Bridge damping ratio and vibration mode identification method based on optimization fitting strategy

Abstract

The invention discloses a bridge damping ratio and vibration mode identification method based on an optimization fitting strategy, and relates to the technical field of bridge damping ratio and vibration mode identification, and the method comprises the steps: building a kinetic equation of an axle system at a static balance position, calculating or measuring the acceleration response of a vehicle, and inverting the acceleration response of a contact point; extracting a target modal acceleration component of the bridge by using empirical wavelet transform, and calculating an envelope line of the target modal acceleration component by using Hilbert transform; dividing the extracted envelope into a left part and a right part in time, and extracting damping characteristic information by calculating the ratio of symmetrical points of the envelope of the left part and the right part; and carrying out logarithm on the envelope ratio to carry out linear fitting so as to identify the damping ratio and extract the distortionless vibration mode. Therefore, by adopting the bridge damping ratio and vibration mode identification method based on the optimization fitting strategy, the identification precision of the vibration mode of the bridge can be improved, the damping ratio and the frequency of the bridge do not need to be directly calculated, and the method has wider applicability.

Description

Technical Field

[0001] The present invention relates to the technical field of bridge damping ratio and mode shape identification, and particularly to a method for identifying bridge damping ratio and mode shape based on an optimized fitting strategy. Background Art

[0002] As an important part of modern transportation infrastructure, the safety and durability of bridges are directly related to public safety and economic operation. Bridge health monitoring (Structural Health Monitoring, SHM) is one of the key technologies to ensure the long-term safe operation of bridges. Among them, the identification of the damping ratio and mode shape of bridges is an important parameter for evaluating the health status of bridge structures. The damping ratio reflects the energy dissipation capacity of the bridge structure, while the mode shape describes the deformation mode of the bridge during vibration. Accurately identifying these parameters is of great significance for bridge damage detection, performance evaluation and maintenance decision-making.

[0003] Traditional methods for identifying bridge damping ratio and mode shape mainly rely on modal analysis techniques, such as frequency domain decomposition method, stochastic subspace method and time domain modal analysis method, etc. These methods usually require a large amount of sensor data and complex calculation processes, and are sensitive to noise and local instantaneous amplitude fluctuations, resulting in insufficient accuracy and robustness of the identification results. In addition, when dealing with the dynamic response of the vehicle-bridge coupling system, traditional methods often have difficulty in accurately separating the vibration signals of vehicles and bridges, further increasing the identification difficulty.

[0004] In recent years, with the continuous development of signal processing technology, methods based on Empirical Wavelet Transform (EWT) and Hilbert Transform (HT) have been gradually introduced into the field of bridge health monitoring. EWT can adaptively divide the frequency band of signals and effectively extract the vibration modes of bridges, while HT can extract the envelope characteristics of signals, providing a new way for the identification of damping ratio. However, existing methods still have the influence of local instantaneous amplitude fluctuations on the damping identification results when dealing with the dynamic response of the vehicle-bridge coupling system, resulting in low identification accuracy.

[0005] To solve the above problems, the present invention proposes a method for identifying bridge damping ratio and undistorted mode shape based on an optimized fitting strategy to weaken the influence of local instantaneous amplitude fluctuations on damping identification, thereby improving the accuracy and stability of the identification results. Summary of the Invention

[0006] The purpose of the present invention is to provide a method for identifying bridge damping ratio and mode shape based on an optimized fitting strategy, which can improve the identification accuracy of bridge mode shapes, does not require direct calculation of the damping ratio and frequency of bridges, and has a wider range of applicability.

[0007] To achieve the above object, the present invention provides a method for identifying the damping ratio and vibration mode of a bridge based on an optimized fitting strategy, including the following steps:

[0008] S1. Establish the dynamic equation of the vehicle-bridge system at the static equilibrium position, calculate or measure the vehicle acceleration response, and invert the acceleration response of the contact point;

[0009] S2. Extract the target modal acceleration component of the bridge from the acceleration response of the contact point based on the empirical wavelet transform, and use the Hilbert transform to extract the corresponding envelope;

[0010] S3. On the time axis, divide the envelope of the target modal acceleration component into left and right parts, and extract the damping characteristic information by calculating the ratio of the symmetric points of the left and right envelopes;

[0011] S4. Based on the damping characteristic information, perform linear fitting by the least squares method to identify the damping ratio of the bridge, and at the same time, combine the envelope to extract the modal vibration mode of the bridge without damping interference.

[0012] Preferably, in step S1, the dynamic equation of the vehicle-bridge system at the static equilibrium position is constructed, including:

[0013] The motion equation of the vehicle:

[0014]

[0015] The dynamic equation of the bridge:

[0016]

[0017] where, m v is the vehicle body mass, k v is the support spring stiffness, m is the mass per unit length of the bridge, c is the damping coefficient, EI is the bending stiffness of the bridge, u(x, t), y v (t) are the vertical displacements of the bridge and the vehicle respectively, u c (t) is the vehicle-bridge contact response, are the first and second derivatives of u(x, t) with respect to time t respectively, u””(x, t) is the fourth derivative of u(x, t) with respect to the longitudinal coordinate x, is the vehicle acceleration response, δ represents the Dirac function, and g is the acceleration due to gravity.

[0018] Preferably, step S2 includes:

[0019] First, perform Fourier transform on the acceleration signal of the contact point, adaptively divide the frequency band according to the local minimum value, and construct the corresponding empirical wavelet filter to decompose the acceleration signal of the contact point by the empirical wavelet transform;

[0020] Then, compare the energy distributions of the decomposed sub-signals, determine the sub-signal containing the modal response of the bridge target, i.e., the target modal acceleration component, and extract its envelope through Hilbert transform.

[0021] Preferably, in step S3, the time coordinates of the characteristic points corresponding to the left and right envelope lines are mirror-symmetric about the axis of symmetry.

[0022] Preferably, extract the damping characteristic information through the ratio of the corresponding characteristic points of the left and right envelope lines.

[0023] Preferably, taking the time t = L / (2v) as the center of symmetry and the time interval Δt on both sides of the axis of symmetry as a variable, reconstruct the left and right envelope line functions of the target modal acceleration component to obtain the reconstruction functions and as follows:

[0024]

[0025] In the formula, is the reconstruction function of the left part of the envelope line, is the reconstruction function of the right part of the envelope line, Δt is the time interval from the current time to the middle axis ξ n is the damping ratio of the bridge, ω bn represents the natural frequency of the nth mode, B Dn represents the initial vibration amplitude of the nth mode, Ω n is the driving frequency.

[0026] Preferably, the ratio of the symmetric points of the left and right envelope lines is expressed as follows:

[0027]

[0028] In the formula, E n (Δt) is the ratio of the symmetric points of the left and right envelope lines.

[0029] Preferably, step S4 includes using the least squares method for linear fitting to obtain the corresponding line slope k and calculating the damping ratio of the bridge; at the same time, multiplying the damping mode by the exponential term e kt / 2 and normalizing it to obtain the modal shape of the bridge without damping interference.

[0030] Preferably, the expression of the bridge damping ratio is as follows:

[0031]

[0032] Among them, ω bn is directly obtained from the Fourier transform of the acceleration signal at the contact point.

[0033] Preferably, the modal vibration mode of the bridge without damping interference is expressed as follows:

[0034]

[0035] In the formula, M n (t) is the modal vibration mode of the bridge without damping interference.

[0036] Therefore, by adopting the above method for identifying the damping ratio and vibration mode of a bridge based on an optimization fitting strategy, the present invention has the following technical effects:

[0037] The present invention combines empirical wavelet transform and Hilbert transform, can decompose and extract features of signals more precisely, reduces the influence of local fluctuations on damping identification by using symmetry analysis, improves the robustness of calculation results, realizes the extraction of undamped modal vibration modes, and does not require direct calculation of the damping ratio and frequency of the bridge.

[0038] Next, through the drawings and embodiments, the technical solutions of the present invention will be further described in detail. Description of the Drawings

[0039] Figure 1 is a vehicle-bridge coupling schematic diagram in an embodiment of a method for identifying the damping ratio and vibration mode of a bridge based on an optimization fitting strategy;

[0040] Figure 2 is a schematic diagram of the left and right division of the damped mode with t as the abscissa in an embodiment of a method for identifying the damping ratio and vibration mode of a bridge based on an optimization fitting strategy, where (a) is the first order and (b) is the second order;

[0041] Figure 3 is a schematic diagram of the left and right division of the damped mode with Δt as the abscissa in an embodiment of a method for identifying the damping ratio and vibration mode of a bridge based on an optimization fitting strategy, where (a) is the first order and (b) is the second order;

[0042] Figure 4 is the fitting result of the logarithmic envelope ratio in an embodiment of a method for identifying the damping ratio and vibration mode of a bridge based on an optimization fitting strategy, where (a) is the first order and (b) is the second order;

[0043] Figure 5 is the undamped mode recovery result in an embodiment of a method for identifying the damping ratio and vibration mode of a bridge based on an optimization fitting strategy, where (a) is the first order and (b) is the second order. Detailed Embodiment

[0044] The present invention can be explained in more detail by the following embodiments. The purpose of disclosing the present invention is to protect all changes and improvements within the scope of the present invention. The present invention is not limited to the following embodiments.

[0045] The present invention provides a method for identifying the damping ratio and vibration mode of a bridge based on an optimized fitting strategy, comprising the following steps:

[0046] S1. As Figure 1 shown, establish the dynamic equation of the vehicle-bridge system at the static equilibrium position, calculate or measure the vehicle acceleration response, and invert the acceleration response at the contact point.

[0047] Among them, the dynamic equation of the vehicle-bridge system at the static equilibrium position includes:

[0048] The motion equation of the vehicle:

[0049]

[0050] The dynamic equation of the bridge:

[0051]

[0052] Among them, m v is the vehicle body mass, k v is the support spring stiffness, m represents the mass per unit length of the bridge, L is the bridge length, c is the damping coefficient, and EI is the bending stiffness of the bridge. The vertical displacements of the bridge and the vehicle are denoted as u(x, t) and y v (t) respectively, where u c (t) represents the vehicle-bridge contact response. are the first and second derivatives of u(x, t) with respect to time t respectively, u””(x, t) is the fourth derivative of u(x, t) with respect to the longitudinal coordinate x, and is the vehicle acceleration response. The symbol δ represents the Dirac function, and g is the acceleration due to gravity.

[0053] For a simply supported beam, the vibration can be expressed as a superposition of multiple sine functions and characterized by the generalized coordinate q n (t), as follows:

[0054]

[0055] In the formula, q n (t) represents the nth-order bridge vibration response.

[0056] Since m v << mL, it can be obtained that:

[0057]

[0058] Among them, ωbn is the natural frequency of the bridge, ξ n is the damping ratio of the bridge, representing the degree of vibration attenuation, ξ n = c / (2mω bn ). In addition, the driving frequency Ω n is defined as Ω n = nπv / L.

[0059] In summary, by calculating or measuring the vehicle acceleration response, the acceleration response of the contact point can be inversely obtained:

[0060]

[0061] where, represents the vehicle body frequency. can be obtained by using the central difference method:

[0062]

[0063] In the formula, τ represents the sampling point, and dt is the sampling interval.

[0064] 2. Use the empirical wavelet transform (EWT) to extract the first few acceleration components of the bridge, that is, the target modal acceleration components, and calculate their envelope characteristics using the Hilbert transform, as follows:

[0065] First, calculate the Fourier transform X(ω) of the contact point acceleration signal , and adaptively divide the frequency band based on the local minimum values in X(ω) and construct the corresponding empirical wavelet filter φ n (ω). Decompose the signal by EWT to obtain each sub-signal c n (t) as:

[0066]

[0067] Then, by comparing the energy distributions of each sub-signal, determine the target sub-signal containing the first few vibration responses of the bridge.

[0068] When the vehicle speed of the measurement vehicle is relatively small, the single-frequency response formula of the bridge can be approximately expressed as:

[0069]

[0070] where, B Dn represents the initial vibration amplitude of the nth mode, which is determined by the bridge structure characteristics and external excitation; is the phase angle, used to characterize the initial phase state of the vibration signal; ω Dn is the damping frequency of the bridge, Ω n << ω Dn, thus determines the main frequency components of the signal, while sin(Ω n t) only affects the amplitude envelope of the signal.

[0071] Perform the Hilbert transform on the above formula to extract the signal envelope as follows:

[0072]

[0073] In the formula, H() is the Hilbert transform, and the absolute value of the sine term Satisfies the relationship within the vehicle-bridge coupling time period [0, L / v] This means that within the vehicle-bridge coupling time, the sine modulation term of the signal is symmetric about the mid-span of the bridge, that is, the periodic modulation part of the envelope is symmetric.

[0074] S3. Based on the characteristic of the symmetry of the periodic modulation part of the envelope, divide the envelope of each target modal acceleration sub-signal into the left and the right two parts, and extract the damping characteristic information by calculating the envelope ratio of the symmetric points as follows:

[0075] Take the time t = L / (2v) as the symmetry center, and use the time interval Δt on both sides of the symmetry axis as a variable to divide the signal into the left (Left) and right (Right) parts, as Figure 2 shown. The time coordinates of the corresponding characteristic points in the signal are mirror-symmetric about the symmetry axis (as and ).

[0076] As Figure 3 shown, construct the expressions of the left (Left) and right (Right) part envelope lines with Δt as the coordinate axis as follows:

[0077]

[0078]

[0079] In the formula, Δt is the time interval between the current time and L / 2v.

[0080] According to the left and right parts to construct a function, calculate the envelope ratio of the left and right parts under the corresponding Δt, and the damping characteristic information can be extracted. The expression form is:

[0081]

[0082] In the formula, E n (Δt) is the envelope ratio of the left and right parts.

[0083] S4. For En Taking the logarithm of (Δt), we get:

[0084] y n (Δt) = ln(E n (Δt)) = -2ξ n ω bn Δt;

[0085] Wherein, y n (Δt) is the linear relationship of the variable Δt, and the slope is determined by the damping ratio ξ n and the natural frequency ω bn jointly. Therefore, by using the least squares method to fit this straight line, the slope k = -2ξ n ω bn can be determined to weaken the local fluctuation interference, as Figure 4 shown. In the actual calculation process, specifically:

[0086] Select a set of (Δt i , ln(E n (Δt i ))) data points, and use the least squares method for linear fitting to obtain:

[0087]

[0088] Wherein, N represents the number of sampling points, is the mean value of all data points Δt i , is the mean value of all data points lnE n (Δt i ), as follows:

[0089]

[0090] Then the calculation formula for the damping ratio is:

[0091]

[0092] Wherein, ω bn can be directly obtained in the Fourier transform in step S2.

[0093] Then, multiply the damping mode T n (t) by the exponential term e kt / 2 and normalize it to obtain the modal vibration shape of the bridge without damping interference, as Figure 5 shown, as follows:

[0094]

[0095] Wherein, M n(t) is the modal vibration mode of the bridge without damping interference, that is, the non-distorted vibration mode of the bridge. This process can achieve the identification of the damping ratio and the extraction of the non-distorted vibration mode of the bridge without directly calculating the damping ratio and frequency of the bridge.

[0096] Therefore, the present invention adopts the above-mentioned method for identifying the damping ratio and vibration mode of a bridge based on an optimization fitting strategy, which can achieve the extraction of the undamped modal vibration mode and improve the robustness of the calculation results.

[0097] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit them. Although the present invention has been described in detail with reference to the preferred embodiments, those of ordinary skill in the art should understand that they can still modify or equivalently replace the technical solutions of the present invention, and these modifications or equivalent replacements cannot make the modified technical solutions deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A bridge damping ratio and vibration mode identification method based on an optimization fitting strategy, characterized in that: The following steps are involved: S1. Establish the dynamic equation of the vehicle-bridge system in the static equilibrium position, calculate or measure the vehicle acceleration response, and invert the acceleration response of the contact point; S2, extracting the target modal acceleration component of the bridge from the acceleration response of the contact point based on empirical wavelet transform, and extracting the corresponding envelope using Hilbert transform; S3. On the time coordinate axis, the envelope of the target modal acceleration component is divided into a left part and a right part, and the damping characteristic information is extracted by calculating the ratio of the symmetric points of the left and right envelopes; S4. Based on the damping characteristic information, the least squares method is used to perform straight line fitting to identify the damping ratio of the bridge. At the same time, the envelope is combined to extract the modal vibration shape of the bridge without damping interference.

2. The bridge damping ratio and vibration mode identification method based on the optimization fitting strategy according to claim 1 is characterized in that: In step S1, a dynamic equation of the vehicle-bridge system at a static equilibrium position is constructed, including: The equation of motion for the vehicle is: The dynamic equation of the bridge is: Among them, m v is the vehicle mass, k v is the support spring stiffness, m is the mass per unit length of the bridge, c is the damping coefficient, EI is the bending stiffness of the bridge, u(x, t), y v (t) are the vertical displacements of the bridge and the vehicle, u c (t) is the bridge contact response, are the first and second derivatives of u(x,t) with respect to time t, respectively; u""(x,t) is the fourth derivative of u(x,t) with respect to the longitudinal coordinate x. is the vehicle acceleration response, δ represents the Dirac function, and g is the gravitational acceleration.

3. The bridge damping ratio and vibration mode identification method based on the optimization fitting strategy according to claim 1 is characterized in that: Step S2 comprises: Firstly, the acceleration signal of the contact point is subjected to Fourier transform, the frequency band is adaptively divided according to the local minimum value, and the corresponding empirical wavelet filter is constructed to decompose the acceleration signal of the contact point by empirical wavelet transform. Then, the energy distribution of each sub-signal after decomposition is compared to determine the sub-signal containing the target modal response of the bridge, namely the target modal acceleration component, and its envelope is extracted through Hilbert transform.

4. The bridge damping ratio and vibration mode identification method based on the optimization fitting strategy according to claim 1 is characterized in that: In step S3, the time coordinates of the feature points corresponding to the left and right envelopes are distributed in a mirror image about the symmetry axis.

5. The bridge damping ratio and vibration mode identification method based on the optimization fitting strategy according to claim 4 is characterized in that: The damping characteristic information is extracted by the ratio of the corresponding characteristic points of the left and right envelope lines.

6. The bridge damping ratio and vibration mode identification method based on the optimization fitting strategy according to claim 5 is characterized in that: Taking time t = L / (2v) as the symmetry center and the time interval Δt on both sides of the symmetry axis as the variable, the left and right envelope functions of the target modal acceleration component are reconstructed to obtain the reconstructed function and as follows: In the formula, is the reconstruction function of the left part of the envelope, is the reconstruction function of the right part of the envelope, Δt is the time from the current time to the middle axis The time interval, ξ n is the bridge damping ratio, ω bn represents the natural frequency of the nth mode, B Dn Represents the initial vibration amplitude of the nth order mode, Ω n The driving frequency.

7. The bridge damping ratio and vibration mode identification method based on the optimization fitting strategy according to claim 6 is characterized in that: The ratio of the symmetric points of the left and right envelopes is expressed as follows: In the formula, E n (Δt) is the ratio of the symmetric points of the left and right envelopes.

8. The bridge damping ratio and vibration mode identification method based on the optimization fitting strategy according to claim 1 is characterized in that: Step S4 includes using the least squares method to perform straight line fitting, obtain the corresponding straight line slope k, and calculate the bridge damping ratio; at the same time, multiply the damping mode by the exponential term e kt / 2 And normalized, and then get the modal vibration shape of the bridge without damping interference.

9. The bridge damping ratio and vibration mode identification method based on the optimization fitting strategy according to claim 8 is characterized in that: The expression of bridge damping ratio is as follows: Among them, ω bn It is directly obtained by Fourier transform of the contact point acceleration signal.

10. The bridge damping ratio and vibration mode identification method based on the optimization fitting strategy according to claim 8, characterized in that: The modal vibration shape of the bridge without damping interference is expressed as follows: Where M n (t) is the modal vibration shape of the bridge without damping interference.

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