Bridge influence line identification method and system based on weighted polynomial chirplet transform

Abstract

The application discloses a bridge influence line identification method and system based on a weighted polynomial frequency modulation wavelet transform, and the method comprises the following steps: performing time-frequency analysis on the acceleration response of a heavy vehicle passing through a bridge, and extracting the instantaneous frequency of the acceleration of the heavy vehicle passing through the bridge by using the weighted polynomial frequency modulation wavelet transform; calculating the vibration mode of the bridge by using the instantaneous frequency of the acceleration of the heavy vehicle passing through the bridge; and calculating the influence line of the bridge by using the vibration mode of the bridge. The application can reflect the characteristics of the frequency change of the vehicle-bridge coupling system when the heavy vehicle is at different positions of the bridge, can realize the extraction of the bridge influence line by using the vertical acceleration of the heavy vehicle passing through the bridge, is an indirect measurement method, has the advantages of simple operation and convenient field measurement, the influence lines of the bridge at different positions extracted by using the method are in good agreement with the reference influence line, and has a wide application prospect in the field of bridge health detection.

Description

Technical Field

[0001] This invention relates to the field of bridge health monitoring technology, specifically to a method and system for identifying bridge influence lines based on weighted polynomial frequency modulated wavelet transform. Background Technology

[0002] Bridge health monitoring is a common method for assessing the health status of bridges. It refers to a system that, during bridge operation, uses sensor systems to collect real-time data on the bridge's response to environmental stimuli, and employs various damage diagnosis techniques to continuously or periodically evaluate the bridge's condition and performance. Bridge health monitoring systems are typically installed on major highway bridges. However, for beam bridges, which constitute the largest number of bridges, cost constraints often prevent the installation of such systems; therefore, regular dynamic and static load tests are necessary to assess their health status.

[0003] Dynamic load testing measures the modal parameters of bridges, while static load testing measures the deflection of bridges under specific loads. These conventional tests typically require traffic closure for extended periods. Therefore, using vehicle detection to measure the bridge's influence line as a rapid detection method has gradually become a research hotspot. The bridge influence line is determined by the bridge's boundary constraints, geometric dimensions, and physical parameters, containing a wealth of information that fully reflects the bridge's structural performance. Numerous methods exist for extracting bridge influence lines, generally categorized into time-domain and frequency-domain methods. Time-domain methods require establishing a vehicle load matrix and a bridge response vector, and then using optimization algorithms to obtain the bridge's influence line. These methods often face difficulties such as the difficulty in constructing the load matrix, high computational cost, and the difficulty in achieving convergence of the optimization algorithm. Frequency-domain methods require performing Fourier transforms on both the vehicle load and the bridge response, followed by an inverse Fourier transform on their quotient. While this effectively reduces computational cost, it can lead to inaccurate peak selection. Furthermore, both time-domain and frequency-domain methods require installing sensors on the bridge and briefly closing traffic when detecting vehicles crossing. Therefore, traditional methods for extracting bridge influence lines are inconvenient and technically challenging in practical engineering applications. Summary of the Invention

[0004] This invention provides a method and system for identifying bridge influence lines based on weighted polynomial frequency modulated wavelet transform. The method extracts bridge influence lines by utilizing the vertical acceleration of heavy-load vehicles crossing the bridge, and the extracted influence lines at different locations on the bridge match the baseline influence lines well.

[0005] The present invention adopts the following technical solution:

[0006] On the one hand, a bridge influence line identification method based on weighted polynomial frequency modulated wavelet transform includes:

[0007] S101, perform time-frequency analysis on the acceleration response of heavy-duty vehicles crossing the bridge, and use weighted polynomial frequency-modulated wavelet transform to extract the instantaneous frequency of the acceleration of heavy-duty vehicles crossing the bridge.

[0008] S102, calculate the mode shape of the bridge based on the instantaneous frequency of the acceleration of heavy vehicles crossing the bridge;

[0009] S103, use the mode shape of the bridge to calculate the influence line of the bridge.

[0010] Preferably, in step S101, the analytical signal of the heavy-duty vehicle's acceleration across the bridge is obtained through Hilbert transform, as follows:

[0011] z(t) = a v (t)+jH(a v (t))#(9)

[0012] Where z(t) represents the analytic signal; j represents the imaginary unit; H(·) represents the Hilbert transform; a v (t) represents the acceleration of a heavily loaded vehicle crossing a bridge;

[0013] Formula (1) is further expressed as follows:

[0014]

[0015] Among them, z i (t) represents the analytic signal of the i-th frequency component; A i (t) represents the instantaneous amplitude of the i-th frequency component at time t; ω i (τ) represents the angular frequency of the i-th frequency component; τ is used as a dummy variable for time, representing a time point in the integration process; θ i This represents the initial phase of the i-th frequency component;

[0016] Weighted polynomial frequency-modulated wavelet transform of z(t) yields:

[0017]

[0018] In the formula,

[0019]

[0020]

[0021] in, Represents the frequency modulation function; bandpass filter w σ (t-t0) represents the signal used to select a specific time window; exp represents the exponential function; α i,k-1 t represents the frequency modulation parameter, and t represents the (k-1)th order frequency modulation coefficient of the i-th frequency component; k-1Used as part of the frequency modulation function to describe how the signal frequency changes over time; It represents a specific point in time; σ represents a parameter used to describe the width of the bandpass filter;

[0022] Extracting the instantaneous frequency of acceleration of heavy-load vehicles crossing a bridge presents the following optimization problem:

[0023]

[0024] in, t represents the coefficient vector; n Representing time raised to the power of n, it is a higher-order term in a polynomial; IF i (t) represents the instantaneous amplitude of the i-th frequency component of z(t) at time t after polynomial frequency-modulated wavelet transform; Amp i (t) represents the instantaneous frequency of the i-th frequency component of z(t) at time t after polynomial frequency-modulated wavelet transform. Let be the coefficient vector to be solved;

[0025] The coefficient vector is obtained through an iterative algorithm. The acceleration a of the heavy-duty vehicle crossing the bridge is calculated using equation (12). v The i-th instantaneous frequency of (t).

[0026] Preferably, in step S102, the mode shape of the bridge is calculated based on the instantaneous frequency of the acceleration of the heavy-load vehicle crossing the bridge, as shown below:

[0027]

[0028] in, The mode shape function represents the mode shape of the bridge at position x; m b The number 'l' represents the mass of the bridge; the number 'l' represents the length of the bridge. This represents the nth instantaneous frequency of the bridge. This represents the nth instantaneous frequency of the bridge at time t=0; express The instantaneous frequency of the bridge at time n; x represents the position on the bridge; v represents the speed of the vehicle; m v This indicates the mass of the vehicle.

[0029] Preferably, in step S103, the influence line of the bridge is calculated using the bridge's vibration modes, as shown below:

[0030]

[0031] Where, l(x) c ,t) represents the influence line function, and x represents the position of the beam in the simply supported beam bridge. cThe influence line of the bridge at time t; t represents time; l represents the bridge length; EI represents the bridge bending moment stiffness; n represents the mode number; Indicates position x c The mode shape of the nth mode at point n; This represents the mode shape of the nth mode at position x.

[0032] On the other hand, a bridge influence line identification system based on weighted polynomial frequency-modulated wavelet transform includes an accelerometer, a signal acquisition module, a time-frequency analysis module, a frequency-modulated wavelet transform module, a bridge vibration mode calculation module, and a bridge influence line calculation module.

[0033] The accelerometer is installed on the heavy-duty vehicle and is used to convert the vibration acceleration of the heavy-duty vehicle into a voltage signal to measure the vibration acceleration of the heavy-duty vehicle.

[0034] The signal acquisition module is used to receive the voltage signal output by the accelerometer, convert the analog voltage signal into a digital signal, and record it;

[0035] The time-frequency analysis module is used to receive the acceleration signal of the heavy-load vehicle output by the signal acquisition module, and to obtain the spectrum of the acceleration of the heavy-load vehicle crossing the bridge using short-time Fourier transform.

[0036] The frequency modulated wavelet transform module is used to receive the spectrum of the heavy-load vehicle's acceleration across the bridge output by the time-frequency analysis module, and to extract the instantaneous frequency of the heavy-load vehicle's acceleration across the bridge using weighted polynomial frequency modulated wavelet transform.

[0037] The bridge vibration mode calculation module is used to receive the instantaneous frequency of the acceleration of the heavy-load vehicle crossing the bridge output by the frequency-modulated wavelet transform module, and calculate the vibration mode of the bridge based on the instantaneous frequency of the acceleration of the heavy-load vehicle crossing the bridge.

[0038] The bridge influence line calculation module is used to receive the bridge vibration mode output by the bridge vibration mode calculation module, and calculate the bridge influence line based on the bridge vibration mode.

[0039] The beneficial effects of this invention are as follows:

[0040] (1) Based on the coupling effect between heavy-duty vehicles and bridges, this invention uses weighted polynomial frequency-modulated wavelet transform to extract the instantaneous frequency of the acceleration of heavy-duty vehicles crossing the bridge. The instantaneous frequency of the acceleration response of heavy-duty vehicles is the time-varying natural frequency of the bridge under the action of heavy-duty vehicles. The latter is directly related to the mode shape of the bridge. Therefore, the mode shape of the bridge can be obtained and the influence line of the bridge can be reconstructed based on the mode shape.

[0041] (2) This invention can reflect the frequency change characteristics of the vehicle-bridge coupling system when heavy-duty vehicles are at different positions on the bridge. It can extract the bridge influence line by using the vertical acceleration of heavy-duty vehicles crossing the bridge. It is a typical indirect measurement method with the advantages of simple operation and convenient on-site measurement. The influence lines of different positions of the bridge extracted by this method match the reference influence line well, and have broad application prospects in the field of bridge health detection.

[0042] The present invention will be further described in detail below with reference to the accompanying drawings and embodiments, but the bridge influence line identification method and system based on weighted polynomial frequency modulated wavelet transform of the present invention are not limited to the embodiments. Attached Figure Description

[0043] Figure 1 This is a flowchart of the bridge influence line identification method based on weighted polynomial frequency modulated wavelet transform according to an embodiment of the present invention;

[0044] Figure 2 This is a time-frequency analysis spectrum of the acceleration response of a heavy-duty vehicle according to an embodiment of the present invention;

[0045] Figure 3 This is a diagram illustrating the effect of the time-varying frequency order on the recognition performance in an embodiment of the present invention.

[0046] Figure 4 This is a structural block diagram of a bridge influence line identification system based on weighted polynomial frequency modulated wavelet transform, according to an embodiment of the present invention. Detailed Implementation

[0047] The present invention will be further described below through specific embodiments. It should be noted that the specific embodiments described herein are only for the convenience of illustrating and explaining the specific implementation of the present invention, and are not intended to limit the present invention.

[0048] To make the objectives and technical solutions of this invention clearer, the invention will be further described below with reference to the accompanying drawings and examples. It should be understood that the examples described herein are for illustrative purposes only and are not intended to limit the invention.

[0049] See Figure 1 As shown in the figure, this embodiment of a bridge influence line identification method based on weighted polynomial frequency modulated wavelet transform includes the following steps.

[0050] S101 performs time-frequency analysis on the acceleration response of heavy-duty vehicles crossing the bridge, and uses weighted polynomial frequency-modulated wavelet transform to extract the instantaneous frequency of the acceleration of heavy-duty vehicles crossing the bridge.

[0051] Specifically, the analytical signal of the acceleration of a heavy-duty vehicle crossing a bridge is obtained through Hilbert transform, as follows:

[0052] z(t) = av (t)+jH(a v (t))#(17)

[0053] Where z(t) represents the analytic signal; j represents the imaginary unit; H(·) represents the Hilbert transform; a v (t) represents the acceleration of a heavily loaded vehicle crossing a bridge;

[0054] Formula (1) is further expressed as follows:

[0055]

[0056] Among them, z i (t) represents the analytic signal of the i-th frequency component; A i (t) represents the instantaneous amplitude of the i-th frequency component at time t; ω i (τ) represents the angular frequency of the i-th frequency component; τ is used as a dummy variable for time, representing a time point in the integration process; θ i This represents the initial phase of the i-th frequency component;

[0057] Weighted polynomial frequency-modulated wavelet transform of z(t) yields:

[0058]

[0059] In the formula,

[0060]

[0061]

[0062] in, Represents the frequency modulation function; bandpass filter w σ (t-t0) represents the signal used to select a specific time window; exp represents the exponential function; α i,k-1 t represents the frequency modulation parameter, and t represents the (k-1)th order frequency modulation coefficient of the i-th frequency component; k-1 Used as part of the frequency modulation function to describe how the signal frequency changes over time; σ represents a specific point in time; σ represents a parameter used to describe the width of the bandpass filter.

[0063] Extracting the instantaneous frequency of acceleration of heavy-load vehicles crossing a bridge presents the following optimization problem:

[0064]

[0065] in, Represents the coefficient vector. and α i,k-1Used in the context of representing frequency modulation functions and polynomial fitting; t n Representing time raised to the power of n, it is a higher-order term in a polynomial; IF i (t) represents the instantaneous amplitude of the i-th frequency component of z(t) at time t after polynomial frequency-modulated wavelet transform; Amp i (t) represents the instantaneous frequency of the i-th frequency component of z(t) at time t after polynomial frequency-modulated wavelet transform. The coefficient vector to be solved

[0066] The coefficient vector is obtained through an iterative algorithm. The acceleration a of the heavy-duty vehicle crossing the bridge is calculated using equation (20). v The i-th instantaneous frequency of (t).

[0067] S102, calculate the mode shape of the bridge based on the instantaneous frequency of the acceleration of heavy-load vehicles crossing the bridge.

[0068] Specifically, the mode shape of the bridge is calculated based on the instantaneous frequency of the acceleration of heavy-load vehicles crossing the bridge, as shown below:

[0069]

[0070] in, The mode shape function represents the mode shape of the bridge at position x; m b The number 'l' represents the mass of the bridge; the number 'l' represents the length of the bridge. This represents the nth instantaneous frequency of the bridge. This represents the nth instantaneous frequency of the bridge at time t=0; express The instantaneous frequency of the bridge at time n; x represents the position on the bridge; v represents the speed of the vehicle; m v This indicates the mass of the vehicle.

[0071] It should be noted that the instantaneous frequency of the heavy-duty vehicle calculated by equation (20) includes the nth instantaneous frequency of the bridge. For example, if four frequencies (arranged from smallest to largest) are obtained by equation (20), since the frequency of the heavy-duty vehicle is much higher than the required first few frequencies of the bridge, it is easy to know that the first three frequencies are the first three instantaneous frequencies of the bridge (the fourth frequency is the frequency of the vehicle itself and is not needed).

[0072] S103, use the mode shape of the bridge to calculate the influence line of the bridge.

[0073] Specifically, the simply supported beam bridge at position x is obtained through mode shape calculation. c The influence line at the location is as follows:

[0074]

[0075] Where, l(x) c ,t) represents the influence line function, indicating the position x c The influence line of the bridge at time t; x c The location of the simply supported beam bridge is indicated by: t (time); l (bridge length); EI (bending moment stiffness); and n (modal number). Indicates position x c The mode shape of the nth mode at point n; This represents the mode shape of the nth mode at position x.

[0076] It should be noted that this is to find x. c The influence line at that point is l(x) c ,t). In this context, 'x' indicates that this is a function related to the spatial coordinate 'x', not the influence line at 'x'. yes Any point in it.

[0077] The following numerical simulation using the finite element software ABAQUS is used for verification.

[0078] A vehicle-bridge coupled model was established using the general-purpose finite element software ABAQUS. The heavy-duty vehicle was modeled using rigid mass points and springs, while the bridge was modeled using two-node beam elements (B23), with simply supported boundary conditions at both ends. The parameters used in the model are as follows: beam length L = 40m, density ρ = 2400kg / m³. 3 The cross-sectional area A = 18m² 2 The moment of inertia of the cross section is I = 13.5m. 4 The material's elastic modulus E = 60 GPa, and its mass per meter is m. b = 43200kg; Vehicle mass m v =100000 kg, stiffness K = 10 GPa, velocity v = 5 m / s. Therefore, the first three natural frequencies of the bridge are 4.251 Hz, 17.004 Hz, and 32.250 Hz, respectively. The dynamic response of the heavy-load vehicle and the bridge was calculated using implicit analysis in ABAQUS with a time step of 0.001 s. A short-time Fourier transform was performed on the acceleration response of the heavy-load vehicle obtained from the numerical simulation; its time-frequency analysis is described in [reference needed]. Figure 2 As shown in the figure, the first three time-varying frequencies of the bridge can be clearly seen. These frequencies can be obtained using a weighted polynomial frequency-modulated wavelet transform, thus revealing the influence lines of the bridge at different locations.

[0079] See Figure 3The diagram compares the bridge influence lines obtained using this method with the baseline influence lines obtained through numerical simulation. When only the first-order time-varying frequency (n=1) is used, the influence line at the mid-span of the bridge almost completely coincides with the baseline influence line, while the influence lines at the quarter-span and third-span locations show significant errors compared to the baseline. When the first three time-varying frequencies are used (n=3), the influence lines at the three different locations of the bridge match the baseline influence line well. Therefore, in actual measurements, if the acceleration response of heavy-load vehicles crossing the bridge includes the first three time-varying frequencies of the bridge, the method proposed in this paper can identify the influence lines at different locations of the bridge.

[0080] See Figure 4 As shown, this embodiment also discloses a bridge influence line identification system based on weighted polynomial frequency-modulated wavelet transform, including an accelerometer 401, a signal acquisition module 402, a time-frequency analysis module 403, a frequency-modulated wavelet transform module 404, a bridge vibration mode calculation module 405, and a bridge influence line calculation module 406.

[0081] The accelerometer 401 is installed on the heavy-duty vehicle and is used to convert the vibration acceleration of the heavy-duty vehicle into a voltage signal to measure the vibration acceleration of the heavy-duty vehicle.

[0082] The signal acquisition module 402 is used to receive the voltage signal output by the accelerometer 401, convert the analog voltage signal into a digital signal, and record it;

[0083] The time-frequency analysis module 403 is used to receive the acceleration signal of the heavy-load vehicle output by the signal acquisition module 402, and to obtain the spectrum of the acceleration of the heavy-load vehicle crossing the bridge using short-time Fourier transform.

[0084] The frequency modulated wavelet transform module 404 is used to receive the spectrum of the heavy-load vehicle's bridge-crossing acceleration output by the time-frequency analysis module 403, and extract the instantaneous frequency of the heavy-load vehicle's bridge-crossing acceleration using weighted polynomial frequency modulated wavelet transform.

[0085] The bridge vibration mode calculation module 405 is used to receive the instantaneous frequency of the acceleration of the heavy-load vehicle crossing the bridge output by the frequency-modulated wavelet transform module 404, and calculate the vibration mode of the bridge based on the instantaneous frequency of the acceleration of the heavy-load vehicle crossing the bridge.

[0086] The bridge influence line calculation module 406 is used to receive the bridge vibration mode output by the bridge vibration mode calculation module 405, and calculate the bridge influence line based on the bridge vibration mode.

[0087] For a detailed implementation of the frequency modulation wavelet transform module and the mode shape and influence line calculation module in a bridge influence line identification system based on weighted polynomial frequency modulation wavelet transform, please refer to a bridge influence line identification method based on weighted polynomial frequency modulation wavelet transform, which will not be repeated in this embodiment.

[0088] It should be understood that those skilled in the art can make improvements and modifications based on the above description, and all such improvements and modifications should fall within the protection scope of the appended claims.

Claims

1. A method for identifying bridge influence lines based on weighted polynomial frequency-modulated wavelet transform, characterized in that, Applicable to simply supported beam bridges, this method indirectly identifies the bridge's influence line by collecting the vertical acceleration response of heavy-load vehicles crossing the bridge and combining it with the vehicle-bridge coupled dynamics relationship. Specifically, it includes the following steps: S101, Time-frequency analysis of the vertical acceleration response of heavy-duty vehicles crossing the bridge: First, the measured acceleration signal is constructed into a complex analytic signal through Hilbert transform. Then, a weighted polynomial frequency-modulated wavelet transform is performed on the complex analytic signal. The instantaneous frequency change law of the signal is fitted by a polynomial function. The signal components within a specific time window are selected by combining the Gaussian window function. The weighted least squares optimization objective is constructed with the instantaneous amplitude of the signal as the weight. By iteratively solving the frequency modulation coefficient vector, the multi-order instantaneous frequencies contained in the signal are extracted. The inherent frequency components of the vehicle itself are removed from the vector, and the instantaneous frequencies of the corresponding bridge vibration are retained. S102, Calculate the bridge vibration modes based on the instantaneous frequencies of the bridge vibration: According to the correspondence between the moving mass and the instantaneous frequency shift of the system in the vehicle-bridge coupling dynamics, combined with the driving speed of the heavy-duty vehicle, the instantaneous frequencies corresponding to different times are mapped to the frequency shifts at different positions of the bridge, and then the distribution functions of the bridge vibration modes along the bridge length direction are obtained. S103, Calculate the bridge influence line using the various modes of vibration of the bridge: Based on the modal superposition principle of simply supported beams, the influence line of the simply supported beam bridge at a specified location is reconstructed by weighted summation according to the fourth power of the reciprocal of the mode order based on the mode function of each mode.

2. The bridge influence line identification method based on weighted polynomial frequency modulated wavelet transform according to claim 1, characterized in that, In step S101, the analytical signal of the acceleration of the heavy-duty vehicle crossing the bridge is obtained through Hilbert transform, as follows: ； in, Indicates an analytical signal; Represents the imaginary unit. Represents the Hilbert transform; This indicates the acceleration of a heavily loaded vehicle crossing a bridge. Formula (1) is further expressed as follows: ； in, This represents the analytic signal of the i-th frequency component; Indicates the i-th frequency component in The instantaneous amplitude at a given moment; τ represents the angular frequency of the i-th frequency component; τ is used as a dummy variable for time, representing a time point in the integration process; This represents the initial phase of the i-th frequency component; right The weighted polynomial frequency-modulated wavelet transform yields: ； In the formula, ； ； in, Frequency modulation function; bandpass filter This indicates the signal used to select a specific time window; Represents an exponential function; Indicates the frequency modulation parameter, indicating the first The first frequency component Frequency modulation coefficient; Used as part of the frequency modulation function to describe how the signal frequency changes over time; Indicates a specific point in time; This represents a parameter used to describe the width of the bandpass filter; Extracting the instantaneous frequency of acceleration of heavy-load vehicles crossing a bridge presents the following optimization problem: ； in, Represents the coefficient vector; Representing time raised to the power of n, it is a higher-order term in a polynomial; express The instantaneous amplitude of the i-th frequency component at time t after polynomial frequency-modulated wavelet transform; express The instantaneous frequency of the i-th frequency component at time t after polynomial frequency-modulated wavelet transform. Let be the coefficient vector to be solved; The coefficient vector is obtained through an iterative algorithm. The acceleration of the heavy-duty vehicle crossing the bridge is calculated using equation (4). The i-th instantaneous frequency.

3. The bridge influence line identification method based on weighted polynomial frequency modulated wavelet transform according to claim 1, characterized in that, In step S102, the mode shape of the bridge is calculated based on the instantaneous frequency of the acceleration of heavy-load vehicles crossing the bridge, as shown below: ； in, The mode shape function represents the mode shape of the bridge at position x. Indicates the mass of the bridge; Indicates the length of the bridge; This represents the nth instantaneous frequency of the bridge. This represents the nth instantaneous frequency of the bridge at time t=0; express The nth instantaneous frequency of the bridge at time t; Indicates the location on the bridge; Indicates the vehicle's speed; This indicates the mass of the vehicle.

4. The bridge influence line identification method based on weighted polynomial frequency modulated wavelet transform according to claim 1, characterized in that, In step S103, the influence line of the bridge is calculated using the bridge's vibration modes, as shown below: ； in, The influence line function represents the location of the influence line in a simply supported beam bridge. The influence line of the bridge at time t; Indicates time; Indicates the length of the bridge; Indicates the bending moment stiffness of a bridge; Indicates the modal number; Indicates the location The first Mode shape; Indicates the location The first Mode shape.

5. A bridge influence line identification system based on weighted polynomial frequency modulated wavelet transform, characterized in that, The method for implementing the method of claim 1 includes an accelerometer, a signal acquisition module, a time-frequency analysis module, a frequency-modulated wavelet transform module, a bridge vibration mode calculation module, and a bridge influence line calculation module; The accelerometer is installed on the heavy-duty vehicle and is used to convert the vibration acceleration of the heavy-duty vehicle into a voltage signal to measure the vibration acceleration of the heavy-duty vehicle. The signal acquisition module is used to receive the voltage signal output by the accelerometer, convert the analog voltage signal into a digital signal, and record it; The time-frequency analysis module is used to receive the acceleration signal of the heavy-load vehicle output by the signal acquisition module, and to obtain the spectrum of the acceleration of the heavy-load vehicle crossing the bridge using short-time Fourier transform. The frequency modulated wavelet transform module is used to receive the spectrum of the heavy-load vehicle's acceleration across the bridge output by the time-frequency analysis module, and to extract the instantaneous frequency of the heavy-load vehicle's acceleration across the bridge using weighted polynomial frequency modulated wavelet transform. The bridge vibration mode calculation module is used to receive the instantaneous frequency of the acceleration of the heavy-load vehicle crossing the bridge output by the frequency-modulated wavelet transform module, and calculate the vibration mode of the bridge based on the instantaneous frequency of the acceleration of the heavy-load vehicle crossing the bridge. The bridge influence line calculation module is used to receive the bridge vibration mode output by the bridge vibration mode calculation module, and calculate the bridge influence line based on the bridge vibration mode.

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