Bridge modal identification method based on time-frequency analysis and index compensation

Abstract

The invention discloses a bridge modal recognition method based on time-frequency analysis and index compensation, and belongs to the technical field of bridge health monitoring, and the method comprises the following steps: S1, building an axle system dynamic model, deducing a bridge vibration equation, and calculating the acceleration of a contact point; s2, deducing a discrete recursive inversion formula of the acceleration response of the contact point based on a vehicle vibration equation; s3, extracting a frequency and a ridge instantaneous amplitude by adopting S transformation, and constructing an approximate expression of the frequency and the ridge instantaneous amplitude; and S4, introducing an index compensation strategy, correcting damping attenuation, identifying a damping ratio and reconstructing a bridge modal shape. According to the bridge modal recognition method based on time-frequency analysis and index compensation, the dynamic model of the axle system is constructed, the acceleration response of the contact point of the bridge is inversely calculated by using the acceleration response of the vehicle, the instantaneous amplitude at the ridge line is extracted in combination with the time-frequency analysis, and the accuracy of bridge modal recognition is improved by adopting the index compensation. And the stability of modal information is improved.

Description

Technical Field

[0001] The present invention relates to the technical field of bridge health monitoring, and particularly to a bridge modal identification method based on time-frequency analysis and exponential compensation. Background Art

[0002] Traditional bridge modal identification methods rely on fixed sensor arrays to obtain the dynamic information of bridges, but this method has obvious limitations. Fixed sensor arrangements are difficult to meet the requirements of long-span bridges or long-term health monitoring, and cannot comprehensively capture the dynamic characteristics of bridges under vehicle action. Especially in complex actual environments, the limitations of sensor layouts may lead to inaccurate identification results.

[0003] In addition, vehicle acceleration signals are easily affected by road unevenness and environmental noise, resulting in instability in modal identification. The interference of noise causes large errors in the process of modal information extraction, thus affecting the accuracy and reliability of identification.

[0004] Existing time-frequency analysis methods also face certain limitations. Methods such as variational mode decomposition (VMD) and empirical mode decomposition (EMD) usually require manual parameter setting, which makes them vulnerable to noise and leads to the problem of modal aliasing. The short-time Fourier transform (STFT) and continuous wavelet transform (CWT) have the problem of balancing time-frequency resolution and are difficult to accurately extract the modal characteristics of bridges, further affecting the identification accuracy.

[0005] The bridge vibration signals generated by vehicle excitation contain damping effects, which cause the modal amplitudes to decay over time and increase the modal identification error. Existing methods usually rely on estimating the damping ratio of bridges, but since it is difficult to accurately obtain the damping ratio under different working conditions, the influence of damping effects on modal identification has also become a difficult point.

[0006] Therefore, the limitations and challenges in many aspects of traditional methods have promoted the exploration and research of new bridge modal identification technologies. Summary of the Invention

[0007] The purpose of the present invention is to provide a bridge modal identification method based on time-frequency analysis and exponential compensation. By constructing a dynamic model of the vehicle-bridge system, using the vehicle acceleration response to back-calculate the contact point acceleration response of the bridge, combining time-frequency analysis to extract the instantaneous amplitude at the ridge line, and adopting exponential compensation to improve the accuracy of bridge modal identification and enhance the stability of modal information.

[0008] To achieve the above purpose, the present invention provides a bridge modal identification method based on time-frequency analysis and exponential compensation, including the following steps:

[0009] S1. Establish a dynamic model of the vehicle-bridge system, derive the bridge vibration equation, and calculate the acceleration at the contact point to characterize the bridge dynamic response;

[0010] S2. For the measured discrete vehicle acceleration signal, based on the vehicle vibration equation, derive the discrete recursive inversion formula for the acceleration response at the contact point;

[0011] S3. Use the S-transform to extract the frequency and the instantaneous amplitude of the ridge line, and construct its approximate expression to characterize the modal characteristics;

[0012] S4. Introduce an exponential compensation strategy to correct the asymmetry of the instantaneous amplitude of the ridge line caused by damping attenuation, and simultaneously identify the damping ratio and reconstruct the bridge modal shape.

[0013] Preferably, in the S1, when establishing the dynamic model of the vehicle-bridge system, the vertical dynamic equation of the vehicle is expressed as:

[0014]

[0015] where m v is the vehicle mass, c v and k v are the damping and stiffness of the suspension system respectively, z v and z c represent the vertical displacements of the contact points between the vehicle and the bridge respectively, and represent the first and second derivatives of z v with respect to time t, that is, the corresponding vertical velocity and acceleration; represents the first derivative of z c with respect to time t.

[0016] Preferably, according to the dynamic model of the vehicle-bridge system, the vibration equation of the bridge is described based on the Euler-Bernoulli beam theory as:

[0017]

[0018] where m b is the mass per unit length of the bridge, z b is the vertical displacement of the bridge, c b and EI are the damping and flexural stiffness of the bridge respectively, g is the acceleration due to gravity, v is the vehicle moving speed, δ(x - vt) is the unit impulse function, is the partial derivative symbol, and x is the coordinate of the bridge along the longitudinal direction;

[0019] Using the modal expansion method, the vertical displacement response of the bridge is expressed as:

[0020]

[0021] where qn (t) is the modal generalized coordinate, n is a positive integer, and L represents the length of the bridge;

[0022] Substitute expression (3) into the bridge vibration equation (2), and multiply both sides of the equation by sin(pπx / L), where p is a positive integer, and then integrate along the bridge interval [0, L]; considering that the vertical acceleration of the vehicle satisfies After simplification, we get:

[0023]

[0024] where, ω n and ξ n correspond to the natural frequency and damping ratio of the bridge respectively, and their expressions are ξ n = c b / (2ωm b ω n ); The driving frequency Ω n is defined as Ω n = nπv / L, where v represents the vehicle moving speed;

[0025] Solve the differential equation (4), and substitute the solution into the bridge modal expansion formula (3) to obtain the vertical vibration response of the bridge:

[0026]

[0027] where, and are undetermined coefficients, represents the characteristic frequency of the attenuation term.

[0028] Preferably, for the vertical acceleration response of the vehicle-bridge contact point, let x = vt and solve its second derivative to obtain:

[0029]

[0030] where, ω Dn·l = ω Dn - Ω n ω Dn·r = ω Dn + Ω n , and are undetermined coefficients of the transient response term, where the transient response term contains the characteristic frequency ω Dn , and is modulated by the driving frequency Ω n to form the left and right side frequency components ω Dn·l and ω Dn·r .

[0031] Preferably, in S2, according to the vehicle vibration equation, the acceleration of the axle contact point is derived and estimated through a vehicle dynamics model:

[0032] First, rewrite the vertical vibration equation (1) of the vehicle as:

[0033]

[0034] Among them, take the second derivative of both sides of equation (7) and organize to get:

[0035]

[0036] Where Use the central difference method to calculate:

[0037]

[0038] Among them, i represents the sampling point, and Δt is the sampling interval;

[0039] Solve the acceleration response of the contact point from it:

[0040]

[0041] Among them, τ represents the sampling point.

[0042] Preferably, generalize formula (10) to t + Δt to get:

[0043]

[0044] Where represents the integral between two sampling points t to t + Δt; use the trapezoidal integration method to calculate the above integral, and the specific expansion form is as follows:

[0045]

[0046] Substitute into formula (11) and simplify to get:

[0047]

[0048] So far, the discrete recursive inversion formula of the contact point acceleration response has been derived.

[0049] Preferably, in S3, perform time-frequency analysis on the signal. Since formula (6) contains the driving frequency and the bridge frequency, and Ω n << ω Dn·r , ω Dn·l , it shows low-frequency and dense-frequency characteristics in the spectrogram; effectively remove the components related to the driving frequency through filtering, and only retain the components related to the first N natural frequencies of the bridge, so as to obtain:

[0050]

[0051] In the above formula, it generally satisfies:

[0052]

[0053] Therefore, the above expression is further simplified to:

[0054]

[0055] It is shown by the structure of the signal that the main frequency of the signal is dominated by sin(ω Dn ), and the low-frequency component sin(Ω n t) applies modulation to make the instantaneous amplitude have a slow-varying characteristic.

[0056] Preferably, in the above S3, the S transform is defined as follows:

[0057]

[0058] Among them, f represents frequency, t represents time, τ represents the transform variable, j represents the imaginary unit, and the window function w(t,f) adopts an adaptive Gaussian window:

[0059]

[0060] The width of the window function w(t,f) is inversely proportional to the frequency, that is, for high-frequency components, the time resolution is high, and for low-frequency components, the frequency resolution is high;

[0061] The S transform is a linear transform. For a multi-frequency signal There is

[0062]

[0063] Among them, S total represents the total S transform of the signal, and S n represents the S transform of the nth frequency component;

[0064] For each mode n, the high-frequency component ω Dn in it dominates the fundamental frequency of the signal, and its spectral energy is concentrated around f≈±ω Dn ; for the modulation term sin(Ω n t), its energy is distributed around f≈±Ω n ; according to the Fourier transform and the modulation theorem, the spectrum of the signal will generate sidebands around the carrier frequency ω Dn , that is:

[0065] f = ω Dn ±Ω n (20);

[0066] Among them, Ωn represents the low-frequency modulation component. Since Ω n << ω Dn , the main energy is still concentrated at f ≈ ω Dn ; the window function w(t, f) of the S-transform has better time resolution for high-frequency components, and its time-frequency spectrum obtains the maximum modulus value at f ≈ ω Dn .

[0067] Preferably, the time-frequency ridge line is defined as the frequency position with the maximum modulus value of the S-transform:

[0068]

[0069] where ω ridge (t) represents the instantaneous frequency, represents finding the frequency f with the maximum modulus value. Based on the S-transform analysis, the energy of the signal is concentrated at f = ω Dn , and the low-frequency modulation component Ω n only affects the amplitude rather than the main frequency; therefore, the instantaneous frequency is expressed as:

[0070] ω ridge (t) ≈ ω Dn (22);

[0071] The instantaneous amplitude A n (t) of each mode is calculated by integrating the modulus value of the S-transform:

[0072]

[0073] where Δf is a small frequency band used to capture the energy of this mode while avoiding interference;

[0074] According to formula (6), the bridge signal in the contact point response is a narrowband signal; therefore, it is further approximated as:

[0075]

[0076] where,

[0077] That is, the instantaneous amplitude is approximately calculated by the modulus value of the S-transform at the main frequency point f = ω Dn , thus simplifying the analysis process and improving the calculation efficiency;

[0078] Preferably, in S4, within the vehicle-bridge coupling time interval [0, L / v], the sine term in the instantaneous amplitude satisfies the following symmetry relationship:

[0079]

[0080] This relationship indicates that, in the theoretically ideal case, this function has symmetric characteristics within the time interval [0, L / v].

[0081] Introduce the compensation factor e αt To eliminate the influence of exponential decay on the instantaneous amplitude, the compensated instantaneous amplitude is expressed as

[0082] Accordingly, define the compensated instantaneous amplitude ratio R α (t) as:

[0083]

[0084] In the ideal case, if the damping effect is completely compensated, that is, set α = ξ n ω n , then it should satisfy R α (t) = 1; that is, the compensated instantaneous amplitude remains strictly symmetric within the time interval [0, L / v];

[0085] For the discretized signal, to optimize the compensation factor α, define the instantaneous amplitude ratio error and minimize its mean square error MSE:

[0086]

[0087] where α * represents the mean square error, t k = kΔt is the discrete time point, Δt is the sampling time interval, and M = L / (vΔt) is the total number of sampling points;

[0088] Multiply the optimal compensation factor α * by the time-frequency ridge line extracted by time-frequency analysis and perform normalization processing to obtain the bridge modal shape without damping decay effect:

[0089]

[0090] where α * represents the mean square error, t k = kΔt is the discrete time point, Δt is the sampling time interval, and N = L / (vΔt) is the total number of sampling points;

[0091] In addition, considering the damping frequency and the damping ratio ξ n is generally below 0.1 in bridge engineering applications, approximately satisfying ω n ≈ ω Dn , since ω Dn can be directly extracted in time-frequency analysis, calculate the damping ratio using the compensation factor α:

[0092]

[0093] Therefore, the present invention adopts the above-mentioned bridge modal identification method based on time-frequency analysis and exponential compensation. By constructing the dynamic model of the vehicle-bridge system, using the vehicle acceleration response to inversely calculate the acceleration response of the bridge contact point, combining time-frequency analysis to extract the instantaneous amplitude at the ridge line, and adopting exponential compensation to improve the accuracy of bridge modal identification and enhance the stability of modal information.

[0094] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Description of the Drawings

[0095] Figure 1 is the vehicle coupling mechanical model diagram of an embodiment of the bridge modal identification method based on time-frequency analysis and exponential compensation of the present invention;

[0096] Figure 2 is the vehicle response curve diagram of an embodiment of the bridge modal identification method based on time-frequency analysis and exponential compensation of the present invention;

[0097] Figure 3 is the contact point acceleration curve diagram of an embodiment of the bridge modal identification method based on time-frequency analysis and exponential compensation of the present invention;

[0098] Figure 4 is the contact point S-transform time-frequency diagram of an embodiment of the bridge modal identification method based on time-frequency analysis and exponential compensation of the present invention;

[0099] Figure 5 is the first-order ridge line instantaneous amplitude curve diagram of an embodiment of the bridge modal identification method based on time-frequency analysis and exponential compensation of the present invention;

[0100] Figure 6 is the second-order ridge line instantaneous amplitude curve diagram of an embodiment of the bridge modal identification method based on time-frequency analysis and exponential compensation of the present invention;

[0101] Figure 7 is the restored first-order modal curve diagram of an embodiment of the bridge modal identification method based on time-frequency analysis and exponential compensation of the present invention;

[0102] Figure 8 is the restored second-order modal curve diagram of an embodiment of the bridge modal identification method based on time-frequency analysis and exponential compensation of the present invention. Detailed Embodiments

[0103] The technical solution of the present invention will be further described below with reference to the accompanying drawings and embodiments.

[0104] Unless otherwise defined, the technical terms or scientific terms used in the present invention shall have the ordinary meanings understood by those of ordinary skill in the field to which the present invention belongs.

[0105] Example 1

[0106] As Figure 1 shown, the present invention provides a bridge modal identification method based on time-frequency analysis and exponential compensation, including the following steps:

[0107] S1. Establish a vehicle-bridge system dynamic model, derive the bridge vibration equation, and calculate the contact point acceleration to characterize the bridge dynamic response;

[0108] First, establish a vehicle-bridge system dynamic model, as Figure 1 shown. The vertical dynamic equation of the vehicle can be expressed as:

[0109]

[0110] where, m v is the vehicle mass, c v and k v are the damping and stiffness of the suspension system respectively, z v and z c represent the vertical displacements of the vehicle and the vehicle-bridge contact point respectively, and represent the first and second derivatives of z v with respect to time t, that is, the corresponding vertical velocity and acceleration; represents the first derivative of z c with respect to time t.

[0111] According to the vehicle-bridge system dynamic model, the vibration equation of the bridge is described based on the Euler-Bernoulli beam theory as:

[0112]

[0113] where, m b is the mass per unit length of the bridge, z b is the vertical displacement of the bridge, c b and EI are the damping and flexural stiffness of the bridge respectively, g is the acceleration due to gravity, v is the vehicle moving speed, δ(x - vt) is the unit impulse function, is the partial derivative symbol, and x is the coordinate of the bridge along the longitudinal direction;

[0114] Using the modal expansion method, the vertical displacement response of the bridge can be expressed as:

[0115]

[0116] where, q n (t) is the modal generalized coordinate, n is a positive integer, and L represents the length of the bridge;

[0117] Substitute expression (3) into the bridge vibration equation (2), and multiply both sides of the equation by sin(pπx / L), where p is a positive integer, and then integrate along the bridge interval [0, L]; considering that the vertical acceleration of the vehicle satisfies After simplification, we get:

[0118]

[0119] where ω n and ξ n correspond to the natural frequency and damping ratio of the bridge respectively, and their expressions are ξ n = c b / (2ωm b ω n ); The driving frequency Ω n is defined as Ω n = nπv / L, where v represents the vehicle moving speed.

[0120] Solve the above differential equation and substitute the solution into the bridge modal expansion formula (3), the vertical vibration response of the bridge can be obtained:

[0121]

[0122] where, and are undetermined coefficients, represents the characteristic frequency of the attenuation term; this solution describes the vibration response of the bridge under the vehicle excitation, including the steady-state vibration component dominated by Ω n and the response component of the transient damped vibration.

[0123] Focusing on the vertical acceleration response of the vehicle-bridge contact point, let x = vt and solve its second derivative to get:

[0124]

[0125] where, ω Dn·l = ω Dn -Ω n , ω Dn·r = ω Dn +Ω n , and are undetermined coefficients of the transient response term.

[0126] This expression reveals the acceleration response characteristics of the vehicle-bridge contact point, where the transient response term contains the characteristic frequency ω Dn , and is modulated by the driving frequency Ω n to form the left and right side frequency components ω Dn·l and ω Dn·rThis indicates that the transient response is not only affected by bridge damping, but also related to the vehicle speed and decays exponentially. In addition, the steady-state response term is dominated by the driving frequency Ω n and is manifested as a harmonic component at twice the driving frequency.

[0127] S2. For the measured discrete vehicle acceleration signal, a discrete recursive inversion formula for the acceleration response at the contact point is derived based on the vehicle vibration equation; the vehicle response is as Figure 2 shown.

[0128] In practical applications, the acceleration at the vehicle-bridge contact point cannot be directly measured. However, it should be noted that according to the vehicle vibration equation, this quantity can be deduced and estimated through a vehicle dynamics model.

[0129] First, rewrite the vertical vibration equation (1) of the vehicle as:

[0130]

[0131] where, taking the second derivative of both sides of equation (7) and arranging it gives:

[0132]

[0133] where can be calculated using the central difference method:

[0134]

[0135] where i represents the sampling point and Δt is the sampling interval;

[0136] From this, the acceleration response at the contact point can be solved as follows, and its graph is as Figure 3 shown.

[0137]

[0138] where τ represents the sampling point.

[0139] Furthermore, generalizing formula (10) to t + Δt gives:

[0140]

[0141] where represents the integral between two sampling points t to t + Δt; this integral can be calculated using various numerical integration methods, such as the simple left rectangle method, right rectangle method, and more accurate trapezoidal method, Simpson's method, and Gaussian quadrature method; using the trapezoidal integration method, the specific expansion form is as follows:

[0142]

[0143] Substituting into formula (11) and simplifying, we get:

[0144]

[0145] So far, the derivation of the discrete recursive inversion formula for the acceleration response of the contact point is completed.

[0146] S3. Use the S-transform to extract the frequency and the instantaneous amplitude of the ridge line, and construct its approximate expression to characterize the modal characteristics;

[0147] Performing time-frequency analysis on the signal, the formula (6) contains the driving frequency and the bridge frequency. Existing research shows that Ω n << ω Dn·r , ω Dn·l , which shows low-frequency and dense-frequency characteristics in the spectrogram; effectively removing the components related to the driving frequency through filtering, and only retaining the components related to the first N natural frequencies of the bridge, thus obtaining:

[0148]

[0149] Existing research shows that the following is usually satisfied in the above formula:

[0150]

[0151] Therefore, the above expression is further simplified to:

[0152]

[0153] It can be seen from the structure of the signal that the main frequency of the signal is dominated by sin(ω Dn ), while the low-frequency component sin(Ω n t) applies modulation, making the instantaneous amplitude show a slow-varying characteristic.

[0154] Taking the S-transform as an example, the present invention analyzes the time-frequency characteristics of the signal, and extracts its time-frequency ridge line and instantaneous amplitude to reveal the dynamic characteristics of the bridge vibration. The S-transform time-frequency diagram of the contact point is as shown in Figure 4 . It should be particularly emphasized that for other time-frequency analysis methods, the theory of the present invention still applies.

[0155] In bridge health monitoring and vibration signal analysis, the actually measured signal is usually composed of the superposition of multiple modal components and is affected by low-frequency modulation. In order to accurately extract the dynamic characteristics of the signal, an appropriate time-frequency analysis method needs to be used to simultaneously obtain the instantaneous amplitude and the instantaneous frequency. The S-transform (S-Transform, ST) is an adaptive time-frequency analysis tool that combines the advantages of the short-time Fourier transform (STFT) and the wavelet transform (WT), and can adaptively adjust the time and frequency resolutions in different frequency ranges. Therefore, the S-transform is applicable to the amplitude-modulated and frequency-modulated signals studied in this paper and can effectively extract the time-frequency ridge line and the instantaneous amplitude.

[0156] The S transform is defined as follows:

[0157]

[0158] where f represents frequency, t represents time, τ represents the transform variable, j represents the imaginary unit, and the window function w(t, f) is an adaptive Gaussian window:

[0159]

[0160] The width of this window function is inversely proportional to the frequency, i.e.:

[0161] For high-frequency components, the time resolution is high, and for low-frequency components, the frequency resolution is high;

[0162] The S transform is a linear transform. For a multi-frequency signal we have

[0163]

[0164] where S total represents the total S transform of the signal, and S n represents the S transform of the nth frequency component.

[0165] Therefore, the S transform can perform separation analysis on multi-frequency superimposed signals and is applicable to the extraction of instantaneous amplitude and instantaneous frequency.

[0166] For each mode n, the high-frequency component ω Dn dominates the fundamental frequency of the signal, and its spectral energy is concentrated around f ≈ ±ω Dn . For the modulation term sin(Ω n t), its energy is mainly distributed around f ≈ ±Ω n . According to the Fourier transform and the modulation theorem, the spectrum of this signal will generate sidebands around the carrier frequency ω Dn , i.e.:

[0167] f = ω Dn ±Ω n (20);

[0168] where Ω n represents the low-frequency modulation component. Since Ω n << ω Dn , the main energy is still concentrated around f ≈ ω Dn .

[0169] In addition, the window function w(t, f) of the S transform has better time resolution for high-frequency components, and its time-frequency spectrum obtains the maximum modulus value mainly at f ≈ ω Dn . Therefore, the main energy of the signal is still concentrated at f = ωDn at

[0170] The time-frequency ridge line is defined as the frequency position with the maximum modulus value of the S-transform:

[0171]

[0172] where ω ridge (t) represents the instantaneous frequency, denotes finding the frequency f with the maximum modulus value. Based on the S-transform analysis, the main energy of the signal is concentrated at f = ω Dn , and the low-frequency modulation component Ω n only affects the amplitude rather than the main frequency. Therefore, the instantaneous frequency can be expressed as:

[0173] ω ridge (t) ≈ ω Dn (22);

[0174] This result shows that the instantaneous frequencies of each mode are stable near their main vibration frequency ω Dn , and the modulation term does not affect the extraction of the instantaneous frequency.

[0175] The instantaneous amplitude A n (t) of each mode is calculated by integrating the modulus value of the S-transform:

[0176]

[0177] where Δf is a small frequency band used to capture the main energy of this mode while avoiding interference from other modes.

[0178] According to formula (6), the bridge signal in the contact point response is usually a narrowband signal. Therefore, it can be further approximated as:

[0179]

[0180] where, The obtained first-order ridge line instantaneous amplitude and second-order ridge line instantaneous amplitude are as Figure 5 and Figure 6 shown.

[0181] That is, the instantaneous amplitude can be approximately calculated by the modulus value of the S-transform at the main frequency point f = ω Dn to simplify the analysis process and improve the calculation efficiency.

[0182] S4. Introduce an exponential compensation strategy to correct the asymmetry of the ridge line instantaneous amplitude caused by damping attenuation, and at the same time identify the damping ratio and reconstruct the bridge modal shape;

[0183] In the vehicle-bridge coupling time interval [0, L / v], the sine term in the instantaneous amplitude satisfies the following symmetry relationship:

[0184]

[0185] This relationship indicates that, in the theoretically ideal case, this function has symmetric characteristics within the time interval [0, L / v]. However, the exponential decay term destroys this symmetry, making the amplitude of the signal in the interval t > L / (2v) significantly lower than that in the interval t < L / (2v), thus affecting the accurate extraction of modal characteristics. To eliminate the influence of exponential decay on the instantaneous amplitude, a compensation factor e αt needs to be introduced for correction. The compensated instantaneous amplitude is expressed as

[0186] Accordingly, the compensated instantaneous amplitude ratio R α (t) can be defined as:

[0187]

[0188] Ideally, if the damping effect is completely compensated, that is, setting α = ξ n ω n , then R α (t) = 1 should be satisfied. That is, the compensated instantaneous amplitude remains strictly symmetric within the time interval [0, L / v].

[0189] For the discretized signal, to optimize the compensation factor α, the instantaneous amplitude ratio error is defined and its mean square error (MSE) is minimized:

[0190]

[0191] where α * represents the mean square error, t k = kΔt is the discrete time point, Δt is the sampling time interval, and M = L / (vΔt) is the total number of sampling points.

[0192] Once the optimal compensation factor α * is obtained, it can be multiplied by the time-frequency ridge line extracted by time-frequency analysis and normalized to obtain the bridge modal shape without damping decay effect:

[0193]

[0194] The restored instantaneous amplitude of the first-order ridge line and the instantaneous amplitude of the second-order ridge line are as Figure 7 and Figure 8 shown.

[0195] In addition, considering the damping frequency and the damping ratio ξ n is generally below 0.1 in bridge engineering applications, approximately satisfying ω n ≈ ωDn , since ω Dn can be directly extracted in time-frequency analysis, the damping ratio can be calculated using the compensation factor α:

[0196]

[0197] This relationship establishes a quantitative connection among the exponential compensation factor, damping characteristics, and the bridge modal frequency. It can not only restore the modal shape without damping influence but also synchronously extract the damping parameters of the bridge, realizing a comprehensive identification of the bridge's dynamic characteristics. It is worth emphasizing that this method directly estimates the damping ratio while compensating the instantaneous amplitude, and the two are carried out synchronously without a sequence, thus improving the efficiency and accuracy of modal identification.

[0198] Grid search, gradient descent, or the bisection method can be used for numerical optimization. To ensure the compensation effect, the constraints of smoothness, stability, and robustness need to be satisfied to avoid signal amplification distortion or high-frequency instability. Finally, by optimizing α, the symmetry of the instantaneous amplitude can be restored, and the accuracy of modal feature extraction can be improved.

[0199] Therefore, the present invention adopts the above-mentioned bridge modal identification method based on time-frequency analysis and exponential compensation. By constructing the dynamic model of the vehicle-bridge system, the acceleration response of the bridge's contact point is inversely calculated using the vehicle acceleration response. Combining time-frequency analysis to extract the instantaneous amplitude at the ridge line, and using exponential compensation to improve the accuracy of bridge modal identification and enhance the stability of modal information.

[0200] 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 do not make the modified technical solutions deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A bridge modal identification method based on time-frequency analysis and exponential compensation, characterized in that: The following steps are involved: S1. Establish a dynamic model of the vehicle-bridge system, derive the bridge vibration equation, and calculate the contact point acceleration to characterize the dynamic response of the bridge; S2. Based on the measured vehicle acceleration discrete signal, the discrete recursive inversion formula of the contact point acceleration response is derived based on the vehicle vibration equation; S3, using S transform to extract the frequency and ridge instantaneous amplitude, and construct their approximate expression to characterize the modal characteristics; S4. An exponential compensation strategy is introduced to correct the instantaneous amplitude asymmetry of the ridge line caused by damping attenuation, and the damping ratio is identified and the modal shape of the bridge is reconstructed.

2. The bridge modal identification method based on time-frequency analysis and exponential compensation according to claim 1 is characterized in that: In S1, a vehicle-bridge system dynamics model is established, and the vertical dynamics equation of the vehicle is expressed as: Among them, m v is the vehicle mass, c v and k v are the damping and stiffness of the suspension system, z v and z c They represent the vertical displacement of the contact point between the vehicle and the axle, and Indicates z v The first and second derivatives with respect to time t, i.e. the corresponding vertical velocity and acceleration; Indicates z c The first derivative with respect to time t.

3. The bridge modal identification method based on time-frequency analysis and exponential compensation according to claim 2 is characterized in that: According to the dynamic model of the vehicle-bridge system, the vibration equation of the bridge is described based on the Euler-Bernoulli beam theory as follows: Among them, m b is the mass per unit length of the bridge, z b is the vertical displacement of the bridge, c b and EI are the damping and bending stiffness of the bridge, g is the acceleration of gravity, v is the vehicle speed, δ(x-vt) is the unit impulse function, is the sign of the partial derivative, x is the coordinate of the bridge along the longitudinal direction; Using the modal expansion method, the vertical displacement response of the bridge is expressed as: Among them, q n (t) is the modal generalized coordinate, n is a positive integer, and L represents the length of the bridge; Substitute expression (3) into the bridge vibration equation (2) and multiply both sides of the equal sign by sin(pπx / L), where p is a positive integer, and then integrate along the bridge interval [0, L]; considering that the vehicle vertical acceleration satisfies After simplification: Among them, ω n and n They correspond to the natural frequency and damping ratio of the bridge respectively, and their expressions are: ξ n =c b / (2vm b ω n ); driving frequency Ω n Defined as Ω n =nπv / L, v represents the vehicle moving speed; Solve the differential equation (4) and substitute the solution into the bridge modal expansion equation (3) to obtain the vertical vibration response of the bridge: in, and is the coefficient to be determined, Represents the characteristic frequency of the attenuation term.

4. The bridge modal identification method based on time-frequency analysis and exponential compensation according to claim 3 is characterized by: The vertical acceleration response of the bridge contact point is set as x = vt, and the second-order derivative is solved to obtain: Among them, ω Dn·l =ω Dn -Ω n ,ω Dn·r =ω Dn +Ω n , and is the unknown coefficient of the transient response term, where the transient response term contains the characteristic frequency ω Dn , while being affected by the driving frequency Ω n Modulation, forming the left and right side frequency components ω Dn·l and ω Dn·r .

5. The bridge modal identification method based on time-frequency analysis and exponential compensation according to claim 2 is characterized by: In S2, the acceleration of the vehicle-bridge contact point is derived and estimated through the vehicle dynamics model according to the vehicle vibration equation: First, the vertical vibration equation (1) of the vehicle is rewritten as: Among them, the second-order derivatives of both sides of equation (7) are obtained: in Calculated using the central difference method: Where i represents the sampling point and Δt is the sampling interval; The acceleration response of the contact point is obtained from it: Among them, τ represents the sampling point.

6. The bridge modal identification method based on time-frequency analysis and exponential compensation according to claim 5 is characterized by: Extending formula (10) to t+Δt, we get: in, Represents the integral between two sampling points t to t+Δt; the trapezoidal integration method is used to calculate the above integral, and the specific expansion form is as follows: Substituting into formula (11) and simplifying it, we get: At this point, the discrete recursive inversion formula for the contact point acceleration response has been derived.

7. The bridge modal identification method based on time-frequency analysis and exponential compensation according to claim 4 is characterized by: In S3, the signal is subjected to time-frequency analysis. Since formula (6) includes the driving frequency and the bridge frequency, and Ω n <<ω Dn·r ,ω Dn·l , in the spectrum graph, it is shown as low-frequency and dense-frequency characteristics; The driving frequency-related components are effectively removed by filtering, and only the components related to the first N-order frequencies of the bridge are retained, thus obtaining: The above formula usually satisfies: Therefore, the above expression can be further simplified to: The structure of the signal shows that the main frequency of the signal is sin(ω Dn ) dominates, while the low-frequency component sin(Ω n t) Modulation is applied to make the instantaneous amplitude change slowly.

8. The bridge modal identification method based on time-frequency analysis and exponential compensation according to claim 7 is characterized by: In S3, the S transformation is defined as follows: Among them, f represents frequency, t represents time, τ represents the transformation variable, j represents the imaginary unit, and the window function w(t,f) adopts an adaptive Gaussian window: The width of the window function w(t,f) is inversely proportional to the frequency, that is, for high-frequency components, the time resolution is high, and for low-frequency components, the frequency resolution is high; The S transform is a linear transform. have Among them, S total represents the total S-transform of the signal, S n represents the S transform of the nth frequency component; For each mode n, the high-frequency component ω Dn The fundamental frequency of the dominant signal, whose spectral energy is concentrated at f≈±ω Dn Nearby; for the modulation term sin(Ω n t), whose energy is distributed in f≈±Ω n Nearby; According to Fourier transform and modulation theorem, the spectrum of the signal will be around the carrier frequency ω Dn Generate sidebands, namely: f=ω Dn ±Ω n (20); Among them, Ω n Represents the low-frequency modulation component, due to Ω n <<ω Dn , the main energy is still concentrated in f≈ω Dn ; The window function w(t,f) of the S transform has better time resolution for high-frequency components, and its time spectrum is f≈ω Dn The maximum modulus value is obtained at .

9. The bridge modal identification method based on time-frequency analysis and exponential compensation according to claim 8 is characterized by: The time-frequency ridge is defined as the frequency position where the S-transform modulus value is the largest: Among them, ω ridge (t) represents the instantaneous frequency, It means finding the frequency f with the largest modulus. Based on S-transform analysis, the energy of the signal is concentrated at f = ω Dn , and the low frequency modulation component Ω n Only the amplitude is affected, not the main frequency; therefore, the instantaneous frequency is expressed as: oh ridge (t)≈ω Dn (22); The instantaneous amplitude A of each mode n (t) is calculated by S-transformation modulus integration: Among them, Δf is a small frequency band used to capture the energy of this mode while avoiding interference; According to formula (6), the bridge signal in the contact point response is a narrowband signal; therefore, it is further approximated as: in, That is, the instantaneous amplitude is transformed by S at the main frequency point f = ω Dn The modulus value at is approximated, thereby simplifying the analysis process and improving the calculation efficiency.

10. The bridge modal identification method based on time-frequency analysis and exponential compensation according to claim 1 is characterized in that: In S4, within the vehicle-bridge coupling time interval [0, L / v], the sinusoidal term in the instantaneous amplitude satisfies the following symmetry relationship: This relationship shows that, under theoretical ideal conditions, the function has symmetric properties in the time interval [0, L / v]; Introducing the compensation factor e αt Eliminating the influence of exponential decay on the instantaneous amplitude, the instantaneous amplitude after compensation is expressed as Based on this, the instantaneous amplitude ratio R after compensation is defined as α (t) is: Ideally, if the damping effect is fully compensated, that is, α = ξ n ω n , then R α (t) = 1; that is, the instantaneous amplitude after compensation remains strictly symmetrical in the time interval [0, L / v]; For the discretized signal, in order to optimize the compensation factor α, the instantaneous amplitude ratio error is defined and its mean square error MSE is minimized: Among them, α * represents the mean square error, t k = kΔt is the discrete time point, Δt is the sampling time interval, M = L / (vΔt) is the total number of sampling points; The optimal compensation factor α * Multiplying it with the time-frequency ridge extracted by time-frequency analysis and normalizing it, we can get the modal shape of the bridge without damping attenuation effect: Among them, α * represents the mean square error, t k = kΔt is the discrete time point, Δt is the sampling time interval, and N = L / (vΔt) is the total number of sampling points; In addition, considering the damping frequency And the damping ratio ξ n In bridge engineering applications, it is generally below 0.1, which approximately satisfies ω n ≈ω Dn , due to ω Dn It can be directly extracted in time-frequency analysis and the damping ratio can be calculated using the compensation factor α:

Images