallel to the side b, is supposed to be present at the cross-section of abscissa s_i, i = 1, 2. Denoting by $\frac{a i}{2}$ the depth of each side crack at s_i, i = 1, 2, the stiffness K_i of the elastic translational spring simulating the damage at s_i, i = 1, 2, is expressed as

K i = \frac{EA}{L δ li}

(90)

where

δ li = 2 \frac{h}{L} (1 - ν 2) (0.7314 α_{i}^{8} - 1.0368 α_{i}^{7} + 0.5803 α_{i}^{6} + 1.2055 α_{i}^{5} - 1.0368 α_{i}^{4} + 0.2381 α_{i}^{3} + 0.9852 α_{i}^{2})

(91)

and

α i = \frac{a i}{h}

is the crack ratio; see, for example, Ruotolo and Surace (2004). Poisson’s coefficient ν was assumed equal to 0.3.

The results of a selected number of cases is presented in the sequel. A first series of tests has been carried out in the absence of errors on the data. Three damage scenarios are considered. The first case, δ_l₁ = 0.000497334 and δ_l₂ = 0.00199979 (corresponding to α₁ = 0.052, α₂ = 0.105, respectively), is characterized by ‘very small–small’ damage, that is, frequency shifts are up to 0.24% of the referential undamaged values. The second case involves ‘moderate–large’ damage, δ_l₁ = 0.00450846 and δ_l₂ = 0.0125024 (α₁ = 0.157, α₂ = 0.262), and it corresponds to variations of the same spectral data of less than 1.63%. Finally, in the third case, the common severity of the two cracks, δ_l₁ = δ_l₂ = 0.00450846 (α₁ = α₂ = 0.157), has been chosen so that the frequency changes are of up to 0.87% (‘moderate–moderate’ damage). Identification results are presented for three sets of damage locations, namely s₁ = L/4, s₂ = 7 L/20 (close cracks, case C), s₁ = 9 L/20, s₂ = 11 L/20 (close and centered cracks, case Cc), and finally s₁ = L/4, s₂ = 13 L/20 (distant cracks, case D3). The eigenvalues of the undamaged beam and their values associated with the cases of damage are shown in Table 4. Natural frequencies and antiresonances are obtained by solving the direct eigenvalue problem exactly in undamaged and damaged conditions. The results of identification are summarized in Table 5. Generally speaking, the solution predicted by the theory is a good estimate of the actual solution, with the exception of the cases in which the two cracks are ‘close’, namely when s₂ − s₁ is less or equal to about

\frac{L}{10}

. For these cases, as happened in the bending vibrating beam (compare the expression (85) of the determinant d with the corresponding expression (46)), the accuracy of the estimates is poor, especially for the damage severity.

Table 4.

First two dimensionless frequencies and dimensionless antiresonances for the undamaged rod ( $f_{n}^{(U)} = L \sqrt{\frac{γ}{E}} \sqrt{λ_{n}^{(U)}}$ ) and their values associated to the cases of damage (free-of-error data). C = close cracks; Cc = close and centered cracks; D3 = distant cracks. Note: in brackets $Δ = 100 \times (f_{n}^{(U)} - f n) / f_{n}^{(U)}$ .

		Very small–small damage			Moderate–large damage			Moderate–moderate damage
f	Undam.	Case C	Case Cc	Case D3	Case C	Case Cc	Case D3	Case C	Case Cc	Case D3
$f_{1}^{F - F}$	3.14159	3.13583	3.13396	3.13583	3.10360	3.09044	3.10381	3.12332	3.11425	3.12340
		(0.18)	(0.24)	(0.18)	(1.21)	(1.63)	(1.20)	(0.58)	(0.87)	(0.58)
$f_{2}^{F - F}$	6.28319	6.27190	6.28169	6.27186	6.20635	6.27290	6.20432	6.23723	6.27774	6.23648
		(0.18)	(0.02)	(0.18)	(1.22)	(0.16)	(1.26)	(0.73)	(0.09)	(0.74)
$f_{0}^{S - F}$	1.57080	1.56785	1.56902	1.56927	1.55083	1.55852	1.55950	1.55971	1.56375	1.56287
		(0.19)	(0.11)	(0.10)	(1.27)	(0.78)	(0.72)	(0.71)	(0.45)	(0.50)
$f_{1}^{S - F}$	4.71239	4.71199	4.70492	4.70270	4.70891	4.66460	4.65141	4.70915	4.69133	4.68823
		(0.01)	(0.16)	(0.21)	(0.07)	(1.01)	(1.29)	(0.07)	(0.45)	(0.51)

Table 5.

Results of damage identification frequency data (cases free of error). Determination of the damage severity $δ l = \frac{EA}{LK}$ and the corresponding damage locations s₁, s₂. C = close cracks; Cc = close and centered cracks; D3 = distant cracks. Percentage of errors are indicated in brackets.

	XS–S Damage δ_l₁ = 0.000497334 δ_l₂ = 0.00199979			M–L Damage δ_l₁ = 0.00450846 δ_l₂ = 0.01250240			M–M Damage δ_l₁ = 0.00450846 δ_l₂ = 0.00450846
Estimated	Case C	Case Cc	Case D3	Case C	Case Cc	Case D3	Case C	Case Cc	Case D3
damage	s₁/L = 0.25	s₁/L = 0.45	s₁/L = 0.25	s₁/L = 0.25	s₁/L = 0.45	s₁/L = 0.25	s₁/L = 0.25	s₁/L = 0.45	s₁/L = 0.25
parameters	s₂/L = 0.35	s₂/L = 0.55	s₂/L = 0.65	s₂/L = 0.35	s₂/L = 0.55	s₂/L = 0.65	s₂/L = 0.35	s₂/L = 0.55	s₂/L = 0.65
δ _l ₁	0.00037	0.00030	0.00050	0.00296	0.00142	0.00446	0.00398	0.00226	0.00446
	(25.60)	(39.67)	(−0.54)	(34.37)	(68.51)	(1.11)	(11.75)	(49.89)	(1.11)
δ_l ₂	0.00212	0.00218	0.00200	0.01371	0.01520	0.01226	0.00494	0.00663	0.00448
	(−6.05)	(−9.05)	(−0.05)	(−9.68)	(−21.60)	(1.92)	(−9.53)	(−47.01)	(0.67)
s ₁	0.23339	0.41932	0.24970	0.22675	0.35296	0.24877	0.24394	0.40899	0.24942
	(6.64)	(6.82)	(0.12)	(9.30)	(21.56)	(0.49)	(2.42)	(9.11)	(0.23)
s ₂	0.34678	0.54447	0.64976	0.34645	0.53524	0.64981	0.34727	0.52803	0.65040
	(0.92)	(1.01)	(0.04)	(1.01)	(2.68)	(0.03)	(0.78)	(3.99)	(−0.06)

The effect of errors on the data was investigated for distant cracks, namely for damage positions s₁ = L/4, s₂ = 9 L/20 (case D1), s₁ = L/4,s₂ = 11 L/20 (D2) and s₁ = L/4,s₂ = 13 L/20 (D3). Exact input data are given in Table 6. Perturbed data were obtained as

(\sqrt{λ_{n}^{F - F}}) pert = \sqrt{λ_{n}^{F - F}} (1 + τ), (\sqrt{λ_{n}^{S - F}}) pert = \sqrt{λ_{n}^{S - F}} (1 + 2 τ)

(92)

where τ is a real random Gaussian variable with zero mean. As in the case of bending vibrations, two normal distributions with standard deviation σ = 0.00033 (Error Level 1) and σ = 0.00067 (Error Level 2) have been considered. Maximum errors on natural frequencies are equal to 0.1% and 0.2% for Error Level 1 and Error Level 2, respectively. Note that, in both cases, the error on antiresonant frequencies is twice that of natural frequencies. This choice is in agreement with experimental results, which show that antiresonant frequency estimates are typically less accurate than natural frequency data; see, for example, Dilena and Morassi (2004).

Table 6.

First two dimensionless frequencies and dimensionless antiresonances for the undamaged rod $f_{n}^{(U)} = L \sqrt{\frac{γ}{E}} \sqrt{λ_{n}^{(U)}}$ ) and their values associated to the cases of damage (free-of-error data). Case D1: s₁ = L/4,s₂ = 9 L/20; Case D2: s₁ = L/4,s₂ = 11 L/20; Case D3: s₁ = L/4, s₂ = 13 L/20. Note: in brackets $Δ = 100 \times (f_{n}^{(U)} - f n) / f_{n}^{(U)}$ .

		Very small–small damage			Moderate–large damage			Moderate–moderate damage
f	Undam.	Case D1	Case D2	Case D3	Case D1	Case D2	Case D3	Case D1	Case D2	Case D3
$f_{1}^{F - F}$	3.14159	3.13470	3.13470	3.13583	3.09689	3.09697	3.10380	3.12084	3.12087	3.12340
		(0.22)	(0.22)	(0.18)	(1.42)	(1.42)	(1.20)	(0.66)	(0.66)	(0.58)
$f_{2}^{F - F}$	6.28319	6.27888	6.27885	6.27186	6.24827	6.24699	6.20432	6.25256	6.25210	6.23648
		(0.07)	(0.07)	(0.18)	(0.56)	(0.58)	(1.26)	(0.49)	(0.49)	(0.74)
$f_{0}^{S - F}$	1.57080	1.56832	1.56881	1.56927	1.55365	1.55664	1.55950	1.56075	1.56184	1.56287
		(0.16)	(0.13)	(0.10)	(1.09)	(0.90)	(0.72)	(0.64)	(0.57)	(0.50)
$f_{1}^{S - F}$	4.71239	4.70947	4.70520	4.70270	4.69309	4.66679	4.65141	4.70345	4.69385	4.38823
		(0.06)	(0.15)	(0.21)	(0.41)	(0.97)	(1.29)	(0.19)	(0.39)	(0.51)

For each damage configuration, a Monte Carlo simulation on a population of 10, 000 samples has been carried out. Table 7 shows the statistical properties of the results of identification. It can be seen that random errors on the data affect the solution in a different way. As was expected, the method gives good results when errors are lower than frequency changes induced by the damage, for example for ‘moderate–large’ and ‘moderate–moderate’ damage. The last two columns of Table 7 also show that the percentage of imaginary values obtained via Monte Carlo simulations increases as the severity of the damage decreases. Finally, identification results generally become better as the distance between the two cracks increases, thus confirming the difficulty in identifying close cracks.

Table 7.

Results of damage identification (cases with random error) corresponding to cases D1, D2 and D3 of Table 6. XS–S = very small–small damage; M–L = moderate–large damage; M–M = moderate–moderate damage.

Case	Statistical property	Damage severity δ				Damage positions				Percentage of complex values
		δ ₋		δ ₊		s₋(δ)/L		s₊(δ)/L		Percentage of complex values
		Err. Lev. 1	Err. Lev. 2	Err. Lev. 1	Err. Lev. 2	Err. Lev. 1	Err. Lev. 2	Err. Lev. 1	Err. Lev. 2	Err. Lev. 1	Err. Lev. 2
$D 1 XS - S$	Mean	0.00091	0.00128	0.00179	0.00182	0.20809	0.16943	0.50537	0.52126
	Std. Dev.	0.00045	0.00064	0.00051	0.00071	0.09951	0.09695	0.07159	0.07962	70.04	82.74
	Mean/Exact	1.82000	2.56000	0.25513	1.65703	0.83320	0.67772	0.91885	1.15835
$D 2 XS - S$	Mean	0.00086	0.00128	0.00193	0.00192	0.18570	0.16025	0.53636	0.54022
	Std. Dev.	0.00042	0.00067	0.00042	0.00056	0.09575	0.09633	0.05902	0.06619	52.07	76.68
	Mean/Exact	1.72000	2.56000	0.96548	0.96480	0.7400	0.64100	0.97520	1.16854
$D 3 XS - S$	Mean	0.00104	0.00156	0.00191	0.00180	0.21238	0.18039	0.64270	0.62561
	Std. Dev.	0.00049	0.00077	0.00551	0.00138	0.08997	0.08582	0.05723	0.08377	47.13	65.79
	Mean/Exact	2.08000	3.12000	0.95047	0.90045	0.84952	0.72156	0.98877	0.96248
$D 1 M - L$	Mean	0.00492	0.00571	0.01176	0.01126	0.24622	0.24764	0.45766	0.47202
	Std. Dev.	0.00207	0.00305	0.00136	0.00321	0.06350	0.08722	0.03787	0.06509	2.20	24.50
	Mean/Exact	1.09091	1.26607	0.94080	0.90096	0.98488	0.99056	1.01702	1.04893
$D 2 M - L$	Mean	0.00466	0.00512	0.01207	0.01200	0.25416	0.25701	0.55336	0.55734
	Std. Dev.	0.00065	0.00150	0.00131	0.01384	0.03691	0.06625	0.01551	0.03669	0.00	5.50
	Mean/Exact	1.03326	1.13525	0.96560	0.96000	1.01664	1.02804	1.00611	1.01334
$D 3 M - L$	Mean	0.00452	0.00488	0.01225	0.01244	0.25362	0.25851	0.65166	0.65681
	Std. Dev.	0.00068	0.00137	0.00074	0.01066	0.03218	0.06416	0.01022	0.02923	0.00	4.44
	Mean/Exact	1.00222	1.08204	0.98000	0.97152	1.01448	1.03404	1.00255	1.01048
$D 1 M - M$	Mean	0.00444	0.00472	0.00475	0.00469	0.24544	0.24940	0.47637	0.49804
	Std. Dev.	0.00126	0.00138	0.00158	0.00209	0.06043	0.07248	0.06710	0.08520	12.09	38.62
	Mean/Exact	0.98448	1.04656	1.05321	1.03991	0.98176	0.99760	1.05860	1.10675
$D 2 M - M$	Mean	0.00455	0.00459	0.00443	0.00496	0.24881	0.25379	0.55468	0.55559
	Std. Dev.	0.00049	0.00100	0.00094	0.02749	0.06383	0.03169	0.03672	0.07178	0.25	11.13
	Mean/Exact	1.00887	1.01774	0.98226	1.09978	0.99524	1.01516	1.00851	1.01016
$D 3 M - M$	Mean	0.00453	0.00477	0.00463	0.00481	0.25379	0.25152	0.65497	0.65856
	Std. Dev.	0.00068	0.00128	0.00850	0.00611	0.03169	0.05374	0.02691	0.05441	0.18	7.68
	Mean/Exact	1.00443	1.05765	1.02661	1.06652	1.01516	1.00608	1.00765	1.01317

4. Conclusions

In this paper we were concerned with the identification of two open small cracks in an initially uniform beam in bending vibration from a knowledge of the damage-induced shifts in the low natural frequencies. Each crack is modeled by a linearly elastic rotational spring located at the damaged cross-section, for a total of four unknowns, namely, the two positions and the two severities of the cracks. It was shown that the inverse problem can be formulated and solved in terms of the first four natural frequencies. This set of data allows for the unique identification of the two cracks, apart from the symmetry with respect to the mid-span cross-section. Closed-form solutions are provided both for the position and the severity of each crack in terms of the data.

Previous results on this problem were available only for the case of two cracks with equal severity. The proposed method enables us to overcome this restriction and to solve the general case of two cracks having different severity.

The method can be extended to deal with double-cracked free–free rods under longitudinal vibration. In this case, each crack is modeled by a translational spring acting along the direction of the beam axis and located at the damaged cross-section. It was shown that the use of the first two natural frequencies and the first two antiresonant frequencies of the driving point FRF evaluated at one end of the rod allows us to exclude the spurious solution due to the symmetry of the undamaged configuration. Even in this case, closed-form solutions are available for the damage parameters in terms of the data.

Analytical results were confirmed by an extended series of numerical simulations carried out on cracked beams and rods under various damage scenarios, with and without errors on the data. When errors on the data are absent, the solution predicted by the theory generally is a good estimate of the actual solution of the diagnostic problem, with the exception of cases in which the two cracks are close, namely, in our experience, when the distance between the two cracks is approximately less than one-tenth of the beam length. It is likely that other approaches based on resonant/antiresonant data must be developed for the identification of neighboring cracks. Numerical results also show that if the frequency data used in identification are affected by errors relatively small with respect to the shifts induced by the cracks, then the method leads to satisfactory estimates of the damage parameters.

Footnotes

Acknowledgement

A Morassi wishes to thank the colleagues of University Carlos III of Madrid, especially Professors L Rubio and J Fernández-Sáez, for the warm hospitality at the Department of Engineering Mechanics.

Funding

The work of A Morassi is supported by University Carlos III of Madrid, Banco de Santander Chairs of Excellence Programme for the 2013–2014 academic year.

References

Adams

Cawley

Pye

(1978) A vibration technique for non-destructively assessing the integrity of structures. Journal of Mechanical Engineering Science 20(2): 93–100.

Attar

(2012) A transfer matrix method for free vibration analysis and crack identification of stepped beams with multiple edge cracks and different boundary conditions. International Journal of Mechanical Sciences 57: 19–33.

Caddemi

Caliò

(2009) Exact closed-form solution for the vibration modes of the Euler–Bernoulli beam with multiple open cracks. Journal of Sound and Vibration 327: 473–489.

Caddemi

Caliò

(2014) Exact reconstruction of multiple concentrated damages on beams. Acta Mechanica 225: 3137–3156.

Caddemi

Morassi

(2013) Multi-cracked Euler-Bernoulli beams: Mathematical modelling and exact solutions. International Journal of Solids and Structures 50(6): 944–956.

Cerri

Vestroni

(2000) Detection of damage in beams subjected to diffused cracking. Journal of Sound and Vibration 234(2): 259–276.

Davini

Morassi

Rovere

(1995) Modal analysis of notched bars: Tests and comments on the sensitivity of an identification technique. Journal of Sound and Vibration 179(3): 513–527.

Dilena

Morassi

(2004) The use of antiresonances for crack detection in beams. Journal of Sound and Vibration 276(1–2): 195–214.

Freund

Herrmann

(1976) Dynamic fracture of a beam or plate in plane bending. Journal of Applied Mechanics 43(1): 112–116.

10.

Gladwell

GML

(2004) Inverse Problems in Vibration, 2nd edn. Dordrecht: Kluwer Academic Publishers.

11.

Greco

Pau

(2012) Damage identification in Euler frames. Computers and Structures 92–93: 328–336.

12.

Gudmundson

(1983) The dynamic behaviour of slender structures with cross-sectional cracks. Journal of the Mechanics and Physics of Solids 31: 329–345.

13.

Hearn

Testa

(1991) Modal analysis for damage detection in structures. Journal of Structural Engineering ASCE 117(10): 3042–3063.

14.

Khiem

Lien

(2004) Multi-crack detection for beam by the natural frequencies. Journal of Sound and Vibration 273: 175–184.

15.

Khiem

Toan

(2014) A novel method for crack detection in beam-like structures by measurements of natural frequencies. Journal of Sound and Vibration 333: 4084–4103.

16.

Liang

Choy

(1992) Quantitative NDE technique for assessing damages in beam structures. Journal of Engineering Mechanics ASCE 118(7): 1468–1487.

17.

Loya

Rubio

Fernández-Sáez

(2006) Natural frequencies for bending vibrations of Timoshenko cracked beams. Journal of Sound and Vibration 290: 640–653.

18.

Mazanoglu

Sabuncu

(2012) A frequency based algorithm for identification of single and double cracked beams via a statistical approach used in experiment. Mechanical Systems and Signal Processing 30: 168–185.

19.

Morassi

(1993) Crack-induced changes in eigenparameters of beam structures. Journal of Engineering Mechanics ASCE 119(9): 1798–1803.

20.

Morassi

(2001) Identification of a crack in a rod based on changes in a pair of natural frequencies. Journal of Sound and Vibration 242(4): 577–596.

21.

Morassi

Rollo

(2001) Identification of two cracks in a simply supported beam from minimal frequency measurements. Journal of Vibration and Control 7(5): 729–740.

22.

Narkis

(1994) Identification of crack location in vibrating simply supported beams. Journal of Sound and Vibration 172(4): 549–558.

23.

Patil

Maiti

(2003) Detection of multiple cracks using frequency measurements. Engineering Fracture Mechanics 70: 1553–1572.

24.

Rubio

(2009) An efficient method for crack identification in simply supported Euler-Bernoulli beams. Journal of Vibration and Acoustics 131: 051001.

25.

Rubio L, Fernández-Sáez J and Morassi A (2013) Identification of two cracks in a rod by minimal natural frequency and antiresonant frequency data. Mechanical Systems and Signal Processing. Submitted.

26.

Ruotolo

Surace

(1997) Damage assessment of multiple cracked beams: Numerical results and experimental validation. Journal of Sound and Vibration 195: 456–477.

27.

Ruotolo

Surace

(2004) Natural frequencies of a bar with multiple cracks. Journal of Sound and Vibration 272: 301–316.

28.

Sekhar

(2008) Multiple cracks effects and identification. Mechanical Systems and Signal Processing 22: 845–878.

29.

Singh

(2009) Transcendental inverse eigenvalue problems in damage parameter estimation. Mechanical Systems and Signal Processing 23: 1870–1883.

30.

Tada

Paris

Irwin

(1985) The Stress Analysis of Cracks Handbook, 2nd edn. St. Louis, MO: Paris Productions.

31.

Vestroni

Capecchi

(2000) Damage detection in beam structures based on frequency measurements. Journal of Engineering Mechanics ASCE 126: 761–768.