1   Introduction

Fluid-carrying pipes have varied applications in engineering fields and play a crucial role in transporting fluids such as oil, gas, and seawater (Wang, 2018). These pipes are often situated in challenging environments, such as offshore platforms, industrial plants, and urban infrastructure, where the risk of damage is significant. Therefore, understanding the dynamic behaviors of both intact and damaged fluid-conveying pipes is critical for ensuring their structural integrity and safe operation. This knowledge is particularly valuable to industries such as energy, petrochemicals, and civil engineering, where the prevention of pipeline failures can prevent costly downtime, environmental hazards, and safety risks. Additionally, developing effective damage detection methods (Song et al., 2018) can aid in proactive maintenance and extend the lifespan of these essential systems. Additionally, many structures are situated on foundations for which researchers used different mathematical models (Belabed et al., 2024; Bouafia et al., 2021; Ma et al., 2022; Lafi et al., 2024; Tounsi et al., 2024; Tounsi et al., 2023a; Tounsi et al., 2023b; Bounouara et al., 2023; Zaitoun et al., 2023; Tahir et al., 2022; Mudhaffar et al., 2023).

The investigation of vibrations in fluid-carrying pipes began in 1950 with the work of Ashley and Haviland (1950). The dynamic behaviors of these pipes were studied both theoretically (Benjamin, 1962) and experimentally (Benjamin, 1961). For example, Fu et al. (2023) investigated the nonlinear vibration analysis of viscoelastic axially functionally graded material pipes conveying pulsating internal flow. They found that elastic modulus, density, and coefficient of viscoelastic damping of the pipe material varied along the axial direction. They also used Euler–Bernoulli beam theory to model the transverse vibration equation of the viscoelastic axially functionally graded material pipe conveying pulsating fluid. Vassilev and Djoundjorov (2006) analyzed the dynamic stability of fluid-carrying pipes using the Galerkin method. Chellapilla and Simha (2007) employed the Fourier series and Galerkin method to study the critical velocity of fluid-carrying pipes on a foundation, examining the problem under three different boundary conditions. The interaction between the fluid and the pipe was explored using the finite element (FE) method (Olson and Jamison, 1997) and the spectral FE method (Lee and Oh, 2003; Lee and Park, 2006). Chu and Lin (1995) presented a general FE formulation employing cubic Hermite interpolation for the dynamic analysis of fluid transmission pipes, considering both shear deformation and rotary inertia. In their work, they considered the dynamic effect of the fluid as external distributed forces acting on the supporting pipe.

Liu et al. (2010) studied the effect of hydrostatic pressure on the vibration dispersion characteristics of fluid-shell coupled structures, considering fluid-loaded cylindrical shells and fluid-filled cylindrical shells. Zhang et al. (2002) investigated the application of three-dimensional (3D) linearized Euler equations and the theory of linearized tension in their research. In this study, an FE formulation based on 3D elasticity theory for the shell was presented, along with the linearized Euler equation for the fluid. Li et al. (2018) investigated the application of the Galerkin method to represent the dynamic response of a fluid-conveying pipe with laterally moving supports at both ends of the pipe. Li et al. (2015a) also examined the nonlinear dynamic behavior of a submerged beam with moving supports at both ends. They derived the equation of motion using a Newtonian approach, after which they added a mass coefficient for the fluid mass attached to the beams. To investigate the dynamic behavior and stability of a multi-mouth fluid transmission pipe, El-Sayed and El-Mongy (2019) presented a new formula based on the variable iteration method. Huang et al. (2010) obtained the natural frequency of fluid-structure interaction in a fluid transmission pipeline using the Reduced-Order-Element-Galerkin method, deriving the natural frequency equations for various boundary conditions. Li and Yang (2017) determined the critical flow velocity and frequency of a fluid transmission pipe using a new semi-analytical method, considering the effects of boundary conditions. Ma et al. (2023) developed the harmonic differential quadrature method to analyze the one-dimensional vibration of pipes conveying fluid under various boundary conditions. This method employed trigonometric functions to formulate the harmonic test function, and the weighting coefficients were calculated explicitly.

Using FE tools, Chatzopoulou et al. (2016) investigated the influence of rotational loading on the mechanical behavior of seamless steel pipes with thick walls in deep water. Ni et al. (2011) presented a semi-analytical method and differential transformation technique to analyze the free vibration of fluid transmission pipes with different boundary conditions. Using the Euler–Bernoulli beam model and the generalized integral transform method, Fu et al. (2024) analyzed the dynamic behavior of an axially functionally graded pipeline conveying gas–liquid two-phase flow. They found that the interactions between structures and fluids refer to the effects that fluids exert on structures, which can manifest through various forces. Such an interaction is often of significant interest to engineers and structural designers.

Meanwhile, Tijsseling (1996) conducted a review of the literature on transient phenomena in fluid-filled pipes by focusing on the history of fluid-structure interaction research in the time domain. Using the FE method, Zhai et al. (2011) obtained the governing equation of the fluid-carrying pipe considering fluid-structure interaction and the effect of shear deformation. To solve this equation, they employed a combination of the perturbation method with the Galerkin method. Li et al. (2015b) provided a broad overview of the literature on the dynamic analysis of fluid-filled pipe systems with a focus on fluid-structure interaction. They compared models and simulation algorithms of varying degrees of sophistication and discussed their range of applications.

The dynamics of fluid-carrying pipes were experimentally investigated by Jendrzejczyk and Chen (1985) under six different types of boundary conditions. Anand (2015) conducted an analytical study of the laminar flow of nanofluids in a circular tube immersed in an isothermal external fluid. Ghadirian et al. (2022) examined nonlinear free vibrations and stability of fluid transmission pipes made of composite materials. They derived the equations of motion for the system using Hamilton's extended principle for open systems based on Timoshenko's beam theory. Meanwhile, Song et al. (2018) used guided waves and conducted a series of experiments on damage detection in large-diameter pipes, both with and without liquid. In their study, two types of liquid‒water and machine oil‒were chosen to fill the pipes and assess the influence of the filler. Meenakumari et al. (2024) investigated the fluid-structure interaction phenomena of a submerged long flexible cylinder conveying two-phase slug flows, considering the geometric and hydrodynamic nonlinearities.

Li et al. (2023) proposed an FE model for analyzing flexible pipes with localized damage in the outer layers. The fundamental concepts of deep learning, including convolutional neural networks, were discussed by Jafari et al. (2020). Additionally, they introduced the use of deep learning methods for preventing pipeline damage through early detection. The results indicated that such an approach can identify damages in the early stages. Gresil et al. (2017) investigated the application of guided wave excitation and damage detection in composite pipes using piezoelectric sensors. They also examined the use of a guided wave-based structural health monitoring method using ultrasonic guided waves. Furthermore, Buethe et al. (2013) presented an approach for structural health monitoring using guided waves in pipe structures, thereby addressing challenges in pattern recognition for damage detection. An algorithm for structural health monitoring of subsea pipeline systems was developed by Bao et al. (2013).

Structural health monitoring is a crucial process in structural engineering that helps prevent structural failure and reduces operational costs. In this regard, various methods for structural health monitoring have been proposed by researchers (Farrar and Worden, 2007). One of the most popular structural health monitoring methods is vibration-based damage detection, which has attracted researchers' attention due to its numerous advantages (Das et al., 2016; Peeters et al., 2001; Saadatmorad et al., 2021). An important advantage of vibration-based methods is their ability to provide global testing. These methods are also cost-effective. In vibration-based tests, the global data of the structure are obtained using numerical or experimental modal analysis. These data are then used in damage identification methods to determine the location and sometimes the severity of the damage (Hou and Xia, 2021; Doebling et al., 1998).

Generally, vibration-based damage detection methods are divided into two main categories: frequency-based damage and mode-shape-based damage detection methods (Khatir et al., 2021). Yang and Wang (2010) investigated the detection of structural damages using natural frequencies. In their work, they introduced an assurance criterion for the natural frequency vector to detect damages in an eight-story structure. By applying this criterion, they were able to successfully identify the location and severity of the damage with high accuracy. A new damage identification formula was introduced by Sotoudehnia et al. (2019). Similarly, Sha et al. (2019) proposed a new damage detection method based on changes in relative natural frequencies for detecting damages in beam structures. The effectiveness of the proposed method in identifying and quantifying damages in beams was demonstrated. To detect the location of damages in beam structures, Seguini et al. (2022) used natural frequencies as input for an artificial neural network. Saadatmorad et al. (2024b) applied the covariance of vibrational mode shape to the continuous wavelet transform to detect damages in beams that were reinforced with nanoparticles.

Furthermore, Pandey et al. (1991) used the derivative of the vibration mode shapes and the curvature of mode shapes for detecting damage in simply supported and cantilever beams. Their method demonstrated effective performance in detecting damages in beams. Wahab and De Roeck (1999) used the derivative of mode shapes in beam and bridge structures to detect damages. Their approach demonstrated high performance in numerical and experimental damage detection scenarios. In their study, Nguyen (2014) used a mode-shape analysis approach to detect cracks in beam structures, concluding that projections of the mode shapes on normal planes served as a good damage detection index. Saadatmorad et al. (2022) proposed a novel damage index called the "Pearson correlation function of mode shapes" to detect cracks in steel beams. Their experimental and numerical results demonstrated the method's effectiveness in identifying the scracks. Nahvi and Jabbari (2005) demonstrated that crack damages tended to disappear when they are located at mode-shape nodes; nevertheless, damage detection based on mode shapes can still be accomplished with high accuracy.

As shown in the literature, damage detection methods based on mode shapes are superior to those based on natural frequency. This is because mode shapes are generally more sensitive to local damage while detecting damage using natural frequencies requires acquiring data from multiple damaged states of the considered structure (Saadatmorad et al., 2024a; Carden and Fanning, 2004).

A review of published studies indicates that, despite the widespread applications of fluid-conveying pipes, damage detection in these structures has not yet been extensively explored. Therefore, the objective of the current study is to fill this research gap. The novelty of the present work lies in suggesting a signal reconstruction-based algorithm called "matching pursuit (MP)" for detecting damages in fluid-conveying pipes. Typically, this algorithm is employed to reconstruct the signal or its approximation. However, this article proposes a novel approach by using the residuals generated from the approximation as effective indicators for identifying damage and errors in the signal. The foundation of this paper rests on the principle that high-frequency components within the signal can be detected in the residuals produced by the optimal atom in the MP algorithm. Initially, we modeled pipe conveying fluid on a Pasternak foundation, and its kinetic and potential energies were obtained. Then, using Hamilton's principle, the governing differential equation for the pipe's deformation was derived. These equations were solved using the FE method, along with the Galerkin method. The computed results were compared with existing results to verify their validity. After this verification, a novel structural damage detection method based on the MP algorithm was proposed for detecting damages in fluid-conveying pipe structures. Finally, the effectiveness of the proposed damage detection method was evaluated numerically and experimentally.

2   Mathematical modeling

Consider the elastic pipe shown in Figure 1, which is placed on a Pasternak foundation. The pipe has a length L, an internal diameter d, an outer diameter D, Young's modulus E, and a second moment of area I. An incompressible fluid with a constant velocity V flows inside the pipe. The elastic stiffness and shear layer stiffness of the Pasternak foundation are denoted by k_f and k_s, respectively. As shown in Figure 1, the origin of the coordinate system is located at the left end of the pipe.

Figure  1  A schematic view of the pipe conveying fluid on the Pasternak foundation

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The kinetic energy of the system consists of two parts that are related to the kinetic energy of the pipe and the fluid, respectively. The kinetic energy of the pipe can be expressed as follows:

The kinetic energy of the fluid is also calculated as follows:

In Eqs. (1) and (2), A and ρ are the area of cross-section and density, respectively; the subscripts p and f stand for the pipe and fluid, respectively; L is the length of the pipe; w (x, t) represents the transverse displacement of the pipe at position x and time t; and V is the constant velocity of the fluid. Thus, the total kinetic energy of the system, which includes the kinetic energy of the pipe and the fluid, can be expressed as follows (Liang et al., 2018):

The strain energy of the pipe and foundation is expressed as follows (Yu et al., 2017; Jafari-Talookolaei and Ahmadian, 2007; Lee and Chung, 2002):

The governing equation of motion for the free vibration of the considered tube was derived using Hamilton's principle as applied to a conservative system. The principle can be written as

where t₁ and t₂ are two specified times, and δ denotes the first variation. Substituting Eqs. (3) and (4) into Eq. (5), and performing the first variation yields the following partial differential equation:

In this equation and in the following ones, a comma denotes differentiation with respect to the variable that immediately follows it.

3   Solution method

In the present work, the FE method was used, along with the Galerkin weighted residual method, to solve Eq. (6). A higher-order pipe element with length L_e, shown in Figure 2, was used in this study, which included two end nodes and one middle node. Each node contains two degrees of freedom of vertical displacement w and slope w_{, x}. Therefore, each element has six degrees of freedom. Thus, the degrees of freedom vector of an element are stated as follows:

Figure  2  Higher-order pipe element used in the present study

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The deflection of the pipe can be interpolated as follows:

in which

and H_i and H_i (i=1, 2, 3) are the Hermite interpolation functions (Logan, 2011). Based on the Galerkin method, we have (Logan, 2011) the following:

where R is a weighted residual, and W is a weight function. Notably, for the Galerkin method used in the FE analysis, the weight function is the same as the shape function. By applying the Galerkin method to Eq. 6, we have the following:

where $N_i=\left(H_1, \bar{H}_1, H_2, \bar{H}_2, H_3, \bar{H}_3\right)$. By using the integral by parts and writing the equations in weak form, the mass, damping, and stiffness matrices of an element can be calculated as follows:

where ξ = x/L_e represents the intrinsic coordinate of an element. Eventually, by assembling the equations of different elements, the final equations can be written in the following form:

where (K, C, M) are the total mass, damping, and stiffness matrices, respectively. Furthermore, $(\ddot{\varDelta}, \dot{\varDelta}, \varDelta)$ are the acceleration, velocity, and degrees of freedom vectors of the pipe, respectively. The eigenvalues of the system can be computed by working out Eq. (15). Assuming the response to be in the form $\varDelta=\underline{\varDelta} \mathrm{e}^{\mathrm{i} \omega t}$ and substituting it into Eq. (15) and removing the terme^iωt, we will have the following:

in which i is the imaginary variable, $\underline{\varDelta}$ is the amplitude of vibration, and ω is the frequency of the system. The above equation has been solved using MATLAB software, and the frequencies and mode shapes, i.e. $\underline{\varDelta}$, have been calculated.

4   Methodology

In this study, a novel mode shape-based damage detection method is proposed, which uses the MP algorithm (Mallat and Zhang, 1993) to detect damages in the pipes conveying fluid on the Pasternak foundation. The MP algorithm is commonly used to approximate or reconstruct a signal. Notably, damage can be detected through the residual obtained from the difference between the original signal and the signal approximated or reconstructed by the MP algorithm. The probability of detecting damage in the residual obtained from the MP algorithm is high, because damage in the pipe conveying fluid manifests as a high-frequency disturbance that is not visible in the mode shapes. The flowchart of our proposed methodology is shown in Figure 3.

Figure  3  Flowchart of the proposed methodology of the current study

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Consider a discrete signal S [j] as follows (Mallat and Zhang, 1993; Chakraborty et al., 2009; Wang and Sun, 2019):

where j is the sampling point number (node number), f_i[j] is the basis function selected from the dictionary D at the ith iteration of the MP, and α_i is the corresponding expansion coefficient.

The energy of the signal can be represented by the following equation (Mallat and Zhang, 1993):

After n iterations, the result can be shown to converge as follows:

The signal can be defined using Eq. 20 (Mallat and Zhang, 1993):

where R_n[j] is the residual signal after n iterations. The steps of the MP algorithm are presented as follows:

By setting R₀[j] = S[j], in ith iteration, i = 0, 1, ⋯, n − 1, the residual R_i[j] is computed on each dictionary atom f ^d[j] ∈ D until the following is obtained:

The atom f_i[j] selected from the dictionary is an atom that has the highest inner product value with the following residual:

The corresponding coefficients are written as follows (Mallat and Zhang, 1993):

The residual in the (i+1)th iteration can be calculated as:

Thus, after n iterations of MP, the residual can be expressed as follows:

As shown in Table 1, the MP algorithm is a method to approximate a signal via a dictionary of functions. This algorithm operates iteratively to select the optimal function in the dictionary that best matches the residual signal at each step. The functions in the dictionary typically have waveforms. The most commonly used dictionaries are Gabor atoms, Fourier basis functions, and wavelet functions. At the first iteration, the first residual is the original signal. The algorithm iteratively selects the atom from the dictionary that is most similar to the current residual. This similarity is often measured using the inner product between the residual and each dictionary atom. The atom with the highest inner product is selected. After satisfying the stopping criterion, the final approximation of the signal is obtained by summing up all the selected atoms and their corresponding scaling coefficients.

5   Weak matching pursuit

In the weak matching pursuit (WMP), the atom selection criterion is limited to a maximum value of inner multiplication less than one to have a computationally efficient method. This criterion is applied as follows:

The MP for the weak condition is stated in Table 2.

6   Results

The results of the current study are presented in this section. First, we verified the accuracy of the present modeling in Subsection 6.1. Then, in Subsection 6.2, we investigated numerical damage detection using six different numerical damage scenarios. After that, we analyzed the effect of noise on the damage indexes in Subsection 6.3. Finally, Subsection 6.4 presents the results of the experimental evaluations.

6.1   Comparison study

In this section, the calculated natural frequencies were compared with other references. We assume that the fluid velocity is zero. The natural frequencies are presented in dimensionless form. Here, results reported in three references (Liang et al., 2018; Ni et al., 2011; Thomson, 1993) were compared with the results of the present study. Table 3 shows the first four dimensionless natural frequencies $\left(\bar{\omega}=\frac{\omega L^2}{\sqrt{\frac{E I}{m_p+m_f}}}\right)$ of the tube with different ditions and V = 0. As shown in this table, the results are in good agreement with the results reported in the literature (Liang et al., 2018; Ni et al., 2011; Thomson, 1993).

6.2   Numerical damage detection

In most studies, the MP algorithms aim to find the bestmatched signal from the original signal. However, in the current study, our aim is to find the best residual using an MP algorithm to detect the location of damages in pipes carrying fluids. Thus, we used the mode shapes of a pipe with the properties listed in Table 4 to evaluate the numerical performance of the proposed MP algorithm. The mode shapes were obtained based on the FE modeling suggested in Section 2 and verified in subsection 3.1. In this method, the atom is the high-frequency component, and the residual represents the low-frequency component.

The atom approximates the signal, and the damage locations are predicted in the residual. In other words, the proposed methodology suggests that instead of using MP for reconstructing the original signal, it can be used as a decomposition tool for damage detection. As shown in Table 5, 10 different damage scenarios are considered to examine the numerical performance of the proposed methodology.

The obtained mode shapes corresponding to the first six numerical damage scenarios are shown in Figure 4. As shown in the figure, detecting damages in most mode shapes is difficult or nearly impossible. Thus, we apply the proposed MP method. Figure 5 compares the damaged mode shapes with the approximation provided by the proposed MP method. Finally, Figure 6 indicates the residuals obtained from the proposed MP method. Evidently, the original signals (mode shapes) in all damage scenarios are well-fitted with the signals approximated by the MP method. This finding shows that the proposed method accurately processes the original signal. As shown in Figure 6, even at low damage percentages at different damage positions, the proposed MP method can accurately detect damage locations in the pipe as a break in the residual signals. Therefore, the suggested method is an effective numerical tool to identify the damages in the mode shapes of the pipe.

Figure  4  Damaged mode shapes for the considered damage scenarios (V = 0, k_f = 0, k_s = 0)

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Figure  5  Comparison of the original signal and the signal approximated using the WPM algorithm (V = 0, k_f = 0, k_s = 0)

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Figure  6  The residual signals obtained from the WPM algorithm and results of damage detection (V = 0, k_f = 0, k_s = 0)

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Figure 6 shows the results of damage detection for static fluid without the elastic foundation and Pasternak foundation. Furthermore, eight different damage scenarios with varying speeds and different values of k_s and k _f were considered to examine the numerical performance of the proposed methodology. The results are shown in Table 6.

As shown in Figure 7, the original signals in all damage scenarios are fitted with the signals approximated by the MP method. The approximated signals from the matching follow-up method are also shown. Figure 8 presents the residual signals and the results of damage detection. As shown in Figure 8, the proposed MP method can detect damage locations with high accuracy, even at low damage percentages at different damage positions.

Figure  7  Comparison of the original signal and the signal approximated using the WPM algorithm

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Figure  8  The residual signals obtained from the WPM algorithm and the results of damage detection

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As shown by the examined damage scenarios, the WMP algorithm can detect damages in all damage scenarios considered in Table 6 (for damages with different severities, in different positions, and with different boundary conditions of the pipe). In the next section, we investigated the effect of noise in the mode shape on the accuracy of damage identification with the proposed WMP.

6.3   Effect of noise on the damage indexes

Due to the noises introduced in operational conditions, the performance of the proposed damage indicators in noisy conditions was examined in four separate scenarios in this section, as shown in Table 5 (Scenarios 7–10). Furthermore, the data were intentionally contaminated with random noise generated by MATLAB, with an intensity of 0.01%. The results obtained for Scenarios 7–10 are shown in Figures 9 and 10. In Figure 9, the original noisy signals in all damage scenarios are well-fitted with the signals approximated by the matching method. In Figure 10, we can see that despite the low damage level at different damage positions, the proposed MP method can detect the damage locations in the pipe as a break in the residual noise signals with high accuracy.

Figure  9  Comparison of the noisy original signal and the signal approximated using the WPM algorithm

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Figure  10  Residual noisy signals obtained from the WPM algorithm and the results of damage detection

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Figure 9 shows the quality of approximating the mode shapes related to Scenarios 7– 10 by the WMP algorithm. The difference between these mode shapes and approximations created by the WMP algorithm is called the "residual", which we proposed as a damage identification index in this paper. The residuals corresponding to damage Scenarios 7–10 are shown in Figure 10. By comparing the results in Figures 9 and 10, we can see that the better the approximation of the signal, the higher the accuracy of damage identification in the residuals.

6.4   Experimental damage detection

In this section, we validated the method presented in operational conditions using modal testing. Figure 11 shows the steps of measuring and applying damage on the steel 304 pipe for the modal test. To conduct the modal test, the pipe-carrying fluid was suspended using two soft strings. As shown in Figure 12, the pipe was divided into 20 segments. Next, to measure acceleration, two one-way piezoelectric accelerometers were installed at points 3 and 12 of the structure. The accelerometer remained fixed at both points. A modal impact hammer equipped with a force gauge was used to apply force to the structure, which was done at all points by moving the hammer along the pipe. Subsequently, the analyzer was used to calculate the force and acceleration signals and obtain the frequency response functions.

Figure  11  Steps of applying damage and measuring the pipe

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Figure  12  Meshing the pipe and displaying the damage location

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Figure 13 shows the laboratory equipment and the test setup, while Figure 14 presents the experimental setup and its different components.

Figure  13  Experimental modal analysis: pipe conveying fluid and equipment for data acquisition system and the modal impact hammer

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Figure  14  Test setup and its different components

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Finally, the obtained frequency response functions were analyzed. The natural frequencies and damping coefficients are reported in Table 7.

The data for the normalized mode shapes of the steel 304 pipe carrying water are presented in Table 7. Eventually, we tested our proposed damage detection methodology for localizing damage in the pipe with the experimental first mode shape. Figure 15 compares the original signal and the approximated signal using the weak MP tracking algorithm. Figure 16 shows the residual obtained from the proposed MP method. As can be seen, the original signal (first mode shape) is well-fitted with the signal approximated by the proposed MP method. This shows that the proposed method processes the original signal correctly. Furthermore, as seen in Figure 16, the proposed MP algorithm detects the location of damage with high accuracy, even with few experimental sampling data.

Figure  15  Comparison of the original signal and approximated signal using the weak matching tracking algorithm

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Figure  16  Residual signal obtained from the WMP algorithm and damage detection results

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7   Conclusions

In this study, numerical and experimental approaches were presented to identify damage in fluid-conveying pipes, which are critical components in various engineering applications, such as oil and gas transportation, water supply systems, and chemical processing. Maintaining the structural integrity of these pipes is crucial, because any damage can result in leaks, reduced efficiency, and catastrophic failures. The dynamic behavior of fluid-conveying pipes is influenced by several factors, including pipe material properties, fluid flow velocity, and external support conditions, such as those provided by a Pasternak foundation. In this study, the governing equation for the pipe supported by a Pasternak foundation is derived based on the Euler–Bernoulli beam theory, and the response is calculated using the finite element method.

This study introduces the potential of a signal approximation algorithm called the MP method for detecting damage and demonstrates its efficacy in accurately locating damage under varying conditions. The damage index proposed in this study is based on the difference between the mode shape of the pipe structure and the approximation signal obtained from the MP algorithm (i. e., the residual signal). This innovative approach enhances traditional damage detection methods by providing a more sensitive and reliable method of monitoring pipe integrity.

Our findings reveal that the MP algorithm effectively identifies damage regardless of boundary conditions, varying damage intensities, and the proximity of damage to support. Given the widespread use of fluid-carrying pipes across various industries, ensuring reliable damage detection in these structures is critical. In the current study, the combination of theoretical modeling and experimental validation enhanced the credibility of the results, making it a robust tool for engineers and researchers in the field. Furthermore, this study provides a robust foundation for future research, serving as a benchmark for further investigations into the dynamic behavior of fluid-carrying pipes and the application of the MP algorithm for signal analysis and damage detection.