Research on rapid identification of bridge damping ratio in vehicle-bridge coupling based on mode shape

Abstract

Based on the fundamental theory of vehicle-bridge coupling vibration, this study proposes a novel method for identifying the bridge damping ratio from the vehicle-bridge interaction by using the bridge’s mode shape. This study conducts theoretical derivation via indirect measurement techniques and performs numerical simulation analysis using a finite element model built in MATLAB. First, acceleration sensors are used to measure the bridge response caused by a single-axle vehicle crossing; subsequently, band-pass filtering is employed to extract the vibration component corresponding to the bridge’s first-order frequency, and the Hilbert transform is used to identify the first-order mode shape curve. Ultimately, the bridge damping ratio is rapidly identified by leveraging the analytical relationship between the damping and the location of the mode shape’s peak. The effectiveness of the method is verified by theoretical derivations and numerical simulations, with a relative error of only 2% under the baseline condition. More importantly, parametric analysis shows that the method has strong robustness, with the relative identification error remaining within 5% even at a higher vehicle speed of 8 m/s and in a strong noise environment of 20 dB. Compared with traditional indirect methods that rely on iterative calculations, the method proposed in this study provides a more efficient, direct analytical identification approach, offering a novel idea for structural health monitoring based on indirect measurement.

Introduction

Bridges, as critical components of transportation networks, bear the responsibility of connecting different regions. Their safety and stability are vital for ensuring smooth traffic flow and promoting economic development, thus rendering their integrity of paramount significance. Consequently, structural health monitoring (SHM) of bridges has received widespread attention in the engineering field.^1–4,5,6 Within the bridge health monitoring system, the accurate identification of modal parameters (natural frequency, damping ratio, and mode shape) is a key step in assessing the structural condition of bridges. Among these, the damping ratio of bridges, as an essential component of modal parameters, plays a crucial role in evaluating the energy dissipation capacity and impact characteristics of bridge structures. Therefore, research and identification of bridge damping ratios are of great importance for assessing the safety condition of bridge structures. ⁷

Traditional methods for obtaining bridge damping ratios mainly rely on collecting and analyzing vibration response data through sensors installed directly on the bridge. ⁸ Li et al. ⁹ through a combination of numerical and experimental approaches, explored the effectiveness and reliability of modal updating methods and vibration measurement data for damping ratio identification, discovering that the method is sensitive to model imperfections but still capable of accurately identifying damping ratios. Kim et al. ¹⁰ analyzed field monitoring system data, employing natural excitation techniques (NExT) combined with eigenvalue realization algorithms (ERA) for modal damping ratio identification. Li et al. ¹¹ evaluated the long-term monitoring data from large-scale structural health monitoring systems, employing Hilbert transform techniques to assess the damping ratio and other parameters of bridge structures. Shang et al. ¹² proposed a time-series decomposition approach for field data, using U-type network models to effectively extract the free decay of bridge acceleration time series under temperature-dependent damping ratio changes, providing new methods and insights for assessing bridge health conditions. However, the reliance on direct sensor installation for data collection presents challenges such as environmental interference, potential damage to sensors, high labor costs, and traffic disruptions, which limit the development of bridge health monitoring systems to some extent. ¹³

To address the above issues, Yang et al. ¹⁴ proposed the Vehicle Scanning Method (VSM) under vehicle-bridge coupling effects. This method requires only a small number of sensors mounted on the vehicle to conduct bridge testing, eliminating the need for sensor installation on the bridge itself. This method offers advantages such as flexibility, cost-effectiveness, and high efficiency, effectively overcoming some of the limitations associated with direct measurement techniques. Based on this, McGetrick et al. ¹⁵ proposed a method using mobile accelerometers installed on vehicles to extract bridge dynamic parameters by analyzing vehicle frequency spectra, providing a cost-effective and efficient solution for bridge health monitoring. Keenanhan et al. ¹⁶ proposed a bridge damping ratio monitoring method based on a car-trailer dynamic system, analyzing the axle acceleration differences between the car and trailer to overcome the influence of road roughness, thereby effectively detecting bridge damping ratio changes and demonstrating the method’s potential as a tool for bridge damage detection. Yang et al. ¹⁷ utilized a single-degree-of-freedom test vehicle and, for the first time, proposed a method for simultaneously extracting the first few self-resonance frequencies and damping ratios of a bridge. This method combines Variational Mode Decomposition (VMD) and Random Decrement Technique (RDT), providing a convenient and cost-effective indirect measurement approach for bridge health monitoring. Tan et al. ¹⁸ proposed a calculation method based on the Hilbert transform, using vehicle-bridge interactions’ acceleration responses to extract bridge modal parameters, including damping ratios and mode shapes, offering a low-cost solution for bridge health monitoring. González et al. ¹⁹ proposed a six-step process for identifying bridge damping ratios, which effectively filters out noise and validates the feasibility of the method through numerical simulations and experimental data. Yang et al. ²⁰ proposed a method for identifying bridge damping ratios using the vertical response signals of dual-axle vehicles, leveraging the frequency domain properties of their corresponding decaying signals to extract the bridge’s damping ratios. Zhang ²¹ utilized a vehicle-trailer system to achieve the simultaneous evaluation of the three key bridge parameters by performing curve fitting on the amplitude spectrum of the residual response. In terms of signal processing, advanced decomposition algorithms have been applied. Li et al. ²² employed the successive variational mode decomposition (SVMD) technique combined with the Random Decrement Technique (RDT) to successfully extract the modal frequencies and damping ratios of a cable-stayed bridge from vehicle responses, with the method being field-tested. He et al. ²³ proposed using the residual responses from a three-connected vehicle system to enhance bridge modal parameter identification; their method combines techniques such as Variational Mode Decomposition (VMD) and Singular Value Decomposition (SVD) to estimate damping ratios and reconstruct the complete bridge mode shapes. Yang et al. ²⁴ utilized the Hilbert Transform (HT) technique to derive a bridge damping formula from the instantaneous amplitudes of the vehicle’s front and rear contact points. Meanwhile, researchers have also recognized that bridge damping is a key factor affecting the accuracy of mode shape identification. To address the distortion issue caused by damping, Xu et al. ²⁵ proposed a novel recursive formula that utilizes a two-axle vehicle to eliminate the distortion effect of bridge damping on mode shape identification. Additionally, Demirlioglu and Erduran ²⁶ proposed a novel reference-based scaling method to improve the accuracy of mode shape identification.

Currently, more researchers are exploring rapid and efficient bridge indirect measurement techniques to meet increasing interest in identifying bridge modal parameters.^27–30 To improve identification accuracy and the level of automation, more advanced indirect identification techniques have emerged. For instance, combined methods based on Variational Mode Decomposition (VMD) and the Random Decrement Technique (RDT) can effectively extract modal parameters, but their identification accuracy depends to some extent on the parameter selection of VMD and the multiple independent samples required by RDT. ¹⁷ Meanwhile, some advanced methods based on Bayesian inference, ¹¹ while theoretically rigorous and capable of quantifying the uncertainty of the results, typically involve complex iterative sampling and are computationally expensive, which contradicts the engineering goal of rapid and efficient detection. Most existing vehicle scanning methods still rely on complex signal processing procedures or iterative calculations to approximate the true damping ratio, lacking a direct analytical identification pathway based on a clear physical phenomenon.^18,19

To address the challenge of balancing identification accuracy, computational efficiency, and on-site applicability in the aforementioned methods, this paper proposes a new method for identifying the bridge damping ratio based on a moving test vehicle. This method is systematically studied through theoretical derivation and numerical simulation analysis. The study demonstrates that the proposed method facilitates the rapid and efficient identification of the bridge damping ratio with minimal human intervention.

The research content of this paper is as follows: In the “Theoretical foundation” section, the vehicle-bridge coupling theory is derived, and the Hilbert transform is used to identify the first-order mode shape of the bridge, demonstrating the relationship between the bridge damping ratio and the location of the maximum value of the first-order mode shape. This initially validates the feasibility of the proposed method. In the “Numerical example” section, numerical examples are used to illustrate the detailed workflow of the proposed method, verifying the correctness of the theoretical formula presented in the “Theoretical foundation” section. In the “Parametric study” section, various influencing parameters are analyzed, and a comparative analysis of different damping ratio identification methods is conducted to demonstrate the effectiveness of the proposed method further. Finally, the “Conclusions and Prospects” section presents the conclusions of this paper.

Theoretical foundation

Theory of vehicle–bridge interaction

Based on the vehicle-bridge coupling theory, ³¹ the bridge is simplified as an Euler–Bernoulli beam, and the vehicle is modeled as a system with concentrated mass and spring stiffness. For the initial derivation, the damping of the test vehicle is assumed to be negligible. The test vehicle passes over the bridge at a constant speed $v$ , exciting the bridge vibration and generating a response. The vertical displacement response of the test vehicle is denoted as $y_v$ , the vertical displacement response at the vehicle-bridge contact point is denoted as $u_c$ , and the vertical displacement response of the beam vibration is denoted as $u_b$ . The relationship between the bridge model and the vehicle model is summarized in Table 1, and the simplified vehicle-bridge coupling model is illustrated in Figure 1.

Table 1.

Technical terms and parameters of the bridge and vehicle model.

Model	Technical term	Symbol
Bridge model	Bridge length	$L$
	Mass per unit length	$\overline{m}$
	Flexural rigidity	$EI$
	Damping coefficient	$c$
Vehicle model	Mass	$m_v$
	Spring stiffness	$k_v$

Figure 1.

Simplified axle coupling model.

Without considering the damping of the test vehicle, the equations of motion for the vehicle and the bridge are expressed as follows

$m_v{\ddot{y}}_v+k_v\left(y_v-u_c\right)=0$

(1)

$\overline{m}{\ddot{u}}_b+c{\dot{u}}_b+{EI{u}_b}^{″″}=\left[k_v\left(y_v-u_c\right)-m_{v}g\right]\delta \left(x-vt\right)$

(2)

In these equations, ( $¨$ ) represents the second derivative with respect to time $t$ , while ( $“”$ ) represents the fourth derivative with respect to position x. g is the gravitational acceleration, and $\delta$ is a delta function, with its arguments preceding it. It is important to note that the vertical displacement response of the vehicle is calculated from the equilibrium position of the vehicle body.

According to the mode superposition principle, ³² the vertical displacement response of the bridge can be expressed as

$u_b={\displaystyle \sum }_n\left[q_{bn}\left(t\right)\sin \left(\frac{n\pi x}{L}\right)\right]$

(3)

Here, $q_{bn}\left(t\right)$ represents the modal coordinate, and ${\varphi }_n\left(x\right)=\sin \left(\frac{n\pi x}{L}\right)$ represents the vibration mode of the bridge. This indicates that the vertical displacement response of the bridge can be described as a series of modal coordinates and mode shapes.

Based on the parameters and structural dynamics-related theory, combining equations (1)–(3), and referring to the literature, ¹⁸ the differential equations can be solved to obtain the vertical displacement response $y_v\left(t\right)$ of the test vehicle. By taking the second derivative with respect to time $t$ , the vertical acceleration response of the test vehicle can be derived as

${\ddot{y}}_{\nu }\left(t\right)={\displaystyle \sum }_n\frac{{\Delta }_{stn}}{2\sqrt{{\left(1-S_n^2\right)}^2+4S_n^2{\xi }_n^2}}\times \{A_{1,n}\hspace{0.17em}\cos \left(\frac{2n\pi \nu t}{L}-\alpha \right)+A_{2,n}\hspace{0.17em}\cos \left({\omega }_{v}t+\alpha \right)+A_{3,n}\hspace{0.17em}\cos \left({\omega }_{v}t-\alpha \right)+e^{-{\xi }_n{\omega }_{bn}t}\left[A_{4,n}\hspace{0.17em}\cos \left({\omega }_{v}t+\beta \right)+A_{5,n}\hspace{0.17em}\sin \left({\omega }_{v}t+\beta \right)+A_{6,n}\hspace{0.17em}\cos \left({\omega }_{bn}t+\frac{n\pi \nu t}{L}-\beta \right)+A_{7,n}\hspace{0.17em}\cos \left({\omega }_{bn}t-\frac{n\mathrm{\pi }vt}{L}-\beta \right)+A_{8,n}\hspace{0.17em}\sin \left({\omega }_{bn}t+\frac{n\mathrm{\pi }vt}{L}-\beta \right)+A_{9,n}\hspace{0.17em}\sin \left({\omega }_{bn}t-\frac{n\mathbf{\pi }vt}{L}-\beta \right)\right]\}$

(4)

In the formulas, ${\xi }_n$ is the damping ratio of the bridge, and ${\omega }_{bn}$ and ${\omega }_v$ are the $n$ th natural frequency of the bridge and the vehicle frequency of the test vehicle, respectively. The vertical acceleration response of the test vehicle, ${\ddot{y}}_{\nu }\left(t\right)$ , includes the vehicle’s own vibration frequency component, the driving frequency component of $2n\pi v/L$ caused by the vehicle’s motion, and the bridge’s own vibration component. Among these, the free vibration component of the bridge is the key for identification; it not only contains the bridge’s damping information which decays over time, but its frequency also undergoes a Doppler shift ( ${\omega }_{bn}\pm n\pi v/L$ ) due to the vehicle’s motion, which is precisely the physical basis for how the vehicle can “scan” the bridge’s information. The expressions for the parameters ${\hspace{0.17em}\Delta }_{stn}$ , $\alpha$ , and $\beta$ are as follows

${\Delta }_{stn}=-\frac{2m_{v}g{L}^3}{EI{n}^4{\pi }^4}$

(5)

$\alpha ={\tan }^{-1}\left(\frac{2S_n{\xi }_n}{1-S_n^2}\right)$

(6)

$\beta ={\tan }^{-1}\left(\frac{2{\xi }_n\sqrt{1-{\xi }_n^2}}{1-S_n^2-2{\xi }_n^2}\right)$

(7)

$S_n=\frac{n\pi v}{L{\omega }_{bn}}$

(8)

Note that in the parameters of equation (4), $A_{6,n}$ , $A_{7,n}$ , $A_{8,n}$ , and $A_{9,n}$ are the amplitude coefficients corresponding to the $n$ th-order frequency vibration component of the bridge. These coefficients are crucial for the subsequent identification of the bridge mode shape using the Hilbert transform, as they determine the magnitude of the bridge’s dynamic response that is coupled into the vehicle’s response. Their values are influenced by the vehicle-bridge frequency ratio $u_n$ , the bridge damping ratio ${\xi }_n$ , and the speed parameter $S_n$ , clearly reflecting the dynamic interaction between the vehicle and the bridge. Here, the coefficients $A_{1,n}$ to $A_{5,n}$ correspond to the components of the vehicle’s own vibration and the forced vibration induced by the driving motion. As the core of this study is to extract the bridge’s free vibration response via band-pass filtering, these components are filtered out in the subsequent signal processing, and thus their detailed expressions are not presented. The expressions for coefficients $A_{6,n}$ to $A_{9,n}$ are as follows

$A_{6,n}=\frac{-S_n{\omega }_{bn}^2}{2\sqrt{1-{\xi }_n^2}}\{\frac{{\mu }_n{\left(1+S_n\right)}^3+{\xi }_n^2{\mu }_n\left(1+S_n\right)+{\left(1+S_n\right)}^2-{\xi }_n^2}{\left[{\left({\mu }_n\left(1-S_n\right)+1\right)}^2+{\xi }_n^2{\mu }_n^2\right]}+\frac{-{\mu }_n{\left(1+S_n\right)}^3-{\xi }_n^2{\mu }_n\left(1+S_n\right)+{\left(1+S_n\right)}^2-{\xi }_n^2}{\left[{\left({\mu }_n\left(1+S_n\right)-1\right)}^2+{\xi }_n^2{\mu }_n^2\right]}\}$

(9)

$A_{7,n}=\frac{-S_n{\omega }_{bn}^2}{2\sqrt{1-{\xi }_n^2}}\{\frac{-{\mu }_n{\left(1-S_n\right)}^3-{\xi }_n^2{\mu }_n\left(1-S_n\right)-{\left(1-S_n\right)}^2+{\xi }_n^2}{\left[{\left({\mu }_n\left(1-S_n\right)+1\right)}^2+{\xi }_n^2{\mu }_n^2\right]}+\frac{{\mu }_n{\left(1-S_n\right)}^3+{\xi }_n^2{\mu }_n\left(1-S_n\right)-{\left(1-S_n\right)}^2+{\xi }_n^2}{\left[{\left({\mu }_n\left(1-S_n\right)-1\right)}^2+{\xi }_n^2{\mu }_n^2\right]}\}$

(10)

$A_{8,n}=\frac{-S_n{\omega }_{bn}^2{\xi }_n^2}{2\sqrt{1-{\xi }_n^2}}\{\frac{-{\mu }_n{\left(1+S_n\right)}^2-{\xi }_n^2{\mu }_n-2\left(1+S_n\right)}{\left[{\left({\mu }_n\left(1-S_n\right)+1\right)}^2+{\xi }_n^2{\mu }_n^2\right]}+\frac{{\mu }_n{\left(1+S_n\right)}^2+{\xi }_n^2{\mu }_n-2{\left(1-S_n\right)}^2}{\left[{\left({\mu }_n\left(1-S_n\right)-1\right)}^2+{\xi }_n^2{\mu }_n^2\right]}\}$

(11)

$A_{9,n}=\frac{-S_n{\omega }_{bn}^2{\xi }_n^2}{2\sqrt{1-{\xi }_n^2}}\{\frac{{\mu }_n{\left(1-S_n\right)}^2+{\xi }_n^2{\mu }_n+2\left(1-S_n\right)}{\left[{\left({\mu }_n\left(1-S_n\right)+1\right)}^2+{\xi }_n^2{\mu }_n^2\right]}+\frac{-{\mu }_n{\left(1-S_n\right)}^2-{\xi }_n^2{\mu }_n+2\left(1-S_n\right)}{\left[{\left({\mu }_n\left(1-S_n\right)-1\right)}^2+{\xi }_n^2{\mu }_n^2\right]}\}$

(12)

where

${\mu }_n=\frac{{\omega }_{bn}}{{\omega }_v}$

(13)

To obtain the $n$ th mode shape of the bridge, the method of applying bandpass filtering technology^33–35,36 can be used. The response corresponding to the $n$ th bridge frequency can be extracted from the measured response. The response expression derived from equation (4) is as follows

$\begin{array}{ll}{R}_b& =e^{-{\xi }_n{a}_{bn}t}\left[\begin{array}{lll}{A}_{6,n}\hspace{0.17em}\cos \left({\omega }_{bn}t+\frac{n\pi \nu t}{L}-\beta \right)& +& A_{7,n}\hspace{0.17em}\cos \left({\omega }_{bn}t-\frac{n\pi \nu t}{L}-\beta \right)\\ +A_{8,n}\hspace{0.17em}\sin \left({\omega }_{bn}t+\frac{n\pi \nu t}{L}-\beta \right)& +& A_{9,n}\hspace{0.17em}\sin \left({\omega }_{bn}t-\frac{n\pi \nu t}{L}-\beta \right)\end{array}\right]\end{array}$

(14)

Using Hilbert Transform to identify the first-order mode shape of the bridge

The method for identifying the bridge mode shape using the Hilbert transform, as proposed by, ³⁷ is a mathematical operation used for signal processing. The Hilbert transform converts a real-valued signal $x\left(t\right)$ into its analytic signal $z\left(t\right)$ , and is expressed as

$z\left(t\right)=x\left(t\right)+j\widehat{x}\left(t\right)$

(15)

Where $\widehat{x}\left(t\right)$ is the Hilbert transform of $x\left(t\right)$ . The instantaneous amplitude $A\left(t\right)$ and the instantaneous phase $\theta \left(t\right)$ of $z\left(t\right)$ are given by

$A\left(t\right)=\left|z\left(t\right)\right|=\sqrt{x^2\left(t\right)+{\widehat{x}}^2\left(t\right)}$

(16)

$\theta \left(t\right)=\arctan \left(\frac{\widehat{x}\left(t\right)}{x\left(t\right)}\right)$

(17)

For more details on the Hilbert transform, refer to. ³⁸ Based on the transformation principle, the Hilbert transform of $R_{b\hspace{0.17em}}$ in equation (14) is ${\widehat{\hspace{0.17em}R}}_b$

${\widehat{R}}_b=e^{-{\xi }_n{\omega }_{bn}t}\left[\begin{array}{l}{A}_{6,n}\hspace{0.17em}\sin \left({\omega }_{bn}t+\frac{n\pi \nu t}{L}-\beta \right)+A_{7,n}\hspace{0.17em}\sin \left({\omega }_{bn}t-\frac{n\pi \nu t}{L}-\beta \right)-\\ A_{8,n}\hspace{0.17em}\cos \left({\omega }_{bn}t+\frac{n\pi \nu t}{L}-\beta \right)-A_{9,n}\hspace{0.17em}\cos \left({\omega }_{bn}t-\frac{n\pi \nu t}{L}-\beta \right)\end{array}\right]$

(18)

Considering that the test vehicle operates at a very slow speed, we can assume $n\pi v/L\ll {\omega }_{bn}$ and thus $S_n\ll 1$ . Therefore, equations (9)–(12) can be simplified

$A_{6,n}=-A_{7,n}=\frac{-S_n{\omega }_{bn}^2}{2\sqrt{1-{\xi }_n^2}}\{\frac{{\mu }_n+{\xi }_n^2{\mu }_n+1-{\xi }_n^2}{\left[{\left({\mu }_n+1\right)}^2+{\xi }_n^2{\mu }_n^2\right]}+\frac{-{\mu }_n-{\xi }_n^2{\mu }_n+1-{\xi }_n^2}{\left[{\left({\mu }_n-1\right)}^2+{\xi }_n^2{\mu }_n^2\right]}\}$

(19)

$A_{8,n}=-A_{9,n}=\frac{-S{\omega }_{bn}^2{\xi }_n^2}{2\sqrt{1-{\xi }_n^2}}\{\frac{-{\mu }_n-{\xi }_n^2{\mu }_n-2}{\left[{\left({\mu }_n+1\right)}^2+{\xi }_n^2{\mu }_n^2\right]}+\frac{{\mu }_n+{\xi }_n^2{\mu }_n-2}{\left[{\left({\mu }_n-1\right)}^2+{\xi }_n^2{\mu }_n^2\right]}\}$

(20)

By combining the above equations, applying equation (16) yields the momentary amplitude $A\left(t\right)$ corresponding to the $n$ th bridge frequency

$A\left(t\right)=\sqrt{R_b^2+{\widehat{R}}_b^2}=2\sqrt{A_{6,n}^2+A_{8,n}^2}{e}^{-{\xi }_n{\omega }_{bn}t}\left|\sin \left(\frac{n\pi vt}{L}\right)\right|$

(21)

Substituting $x=vt$ into the equation gives

$A\left(\frac{x}{v}\right)=\sqrt{R_b^2+{\widehat{R}}_b^2}=2\sqrt{A_{6,n}^2+A_{8,n}^2}{e}^{-{\xi }_n{\omega }_{bn}t}\left|\sin \left(\frac{n\pi x}{L}\right)\right|$

(22)

In the actual vibration testing process, the first mode shape of the bridge is the easiest to measure and is most accurately identified [30]. The momentary amplitude corresponding to the first bridge frequency can be obtained from equation (22)

$A\left(\frac{x}{v}\right)=2\sqrt{A_{6,1}^2+A_{8,1}^2}{e}^{-{\xi }_n{\omega }_{b1}\frac{x}{v}}\hspace{0.17em}\sin \left(\frac{\pi x}{L}\right)$

(23)

From equation (23), the instantaneous amplitude corresponding to the first bridge frequency is related to the first mode shape, but its maximum amplitude no longer occurs at the mid-span; instead, it is shifted towards the starting point of the test vehicle. The specific physical explanation is as follows: as the vehicle travels (as time $increases$ ), the overall amplitude of the bridge’s vibration will continuously decay due to damping. Therefore, for the spatial mode shape that should originally peak at the mid-span, its peak point is instead shifted towards a position closer to the vehicle’s starting point, where the effect of damping has acted for a shorter duration. This paper makes good use of this phenomenon to identify the damping ratio of the bridge efficiently.

Identification of bridge damping ratio

The method of differentiation is used to obtain the location $x_{\mathit{max}}$ of the maximum value of the first mode shape, and the mode shape curve is plotted as shown in Figure 2:

Figure 2.

Schematic diagram of the first mode shape of the bridge.

From Figure 2, it can be seen that the maximum value of the mode shape occurs at $x_{\mathit{max}}$ , and it shifts towards the starting point of the test vehicle. The expression for $x_{\mathit{max}}$ is

$x_{\max }=\frac{L}{\pi }\arctan \left(\frac{\pi v}{{\omega }_{b1}L{\xi }_n}\right)$

(24)

Equation (24) shows the relationship between the location of the maximum value $x_{\mathit{max}}$ and the damping ratio ${\xi }_n$ . From equation (24), it can be seen that when the bridge has damping (i.e., ${\xi }_n>0$ ), the range of the arctangent function $\arctan \left(\frac{\pi v}{{\omega }_{b1}L{\xi }_n}\right)$ is within the interval (0, $\pi /2$ ). From this, the range for $x_{\mathit{max}}$ can be deduced as $0<x_{\max }<L/2$ . This mathematically proves that as long as damping exists, the peak point of the mode shape will inevitably occur to the left of the bridge’s mid-span ( $L/2$ ), shifting towards the vehicle’s starting point. Therefore, with the rest of the parameters determined, the damping ratio ${\xi }_n$ can be obtained using the following formula

${\xi }_n=\frac{\pi v}{{\omega }_{b1}L\hspace{0.17em}\tan \left(\frac{x_{\max }\pi }{L}\right)}$

(25)

From the equation on the right, $x_{\mathit{max}}$ can be obtained using the aforementioned method of identifying the mode shapes, and the rest of the parameters are known during the test, so the bridge damping ratio ${\xi }_n$ can be obtained relatively easily.

Through the guidance mentioned above, this paper proposes a method that only requires a brief test for the vehicle to pass through the bridge, which allows the fast identification of the damping ratio of the bridge. The method is efficient and does not require excessive personnel for assistance.

Process of bridge damping ratio identification

This research analyzes the response of the test vehicle using the Hilbert transform to identify the location of the maximum value $x_{\mathit{max}}$ of the first mode shape of the bridge under damping conditions. By utilizing the relationship between $x_{\mathit{max}}$ and the damping ratio of the bridge, the damping ratio can be quickly identified, as illustrated in the identification process shown in Figure 3.

Figure 3.

Flow chart of bridge damping ratio identification.

Numerical example

Numerical model

Based on the actual testing conditions, a bridge-vehicle coupled model was established, as shown in Figure 1. The specific parameters are set as follows: The bridge beam is simplified as a simply supported beam with a span length of L = 30 m, an elastic modulus of E = 2.75×10¹⁰ N/m², a linear mass density of $\overline{m}$ = 8000 kg/m, and a moment of inertia of the beam’s cross-section of I = 0.15 m⁴. The test vehicle is modeled as a single-degree-of-freedom mass-spring-damper system, with a mass of $m_v$ = 2000 kg, a stiffness of $k_v$ = 500000 N/m, and negligible damping. Therefore, the natural frequency of the bridge beam is 1.25 Hz, and the vehicle frequency is 2.51 Hz. The test vehicle crosses the bridge from the left support at a constant speed of v = 1 m/s. Regarding the vehicle speed, a relatively low speed of 1 m/s was selected as the baseline to mitigate the adverse effects of high vehicle speeds on identification accuracy, a phenomenon confirmed in numerous studies.^18,37 In this study, the bridge-vehicle interaction (VBI) was analyzed numerically using MATLAB software. The sampling frequency was set at 1000 Hz. Regarding the bridge damping ratio, a value of 1% was chosen as the baseline because it is a typical damping ratio for small to medium-span bridges in China. This is supported by related studies: for instance, Yang et al. ²⁰ selected a 1% damping ratio as the baseline in their numerical simulations; Tan et al. ¹⁸ measured a baseline damping ratio of 1.6% in their laboratory model test; meanwhile, Yang et al. ⁷ measured a damping ratio of 0.7% in a field test on an actual bridge. These references indicate that selecting a damping ratio within the 1%-2% range for a preliminary parametric study is a reasonable and representative approach.

Following the flowchart in Figure 3, the numerical analysis steps include first extracting the response amplitude of the first-order frequency of the bridge due to the vehicle acceleration signal during steady-speed crossing. The maximum amplitude location $x_{\mathit{max}}$ is determined using the Hilbert transform to identify the first-order vibration mode of the bridge under damping conditions. Finally, Equation (25) is used to quickly identify the bridge damping ratio. The steps of numerical analysis will be specifically demonstrated using the parameters of the numerical model as an example.

Specific identification process

Following the flowchart in Fig. 3Fig. 3 and the model parameters in the “Numerical model” subsection, the single-axis acceleration response signal of the test vehicle after crossing the bridge is obtained, as shown in Figure 4.

Figure 4.

Acceleration response of the test vehicle.

As shown in Figure 4, when the bridge has a damping ratio, the amplitude of the acceleration response collected by the test vehicle is not at its maximum at mid-span but is shifted toward the starting position. The collected vehicle signal is processed using a band-pass filter to extract the response component corresponding to the bridge’s first-order frequency (1.25 Hz). The first-order vibration mode of the bridge is then identified using the Hilbert transform, with the mode shape curve shown in Figure 5.

Figure 5.

Identified first mode shape curves of the bridge.

In the first-order vibration mode shape curve of the bridge shown in Figure 5, the shift of the mode shape can be clearly observed. The location of the maximum mode shape, $x_{\mathit{max}}$ can also be obtained from the curve, with a value of 8.753 m.

Based on equation (25) and the previously described model parameters, the identified damping ratio of the bridge is shown in Table 2.

Table 2.

Bridge damping ratio identification results.

Identified value	Theoretical value	Absolute error	Relative error
1.02%	1%	0.0002	2%

From the identification results in Table 2, it can be seen that the proposed method accurately identifies the damping ratio of the bridge. The relative error with respect to the theoretical value is only 2%, meeting the precision requirements of dynamic testing in actual engineering applications. Furthermore, the identification process is free from human interference, further demonstrating the feasibility of the method proposed in this study.

Parametric study

In actual testing processes, data acquisition and processing are often inevitably disturbed by various factors, such as test damping, bridge damping ratio, test vehicle speed, and environmental noise changes. These factors may impact the accuracy of the test results. To validate the applicability of the proposed method, this paper designs a simulation analysis focusing on different potential disturbance factors. Through comprehensive evaluation and discussion, the reliability and accuracy of the bridge damping ratio identification method under complex conditions are further explored.

Effect of vehicle damping

In the bridge-vehicle coupled system, the test vehicle acts as the medium between the sensor and the bridge. The damping $c_v$ of the test vehicle affects the transmission of the vertical acceleration signal of the bridge, making it an important factor influencing the identification of the bridge damping ratio. Four test vehicle damping conditions are set, namely 100N·s/m, 400N·s/m, 800N·s/m, and 1000N·s/m. To investigate the effect of vehicle damping, four scenarios were simulated based on the baseline model from the “Numerical model” subsection. The first-order bridge mode shape curve is shown in Figure 6, and the results of the bridge damping ratio identification are presented in Table 3.

Figure 6.

Results of mode shape identification under different test vehicle damping.

Table 3.

Bridge damping ratio identification results with different test vehicle damping.

Test vehicle damping	Identified value (%)	Theoretical value (%)	Absolute error	Relative error (%)
100	1.02	1	0.0002	2
400	1.02	1	0.0002	2
800	1.02	1	0.0002	2
1000	1.02	1	0.0002	2

As demonstrated in Figure 6 and Table 3, the damping of the test vehicle has minimal effect on the identification results of the bridge mode shape and bridge damping ratio. The relative error of the identified damping ratio is consistently 2%, which is in accordance with the accuracy requirements for practical engineering dynamic tests.

Effect of bridge damping ratio

As previously outlined, the damping ratio of small to medium-span bridges typically does not exceed 2%. Consequently, four distinct bridge damping ratio conditions are established: 0.6%, 0.8%, 1.0%, and 1.5%. The present study aims to evaluate the efficacy of the proposed method in identifying these various bridge damping ratios. Building upon the baseline model, the influence of the bridge’s own damping ratio is then investigated. Four distinct damping conditions are established, with all other parameters remaining as defined in the “Numerical model” subsection. The first-order bridge mode shape curve is shown in Figure 7, and the results of bridge damping ratio identification are presented in Table 4.

Figure 7.

Results of mode shape identification under different bridge damping ratios.

Table 4.

Bridge damping ratio identification results.

Bridge damping ratio (%)	Identified value (%)	Absolute error	Relative error (%)	Identified $x_{\mathit{max}}$
0.6	0.58	0.0002	3.33	11.08
0.8	0.81	0.0001	1.25	9.79
1.0	1.02	0.0002	1.96	8.76
1.5	1.53	0.0003	2.00	6.84

As demonstrated in Figure 7 and Table 4, it can be observed that the bridge damping ratio has a significant impact on the location of the maximum value of the mode shape curve, which shifts closer to the starting point as the ratio increases. This is quantitatively supported by the new “Identified $x_{\mathit{max}}$ ˮ column in Table 4, which shows that as the damping ratio increases from 0.6% to 1.5%, the value of $x_{\mathit{max}}$ decreases from 11.08 m to 6.84 m, providing clear data support for this physical phenomenon of peak point shift. The proposed method successfully identifies the bridge damping ratio under various conditions, with a relative error of only 3.33%, meeting the accuracy requirements for practical engineering dynamic tests. The findings underscore the broad applicability of the proposed method to a wide range of small-to medium-span simply supported bridges, thereby reinforcing its generalizability. ³⁹

The parametric analysis in this section has primarily validated the effectiveness of the proposed method within the typical damping ratio range (0.6% to 1.5%) of conventional small to medium-span bridges. It is important to note that, in the field of bridge engineering, there is a significant class of structures specifically designed as high-damping systems to enhance seismic performance. For instance, the equivalent damping ratio of a bridge can be substantially increased by equipping it with seismic isolation systems like High Damping Rubber Bearings (HDRB) or Friction Pendulum Systems (FPS), or by retrofitting it with supplemental devices such as Clutching Inerter Dampers (CID).^5,40 These high-damping characteristics will undoubtedly introduce new challenges for the application of our method, and its applicability in such cases will be further explored in future research.

Effect of vehicle speed

In order to investigate the effect of different test vehicle speeds on the proposed method, five test speeds were set: 1 m/s, 2 m/s, 4 m/s, 6 m/s, and 8 m/s. The remaining parameters were configured as outlined in the “Numerical model” subsection. Following the process outlined in Figure 3, simulations were conducted for the five test speed scenarios. The first-order bridge mode shape curve is shown in Figure 8, and the bridge damping ratio identification results are presented in Table 5.

Figure 8.

Results of mode shape identification under different test vehicle speeds.

Table 5.

Bridge damping ratio identification results with different test vehicle speeds.

Test vehicle speed	Identified value (%)	Theoretical value (%)	Absolute error	Relative error (%)	Identified $x_{\mathit{max}}$
1 m/s	1.02	1	0.0002	2	8.76
2 m/s	0.96	1	0.0004	4	11.70
4 m/s	0.97	1	0.0003	3	13.28
6 m/s	0.97	1	0.0003	3	13.85
8 m/s	1.03	1	0.0003	3	14.08

As demonstrated in Figure 8, it is evident that an increase in the velocity of the test vehicle results in a shift in the location of the maximum value of the mode shape towards the center of the span. The new data column in Table 5 also clearly reflects this trend: as the vehicle speed increases from 1 m/s to 8 m/s, the identified $x_{\mathit{max}}$ increases from 8.76 m to 14.08 m, gradually approaching the bridge’s mid-span (15 m). This phenomenon can be elucidated by equation (24), where $x_{\mathit{max}}$ is proportional to the velocity $v$ . As illustrated in Table 5, the magnitude of the test vehicle speed does not exert a substantial influence on the identification results. For instance, at a speed of 8 m/s, the relative error is recorded as 3%, with a maximum error of 4.00%, thus meeting the 5% error requirement for practical engineering dynamic tests.

It is worth noting that in the mode shape curves in Figure 8, higher speeds result in more noticeable disturbances near the support locations. However, the mode shape remains smooth near the maximum curve value, with minimal errors. Therefore, the speed has little effect on the proposed method. In contrast, the damping ratio identification method proposed by Tan et al. [16] is greatly limited by speed. The proposed method demonstrates a significant advantage in this aspect.

Effect of environmental noise

Noise in field measurements cannot be avoided, so white noise is added to the raw acceleration to simulate ambient noise. The signal-to-noise ratio (SNR) is defined as

$\text{SNR}=10\times \log \left(\frac{\frac{1}{N}{\sum }_{i=1}^N y_i^2}{\frac{1}{N}{\sum }_{i=1}^N {{\delta }_i}^2}\right)$

(26)

where $y_i$ is the acceleration at the ith sampling time step, $N$ is the total number of data, and ${\delta }_i$ is the noise at the sampling time step. From the above equation, the SNR decreases as the noise increases.

The test vehicle body acceleration response was simulated under four noise conditions: no noise, 40 dB, 30 dB, and 20 dB. The other parameters of the test vehicle and bridge were set as described in the “Numerical model” subsection. Following the process shown in Figure 3, the first-order bridge mode shape curve was obtained, as shown in Figure 9, and the bridge damping ratio identification results are presented in Table 6.

Figure 9.

Results of mode shape identification under different noise levels.

Table 6.

Bridge damping ratio identification results with different noise levels.

Environmental noise	Identified value	Theoretical value	Absolute error	Relative error
No noise	1.02%	1%	0.0002	2%
40 dB	1.02%	1%	0.0002	2%
30 dB	1.03%	1%	0.0003	3%
20 dB	1.05%	1%	0.0005	5%

As shown in Figure 9, the mode shape curves remain consistent under different noise levels. From the bridge damping ratio identification values in Table 6, it can be concluded that the proposed method effectively identifies the bridge damping ratio under different noise conditions. Even under 20 dB noise, the method maintains good performance with an error of only 5%, highlighting the strong noise resistance of the proposed method.

Comparison of different methods

To further highlight the advantages of the proposed method, this study compares it with the method proposed by Tan et al. ¹⁸ in terms of accuracy and time efficiency. The vehicle speed was set to 4 m/s, the noise level to 30 dB, and other parameters were the same as described in the “Numerical model” subsection. It is worth noting that the method in Tan et al. ¹⁸ also uses the Hilbert transform to identify the first-order mode shape of the bridge, resulting in identical mode shape curves, as shown in Figure 10.

Figure 10.

Identified mode shape curve.

In the subsequent process of identifying the damping ratio, as mentioned in the introduction, the method proposed by Tan et al. ¹⁸ requires continuously adjusting the assumed values of the bridge damping ratio to approximate the actual value. This process involves numerous repeated simulations, which inevitably reduces identification efficiency. The comparison of the identification results of the two methods is shown in Table 7.

Table 7.

Identification results using different methods.

Identification method	Identified value (%)	Theoretical value (%)	Relative error (%)	Identification time/s
Proposed method	0.97	1	3	2.30
Tan et al. 18 method	1.05	1	5	15.7

From the identification results in Table 7, it can be seen that the proposed method outperforms the method proposed by Tan et al. ¹⁸ in terms of both accuracy (relative error) and identification time. This highlights the proposed method’s strong identification capability and high efficiency, making it suitable for rapid detection of small-to medium-span bridge clusters.

Conclusions and prospects

Conclusions

This study proposed and validated a novel method for the rapid identification of bridge damping ratios based on the phenomenon of peak point shift in the bridge’s first-order mode shape. Its most significant contribution lies in establishing a direct analytical identification approach that bypasses traditional iterative computations. Through comprehensive theoretical derivation and numerical simulation, the main conclusions of this study are as follows.

(1) The core of the research is the derivation of a direct mathematical relationship between the bridge damping ratio ${\xi }_n$ and the peak position of the first-order mode shape $x_{\mathit{max}}$ : ${\xi }_n=\frac{\pi v}{{\omega }_{b1}L\hspace{0.17em}\tan \left(\frac{x_{\max }\pi }{L}\right)}$ . The establishment of this formula simplifies the complex problem of damping ratio identification into a direct solution process for a spatial position parameter, greatly improving identification efficiency. In comparison with existing methods, the identification time of this method is shortened by nearly 7 times.

(2) Numerical examples show that this method has high identification accuracy; under the baseline condition, the relative error between the identified value and the theoretical value is only 2%. Comprehensive parametric analysis further confirms the strong robustness of this method:

• The method is almost unaffected by changes in the test vehicle’s own damping.

• It remains stable at different driving speeds; even at a higher speed of 8 m/s, the maximum relative error can be kept within 4%.

• The method has strong noise resistance; even in a strong noise environment with a signal-to-noise ratio as low as 20 dB, the identification error is only 5%, meeting the accuracy requirements of practical engineering.

• The method successfully identified the typical damping ratio range (0.6% to 1.5%) for small to medium-span bridges, with all relative errors being lower than 3.5%, indicating its applicability to small and medium-span bridges over a wide range of damping ratios.

(3) This method utilizes a moving test vehicle to indirectly acquire bridge information, offering the advantages of low cost and rapid implementation. Its characteristics of high efficiency, high accuracy, and strong robustness make it particularly suitable for the rapid, non-destructive health condition screening of a large number of small and medium-span bridges, showing clear engineering application prospects.

Prospects

This study has initially confirmed the feasibility and robustness of the proposed method through theoretical derivation and numerical analysis, laying a solid foundation for subsequent research. In the future, we will deepen our exploration from multiple dimensions. On the one hand, we will focus on on-site tests of medium and small-span Bridges, verifying the accuracy and reliability of the methods in actual scenarios such as road unevenness, mixed traffic flow of multiple vehicles, and changes in environmental temperature and humidity. At the same time, we will expand from simply supported beam and single-axle vehicle models to continuous beam Bridges and multi-axle vehicle models. And in view of the characteristics of large-span, cable-stayed Bridges and irregular damping Bridges, such as dense high-order modes and significant aerodynamic damping effects, the limitations of the methods and improvement plans are explored. On the other hand, in order to expand its application in health monitoring, the integration of high-order mode shape information that is more sensitive to local damage will be explored. The promotion method will be upgraded from a single global parameter identification tool to a diagnostic solution that can assist in identifying and locating local stiffness changes. The effectiveness of this method in high-damping-ratio Bridges (such as Bridges with isolation devices) will also be verified, and the identification accuracy and applicable boundaries under different damping levels will be analyzed to further improve the relevant theoretical model.