# A Novel Low-Velocity Impact Region Identification Method for Cantilever Beams Using a Support Vector Machine

Authors: Fengde Wang; Yongtian Kang; Wensheng Xiao; Changjiang Li; Qi Liu
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Damage induced by a low-velocity impact can reduce the stability and reliability of structures. In this study, a novel low-velocity impact region identification method based on the spectral peak frequency (SPF) and support vector machine (SVM) is proposed to identify the low-velocity impact regions on a steel cantilever beam. A low-velocity impact region identification system of the cantilever beam is established by applying fiber Bragg grating (FBG) sensors, and only 2 sensors are used in this system. The power spectral density functions of the impact response signal are smoothed using the linear weighting method to remove pseudospectral peak frequencies, and then, SPFs are extracted as the features. For 25 low-velocity impact regions with dimensions of 30 mm × 10 mm, the results show that the recognition rate obtained by the proposed method is 100% and the feature vector consisting of the first two SPFs with the largest amplitude has the highest recognition rate. Through the comparative study, it is found that the recognition rate of SVM is higher than that of the probabilistic neural network (PNN) and extreme learning machine (ELM) for low-velocity impact area recognition of cantilever beams. As a result, the low-velocity impact region identification method of this paper can be applied to the real-time health monitoring of cantilever beam structures.


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## Body

## 1. Introduction

A cantilever beam is the basic component of complex structural systems, and damages induced by the low-velocity impact will weaken the strength of the structure and shorten its working life. To ensure the reliability and safety of structures, structural health monitoring technology has been developed to monitor the working condition of structures in real time. The structural health monitoring system mainly consists of sensors and data analysis systems, which can identify the damage area and quantify damage. For cantilever beam structural health monitoring, modal parameters and their spatial derivatives, structural parameters, and vibration response signals are frequently used to achieve the damage detection, localization, and quantitative analysis. Vibration mode changes are more correlated with structural damage; however, since high-order modes are difficult to measure accurately, Vafaei and Alih [1] proposed a damage identification method based on the artificial neural network and the first-order mode. To overcome the disadvantage of mode shapes being more difficult to be measured, Oliveira Filho et al. [2] performed some work to study the optimization of the sensor installation position on beam structures. Ghannadi and Kourehli [3] proposed a modal test analysis model to estimate unmeasured mode shapes for damage assessment of the cantilever beam. For noisy signals, a stationary wavelet transform was applied to separate the feature and noise, and the differences between modes were determined by continuous wavelet transform [4]. To eliminate irregularities and noises of the beam damage detection process, Masoumi and Ashory [5] proposed a uniform load surface (ULS) based on the flexibility matrix and improved the obtained ULS by stationary wavelet transform. Derived from the fundamental mode shape, a curvature mode shape has been widely used in damage identification of beams, to overcome its susceptibility to measurement noise, Cao et al. [6] developed a synergy method using a wavelet transform and a Teager energy operator, and a mode shape curvature estimation method based on the damaged structure data has been developed to locate damage [7]. Since the natural frequency is much easier to measure than the mode and curvature, it has been used by some scholars to investigate the crack detection of cantilever beams [8–10]. More scholars applied the mode shape and natural frequency together for damage identification and location [11–15]. Flexibility is another key parameter for structure damage identification, the comparative study of vibration and wave propagation approaches based on the flexibility matrix of cantilever beams has been implemented [16], and a damage detection method based on a modal flexibility matrix has been proposed for the cantilever beam structures [17]. By combining natural frequency and flexibility, Dahak et al. [18] developed an algorithm to identify the damage regions in the cantilever beam. According to the experimental data of damaged and undamaged beams, Khatir et al. [19] formed an objective function with the changes of natural frequencies and proposed a damage detection method based on the particle swarm optimization algorithm, and their method can also quantify damage. Hu and Zhang [20] calculated the stiffness reduction of the damage beam using the improved beetle antennae search algorithm and constructed a damage index that can track the curvature changes of modes.Due to the basic idea that the damage-induced changes of structures will cause detectable signals of vibration responses, some scholars have developed several damage detection methods using the measured signal [21–25]. Zai et al. [26] investigated the changes in frequency, damping, and vibration response amplitude under thermomechanical loads and proposed a damage quantification method for cantilever beams. With the development of intelligent algorithms and signal processing technologies, bispectral analysis [27], neural network [27, 28], spatial wavelet analysis [29], full wavelet scalogram [28], metaheuristic algorithm [30], Bayesian parameter estimation [31], and deep learning [32] have shown significant promise in the damage detection of cantilever beams.Impacts occurring at velocities 1 to 10 m/s are treated as a low-velocity impact (LVI) by Cantwell and Morton [33], and LVI often creates damage with dent depths lower than barely visible impact damage, which attracted some scholars to investigate it. Jang et al. [34] proposed a real-time impact identification algorithm for the composite flat plate under the low-velocity impact load, and Qi et al. [35] investigated the effects of impact velocity on the performance of an expansion tube by the experiment method and numerical simulation. In view of the application of the support vector machine in pattern recognition problems [36–38], we put our focus on the low-velocity impact region identification method of cantilever beams. In this study, two fiber Bragg grating (FBG) sensors are installed on the cantilever beam to collect impact response signals and the power spectral density (PSD) is obtained using the Fourier transform method. Based on the spectral peak frequencies (SPFs) of PSD and SVM, we proposed a low-velocity impact region identification method. The identification rate of this method is 100%, which can be applied to real-time health monitoring of cantilever beams.The paper is structured as follows: Section2 illustrates the relative theories of the research. The details of a low-velocity impact identification system of cantilever beams are presented in Section 3. In Section 4, the proposed algorithm is introduced briefly. Then, the method is used to identify the impact regions, and the results are discussed in Section 5. Finally, Section 6 gives the conclusions.


## 2. Relative Theories

### 2.1. Working Principles of an FBG Sensor

An FBG sensor satisfies the Bragg condition as follows:(1)λ=2nneffΛ,where λ is the Bragg wavelength, nneff is the effective refractive index of fiber grating, and Λ is the Bragg grating spacing.When the measured physical quantities (e.g., temperature and strain) acting on the fiber grating change, the corresponding changes ofnneff and Λ will result in the drift of λ, and it changes linearly. Therefore, the information of the measured physical quantity can be obtained by measuring the drift of λ. In the laboratory environment, the temperature does not change much and can be regarded as a constant value. Then, the variation of λ is generated only by the change of strain. Thus, the shift of λ can be formulated as follows:(2)Δλε=λε1−Pε,where Pε is the effective elasto-optical coefficient, and ε is strain.When an FBG sensor is used for structural health monitoring of the cantilever beam, the strain response pattern of the beam changes as the impact region changes, which subsequently causes the change ofλ.


### 2.2. Support Vector Multiple Classification Algorithm

SVM is a structural risk minimization method developed based on statistical learning theory, which can solve classification and pattern recognition problems. Since the size of the sample has little significant effect on the performance of SVM, the support vector machine also has high efficiency and accuracy when processing large sample data.The sample data and categories are labeled as {pi, qi}, i = 1, 2, …, s, qi∈{−1, 1}, where s is the number of the sample data. For soft margin classification, the original problem of SVM is determined as follows:(3)min12ω2+c∑i=1sεis.t.qiωp+b∗≥1−εiεi≥0.In (3), ω is the weight vector, b∗ is the bias vector, εi is the slack factor, and c is the penalty parameter. Then, the classification problems can be expressed as the dual optimization problem:(4)min12∑i=1j∑j=1sqiqjαiαjKpi,pj−∑i=1sαis.t.∑i=1sαiqi=00≤αi≤c.In (4), αi is the Lagrange multiplier and K (pi, pj) is a kernel function. The main forms of the kernel function are linear function, polynomial function, Gaussian function, and radial basis function (RBF). Since RBF has faster learning speed and stronger classification ability, we choose it as the kernel function in this study, and the expression of RBF is presented in (5), where ξ is the kernel parameter.(5)Kpi,pj=exp−ξpi−pj2.According to the sample data, the classification results can be obtained by the following decision function:(6)fp=sgn∑i=1sαi∗qiKpi,pj+b∗.


## 2.1. Working Principles of an FBG Sensor

An FBG sensor satisfies the Bragg condition as follows:(1)λ=2nneffΛ,where λ is the Bragg wavelength, nneff is the effective refractive index of fiber grating, and Λ is the Bragg grating spacing.When the measured physical quantities (e.g., temperature and strain) acting on the fiber grating change, the corresponding changes ofnneff and Λ will result in the drift of λ, and it changes linearly. Therefore, the information of the measured physical quantity can be obtained by measuring the drift of λ. In the laboratory environment, the temperature does not change much and can be regarded as a constant value. Then, the variation of λ is generated only by the change of strain. Thus, the shift of λ can be formulated as follows:(2)Δλε=λε1−Pε,where Pε is the effective elasto-optical coefficient, and ε is strain.When an FBG sensor is used for structural health monitoring of the cantilever beam, the strain response pattern of the beam changes as the impact region changes, which subsequently causes the change ofλ.


## 2.2. Support Vector Multiple Classification Algorithm

SVM is a structural risk minimization method developed based on statistical learning theory, which can solve classification and pattern recognition problems. Since the size of the sample has little significant effect on the performance of SVM, the support vector machine also has high efficiency and accuracy when processing large sample data.The sample data and categories are labeled as {pi, qi}, i = 1, 2, …, s, qi∈{−1, 1}, where s is the number of the sample data. For soft margin classification, the original problem of SVM is determined as follows:(3)min12ω2+c∑i=1sεis.t.qiωp+b∗≥1−εiεi≥0.In (3), ω is the weight vector, b∗ is the bias vector, εi is the slack factor, and c is the penalty parameter. Then, the classification problems can be expressed as the dual optimization problem:(4)min12∑i=1j∑j=1sqiqjαiαjKpi,pj−∑i=1sαis.t.∑i=1sαiqi=00≤αi≤c.In (4), αi is the Lagrange multiplier and K (pi, pj) is a kernel function. The main forms of the kernel function are linear function, polynomial function, Gaussian function, and radial basis function (RBF). Since RBF has faster learning speed and stronger classification ability, we choose it as the kernel function in this study, and the expression of RBF is presented in (5), where ξ is the kernel parameter.(5)Kpi,pj=exp−ξpi−pj2.According to the sample data, the classification results can be obtained by the following decision function:(6)fp=sgn∑i=1sαi∗qiKpi,pj+b∗.


## 3. Low-Velocity Impact Region Identification System

The low-velocity impact region identification system consists of a fiber grating sensing interrogator (Micron Optics si155 of Micron Optics Inc), a steel cantilever beam, two FBG sensors, and a computer, and Figure1 illustrates the experimental site. With a wavelength accuracy of 2pm/3pm, Micron Optics si155 has two parallel channels, each with a bandwidth of 80 nm, which enables simultaneous full-spectrum scanning at 5 kHz. si155 is equipped with a high-power, low-noise, ultralarge bandwidth scanning laser source to ensure wavelength accuracy of scanning. It enables fast full-spectrum data acquisition and low latency access to data in closed-loop feedback applications, and it is suitable for Bragg gratings, long-period gratings, and Fabry–Perot optical fiber sensors. si155 is equipped with special demodulation software ENLIGHT. Its functions include data acquisition, calculation, and analysis of fiber grating sensor networks, and the software can convert optical wavelength parameters into strain, temperature, acceleration, pressure, etc.Figure 1
A site diagram of the low-velocity impact test of the steel cantilever beam.Figure2 shows a schematic diagram of the low-velocity impact localization system. The dimensions of the steel cantilever beam are 750 mm × 10 mm × 2 mm, and it is evenly divided into 25 small regions of 30 mm × 10 mm along the axial direction in the experiment. The grating lengths of the two FBG sensors are 10 mm, and the Bragg wavelength values are 1549.859 nm and 1549.958 nm, respectively. FBG sensors are mounted on each end of the beam at coordinates (0, 105 mm) and (0, 645 mm). A steel ball weighing 36 g was used as the impact device, and in the experiment, it hit the beam vertically in a free-fall manner.Figure 2
A schematic diagram of the low-velocity impact localization system.


## 4. Low-Velocity Impact Region Identification Method

### 4.1. Feature Extraction

To identify the impact regions of the cantilever beam, the SPFs of the PSD of the impact response signal are extracted as features. The impact response signals collected by two FBG sensors are shown in Figure3 Then, the PSD of the impact response signals are obtained by the Fourier transform, as shown in Figures 4 and 5. Obviously, there are multiple pseudospectral frequencies near spectral peak frequencies. To extract the main features and ensure the validity of the features, PSD needs to be processed to remove the pseudospectral peak frequencies. In this research, we chose the smoothing process method to filter the pseudospectral peak frequencies. Through the comparative study, it is found that the weighted linear fitting method has a better processing effect on PSD.Figure 3
Original signals of FBG sensors in region 13: (a) FBG 1 signal and (b) FBG 2 signal.
(a)(b)Figure 4
Power spectral density of the FBG 1 signal in region 13.Figure 5
Power spectral density of the FBG 2 signal in region 13.For the weighted linear fitting method, the window length has a significant effect on data processing results, which indicates that the processing results are severely distorted (Figure6) when the window length is less than 1. Therefore, the window length should be greater than 1 when weighted linear fitting is used to process impact response signals. The window length is set to 15 after executing extensive tests, and the power spectral density of impact response signals after the smoothing process are presented in Figures 4 and 5.Figure 6
Power spectral density of the FBG 1 signal in region 13 at different window lengths.The impact signals collected by two FBG sensors in each region were randomly selected to study the relationship between the characteristics of the signal and the impact regions. PSD of signals in region 2, region 5, region 8, region 11, region 14, region 17, region 20, and region 23 were selected to illustrate our method, as shown in Figures7–9. PSD shows that the main frequencies vary with the impact region, and the FBG 2 sensor collects low-order spectral peak frequencies of the signal.Figure 7
Amplitude-frequency characteristics of the FBG 1 sensor signal.Figure 8
Amplitude-frequency characteristics of the FBG 2 sensor signal.Figure 9
Amplitude-frequency characteristics impact signal in region 23. (a) Power spectrum of the FBG 1 signal and (b) power spectrum of the FBG 2 signal.
(a)(b)As shown in Figure8, the main frequency of the impact response signal collected by using the FBG 2 sensor is 5.49 Hz in region 2, region 5, region 8, region 11, and region 14, and 34.79 Hz is the main frequency in region 17 and region 20. Therefore, according to spectral peak frequency characteristics of impact response signals, the impact range can be roughly determined: when the main frequency is 5.49 Hz, the impact region is close to the free end; when the main frequency is 34.79 Hz, the impact region is located in the middle of the cantilever beam; when the main frequency is 881.4 Hz and other high spectral peak frequencies, the impact region is distributed near the fixed end of the cantilever beam, and Figure 9 illustrates the relevant results.Comparing Figures7–9, it is found that more spectral peak frequencies of the impact response signal are excited in the region close to the fixed end, which indicates that it is easier to achieve high-precision location of impact regions. Based on the above research results, the first three spectral peak frequencies with the highest power amplitude of the FBG 1 signal and the first two spectral peak frequencies with the highest power amplitude of the FBG 2 signal were selected as features to identify the impact regions.


### 4.2. Identification Method Based on SVM

Aiming at identifying the low-velocity impact regions on the cantilever beam,m regions are divided on the cantilever beam as the impact regions to be identified, and an SVM model is established by treating each region as one category. The m categories are labeled as R = {1, 2, …, m}, and then, the low-velocity impact region identification problem is transformed into a classification problem with m categories. The sample data are defined as {(xi, yi), i = 1, 2, …, m} and yi∈R. xi represents the signal characteristics monitored by n sensors, xi = {Fi1, Fi2, …, Fin}, and the signal monitored using a single FBG sensor is characterized by Fin = {f1, f2, …, fk}. In this study, we take the PSD of the strain response signal as the characteristic and fk is the amplitude of PSD. The features of the impact response signals from different regions are used as input to the SVM model, and the impact regions are output. Then, the training and testing processes of the SVM model are constructed. In [39], we can note that the computational complexity of SVM in the training process is O (k·m2) and the computational complexity of SVM in the testing process is O (k·Ns), where Ns is the number of the support vectors.


### 4.3. Low-Velocity Impact Region Identification Procedure

Based on SPF and SVM, the low-velocity impact region identification method of the cantilever beam is established in this study, and our method mainly has three steps: (a) signal preprocessing; (b) feature extraction; (c) impact region identification. Figure10 illustrates the flowchart of our method.Figure 10
The flowchart of the low-velocity impact region identification procedure for the cantilever beam.The details of the low-velocity impact region identification method are given as follows:Step 1: The impact response signal obtained from the fiber grating sensing system is processed by applying Fourier transformation to get the PSD of the signal. Then, PSD is smoothed by the weighted linear fitting method.Step 2: The SPFs of smoothed PSD are selected as features. Each region is labeled by four features which are composed of the first two peak frequencies with the maximum amplitude of the smoothed PSD of the FBG 1 signal and the first two peak frequencies with the maximum amplitude of the smoothed PSD of the FBG 2 signal. Then, the feature matrixFM can be expressed as follows:(7)FM=FM1FM2⋮FMNum=SPF1×21SPF1×22SPF2×21SPF2×22⋮⋮SPFNum×21SPFNum×22.Step 3: Low-velocity impact tests were carried out in 25 regions on the cantilever beam, respectively, and one group of feature samples obtained in the experiment was randomly selected as test samples, while the other samples were used as training samples. We take the SPFs of signals as the input to the SVM, and then, impact regions will be generated as the output of the SVM. The steel ball was used to impact 25 regions of the cantilever beam respectively, each area was impacted 3 times, and 3 groups of impact signals were obtained from each region. The SVM model was trained with the training sample set, and then, the recognition rate of the low-velocity impact region of the trained model was verified with the test sample set.


## 4.1. Feature Extraction

To identify the impact regions of the cantilever beam, the SPFs of the PSD of the impact response signal are extracted as features. The impact response signals collected by two FBG sensors are shown in Figure3 Then, the PSD of the impact response signals are obtained by the Fourier transform, as shown in Figures 4 and 5. Obviously, there are multiple pseudospectral frequencies near spectral peak frequencies. To extract the main features and ensure the validity of the features, PSD needs to be processed to remove the pseudospectral peak frequencies. In this research, we chose the smoothing process method to filter the pseudospectral peak frequencies. Through the comparative study, it is found that the weighted linear fitting method has a better processing effect on PSD.Figure 3
Original signals of FBG sensors in region 13: (a) FBG 1 signal and (b) FBG 2 signal.
(a)(b)Figure 4
Power spectral density of the FBG 1 signal in region 13.Figure 5
Power spectral density of the FBG 2 signal in region 13.For the weighted linear fitting method, the window length has a significant effect on data processing results, which indicates that the processing results are severely distorted (Figure6) when the window length is less than 1. Therefore, the window length should be greater than 1 when weighted linear fitting is used to process impact response signals. The window length is set to 15 after executing extensive tests, and the power spectral density of impact response signals after the smoothing process are presented in Figures 4 and 5.Figure 6
Power spectral density of the FBG 1 signal in region 13 at different window lengths.The impact signals collected by two FBG sensors in each region were randomly selected to study the relationship between the characteristics of the signal and the impact regions. PSD of signals in region 2, region 5, region 8, region 11, region 14, region 17, region 20, and region 23 were selected to illustrate our method, as shown in Figures7–9. PSD shows that the main frequencies vary with the impact region, and the FBG 2 sensor collects low-order spectral peak frequencies of the signal.Figure 7
Amplitude-frequency characteristics of the FBG 1 sensor signal.Figure 8
Amplitude-frequency characteristics of the FBG 2 sensor signal.Figure 9
Amplitude-frequency characteristics impact signal in region 23. (a) Power spectrum of the FBG 1 signal and (b) power spectrum of the FBG 2 signal.
(a)(b)As shown in Figure8, the main frequency of the impact response signal collected by using the FBG 2 sensor is 5.49 Hz in region 2, region 5, region 8, region 11, and region 14, and 34.79 Hz is the main frequency in region 17 and region 20. Therefore, according to spectral peak frequency characteristics of impact response signals, the impact range can be roughly determined: when the main frequency is 5.49 Hz, the impact region is close to the free end; when the main frequency is 34.79 Hz, the impact region is located in the middle of the cantilever beam; when the main frequency is 881.4 Hz and other high spectral peak frequencies, the impact region is distributed near the fixed end of the cantilever beam, and Figure 9 illustrates the relevant results.Comparing Figures7–9, it is found that more spectral peak frequencies of the impact response signal are excited in the region close to the fixed end, which indicates that it is easier to achieve high-precision location of impact regions. Based on the above research results, the first three spectral peak frequencies with the highest power amplitude of the FBG 1 signal and the first two spectral peak frequencies with the highest power amplitude of the FBG 2 signal were selected as features to identify the impact regions.


## 4.2. Identification Method Based on SVM

Aiming at identifying the low-velocity impact regions on the cantilever beam,m regions are divided on the cantilever beam as the impact regions to be identified, and an SVM model is established by treating each region as one category. The m categories are labeled as R = {1, 2, …, m}, and then, the low-velocity impact region identification problem is transformed into a classification problem with m categories. The sample data are defined as {(xi, yi), i = 1, 2, …, m} and yi∈R. xi represents the signal characteristics monitored by n sensors, xi = {Fi1, Fi2, …, Fin}, and the signal monitored using a single FBG sensor is characterized by Fin = {f1, f2, …, fk}. In this study, we take the PSD of the strain response signal as the characteristic and fk is the amplitude of PSD. The features of the impact response signals from different regions are used as input to the SVM model, and the impact regions are output. Then, the training and testing processes of the SVM model are constructed. In [39], we can note that the computational complexity of SVM in the training process is O (k·m2) and the computational complexity of SVM in the testing process is O (k·Ns), where Ns is the number of the support vectors.


## 4.3. Low-Velocity Impact Region Identification Procedure

Based on SPF and SVM, the low-velocity impact region identification method of the cantilever beam is established in this study, and our method mainly has three steps: (a) signal preprocessing; (b) feature extraction; (c) impact region identification. Figure10 illustrates the flowchart of our method.Figure 10
The flowchart of the low-velocity impact region identification procedure for the cantilever beam.The details of the low-velocity impact region identification method are given as follows:Step 1: The impact response signal obtained from the fiber grating sensing system is processed by applying Fourier transformation to get the PSD of the signal. Then, PSD is smoothed by the weighted linear fitting method.Step 2: The SPFs of smoothed PSD are selected as features. Each region is labeled by four features which are composed of the first two peak frequencies with the maximum amplitude of the smoothed PSD of the FBG 1 signal and the first two peak frequencies with the maximum amplitude of the smoothed PSD of the FBG 2 signal. Then, the feature matrixFM can be expressed as follows:(7)FM=FM1FM2⋮FMNum=SPF1×21SPF1×22SPF2×21SPF2×22⋮⋮SPFNum×21SPFNum×22.Step 3: Low-velocity impact tests were carried out in 25 regions on the cantilever beam, respectively, and one group of feature samples obtained in the experiment was randomly selected as test samples, while the other samples were used as training samples. We take the SPFs of signals as the input to the SVM, and then, impact regions will be generated as the output of the SVM. The steel ball was used to impact 25 regions of the cantilever beam respectively, each area was impacted 3 times, and 3 groups of impact signals were obtained from each region. The SVM model was trained with the training sample set, and then, the recognition rate of the low-velocity impact region of the trained model was verified with the test sample set.


## 5. Results and Discussion

### 5.1. Parameter Setting

In the SVM, the form of the kernel mapping distribution of samples is mainly determined by the kernel parameter, while the penalty factor is mainly used to weigh the fitting ability and prediction ability of the algorithm. When the value of the kernel parameter is smaller and the penalty parameter is larger, the nonlinear fitting ability of the algorithm is stronger, but the training time increases. Smoothness decreases with the decrease of the kernel parameter, and generalization ability decreases with the increase of the penalty parameter. Therefore, it is necessary to determine the optimal combination of the kernel parameter and the penalty factor because the two parameters significantly affect the identification accuracy of SVM. The cross-validation method is an unbiased estimation of generalization errors, and it can effectively prevent overlearning phenomenon. It has both certain training accuracy and good generalization performance. Researchers have conducted many experiments using different learning techniques and a large data set, and the results showed that 10-fold is an appropriate choice to obtain the best error estimate [40–42]. In this study, based on the 10-fold cross-validation strategy, the grid search method was used to optimize the penalty factor, and the value of the penalty factor is determined to be 2.


### 5.2. Effect of the Number of FBG Sensors and the Number of Features

The number of FBG sensors and features was investigated in this research. First, only the SPFs of the FBG 1 sensor signal were used for identification, and the identification rate was 44% when we only used the SPF with the maximum amplitude; when the first two SPFs with the largest amplitude were used, the identification rate achieved 84%; the first three SPFs with the largest amplitude can achieve the 56% identification rate. The results show that it is reasonable to use the first two SPFs with the largest signal amplitude of the FBG 1 sensor for identification. According to the results of feature extraction mentioned above and combined with FBG 2 sensor signals, the identification rates under different numbers of features are studied, respectively. The research results are shown in Figure11. The results show that the feature vectors constructed by using the first two SPFs with the largest amplitude of the FBG 1 signal and the first two SPFs with the largest amplitude of the FBG 2 signal can achieve the 100% identification rate. Therefore, the number of features corresponding to each region is 4.Figure 11
Recognition rate under different numbers of features.


### 5.3. Comparison of SVM and Other Algorithms

In order to verify the performance of SVM in the identification of low-velocity impact regions of cantilever beams, we used the SVM, least squares support vector machines (LS-SVM), probabilistic neural networks (PNN), and extreme learning machines (ELM) to identify impact regions, respectively. Figure12 shows the recognition results of the above algorithms.Figure 12
Recognition results of different algorithms.As shown in Figure12, the identification rate of PNN is 92% and the identification rate of ELM is 84%, while both SVM and LS-SVM achieve the 100% identification rate. According to the identification results of PNN and ELM, it can be concluded that the impact regions in the middle of the cantilever beam are more difficult to be identified. Based on the research results, it is speculated that the recognition accuracy of the impact regions can be further improved by adding sensors in the middle of the cantilever beam. The identification results in Figure 12 show that those areas near the fixed end of the cantilever beam are easier to identify, verifying the preliminary judgment in Section 4.1. For engineering applications, the calculation time of the algorithm should also be considered, and for 25 regions in this study, the average calculation time of SVM is 0.03987 s (30 times) and the average calculation time of LS-SVM is 0.28454 s (30 times). Therefore, we choose SVM to fulfill impact region identification. The results in this study are obtained based on the experimental data, and the quality of sample signals is relatively high. In practical application, many uncertain working conditions will be encountered, such as the sudden change in signal amplitude and the change in the signal type [43, 44]. The problems that may occur in the real-time application of the proposed algorithm are multisource impact and high-speed impact. Multisource impact can still be assumed as a single point impact in essence, and high-speed impact does not belong to the research scope of this paper.


## 5.1. Parameter Setting

In the SVM, the form of the kernel mapping distribution of samples is mainly determined by the kernel parameter, while the penalty factor is mainly used to weigh the fitting ability and prediction ability of the algorithm. When the value of the kernel parameter is smaller and the penalty parameter is larger, the nonlinear fitting ability of the algorithm is stronger, but the training time increases. Smoothness decreases with the decrease of the kernel parameter, and generalization ability decreases with the increase of the penalty parameter. Therefore, it is necessary to determine the optimal combination of the kernel parameter and the penalty factor because the two parameters significantly affect the identification accuracy of SVM. The cross-validation method is an unbiased estimation of generalization errors, and it can effectively prevent overlearning phenomenon. It has both certain training accuracy and good generalization performance. Researchers have conducted many experiments using different learning techniques and a large data set, and the results showed that 10-fold is an appropriate choice to obtain the best error estimate [40–42]. In this study, based on the 10-fold cross-validation strategy, the grid search method was used to optimize the penalty factor, and the value of the penalty factor is determined to be 2.


## 5.2. Effect of the Number of FBG Sensors and the Number of Features

The number of FBG sensors and features was investigated in this research. First, only the SPFs of the FBG 1 sensor signal were used for identification, and the identification rate was 44% when we only used the SPF with the maximum amplitude; when the first two SPFs with the largest amplitude were used, the identification rate achieved 84%; the first three SPFs with the largest amplitude can achieve the 56% identification rate. The results show that it is reasonable to use the first two SPFs with the largest signal amplitude of the FBG 1 sensor for identification. According to the results of feature extraction mentioned above and combined with FBG 2 sensor signals, the identification rates under different numbers of features are studied, respectively. The research results are shown in Figure11. The results show that the feature vectors constructed by using the first two SPFs with the largest amplitude of the FBG 1 signal and the first two SPFs with the largest amplitude of the FBG 2 signal can achieve the 100% identification rate. Therefore, the number of features corresponding to each region is 4.Figure 11
Recognition rate under different numbers of features.


## 5.3. Comparison of SVM and Other Algorithms

In order to verify the performance of SVM in the identification of low-velocity impact regions of cantilever beams, we used the SVM, least squares support vector machines (LS-SVM), probabilistic neural networks (PNN), and extreme learning machines (ELM) to identify impact regions, respectively. Figure12 shows the recognition results of the above algorithms.Figure 12
Recognition results of different algorithms.As shown in Figure12, the identification rate of PNN is 92% and the identification rate of ELM is 84%, while both SVM and LS-SVM achieve the 100% identification rate. According to the identification results of PNN and ELM, it can be concluded that the impact regions in the middle of the cantilever beam are more difficult to be identified. Based on the research results, it is speculated that the recognition accuracy of the impact regions can be further improved by adding sensors in the middle of the cantilever beam. The identification results in Figure 12 show that those areas near the fixed end of the cantilever beam are easier to identify, verifying the preliminary judgment in Section 4.1. For engineering applications, the calculation time of the algorithm should also be considered, and for 25 regions in this study, the average calculation time of SVM is 0.03987 s (30 times) and the average calculation time of LS-SVM is 0.28454 s (30 times). Therefore, we choose SVM to fulfill impact region identification. The results in this study are obtained based on the experimental data, and the quality of sample signals is relatively high. In practical application, many uncertain working conditions will be encountered, such as the sudden change in signal amplitude and the change in the signal type [43, 44]. The problems that may occur in the real-time application of the proposed algorithm are multisource impact and high-speed impact. Multisource impact can still be assumed as a single point impact in essence, and high-speed impact does not belong to the research scope of this paper.


## 6. Conclusions

A novel method to identify low-velocity impact regions of cantilever beams is proposed based on the spectral peak frequencies of PSD and SVM, and a low-velocity impact region identification system of cantilever beams based on an FBG sensor was established. The results show that the proposed method can accurately identify 25 regions of 30 mm × 10 mm on the cantilever beam with only two FBG sensors, and the recognition rate of this system is 100%. Therefore, it is feasible to use an FBG sensor to achieve low-velocity impact region identification of steel cantilever beams based on SVM with the spectral peak frequency feature. The proposed method in this paper can be used for real-time health monitoring of cantilever beam structures; however, the identification accuracy of the proposed method is not high enough, and it is more suitable for impact identification of large-scale structures. Aiming at the defect of insufficient precision, the optimization of sensor quantity and installation position should be further carried out. This study was conducted under laboratory conditions, where the experimental environment was more deterministic, making the signal characteristics more stable. However, in actual working conditions, both randomness and noise of the signal are enhanced, which makes it more difficult to identify the features of the signal. Therefore, in order to make our method applied and promoted in the field, it is necessary to carry out field experiments and research on impact region identification based on the field experimental data.



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*Source: 2906535-2022-10-07.xml*