Damping and modal restoration method for any support bridge based on left and right running vehicles

Abstract

The application discloses a left and right driving vehicle-based arbitrary support bridge damping and modal recovery method and relates to the technical field of bridge health monitoring. * The application comprises the following steps: establishing a vehicle-bridge coupling mechanical model; calculating left and right driving vehicle-bridge contact point acceleration responses; extracting main modal characteristics of the bridge and separating modal instantaneous amplitudes; constructing a mixed signal based on two-vehicle-bridge contact point instantaneous amplitudes, solving an optimal parameter alpha * , calculating a bridge damping ratio, compensating the modal instantaneous amplitude signals of each order to eliminate damping attenuation effects, then normalizing the amplitudes, and extracting modal shapes without damping attenuation. The application adopts the left and right driving vehicle-based arbitrary support bridge damping and modal recovery method, which not only effectively compensates signal distortion caused by damping, but also accurately identifies damping and modal parameters of an arbitrary support bridge, thereby providing a technical solution for dynamic performance evaluation and health monitoring of complex bridge structures.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of bridge health monitoring, in particular to a method for restoring damping and modal of arbitrary support bridge based on left and right driving vehicles. BACKGROUND

[0002] As a key part of traffic infrastructure, the structural health monitoring of bridge is of great importance. Accurate acquisition of damping and modal parameters of bridge is of great significance for evaluating the working state of bridge, predicting potential diseases and ensuring the safe operation of bridge. In actual monitoring, the vibration response signals generated by the interaction between vehicle and bridge contain rich bridge structural information. With the increase of traffic flow and the growth of bridge service life, the traditional monitoring method has exposed many limitations in processing bridge vibration signals in complex environment.

[0003] The existing vehicle scanning method mainly focuses on modal identification of simply supported beam, and its theoretical premise depends on symmetric boundary conditions and uniform damping distribution. However, in actual engineering, bridges often have asymmetric support, complex geometric shape or heterogeneous material characteristics, resulting in significant asymmetry of structural response affected by damping attenuation. This asymmetry not only destroys the theoretical symmetry of the signal, but also greatly reduces the accuracy of traditional methods in extracting instantaneous amplitude and modal parameters, and even cannot be applied to complex support system of non simply supported beam.

[0004] When facing the bridge with asymmetric support or complex boundary conditions, the existing method often leads to the decline of modal feature extraction accuracy due to damping attenuation and asymmetric structural response. Therefore, it is urgent to develop a method that can be applied to arbitrary support bridge, effectively compensate the influence of damping attenuation, and accurately separate and analyze the modal information. SUMMARY

[0005] The purpose of the present application is to provide a method for restoring damping and modal of arbitrary support bridge based on left and right driving vehicles, to solve the limitation problem that the traditional vehicle scanning method is mainly limited to simply supported beam modal identification, and to restore the symmetry of the signal by adjusting the exponential decay effect of the signal, thereby improving the accuracy of modal feature extraction.

[0006] To achieve the above purpose, the present application provides a method for restoring damping and modal of arbitrary support bridge based on left and right driving vehicles, comprising the following steps:

[0007] Step S1, a vehicle-bridge coupling mechanical model is established, based on the difference of vehicle driving direction, the dynamic equations of left and right driving vehicles are established respectively, and combined with the bridge vibration equation, a complete coupling system is formed;

[0008] Step S2, the left and right vehicle-bridge contact point acceleration responses are calculated, and the left and right vehicle-bridge contact point response inversion recursive formula is obtained.

[0009] Step S3, extracting the main modal characteristics of the bridge, separating the modal instantaneous amplitudes;

[0010] Step S4, constructing the mixed signal based on the instantaneous amplitudes of the left-vehicle-bridge contact point, and then introducing a penalty term e αt to restore the symmetry of the mixed signal, where a is an optimization parameter, and a is recorded as a when the signal restores the complete symmetry * , solving the optimal parameter a * ;

[0011] Step S5, calculating the damping ratio of the bridge based on the optimal parameter a * , compensating the modal instantaneous amplitude signals of each order to eliminate the damping attenuation effect, and then normalizing the amplitudes to extract the modal shapes without damping attenuation.

[0012] Preferably, in the step S1, the motion equation of the left-vehicle is as follows:

[0013]

[0014] The motion equation of the right-vehicle is as follows:

[0015]

[0016] where the superscripts and represent the left-vehicle and the right-vehicle, respectively, and represent the mass, the damping coefficient and the spring stiffness of the vehicle, respectively, and j = 1 or and are the first and second order derivatives of the relevant displacement with respect to time t, respectively, and represent the vertical acceleration of the left-vehicle and the right-vehicle, respectively, and represent the vertical velocity of the left-vehicle and the right-vehicle, respectively, and represent the left-vehicle-bridge contact point velocity and the right-vehicle-bridge contact point velocity, respectively, and v represents the vehicle running speed.

[0017] Preferably, the bridge vibration equation in the step S1 is as follows:

[0018]

[0019] where m represents the unit length mass of the bridge, c is the damping coefficient of the bridge, EI is the bending stiffness of the bridge, u(x, t) is the vertical displacement of the bridge, and respectively, u" "(x, t) is the fourth order derivative with respect to the longitudinal coordinate x, f c (x, t) is the external load on the bridge, and is related to the left and right moving vehicles.

[0020] Preferably, the contact point response of the left moving vehicle in step S2 is as follows:

[0021]

[0022] The contact point response of the right moving vehicle is as follows:

[0023]

[0024] wherein, and respectively represent the vehicle-bridge contact point acceleration response of the left moving vehicle and the right moving vehicle, Δt is the sampling interval, and respectively represent the second order derivative of the vertical vibration acceleration of the left moving vehicle and the right moving vehicle with respect to time t.

[0025] Preferably, the specific steps of step S3 are as follows:

[0026] Step S31, band-pass filtering the contact point response obtained by inversion, removing the influence of vehicle movement and irrelevant frequency components, and retaining the vibration characteristics of the first few modes of the bridge;

[0027] Irrelevant frequency components are removed by band-pass filtering, and after filtering, the contact point acceleration and is approximately expressed as:

[0028]

[0029] wherein, N is the finite modal order, which is related to the filtering interval range, and is usually an integer greater than 2 to ensure that the key modal information is not lost, represents the bridge damping frequency of the nth mode, φ n (β n vt) and φ n (β n L-β n vt) represent the low-frequency envelope function, β n is a modal related parameter, which is closely related to the bridge mode shape, and respectively represent the coefficients related to the sine and cosine vibration components, represents the decay factor considering damping;

[0030] Step S32, using time-frequency analysis combined with ridge tracking method, through wavelet transform to separate the instantaneous amplitude of each mode, when the wavelet transform locks the mode main frequency, the instantaneous amplitude can be expressed as:

[0031]

[0032] For the right contact point signal:

[0033]

[0034] In the formula, Π n is a constant, determined by the coefficient and ;

[0035] Step S33, construct a signal symmetrical on the time axis, define the new instantaneous amplitude as:

[0036]

[0037] Where

[0038]

[0039] In the time interval [0, L / v] of axle coupling, satisfy the following symmetry relationship:

[0040]

[0041] However, due to the exponential decay term Monotonically decreasing on the time axis, so that:

[0042] A n (t)≠A n (L / v-t);

[0043] The theoretical symmetry of the signal is destroyed by the exponential decay.

[0044] Preferably, the specific steps of step S4 are as follows:

[0045] In order to restore the symmetry of the signal, a penalty term e αt is introduced, so that:

[0046]

[0047] Where, α is a parameter that needs to be determined by optimization, when the following formula is satisfied:

[0048] α=ξ n ω n ;

[0049] The signal restores the complete symmetry, and at this time α is denoted as α *Solve the optimal parameter a * The steps are as follows:

[0050] Step S41, evaluate the signal symmetry by DTW, calculate the DTW distance between the signal and the symmetric signal, the specific steps are as follows:

[0051] Step S411, local distance calculation: first, calculate the local distance d(i,j) between each time step t i And t j :

[0052]

[0053] Step S412, construct a distance matrix: then, fill in a distance matrix D(i,j) by dynamic programming, which represents the cumulative minimum distance from the starting point to the (i,j) point:

[0054] D(i,j)=d(i,j)+min(D(i-1,j),D(i,j-1),D(i-1,j-1));

[0055] Wherein, D(i-1,j) represents the cumulative distance of the last time point; D(i,j-1) represents the cumulative distance of the last time inversion point; D(i-1,j-1) represents the cumulative distance on the diagonal line;

[0056] Each element D(i,j) of the matrix is calculated by selecting the smallest value in the previous element, and then adding the distance of the current point. The first element D(0,0) of the matrix is initialized to 0;

[0057] Step S413, calculate the final DTW distance: the right lower corner element D(N,N) of the matrix D is the total cost of the optimal alignment path between the signals, that is, the DTW distance between the two signals:

[0058] D final =D(N,N);

[0059] Step S42, optimize in the given interval by using bisection method to make the DTW distance minimum, the specific steps are as follows:

[0060] Step S421, initialization: set the initial interval [a min , a max ], and calculate the midpoint

[0061] Step S422, calculate the DTW distance: for the current a mid , calculate the corresponding DTW distance D(a mid ):

[0062] Step S423, adjusting interval: according to the size of D(α mid ), the search interval is reduced:

[0063] If D(α mid ) is less than the target value or the previous distance, it indicates that the current α mid is closer to the optimal solution, and α min is updated to α mid ;

[0064] If D(α mid ) is greater than the target value, α max is updated to α mid ;

[0065] Step S43, setting convergence criteria, when the interval length is less than the preset precision threshold, stopping optimization, outputting the optimal parameter α * , the expression is as follows:

[0066] |α max -α min |≤∈ threshold ;

[0067] In the formula, ∈ threshold represents the preset precision threshold, and |α max -α min | represents the interval length.

[0068] Preferably, the calculation formula of the bridge damping ratio in the step S5 is as follows:

[0069]

[0070] In the formula, ω n represents the bridge frequency, generally ω n = ω Dn .

[0071] Preferably, the signal compensated and normalized in the step S5 is as follows:

[0072]

[0073] By introducing the time-space transformation t=x / v and performing normalization, we get and respectively represent the recovery modal shape of the bridge structure along the left-to-right and right-to-left directions, in order to obtain the modal shape conforming to the conventional definition, the mirror flip operation along the x-axis is performed on , so as to realize the standardized representation of the modal shape.

[0074] Therefore, the present application has the following beneficial effects by adopting the above-mentioned arbitrary support bridge damping and modal recovery method based on left and right driving vehicles:

[0075] (1) The method innovatively utilizes the differential dynamic response induced by left and right driving vehicles on the bridge, and accurately captures and compensates for the distortion effect of the signal shape caused by the damping effect through the introduction of exponential decay correction technology.

[0076] (2) The core of the method is based on the symmetry optimization of the signal according to the driving direction of the vehicle, and a variety of mathematical strategies are combined to realize the symmetry recovery of the bridge structure response. This symmetry recovery not only eliminates the interference of damping attenuation on modal characteristics, but also significantly improves the identification accuracy of modal parameters under any support conditions.

[0077] (3) Compared with the traditional method, the method breaks through the limitation of simply supported beam and for the first time realizes the comprehensive analysis of the damping characteristics and modal information of the asymmetrically supported bridge (such as cantilever beam, continuous beam or cable-stayed bridge, etc.). It provides a more universal, robust and efficient technical path for dynamic performance evaluation, health monitoring and long-term service safety analysis of complex bridge structures.

[0078] The technical solutions of the present application will be further described in detail below through the drawings and examples. BRIEF DESCRIPTION OF DRAWINGS

[0079] Figure 1 It is a practical model diagram of the arbitrary support bridge damping and modal recovery method based on left and right driving vehicles of the present application;

[0080] Figure 2 It is a physical model diagram of the arbitrary support bridge damping and modal recovery method based on left and right driving vehicles of the present application.

[0081] Figure 3 It is a time domain diagram of vehicle contact point acceleration, wherein (a) is a time domain diagram of contact point acceleration of left driving vehicle, and (b) is a time domain diagram of contact point acceleration of right driving vehicle;

[0082] Figure 4 It is a wavelet transform time-frequency diagram of vehicle contact point, wherein (a) is a wavelet transform time-frequency diagram of contact point of left driving vehicle, and (b) is a wavelet transform time-frequency diagram of contact point of right driving vehicle;

[0083] Figure 5 It is the first-order instantaneous amplitude of contact point of left driving vehicle, (a) is the first-order instantaneous amplitude of contact point of left driving vehicle extracted, and (b) is the second-order instantaneous amplitude of contact point of left driving vehicle extracted;

[0084] Figure 6 It is the instantaneous amplitude of contact point of right driving vehicle, wherein (a) is the first-order instantaneous amplitude of contact point of right driving vehicle extracted, and (b) is the second-order instantaneous amplitude of contact point of right driving vehicle extracted;

[0085] Figure 7is the compensated first order mixed signal instantaneous amplitude, and (b) is the compensated second order mixed signal instantaneous amplitude.

[0086] Figure 8 is the compensated first order mixed signal instantaneous amplitude, and (b) is the compensated second order mixed signal instantaneous amplitude.

[0087] Figure 9 is the comparison diagram of the theoretical modal and the recovered modal, wherein (a) is the comparison diagram of the recovered first order vibration mode and the theoretical modal, and (b) is the comparison diagram of the recovered second order vibration mode and the theoretical modal. DETAILED DESCRIPTION

[0088] The technical solutions of the present application are further described below by means of the accompanying drawings and examples.

[0089] Unless otherwise defined, the technical terms or scientific terms used in the present application shall have the usual meanings understood by those with ordinary skills in the art to which the present application pertains. The terms "first", "second", and similar words used in the present application do not represent any order, number, or importance, but are only used to distinguish different components. The terms "include", "contain", and similar words mean that the elements or objects appearing before the words cover the elements or objects listed after the words and their equivalents, without excluding other elements or objects. The terms "connect" or "connected" and similar words are not limited to physical or mechanical connections, but can include electrical connections, whether direct or indirect. The terms "upper", "lower", "left", "right", and the like are only used to represent relative positional relationships, which can change accordingly when the absolute positions of the described objects change.

[0090] EMBODIMENT

[0091] Referring to Figures 1-3 The present application provides a method for restoring the damping and modal of any support bridge based on left and right driving vehicles, which comprises the following steps:

[0092] Step S1, a vehicle-bridge coupling mechanical model is established. Based on the different driving directions of the vehicles, the dynamic equations of the left and right driving vehicles are established respectively, and the complete coupling system is formed by combining the bridge vibration equation.

[0093] Step S2, the acceleration response of the left and right vehicle-bridge contact points is calculated, and the response inversion recursive formula of the left and right vehicle-bridge contact points is obtained.

[0094] Step S3, the main modal characteristics of the bridge are extracted, and the instantaneous amplitudes of each modal are separated.

[0095] Step S4, based on the left lane vehicle-bridge contact point instantaneous amplitude, construct the mixed signal, and then introduce the penalty term e αt To restore the symmetry of the mixed signal, where a is the optimization parameter, and a is denoted as a when the signal restores the complete symmetry * , solve the optimal parameter a * ;

[0096] Step S5, based on the optimal parameter a * , calculate the bridge damping ratio, compensate the instantaneous amplitude signal of each mode to eliminate the damping attenuation effect, and then normalize the amplitude to extract the mode shape without damping attenuation.

[0097] The method is further illustrated by the following embodiment:

[0098] In practical applications, it is not required that the left and right vehicles enter the bridge at the same time. As long as the operating speeds of the two vehicles are the same, the time axis can be aligned through data processing to achieve the required signal characteristic analysis.

[0099] A vehicle-bridge coupling dynamics model is constructed.

[0100] The interaction between the vehicle and the bridge can be described by an actual model and a physical model. Based on the different directions of vehicle travel, the dynamic equations of left and right traveling vehicles are established respectively, and combined with the bridge vibration equation to form a complete coupling system.

[0101] Left lane vehicle motion equation:

[0102]

[0103] Right lane vehicle motion equation:

[0104]

[0105] where the superscripts and represent the left and right lane vehicles, and represent the mass, damping coefficient and spring stiffness of the vehicle, or and are the first and second order derivatives of the relevant displacement with respect to time t, and represent the vertical acceleration of the left and right lane vehicles, and represent the vertical velocity of the left and right lane vehicles, and represent the left and right lane vehicle-bridge contact point velocity, and v represents the vehicle operating speed.

[0106] The bridge structure is described by the Euler-Bernoulli beam model, and its vibration equation can be expressed as:

[0107]

[0108] where m represents the unit length mass of the bridge, c is the damping coefficient of the bridge, EI is the bending stiffness of the bridge, u(x, t) is the vertical displacement of the bridge, and denote the first and second order derivatives of the relevant displacement with respect to time t, respectively, u””(x, t) is the fourth order derivative with respect to the longitudinal coordinate x, f c (x, t) is the external load borne by the bridge, and is related to the left and right moving vehicles.

[0109] f c (x, t) is determined by the sum of the forces of the left and right vehicles, and its expression is as follows:

[0110]

[0111] where δ(x) is the Dirac function, g is the acceleration of gravity, L is the length of the bridge, and v is the running speed of the vehicle.

[0112] Based on the Euler beam theory, the vertical displacement u(x, t) of the bridge can be expanded into a modal superposition form:

[0113]

[0114] where q n (t) represents the time evolution characteristics, and φ n (β n x) describes the spatial modal distribution, β n is the modal related parameter.

[0115] Substitute the vehicle-bridge coupling force term (equation (4)) and the bridge modal expansion (equation (5)) into the dynamic equation (equation (3)), and multiply the modal function φ p (β p x) on both sides of the equation, and then integrate over the interval [0, L]. By using the orthogonality of the mode shape functions, and assuming that the vertical acceleration of the vehicle is much smaller than the acceleration of gravity (i.e. ), the approximate differential equation of the modal coordinate q n (t) can be obtained:

[0116]

[0117] where F n (t) is the external load term, and its expression is:

[0118]

[0119] The equation consists of a homogeneous solution (the free vibration response of the bridge itself) and a particular solution (the forced vibration generated by external vehicle excitation). The particular solution can be denoted as P. n (β n The frequency of the modal response (vt) is consistent with that of the external load term. The homogeneous solution depends only on the modal characteristics of the bridge itself. After derivation, the final modal response is expressed as:

[0120]

[0121] In the formula, and The parameter that determines the magnitude of the sinusoidal and cosine vibration components in free vibration;

[0122] Based on modal expansion, the contact point responses of the left and right moving vehicles can be expressed as follows:

[0123]

[0124] in and For low-frequency components, the expression is:

[0125]

[0126] Furthermore, in formulas (9) and (10), the bridge frequency ω is involved. Dn In the item, φ n (β n vt) and φ n (β n L-β n vt) is a low-frequency envelope function, whose rate of change over time is much smaller than that of the main oscillation term sin(ωt). Dn t) and cos(ω) Dn The characteristic frequencies of t). In particular, according to research, β n v << ω Dn This indicates that the changes of these envelope functions are slow, and the magnitudes of their first and second derivatives are orders of magnitude smaller than the second derivative of the main oscillation term.

[0127] From the perspective of magnitude analysis, φ′ n (β n The magnitude of the contribution of vt) is approximately O(β). n v), and its second derivative φ” n (β n The magnitude of the contribution of vt) is approximately O(β). n vt) 2 Similarly, φ′ n (β n L-β n vt) and φ”n (β n L-β n vt) also exhibits the same variation characteristics. In contrast, the second derivative of the main oscillation term is on the order of O(ωt). Dn ) 2 And in β n v<<ω Dn Under the condition that, satisfy This indicates that the derivative term of the envelope function has a weak effect on acceleration and can be considered a higher-order small quantity compared to the dominant term, so it can be ignored in approximate calculations.

[0128] Ignoring minor terms, the acceleration in displacement formulas (9) and (10) can be approximated as follows:

[0129]

[0130] in, and It is determined by the following relationship:

[0131]

[0132] and Some of these components are mainly controlled by the driving frequency. Studies have shown that these components usually belong to the low-frequency dense region (below 0.5Hz), and have little impact on the main vibration modes of the bridge. Therefore, they can be removed by low-pass filtering.

[0133] Therefore, it can be seen that the main component of the contact point acceleration is the exponential decay term. And the main oscillation term sin(ω) Dn t) and cos(ω) Dn It consists of t), and the envelope function φ n (β n vt) and φ n (β n L-β n vt) modulates only the amplitude. Ignoring the first and second derivative terms of the envelope function is not only reasonable, but also effectively simplifies the calculation, making the expression clearer.

[0134] Furthermore, while the contact point response can be transmitted to the vehicle body, existing research has shown that the contact point response is more advantageous than the vehicle response in bridge modal parameter extraction. Therefore, this patent focuses on the contact point response, although the same method is still applicable to the extraction and analysis of vehicle body signals.

[0135] like Figure 3 As shown, the time-domain plot of the contact point acceleration of a left-hand vehicle is as follows. Figure 3 As shown in (a), the time-domain plot of the acceleration at the contact point of the right-hand vehicle is as follows:Figure 3 (b) shown. The contact point acceleration calculation method is as follows:

[0136] Taking the left vehicle contact point response as an example, first, the second derivative of the vertical vibration equation (1) of the vehicle is taken, and is rewritten as:

[0137]

[0138] Wherein, The central difference method can be used for calculation, and the expression is as follows:

[0139]

[0140] Wherein, i represents the sampling point, and Δt is the sampling interval.

[0141] Integrating formula (17) in the time interval [0, t], the acceleration response of the left contact point is obtained:

[0142]

[0143] After arrangement,

[0144]

[0145] Further, formula (10) is extended to And the recursive formula is arranged:

[0146]

[0147] The integral term is used to calculate the value between two sampling points, and a variety of integral methods can be used to solve, such as the left rectangle method, the right rectangle method, and higher precision trapezoidal method, Simpson method and Gauss quadrature method, etc. The trapezoidal integral method is selected in this patent example, and its specific form is as follows:

[0148]

[0149] Substituting formula (11) and simplifying, the contact point response inversion recursive formula of the left vehicle is obtained, as shown in the following formula:

[0150]

[0151] Similarly, the contact point response inversion recursive formula of the right vehicle is:

[0152]

[0153] Wherein, And Respectively represent the vehicle-bridge contact point acceleration response of the left vehicle and the right vehicle, Δt is the sampling interval, And respectively, denote the second derivative of the vertical vibration acceleration of the left and right vehicles with respect to time t.

[0154] To extract the main modal characteristics of the bridge, the calculated contact point responses need to be filtered to remove the effects of vehicle motion and retain only the vibration characteristics of the first few modes of the bridge. Since the frequencies of interest for most bridges are mainly below 30 Hz, a band-pass filter can be used to remove irrelevant frequency components. After filtering, the contact point acceleration and can be approximated as:

[0155]

[0156] where N is the number of finite modal orders, which is related to the range of the filter interval, and is usually an integer greater than 2 to ensure that the key modal information is not lost, denotes the bridge damping frequency of the nth mode, and n denotes the bridge frequency of the nth mode, and n denotes the low-frequency envelope function, and n denotes the bridge frequency of the nth mode, and n denotes the low-frequency envelope function, and n denotes the bridge frequency of the nth mode, and n denotes the bridge frequency of the nth mode, and and denote the coefficients related to the sine and cosine vibration components, respectively, denotes the damping factor considering damping.

[0157] Although the filter removes some non-modal components, the signal still contains multiple modal components. Therefore, it is necessary to further separate the instantaneous amplitude of each mode to characterize its time-varying characteristics. Time-frequency analysis combined with the ridge tracking method can effectively extract the envelope line (ridge) of each mode to depict the time-varying evolution of the modal characteristics.

[0158] To achieve accurate modal component extraction, wavelet transform can be used for time-frequency analysis (other time-frequency methods are also applicable), as shown in Figure 4 the wavelet transform time-frequency diagram of the contact point of the left vehicle is shown in Figure 4 (a), and the wavelet transform time-frequency diagram of the contact point of the right vehicle is shown in Figure 4 (b). Wavelet transform has good localization characteristics in both time and scale (corresponding to frequency) domains, and its mother wavelet is defined as:

[0159]

[0160] where ω0 is the center angular frequency, and j is the imaginary unit.

[0161] The continuous wavelet transform (CWT) of the signal is performed, and its mathematical expression is:

[0162]

[0163] where a > 0 is a scale parameter corresponding to the characteristic frequency of the signal; b is a time parameter representing the position of the signal in time domain; ψ * is the complex conjugate of Morlet wavelet.

[0164] After wavelet transform, the time-varying frequency characteristics of the signal can be observed on the time-scale plane (b, a), and different modal components correspond to different characteristic scales on the plane. Define the wavelet modulus value:

[0165]

[0166] At each time point b, the wavelet modulus value distribution is different at different scales a. The ridge line extracts the evolution track of the main modal component, and is defined as:

[0167]

[0168] Along the ridge line Reading the wavelet modulus value, the instantaneous envelope of the nth order modal can be obtained:

[0169]

[0170] The center frequency ω0 of Morlet wavelet is related to the modal frequency ω Dn of the signal:

[0171]

[0172] Therefore, in order to accurately capture the modal component Select the matching scale:

[0173]

[0174] At the scale a n , the result of wavelet transform can be approximately expressed as:

[0175]

[0176] where C n reflects the scale normalization constant of wavelet transform, which contains the normalization coefficient of wavelet kernel and the scaling factor of modal amplitude, θ n (b) represents the phase term. Taking the modulus value of the above formula, the instantaneous amplitude can be obtained, as shown in Figure 5 The first-order instantaneous amplitude of the contact point of the left vehicle extracted in this embodiment is shown in Figure 5 (a), the second-order instantaneous amplitude of the contact point of the left vehicle extracted is shown in Figure 5 (b); asFigure 6 The first-order instantaneous amplitude of the extracted contact point of the right vehicle is shown in Fig. 3(a). Figure 6 The second-order instantaneous amplitude of the extracted contact point of the right vehicle is shown in Fig. 3(b). Figure 6

[0177] Further, when the wavelet transform locks the modal frequency , the original time variable b = tb of the recovered signal is restored, and the instantaneous amplitude can be expressed as:

[0178]

[0179] For the right contact point signal:

[0180]

[0181] where Π n is a constant determined by the coefficients and ;

[0182] In order to construct a signal symmetric on the time axis, a new instantaneous amplitude is defined as:

[0183]

[0184] The first-order instantaneous amplitude of the mixed signal is shown in Fig. 4(a). Figure 7 The second-order instantaneous amplitude of the mixed signal is shown in Fig. 4(b). Figure 7 Figure 7

[0185] where

[0186]

[0187] In the coupling time interval [0, L / v] of the vehicle bridge, the following symmetry relationship is satisfied:

[0188]

[0189] However, the exponential decay term is monotonically decreasing on the time axis, so that

[0190] A n (t)≠A n (L / v-t) (40)

[0191] That is, the theoretical symmetry of the signal is destroyed by the exponential decay.

[0192] In order to restore the symmetry of the signal, a penalty term e αt is introduced, so that

[0193] ​​​

[0194] where a is a parameter that needs to be determined by optimization, aiming to compensate the symmetry breaking caused by the exponential decay and restore the symmetry of the signal, as shown in Figure 8 , the compensated first-order mixed signal instantaneous amplitude is as shown in Figure 8 (a), and the compensated second-order mixed signal instantaneous amplitude is as shown in Figure 8 (b). When the following condition is met:

[0195] a = ξ n ω n (42)

[0196] the signal restores the complete symmetry, and a is denoted as a * . The following describes a * specific solving process:

[0197] In actual processing, in order to calculate the optimal a, we use dynamic time warping (DTW) to evaluate the symmetry of the signal. First, calculate the DTW distance between the signal and the symmetric signal . The core of the DTW calculation process is to construct a distance matrix that measures the optimal path of aligning two signals on the time axis.

[0198] The calculation of the DTW distance can be completed by the following steps:

[0199] Local distance calculation: First, calculate the local distance d(i,j) between each time step t i and t j .

[0200]

[0201] Construct a distance matrix: Then, fill in a distance matrix D(i,j) by dynamic programming, which represents the cumulative minimum distance from the starting point to the (i,j) point:

[0202] D(i,j) = d(i,j) + min(D(i-1,j), D(i,j-1), D(i-1,j-1)) (44)

[0203] where D(i-1,j) represents the cumulative distance of the previous time point; D(i,j-1) represents the cumulative distance of the previous time inversion point; D(i-1,j-1) represents the cumulative distance on the diagonal.

[0204] Each element D(i,j) of the matrix is calculated by selecting the smallest value from the previous elements (above, left, or upper left), and then adding the distance of the current point. The first element D(0,0) of the initialization matrix is 0.

[0205] The final DTW distance is calculated as the right-bottom element of matrix D, D(N,N), which is the total cost of the optimal alignment path between the two signals, i.e., the DTW distance between the two signals:

[0206] D final = D(N,N) (45)

[0207] The bisection optimization procedure is as follows:

[0208] Optimize α in the interval [α min ,α max ] by bisection to minimize the DTW distance. The basic steps of bisection are as follows:

[0209] (1) Initialization: Set the initial interval [α min ,α max ] and compute the midpoint

[0210] (2) Calculate the DTW distance: For the current α mid , calculate the corresponding DTW distance D(α mid ).

[0211] (3) Adjust the interval: According to the size of D(α mid ), reduce the search interval:

[0212] If D(α mid ) is less than the target value (or the previous distance), it means that the current α mid is closer to the optimal solution, and update α min to α mid .

[0213] If D(α mid ) is greater than the target value, update α max to α mid .

[0214] (4) Convergence criterion: When the length of the interval is less than the pre-set precision threshold, stop optimization and output the optimal α value.

[0215] The convergence criterion is as follows:

[0216] In order to determine whether convergence has been reached, a small threshold value ∈ threshold is set in this embodiment, and when the length of the interval is less than this value, it is considered that the optimal solution has been found, and the optimal solution α * is output.

[0217] |α max -α min |≤∈ threshold (46)

[0218] wherein, ∈ threshold denotes a preset precision threshold, |a max -a min | denotes the interval length.

[0219] Generally, ∈ threshold can take a smaller value to ensure high precision.

[0220] By optimizing the solution of a * , the damping ratio can be further calculated, and the expression is as follows:

[0221]

[0222] wherein, ω n denotes the bridge frequency, which can generally take ω n = ω Dn .

[0223] Further, the compensated and normalized signal is obtained, and the expression is as follows:

[0224]

[0225] By introducing the time-space transformation t=x / v and performing normalization, the following is obtained: and respectively represent the modal shape of the bridge structure along the left-to-right and right-to-left directions. To obtain the modal shape conforming to the conventional definition (i.e., the modal form propagating from left to right), the mirror flip operation along the x-axis can be performed on , so as to realize the standardized representation of the modal shape.

[0226] As shown in Figure 9 , the comparison chart of the recovered first-order mode shape and the theoretical modal is shown in Figure 9 (a), and the comparison chart of the recovered second-order mode shape and the theoretical modal is shown in Figure 9 (b).

[0227] Therefore, the present application adopts the above-mentioned arbitrary support bridge damping and modal recovery method based on left and right driving vehicles, which fully utilizes the structural response characteristics induced by left and right driving vehicles, combines the exponential decay correction and signal symmetry optimization strategy, not only effectively compensates the signal distortion caused by damping, but also accurately identifies the damping and modal parameters of the arbitrary support bridge, thereby providing an innovative and universal technical solution for dynamic performance evaluation and health monitoring of complex bridge structures.

[0228] It should be pointed out finally that the above examples are only used to illustrate the technical solutions of the present application but not to limit it, and although the present application has been described in detail with reference to the preferred embodiments, it should be understood by those skilled in the art that the technical solutions of the present application can still be modified or replaced equivalently, and these modifications or equivalent replacements should not make the modified technical solutions deviate from the spirit and scope of the technical solutions of the present application.

Claims

1. A method for damping and modal restoration of any supported bridge based on left and right traveling vehicles, characterized by, The method comprises the following steps: Step S1, a vehicle-bridge coupling mechanics model is established, dynamic equations of left and right driving vehicles are respectively established based on different vehicle driving directions, and a complete coupling system is formed by combining a bridge vibration equation; Step S2, left and right vehicle-bridge contact point acceleration responses are calculated to obtain left and right vehicle-bridge contact point response inversion recursive formulas; Step S3, main modal characteristics of the bridge are extracted, each modal instantaneous amplitude is separated, and a mixed signal is constructed based on left and right vehicle-bridge contact point instantaneous amplitudes; Step S4, introducing a penalty term to restore the symmetry of the mixed signal, wherein, is an optimization parameter, when the signal restores the complete symmetry, is denoted as , the optimal parameter is solved; Step S5, based on the optimal parameters The bridge damping ratio is calculated, the instantaneous amplitude signal of each order mode is compensated to eliminate the damping attenuation effect, and then the amplitude is normalized to extract the mode shape without damping attenuation.

2. The arbitrary support bridge damping and modal restoring method for a left-and-right traveling vehicle according to claim 1, characterized by, In the step S1, the dynamic equation of the left driving vehicle is as follows: ； The dynamic equation of the right driving vehicle is as follows: ； where the superscripts and denote the left and right lane vehicles, respectively, , and denote the mass, damping coefficient and spring rate of the vehicle, respectively, ; and are the first and second derivatives of the relevant displacement with respect to time t, respectively, and denote the vertical accelerations of the left and right lane vehicles, respectively, and denote the vertical velocities of the left and right lane vehicles, respectively, and denote the velocities of the left and right lane vehicle-bridge contact points, respectively, denotes the vehicle running speed.

3. The arbitrary support bridge damping and modal restoring method for a left-and-right traveling vehicle according to claim 2, characterized by, The bridge vibration equation in the step S1 is as follows: ； wherein, denotes the mass per unit length of the bridge, is the damping coefficient of the bridge, is the bending stiffness of the bridge, is the vertical displacement of the bridge, and denote the first and second order derivatives of the relevant displacement quantity with respect to time , respectively, is the fourth order derivative with respect to the longitudinal coordinate , and is the external load on the bridge, and is related to the moving vehicles.

4. The arbitrary support bridge damping and modal restoring method for a left-and-right traveling vehicle according to claim 3, characterized by, The contact point response inversion recursive formula of the left driving vehicle in the step S2 is as follows: ； The contact point response inversion recursive formula of the right driving vehicle is as follows: ； wherein and respectively represent the vehicle-bridge contact point acceleration response of the left and right vehicles, is the sampling interval, and respectively represent the second order derivative of the vertical vibration acceleration with respect to time t of the left and right vehicles.

5. The arbitrary support bridge damping and modal restoring method for left-and-right travel vehicle based on claim 4, characterized by, The specific steps of the step S3 are as follows: Step S31, band-pass filtering is performed on the calculated contact point response to remove vehicle motion influence and irrelevant frequency components and retain bridge first-order modal vibration characteristics; The band-pass filter is used to remove irrelevant frequency components, and after the filtering process, the acceleration of the contact point and is approximately expressed as: ； ； wherein, is the limited modal order, which is related to the filter interval range, and is an integer greater than 2 to ensure that the key modal information is not lost, represents the bridge damping frequency of the nth modal, and represents the low-frequency envelope function, is the modal-related parameter, which is closely related to the bridge mode shape, and respectively represent the coefficients related to the sine and cosine vibration components, represents the damping factor considering damping; Step S32, time-frequency analysis is used in combination with a ridge line tracking method, wavelet transform is used to separate the instantaneous amplitudes of each mode, and when the wavelet transform locks the mode main frequency, the instantaneous amplitude can be expressed as: ； For the right contact point signal: ； wherein is a constant determined by the coefficient and and Step S33, a signal symmetrical on the time axis is constructed, and a new instantaneous amplitude is defined as: ； Wherein ； In the axle coupling time interval Within, The following symmetry relationship is satisfied: ； However, due to the exponential decay term monotonically decreasing in the time axis, such that: ； The theoretical symmetry of the signal is destroyed by exponential decay.

6. The arbitrary support bridge damping and modal restoration method for a left-and-right travel vehicle according to claim 5, characterized by, The specific steps of the step S4 are as follows: To restore the symmetry of the signal, a penalty term is introduced such that: ； wherein is a parameter to be determined by optimization, when the following is satisfied: ； wherein denotes the bridge damping ratio, denotes the bridge frequency; The signal recovers the full symmetry, in which case is written as The optimal parameters are found by solving the following steps: Step S41, symmetry of the signal is evaluated by DTW, and a DTW distance between the signal and the symmetrical signal is calculated, and the specific steps are as follows: Step S411, local distance calculation: First, calculate the local distance between each time step and between each time step : ； Step S412, constructing a distance matrix: Then, a distance matrix is filled by dynamic programming , representing the cumulative minimum distance from the starting point to the point. ； wherein, cumulative distance representing the time point of the previous step; cumulative distance representing the time point of the previous step; cumulative distance representing the diagonal line; Each element of the matrix It is calculated by selecting the smallest value among the previous elements and adding the distance to the current point, initializing the first element of the matrix. ; Step S413, the final DTW distance is calculated: the right lower corner element D(N,N) of the matrix D is the total cost of the optimal alignment path between the signals, that is, the DTW distance between the two signals: ； Step S42, bisection method is used to optimize in a given interval to minimize the DTW distance, and the specific steps are as follows: Step S421, initialization: set initial interval and calculate midpoint ; Step S422, calculating the DTW distance: for the current , calculating the corresponding DTW distance ; Step S423, Adjust the interval: According to The size of the search range is reduced. If less than the target value or the previous distance, it means that the current solution is closer to the optimal solution, update for ; If greater than the target value, update for ; Step S43, setting a convergence criterion, when the interval length is less than a preset precision threshold, stopping optimization, and outputting the optimal parameters The expression is as follows: ； In the formula, denotes a preset precision threshold, denotes an interval length.

7. The arbitrary support bridge damping and modal restoring method for a left-and-right traveling vehicle according to claim 6, characterized by, The calculation formula of the bridge damping ratio in the step S5 is as follows: ； In the formula, represents the bridge frequency, taken as .

8. The arbitrary support bridge damping and modal restoring method for left-and-right travel vehicle based on claim 7, characterized by, In the step S5, the instantaneous amplitude signals of each order mode are compensated to eliminate the damping decay effect, and then the amplitude is normalized to extract the mode shape without damping decay, as follows: ； ； By introducing a time-space transformation and normalizing it gives and represent the recovery modal shape of the bridge structure along the left-to-right and right-to-left directions, respectively. To obtain the modal shape in accordance with the conventional definition, a mirror flip operation along the x-axis is performed on , thus achieving the standard representation of the modal shape.

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