Assessment of dynamic responses sensitivity in relation to moving load identification quality

Abstract

The quality of measured responses in terms of displacement or deformation plays an important role in structural health monitoring (SHM). The aim of this article is to identify the impact force that takes the form of a moving signal, such as the passage of trains and vehicles over a bridge. The structure considered in this work is a beam resting on elastic and/or rigid supports, the aim of which is to study the influence of the moving force while taking into account different support conditions. The effect of sensor position on reconstruction quality will be discussed in this article. Since the inverse problem is ill-posed, regularization methods are essential. The Tikhonov-SVD technique is considered in this work. This regularization technique will be tested by adding noise to the measured responses.

Introduction

The study of impact forces on structures is very essential to understanding their dynamic behavior under instantaneous loads, especially when the structure is under dynamic forces. In this context, beams are among those basic structural elements that play a vital role in many civil engineering structures in terms of load transfer and dissipation. When beams are subjected to dynamic loads, such as those caused by vehicles and trains crossing a bridge or machinery working inside a factory, understanding the actions of such loads on their structural integrity and strength becomes a must. The study of the influence of a moving force applied to a beam includes the description of the dynamics related to the motion of the force along the structure and consideration of the support conditions of the beam, which can be either rigid or elastic.^1–5 These two different classes of supports significantly affect the force distribution and the dynamic response of the beam.

Rigid supports are those that will not allow any form of deformation, ⁶ hence providing infinite rigidity. On the other hand, the elastic supports do allow some amount of deformation once an external load is applied to them.^7,8 Battini and Ülker-Kaustell ⁶ proposed an approach to study the nonlinear influence of ballast in vertical dynamic analyses of railway bridges for a beam sitting on rigid supports. Liao et al. ⁷ used fiber Bragg grating (FBG) sensors to collect deformation data, enabling the construction of neural networks. A comparison was made between the data collected by FBG and that obtained using the finite element method. Moliner et al. ⁸ proposed a solution that relies on the modernization of the bridge with a set of discrete viscoelastic dampers (VED) connected to the slab and to an auxiliary structure, placed under the bridge deck and resting on the abutments. Yu and Chan ⁹ presented the study focused on time-frequency domain to identify an axle force acting the bridge with rigid support. Studying the reaction of a beam under these two support conditions is an integral part of analyzing its capacity to absorb impact forces, which could either be concentrated forces at a given point or forces that travel along the length of the beam.^10–12 The importance of this study is even more significant in areas where the structures’ safety and durability are paramount.

The identification of impact forces is a fundamental technique in structural health monitoring (SHM). It allows for the assessment of structural damage by detecting and quantifying the applied forces. Typically, this technique relies on the inversion transfer matrix, which encapsulates the dynamic and mechanical properties of materials. However, this inversion is highly sensitive to even minor perturbations in the transfer matrix parameters, which can lead to unstable or physically inconsistent solutions. Zhu and Law ¹⁰ developed a method based on modal superposition and regularization techniques in order to identify the moving load. They studied the effect of different parameters, namely, the measurement noise, the vertical and rotational stiffness of the supports, as well as the speed of displacement of the forces. Zhang et al. ¹³ proposed an approach based on intelligent visual perception technology to identify a moving load occurring on bridges. They evaluated the efficiency and robustness of the identification method, which depended on the position of the cameras and the number of points required to reconstruct the impact force. He et al. ¹⁴ proposed a time-domain identification method using the finite element method (FEM), based on the second-order Taylor expansion. They discuss the sensitivity of the method to factors such as the number of sensors, noise, and vehicle speed.

In this context, the identification problem is known to be ill-posed, and solving this ill-posedness requires the use of regularization methods. ¹⁵ The utilization of regularization techniques is essential to ensure the stability of the resulting data. These methods, including Tikhonov regularization, truncated singular value decomposition (TGSVD),^16–18 and other variants, help control the impact of perturbations on the obtained solutions. The effectiveness of these approaches has been demonstrated in various inverse problems.^19,20 Qiao et al. ²¹ proposed a sparse regularization approach to solve the underdetermined problem of impact-force reconstruction. Pan et al. ²² identified a moving load using sparse regularization, considering a constraint of stable mean values for the time histories of moving forces. They discuss the uncertainty in dynamic responses due to sensor calibration errors.

It is therefore essential to select appropriate regularization techniques and test their robustness against disturbances and noise to ensure reliable identification of the impact force. Sensor measurements are often affected by external interference or internal disturbances, such as electromagnetic noise or calibration errors, which degrade the quality of recorded data.^23–25 Such disturbances complicate the accurate reconstruction of the impact force, leading to reduced reliability of the results. To address these challenges, the study employs regularization methods to filter out noise and preserve the integrity of the reconstruction.

This research is crucial for the field of structural health monitoring (SHM), where the precision of impact force measurements and the ability to minimize disturbances are vital for ensuring the safety and functionality of structures. The primary objective of this study is to examine how sensor placement and regularization methods influence the precision of impact force history reconstruction in beams. Additionally, the study investigates the optimal sensor placement configuration, considering the propagation of elastic waves through the structure, and evaluates the effect of noise on the reconstruction process. The study also explores the impact of excitation frequency and the effect of the stiffness of the elastic support.

Theory of moving load identification

Dynamic behavior of structure

A homogeneous and isotropic elastic beam with a uniform rectangular cross-section is considered. The beam has length L, width b, and height e, and its mechanical properties are characterized by the Young’s modulus E and the Poisson’s ratio $\upsilon$ . A moving load, with a velocity $v$ , is applied along the beam’s upper face, and two support conditions are examined. In the first case, the beam is supported by rigid supports at both ends. In the second case, the rigid supports are replaced with linear elastic springs (Figure 1). Sensors are positioned at specific points along the upper face of the beam to measure the dynamic response, which can be either displacement, acceleration, or strain. These sensors may include strain gauges or accelerometers. The dynamic response is analyzed at a given point along the beam, referenced on the origin at the left-hand end of the mean fiber. The reference axis is oriented to the right, as illustrated in Figure 1. Under the action of a moving load $f(t)$ , the transient dynamic response is measured by one or more sensors located at predetermined points along the beam.

Figure 1.

An elastic beam with a uniform rectangular cross-section, elastically supported and subjected to a moving load.

Based on the above hypotheses, the transverse displacement of the beam occurs in the vertical plane of symmetry. Taking into consideration the damping effect, the Euler–Bernoulli model describing the transient dynamic response of the beam following an impact is given by the following equation:

$$\rho A\frac{{\partial }^{2}w(x,t)}{\partial t^2}+c\frac{\partial w(x,t)}{\partial t}+\mathrm{EI}\frac{{\partial }^{4}w(x,t)}{\partial x^4}=f(t)\delta (x-x_p(t))$$

(1)

Where $\rho$ the mass per unit length, A is the cross-sectional area, $I$ the second moment of inertia of the beam cross-section, $c$ is the damping coefficient $f(t)$ is moving load acting on a beam at time $t$ , $x_p(t)=\mathrm{vt}$ is applied point moving loads, and $\delta (t)$ is the Dirac delta function.

The corresponding boundary conditions, defined in terms of displacement and its derivatives with respect to the spatial coordinate $x$ and time $t$ for each type of support, are as follows:

Rigid supports:

$$\begin{array}{c}w(0,t)=w(L,t)=0\hfill \\ \frac{{\partial }^{2}w}{\partial x^2}(0,t)=\frac{{\partial }^{2}w}{\partial x^2}(L,t)=0\hfill \end{array}$$

(2)

Elastic supports:

$$\begin{array}{c}{\frac{{\partial }^{2}w(x,t)}{\partial x^2}|}_{x=0}={\frac{{\partial }^{2}w(x,t)}{\partial x^2}|}_{x=L}=0\hfill \\ \mathrm{EI}{\frac{{\partial }^{3}w(x,t)}{\partial x^3}|}_{x=0}=-K_{L}w(0,t)\hfill \\ \mathrm{EI}{\frac{{\partial }^{3}w(x,t)}{\partial x^3}|}_{x=L}=K_{L}w(L,t)\hfill \end{array}$$

(3)

Expressing the transverse displacement $w(x,t)$ in modal coordinates, we have:

$$w(x,t)=\sum _{i=1}^{\infty }{q}_i(t){\phi }_i(x)$$

(4)

where ${\phi }_i(x)$ denotes the mode shape function of the $i\mathrm{th}$ mode; and $q_i(t)$ is the $i\mathrm{th}$ modal amplitude which is obtained by studying the eigenvalues and eigenfunctions mentioned in following section. Substituting equation (4) into equation (1) and multiplying by ${\phi }_i(x)$ , integrating with respect to $x$ between 0 and $L$ , and applying the orthogonality conditions, we obtain:

$$\frac{d^2{q}_i(t)}{d{t}^2}+2{\xi }_i{\omega }_i\frac{d{q}_i(t)}{\mathrm{dt}}+{\omega }_i^2{q}_i(t)=\frac{1}{M_i}f(t)\phi (x_p(t))$$

(5)

where ${\omega }_i$ is the modal frequency, ${\xi }_i$ is the damping ratio and $M_i$ is the modal mass of the $\mathrm{ith}$ mode, with:

$$M_i=\underset{0}{\overset{L}{\int }}\rho A{\phi }_i^2(x)\mathrm{dx}$$

(6)

This is a second-order linear differential equation with a second member. Given the initial conditions at rest, these can be expressed as follows:

$$w(x,0)=\frac{\mathrm{dw}(x,0)}{\mathrm{dt}}=0$$

(7)

The general solution of differential equation (5) is given by the following Duhamel integral:

$$q_i(t)=\frac{1}{M_i}{\int }_0^t{h}_i(t-\tau )f(\tau ){\phi }_i(x_l)d\tau$$

(8)

where:

$$\begin{array}{c}{h}_i(t-\tau )\hfill \\ =\{\begin{array}{cc}\frac{1}{{\omega }_i\sqrt{1-{{\xi }_i}^2}}\exp (-{\xi }_i{\omega }_{i}t)\sin \left({\omega }_i\sqrt{1-{\xi }_i^2}t\right)\hfill & \mathrm{si}\tau \ge t\hfill \\ \multicolumn{1}{c}{0}& \mathrm{si}t<\tau \hfill \end{array}\hfill \end{array}$$

(9)

The eigenvalue problem’s solution

The general expression for the eigenfunction of the Euler-Bernoulli beam is given as:

$${\phi }_i(x)=C_1\sin ({\beta }_{i}x)+C_2\cos ({\beta }_{i}x)+C_3\mathrm{sh}({\beta }_{i}x)+C_4\mathrm{ch}({\beta }_{i}x)$$

(10)

where $C_1;C_2;C_3\mathrm{and}{C}_4$ are coefficients determined from the boundary conditions for each type of support (equations (2) and (3)), and ${\beta }_i$ is the eigenvalue defined as:

$${\beta }_i^4=\frac{\rho A}{\mathrm{EI}}{{\omega }_i}^2$$

(11)

For a beam with rigid supports, the eigenvalue and eigenfunction are expressed as follows ²⁶ :

The eigenvalue is defined as:

$${\beta }_i=\frac{i\pi }{L}$$

(12)

And the eigenfunction is defined as:

$${\phi }_i(x)=\sin ({\beta }_{i}x)$$

(13)

For a beam with elastic supports, the eigenvalue and eigenfunction are expressed as follows ⁵ :

The eigenvalue is the solution of the following equation:

$$\begin{array}{c}\frac{{\beta }_i^6}{2\kappa }[\cos ({\beta }_{i}L)\mathrm{ch}({\beta }_{i}L)-1]-\kappa \sin ({\beta }_{i}L)\mathrm{sh}({\beta }_{i}L)\hfill \\ +{\beta }_i^3[\sin ({\beta }_{i}L)\mathrm{ch}({\beta }_{i}L)-\cos ({\beta }_{i}L)\mathrm{sh}({\beta }_{i}L)]=0\hfill \end{array}$$

(14)

where $\kappa =\frac{K_L}{\mathrm{EI}}$ , is the inversion of stiffness ration of the beam to the elastic bearings and the eigenfunction is written as follows:

$${\phi }_i(x)=\sin ({\beta }_{i}x)+{\zeta }_2\mathrm{sh}({\beta }_{i}x)+{\zeta }_1[\cos ({\beta }_{i}x)+\mathrm{ch}({\beta }_{i}x)]$$

(15)

where:

$$\begin{array}{c}{\zeta }_1=\frac{{\beta }_i^3[\mathrm{sh}({\beta }_{i}L)-\sin ({\beta }_{i}L)]}{2\kappa \mathrm{sh}({\beta }_{i}L)-{\beta }_i^3[\cos ({\beta }_{i}L)-\mathrm{ch}({\beta }_{i}L)]}\mathrm{and}\hfill \\ {\zeta }_2=\frac{2\kappa \sin ({\beta }_{i}L)+{{\beta }_i}^3[\cos ({\beta }_{i}L)-\mathrm{ch}({\beta }_{i}L)]}{2\kappa \mathrm{sh}({\beta }_{i}L)-{\beta }_i^3[\cos ({\beta }_{i}L)-\mathrm{ch}({\beta }_{i}L)]}\hfill \end{array}$$

Identification of the moving load

The displacement of the beam at point x and time t can be found from equations (4) to (8) as:

$$w(x,t)=\sum _{i=1}^{\infty }\{\frac{{\phi }_i(x)}{M_i}{\int }_0^t{h}_i(t-\tau )f(\tau ){\phi }_i(x_p(t))d\tau \}$$

(16)

The strain in the beam at point x and time t can be written as:

$${\epsilon }_x=-\frac{e}{2}\frac{{\partial }^{2}w(x,t)}{\partial x^2}$$

(17)

Substituting equation (17) into equation (16) and rewriting it in matrix form gives:

$$\begin{array}{c}{\epsilon }_x(x_s,m)\approx \hfill \\ -\sum _{i=1}^N\left\{\frac{e\Delta t{\phi }_i^{″}(x_s)}{2M_i}\sum _{j=0}^m[{\phi }_i(x_p(j\Delta t))f(j\Delta t)h_i((m-j)\Delta t)]\right\}\hfill \end{array}$$

(18)

with $(m=1,2,3,...,N_t)$ and $(j=0,1,2,3,...,N_t-1)$ .

where $N$ is the number of vibration mode; x_s is the location of the sensor; $N_s$ is the number of sensors; $N_t$ is the number of data points and $\Delta t=\frac{T_{\mathrm{ob}}}{N_t-1}$ is the time interval, e denotes the height of the cross-section, $M_i$ is the modal mass of the $\mathrm{ith}$ , as given in equation (6), ${\phi }_i(x)$ represents the mode shape function corresponding to the $\mathrm{ith}$ mode, as given in equation (15), and the expression of $h_i$ is given as:

$$\begin{array}{c}{h}_i[(m-j)\Delta t)]\hfill \\ =\{\begin{array}{cc}\frac{1}{{\omega }_i\sqrt{1-{\xi }_i^2}}\exp [-{\xi }_i{\omega }_i(m-j)\Delta t]\sin [{\omega }_i\sqrt{1-{\xi }_i^2}(m-j)\Delta t]\hfill & \mathrm{si}\mathrm{j}\le m\hfill \\ \multicolumn{1}{c}{0}& \mathrm{si}\mathrm{m}<j\hfill \end{array}\hfill \end{array}$$

(19)

Equation (18) can be rewritten in compact form as follows:

$$Y=\mathrm{GF}$$

(20)

where $Y$ is $N_t*N_s$ matrix representing the dynamic response in terms of deformations, $G$ is $(N_t*N_s)\times (N_t*1)$ matrix representing the transfer matrix, and $F$ is $N_t*1$ matrix representing the input signal, with the expressions of these matrices being rewritten in detailed form:

$$\begin{array}{c}Y={[\epsilon \left(\Delta t\right),\epsilon \left(2\Delta t\right),...,\epsilon \left(N_t\Delta t\right)]}^T;F={[f(0),f(1\Delta t),.....,f((N_t-1)\Delta t)]}^T,\mathrm{and}\hfill \\ G={\left[\begin{array}{cccc}{G}_{10}\hfill & 0\hfill & \cdots \hfill & 0\hfill \\ \multicolumn{1}{c}{G_{20}}& G_{21}\hfill & \cdots \hfill & 0\hfill \\ \multicolumn{1}{c}{⋮}& ⋮\hfill & ⋮\hfill & ⋮\hfill \\ \multicolumn{1}{c}{G_{N_{t}0}}& G_{N_{t}1}\hfill & \cdots \hfill & G_{N_t{N}_t-1}\hfill \end{array}\right]}_{(N_s*N_t)\times (1*N_t)},G_{\mathrm{mj}}={\left[\begin{array}{c}{g}_{11}\hfill \\ \multicolumn{1}{c}{g_{21}}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{g_{N_{s}1}}\end{array}\right]}_{N_s\times 1}\hfill \end{array}$$

Where $g_{s1}$ expression can be defined as follows:

$$\begin{array}{c}{g}_{s1}=-\sum _{i=1}^N\left\{\frac{e\Delta t{\varphi }_i^{″}(x_s)}{2M_i}{\phi }_i({\hat{x}}_p(j\Delta t))h_i((m-j)\Delta t)\right\},\hfill \\ (s=1,2,3,...,N_s)\hfill \end{array}$$

(21)

Regularization

The transfer matrix G associated with the moving load acting on a beam is ill-conditioned, so we use Tikhonov’s method coupled with singular value decomposition (SVD) ¹⁹ to regularize it. This reformulates the ill-posed problem as a constrained optimization problem expressed as:

$$\underset{F}{\min }(||\mathrm{GF}-Y|{|}^2+\lambda ||F|{|}^2)$$

(22)

Where $\lambda$ is the regularization parameter that balances the stability of the solution.

By differentiating equation (22) and applying the optimality condition, which consists of setting this derivative to zero, we obtain the modified normal equation. This equation provides a mathematically rigorous framework for solution stabilization and is written as follows:

$$(G^{T}G+\lambda I)F=G^{T}Y$$

(23)

The technique employs the singular value decomposition (SVD) of the transfer matrix, expressed as $G=U\Sigma V^T$ where $U$ and $V$ are orthogonal matrices and the singular values are ordered in descending order in the matrix $\Sigma$ . This decomposition allows the regularized solution to be stated as:

$$F_{\mathrm{reg}}=\sum _{i=1}^k\frac{{\sigma }_i}{{\sigma }_i^2+\lambda }(u_i^T\epsilon )v_i$$

(24)

The filtering factors, $f_i=\frac{{\sigma }_i}{{\sigma }_i^2+\lambda }$ , play a crucial role in minimizing the effect of small singular values, which may influence the stability of the solution. The effectiveness of this regularization method essentially depends on the precise selection of the regularization parameter $\lambda$ .

In this article, we will examine two selection methods, $\lambda$ under the effect of digital noise, in order to choose the method that gives the best possible results in future studies.

The first is direct minimization of the mean square error called the S-curve error, ²⁷ based on a comparison between the regularized force and the actual force applied, which allows us to plot an error curve S(λ) for different values of λ. This represents the relative error between the actual force and the regularized force, calculated as follows:

$$S(\lambda )=\frac{||F_{\mathrm{reg}}(\lambda )-F_{\mathrm{real}}||}{||F_{\mathrm{real}}||}$$

(25)

this method is limited by the requirement to know the actual values, which are often inaccessible in practical applications. The second, called generalized cross-validation (GCV) criterion, ²⁸ based on statistical considerations and on the qualities of the solution prediction in relation to the data. A good value of the regularization parameter should predict missing values. This predictive method seeks to minimize the mean square error of estimation. The GCV function is defined as follows:

$$\mathrm{GCV}(\lambda )=\frac{\parallel Y-\mathrm{GV}{\Sigma }_{\lambda }^{-1}{U}^{T}Y{\parallel }^2}{{(n-\sum _{i=1}^r\frac{{\sigma }_i^2}{{\sigma }_i^2+{\lambda }^2})}^2}$$

(26)

Where $n$ is number of observations and ${\Sigma }_{\lambda }^{-1}$ is the diagonal matrix defined as ${\Sigma }_{\lambda }^{-1}=\mathrm{diag}\left(\frac{{\sigma }_i}{{\sigma }_i^2+{\lambda }^2}\right)$ . The optimal λ value selection is the $min\left(\mathrm{GCV}\left(\lambda \right)\right)$ .

Simulation and results

Numerical verification of the proposed approach was performed on an aluminum beam having the following parameters: length $L=0.9\mathrm{m}$ , rectangular cross-section with width $b=0.02\mathrm{m}$ and height $e=0.02\mathrm{m}$ ; elastic supports having stiffness $K_L=1000\mathrm{N}/\mathrm{m}$ . The material properties of the aluminum beam were given as: Young’s modulus $E=70\mathrm{GPa}$ , Poisson’s ratio $\upsilon =0.33$ , mass density $\rho =2800\mathrm{kg}/3$ , and a modal damping ratio ${\mathrm{\xi }}_{\mathrm{i}}=0.02$ applicable to all modes. The simulation time was set to $T_{\mathrm{ob}}=0.5\mathrm{s}$ and Δt $=0.001\mathrm{s}$ . To evaluate the influence of support elasticity, the identical geometric and material properties of a beam with rigid boundary conditions were utilized in a comparative analysis with a beam that was supported by elastic springs. This comparison demonstrates the influence of elastic supports on the reconstruction of the applied load. In this study, we focus on the case of a moving load with a speed of $v=2{\mathrm{ms}}^{-1}$ . His expression, that is, the subject of this paper is as follows:

$$f(t)=250(1+0.1\sin (70\pi \mathrm{t})+0.05\sin (120\pi \mathrm{t}))$$

(27)

Direct problem analysis

Analysis of the frequency response function (FRF) enables us to assess the impact of boundary conditions, such as rigid or elastic supports, on the dynamic behavior of a structure. Figure 2 shows that, in the case of rigid supports, the response is dominated by a marked resonance peak around 56 Hz, concentrating vibratory energy at low frequencies. Conversely, for elastic supports, a second major peak appears around 350 Hz, revealing the effect of the support’s flexibility, which introduces higher natural frequencies and a broader response. This comparison shows that the nature of the support influences not only resonance amplitudes but also the distribution of vibratory modes in the frequency spectrum. It should be noted that the first resonance frequency, in the case of an elastically supported system, decreases significantly to 7 Hz. The Table 1 presents the different frequency values for the first 10 modes, whether for a rigid support or an elastic support.

Figure 2.

Frequency response functions for rigid and elastic supports. (a) Rigid supports case, (b) elastic supports case.

Table 1.

The frequencies of the first 10 modes.

Modes	Frequency (Hz)
	Rigid supports	Elastic supports
1	56	7
2	224	128
3	504	350
4	896	686
5	1400	1134
6	2015	1693
7	2743	2365
8	3583	3149
9	4534	4045
10	5598	5052

Effect of noise

The effect of noise on the measured response can be written in equation (28), which represents the proportionality of the standard deviation of the calculated response. To evaluate the sensitivity of the regularization method to noise, white noise is added to the calculated beam responses to simulate the contaminated measurements, and 0.5%, 1%, 3%, and 5% noise levels are examined separately:

$$Y=Y_{\mathrm{calculated}}+E_p\sigma (Y_{\mathrm{calculated}})$$

(28)

where $Y$ signifies the strain and $E_p$ denotes the noise level. The noise is characterized as a standard normal distribution vector, exhibiting a mean of zero and a standard deviation of one. $Y_{\mathrm{calculated}}$ indicates the computed strain, whereas $\sigma (Y_{\mathrm{calculated}})$ denotes the corresponding standard deviations. The sensor is positioned at the middle of the beam ( $x_s=0.45\mathrm{m}$ ), and the first seven modes are included.

The quality of regularization is assessed by calculating relative errors as follows ²⁴ :

$$\mathrm{RPE}=\frac{||F_{\mathrm{reg}}-F_{\mathrm{real}}||}{||F_{\mathrm{real}}||}\times 100$$

(29)

Table presents the results of a comparative study of S-curve and GCV methods for estimating the optimal regularization parameter ( ${\lambda }_{\mathrm{optimal}}$ ) and relative percentage error (RPE) at different noise levels in rigid and elastic support configurations. Regarding the estimation of ${\lambda }_{\mathrm{optimal}}$ , the GCV method offers results close to those of the S-curve method in the case of rigid supports, with acceptable deviations despite the presence of noise. In contrast, for elastic supports, the GCV method proves more sensitive to increasing noise levels, while the S-curve method maintains better stability under these conditions. Thus, this study will favor the use of the S-curve method for estimating the optimal regularization parameter. Concerning the effect of noise on the relative percentage error, it is observed that the RPE increases with noise levels in both configurations, which is consistent with theoretical expectations that an increase in noise leads to a reduction in the precision of applied force reconstruction (Table 2).

Table 2.

The values of optimal regularization parameter and the relative errors RPE for each noise level by the S-curve method and the GCV method for both types of supports.

Level noise (%)	Rigid supports case				Elastic supports case
	S-curve method		GCV method		S-curve method		GCV method
	${\mathrm{\lambda }}_{\mathrm{optimal}}$	$RPE(\%)$	${\mathrm{\lambda }}_{\mathrm{optimal}}$	$RPE(\%)$	${\mathrm{\lambda }}_{\mathrm{optimal}}$	$RPE(\%)$	${\mathrm{\lambda }}_{\mathrm{optimal}}$	$RPE(\%)$
0	6.88 × 10−12	4.88	1.00 × 10−12	4.88	2.17 × 10−10	0.00	1.00 × 10−10	0.00
0.50	1.53 × 10−4	5.21	1.00 × 10−4	5.24	2.15 × 10−3	4.31	1.00 × 10−3	4.9
1	2.32 × 10−4	5.88	1.00 × 10−4	6.15	4.10 × 10−3	5.55	1.44 × 10−3	7.33
3	4.63 × 10−4	9.51	2.09 × 10−5	10.51	9.77 × 10−3	8.11	2.55 × 10−3	14.28
5	9.25 × 10−4	13.05	5.65 × 10−4	31.31	1.31 × 10−3	9.80	3,17 × 10−3	20.24

However, the RPE is systematically higher in the case of elastic supports compared to rigid supports, despite a better overall quality of force identification in the rigid configuration, as shown in the Figures 3 and 4. This disparity can be attributed to differences in initial boundary conditions. Indeed, for rigid supports, the initial deformation at time is zero, while the initial applied force is non-zero, which begins force identification from zero. In order to evaluate the effectiveness of the proposed approach on the quality of impact force reconstruction, three regularization methods were selected and applied to identify the impact force. This comparison aims to highlight the respective performances of each method within the context of the considered inverse problem. To better clarify the methodological choice of coupling Tikhonov with SVD, Table 3 presents the performance evolution of each method across different noise levels.

Figure 3.

Identification of the moving force in rigid supports case with a noise level: (a) 0%, (b) 1%.

Figure 4.

Identification of the moving force in elastic supports case with a noise level: (a) 0%, (b) 1%.

Table 3.

RPE for different regularization methods under varying noise levels in rigid and elastic support configurations.

Level noise (%)	Rigid supports case			Elastic supports case
	Tikhonov-SVD (%)	Tikhonov (%)	TGSVD (%)	Tikhonov-SVD (%)	Tikhonov (%)	TGSVD (%)
0	4.88	3.39	12.01	0.00	1.82	4.91
0.5	5.21	6.85	12.09	4.31	5.39	5.94
1	5.88	10.27	12.32	5.55	6.45	6.40
3	9.51	18.76	14.67	8.11	11.72	9.27
5	13.05	24.91	18.45	9.80	16.79	10.02

RPE: relative percentage error.

Based on the results presented in Table 3, it is clear that the Tikhonov method combined with SVD shows a significant advantage over standalone Tikhonov and TGSVD approaches. Even at a relatively high noise level of around 5%, the method proposed in this study maintains an acceptable reconstruction quality, with a relative error of 9.80% for the elastic support and 13.05% for the rigid support. Consequently, the Tikhonov method combined with SVD is adopted for the remainder of this work.

In contrast, for elastic supports, the initial deformation is non-zero and measurable, which improves initial identification precision but leads to an overall elevation of the RPE. Despite this, the rigid configuration maintains a better precision in force identification, as illustrated in Figure 5, which compares the differences between real and identified forces for both configurations. These results highlight the influence of boundary conditions and initial dynamics strain on the quality of impact force identification and relative error behavior. It is worth noting that the stability of the force generated by the train passage after 0.45 s is due to the fact that the time required to cross the structure is 0.45 s. The relationship that allows visualizing this effect is $t=L/v=0.45s$ .

Figure 5.

The difference between the real force and the force identified with the 1% noise level. (a) Rigid supports case, (b) elastic supports case.

Effect of modal analysis

In this study, we aim to examine the effect of mode number truncation on the improvement in regularization quality for both beam cases. The same moving force given in equation (27) is applied, and the sensor is placed at the same position ( $x_s=0.45m$ ). The number of modes is varied from 3 to 15, and the relative error RPE is recorded for each level of study.

The Figure 6 shows the variation in relative error (RPE) as a function of the number of modes tested for each noise level. In the case of rigid supports, there is a significant decrease in relative errors (RPE) as the number of modes increases, with errors stabilizing from 10 to 12 modes. However, errors remain higher at higher noise levels (5%), and gradually decrease as noise is reduced. This highlights an improvement in reconstruction accuracy and stability as soon as a sufficient number of modes are taken into account, beyond 12.

Figure 6.

Evolution of relative errors as a function of truncation modes. (a) Rigid supports case, (b) elastic supports case.

On the other hand, for elastic supports, although the trend is similar, errors are generally lower when the number of modes is low. Errors stabilize at 8–10 modes. Noise levels also influence errors in this case, but their effect seems somewhat less pronounced than that observed for rigid supports. Figures 7 to 10 illustrate the optimal force reconstructions for varying noise levels in both elastic and rigid support configurations. Rigid supports, while requiring a higher number of modes to attain comparable accuracy, exhibit greater overall robustness in reconstruction. Elastic supports, by contrast, display slightly increased sensitivity to noise at higher mode counts but achieve satisfactory performance with a relatively smaller number of modes.

Figure 7.

Optimal modes for force reconstruction with a noise levels of 0.5%. (a) Rigid supports case, (b) elastic supports case.

Figure 8.

Optimal modes for force reconstruction with a noise level of 1%. (a) Rigid supports case, (b) elastic supports case.

Figure 9.

Optimal modes for force reconstruction with a noise levels of 3%. (a) Rigid supports case, (b) elastic supports case.

Figure 10.

Optimal modes for force reconstruction with a noise levels of 5%. (a) Rigid supports case, (b) elastic supports case.

Effect of sensor location analysis

In this test, we examine the effect of the position of the sensor on the regularization of the moving load in the two beam cases. Applying the same moving force and taking the first 15 modes for the rigid supports case and 13 modes for the elastic support case, we vary the location of the $x_s$ sensor along the length of the beam, compare the variation in errors, and discover the best position for the transducer, should there be any effects.

The two graphs presented, corresponding, respectively, to the case of rigid supports (Figure 11(a)) and the case of elastic supports (Figure 11(b)), illustrate the evolution of relative errors (RPE (%)) as a function of the normalized sensor position $x_s/L$ , under the influence of different noise levels (0.5%, 1%, 3%, and 5%). In the rigid case, the errors increase significantly for high noise levels, particularly 5%, where the peaks reach up to 38%, indicating a strong sensitivity to the sensor position. In contrast, for low noise levels (0.5%), the errors remain generally stable, oscillating around 5%, with a slight tendency to decrease for specific positions (e.g. around $x_s/L=0.5$ ). For elastic supports, the errors are generally lower and better controlled, with a notable reduction for positions around $x_s/L=0.3$ , particularly for low to moderate noise levels. However, the errors increase slightly as the sensor approaches the ends. These results highlight a better robustness of the elastic case against noisy disturbances, compared to the rigid case, where the sensor’s location plays a predominant role in the accuracy of the reconstruction.

Figure 11.

Evolution of RPE as a function of the sensor position parameter. (a) Rigid supports case, (b) elastic supports case.

Effect of number of measuring points

The number of measurement points is selected in turn as follows: 1, 2, 3, and 4, respectively. The measurement points are evenly distributed along the beam. The following Table 4 presents the relative errors based on the chosen measurement points for each noise level.

Table 4.

The relative error values for each noise level at different measurement points for the two types of supports.

Number of sensors	Sensors location	RPE (%)
		Rigid supports case				Elastic supports case
		0.5%	1%	3%	5%	0.5%	1%	3%	5%
1	L/2	4.96	5.19	7.12	9.71	4.59	5.30	7,57	9.38
2	L/3–2L/3	4.98	5.27	7.48	9.90	1.88	3.44	6.70	8.59
3	L/4–L/2–3L/4	4.91	4.99	5.72	6.84	2.01	3.87	6.09	7.79
4	L/5–2L/5–3L/5–4L/5	4.90	4.95	5.48	6.40	1.50	2.99	6.04	7.76

The results show that as the number of measurements increases, the relative error (RPE) decreases. Therefore, we conclude that adding sensors significantly improves accuracy, as with four measurement points the error stabilizes, even in the presence of noise, which indicates the importance of adequate coverage to improve identification for both cases of supports. Figures 12 to 15 show the optimal results for identifying the moving load with four measurement points for each type of noise for both cases of supports. As we can see, there is a significant improvement in the quality of identification when compared to the results obtained in Figures 7 to 10, which correspond to a single measurement point.

Figure 12.

Optimal number of measuring points for force reconstruction with a noise levels of 0.5%. (a) Rigid supports case, (b) elastic supports case.

Figure 13.

Optimal number of measuring points for force reconstruction with a noise levels of 1%. (a) Rigid supports case, (b) elastic supports case.

Figure 14.

Optimal number of measuring points for force reconstruction with a noise levels of 3%. (a) Rigid supports case, (b) elastic supports case.

Figure 15.

Optimal number of measuring points for force reconstruction with a noise levels of 5%. (a) Rigid supports case, (b) elastic supports case.

The results shown in the figures above illustrate the effect of the number of sensors on the quality of the reconstruction of the impact force history, which takes the form of a moving force. The stability ensured by using four sensors at a time is due to the quality of the measurements and their coverage of the entire beam. Additionally, when the noise level exceeds 3%, the quality of the reconstruction begins to deteriorate, especially in the case of an elastic support.

Influence of excitation frequency

In this section, we aim to study the effect of the excitation frequency of the moving force on the regularization of the moving load, applied over the entire length of the beam. The sensor is located in the middle of the beam ( $x_s=0.45m$ ) and the first 15 modes are considered for the case of rigid supports, and for the case of elastic supports, the sensor is located at $x_s=0.27m$ and 13 modes are considered. In order to have the optimal identifications studied in the previous sections for each type of support. The expression of the moving load with a speed of $v=2m{s}^{-1}$ is as follows:

$$f(t)=250(1+0.1\sin (2{\mathrm{f}}_{\mathrm{ext}1}\pi \mathrm{t})+0.05\sin ({\mathrm{f}}_{\mathrm{ext}1}\pi \mathrm{t}))$$

(30)

Where $f_{\mathrm{ext}1}$ is the excitation frequency of the moving force.

Figure 16 shows the variation of the RPE as a function of the excitation frequency. The results show that, for the rigid case, the variation of relative errors as a function of excitation frequencies is generally stable (the variations are small), but at the frequency $f_{\mathrm{ext}1}=60\mathrm{Hz}$ , the RPE shows a significant peak, particularly at high noise levels, indicating increased sensitivity at this frequency. On the other hand, in the elastic case, the same observation applies: there is a slight and unremarkable variation without a peak, as in the rigid case.

Figure 16.

Evolution of the RPE as a function of the excitation frequency. (a) Rigid supports case, (b) elastic supports case.

Effect of spring stiffness

We have discussed the boundary conditions for both a rigid support and an elastic support with a moderate stiffness of 1000 N/m. Next, we attempted to reduce the stiffness to a value of 1 N/m, which is close to a free edge in reality. In this context, we aim to demonstrate that, for an acceptable reconstruction, the stiffness must be greater than a certain threshold. It should be noted that for very low stiffness values, the reconstruction quality is poor, especially when the signal is noisy and the noise level is low. Figure 17 shows the evolution of the impact force as a function of time for a noise level of 0.5%.

Figure 17.

Reconstruction of impact force as a function of time with a noise level of 0.5%.

To illustrate the overall effect of the stiffness of the elastic support on the quality of impact force identification, the relative error shown in Figure 18 reaches its maximum for low stiffness, particularly for an elastic support with a stiffness of 1 N/m. This error begins to decrease as the stiffness increases and stabilizes once the stiffness becomes large enough for the support to behave as rigid.

Figure 18.

Evolution of the RPE as a function of stiffness with a noise level varied from 0.5% to 5%.

Conclusions

The identification problem of impact force is an essential approach for monitoring the health of structures. Generally, this inverse problem is considered ill-posed, which has led to the use of regularization methods to stabilize the measurements obtained during the inversion of the matrix system. In this context, the selected regularization methods Tikhonov, TGSVD, and Tikhonov-SVD were tested against several parameters in order to select the most suitable one. First, sensitivity to noise was analyzed. Noise, as a disturbing factor, is crucial for testing the robustness of the regularization methods. Compared to the results obtained with standard Tikhonov and TGSVD, the Tikhonov-SVD regularization proves to be more applicable to moving load identification problems. An optimization of the $\mathrm{\lambda }$ parameter was performed, and the S-curve method showed advantages over GCV. In this work, the sensitivity of the moving load identification approach has been analyzed with respect to the sensor positioning, modal analysis, and excitation frequency, each under the influence of noise. The study also takes into account the boundary conditions and their influence on the quality of the reconstruction of the impact force history. A comparison incorporating these different elements shows that the effect of elastic supports on the quality of the reconstruction is particularly significant, as they play an important role in energy absorption. The regularization method employed proved to be very effective in addressing this problem. However, it is important to test the effectiveness of this regularization method under nonlinear boundary conditions.