1. Introduction
In recent decades, significant research has been conducted on Structural Health Monitoring (SHM) by using many techniques which are applied in various fields such as aerospace, civil and mechanical engineering. SHM aims to provide maintenance services throughout the life of the structure. Vibration based structure health monitoring (VBSHM) is widely used as a structure diagnostic technique over the years because of its ability to monitor and detect damage from global testing of the structure [
1,
2]. Worden and Dulieu-Barton [
3] gave an overview strategy for damage detection such as the definition based on a taxonomy of damage, operational evaluation with the use of a hierarchical damage identification scheme, an approach to sensor prescription and optimization and a data processing methodology. More recently, Kong et al. [
4] reviewed the state of the art on the framework of vibration-based damage identification in different levels including the prediction of the remaining useful life of structures and the decision making for proper actions. Toh and Park [
5] provided a summary of studies applying machine learning algorithms for health monitoring using vibration factors to categorize the studies.
The first step for damage identification using modal analysis relies on changes in the properties of the structure, mainly their eigenvalues. Salawu [
6] proposed a review on methods using eigenfrequencies only for the localization and identification of damage in structure. Several works are described more precisely here, and more recent research works are added. Cawley and Adams [
7] developed the first study for damage localization and estimation, using changes in natural frequencies for 1D and 2D structures and relying on a FE model of the structures. Messina [
8] proposed Damage Location Assurance Criterion (DLAC) and after extended to the statistical correlation coefficient, which is called Multiple Damage Location Assurance Criterion (MDLAC) [
9]. The two methods rely on changes in natural frequencies for estimating the position and size of defects in complex structures. Hassiotis and Jeong [
10] developed a method for the estimation of damage using measured changes in the natural frequencies. They presented eigenvalues variation to localize damage using reductions in the stiffness by derived equations from the first-order perturbation. They have also introduced a criterion to solve these equations by minimizing the difference between eigenvalue problems for the undamaged and damaged structures. It was concluded that few frequencies allows the estimation and location of single or multiple defect in structures such as beams but that the results are sensitive to uncertainty on the natural frequencies. Meneghetti and Maggiore [
11] made a sensitivity analysis on a cracked beam and used a local modal crack sensitivity in order to localize damage using natural frequency measurement only. According to Doebling [
12], Silva and Gomes [
13] proposed a technique to detect the crack size and position using a so-called Frequency Shift Coefficient.
Sinha et al. [
1] use a model for stiffness reduction due to a crack and a minimization involving the sensitivity matrix from a Finite Element (FE) model and a set of five measured natural frequencies for finding crack depth and location on a cantilever beam. Khiem and Toan [
14] examined natural frequencies from the Rayleigh quotient for a clamped beam with an arbitrary number of cracks. The cracks are represented as spring stiffnesses. Through an iterative procedure, three experimentally measured frequencies of a beam with fixed ends are sufficient to locate cracks and estimate their severity using the explicit expression of the Rayleigh quotient. Le et al. [
15] introduced a method based on the first-order analytical estimation of the variation of natural frequencies due to structural modifications. It allows, with the help of a minimization procedure, to quantify and localize several structural modifications i.e., damage on a Euler–Bernoulli beam subject to axial constraint, using only modal shapes of the intact structure and natural frequencies of both intact and modified structures. Dahak and Benseddiq [
16] used normalized frequencies method for a specifical damage location in the cantilever beam. The damaged zone is determined only by the classification of the normalized frequencies of the structure. The change in stiffness caused by the damage is reflected locally in the frequencies. Yang and Oyadiji [
17] have presented a theoretical and experimental study of a frequency-based damage detection method. The method uses the modal frequency curve of a beam with moving point mass and discrete wavelet transform in order to build a damage index. Gillich et al. [
18] used the link between natural frequency shift and modal curvature to localize and identify defects in a multispan beam. Sha et al. [
19] presented a study of relative natural frequency change curves for characterization of damage in a fixed-fixed beam. They detail the relationship between these curves and mode shapes. They introduced a novel probabilistic damage indicator for localizing single and multiple damages and severity factor to quantify the damage. Serra and Lopez [
20] proposed a new methodology based on combined modal wavelet transform strategy to cope with noisy signals, while at the same time, able to extract the relevant information from each mode shape. Pan et al. [
21] presented a noise response rate to evaluate the sensitivity of each mode of the frequency shift to noise. The authors developed a novel method of mode selection for improving the accuracy of frequency-based damage detection.
In some experimental studies, Niemann et al. [
22] presented an optimization approach for damage localization based on the correlation of a local stiffness loss and the change in experimental modal parameters. They applied this approach to a composite laminate beam. Masoumi and Ashory [
23] proposed a method based on the flexibility matrix and the use of stationary wavelet transform to localize cracks in cantilever beams. Gautier et al. [
24] use a 4SID technique combined with a FE model updating procedure coupled and an iterative domain partitioning procedure to localize damages. Altunişik et al. [
25] considered multiple crack effects on a cantilever beam. In their work, Enhanced Frequency Domain Decomposition and Stochastic Subspace Identification were implemented to extract the dynamic characteristics; the Modal Assurance Criterion, the Coordinated Modal Assurance Criterion and the natural frequencies are used to update a FE model and the approach was validated by experiment.
From this literature overview, it can be concluded that for simple structures at least, natural frequencies may be sufficient for the localization and estimation of single or multiple damage. Several indicators exist and the use of modal shapes may provide more accuracy. Yet, few studies make the link between natural frequencies variation and the geometry of damage; this paper is going beyond a severity indicator by estimating the geometry of single or multiple damages. To achieve this goal, a novel VBSHM strategy is developed based only on frequency changes for the identification of geometrical damage properties, i.e., location, severity, and size. In this matter, the focus is made on the development of a damage library in order to estimate the geometry damage parameters. For the sake of simplicity, a cantilever beam is chosen. The proposed strategy uses the Frequency Shift Coefficient (FSC) as cost function to correlate position, element stiffness reduction and geometry of damage between different FE models of the structure, using a minimization algorithm. The correlation between 2D and 3D FE models allows us to build a numerical damage library for the beam. As the FSC only uses natural frequencies, it will also be used to update a 2D FE model using a small set of natural frequencies of a tested structure. We can thus solve the damage identification problem (single and multiple damages) as a function of damage position and severity. Rectangular geometry damage cases are tested, and their properties are identified based on obtained damage library. The proposed strategy is demonstrated on numerical cases. Finally, the experimental test is carried out by considering the geometry damage in a real beam in order to validate the efficiency of the strategy.
The rest of paper is organized as follows. In
Section 2, the dynamics of the specific case of a cantilever beam in flexural vibration are recalled, and FE models (2D and 3D) are introduced to extract the dynamic parameters of a structure. In
Section 3, the proposed strategy for damage identification is given, and an approach that involves minimizing the FSC as a function of position and severity is proposed. In
Section 4, 2D and 3D FE models of beams are used to build a so called damage library. In addition, this section contains numerical examples that are used to validate FSC based approach. Different test cases are examined to localize and estimate the damages which concern (i) four numerical cases, consisting of 2D vs. 2D FE model correlation that contains single and multiple damages, and (ii) two numerical cases for rectangular geometry damage, consisting of 3D vs. 2D FE model correlation. Finally, in
Section 5, an experimental beam is tested to identify damage properties.
3. Strategy and Objective Function for Damage Identification
It is well known that the presence of damages modifies dynamic parameters and behavior of the structure. In the proposed strategy, numerical cases and experimental validation is performed with only using frequencies changes between healthy and damaged structure. In this work, the novelty consists in using an existing severity indicator building a damage library for estimating properties of rectangular damage.
3.1. Proposed Strategy
In this section, a methodology for damage properties identification is described in three main steps.
Figure 4 represents a complete flowchart of the proposed strategy for geometry damage properties identification.
Step 1: A set of healthy state frequencies are identified from the experimental beam. 2D and 3D FE healthy model are built, and Young’s modulus of FE models are updated to closely match the frequencies with the healthy beam (the cost function is described later on). Then, in the 3D FE model, the damage is materialized as a rectangular discontinuity considering the position, width and depth of the damage. At the same time, the damage is materialized as a local reduction of bending stiffness in one element in the 2D FE model. Numerical correlation using FSC minimization between 2D and 3D FE models is performed, that is, for each set of physical parameter, a bending stiffness reduction is sought in the 2D model. At the end of this step, damage library is obtained, where 3D rectangular damage provides the width and depth, and 2D model gives bending stiffness reduction in percentage value of severity at particular damage position. More precisely, the damage library is a relation between physical parameters and bending stiffness reduction. Moreover, these parameters can be extracted in the form of one item from the damage library.
Step 2: An updating of the 2D FE model of damaged beam is performed using FSC to identify location and severity of single or multiple damage. Typically, the basis for the model update is a set of frequencies issued from a 2D model, a 3D model or an experimental measurement. For a given updating procedure, four beam models with the same characteristics (healthy beam case, damaged beam case, 2D healthy model and 2D stiffness updated damaged model) are used. The stiffness reduction coefficient (or severity) and the localization of the reduction are updated during this procedure. The minimum value of the FSC is sought during the updating process. The argument of the minimum yield wanted location and severity. Practically, the minimization is performed using particle swarm optimization (PSO) [
26] in MATLAB global optimization toolbox. If the damage is identified, then we can proceed to the next step for geometry investigation.
Step 3: In this step, based on the results of Step 2 and Step 3, the identification of damage physical parameters can be done. The estimated damage severity from Step 2 is searched in the damage library. It corresponds to a (potentially nonunique) set of width and depth for a rectangular geometry. Note that curve fitting may be necessary if the library is not precise enough and that a tolerance can be introduced. The nonuniqueness of the solution and the way to go beyond will be discussed later on.
Remark: The interest of using a damage library instead of directly correlating the tested structure to a 3D model is indeed to reduce the computational time.
3.2. Initial Model Updating
According to
Figure 4, an initial updating of the numerical models (2D and 3D) with respect to an experimental healthy beam is done first. Only the Young’s modulus is updated. A maximization problem is formulated which uses the inverse of the statistical error on the natural frequencies. The updated Young’s modulus is then given by:
where
is the
ith frequency of 2D/3D FE model of the beam,
is the
ith healthy beam frequency which is obtained from the experimental modal analysis.
3.3. Frequency Shift Coefficient (FSC)
The modal method for damage detection relies on changes in dynamic properties of the structure and particularly natural frequencies. Any changes in the structure, such as the reduction in stiffness, will cause changes in the natural frequencies. One of the important advantages of natural frequency is that it can be easily measured through vibration measurements. Classical measurements schemes can be used for the determination of experimental resonant frequencies. In this context, the Frequency Shift Coefficient is first presented by Silva and Gomes [
13] for damage identification problems. This coefficient uses four sets of natural frequencies. We consider on the one hand so called experimental frequencies denoted
X (issues from a numerical model or an experiment) and on the other hand frequencies from a model, denoted
A. In each category, there can be frequencies from a healthy or reference structure, denoted
h and frequencies from an unknown, or damaged structure, denoted
d. To be clear, there is a set of experimental healthy frequencies
, a set of experimental unknown frequencies
, a set of frequencies obtained with a healthy model
and a set obtained with a damaged model
. The frequency shift coefficient is defined as:
where
m is number of modes taken into account,
X is the damaged case,
A is the 2D beam model (reference),
is the damaged beam frequencies,
is healthy beam frequencies and
i denotes modes’ indices.
The FSC is then used to qualify by updating using the shifts with respects to the sets and . Remember that was first updated with respect to . In other words, the location and severity of damage in are given by minimizing the value of FSC.