Anurag Dubey
email: anurag.dubey@insa-cvl.fr

Vivien Denis

François Hennebelle

Roger Serra

Localization and quantification of geometry damage and their size estimation using frequency shift coefficient and damage library

come

Introduction

Structural health monitoring (SHM) aims to maintain the integrity of an entire structure by performing a variety of activities such as monitoring, inspection and damage detection. Over the years, vibration-based structural health monitoring (VBSHM) has become popular as a structural diagnostic tool due to its ability to monitor and diagnose damage in structures [START_REF] Sinha | Simplified models for the location of cracks in beam structures using measured vibration data[END_REF][START_REF] Chiementin | Experimental modal analysis of hand-arm vibration in golf: influence of grip strength[END_REF]. Several reviews on VBSHM have been provided in Doebling et al. [START_REF] Doebling | Damage identification and health monitoring of structural and mechanical systems from changes in their vibration characteristics: A Literature Review[END_REF] and Montalvao et al. [START_REF] Montalvao | A review of vibration-based structural health monitoring with special emphasis on composite materials[END_REF] based on advances in damage detection methods and SHM with a concentration on structures and dynamic parameters. According to Rytter [START_REF] Rytter | Vibrational based inspection of civil engineering structures[END_REF], damage detection and assessment are divided into four levels: (i) detection (ii) localization (iii) quantification (iv) prediction of the remaining life of the structure. Damage identification based on natural frequencies can be reliable due to the sensitivity of frequency changes between healthy and damaged states. Several vibration methods have been given in the literature for detection, localization and quantification of damage by considering the frequency changes. Salawu [START_REF] Salawu | Detection of structural damage through changes in frequency: A review[END_REF] discovered that natural frequency is a sensitive parameter for damage detection. Messina et al. [START_REF] Messina | Damage detection and localization using natural frequency changes[END_REF] introduced damage location assurance criterion (DLAC) and later included multiple damage location assurance criterion (MDLAC) [START_REF] Messina | Structural damage detection by a sensitivity and statisticalbased method[END_REF] for damage identification. Silva and Gomes [START_REF] Messina | Structural damage detection by a sensitivity and statisticalbased method[END_REF] first introduced the frequency shift coefficient (FSC) to detect the damage position and estimate the crack length. Dubey et al. [START_REF] Dubey | A damage identification strategy in beams based on natural frequencies shift[END_REF][START_REF] Dubey | Sensitivity and efficiency of the frequency shift coefficient based on the damage identification algorithm: modeling uncertainty on natural frequencies[END_REF] presented an FSC-based damage identification algorithm and its efficiency with modeling uncertainties on natural frequency by testing the damaged structure. The authors efficiently tested the FSC minimization algorithm on an experimental beam with double damage. Additionally, Dubey et al. [START_REF] Dubey | A novel VBSHM strategy to identify geometrical damage properties using only frequency changes and damage library[END_REF][START_REF] Dubey | Localization and quantification of damage by frequency based methods: numerical and experimental applications on bending vibration beams[END_REF] introduced a concept of correlation of 2D and 3D FE models to build a damage library using FSC minimization, and further detailed work was presented to identify properties of geometry damage (rectangular and hole) with experimental results [START_REF] Dubey | Damage identification and estimation of geometry damage properties using vibration based structure health monitoring[END_REF]. Serra and Lopez [START_REF] Serra | Damage detection methodology on beam-like structures based on combined modal wavelet transform strategy[END_REF] evaluated many traditional indicators for their effectiveness and their ability to identify and locate damage in various damaged cases.

Yet, few studies have introduced a relationship between natural frequency change and damage geometry. In this research, a VBSHM technique based on frequency changes is proposed to estimate geometry damage parameters, such as position, severity and sizes. In this context, the focus is on the development of a damage library to estimate geometry damage parameters. The proposed strategy uses the FSC minimization technique to correlate damage location, elemental stiffness reduction, and damage geometry using the correlation of 2D and 3D FE models of the structure. A numerical damage library is built for the cantilever beam structure through the correlation between the 2D and 3D FE models. The proposed strategy is demonstrated by performing an experimental test and identifying the geometry properties of damage (position, severity, and size) in a real beam to assess the effectiveness of the strategy.

The rest of the paper is organized as follows: Section 2 recalls the dynamic of a cantilever beam structure in bending vibration, and FE models (2D and 3D) are introduced to extract the dynamic properties of a structure; Section 3 presents the proposed a new strategy to identify the damage, as well as a technique to minimize FSC as a function of position and severity; Finally, Section 4, an experimental beam is tested to determine the position, severity and size of the real geometry damage.

Cantilever beam vibration background and finite element models

The equation of motion for the flexural vibration of a cantilever Euler-Bernoulli beam can be expressed as:

M (x) ∂ 2 w(x, t) ∂t 2 + C(x) ∂w(x, t) ∂t + ∂ 2 ∂x 2 EI(x) ∂ 2 w(x, t) ∂x 2 = F (x, t), (1)
where M (x) is the mass per unit length, C(x) is the viscous damping coefficient, EI(x) is the bending stiffness, w(x, t) is the transverse deviation, x is the coordinate along the main direction of the beam, t is the time, and F (x, t) is the external force per unit length of the beam. When discretized, for example using the finite element method, the equation of motion for each degree of freedom can be given as a matrix:

[M ]{ Ẍ} + [C]{ Ẋ} + [K]{X} = {F }, (2)
where [M ], [C], and [K] are the mass, damping, and stiffness matrices, respectively. In this case, { Ẍ}, { Ẋ}, and {X} are vectors containing acceleration, velocity and displacement in all degree of freedom of the model. The eigenvalue equation for the ith vibration mode of a dynamic structure with no external force or damping, is expressed as follows:

([K] -λ i [M ])ϕ i = 0, (3)
where λ i and ϕ i are the ith eigenvalue and eigenvector, and

λ i = ω 2 i
, where ω i is the ith natural frequency. The natural frequencies (in Hz) are given by f i = ω i 2π .

2D finite element model of beam

As shown in Figure 1, the cantilever beam is divided into N equal size elements, and the model includes N+1 nodes. A healthy and damaged 2D FE model (stiffness updating) of the beam is developed in this work. It is considered that the damage is placed at location x l in the element with the nodes i and i + 1 of the beam. The beam is divided into 100 finite elements, and the size of each element is 10 mm. The total number of degrees of freedom (DOF) of the 2D FE model is 202, with each node having two DOF: displacement (y) and rotation (θ z ). The severity of the damage is indicated by the stiffness reduction coefficient β c , which is the ratio of the stiffness reduction to the initial stiffness. The stiffness matrix of the damaged beam is written by the following equation:

[K d ] = n c=1 (1 -β c )[K c ], (4)
where [K d ] is the global stiffness matrix for a damaged beam, n is the number of elements, [K c ] is the stiffness matrix of element c, and β c is a bending stiffness reduction parameter for the damaged structure ranging from 0 to 1. A value of β c = 0 indicates healthy element.

3D finite element model of the beam

3D FE models of the healthy and damaged cantilever beam structure are developed using COMSOL Multiphysics software. The beams are modeled by a fine mesh using tetrahedral elements (6 DOF per node), and the mesh density is controlled by COMSOL. Typically, converged damage models have 22,000 DDLs. A high mesh density is used near the damaged area. Figure 2 shows a zoomed view of the damaged rectangular geometry and its physical parameters (width, depth, and height). The natural frequencies of healthy and damaged 3D models are estimated using COMSOL Multiphysics own solver.

Strategy and objective function for damage identification

The novel objective of this study is to build a damage library using an existing severity indicator to estimate the properties of rectangular geometry damage [START_REF] Dubey | A novel VBSHM strategy to identify geometrical damage properties using only frequency changes and damage library[END_REF][START_REF] Dubey | Localization and quantification of damage by frequency based methods: numerical and experimental applications on bending vibration beams[END_REF][START_REF] Dubey | Damage identification and estimation of geometry damage properties using vibration based structure health monitoring[END_REF]. The strategy for identifying damage properties is described in three main steps:

Step 1: In the beginning, the experimental beam is used to identify a set of healthy beam frequencies. The healthy 2D and 3D FE models are developed, and Young's modulus of the FE models is updated using the maximization function to closely match the frequencies with the healthy experimental beam. Then, the damage is represented as a rectangular discontinuity in the 3D FE model, considering position, width and depth of the damage. At the same time, the damage is simulated in the 2D FE model as a local elemental reduction of bending stiffness. Numerical correlation is performed between 2D and 3D FE models by FSC minimization, i.e. for each set of physical parameters, a reduction in bending stiffness is performed in the 2D model. At the end of this step, a damage library is built, with 3D rectangular damage providing the width and depth, and a 2D model providing the bending stiffness reduction value (%) of the severity at a position specific to the damage. Note that the damage library is numerically validated once built.

Step 2: FSC is used to update the 2D FE model of a damaged beam to identify the position and severity of the damage. Four beam models with similar characteristics (healthy beam, damaged beam, healthy 2D model and damaged 2D model with stiffness updating) are used for a given FSC update procedure. During this procedure, the stiffness reduction coefficient (or severity) and the location of the reduction are updated. The minimum value of the FSC criterion gives the desired location and severity. In practice, the FSC minimization is calculated using Particle Swarm Optimization (PSO) [START_REF] Kennedy | Particle swarm optimization[END_REF] solver in MATLAB global optimization toolbox.

Step 3: The damage severity from Step 2 is searched in the damage library for the identification of physical geometry parameters. In the case of rectangular damage, the severity is linked to a (potentially non-unique) set of width and depth. It should be noted that curve fitting may be required if the library is not accurate enough, and tolerance can be applied. The non-uniqueness of the solution from the rectangular damage library as well as the results are discussed in the experimental part.

Frequency shift coefficient

The frequency shift coefficient (FSC) is used for the minimization procedure to determine the position and severity (%) of the damage and to build a damage library. Silva and Gomes [START_REF] Silva | Crack identification on simple structural elements through the use of natural frequency variations: The inverse problem[END_REF] were first proposed FSC for damage identification. The FSC equation is written as follows:

FSC = 1 m m i=1 (R i ) X -(R i ) A (R i ) X et R i = f d i f h i ( 5
)
where R i represents the ratio of the frequencies of the damaged beam (f d i ) to those of the healthy beam (f h i ), m is the total number of modes, D denotes the damaged case, X refers to the tested case, A denotes to the reference, i.e. the healthy FE model, and i denotes the frequency indices. For the FSC to be calculated, at least two vibration frequencies must be considered.

Identification of damage on an experimental beam

The efficiency of the proposed strategy based on the FSC approach is examined by testing an experimental cantilever beam. Figure 3 shows a damaged steel beam with two accelerometers (length L = 1000 mm, width W = 25 mm, and thickness T = 5.3 mm). The material characteristics of the steel beam are considered with the standard values (Young's modulus E = 210 GPa, ρ = 7850 kg/m 3 , v = 0.33). A rectangular saw cut is created on the damaged beam with a depth of 1.35 mm and a width of 1.00 mm. The hammer impact test is performed, and vibration signals are recorded with an LMS multi-analyzer (Siemens LMS software, Plano, TX, USA) to extract natural frequencies from healthy and damaged beams. The hammer generated the input excitation, while the monoaxial accelerometer (Brüel and Kjaer-DeltaTron Type 4507) recorded the output data. The experimental device is used to extract natural frequencies from cantilever beams (healthy and damaged). The frequency response functions of healthy and damaged cantilever beams are used to obtain the natural frequencies using the circle fit method and the application of the EasyMod package [START_REF] Kouroussis | Easymod: A matlab/scilab toolbox for teaching modal analysis[END_REF]. The first seven measured natural frequencies of healthy and damaged beams are listed in Table 1.

Results

FSC is used to evaluate and compare the real damaged beam case (see Figure 3) to the numerical healthy and damaged 2D FE models. The mass of the accelerometers is not taken into account, as the localized mass in the FE models when identifying natural frequencies and damage in a beam. The FE model is used to update based on the natural frequencies of the experimental beam with the accelerometers. The damage is first located and quantified, and then its size is extracted from the damage library according to the estimated severity.

Step 2 is used to calculate FSC, where position and severity are estimated as shown in Figure 4(a). The colorbar represents the FSC value, while the white circle represents (obtained by particleswarm() function in MATLAB Global Optimization Toolbox) the position and severity based on FSC minimization. In the real case, the damage is estimated with the value of 27.75% at the position of 0.238 m from the clamped part of the beam, where the FSC minimum value is 0.0494. It is important to note that the estimation of the position of the damage (0.238 m) is obtained close to the real position (0.245 m). The FSC minimization problem is solved in 5-10 seconds. Finally, the determined damage severity (27.75%) is searched in the damage library with a tolerance of 1% (see Step 3) to extract the width and depth of the damage. As shown in Figure 4(b), multiple damage sizes correspond to a single severity. For demonstration purpose, we assume that the width is equal to the real width value (1.0 mm). It is noted that the depth is obtained by the value of 1.408 mm for a damage width of 1 mm (see Figure 4(b)). This estimated value is close to the real depth value (1.350 mm). This means that the damage library is suitable to provide the physical properties of geometry type damage. Therefore, the proposed strategy is an efficient identification tool to estimate the characteristics of geometry damage in a real problem.

Conclusion

In this research, a strategy based on bending stiffness reduction is developed to identify damage properties (position, severity and size) in beams. The approach uses only natural frequencies and FE models of the structure. The method is tested on a steel cantilever beam. In this context, minimization of the frequency shift coefficient is used in two ways. First, a damage library is built by correlating the 2D and 3D FE models of the beams. The library is designed for rectangular damage which combines depth and width of the 3D FE model with bending stiffness reduction of the 2D FE model. The damage library does not influence by the position of the beam. Second, a FE model of the damaged beam is updated to determine the position and severity of the damage. At last, the damage library is used to determine more about the damage geometry size. The method has been successfully tested and estimated geometry damage properties (position, severity and size) on an experimental cantilever beam in which saw-cut type damage was incorporated. In the future, this strategy will be validated by testing it with other types of structures and other damage geometries such as hole, oblique or triangular cut.

Figure 1 :

 1 Figure 1: 2D FE model of a cantilever beam and cross-section area.

Figure 2 :

 2 Figure 2: 3D finite element mesh of the beam with zoomed damage

Figure 3 :

 3 Figure 3: Experimental damaged cantilever beam and illustration of measurement input and output points.

Figure 4 :

 4 Figure 4: (a) FSC as a function of position and severity where white circle represents 27.75% estimated damage severity at 0.238 m position; (b) width 1.0 mm and depth 1.408 mm for a severity of 27.75% extracted from the damage library (blue dots: raw extraction, red line: cubic fit, black cross: retrieved parameters).

Table 1 :

 1 Experimentally identified natural frequencies of healthy and damaged beams.

	Experimental beam	Natural frequencies (Hz)
		1	2	3	4	5	6	7
	fhealthy	4.23 26.34 73.26 142.34 240.13 355.67 499.67
	fdamaged	4.21 26.35 73.14 141.87 240.02 357.80 498.78

Acknowledgements

The research project was funded by the Region Center Val de Loire (CVL) France as part of a doctoral contract (2017-2020). The authors are grateful for financial support.


Anurag Dubey
email: anurag.dubey@insa-cvl.fr

Vivien Denis

Roger Serra

A damage identification strategy in beams based on natural frequencies shift

This paper presents a damage identification technique based on the frequency changes between healthy and damaged beam structure. An algorithm based on frequency shift coefficient (FSC) is proposed, and the sensitivity of eigenfrequencies is studied. The method is tested by introducing an artificial damage, which depends on local stiffness reduction using five frequencies. The modal analysis technique is employed for healthy and damaged 2D Euler-Bernoulli beam modeled with Finite Element by using mass and stiffness matrices in order to provide natural frequencies. The effect of uncertainty on natural frequencies (0.1%, 0.2% and 0.3%) in damaged case is considered, and samples are used to estimate the average damage position and severity. In the beginning, the damage is localized and quantified. Then, the sensitivity of eigenfrequencies is investigated based on added uncertainty. The conclusion of analyses indicates that the algorithm correctly localizes and accurately estimates the damage by requiring a short amount of computational time.

Introduction

Structural integrity evaluation based on vibration analysis is an important aspect for researchers and scientific committee. There are many critical mechanisms in a structure that can cause for damage. However, the presence of damage can causes changes in the physical parameters (mass, stiffness and damping) and geometry parameters (thickness, length, height, etc.). Therefore, changes in the natural frequencies and mode shapes of the structure. One of the most suitable techniques to identify the presence of damage and assess the structure is vibration-based structure health monitoring (VBSHM). Doebling et al. [START_REF] Doebling | Damage identification and health monitoring of structural and mechanical systems from changes in their vibration characteristics: a literature review[END_REF], Montalvao [START_REF] Montalvao | A review of vibration-based structural health monitoring with special emphasis on composite materials[END_REF] and Sohn et al. [START_REF] Sohn | A review of structural health review of structural health monitoring literature 1996-2001[END_REF] presented reviews covering on VBSHM based on advances in damage detection and SHM with an emphasis on structures. Rytter [START_REF] Rytter | Vibrational based inspection of civil engineering structures[END_REF] defined four levels for damage detection and assessment: (i) detection (ii) localization (iii) quantification (iv) prediction of the remaining life of the structure. The first step for structural identification is to compare the natural frequencies of healthy beam and unknown beam. If the frequency shift remains, then there is a chance that damage is present in the structure. The frequencies are commonly used modal parameters for identification of defect as they are sensitive for different types of damage. Cawley and Adams [START_REF] Cawley | The location of defects in structures from measurements of natural frequencies[END_REF] introduced sensitivity analysis using natural frequencies to detect, locate and quantify damage. Salawu [START_REF] Salawu | Detection of structural damage through changes in frequency: a review[END_REF] mentioned that natural frequency is a sensitive parameter and measurements of frequencies can be quickly conducted and are reliable. Messina et al. [START_REF] Contursi | A multiple-damage location assurance criterion based on natural frequency changes[END_REF] presented damage location assurance criterion (DLAC) and after multiple damage location assurance criterion (MDLAC) based on natural frequency changes. Silva and Gomes [START_REF] Silva | Crack identification on simple structural elements through the use of natural frequency variations: The inverse problem[END_REF] introduced frequency shift coefficient to detect damage position and estimate crack length. Shukla and Harsha [START_REF] Shukla | Vibration response analysis of last stage lp turbine blades for variable size of crack in root[END_REF] acknowledged that the changes in natural frequency are an indication of cracks in the blade geometry. Zhang et al. [START_REF] Zhang | Sensitivity analysis of inverse algorithms for damage detection in composites[END_REF] presented techniques based on vibration methodologies by the frequency shifts from three inverse algorithms based on graphical technique, surrogate-assisted optimization and artificial neural network, to predict the location, size and interface of damages in the composite beam. Le et al. [START_REF] Le | Structural changes assessment in axial stressed beams through frequencies variation[END_REF] employed a method based on the first-order analytical estimation value of the relative variations of eigenfrequencies due to the change in mass and/or rigidity from the estimation of the modal parameters. Serra et al. [START_REF] Serra | Tentative of damage estimation for different damage scenarios on cantilever beam using numerical library[END_REF] examined numerical damage case using different indicators and a damage library was created from 3-D and 2-D Finite Element (FE) model of a cantilever beam from considering bending frequencies. Serra and Lopez [START_REF] Serra | Damage detection methodology on beam-like structures based on combined modal wavelet transform strategy[END_REF] compared many classical indicators to evaluate their performance and capacity to detect and localize damage from different cases.

In this paper, a FE based algorithm using modal analysis technique is proposed for a cantilever beam structure. The vibratory response of the structure is tested with two states: the healthy state and the damaged state. Localization and quantification of damage are calculated from these two states using frequency variations. The main objective of this paper is to implement the proposed algorithm which can be used for damage identification based on the frequency shift between healthy and damaged structure. In order to validate the proposed damage identification algorithm, at first, the algorithm is implemented between FE models to identify damage. Then noise levels or uncertainty levels on natural frequencies (0.1%, 0.2% and 0.3%) are tested by adding random noise in damaged beam frequencies. In each noise level, ten samples are used in damaged beam frequencies to estimate the average damage position and severity. Finally, the sensitivity of the algorithm is mentioned through the identification of damage case with or without noise levels.

The paper is organized as follows. In Section 2 , a brief introduction about beam bending vibration theory for 2D FE model is presented to extract the modal parameters from a structure. In Section 3 , the steps of proposed strategy and variation in frequencies using random noise function are illustrated. In Section 4 , a numerical case is tested, and results are analyzed employing noise levels in damaged beam frequencies. In Section 5 , the conclusion is drawn based on the proposed algorithm using the different noise levels.

Beam vibration model

The study model focuses on a Euler's-Bernoulli cantilever beam structure, as shown in Fig. 1 . The beam is modeled with 100 elements, and each element size is 10 mm. The 2D FE model has a the total number of 202 degree of freedoms (DOFs), where each node has 2 DOFs: displacement along y-axis and rotation (θz) along the z-axis. The formulation of the equation of motion for each degree of freedom can be described in matrix form:

[M ]{ Ẍ} + [C]{ Ẋ} + [K]{X} = {F } (1)
where [M ], [C] and [K] are the mass, damping and stiffness matrices, respectively. Here, X stands for a nodal displacements vector. Considering no external force and damping, the eigenvalue equation for the ith vibration mode of a dynamic structure is expressed as:

([K] -λ i [M ])ϕ i = 0 (2)
where λ i and ϕ i are the ith eigenvalue and eigenvector, and where λ i = ω 2 i in which ω i is the ith natural frequency. In this particular 2D FE approach, it is assumed that the damage only affects the stiffness of selected elements. In this study, an artificial damage is simulated by reducing the bending stiffness of the selected element. Damage severity is represented by the stiffness reduction coefficient β e , which is the ratio of stiffness reduction to the initial stiffness.

The following equation defines the stiffness matrix of a damaged beam:

[K d ] = n e=1 (1 -β e )[K e ] (3)
where [K d ] is global stiffness matrix for a damaged beam, [K e ] is the element stiffness matrix for element e, n is the number of elements, and β e is a bending stiffness reduction parameter, which varies from 0 to 1 for the damaged structure. The value of β e = 0 indicates a healthy element.

Proposed damage identification strategy

This work presents a strategy based on natural frequencies for locating and estimating damage in a beam discretized using FE. The strategy focuses on the shift of natural frequencies between healthy and damaged beam, whereas 2D FE models are simulated in order to perform damage identification. To achieve this goal, the damage is identified by updating a FE model using a criterion based on natural frequencies. The identification is performed by the use of four sets of natural frequencies : on the one hand, a healthy beam and a damaged one, the latter being to identify, on the other hand, a 2D healthy beam model and damaged one, the latter being updated. Here, the strategy uses numerical computation of frequencies along with the damage position and 1 to 99% stiffness reduction, as the study is carried out using the FE model updating.

To sum up, the strategy consists in updating the stiffness of an FE model, based on frequency shifts. The numerical solution of damage case is obtained with the use of FE model updating approach. Finally, the strategy is carried out in order to investigate the sensitivity of the eigenfrequencies.

Objective function and minimization procedure

In this study, a Frequency Shift Criterion (FSC) is employed during minimization procedure to find the position and severity percentage. This FSC is first presented by Silva and Gomes [START_REF] Silva | Crack identification on simple structural elements through the use of natural frequency variations: The inverse problem[END_REF] for damage identification problems. The FSC is expressed as:

FSC = 1 m || m i=1 (R i ) X -(R i ) A (R i ) X || and R i = ω d i ω h i ( 4
)
where R i is the ratio of damaged (ω d i ) and healthy (ω h i ) beam frequencies, m is the total number of frequencies, D refers to the damaged case, A refers to the reference i.e. safe FE model to be updated, and i denotes frequencies indices. At least two vibration frequencies must be considered for the evaluation of the FSC.

In the minimization procedure, the updating parameters are set by the vectors of positions [p 1 , p 2 ....p N ] and severities [β 1 , β 2 .....β N ]. The objective of 2D FE updating is to obtain the best-fit value through the minimization procedure between 2D FE model to be updated and tested damaged case. The parameters minimizing FSC correspond to the position and severity of the damaged case.

Practically, the minimization problem is solved using Particle Swarm Optimization (PSO) [START_REF] Kennedy | Particle Swarm Optimization[END_REF]. Several other techniques may be employed, but the main reasons behind this choice is that a global solution must be found and the gradient of the cost function i.e. the FSC, is difficult to define or to estimate. In practice, the PSO (as implemented in the particleswarm() function in MATLAB Global Optimization Toolbox) proves to be adequate to solve the problem.

There are many parameters in PSO, i.e., swarm size, number of iterations, inertia weight, learning factors, etc. The influence of PSO parameters are not considered during minimizing procedure. Only the swarm size and the number of maximum iterations are explicitly specified during FSC minimization, with values of 10 and 50, respectively.

Algorithm

The main steps for the proposed algorithm are summarized as follows:

Step-1: Generate damage case with the bending stiffness reduction ranging between 0.01 and 0.99 with a step of 0.01.

Step-2: Develop the FE models of the beam structure in order to compute natural frequencies as for healthy and given damaged case from step-1. Optionally, add a perturbation to the set of frequencies of the damaged case in order to simulate measurement imprecision.

Step-3 : Using the FSC and the minimization algorithm, update a beam FE model with a generic defect, for each set of tested frequencies.

Step-4 : Consider the updated parameters as the damage parameters. In the case where perturbations are introduced, consider the mean and standard deviation of the results.

Modeling uncertainty on natural frequencies

Measurement imprecision always exists in the experimentally measured frequencies as well as other modal parameters. In this study, artificial uncertain cases, i.e. cases whose natural frequencies are incorrectly estimated, are generated numerically, and the effect on the results are analysed based on the proposed algorithm. The uncertain or noisy cases are introduced in damaged beam frequencies by the addition of percentage noise levels with assumed normally distributed random variables. In this regard, the uncertain or noisy damaged beam frequencies can be expressed as:

ωd i = ω d i (1 + ηγ i ) and i = 1, 2..., n m (5)
where ωd i is ith damaged beam frequency after noise addition, η is the percentage of noise, and γ i is a Gaussian random number between -1 to 1, different for each i.

Results and discussions

To investigate the proposed strategy and evaluate the performance of the algorithm, we propose to identify a damage location and severity on a Euler-Bernoulli cantilever beam. Furthermore, on a given damaged case, we will estimate the sensitivity of the algorithm by introducing a perturbation on the natural frequencies as presented in Eq. 5.

The steel beam properties and dimensions are listed in Table 1 . The artificial damage is introduced by reducing the stiffness of the individual element. Here, the damage case is set at 0.8 m beam position by 50% stiffness reduction. The first five lowest natural frequencies of the healthy and damaged beam are extracted from the respective FE models. The result may vary with consideration of more than five frequencies. However, it is also essential to reduce the computation time. Hence, five frequencies prove to be sufficient to identify the considered structural modifications. The frequencies of healthy and damaged beam are reported in Table 2. Three significant digits are kept in the natural frequencies, which is enough for the algorithm to works accurately and identify a damage.

In a first step, the damage of the tested case is identified based on the exact natural frequencies. The Frequency Shift Criterion is minimized in order to identify position and severity of the damage. In Fig. 2, the FSC is plotted in color levels as a function of position and severity. The red circle denotes real damage position and severity (%) while the cross symbol is obtained from minimizing the FSC using PSO; The locus of the FSC lowest value corresponds to the result of the minimization and also to the exact parameters. Essentially, it shows that the criterion is valid and that the minimization yielded the expected result. Note that the computational time for minimization using PSO is approximately 10 s. In a second step, the natural frequencies of the tested case are perturbed by adding noise with η = 0.1, 0.2 and 0.3 %. It is important to notice that the algorithm does not affect by the addition of the noise levels. By these noise levels, the algorithm works accurately for the localization and estimation of the damage. Using Eq. 5, 10 random samples are generated by randn MATLAB function based on the tested damaged case. Each sample i.e. each set of five natural frequencies is used in the minimization procedure in order to estimate damage parameters.

The results are represented in Fig. 3, where the red dot represent real damage position and severity i.e. without perturbation. Green, blue and black stars denote identified damage parameters for 0.1%, 0.2% and 0.3%, perturbation levels, respectively. Average values of positions and severities are 0.806 m, 0.808 m, 0.811 m and 50.37%, 50.07%, 51.54%, respectively. The standard deviation values for obtained positions and severities are 0.006 m, 0.022 m, 0.024 m and 1.56%, 3.65%, 5.09%, respectively.

The average identified damage parameters for 0.1%, 0.2%, and 0.3% perturbation levels are really close to real identified damage parameters. These results are a good way to represent the sensitivity of the criterion and algorithm. Sensitivity is not negligible and show that a precise estimation of natural frequencies is needed.

Conclusion

The present work focused on damage localization and quantification with an algorithm using FE models and the minimization of a Frequency Shift Criterion. As small errors are inevitable in the estimation of experimental frequencies, the sensitivity of the criterion to the natural frequencies of the beam is investigated by considering the perturbation of these frequencies at different levels. Each random perturbation provides a different set of damage parameters. On average, these parameters depends on the perturbation level but are close to the real ones; sensitivity is given by the standard deviation on a number of sample. Sensitivity is not negligible. The numerical results demonstrated that damage positions and severities may be correctly identified by minimizing the FSC with or without perturbation, within a small computational time. The results are validated for several cases (different damage levels and position, several perturbation levels). Future investigation on the identification of multiple damages, and damages on other structure needs to be conducted.

Figure 1 :

 1 Figure 1: 2D FE model of a cantilever beam and cross-section area.

Figure 2 :

 2 Figure 2: FSC as a function of position and damage (%). The red circle denotes a real damage case, where cross symbol and colorbar indicate the lowest minimized value corresponding to identified 0.8 m beam position with 50% damage severity.

Figure 3 :

 3 Figure3: Real and identified damage positions and damage (%) concerning for ten samples with 0.1%, 0.2% and 0.3% random noise levels.

Table 1 :

 1 Steel beam properties and dimensions.

	Properties	Value
	Length (l)	1 m
	Width (w)	0.0249 m
	Height (h)	0.0054 m
	Poisson's ratio (v)	0.33
	Mass density (ρ)	7850 kg/m 3
	Young's modulus (E) 210 GPa

Table 2 :

 2 Numerically identified natural frequencies of healthy and damaged beam.

	Beam case	Natural frequencies (Hz)
		1	2	3	4	5
	Healthy	4.554 28.537 79.904 156.579 258.837
	Damaged ( 0.8 m with 50% ) 4.553 28.480 79.294 154.845 256.882

Acknowledgements

The authors express their thanks for the financial support provided by Région Centre Val de Loire, France through a doctoral contract.