Research on Indirect Influence-Line Identification Methods in the Dynamic Response of Vehicles Crossing Bridges

Abstract

The bridge influence line can effectively reflect its overall structural stiffness, and it has been used in the studies of safety assessment, model updating, and the dynamic weighing of bridges. To accurately obtain the influence line of a bridge, an Empirical and Variational Mixed Modal Decomposition (E-VMD) method is used to remove the dynamic component from the vehicle-induced deflection response of a bridge, which requires the preset fundamental frequency of the structure to be used as the cutoff frequency for the intrinsic modal decomposition operation. However, the true fundamental frequency is often obtained from the picker, and the testing process requires the interruption of traffic to carry out the mode decomposition. To realize the rapid testing of the influence lines of bridges, a new method of indirectly identifying the operational modal frequency and deflection influence lines of bridge structures from the axle dynamic response is proposed as an example of cable-stayed bridge structures. Based on the energy method, an analytical solution of the first-order frequency of vertical bending is obtained for a short-tower cable-stayed bridge, which can be used as the initial base frequency to roughly measure the deflection influence line of the cable-stayed bridge. The residual difference between the deflection response and the roughly measured influence line under the excitation of the vehicle is operated by Fast Fourier Transform, from which the operational fundamental frequency identification of the bridge is realized. Using the operational fundamental frequency as the cutoff frequency and comparing the influence-line identification equations, the empirical variational mixed modal decomposition, and the Tikhonov regularization to establish a more accurate identification of the deflection influence line, the deflection influence line is finally identified. The accuracy and practicality of the proposed method are verified by real cable-stayed bridge engineering cases. The results show that the relative error between the recognized bridge fundamental frequency and the measured fundamental frequency is 0.32%, and the relative error of the recognized deflection influence line is 0.83%. The identification value of the deflection influence line has a certain precision.

1. Introduction

By the end of 2023, there were 1,079,300 highway bridges in China, representing an increase of 46,100 over the end of the previous year, including 10,239 special bridges and 177,700 bridges. Bridges inevitably produce structural cracking, performance degradation, and other damage during long-term operation due to creep, corrosion, and cyclic loading [1]. Therefore, it is necessary to establish a perfect structural health-monitoring system to monitor the structural condition of bridges in real time and extend the service life of bridges through appropriate management and maintenance.

The influence line and frequency, as an important performance index for bridge structural condition assessment [2], reflects the response of displacement, strain, and cornering of a specific cross-section under the action of moving load and have been widely used in the field of bridge health monitoring [3], such as model updating [4,5], damage identification [6,7], and load-carrying capacity assessment [8,9]. Lin Shiwei et al. [10] proposed a system model-updating theory using an influence line with a global optimization method based on adaptive meta-models to fit iteratively, automatically adding sample data through the selection of three different objective functions and model-updating parameters so that the updated model better meets the engineering requirements. Wang Zhen et al. [11] used co-integration analysis and the Johansen program to remove environmental trends and obtain multivariate co-integration residuals. In this paper, a new local frequency co-integration method is proposed, which eliminates the false influence of environmental change and then constructs an exponential weighted moving average to warn about subtle damage to a bridge. Zheng Xu et al. [12] proposed the design concept of a structural health-monitoring system for small- and medium-span bridges based on the line of influence, established a corresponding design method, and realized the lightweight bridge load-carrying capacity assessment. The application of bridge influence line and vibration frequency is promising, but the accuracy and operability of the identification method of quasi-static influence line and vibration frequency in practical operation needs to be investigated.

The identification of bridge vibration modal parameters can be divided into two categories: frequency-domain methods and time-domain methods [13]. Most of the frequency-domain methods utilize classical spectral estimation, which is measured using nonparametric methods. Time-domain methods belong in the category of modern spectral analysis, and they are parametric methods. Ye Xijun et al. [14] decoupled the power spectral density function of a multi-degree-of-freedom system into a series of single-degree-of-freedom power spectral density functions by singular value decomposition of the power spectral density and subsequently used the peaking method to find the frequency; Yang Yongbin et al. [15] proposed for the first time an indirect measurement method for bridges based on the response of the vehicle, which identifies the modal frequency of the bridge and has advantages of maneuverability, efficiency, economy, etc. There is no need to close the road, and there is also a stationary operation that realizes the rapid detection of grouped bridges. He Wenyu et al. [16] proposed a method for identifying the modal parameters of bridges under vehicle static, where the identification of vibration modes is replaced by frequency identification and a small number of vibration pickers are used to quickly identify the normalized vibration modes of the bridge mass. The above-proposed methods can obtain modal parameters with high accuracy, but all of them require the use of vibration pickers and expensive equipment.

In practical engineering, the response data of a certain cross-section is usually obtained quickly by moving the load of a single heavy vehicle, and the bridge influence line is obtained after conversion [17]. However, the measured influence line is often mixed with bridge vibration and noise fluctuation. Chen Zhiwei et al. [18] proposed a new method of influence-line identification based on an adaptive B-spline-basis dictionary and sparse regularization technique, which constructs the influence line by establishing basis function representations, and the curvature-based adaptive node optimization method, which is integrated with a redundant B-spline basis dictionary to ensure the sparsity of the solution and improve the accuracy of influence-line identification. Zheng Xu et al. [19] used Empirical Mode Decomposition (EMD) to eliminate dynamic fluctuations in the response of the bridge induced by high-speed vehicles, established influence-line identification pathology with the help of Tikhonov regularization equations, and constructed the vehicle-information matrix to invert quasi-static influence lines. Zhou Yu et al. [20] proposed a new method of mixed Empirical and Variational Modal Decomposition, which avoids the distortion of the Variational Modal Decomposition (VMD) method under the high-speed excitation of the vehicle and improves the stability and accuracy of the recognition of influence lines. There is a mature system for removing dynamic perturbation contained in the time response of the influence line, but the proposed methods require dynamic testing to obtain the fundamental frequency of the bridge vibration as the threshold value for decomposition, which is cumbersome. Therefore, based on the axle-coupling system, this paper proposes a non-contact identification method based on the frequency update of the bridge-operation mode to obtain a quasi-static influence line, which is indirectly identified from the vibration displacement response of the axle coupling without the need to deploy a vibration picker, which shortens the testing time of the inspection vehicle and achieves feasibility verification using engineering examples. The technology roadmap is shown in Figure 1.

2. Theoretical Approach Establishment

2.1. Frequency Derivation Based on Rayleigh Method

Neglecting the effect of structural damping, the sum of kinetic and potential energies of the structure at any moment is constant according to the law of the conservation of energy. In the case of undamped intrinsic vibration, the instantaneous transverse vibration displacement $y\left(x,t\right)$ of the bridge can be expressed as

$$y\left(x,t\right)=\eta \left(x\right)\sin \left(wt+\theta \right)$$

(1)

where $\eta \left(x\right)$ is the approximate shape function that satisfies the bridge displacement boundary condition, $w$ is the corresponding frequency, and $\theta$ is the phase difference.

According to the law of conservation of energy, the approximate value of frequency $w$ can be deduced as

$$w=\sqrt{{\displaystyle \frac{E_{p\max }}{E_{k\max }}}}={\displaystyle \frac{{\displaystyle {\int }_0^{l}EI\left(x\right){\left[{\eta }^{″}\left(x\right)\right]}^{2}dx}}{{\displaystyle {\int }_0^{l}m\left(x\right){\eta }^2\left(x\right)dx}}}$$

(2)

where E_p and E_k are the potential and kinetic energies of the structural system, respectively, $EI\left(x\right)$ is the bending stiffness, and $m\left(x\right)$ is the mass distribution value.

According to the derivation formula [21], to achieve positive-symmetry double-tower–beam cementation, using a pier-supported short-tower cable-stayed bridge as an example, such as the bridge in the vibration process of the overall kinetic energy for Equation (3), the overall potential energy for the formula shown in Equation (4), due to the cable-stayed bridge main girder stiffness, is larger. The main tower, the potential energy of the cable-stayed cable, the cable-stayed cable’s secondary potential energy, and the kinetic energy of the main tower can be ignored. At the same time, the cable-stayed cable mass is equivalent to the small main beam mass, so the cable-stayed cable kinetic energy is also ignored through deformation. The coordination equation can be deduced from the short-tower cable-stayed bridge first-order vertical bending vibration mode frequency analytical formula for the formula shown in Equation (5).

$$E={\displaystyle \frac{w^2}{2}}{\displaystyle {\int }_0^l{m}_G}{\eta }^{2}dx+{\displaystyle \sum _{i=1}^n\frac{w^2}{6}}{m}_{ci}{{\eta }^2}_{ci}{L}_{ci}+{\displaystyle \frac{w^2}{2}}{\displaystyle \sum _{i=1}^n{\displaystyle {\int }_0^{hi}{m}_T}{\xi }^{2}dz}={\displaystyle \frac{w^2}{2}}\left({\displaystyle {\int }_0^l{m}_G}{\eta }^{2}dx+{\displaystyle \sum _{i=1}^n\frac{1}{3}}{m}_{ci}{{\eta }^2}_{ci}{L}_{ci}+{\displaystyle \sum _{i=1}^n{\displaystyle {\int }_0^{hi}{m}_T}{\xi }^{2}dz}\right)$$

(3)

$$E_p={\displaystyle \frac{1}{2}}{\displaystyle {\int }_0^l{E}_G}{I}_G{\left({\displaystyle \frac{d^2\eta }{d{x}^2}}\right)}^{2}dx+{\displaystyle \sum _{i=1}^n\left(\frac{1}{2}{{\displaystyle {\int }_0^{hi}{E}_T{I}_T\left(\frac{d^2\xi }{d{x}^2}\right)}}^{2}dz\right)}+{\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^n\frac{E_{ci}{A}_{ci}{h_{ci}}^2{\sin }^2{a}_{ci}}{L_{ci}}}+{\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^n∆F\frac{\cos a_{ci}}{L_{ci}}}{{\eta }^2}_{ci}$$

(4)

$$w=\sqrt{{\displaystyle \frac{E_G{I}_G{\pi }^4\left(\frac{l_c}{l_s}+\frac{1}{2}\right)}{{l^4}_C\left(\frac{V{G}_S}{g}\frac{{l^3}_S}{{l^3}_C}+\frac{V{G}_C}{2g}\right)}}}$$

(5)

2.2. Influence-Line Identification Methods

2.2.1. E-VMD Method Establishment

We use EMD to adaptively decompose the measured deflection time–range response x(n) to derive the IMF components of the intrinsic modal function, calculate the correlation coefficients between each IMF component where x(n), l₁(m), (m = 1,2 … M), and M is the number of IMFs, and define the correlation thresholds l_limt.

$$l_1\left(m\right)={\displaystyle \frac{{\displaystyle \sum _{n=1}^N\left(x\left(n\right)-\overline{x}\left(n\right)\right)\left(c_k\left(n\right)-{\overline{c}}_k\left(n\right)\right)}}{\sqrt{{\displaystyle \sum _{n=1}^N{\left(x\left(n\right)-\overline{x}\left(n\right)\right)}^2{\displaystyle \sum _{n=1}^N{\left(c_k\left(n\right)-{\overline{c}}_k\left(n\right)\right)}^2}}}}}$$

(6)

$$l_{\lim t}=-\ln \left[\xi c/\left(1+\xi c\right)\right]$$

(7)

where $\xi$ is the resolution factor and c is an empirical constant. When $l_1\left(m\right)\ge l_{\lim t}$ is satisfied, the corresponding m pieces IMF component is defined as the useful component; otherwise, it is the noise component. Summing all the useful components yields the preprocessed signal, denoted as $x_0\left(n\right)$.

The VMD decomposition of signal $x_0\left(n\right)$ is performed at different values of K. A set of narrowband IMF components is obtained for each decomposition. The narrowband IMFs are determined by calculating the correlation coefficient $l_2(k)$ in relation to $x_0\left(n\right)$. The reconstructed signal $x_1\left(n\right)$ is obtained by accumulating all the narrowband IMF components that satisfy $l_2(k)\ge l_{\lim t}(k=1,2,\cdots K)$. The mutual information relationship between the preprocessed signal $x_0\left(n\right)$ and the reconstructed signal $x_1\left(n\right)$ is calculated by the mutual information method, and the mutual information (MI) index is proposed.

$$MI\left[x_1,x_0\right]=H(x_1)+H(x_0)-H(x_1,x_0)$$

(8)

where $H(x_1)$, $H(x_0)$ and $H(x_1,x_0)$ are the information entropy and joint entropy of $x_0\left(n\right)$ and $x_1\left(n\right)$, denoted as:

$$H(x_1)=-{\displaystyle \sum _i{p}_{x_1}\left(x_{1i}\right){\log }_2{p}_{x_1}}\left(x_{1i}\right)$$

(9)

$$H(x_0)=-{\displaystyle \sum _j{p}_{x_0}\left(x_{0j}\right){\log }_2{p}_{x_0}}\left(x_{0j}\right)$$

(10)

$$H(x_1,x_0)=-{\displaystyle \sum _{i,j}{p}_{x_1,x_0}\left(x_{1i},x_{0j}\right){\log }_2{p}_{x_1{x}_0}}\left(x_{1i},x_{0j}\right)$$

(11)

where $p_{x_1}\left(x_{1i}\right)$, $p_{x_0}\left(x_{0j}\right)$ and $p_{x_1,x_0}\left(x_{1i},x_{0j}\right)$ are the probability mass function and the joint probability mass function of $x_0\left(n\right)$ and $x_1\left(n\right)$, respectively. When the mutual information index MI reaches the maximum value, the corresponding K value is the optimal number of modes obtained from the VMD of the preprocessed signal $x_0\left(n\right)$. The quasi-static component of the vehicular time–range response is obtained by reconstructing all the narrowband IMFs in the optimal number of modes $x_0\left(n\right)$ that satisfy $l_2(k)\ge l_{\lim t}(k=1,2,\cdots K)$. The method flow is shown in Figure 2.

2.2.2. Rejection of Axle Effects

To effectively eliminate the vehicle multi-axis effect in the quasi-static response of the bridge, Tikhonov regularization is introduced to establish the influence-line identification model and error e is introduced to establish the influence-line identification model after the E-VMD decomposition to process the time–range response Rs, where Rs is the quasi-static response of the bridge, L is the vehicle-information matrix, and Φ is the influence-line coefficient of the bridge node.

$${\mathbf{R}}_{\mathbf{s}}=\mathbf{L}\mathbf{\Phi }+\mathbf{e}$$

(12)

The Tikhonov regularization method can effectively solve the pathologization of the mathematical equations of the loud line-identification model due to the error term by restricting the least squares expression through the L2 paradigm as a penalty function. The regularization equation is shown in Equation (13), where λ is the regularization coefficient.

$$\underset{\mathrm{\Phi }\in R}{\min }[{‖{\mathbf{R}}_{\mathbf{s}}-\mathbf{L}\mathbf{\Phi }‖}_2^2+\lambda {‖\mathbf{T}\mathbf{\Phi }‖}_2^2]$$

(13)

The regularized empirical matrix is:

$$\mathbf{T}={\left(\begin{array}{ccccccc}1& -2& 1& & & & \\ & 1& -2& 1& & & \\ & & \ddots & \ddots & \ddots & & \\ & & & & 1& -2& 1\end{array}\right)}_{\left(n-2\right)\times n}$$

(14)

Following the establishment of the axle information matrix, the vehicle-information matrix is determined by sampling frequency, vehicle axle weight, number of axles, axle distance, vehicle speed, and other parameters. Vehicle speed and sampling frequency determine the number of columns of the vehicle-information matrix, and the sensor sampling frequency determines the number of its rows. Take k-axle vehicles as an example. For the vehicle front axle on the bridge and the last axle out of the bridge at the start and end of the timing, the vehicle-information matrix expression is shown in Equation (15), where A_i is the vehicle axle weight.

$${\mathbf{L}}^{\mathrm{T}}={\left(\begin{array}{ccccccccccc}{A}_1& 0& \cdots & A_2& 0& \cdots & A_k& 0& \cdots & \cdots & 0\\ 0& A_1& \ddots & \ddots & A_2& \ddots & \ddots & A_k& \ddots & \ddots & ⋮\\ ⋮& \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & ⋮\\ ⋮& \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & 0\\ 0& \cdots & \cdots & 0& A_1& 0& \cdots & A_2& 0& \cdots & A_k\end{array}\right)}_{n\times m}$$

(15)

Substituting the regularized empirical matrix into the regularized equation and solving Φ with the inverse of the first derivation, then the expression of the influence line is as follows:

$$\mathbf{\Phi }={({\mathbf{L}}^{\mathbf{T}}\mathbf{L}+{\lambda }^2{\mathbf{T}}^{\mathbf{T}}\mathbf{T})}^{-\mathbf{1}}{\mathbf{L}}^{\mathbf{T}}{\mathbf{R}}_{\mathbf{s}}$$

(16)

3. Numerical Simulation Analysis

3.1. Case I

Midas/Civil is used to establish a symmetrical double-tower tower–beam consolidation, pier-supported, three-span low-tower cable-stayed bridge for numerical analysis. The total length of the bridge is 728 m, the span combination is (184 + 360 + 184) m, the bridge width is 14.2 m, the main girder of the example is a single box with two chambers, and the structural material is selected as C50 concrete, and the nonlinear material parameters are shown in

Table 1. Nonlinear material parameters.

Material Parameter	Data	Material Parameter	Data
Modulus of elasticity of concrete	3.45 × 107 KN/m2	Bending moment of inertia of main beam	19.0956 m4
Concrete capacity	25 KN/m3	Main beam section area	14.1195 m2
Poisson’s ratio	0.2	Main beam section height	1.5 m

. Through the stochastic subspace algorithm analysis of the bridge first-order vibration frequency of 0.266 Hz, the example does not take into account the unevenness of the bridge deck. The bridge model and the first-order calculation of the vibration pattern are shown in Figure 3.

The analytical value of the model bridge vibration fundamental frequency is calculated as 0.190 Hz by Equation (5), while the calculated value of the vertical bending frequency of the finite-element model is 0.266 Hz, with a relative error of 29.2%, which verifies the feasibility of the formula.

3.2. Case II

Adopting the data reported by the authors of [22,23], the Yonghe Bridge in Tianjin, China, is analyzed. The Tianjin Yonghe Bridge was opened to traffic in 1987. It has a main span of 260 m, double towers, double cable-stayed, tower–pier cementation, and a continuous floating system of prestressed concrete as a cable-stayed bridge. The total length of the bridge is 512.4 m, the beam height is 2 m, the tower height is 55.5 m, and the bridge width is 13.6 m. The diagonal leg section of the tower column is a steel-skeleton hollow concrete column, the main pier is a sunken-well foundation, the rest of the pier is a tubular-pile foundation, and the auxiliary piers are equipped with tension pendulums. Bridge information is shown in Figure 4.

As reported by the authors of [24], data can be seen by the bridge-free vibration test to obtain the Yonghe Bridge measured vibration fundamental frequency of 0.407 Hz. Equation (5) is used to calculate the Yonghe Bridge theoretical analytical fundamental frequency of 0.299 Hz, with a relative error of 26.5%. The same verified the feasibility of the formula.

4. Engineering Examples

A three-span short-tower cable-stayed bridge was used as an engineering case for test verification. The bridge for the waveform steel web has a PC combination box girder, a span combination of (105 + 180 + 105) m = 390 m, a main beam using C50 concrete, and a top plate width of 35.5 m. To obtain the deflection influence line of the bridge structure via a test through the idling vehicle loading, the loading path is selected from the centerline of the bridge so that the full load of the 35T four-axle test is employed. The vehicle traveled according to the loading route, which measured the deflection response data of the bridge under the idling loading test and identified the deflection influence line of the bridge structure under unit load. The main geometric dimensional information of the bridge section was further reviewed on site in conjunction with the design drawings. The bridge dimensional information and the arrangement of measurement points are shown in Figure 5, Figure 6 and Figure 7. A non-contact bridge dynamic deflectometer was used in the field to develop the deflection influence line test on the established measurement points of the cable-stayed bridge, and the bridge modal test was carried out using 941B acceleration sensors. The measured bridge fundamental frequency was 0.843 Hz.

5. Bridge Influence-Line Identification

5.1. Measured Influence-Line Identification

Midas/Civil is used to establish a finite-element analysis model. The first-order vertical bending frequency of the bridge is 0.839 Hz, and the measured fundamental frequency of the bridge is 0.843 Hz. The vibration test results show that the measured first three orders of the self-oscillation frequency of the bridge are greater than the theoretical calculation value, which indicates that the overall stiffness of the structure of the test span is better [25]. The measured and modeled bridge vibration patterns are shown in Figure 8. The vibration fundamental frequency of the bridge is measured in the field as the Intrinsic Mode Function (IMF) decomposition threshold for the measured deflection response E-VMD modal decomposition to obtain the quasi-static deflection time response. The establishment of the influence-line identification equation, excluding the vehicle axle effect, to restore the quasi-static measured deflection influence line is shown in Figure 9 and Figure 10.

5.2. Bridge Influence-Line Identification Update

Using the derived first-order vertical bending frequency of the cable-stayed bridge as shown in Equation (5), the theoretical analytical frequency of the test bridge is calculated to be 0.707 Hz, and the error quantization formula is introduced as shown in Equation (17) [26], which shows that the relative error between the vertical bending frequency derived from the empirical formula and the measured bridge fundamental frequency and the computed frequency of the finite-element model is controlled at about 15%. The error accuracy is not high. To obtain a more accurate vertical bending frequency of the bridge, the empirically derived frequency is used as the threshold value of IMF decomposition, and the E-VMD method is used to strip the IMF components, whose main frequency is larger than the fundamental frequency, and Fast Fourier Transform is performed on the stripped power components [27], as shown in Figure 11. The operational modal frequency of the bridge is identified to be 0.84034 by stripping the power components that are doped in the influence line. The relative error between the identified operational modal frequency and the theoretical frequency is improved to 0.16%, and the error with the measured frequency is 0.32%.

$$d={\displaystyle \frac{{\displaystyle \sum \left|{\epsilon }_a-{\epsilon }_m\right|}}{{\displaystyle \sum \left|{\epsilon }_a\right|}}}$$

(17)

where d denotes the relative error to the measured data, ${\epsilon }_a$ denotes the measured data, and ${\epsilon }_m$ denotes the data identified by the method in this paper.

From the error analysis in

Table 2. Error analysis of different methods to identify bridge vibration frequency.

	Measured Frequency	Frequency of Finite-Element Model Calculations	Equation Resolution Frequency	The Method Proposed in This Paper Recognizes the Frequency
Vibration fundamental frequency/Hz	0.843	0.839	0.707	0.84034
Relative error to measured fundamental frequency	0	0.47%	16.13%	0.32%

, it can be seen that the method proposed in this paper identifies the bridge vibration fundamental frequency. The accuracy meets the general engineering needs, avoiding the actual measurement steps of the fundamental frequency at the engineering site and providing a new idea for the rapid testing of bridge inspection vehicles.

The measured deflection response is preprocessed, and the empirically derived value is used as the cutoff frequency of the E-VMD method to obtain the deflection curve, which is combined with the Tikhonov regularization to construct the pathological equation for the identification of the influence line. The quasi-static deflection influence line is identified through the relative error analysis with the measured base frequency as the decomposition threshold to identify the deflection line of influence. It is found that the error is 14.02%, and the accuracy also does not meet the requirements of load-carrying capacity assessment, model correction, and other research accuracy, with the above identification of the bridge-operation modal frequency to re-use as the threshold value of IMF decomposition, re-peeling the dynamic components, restoring the quasi-static time curve, and then removing the axle effect, to obtain the updated deflection line of influence, at the same time, the measured deflection influence line is compared. The relative error of the measured deflection line is only 0.83%. The recognition accuracy is greatly improved compared with the previous period, which meets the subsequent research requirements, as shown in Figure 12 and Figure 13.

6. Conclusions

This paper takes a three-span short-tower cable-stayed bridge as an example, proposes a new method of the common identification of modal frequency and the deflection influence line of cable-stayed bridge operation under vehicle-caused excitation based on vehicle–bridge coupled vibration theory, and verifies the feasibility of the proposed method through the measured data of dynamic testing.

Based on the energy method, the analytical expression of the first-order vertical bending frequency of the short-tower cable-stayed bridge is derived by the energy method. The relative error between the measured bridge vibration frequency and the finite-element model-computed frequency is 16.13%, and the relative error between the measured bridge vibration frequency and the finite-element model-computed frequency is 15.7%. The analytical value is less accurate and cannot accurately describe the modal frequency of the structure. However, the base frequency of the bridge vibration based on the identification of the dynamic response of the vehicle crossing the bridge has high accuracy, and the relative error of the measured bridge vibration frequency is only 0.32%, avoiding the traditional modal-test time-domain method, which has cumbersome steps.

Having identified the bridge-operation fundamental frequency as the threshold and the deflection response modal decomposition, we establish the Influence-Line Identification Methods and eliminate the axle effect. Compared with the traditional Influence-Line Identification Methods based on the measured fundamental frequency, the relative error is only 0.83%, which eliminates traditional Influence-Line Identification Methods needing modal testing steps and has the superiority of higher accuracy with simple operation and convenience for on-site measurement. Compared with the traditional method of installing a vibration picker on the bridge, the proposed bridge vibration fundamental frequency and deflection line of influence common identification method only needs to set up a non-contact deflectometer under the bridge, which greatly shortens the duration of traffic closure to the testing of a sports car, and achieves a “non-contact” bridge vibration frequency and line influence of the common identification, realizing “non-contact” bridge inspection and evaluation. The bridge testing and evaluation is lightweight, providing the rapid detection of bridges and offering more economical new ideas.