1. Introduction
Damage identification has drawn increasing interest in the sectors of aerospace, civil engineering and mechanical structures. The basic components of a structure make it very sensitive to damage that then require techniques for detecting damage using efficient methodologies. Damage can occur during manufacturing and in-service loading, such as fatigue and other object impacts. There are many methods to detect and determine the severity of damage based on Structure Health Monitoring (SHM). Many SHM methods, such as ultrasonic [
1], guided wave [
2], eddy current [
3], scaling subtraction method [
4,
5,
6] and nonlinear vibro-acoustic wave modulation technique [
6], etc., have been developed to identify structural damage, which are used for various purposes. Although several Vibration Based Structure Health Monitoring (VBSHM) methods have been also proposed, they rely on vibration characteristics such as natural frequencies, mode shapes, etc. Some important VBSHM indicators are Damage Location Assurance Criterion (DLAC) [
7], Multiple Damage Location Assurance Criterion (MDLAC) [
8] mode shape curvature method [
9] and the flexibility based method [
10], which are effective and widely accepted in order to identify damage and their characteristics. Doebling et al. [
11] presented a comprehensive review of different damage identification methods and health monitoring of structures from changes in the vibration characteristics. The damage conditions of the system can be described in five steps, as discussed in Rytter [
12]. These steps are: (1) detection; (2) localization; (3) classification; (4) assessment; and (5) prediction. Dubey et al. [
13] introduced a novel VBSHM strategy for geometry damage identification and size estimation using a damage library. The strategy was employed to estimate the size of rectangular geometry damage both numerically and experimentally in a tested cantilever beam. Vibration-based damage identification was used for a three-span continuous beam, a two-span steel grid [
14] and a reinforced concrete beam [
15], considering the effects of temperature variations. More recently, Toh and Park [
16] provided a review applying machine learning algorithms for damage monitoring using vibration factors and interpretation of deep neural networks in order to guide further applications for structural vibration analysis.
In recent decades, natural frequencies have been used as an identification parameter for the detection, localization and quantification of damage. Salawu [
17] acknowledged that natural frequency is a sensitive indicator to detect damage in the structure. Cawley and Adams [
18] proposed a method based on the frequency shift that identifies the position of the damage in a plane structure. Narkis [
19] analyzed the inverse problem for identification of crack position from frequency measurements. Silva and Gomes [
20] proposed a technique using the frequency shift coefficient (FSC) to detect the crack size and position. Brincker [
21] used a statistical analysis indicator to detect damage by changes in the measured natural frequencies. Kim et al. [
22] developed algorithms to locate and quantify the damage through changes in the natural frequency. They addressed the damage sizing algorithms to quantify the size of the damage from a natural frequency perturbation. Armon et al. [
23] introduced the rank ordering of natural frequency shifts for localization of damage. The development of a damage detection method by Zhang et al. [
24] based on the frequency shift curve caused by auxiliary mass with both the natural frequency and mode shape information for cylindrical shell structures. Gillich et al. [
25] performed crack identification based on natural frequency change. They established a mathematical model and signal processing algorithm, which can predict frequency changes for any boundary conditions with the identification of cracks on multi-span beams. Shukla and Harsha [
26] presented a view that the change in natural frequency is an indication of cracks in the blade geometry. Keye [
27] investigated the advantages of Finite Element (FE) model updating in association with a model-based method for structural damage localization. Gautier et al. [
28] used a 4SID technique in combination with FE model updating procedure including an iterative domain partitioning procedure to localize damages. Dahak and Benseddiq [
29] presented a normalized natural frequencies based method for a specific damage position in order to locate the damage in the cantilever beam. Therefore, the use of the unchanged frequency also gives us more accuracy when the damage is symmetric to the mode shape node. However, this method is independent of the beam dimension, material propriety or the severity of the damage.
Khiem and Toan [
30] investigated natural frequency changes from the Rayleigh quotient that are derived for a clamped free beam with an arbitrary number of cracks. The authors compared natural frequencies calculated using the Rayleigh quotient and measured through an experiment, and they showed that the Rayleigh formula is a simple and consistent tool for modal analysis of cracked structures. Moreover, a crack detection procedure based on natural frequencies was introduced using Rayleigh quotient parameters. Le et al. [
31] presented a method for the localization and quantification of simultaneous structural modifications based on the dynamic analysis in Euler Bernoulli beams with or without axial force. This method employs first-order estimation of frequency relative variation, which is derived from the continuous formulation. With this method, the damage position was identified, and then the damage was quantified by the relative variations of axial force, density and bending stiffness with nonlinear coefficients depending on the location of density and bending stiffness modifications. Khatir et al. [
32] presented an approach for damage identification based on model reduction where an optimization algorithm is used to minimize the normalized difference between a frequency vector of the tested structure and its numerical model. Yam et al. [
33] investigated the occurrence of damage in plate-like structures using sensitivities of static and dynamic parameters. The authors suggested two damage indices for damage identification based on the curvature mode shape and the strain frequency response function.
Serra et al. [
34] proposed a strategy to detect and localize damage using various classical indicators by testing different damage cases. Eraky et al. [
35] focused on the damage index method (DIM) as a tool for determining elemental local damage that occurred in beam and plate structures. However, this technique depends on an experiment based on comparing modal strain energies at different degradation stages. More recently, Serra and Lopez [
36] presented a combined modal wavelet strategy. They compared it with the most frequently used indicators and widely studied methods in order to identify the damages. The performance of each method is evaluated and the capacity to detect and localize damage are tested through different cases. Hu et al. [
37] used a statistical based damage-sensitive indicator for the health monitoring of a wind turbine system by considering environmental and operational influences on the structural dynamic properties. Karbhari and Lee [
38] used a dynamic structural analysis to detect damage by applying a cosine based indicator and a model assurance criterion for an eight degrees of freedom structure in order to perform effectively in identifying damages to the structure.
Finally, several studies on damage identification are based on the use of the Frequency Response Function (FRF) and Particle Swarm Optimization (PSO). Porcu et al. [
39] proposed an FRF-curvature based technique (FRF-curvature damage indicator) for damage identification in structural components and tested it both experimentally and numerically. The authors show that the FRF curvature method is more effective, compared to other methods (e.g., natural frequencies, mode shapes or mode-shape curvatures). Furukawa and Kiyono [
40] introduced a technique for the detection of damage in structures that use FRF data as generated from the harmonic excitation force. The method is based on the fact that structural damage usually causes a decrease in structural stiffness and an increase in structural damping, thereby producing changes in vibration characteristics. Mohan et al. [
41] used FRFs with the help of the PSO technique for damage detection and quantification. The robustness and efficiency of this method are acknowledged after comparing the results between two methods: Genetic Algorithm (GA) and PSO. Khatir et al. [
32] presented an inverse problem with an optimization algorithm for minimizing the cost function for damage detection and localization. They implemented FEM with PSO and GA to perform the inverse computations. Huang et al. [
42] proposed an optimization approach, known as bare bones PSO with a double jump, in order to come up with a solution for the damage identification. The authors implemented a
regularization function for detecting damage cases especially in a noisy environment. Li et al. [
43] used the standard PSO-FEM to compare the performance of fitness functions using natural frequencies. Later, the authors proposed an algorithm based on multi-component PSO with a cooperative leader learning mechanism for structural damage detection and further compared with other recent optimization algorithms [
44]. Alamdari et al. [
45] implemented FRFs in a damaged structure and a damage sensitive shape was generated by taking the derivatives of operational mode shapes with the anti-symmetric extension and shape signals that are normalized at different natural frequencies. Moreover, these studies focused on frequency-based damage detection strategies.
From this literature review, it is found that several VBSHM techniques using natural frequencies have been considered for structural damage detection. A few techniques (wavelet transform, artificial neural network, etc.) have shown reliable results with the consideration of measurement errors or uncertainties. As we know, uncertainty or noise is always present on natural frequencies and other modal parameters that can lead to inadequate structural damage detection. An FSC-based algorithm is introduced, and different cases were investigated with or without consideration of uncertainty or measurement errors on natural frequencies. The algorithm was employed by minimizing FSC using PSO, where damages are localized and quantified by updating the FE model from the FSC algorithm that is based on natural frequency shifts. The damage identification technique was performed based on bending stiffness reduction using the FE models. For that, 2D FE models were developed for the healthy and damaged beams, and numerical damage cases were built artificially to test the proposed algorithm. The difference between healthy and damaged models were weighted depending on the shift of natural frequencies. It means the damage localization and quantification can be accurate based on the sensitivity of frequencies shift to the damage states. The paper is intended to further investigate the efficiency of the FSC based method by evaluating the identification capacity in uncertain damaged cases. The FSC-based method is demonstrated by testing a real beam that has been double damaged using an experimental test.
The paper is organized as follows:
Section 2 illustrates the modeling of the beams and bending vibration theory;
Section 3 presents the proposed damage identification strategy using FSC minimizing algorithm;
Section 4 shows different numerical examples in order to verify the effectiveness of the method and discussion about the influencing factors, i.e., influence of the damage positions and severity. The artificially damaged cases are investigated in order to localize and estimate the severity along the cantilever beam. The effect of the modeling uncertainty on natural frequencies is considered and the test cases are examined by considering different noise levels;
Section 5 shows a simple laboratory experiment for the vibration measurements in order to find positions and severities of damage in a real beam structure.
2. Beam Vibration Theory
It is assumed that the simplest damage detection problem can be explained by testing beams using a linear equation of motion with undamped free vibration. The equation of motion for free vibration analysis of an Euler–Bernoulli beam is given by:
where
is the transverse deflection of the beam base axis,
is the mass per unit length of the beam,
is the bending stiffness of the beam. Here, a harmonic time dependency is assumed, and the cantilever beam is taken into consideration that is clamped in
x = 0 and free in
; then the solution would satisfy
= 0 and
= 0. To calculate the normal modes we have to consider the linear homogeneous equation related to Equation (
1). Then, the differential equation of eigenvalue problem is written as:
where
and
are the associated natural frequencies and the normal modes. In order to determine the Rayleigh quotient using normal mode shapes
of the undamped problem the normal modes
are considered as the functions, represented by
, which is the square integral on
(i.e.,
and
denotes for a square-integrable function) as well as
and
. Multiplying Equation (
2) by any function
with
and taking the partial integration, we obtain:
where
is a vanishing term representing the boundary conditions. The quantity of
is equal to zero for any function
by verifying the same boundary condition as the modes. Similarly, natural frequency (Hz) calculation for the beam is given by:
2.1. 2D Finite Element Models of Beam
The studied model is considered to be an Euler–Bernoulli cantilever beam with a uniform cross-section area with 2D healthy and damaged FE beam models. The 2D FE beam models are discretized in
N elements and
nodes.
Figure 1 shows a 2D FE damaged model of a cantilever beam and its cross-section area. Each node of the FE models has two degrees of freedom, a vertical translation
V and a bending rotation
.
Natural frequencies and mode shapes may be obtained by solving an eigenvalue problem from the FE model as described by the following equation:
where
is the
mass matrix of the system, and
is the
stiffness matrix of the system, where
are natural frequencies and
are modal shapes.
Here, the damage is assumed at position within node i to of the beam. If a defect is introduced in a beam structure, it reduces the stiffness of the beam structure at a particular element.
2.2. Numerical Modeling of Damage
In the 2D FE model, damage severity is represented by an elemental stiffness reduction coefficient
, which is the ratio of the stiffness reduction to the base stiffness. The stiffness matrix of a numerical damaged FE model is defined as the sum of elemental matrices multiplied by the reduction coefficient:
where
is the global stiffness matrix for a damaged beam,
is the elemental stiffness matrix,
N is the number of elements and
is a reduction coefficient, which varies from 0 to 1 for the damaged structure. A value of
= 0 indicates a healthy element.