Silicone Rubber-Packaged FBG Sensing Information and SSI-COV-Recognized Modal Parameters Motivated Damage Identification in Pipe Structures

Abstract

Pipes are the main structures serving as the lifeline for oil and gas transportation. However, they are prone to cracks, holes and other damages due to harsh working environments, which can lead to leakage incidents and result in significant economic losses. Therefore, the development of structural health monitoring systems with advanced online diagnostic methods is of great importance for identifying local damages and assessing the safety state of pipe structures. These efforts can guide rapid repairs and ensure the continuous, efficient and cost-effective transportation of oil and gas resources. To address this problem, this paper proposes the development of a pipe monitoring system based on quasi-distributed fiber Bragg grating (FBG) sensing technology. The SSI-COV method is employed to process the sensor responses and extract the modal parameters of the structure. Based on this foundation, an enhanced damage identification index is proposed, which mitigates the effects of support and excitation positions on damage identification. The pipe structure can be regarded as a continuous super-statical beam, and based on its structural symmetry, a unit structure, specifically a stainless-steel pipe with fixed ends, is regarded as the experimental subject. Impact experiments have been conducted to analyze its behavior in both undamaged and damaged states. The research indicates that by using the proposed modal parameter identification method and the $ASMDI$ damage index, $ASMDI$ exhibits peak values at damage locations of the pipe structure. This allows for the identification of structural damage with high accuracy, fast processing efficiency and strong robustness. The study provides an effective and reliable damage diagnosis method, which can contribute to the refinement and visualization of pipe structural health monitoring systems.

1. Introduction

Pipes are widely used in industries that involve the transportation of energy [1,2] (oil and gas), livelihood security [3] (tap water), industrial production (hydraulic systems) and aerospace [4]. Pipe structures are susceptible to self-weight stresses, temperature stresses and fluid pressures within the pipe. Additionally, buried pipes are also subjected to earth pressure; deep-sea pipes are impacted by water pressure, seawater movement and vibration in their highly corrosive environment; and pipes serviced in the aerospace industry are subjected to cosmic radiation. All these actions may contribute to the damage and leakage of their transport medium. For example, an oil pipe belonging to China Petrochemical Corporation Company exploded in Qingdao City on 12 November 2013, and 55 Chinese people lost their lives [5]. An Enbridge natural gas pipe ruptured and exploded in Lincoln County, Kentucky, U.S., on 1 August 2019, which led to one person dead and five persons injured, and nearby railroad tracks and several buildings were destroyed [6]. Damage to the miniature, lightweight pipes widely used in hydraulic systems for aerospace engineering can cause systematic damage to the structure and result in significant economic losses [7]. Therefore, it is crucial to develop a real-time health monitoring system for pipe structures and carry out daily maintenance of the structures based on the monitoring data. Thus, pipe damage can be detected in time and quickly repaired to ensure the normal, safe, efficient and low-carbon operation of the global structure system.

Traditional pipe structures are detected periodically by using a magnetic leakage detection method, eddy current detection method, ultrasonic detection method, etc. These methods focus on detecting damaged pipe structures but are unable to track and predict the process in a real-time manner before the damage occurs. Additionally, using these methods, it is much more difficult to achieve long-term continuous monitoring of structural damages with temporal and spatial characteristic evolution, online damage diagnosis and the assessment of the structural safety status. The monitoring system that is integrated using piezoelectric sensors is highly affected by electromagnetic interference, which is not suitable for harsh environments. Moreover, its additional mass changes the modal parameters of the structures, which affects the effect of modal parameter identification [7,8,9]. A fiber Bragg grating (FBG) sensing element has the superior characteristics of light weight, small volume, high-temperature resistance, corrosion resistance, electromagnetic interference resistance, flexible deployment method, easy network integration, etc. Thus, it is suitable for long-term, continuous and high-precision measurements and has the ability to build a long-term structural health monitoring system in harsh environments [10,11]. Numerous scholars have explored FBG sensor-based structural health monitoring. Wang et al. [7] proposed an operational modal analysis (OMA) method based on FBG test information and an enhancement index based on a weighted intrinsic frequency rate of variation to successfully identify cracks in small lightweight pipes, but the applicability to large transport pipelines was not discussed. Gan et al. [12] proposed a pipe leakage localization method based on the peak-to-peak slope and threshold to judge the mutation point based on the micro-strain sensing technique of w-FBG with a localization error around 0.5 m, but selecting appropriate thresholds with high automation was challenging. Kou et al. [13] arranged FBG sensors on prestressed high-strength concrete (PHC) pipe piles and demonstrated the effectiveness of FBG sensors in pile structures. Wang et al. [14] proposed a moving surface spline algorithm capable of accurately reconstructing the strain field of carbon fiber-reinforced plastic (CFRP) plates with a fast response time, which is an important advancement for the construction of a real-time structural health monitoring (SHM) system and visual characterization of damage. Wang et al. [15] proposed a shape configuration algorithm for CFRP pipes based on quasi-distributed optical fiber sensing technology, which accurately identified pipeline buckling and provided suggestions for sensor placement, guiding the construction of structural health monitoring systems for pipelines.

Based on the structural strains sensed by the FBG sensors, the structural modal parameters can be identified. A real-time monitoring system can be set up without the need for additional excitation equipment or the interruption of the structure in normal operation by using the structural modal parameter identification method based on environmental excitation. Modal analysis under environmental excitation is a type of OMA with no excitation information but only response information. Hence, it has the advantages of economy and safety without affecting the normal operation of the structures [16]. Common identification methods of modal parameters based on environmental excitation are stochastic subspace identification (SSI), Peak Picking (PP), frequency domain decomposition (FDD), Hilbert–Huang transform (HHT), etc. [17,18]. Chen et al. [19] used PP, FDD and SSI-DATA to extract the modal parameters of ramp bridges under environmental excitation. The PP method identified the most frequencies, while the SSI method produced the highest-quality vibration shapes. Combining SSI with PP offers quick preliminary results, whereas SSI with FDD enables detailed analysis. Wang et al. [20] combined HHT with convolutional neural networks (CNNs), improving damage identification accuracy by over 10% and enhancing robustness, although this approach was also time consuming. SSI was firstly proposed by Peters [21] in 1995, and it can be categorized into SSI-DATA and SSI-COV. SSI-DATA uses the time-domain response directly to construct the Hankel matrix, whereas SSI-COV uses the time-domain response covariance matrix before constructing the Toeplitz matrix. Although the quality of the identification results is similar for both methods, the SSI-COV calculation is faster because the time series are compressed in the covariance matrix [22]. SSI-COV has been widely developed by many scholars due to its advantages of mature theory, fast computation speed, high recognition accuracy and the ability to recognize dense modes. Cheynet et al. [23] successfully identified the modal parameters of the Lysefjord bridge by using the SSI-COV method. Liu et al. [24] introduced the mathematical features of modal parameter identification in SSI into a deep neural network (DNN) and transformed the identification problem of modal parameters into an optimization problem of the DNN to improve the accuracy of the identification of modal parameters. Fan et al. [25] proposed an automatic structural modal parameter identification method based on SSI and clustering methods, which clusters the modal parameters obtained from SSI to eliminate erroneous modes and accurately identify structural modal parameters. However, some parameters, including the system order, rely on the user’s understanding of structural vibrations and testing conditions, significantly impacting identification quality and computational efficiency. Sun et al. [26] proposed a fast online method for SSI-COV. With the help of using incremental aggregation and the Lanczos algorithm to speed up the computation, the two most important steps in the SSI-COV method are the computation of the covariance matrix and the singular value decomposition (SVD) of the Toeplitz matrix. This results in a near real-time SSI-COV method with a negligible loss of accuracy. It is designed for online use, and further research is needed for offline applications.

Commonly used modal parameters for identifying structural damage are the intrinsic frequency, displacement mode shapes, strain modes, mode shape curvature, etc. [7]. The intrinsic frequency is insensitive to localized and minor injuries, and the same frequency variation can correspond to different injury locations, which makes it impossible to localize the injury [27]. Displacement mode shapes and strain modes essentially correspond to each other, where the displacement can be derived from the strain using differential operations. However, the strain modes are more sensitive to local damage, having a better sensitivity and robustness for damage identification [28].

The pipe structure can transport resources over long distances, requiring supports at regular intervals for stabilization. Therefore, the entire pipe system can be viewed as a continuous (multi-span) super-statical beam. Based on its structural symmetry, a unit structure, specifically a pipe with fixed ends, is explored. Accordingly, this paper proposes a method to identify the damage to pipe structures based on the strain data obtained from FBG sensors. The modal parameters, such as the intrinsic frequency and strain modes of the structure, recognized using the SSI-COV method and a proposed damage indicator based on the variation in the intrinsic frequency and strain mode have also been established. This method accurately identifies damage in pipe structures, providing high accuracy, fast processing efficiency and strong robustness. It offers an effective and reliable damage diagnosis approach for the refinement and visualization of pipeline monitoring systems.

2. Description of the Involved Theory and Principle

To conduct the investigation, the basic concept of the FBG sensing element, the SSI-COV method and the function of the stabilization diagram have been explained.

2.1. Working Principle of FBG Sensors

The optical fiber of the FBG sensor is distributed with grating regions, which have a periodic refractive index distribution produced via ultraviolet light irradiation of the fiber. When a wide band of light propagates in an optical fiber and passes through the grating region, a narrow band of light that satisfies the Bragg condition is reflected with a center wavelength called the Bragg wavelength ${\lambda }_B$ [29], as shown in Figure 1a. The relationship can be expressed as

(1)${\lambda }_B=2n_{eff}(\epsilon ,T)\Lambda (\epsilon ,T)$

where ${\lambda }_B$ is the reflected center wavelength, $n_{eff}$ is the effective refractive index of the fiber core, $\Lambda$ is the grating modulation periods, $\epsilon$ is the axial strain, and $T$ is the temperature. It is observed that the Bragg wavelength ${\lambda }_B$ is directly related to the grating modulation periods $\Lambda$ and the effective refractive index $n_{eff}$, both of which are affected by axial deformation and temperature. Strain and temperature affect $n_{eff}$ and $\Lambda$ of the fiber core, which results in the Bragg wavelength deviating from the original wavelength. The wavelength offset $\Delta {\lambda }_B$ is:

(2)$\Delta {\lambda }_B=2n_{eff}\Lambda +2n_{eff}\Delta \Lambda$

where $\Delta n_{eff}$ is the effective refractive index change of the fiber core; and $\Delta \Lambda$ is the optical fiber modulation period variation. The wavelength offset $\Delta {\lambda }_B$ has an approximately linear relationship with temperature and strain variations over the measurement range, specifically as

(3)${\displaystyle \frac{\Delta {\lambda }_B}{{\lambda }_B}}=k_{\epsilon }\Delta \epsilon +k_T\Delta T$

where $k_{\epsilon }$ is the strain coefficient; $\Delta \epsilon$ is the amount of strain variation; $k_T$ is the temperature coefficient; and $\Delta T$ is the amount of temperature variation. FBG sensors are highly sensitive and accurate to strains of up to 1 με with a response time of less than 0.01 s for a single FBG and strain measurement amplitudes of 10,000 με. In practical testing, FBGs are often packaged with flexible silicone rubber to measure the strain information of rigid structures, as shown in Figure 1b,c.

2.2. SSI-COV Operating Principle

The theory of using stochastic subspace methods to solve for strain modes is described, in detail, in the references [21,30]. A flowchart of SSI-COV is shown in Figure 2. A brief description of the main ideas is presented as follows:

• (1). Establish a model of the stochastic state space with strain representation as:

  (4)$\begin{array}{c}{x}_{k+1}=A_{\epsilon }\cdot x_k+w_k\\ y_k=C_{\epsilon }\cdot x_k+v_k\end{array}$

  where $x_k$ is the vector of discrete time states at moment $k$; $y_k$ is the output vector of the sample; $A_{\epsilon }$ is the discrete state matrix; $C_{\epsilon }$ is the discrete output matrix; and $w_k$ and $v_k$ represent noise due to interference and modeling inaccuracies as well as measurement noise due to the inaccuracy of sensor measurements, respectively.

• (2). Obtain the covariance matrix of the measured structural response time series according to

  (5)$R_i={\displaystyle \frac{1}{N-1}}{\displaystyle \sum _{k=0}^{N-i-1}{y}_{k+1}\cdot y_k^T}$

  where $N$ is the number of points in the time series, and $T$ indicates a transpose.

• (3). Use the resulting covariance matrix to form the Toeplitz matrix as

  (6)$T_{1\left|i\right.}=\left[\begin{array}{cccc}{R}_i& R_{i-1}& \cdots & R_1\\ R_{i+1}& R_i& \cdots & R_2\\ ⋮& ⋮& \cdots & ⋮\\ R_{2i-1}& R_{2i-2}& \cdots & R_i\end{array}\right]$

• (4). Solve the Toeplitz matrix by using singular value decomposition and least squares to identify the state space model to solve the matrices $A_{\epsilon }$ and $C_{\epsilon }$, which, in turn, will be used to extract the modal parameters from it. The pseudocode for the SVD decomposition process of the Toeplitz matrix in Algorithm 1 is attached for reference.

The identification of state space models requires defining the model order, where the choice of order often depends on conservative empirical estimates. Estimating the correct system order is challenging, and an overly high estimation can result in false modes. Therefore, a stabilization diagram method is used in this paper for modal validation with a way of removing spurious modes.

Algorithm 1. Pseudocode for SVD decomposition of Toeplitz matrix. SVD of Toeplitz(h):
Input: h is a three-dimensional matrix representing the cross-correlation function.
Output: U, S and V are the SVD results of H0
1	IF size(h, 1) ≠ size(h, 2) OR ndims(h) ≠ 3 THEN
2	ERROR “the IRF must be a 3D matrix with dimensions <M x M x N>”
3	END IF
4	N1 = ROUND(size(h, 3)/2) − 1
5	M = size(h, 2)
6	INITIALIZE H0 AS EMPTY
7	FOR oo FROM 1 TO N1 DO
8	FOR ll FROM 1 TO N1 DO
9	T1[(oo-1) × M + 1:oo × M, (ll − 1) × M + 1:ll × M] = h(:,:,N1 + oo-ll + 1)
10	END FOR
11	END FOR
12	IF ANY(isinf(T1)) OR ANY(isnan(T1)) THEN
13	WARNING “Input to SVD must not contain NaN or Inf.”
14	U = NaN
15	S = NaN
16	V = NaN
17	RETURN
18	ELSE
19	TRY
20	[U, S, V] = svd(T1)
21	CATCH exception
22	WARNING “SVD of the block-Toeplitz failed”
23	U = NaN
24	S = NaN
25	V = NaN
26	RETURN
27	END TRY
28	END IF
29	END FUNCTION

2.3. The Principle of the Stabilization Diagram

The stabilization diagram is a common method for identifying the system poles, in which horizontal coordinates represent the frequency, and vertical coordinates represent the system order. It accomplishes modal identification by choosing different orders and determines the veracity of the poles by identifying the stability of the system’s natural pole frequency, normalized vibration mode and damping ratio [31]. The stabilization diagram labels the poles corresponding to each order in the same diagram. A mode is recognized as stable when the interpolated values between the parameters of the higher order compared to the parameters of the lower order are within a certain range of tolerance.

3. Presentation of Damage Identification Parameters

In this paper, a damage identification index $SMDI$ proposed by Wang et al. [7] is optimized to achieve a better effect. A strain mode correction factor ${\kappa }_r(i)$ is proposed based on the effect of the support on the strain modes, which, in turn, leads to an augmented damage indicator $ASMDI$. Both of these indicators of damage recognition are described as follows:

3.1. Indicators of Damage in the Literature [7]

The strain modal difference of each unit at the same order of the pipe structure is:

(7)$\Delta {\varphi }_r^{\epsilon }(i)=\left|{\varphi }_{r,h}^{\epsilon }(i)-{\varphi }_{r,d}^{\epsilon }(i)\right|$

where ${\varphi }_{r,h}^{\epsilon }(i)$ is the value of the strain mode of order $r$ for the cell $i$ of the pipe structure in a healthy state; and ${\varphi }_{r,d}^{\epsilon }(i)$ is the value of the strain mode of order $r$ for the cell $i$ of the pipe structure in a damaged state. The intrinsic frequency of the pipe structure decreases when damage occurs. The rate of variation is expressed as

(8)$\delta {\omega }_r={\displaystyle \frac{\left|{\omega }_{r,h}-{\omega }_{r,d}\right|}{{\omega }_{r,h}}}$

where ${\omega }_{r,h}$ is the intrinsic frequency of the pipe structure in the healthy state mode in order $r$; and ${\omega }_{r,d}$ is the intrinsic frequency of the pipe structure in the damaged state mode in order $r$. Therefore, the structural damage indicator $SMDI$, based on the variation of the intrinsic frequency and strain modes of the structure, can be given by

(9)$SMDI(i)={\displaystyle \frac{1}{M}}{\displaystyle \sum _{r=1}^M\delta {\omega }_r\cdot \Delta {\varphi }_r^{\epsilon }(i)}$

where $M$ is the order of the identified structural modal parameters.

3.2. Augmented Structural Damage Indicators

Based on the mechanical theory, it can be seen that the two ends of the solidly supported beam cannot rotate or be displaced due to the existence of the fixed support constraints. Its axial deformation is 0, and the axial strain is also 0. Therefore, its strain variation is also 0 before and after damage. For the slopes of the unit strain modes in the healthy and damaged states, the slopes of the unit strain modes of the damaged units will change greatly when the structure is damaged, while the other healthy units should be in good agreement. However, the unit strain modal slopes of the healthy unit part are not in good agreement, and peaks even appear in some of the measurement points near the support, although they remain unchanged before and after the damage due to the effects of the support and random natural excitation. Additionally, its effects cannot be eliminated by averaging multiple measurements. Thus, this paper proposes a strain modal correction factor ${\kappa }_r(i)$ to reduce the influence of the support on structural damage identification. The derivation process is as follows:

The unit strain slopes for the healthy and damaged states are:

(10)$\begin{array}{c}{k}_{r,h}^{\epsilon }(i,i+1)=\left|{\displaystyle \frac{{\varphi }_{r,h}^{\epsilon }(i)-{\varphi }_{r,h}^{\epsilon }(i+1)}{l_i-l_{i+1}}}\right|\\ k_{r,h}^{\epsilon }(i,i+1)=\left|{\displaystyle \frac{{\varphi }_{r,d}^{\epsilon }(i)-{\varphi }_{r,d}^{\epsilon }(i+1)}{l_i-l_{i+1}}}\right|\end{array},i=1,\cdots ,n-1$

where $k_{r,h}^{\epsilon }(i,i+1)$ and $k_{r,d}^{\epsilon }(i,i+1)$ denote the slopes of the straight lines joining the values of the strain modes of order $r$, which are measured by the $i$-th and $(i+1)$-th FBG sensors in the healthy and damaged states, respectively; $l_i$ denotes the location where the $i$-th sensor is arranged on the pipe structure; and $n$ denotes the number of FBG sensors arranged on the pipe. The strain slope ratio $k_{ratio}$ of the structural unit before and after damage is:

(11)$k_{ratio,r}(i,i+1)=\left|{\displaystyle \frac{MAX(k_{r,h}^{\epsilon }(i,i+1),k_{r,d}^{\epsilon }(i,i+1))}{MIN(k_{r,h}^{\epsilon }(i,i+1),k_{r,d}^{\epsilon }(i,i+1))}}\right|,i=1,\cdots ,n-1$

where $k_{ratio,r}(i,i+1)$ denotes the strain mode slope ratio of cell $(i,i+1)$ before and after damage. The threshold value of the structural unit strain slope is

(12)$k_{\lim it}=\underset{i,r}{\underbrace{1.5\times \mathrm{mode}\left(\mathrm{round}\right(k_{ratio,r}(i,i+1\left)\right))}},\begin{array}{c}i=1,\cdots ,n-1\\ r=1,\cdots ,M\end{array}$

where $round(X)$ indicates that each value in the matrix $X$ is taken as an integer not less than its original value; $\mathrm{mode}(X)$ denotes the plurality of all values taken from the matrix $X$; and $M$ denotes the order of the identified modal parameters.

The value of the strain modal correction factor ${\kappa }_r(i)$ is taken as follows:

When $k_{ratio,r}(i,i+1)<k_{\lim it}$:

(13)${\kappa }_r(i)=\left\{\begin{array}{cc}\left|{\displaystyle \frac{\Delta {\varphi }_r^{\epsilon }(i)-(\Delta {\varphi }_r^{\epsilon }(1)+\Delta {\varphi }_r^{\epsilon }(2))/2}{\Delta {\varphi }_r^{\epsilon }(i)}}\right|& ,i=1,2\hfill \\ \left|{\displaystyle \frac{\Delta {\varphi }_r^{\epsilon }(i)-(\Delta {\varphi }_r^{\epsilon }(n-1)+\Delta {\varphi }_r^{\epsilon }(n))/2}{\Delta {\varphi }_r^{\epsilon }(i)}}\right|& ,i=n-1,n\hfill \\ 1& ,others\hfill \end{array}\right.$

When $k_{ratio,r}(i,i+1)>k_{\lim it}$:

(14)${\kappa }_r(i)=\left\{\begin{array}{cc}{\displaystyle \frac{MIN(\Delta {\varphi }_r^{\epsilon }(1),\Delta {\varphi }_r^{\epsilon }(2))}{\Delta {\varphi }_r^{\epsilon }(i)}}& ,i=1\hfill \\ {\displaystyle \frac{MIN(\Delta {\varphi }_r^{\epsilon }(n-1),\Delta {\varphi }_r^{\epsilon }(n))}{\Delta {\varphi }_r^{\epsilon }(i)}}& ,i=n\hfill \\ 1& ,others\hfill \end{array}\right.$

where ${\kappa }_r(i)$ is the correction factor for the strain modes of order $r$ with respect to transducer $i$. The proposed augmented damage indicator is given by

(15)$ASMDI(i)={\displaystyle \frac{1}{M}}{\displaystyle \sum _{r=1}^M\delta {\omega }_r\cdot {\kappa }_r(i)\cdot \Delta {\varphi }_r^{\epsilon }(i)}$

4. Experimental Investigation

To check the ability of the proposed SSI-COV method and the proposed damage indicators in recognizing the defects, pipe structures with micro-defects under different impact actions have been studied. Silicone rubber-packaged FBGs in series have been attached on the pipe to measure the dynamic response.

4.1. Experimental Description

As shown in Figure 3, the experimental object is a 304 stainless-steel pipe with an axial length of 1.5 m, an outer diameter of 25 mm and an inner diameter of 23 mm. The pipe structure is lightweight, corrosion resistant and easy to handle during experiments. The two ends of the steel pipe are fixed to the test bench using clamps. The series of nine bare FBG (B−FBG) sensors and two packaged FBG (P−FBG) sensors, each with six measurement points, are fixed to the surface of the stainless-steel pipe using cured liquid silicone rubber. Among them, B−FBG1 to B−FBG9 are bare optical fibers placed on the upper surface of the pipe; P−FBG1 to P−FBG6 are semicircular packaged FBGs arranged on the lower surface of the pipe; P−FBG7 to P−FBG12 are circular packaged FBGs arranged on the upper side surface of the pipe. The sensors were securely attached to the pipe wall to ensure that the sensors deformed synchronously with the experimental pipe. The encapsulation and fixing process used Dow Corning SYLGARD 184 liquid silicone rubber, which is non-corrosive, highly elastic, radiation resistant and depolymerization resistant. After curing, it becomes transparent with minimal shrinkage. It maintains high elasticity and stability in various harsh environments within a temperature range of −55 °C to 200 °C. This material effectively transmits external micro-strains to the internal grating, ensuring that the FBG sensors operate with high sensitivity, precision and stability. The sensor-specific packaging methods are detailed in references [32,33]. Since this paper focuses on damage identification in pipes, only the data obtained from B−FBG sensors are analyzed. The placement locations of the B−FBG sensors are shown in Table 1. The sensor arrangement is designed to ensure an even distribution across the surface of the pipeline structure, enabling accurate monitoring and analysis of the vibration behavior of the pipeline while also facilitating the identification of damage locations.

Since SSI-COV is a method for identifying the modal parameters of a structure under environmental excitation, it does not require knowledge of the magnitude of the excitation. Environmental excitation is caused by natural factors affecting the engineering structure. Its frequency band is approximately uniformly distributed, which can be equivalently treated as Gaussian white noise (with a self-spectrum that is a constant value over an infinite bandwidth). Since it is not possible to generate true natural excitation in the laboratory, artificial excitation using a hammering method is employed to simulate natural excitation. To simulate natural excitation using hammers, tests were conducted with one to three hammers at excitation points 23 cm, 53.5 cm and 94 cm from the left end of the pipe. It was found that using a single hammer, regardless of the excitation point, often resulted in insufficient excitation of the pipe and missing modal parameters. However, using two hammers for excitation at 53.5 cm and 94 cm from the left end of the pipe provided the most complete and stable modal parameters. While the effect of using three hammers was similar to that of using two, it was less convenient to operate. Therefore, in this experiment, two hammers were used to apply random velocities and forces at 53.5 cm and 94 cm from the left end of the pipe to simulate natural excitation.

The experiment was conducted in a temperature-controlled environment. During the experiments, the optical fiber demodulator was set to a frequency of 5000 Hz, and each set of excitation was applied for 5 min. Bare FBG sensors can accurately monitor the strain of a structure by measuring the shift in the center wavelength [34]. Calibration tests show that at time $t$, the wavelength data ${\lambda }_B(t)$ output by the FBG and the structural strain $\epsilon (t)$ are related by the following equation:

(16)$\epsilon (t)={\displaystyle \frac{1000\times \Delta {\lambda }_B(t)}{1.2}}$

The experimental setup is shown in Figure 4.

4.2. Modal Parameter Identification

4.2.1. Modal Parameters of the Healthy Pipe

Under healthy pipe conditions, we conducted three sets of parallel experiments and processed multiple sets of data to ensure that the identified modal parameters are highly representative. Time-domain data for B−FBG1 to B−FBG9 on the pipe during the first experiment under healthy conditions are shown in Figure 5. As shown in Figure 5, the measurement points closest to the excitation point (B−FBG3 and B−FBG6) experience continuous impact from the force hammer, which restricts part of the deformation. As a result, their strain amplitudes are smaller than those of the adjacent points (B−FBG2, B−FBG4, B−FBG5 and B−FBG7). For the remaining points, the strain amplitude decreases as the distance from the excitation point increases. Strain modal analysis of the structural strain responses was performed using SSI-COV. The stable modal parameters for the first experiment under healthy pipe conditions are shown in Figure 6. From Figure 6, it can be observed that the proposed method accurately identifies the first five-order modal parameters of the pipe structure. Subsequently, two additional parallel experiments were conducted. The obtained natural frequencies of the first five-order modes are listed in Table 2, and the variation rates of the natural frequency under different loading cases are listed in Table 3. The identified strain modes for the first five-order modes are shown in Figure 7.

It can be noted from Figure 6 that the vibration majorly inspires the first five-order natural frequencies. The correlated strain modes presented in Figure 7 indicate that the proposed method can much accurately reflect the strain shapes based on the three testing results.

4.2.2. Modal Parameters of the Damaged Pipe

The damage location in the pipe is shown in Figure 3, with the damage center located 86 cm from the left end of the pipe. The physical details of the damage are illustrated in Figure 4b,c. Under damaged conditions, three sets of parallel experiments were conducted similarly. Using SSI-COV for modal parameter extraction, the identified natural frequencies of the damaged pipe are listed in Table 4, and the variation rates of the natural frequency under different loading cases are listed in Table 5.

The strain modes for the first five modes of the damaged pipe are shown in Figure 8. It can be noted that the vibration shapes are very different in different orders. Generally, the higher order brings relatively larger deformation, which should be paid considerable attention.

As shown in Table 2 and Table 4, the computation time for a single operation by using the proposed method is less than 30 s, demonstrating its efficiency. Furthermore, Table 3 and Table 5 indicate that the variation in the first five natural frequencies identified using this method is less than 5%, proving the accuracy of the modal parameter identification.

4.3. Comparison of Modal Parameters before and after Damage

Due to the use of two hammers to excite the pipe structure at fixed positions to simulate natural excitation as well as the variation in the impact force and actual excitation points for each hammer strike, it is not possible to ensure that the modal parameters identified each time are exactly consistent. To verify the effectiveness and robustness of the proposed method and to eliminate the effects of excitation on strain modal shapes, multiple excitations were performed before and after the pipe structure was damaged. The identified modal parameters were then averaged to determine the structural modal parameters.

In this study, the modal parameters obtained from the first two experiments for both healthy and damaged pipes were averaged, and this set of data was defined as Condition 1. The modal parameters of the three healthy experiments and the first three damaged experiments are averaged and defined as Condition 2. Similarly, the modal parameters of the three healthy experiments and the last three damaged experiments are averaged and defined as Condition 3. Due to the difficulty of applying specific controllable noise in the experiment, the noise level is evaluated based on the amount of data collected. Since both Condition 2 and Condition 3 involve averaging a set of data, they are less affected by differences in pipeline excitation and can be considered as having similar low noise levels.

4.3.1. Condition 1

Under Condition 1, the natural frequencies identified before and after the structural damage are shown in Table 6.

The strain modes of the pipe under healthy and damaged conditions for Condition 1 are shown in Figure 9.

4.3.2. Condition 2

Under Condition 2, the natural frequencies identified before and after the structural damage are shown in Table 7.

The strain modes of the pipe under healthy and damaged conditions for Condition 2 are shown in Figure 10.

4.3.3. Condition 3

Under Condition 3, the natural frequencies identified before and after the structural damage are shown in Table 8.

Strain modes of the pipe under healthy and damaged conditions for Condition 3 are shown in Figure 11.

From Table 6, Table 7 and Table 8, it can be observed that the SSI-COV method is quite sensitive in detecting damage. When the structure is damaged, except for the fourth natural frequency under Condition 2 and Condition 3, the natural frequencies of various modes decrease to different extents. This is consistent with the theoretical expectation that a reduction in structural stiffness due to damage leads to a decrease in the natural frequencies of the structure.

The anomaly in the fourth natural frequency is related to the excitation method. Since it cannot be ensured that each excitation is identical, the resulting data may not be perfectly consistent. Additionally, the strain modal damage identification method is mainly influenced by the first two modes, while the fourth mode has relatively low importance for damage detection. Therefore, the fourth mode can be regarded as noise affected, but the data from this condition are still used to validate the robustness and feasibility of the proposed method.

From Figure 9, Figure 10 and Figure 11, it can be seen that the first strain mode is the most sensitive to damage. The other four strain modes also show changes near the damage location. Additionally, the strain modes at the excitation points, apart from the damage location, exhibit varying degrees of change. Low-order modes are greatly influenced by the pipe’s fixed supports, resulting in poor consistency near the supports before and after damage. In contrast, higher-order modes are less affected by the supports.

4.4. Damage Identification

The strain mode data of the pipe under the three conditions were processed by using the two damage identification indicators proposed in Section 3. The results are shown in Figure 12, Figure 13 and Figure 14, respectively.

From Figure 12, it can be observed that under Condition 1, using only two sets of data, the impact of load excitation is significant. Both indicators, $SMDI$ and $ASMDI$, identified the damage at B−FBG7. Additionally, index $ASMDI$ is less affected by the supports, but $SMDI$ shows a peak near the supports. This indicates that with a limited amount of data, the proposed method does not provide very accurate identification, but it can roughly locate the position of the damage.

From Figure 13 and Figure 14, it can be seen that with sufficient data, both indices, $SMDI$ and $ASMDI$, accurately identify the damage location at B−FBG6. However, index $SMDI$ also shows an abnormal peak near the supports. When using this method for constructing a structural health monitoring system, it is necessary to select damage locations away from the supports. On the other hand, index $ASMDI$ can effectively reduce the influence of the supports on damage identification, pinpointing the exact location of the damage. Therefore, index $ASMDI$ provides a faster, more convenient and more accurate method for identifying structural damage in the development of a structural health monitoring system compared to index $SMDI$. In addition, although $ASMDI$ shows a peak at B−FBG6, the nearby measurement point B−FBG7 also produces a relatively similar value. This may be because B−FBG7 is close to both the damage location and the excitation point, resulting in a higher $ASMDI$ index. This experiment demonstrated that when sufficient data are available, the proposed method can accurately identify the location of damage in the pipe structure. It is characterized by a fast identification speed, high accuracy and strong robustness to noise. In the subsequent development of a structural health monitoring system, multiple sets of data from the pipe structure under natural excitation can be processed to further mitigate the impact of varied excitations on the identified modal parameters, thereby enhancing the accuracy of damage detection.

5. Conclusions and Future Outlook

This paper uses FBG-based sensing information and SSI-COV methods to identify the modal parameters of pipes and introduces a strain modal correction factor ${\kappa }_r(i)$ to mitigate the effects of excitation positions and supports on the identified modal parameters. Additionally, a structural damage identification index $ASMDI$ is proposed, which demonstrates high accuracy and robustness. The feasibility and effectiveness of the proposed methods were validated through experiments. However, this method tends to produce damage responses at measurement points surrounding the damage location, resulting in a broader identified damage area. Further refinement is needed to improve the accuracy of damage localization.

Based on the method proposed, encapsulated FBG sensors can be added to existing pipeline monitoring systems, integrating FBG data processing software into the current data acquisition and processing platform. Utilizing the corrosion resistance, high accuracy and real-time measurement capabilities of FBG sensors can enhance the precision, timeliness and durability of the monitoring system. However, for industrial-scale applications, economic considerations such as the costs of system upgrades and operator training, as well as the expected return on investment, need to be addressed. Additionally, technical aspects like optimizing the sensor placement, improving the data processing efficiency and ensuring the long-term stable operation of the monitoring system must be taken into account.

The main contributions of this paper can be summarized as follows:

• (1). Silicone rubber-packaged FBGs in series have been adopted to characterize the response information of pipes under different impact actions. The test results indicate that the polymer-based flexible elastic rubber shows good performance and can be adopted to package high-precision sensing elements.

• (2). Development of an FBG sensing information- and SSI-COV method-motivated modal parameter identification method: This method is fast, efficient and highly accurate, facilitating the construction of low-latency, real-time monitoring systems.

• (3). Introduction of strain modal correction factor${\kappa }_r\left(i\right)$and structural damage identification index$ASMDI$: These innovations mitigate the effects of supports and load positions on strain modal identification, enabling more precise damage location detection with high accuracy and robustness.

• (4). Simulation of natural excitation using impact hammers: A method using two impact hammers to simulate natural excitation in the laboratory was proposed and experimentally validated. This approach effectively excites the pipe structure and accurately identifies the first five-order modal parameters of the pipe.

In future research, the following two directions can be given considerable attention:

• (1). Additional axial and radial measurement points on the pipeline can be arranged, and finite element and morphological reconstruction methods can be adopted. This involves slicing the pipe structure from the top surface, unfolding it into a plate-like structure, meshing it and studying the modal parameter variations in each small element. This approach will facilitate the identification of damaged elements in the plate structure and extend to 3D damage localization in the pipe structure.

• (2). Various damage scenarios at multiple locations on the pipe can be explored, ranging from minor to severe damage, by using the proposed method to detect both the location and extent of the pipe damage. Subsequently, the entire identification process can be integrated into MATLAB code, and a real-time health monitoring system for the pipe structure can be constructed by using MATLAB GUI.

Author Contributions

Conceptualization, C.Z. and H.-P.W.; methodology, C.Z.; software, C.Z.; validation, C.Z., G.-Z.L. and M.W.; formal analysis, C.Z.; investigation, G.Y.; resources, G.-Z.L.; data curation, C.Z.; writing—original draft preparation, C.Z. and G.-Z.L.; writing—review and editing, C.Z. and H.-P.W.; visualization, M.W. and G.Y.; supervision, H.-P.W. and C.-L.Y.; project administration, H.-P.W. and C.-L.Y.; funding acquisition, H.-P.W. and C.-L.Y. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors on request.

Acknowledgments

Special thanks are given to Jinping Ou and Zhi Zhou of Dalian University of Technology and Youhe Zhou and Xingzhe Wang of Lanzhou University. The findings and opinions expressed in this article are only those of the authors and do not necessarily reflect the views of the sponsors.

Conflicts of Interest

The author Ge Yan and Chang-Lin Yan are employed by Lanzhou Vacuum Equipment Limited Liability Company. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1. FBG sensors: (a) working principle; (b) cylinder silicone rubber-packaged FBGs in series; (c) semi-cylinder silicone rubber-packaged FBGs in series.

Figure 2. Flowchart of SSI-COV.

Figure 3. Diagram of FBG sensor placement and damage locations in the pipe experiment.

Figure 4. Experimental setup: (a) sensing system; (b) defect of the pipe; (c) FBG sensors.

Figure 5. Strain time-domain data for pipe measurement points under healthy conditions: (a) B−FBG1, (b) B−FBG2, (c) B−FBG3, (d) B−FBG4, (e) B−FBG5, (f) B−FBG6, (g) B−FBG7, (h) B−FBG8 and (i) B−FBG9.

Figure 5. Strain time-domain data for pipe measurement points under healthy conditions: (a) B−FBG1, (b) B−FBG2, (c) B−FBG3, (d) B−FBG4, (e) B−FBG5, (f) B−FBG6, (g) B−FBG7, (h) B−FBG8 and (i) B−FBG9.

Figure 6. Stability diagram for the pipe during the first experiment under healthy conditions.

Figure 7. Strain modes identified from three experiments under healthy pipe conditions: (a) first−order strain mode; (b) second−order strain mode; (c) third−order strain mode; (d) fourth−order strain mode; (e) fifth−order strain mode.

Figure 7. Strain modes identified from three experiments under healthy pipe conditions: (a) first−order strain mode; (b) second−order strain mode; (c) third−order strain mode; (d) fourth−order strain mode; (e) fifth−order strain mode.

Figure 8. Strain modes identified from four experiments under damaged pipe conditions: (a) first−order strain mode; (b) second−order strain mode; (c) third−order strain mode; (d) fourth−order strain mode; (e) fifth−order strain mode.

Figure 8. Strain modes identified from four experiments under damaged pipe conditions: (a) first−order strain mode; (b) second−order strain mode; (c) third−order strain mode; (d) fourth−order strain mode; (e) fifth−order strain mode.

Figure 9. Comparison of strain modes under healthy and damaged conditions for Condition 1: (a) first−order strain mode; (b) second−order strain mode; (c) third−order strain mode; (d) fourth−order strain mode; (e) fifth−order strain mode.

Figure 9. Comparison of strain modes under healthy and damaged conditions for Condition 1: (a) first−order strain mode; (b) second−order strain mode; (c) third−order strain mode; (d) fourth−order strain mode; (e) fifth−order strain mode.

Figure 10. Comparison of strain modes under healthy and damaged conditions for Condition 2: (a) first−order strain mode; (b) second−order strain mode; (c) third−order strain mode; (d) fourth−order strain mode; (e) fifth−order strain mode.

Figure 11. Comparison of strain modes under healthy and damaged conditions for Condition 3: (a) first−order strain mode; (b) second−order strain mode; (c) third−order strain mode; (d) fourth−order strain mode; (e) fifth−order strain mode.

Figure 12. Comparison of damage identification effectiveness between [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] under Condition 1.

Figure 13. Comparison of damage identification effectiveness between [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] under Condition 2.

Figure 14. Comparison of damage identification effectiveness between [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] under Condition 3.

Table 1

FBG positions and wavelength information.

FBG Number	Initial Wavelength (nm)	Distance from the Left End of the Pipe (mm)	Adhesive Surface
B−FBG1	1542	150	Upper surface
B−FBG2	1530	300
B−FBG3	1536	450
B−FBG4	1539	600
B−FBG5	1542	750
B−FBG6	1546	900
B−FBG7	1549	1050
B−FBG8	1551	1200
B−FBG9	1554	1350

Table 2

Natural frequencies under healthy pipe conditions.

No. of Experiments	First Order (Hz)	Second Order (Hz)	Third Order (Hz)	Fourth Order (Hz)	Fifth Order (Hz)	Time(s)
First time	38.33	108.96	264.45	457.07	659.80	7.809
Second time	38.11	107.81	263.88	457.64	662.29	10.414
Third time	37.85	103.64	263.52	455.56	655.45	16.684

Table 3

Variation rates of natural frequencies from different experiments under healthy pipe conditions.

No. of Experiments	First Order (%)	Second Order (%)	Third Order (%)	Fourth Order (%)	Fifth Order (%)
1 and 2	0.57	1.06	0.22	0.12	0.38
2 and 3	0.68	3.87	0.14	0.46	1.03
1 and 3	1.25	4.88	0.35	0.33	0.66

Table 4

Natural frequencies under damaged pipe conditions.

No. of Experiments	First Order (Hz)	Second Order (Hz)	Third Order (Hz)	Fourth Order (Hz)	Fifth Order (Hz)	Time(s)
First time	37.41	106.78	263.14	456.38	659.75	23.88
Second time	37.28	105.70	262.61	456.76	658.68	29.98
Third time	36.22	105.68	265.17	460.22	657.93	10.041
Fourth time	36.83	106.79	263.73	457.05	658.44	22.547

Table 5

Variation rates of natural frequencies from different experiments under damaged pipe conditions.

No. of Experiments	First Order (%)	Second Order (%)	Third Order (%)	Fourth Order (%)	Fifth Order (%)
1 and 2	0.35	1.015	0.20	0.08	0.16
2 and 3	2.84	0.02	0.97	0.75	0.11
1 and 3	3.18	1.03	0.77	0.83	0.28
2 and 4	1.21	1.02	0.42	0.063	0.036
3 and 4	1.66	1.04	0.54	0.69	0.077

Table 6

Natural frequencies of the pipe before and after damage under Condition 1.

	First Order (Hz)	Second Order (Hz)	Third Order (Hz)	Fourth Order (Hz)	Fifth Order (Hz)
Healthy condition	38.22	108.38	264.17	457.36	661.04
Damaged condition	37.35	106.24	262.88	456.57	659.22
Frequency variation rate (%)	2.27	1.97	0.49	0.17	0.28

Table 7

Natural frequencies of the pipe before and after damage under Condition 2.

	First Order (Hz)	Second Order (Hz)	Third Order (Hz)	Fourth Order (Hz)	Fifth Order (Hz)
Healthy condition	38.10	106.80	263.95	456.76	659.18
Damaged condition	36.97	106.05	263.64	457.79	658.79
Frequency variation rate (%)	2.97	0.70	0.12	-0.23	0.06

Table 8

Natural frequencies of the pipe before and after damage under Condition 3.

	First Order (Hz)	Second Order (Hz)	Third Order (Hz)	Fourth Order (Hz)	Fifth Order (Hz)
Healthy condition	38.10	106.80	263.95	456.76	659.18
Damaged condition	37.00	105.72	263.84	458.01	658.35
Frequency variation rate (%)	2.89	1.01	0.042	0.27	0.13

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