Bridge response reconstruction method based on main contribution mode superposition

Abstract

A bridge response reconstruction method based on main contribution mode superposition comprises the following steps: firstly, testing a bridge influence line, firstly obtaining a deflection influence line of a bridge, calculating a rigidity distribution function of the bridge through the deflection influence line, and then obtaining a vibration mode matrix of the bridge through a characteristic value method by combining a quality function obtained by a bridge design drawing. And analyzing the order of the main contribution mode by using the responded singular value spectrum, obtaining the vibration mode of the main contribution mode in combination with the vibration mode matrix, and substituting the measurement point vibration mode of the main contribution mode into a measurement point bridge response calculation formula based on mode superposition to obtain modal response. And substituting the vibration mode of the unmeasured area of the main contribution mode and the modal response into a bridge response calculation formula based on modal superposition, and reconstructing the response of the unmeasured area of the bridge. In the bridge unmeasured area response reconstruction theory, the error between the modal shape obtained by the finite element model and the actual modal shape is solved, and the precision of the modal response and the reconstruction response of the unmeasured area is improved.

Description

Technical Field

[0001] The present invention belongs to the technical field of engineering structure detection data analysis, and relates to a bridge response reconstruction method based on the superposition of main contribution modes. Background Art

[0002] Structural testing is an important means of regularly maintaining and ensuring structural safety. This often requires analyzing bridge health by collecting sensors to measure bridge responses. However, in bridge testing, the number of sensors that can be placed on the bridge is limited. Furthermore, given the often complex environments of bridge structures, some critical locations may not be accessible for sensor placement. Therefore, reconstructing the responses of unknown points from the responses of a limited number of measurement points is crucial for bridge health monitoring.

[0003] The core idea of response reconstruction is to first select an appropriate modeling method to establish a mathematical model of the bridge, and then use mathematical models such as the sensor's measurement point response and the finite element model to estimate the response of the unknown point of interest. It has been widely used in bridge dynamics research such as damage identification and model updating.

[0004] There are many methods for response reconstruction in actual tests. In 2021, Zhang Xiaohua et al. proposed a structural response reconstruction method based on a moving window Kalman filter algorithm, and effectively estimated the measurement noise variance in real time; in 2022, Zou Yunfeng et al. proposed a dynamic response reconstruction method combining empirical mode decomposition and model reduction; in 2021, Li et al. proposed a method for estimating external inputs and reconstructing structural responses using a two-dimensional multi-scale model, and demonstrated the high accuracy of this method through simulation of a complex bridge finite element model and field experiments; in 2023, Yang et al. proposed a time-domain reconstruction method combining the modal synthesis method and empirical mode decomposition with a discontinuity criterion, which can quickly obtain the overall response of the structure. At present, the problem with response reconstruction theory is that there is an error between the modal vibration shape obtained by the finite element model and the actual modal vibration shape, and it is difficult to accurately identify the modal response. Summary of the Invention

[0005] The present invention aims to provide a bridge response reconstruction method based on the superposition of main contributing modes, so as to solve the problem that there is an error between the vibration mode of the finite element model and the actual mode vibration mode during the bridge response reconstruction process, and it is difficult to accurately identify the modal response.

[0006] The technical solution of the present invention:

[0007] A bridge response reconstruction method based on the superposition of major contributing modes begins with bridge influence line testing to obtain the bridge's deflection influence line. The bridge's stiffness distribution function is then calculated using the deflection influence line. The bridge's modal matrix is then derived using the eigenvalue method, combining the mass function obtained from the bridge design drawings. The order of the major contributing modes is analyzed using the response singular value spectrum, and the modal shapes of the major contributing modes are obtained using the modal matrix. The modal shapes of the measured points of the major contributing modes are then incorporated into the bridge response calculation formula based on modal superposition to obtain the modal response. The modal shapes and modal responses of the unmeasured regions of the bridge are then incorporated into the bridge response calculation formula based on modal superposition to reconstruct the response of the unmeasured regions.

[0008] A bridge response reconstruction method based on the superposition of major contributing modes is proposed. The steps are as follows:

[0009] (1) Identification of bridge influence lines and mid-span bending moments

[0010] By exciting the bridge with a unit moving load, the corresponding relationship between the unit moving load and the mid-span deflection of the bridge is obtained, which is the deflection influence line DIL of the bridge. A (x); Apply a virtual load to the mid-span position of the bridge through structural mechanics theory to calculate the bridge bending moment distribution corresponding to the bridge deflection influence line. When a virtual vertical unit force is applied to the mid-span position of the bridge, the support reaction force F and the distance x between the calculation moment point and the support are used to solve the bending moment distribution M(x) = F·x;

[0011] (2) Obtaining the vibration matrix of the bridge

[0012] The second derivative of the bridge deflection influence line And the bridge bending moment distribution M(x) when the vertical unit force acts at the mid-span position of the bridge is brought into the bridge stiffness calculation formula Calculate the bridge stiffness function EI(x); Combine the bridge stiffness function with the mass function obtained from the bridge drawings and obtain the modal vibration matrix through eigenvalue decomposition in is the vibration shape of different modes, N is the total number of modes;

[0013] (3) Calculation of bridge power spectrum density function matrix

[0014] Obtain the displacement response of the bridge measuring point y=[y1(t),y2(t),y3(t)…y I (t)], I is the number of sensors, and the displacement response is used to obtain its correlation function R y (τ), perform Fourier transform on the correlation function to obtain the power spectrum density function, and then obtain the power spectrum density function matrix G yy (ω);

[0015] (4) Analyze the vibration mode of the measuring point through the response of the bridge measuring point

[0016] By decomposing the power spectrum density function matrix of the displacement response of the bridge measurement point by singular value decomposition, the curve s(ω) of the maximum singular value of the bridge with respect to frequency is obtained, where the peak value of the curve corresponds to the natural frequency ω of the bridge n (n=1~N), determine the modal order n=1~N corresponding to the natural frequency in the order of frequency from small to large;

[0017] (5) Selection of main contributing modes

[0018] Due to the limitation of the number of bridge measurement points, the first I peaks s(ω i )(i=1~I), the corresponding mode is the main contribution mode, and the number of modes selected as the main contribution mode is the same as the number of sensors; the corresponding modal order is determined according to the natural frequency of the main contribution mode, and the vibration shape of the main contribution mode is determined in combination with the modal vibration shape matrix Φ

[0019] (6) Modal response calculation

[0020] The vibration shape of the main contributing mode of the bridge Extract the vibration mode values corresponding to the bridge measuring points and form the vibration mode of the bridge measuring points with the main contributing modes. The displacement response of the bridge measuring point y=[y1(t),y2(t),y3(t)…y I (t)] is brought into the equation formed by the mode superposition method to solve the modal response of the main contributing mode q(t)=[q1(t),q2(t),q3(t)…q I (t)]:

[0021]

[0022] (7) Reconstruction of responses in unmeasured areas

[0023] From the vibration shape of the main contributing mode Extract the vibration mode value of the unmeasured point The modal response of the main contributing mode obtained by the solution is q(t)=[q1(t),q2(t),q3(t)…q I (t)] is brought into the formula formed by the mode superposition method to calculate the response y of the unmeasured point w , reconstruct the response of the unmeasured area of the bridge:

[0024]

[0025] A response reconstruction device for an unmeasured area of a bridge comprises: an acquisition module for acquiring measured response data of the structure;

[0026] A memory device for storing measured data and a data processing program;

[0027] A processor is configured to execute a data processing program stored in the memory. When the data processing program is executed, the processor is configured to:

[0028] The measured response data of the structure is read, wherein the response data is collected and stored over a period of time; and a data processing program is used to analyze the measured response data to reconstruct the bridge response of the unmeasured area.

[0029] The beneficial effects of the present invention are as follows: in the theory of reconstruction of the response of unmeasured areas of bridges, the error between the modal vibration shape obtained by the finite element model and the actual modal vibration shape is resolved, and the accuracy of the modal response and the reconstructed response of the unmeasured area is improved. DETAILED DESCRIPTION

[0030] The specific implementation of the present invention is further explained below in conjunction with the technical solutions and drawings.

[0031] Take a uniform cross-section simply supported beam model as an example. The length of the uniform cross-section simply supported beam is 20m, the mass is 150Kg / m, and the stiffness is 3×10 6 N / m 2 The damping ratio of each mode is set to 0.005, and a sensor is placed every 2.5 m to collect the response, with a total of 7 sensors.

[0032] The excitation is in the form of 20 Hz sinusoidal excitation, and the response signal is the displacement response collected by the sensor.

[0033] (1) By exciting the bridge with a unit moving load, the corresponding relationship between the load position and the mid-span deflection of the bridge is obtained, which is the deflection influence line DIL of the bridge. A (x), and calculate the bridge bending moment distribution M(x) when a vertical unit load acts at the mid-span using structural mechanics theory.

[0034] (2) The second-order derivative of the bridge deflection influence line And the bridge bending moment distribution M(x) when the unit load acts on the bridge span is brought into the bridge stiffness calculation formula The stiffness function EI(x) of the bridge is calculated, and the vibration mode matrix is obtained by combining the stiffness function of the bridge with the mass function obtained from the bridge drawings through eigenvalue decomposition. in are the vibration shapes of different modes.

[0035] (3) Obtain the bridge measurement point response y = [y1(t), y2(t), y3(t)…y6(t)] and use the displacement response to obtain its correlation function Perform Fourier transform on the correlation function to obtain the power spectrum density function, and then obtain the power spectrum density function matrix The diagonal line is the auto-power spectral density function of each degree of freedom, and the off-diagonal line is the cross-spectral density function between different degrees of freedom;

[0036] (4) Singular value decomposition of the power spectrum matrix of the bridge measurement point response is used to obtain the curve s(ω) of the bridge's maximum singular value with respect to frequency, where the peak of the maximum singular value curve corresponds to the bridge's natural frequency ω n (n=1~N), determine the modal order n=1~N corresponding to the natural frequency in order from small to large frequency.

[0037] (5) Select the first 6 peaks s(ω) according to the maximum singular value peak of the bridge response from high to low i )(i=1~6), the corresponding mode is the main contribution mode, and the corresponding modal order is determined according to the natural frequency of the main contribution mode, and the vibration shape of the main contribution mode is determined in combination with the modal vibration shape matrix Φ

[0038] (6) From the main contributing modal vibration mode of the bridge Extract the vibration mode value corresponding to the measuring point to form the vibration mode of the measuring point that mainly contributes to the mode Together with the measurement point response y = [y1(t), y2(t), y3(t)…y6(t)], the following equation constructed by the mode superposition method is solved to obtain the modal response q(t) = [q1(t), q2(t), q3(t)…q6(t)] of the main contributing mode:

[0039]

[0040] (7) Select the 7th measurement point as the unmeasured point for response reconstruction, and compare the sensor measured response with the reconstructed response, and analyze the main contribution mode vibration shape. The vibration mode value of the unmeasured point Extract and solve the modal response q(t) = [q1(t), q2(t), q3(t)…q6(t)] and put it into the following formula constructed by the mode superposition method to calculate the response y of the unmeasured point w , reconstruct the response of the unmeasured area of the bridge:

[0041]

[0042] Figure 1 Shown is the reconstructed response compared to the sensor's measured response.

[0043] From the analysis results, it can be obtained that the present invention can effectively reconstruct the response of the unmeasured area of the bridge, and the error with the actual response of the bridge is small.

Claims

1. A bridge response reconstruction method based on the superposition of main contribution modes, characterized by: Here are the steps: (1) Identification of bridge influence lines and mid-span bending moments By exciting the bridge with a unit moving load, the corresponding relationship between the unit moving load and the mid-span deflection of the bridge is obtained, which is the deflection influence line DIL of the bridge. A (x); Apply a virtual load to the mid-span position of the bridge through structural mechanics theory to calculate the bridge bending moment distribution corresponding to the bridge deflection influence line. When a virtual vertical unit force is applied to the mid-span position of the bridge, the support reaction force F and the distance x between the calculation moment point and the support are used to solve the bending moment distribution M(x) = F·x; (2) Obtaining the vibration matrix of the bridge The second derivative of the bridge deflection influence line And the bridge bending moment distribution M(x) when the vertical unit force acts at the mid-span position of the bridge is brought into the bridge stiffness calculation formula Calculate the bridge stiffness function EI(x); Combine the bridge stiffness function with the mass function obtained from the bridge drawings and obtain the modal vibration matrix through eigenvalue decomposition in is the vibration shape of different modes, N is the total number of modes; (3) Calculation of bridge power spectrum density function matrix Obtain the displacement response of the bridge measuring point y=[y1(t),y2(t),y3(t)…y I (t)], I is the number of sensors, and the displacement response is used to obtain its correlation function R y (τ), perform Fourier transform on the correlation function to obtain the power spectrum density function, and then obtain the power spectrum density function matrix G yy (ω); (4) Analyze the vibration mode of the measuring point through the response of the bridge measuring point By decomposing the power spectrum density function matrix of the displacement response of the bridge measurement point by singular value decomposition, the curve s(ω) of the maximum singular value of the bridge with respect to frequency is obtained, where the peak value of the curve corresponds to the natural frequency ω of the bridge n (n=1~N), determine the modal order n=1~N corresponding to the natural frequency in the order of frequency from small to large; (5) Selection of main contributing modes Due to the limitation of the number of bridge measurement points, the first I peaks s(ω i )(i=1~I), the corresponding mode is the main contribution mode, and the number of modes selected as the main contribution mode is the same as the number of sensors; the corresponding modal order is determined according to the natural frequency of the main contribution mode, and the vibration shape of the main contribution mode is determined in combination with the modal vibration shape matrix Φ (6) Modal response calculation The vibration shapes of the main contributing modes of the bridge Extract the vibration mode values corresponding to the bridge measuring points and form the vibration mode of the bridge measuring points with the main contributing modes. The displacement response of the bridge measuring point y=[y1(t),y2(t),y3(t)…y I (t)] is brought into the equation formed by the mode superposition method to solve the modal response of the main contributing mode q(t)=[q1(t),q2(t),q3(t)…q I (t)]: (7) Reconstruction of responses in unmeasured areas From the vibration shape of the main contributing mode Extract the vibration mode value of the unmeasured point The modal response of the main contributing mode obtained by the solution is q(t)=[q1(t),q2(t),q3(t)…q I (t)] is brought into the formula formed by the mode superposition method to calculate the response y of the unmeasured point w , reconstruct the response of the unmeasured area of the bridge:

2. A response reconstruction device for an unmeasured area of a bridge, characterized in that: The response reconstruction device for an unmeasured region of a bridge includes: an acquisition module for acquiring measured response data of the structure; A memory device for storing measured data and a data processing program; A processor is configured to execute a data processing program stored in the memory. When the data processing program is executed, the processor is configured to: The measured response data of the structure is read, wherein the response data is collected and stored over a period of time; and a data processing program is used to analyze the measured response data to reconstruct the bridge response of the unmeasured area.

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