Arbitrary supporting bridge damping and modal recovery method based on vehicles running left and right

Abstract

The invention discloses an arbitrary supporting bridge damping and modal recovery method based on a left-right driving vehicle, and relates to the technical field of bridge health monitoring, and the method comprises the steps: building a vehicle-bridge coupling mechanical model; calculating the acceleration response of a left-driving and right-driving-bridge contact point; extracting main modal characteristics of the bridge, and separating instantaneous amplitude of each modal; constructing a mixed signal based on the instantaneous amplitudes of the two vehicle-bridge contact points, and solving an optimal parameter alpha * to recover the symmetry of the mixed signal; and based on the optimal parameter alpha *, calculating a bridge damping ratio, compensating each-order modal instantaneous amplitude signal to eliminate a damping attenuation effect, then normalizing the amplitude, and extracting a modal shape without damping attenuation. According to the damping and modal recovery method for any supporting bridge based on the vehicles running left and right, signal distortion caused by damping is effectively compensated, the damping and modal parameters of any supporting bridge can be accurately recognized, and a technical solution is provided for dynamic performance evaluation and health monitoring of a complex bridge structure.

Description

Technical Field

[0001] The present invention relates to the technical field of bridge health monitoring, and particularly to a method for restoring the damping and mode of an arbitrarily supported bridge based on left and right moving vehicles. Background Art

[0002] As a key part of transportation infrastructure, the structural health monitoring of bridges is of great importance. Accurately obtaining the damping and modal parameters of bridges is of great significance for evaluating the working state of bridges, predicting potential diseases, and ensuring the safe operation of bridges. In actual monitoring, the vibration response signals generated by the interaction between vehicles and bridges contain rich bridge structure information. With the increase in traffic flow and the growth of the service life of bridges, traditional monitoring methods have exposed many limitations when dealing with bridge vibration signals in complex environments.

[0003] The existing vehicle scanning method mainly focuses on the modal identification of simply supported beams, and its theoretical premise depends on symmetric boundary conditions and uniform damping distribution. However, in actual engineering, bridges often have asymmetric supports, complex geometric shapes, or heterogeneous material properties, resulting in significant asymmetry in the structural response affected by damping attenuation. This asymmetry not only destroys the theoretical symmetry characteristics of the signals but also causes a significant decrease in the accuracy of traditional methods in extracting instantaneous amplitudes and modal parameters, and even makes them inapplicable to complex support systems of non-simply supported beams.

[0004] When the existing methods face bridges with asymmetric supports or complex boundary conditions, the accuracy of modal feature extraction often decreases due to damping attenuation and asymmetric structural response. Therefore, it is urgent to develop a method that can be applied to arbitrarily supported bridges, effectively compensate for the influence of damping attenuation, and accurately separate and analyze each modal information. Summary of the Invention

[0005] The purpose of the present invention is to provide a method for restoring the damping and mode of an arbitrarily supported bridge based on left and right moving vehicles, to solve the limitation problem that the traditional vehicle scanning method is mainly limited to the modal identification of simply supported beams, and to restore the symmetry of the signals by adjusting the exponential decay effect of the signals, thereby improving the accuracy of modal feature extraction.

[0006] To achieve the above purpose, the present invention provides a method for restoring the damping and mode of an arbitrarily supported bridge based on left and right moving vehicles, including the following steps:

[0007] Step S1: Establish a vehicle-bridge coupling mechanical model, and respectively establish the dynamic equations of left and right moving vehicles based on the different driving directions of the vehicles, and combine them with the bridge vibration equation to form a complete coupling system;

[0008] Step S2: Calculate the acceleration responses of the left and right vehicle-bridge contact points, and obtain the inverse recurrence formula for the responses of the left and right vehicle-bridge contact points;

[0009] Step S3: Extract the main modal characteristics of the bridge and separate the instantaneous amplitudes of each mode;

[0010] Step S4: Based on the instantaneous amplitudes of the left vehicle-bridge contact points, construct a mixed signal, and then introduce a penalty term e αt to restore the symmetry of the mixed signal, where α is an optimization parameter. When the signal restores complete symmetry, α is denoted as α * , and solve for the optimal parameter α * ;

[0011] Step S5: Based on the optimal parameter α * , calculate the bridge damping ratio, compensate the instantaneous amplitude signals of each order of the mode to eliminate the damping attenuation effect, and then normalize the amplitudes to extract the modal shape 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] In the formula, the superscripts and represent the left and right vehicles respectively, and represent the mass, damping coefficient, and spring stiffness of the vehicle respectively. j = l or and ) are the first and second derivatives of the relevant displacement with respect to time t respectively, and represent the vertical accelerations of the left and right vehicles respectively, and represent the vertical velocities of the left and right vehicles respectively, and represent the velocities of the left and right vehicle-bridge contact points respectively. v represents the vehicle running speed.

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

[0018]

[0019] In the formula, m represents the mass per unit length of the bridge, c is the damping coefficient of the bridge, EI is the flexural stiffness of the bridge, u(x, t) is the vertical displacement of the bridge, and respectively represent the first and second derivatives of the relevant displacement with respect to time t, u””(x,t) is the fourth derivative with respect to the longitudinal coordinate x, and f c (x,t) is the external load borne by the bridge, related to the left - and right - moving vehicles.

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

[0021]

[0022] The inversion recurrence formula for the contact point response of the right - moving vehicle is as follows:

[0023]

[0024] In the formula, and respectively represent the acceleration responses of the vehicle - bridge contact points of the left - moving vehicle and the right - moving vehicle, Δt is the sampling interval, and respectively represent the second derivatives of the vertical vibration accelerations 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: Perform band - pass filtering on the back - calculated contact point response to remove the influence of vehicle movement and irrelevant frequency components, and retain the vibration characteristics of the first few modal vibrations of the bridge;

[0027] After using band - pass filtering to remove irrelevant frequency components, the contact point accelerations and are approximately expressed as:

[0028]

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

[0030] Step S32: Using time-frequency analysis combined with ridge tracking method, separate the instantaneous amplitudes of each mode through wavelet transform. When the wavelet transform locks the main frequency of the mode, 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 coefficients and ;

[0035] Step S33: Construct a signal symmetric on the time axis and define the new instantaneous amplitude as:

[0036]

[0037] where

[0038]

[0039] Within the vehicle-bridge coupling time interval [0, L / v], satisfies the following symmetry relationship:

[0040]

[0041] However, due to the exponential decay term monotonically decreasing on the time axis, it results in:

[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] To restore the symmetry of the signal, introduce the penalty term e αt , so that:

[0046]

[0047] where α is a parameter to be determined by optimization. When the following formula is satisfied:

[0048] α = ξ n ω n ;

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

[0050] Step S41: Evaluate the signal symmetry through DTW, and 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 the distance matrix: Then, fill a distance matrix D(i, j) through dynamic programming, representing the cumulative minimum distance from the starting point to the point (i, j):

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

[0055] where D(i - 1, j) represents the cumulative distance at the previous time point; D(i, j - 1) represents the cumulative distance at the previous time reversal point; D(i - 1, j - 1) represents the cumulative distance on the diagonal;

[0056] Each element D(i, j) of the matrix is calculated by selecting the smallest value among the previous elements and then adding the distance of the current point. Initialize the first element of the matrix D(0, 0) = 0;

[0057] Step S413: Calculate the final DTW distance: The lower - right 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 within a given interval using the bisection method to minimize the DTW distance. The specific steps are as follows:

[0060] Step S421: Initialization: Set the initial interval [α min , α max , and calculate the mid - point

[0061] Step S422: Calculate the DTW distance: For the current α mid , calculate the corresponding DTW distance D(α mid );

[0062] Step S423, Adjustment Interval: According to the magnitude of D(α mid ), narrow the search interval:

[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 update α min to α mid ;

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

[0065] Step S43, Set the convergence criterion. When the interval length is less than the preset precision threshold, stop the optimization and output the optimal parameter α * , and 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 for the bridge damping ratio in step S5 is as follows:

[0069]

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

[0071] Preferably, the signal for compensation and normalization in step S5 is as follows:

[0072]

[0073] By introducing the time - space transformation t = x / v and performing normalization, we obtain and respectively represent the restoring modal shapes of the bridge structure along the left - to - right and right - to - left directions. To obtain a modal shape that conforms to the conventional definition, perform a mirror flip operation of along the x - axis, thereby realizing the standardized representation of the modal shape.

[0074] Therefore, the present invention adopts the above - mentioned method for arbitrary - support bridge damping and modal restoration based on left - and right - driving vehicles, and has the following beneficial effects:

[0075] (1) This method innovatively utilizes the differential dynamic responses induced by left - and right - driving vehicles on the bridge. By introducing the exponential decay correction technique, it accurately captures and compensates for the distortion effect of the damping effect on the signal shape.

[0076] (2) The core of this method lies in the optimization of signal symmetry based on the vehicle driving direction. Combining multiple mathematical strategies, it realizes the symmetry restoration of the bridge structure response. This symmetry restoration not only eliminates the interference of damping decay on modal characteristics but also significantly improves the identification accuracy of modal parameters under any support conditions.

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

[0078] 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

[0079] Figure 1 It is the actual model diagram of the damping and modal restoration method for arbitrary - supported bridges based on left - and right - driving vehicles of the present invention;

[0080] Figure 2 It is the physical model diagram of the damping and modal restoration method for arbitrary - supported bridges based on left - and right - driving vehicles of the present invention.

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

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

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

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

[0085] Figure 7is the instantaneous amplitude of the mixed signal, where (a) is the instantaneous amplitude of the first-order mixed signal and (b) is the instantaneous amplitude of the second-order mixed signal;

[0086] Figure 8 is the instantaneous amplitude of the compensated mixed signal, where (a) is the instantaneous amplitude of the compensated first-order mixed signal and (b) is the instantaneous amplitude of the compensated second-order mixed signal;

[0087] Figure 9 is the comparison diagram between the theoretical mode and the restored mode, where (a) is the comparison diagram between the restored first-order vibration mode and the theoretical mode, and (b) is the comparison diagram between the restored second-order vibration mode and the theoretical mode. Detailed implementation manners

[0088] The technical solutions of the present invention will be further described below with reference to the accompanying drawings and embodiments.

[0089] 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. The "first", "second" and similar terms used in the present invention do not denote any order, quantity or importance, but are only used to distinguish different components. The terms such as "comprising" or "including" mean that the elements or objects appearing before this term cover the elements or objects listed after this term and their equivalents, without excluding other elements or objects. The terms such as "connected" or "coupled" are not limited to physical or mechanical connections, but may include electrical connections, whether direct or indirect. The terms such as "upper", "lower", "left" and "right" are only used to represent relative positional relationships, and when the absolute position of the object being described changes, the relative positional relationship may also change accordingly.

[0090] Embodiment

[0091] Please refer to Figures 1-3 , the present invention provides a method for damping and modal restoration of any supported bridge based on left- and right-traveling vehicles, including the following steps:

[0092] Step S1: Establish a vehicle-bridge coupling mechanical model, and respectively establish dynamic equations of left- and right-traveling vehicles based on different vehicle traveling directions, and combine with the bridge vibration equation to form a complete coupling system;

[0093] Step S2: Calculate the acceleration responses of the left- and right-traveling vehicle-bridge contact points, and obtain the inverse recurrence formula for the responses of the left- and right-traveling vehicle-bridge contact points;

[0094] Step S3: Extract the main modal characteristics of the bridge and separate the instantaneous amplitudes of each mode;

[0095] Step S4: Based on the instantaneous amplitude of the left vehicle-bridge contact point, construct a mixed signal, and then introduce a penalty term \(e\) αt to restore the symmetry of the mixed signal, where \(\alpha\) is an optimization parameter. When the signal restores complete symmetry, \(\alpha\) is denoted as \(\alpha\) * , and solve for the optimal parameter \(\alpha\) * ;

[0096] Step S5: Based on the optimal parameter \(\alpha\) * , calculate the bridge damping ratio, compensate the instantaneous amplitude signals of each order mode to eliminate the damping attenuation effect, and then normalize the amplitude to extract the modal shape without damping attenuation.

[0097] The following further illustrates this method through this embodiment:

[0098] In practical applications, it is not required that vehicles traveling in both directions enter the bridge simultaneously. As long as the running speeds of the two vehicles are the same, and the time axis is aligned through data processing, the required signal characteristic analysis can be achieved.

[0099] Construct a vehicle-bridge coupling dynamics model.

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

[0101] Equation of motion for the left-going vehicle:

[0102]

[0103] Equation of motion for the right-going vehicle:

[0104]

[0105] Among them, the superscripts and represent the left-going and right-going vehicles respectively, and represent the mass, damping coefficient, and spring stiffness of the vehicle respectively, or and are the first and second derivatives of the relevant displacement with respect to time \(t\) respectively, and represent the vertical accelerations of the left-going and right-going vehicles respectively, and represent the vertical velocities of the left-going and right-going vehicles respectively, and represent the velocities of the left-going and right-going vehicle-bridge contact points respectively, and \(v\) represents the vehicle running 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 mass per unit length of the bridge, c is the damping coefficient of the bridge, EI is the flexural rigidity of the bridge, u(x,t) is the vertical displacement of the bridge, and represent the first - order 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, and f c (x,t) is the external load borne by the bridge, which is related to the left - moving and right - moving vehicles.

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

[0110]

[0111] where δ(x) is the Dirac function, g is the acceleration due to 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 in the form of modal superposition:

[0113]

[0114] where q n (t) characterizes the time - evolution characteristics, while φ n (β n x) describes the spatial modal distribution, and 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 both sides of the equation by the modal function φ p (β p x), and then integrate over the interval [0,L]. Using the orthogonality of the mode - shape functions and assuming that the vertical acceleration of the vehicle is much smaller than the acceleration due to 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 the expression is:

[0118]

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

[0120]

[0121] In the formula, and represent the parameters that determine the amplitudes of the sine and cosine vibration components in the free vibration;

[0122] Based on modal expansion, the responses of the contact points of the vehicles traveling in both directions can be respectively expressed as:

[0123]

[0124] where and are the low-frequency components, and the expressions are:

[0125]

[0126] In addition, in formulas (9) and (10), in the terms involving the bridge frequency ω Dn , φ n (β n vt) and φ n (β n L-β n vt) are low-frequency envelope functions, and the rate of their time variation is much smaller than the characteristic frequencies of the main oscillation terms sin(ω Dn t) and cos(ω Dn t). In particular, according to the research, β n v << ω Dn , indicating that the changes of these envelope functions are slow, and the amplitudes of their first and second derivatives are of a smaller order of magnitude compared to the second derivative of the main oscillation term.

[0127] From the perspective of magnitude analysis, the magnitude contributed by φ′ n (β n vt) is approximately O(β n v), while the magnitude contributed by its second derivative φ” n (β n 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 has the same variation characteristics. In contrast, the order of magnitude of the second derivative of the main oscillation term is O(ω Dn ) 2 , and under the condition of β n v << ω Dn , it satisfies This indicates that the derivative term of the envelope function has a weak influence on acceleration and can be regarded as a higher-order small quantity compared to the dominant term. Therefore, it can be ignored in approximate calculations.

[0128] After ignoring the small quantity terms, the accelerations of displacement formulas (9) and (10) can be approximately expressed as

[0129]

[0130] where and are determined by the following relationships:

[0131]

[0132] and The part is mainly controlled by the driving frequency. Research shows that these components usually belong to the low-frequency dense area (below 0.5 Hz), and their influence on the main vibration mode of the bridge is small. Therefore, they can be removed by low-pass filtering.

[0133] It can be seen that the main components of the contact point acceleration consist of the exponentially decaying term as well as the main oscillation terms sin(ω Dn t) and cos(ω Dn t), while the envelope functions φ n (β n vt) and φ n (β n L-β n vt) only modulate the amplitude. Ignoring the first and second derivative terms of the envelope function is not only reasonable but also can effectively simplify the calculation and make the expression clearer.

[0134] In addition, the contact point response can be transmitted to the vehicle body. However, existing research shows that during the extraction of bridge modal parameters, the contact point response has more advantages than the vehicle response. Therefore, this patent takes the contact point response as the main research object, although the same method is still applicable to the extraction and analysis of vehicle body signals.

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

[0136] Taking the contact point response of a left - moving vehicle as an example, first, take the second - derivative of the vertical vibration equation (1) of the vehicle and rewrite it as:

[0137]

[0138] where, can be calculated by the central difference method, and the expression is as follows:

[0139]

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

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

[0142]

[0143] After arrangement,

[0144]

[0145] Furthermore, generalize formula (10) to and arrange the recurrence formula:

[0146]

[0147] The integral term is used to calculate the value between two sampling points, and various integration methods can be used to solve it, such as the left - rectangle method, the right - rectangle method, and higher - precision methods like the trapezoidal method, Simpson's method, and Gaussian quadrature method, etc. In this patent example, the trapezoidal integration method is selected, and its specific form is as follows:

[0148]

[0149] Substitute it into formula (11) and simplify to obtain the inversion recurrence formula of the contact point response of the left - moving vehicle, as shown in the following formula:

[0150]

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

[0152]

[0153] where, and respectively represent the vehicle - bridge contact point acceleration responses of the left - moving vehicle and the right - moving vehicle, Δt is the sampling interval, and respectively represent the second derivatives of the vertical vibration accelerations of left - moving vehicles and right - moving vehicles with respect to time t.

[0154] To extract the main modal characteristics of the bridge, it is necessary to filter the contact point responses obtained by back - calculation to remove the influence of vehicle motion and only retain the vibration characteristics of the first few modes of the bridge. Given that the concerned frequencies of most bridges mainly concentrate below 30 Hz, band - pass filtering can be used to remove the irrelevant frequency components. After filtering, the contact point accelerations and can be approximately expressed as:

[0155]

[0156] where N is the finite modal order, which is related to the filtering interval range. Usually, an integer greater than 2 is taken to ensure that the key modal information is not lost. represents the damped natural frequency of the n - th mode of the bridge, and φ n (β n vt) and φ n (β n L - β n vt) represent the low - frequency envelope functions, β n is a modal - related parameter, which is closely related to the bridge vibration mode. and respectively represent the coefficients related to the sine and cosine vibration components. represents the attenuation factor considering damping.

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

[0158] To achieve accurate extraction of modal components, 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 - moving vehicle is shown in Figure 4 (a), and the wavelet transform time - frequency diagram of the contact point of the right - moving vehicle is shown in Figure 4 (b). Wavelet transform has good localization characteristics in both the time domain and the scale domain (corresponding to the frequency domain). Its mother wavelet is defined as:

[0159]

[0160] where ω 0 is the central angular frequency and j is the imaginary unit.

[0161] For the signal Perform continuous wavelet transform (CWT), whose mathematical expression is:

[0162]

[0163] where a > 0 is the scale parameter, corresponding to the characteristic frequency in the signal; b is the time parameter, indicating the position of the signal in the time domain; ψ * (·) is the complex conjugate of the 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 this plane. Define the wavelet modulus value:

[0165]

[0166] At each time point b, the wavelet modulus value distributions at different scales a are different. The ridge extracts the evolution trajectory of the main modal component, defined as:

[0167]

[0168] Along the ridge Read the wavelet modulus value, and the instantaneous envelope of the nth order mode can be obtained:

[0169]

[0170] The central frequency ω of the Morlet wavelet 0 is approximately satisfied with the modal frequency ω of the signal Dn as:

[0171]

[0172] Therefore, 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, including the normalization coefficient of the wavelet kernel and the scaling factor of the modal amplitude, and θ n (b) represents the phase term. Taking the modulus value of the above formula can obtain the instantaneous amplitude. As Figure 5 shown, the first-order instantaneous amplitude of the contact point of the left-moving vehicle extracted in this embodiment is as Figure 5(a) As shown, the second-order instantaneous amplitude of the contact point of the left-going vehicle extracted is as Figure 5 (b) shown; as Figure 6 shown, the first-order instantaneous amplitude of the contact point of the right-going vehicle extracted in this embodiment is as Figure 6 (a) shown, and the second-order instantaneous amplitude of the contact point of the right-going vehicle extracted is as Figure 6 (b) shown.

[0177] Furthermore, when the main frequency of the locked mode by wavelet transform is , the original time variable b of the restored signal is b = tb, and the instantaneous amplitude can be expressed as:

[0178]

[0179] For the right contact point signal:

[0180]

[0181] In the formula, 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] As Figure 7 shown, the instantaneous amplitude of the first-order mixed signal is as Figure 7 (a) shown, and the instantaneous amplitude of the second-order mixed signal is as Figure 7 (b) shown.

[0185] Among them

[0186]

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

[0188]

[0189] However, due to the exponentially decaying term monotonically decreasing on the time axis, it makes

[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] To restore the symmetry of the signal, a penalty term e is introduced αt , so that

[0193]

[0194] where α is a parameter to be determined by optimization, aiming to compensate for the symmetry breaking caused by exponential decay and restore the symmetry of the signal. As shown in Figure 8 , the instantaneous amplitude of the first-order mixed signal after compensation is as shown in Figure 8 (a), and the instantaneous amplitude of the second-order mixed signal after compensation is as shown in Figure 8 (b). When the condition

[0195] α = ξ n ω n (42)

[0196] is satisfied, the signal restores complete symmetry, and at this time α is denoted as α * . The following introduces the specific solution process of α * :

[0197] In actual processing, to calculate the optimal α, 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, which measures the optimal path for aligning two signals on the time axis.

[0198] The calculation of the DTW distance can be completed through 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] Constructing the distance matrix: Then, fill a distance matrix D(i,j) through dynamic programming, representing the cumulative minimum distance from the starting point to the point (i,j):

[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 at the previous time point; D(i,j - 1) represents the cumulative distance at the previous time reversal 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 among the previous elements (above, to the left, or upper left), and then adding the distance of the current point. Initialize the first element of the matrix D(0,0) = 0.

[0205] Calculate the final DTW distance: The bottom-right 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:

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

[0207] The optimization steps of the bisection method are as follows:

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

[0209] (1) Initialization: Set the initial interval [α min , α max , and calculate 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 ), narrow 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 judgment: When the length of the interval is less than the preset precision threshold, stop the optimization and output the optimal α value.

[0215] The convergence criterion is as follows:

[0216] To determine whether convergence has occurred, this embodiment sets a small threshold ∈ threshold . When the interval length 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] In the formula, ∈ threshold represents a preset precision threshold, and |α max -α min | represents the interval length.

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

[0220] By optimizing and solving for α * the damping ratio can be further calculated, and the expression is as follows:

[0221]

[0222] In the formula, ω n represents the bridge frequency, and generally ω n =ω Dn .

[0223] Furthermore, 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, and respectively represent the restored modal shapes of the bridge structure along the left - to - right and right - to - left directions. To obtain the modal shape that conforms to the conventional definition (i.e., the modal form propagating from left to right), the mirror - flipping operation along the x - axis can be performed on

[0226] such as Figure 9 shown, the comparison diagram between the restored first - order vibration mode and the theoretical mode is as shown in Figure 9 (a), and the comparison diagram between the restored second - order vibration mode and the theoretical mode is as shown in Figure 9 (b).

[0227] Therefore, the present invention adopts the above - mentioned method for damping and modal restoration of any - supported bridges based on left - and right - driving vehicles. By making full use of the structural response characteristics induced by left - and right - driving vehicles and combining the exponential decay correction and signal symmetry optimization strategy, it not only effectively compensates for the signal distortion caused by damping but also accurately identifies the damping and modal parameters of any - supported bridges, providing an innovative and universal technical solution for the dynamic performance evaluation and health monitoring of complex bridge structures.

[0228] 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 the technical solutions of the present invention or make equivalent replacements, and these modifications or equivalent replacements cannot make the modified technical solutions deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. The damping and modal recovery method of an arbitrary supported bridge based on left and right vehicles is characterized in that: The following steps are involved: Step S1, establishing a vehicle-bridge coupling mechanical model, based on the different driving directions of the vehicles, respectively establishing the dynamic equations of the left and right vehicles, and combining the bridge vibration equation to form a complete coupling system; Step S2, calculating the acceleration response of the left-traveling and right-traveling vehicle-bridge contact point, and obtaining the inverse recursive formula of the left-traveling and right-traveling vehicle-bridge contact point response; Step S3, extracting the main modal features of the bridge and separating the instantaneous amplitude of each modal; Step S4: construct a mixed signal based on the instantaneous amplitude of the left-traveling vehicle-bridge contact point, and then introduce a penalty term e αt To restore the symmetry of the mixed signal, α is the optimization parameter. When the signal restores the complete symmetry, α is recorded as α * , solve for the optimal parameter α * ; Step S5: Based on the optimal parameter α * , 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.

2. The damping and modal recovery method of an arbitrary supported bridge based on left and right traveling vehicles according to claim 1 is characterized in that: In step S1, the motion equation of the left-moving vehicle is as follows: The equation of motion for the right-moving vehicle is as follows: In the formula, the superscript l and Represent left and right vehicles respectively. and Respectively represent the mass, damping coefficient and spring stiffness of the vehicle, j = l or (·) and (··) are the first and second order derivatives of the displacement with respect to time t, respectively. and Respectively represent the vertical acceleration of the left and right vehicles, and Respectively represent the vertical speeds of left and right vehicles, and They represent the contact point speeds of the left- and right-moving vehicles-bridge respectively, and v represents the vehicle running speed.

3. The damping and modal recovery method of an arbitrary supported bridge based on left and right traveling vehicles according to claim 2 is characterized in that: The bridge vibration equation in step S1 is as follows: Where m represents the mass per unit length 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 (··) represent the first and second order derivatives of the displacement with respect to time t, respectively, u””(x, t) is the fourth order derivative with respect to the longitudinal coordinate x, and f c (x, t) is the external load on the bridge, which is related to the vehicles moving left or right.

4. The damping and modal recovery method of an arbitrary supported bridge based on left and right traveling vehicles according to claim 3 is characterized in that: The inverse recursive formula for the contact point response of the left-moving vehicle in step S2 is as follows: The inverse recursive formula for the contact point response of a right-moving vehicle is as follows: In the formula, and They represent the acceleration response of the vehicle-bridge contact point of the left-moving vehicle and the right-moving vehicle respectively, Δt is the sampling interval, and Y l (t) and They represent the second-order derivatives of the vertical vibration acceleration of the left-moving vehicle and the right-moving vehicle with respect to time t respectively.

5. The damping and modal recovery method of an arbitrary supported bridge based on left and right traveling vehicles according to claim 4 is characterized in that: The specific steps of step S3 are as follows: Step S31, band-pass filtering is performed on the contact point response obtained by back-calculation to remove the influence of vehicle motion and irrelevant frequency components, and retain the vibration characteristics of the first few modal orders of the bridge; Bandpass filtering is used to remove irrelevant frequency components. After filtering, the contact point acceleration and It can be expressed approximately as: Where N is the finite mode order, which is related to the filtering range and is usually an integer greater than 2 to ensure that key mode 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) represents the low-frequency envelope function, β n is a modal-related parameter, which is closely related to the vibration mode of the bridge. and denote the coefficients associated with the sine and cosine vibration components, respectively, represents the attenuation factor considering damping; Step S32: using time-frequency analysis combined with ridgeline tracing method, the instantaneous amplitude of each mode is separated by wavelet transform. When the wavelet transform locks the main frequency of the mode, the instantaneous amplitude can be expressed as: For the right contact point signal: In the formula, Π n is a constant, and the coefficient and Decide; Step S33: construct a signal symmetrical on the time axis and define a new instantaneous amplitude as: in In the vehicle-bridge coupling time interval [0,L / v], The following symmetry relations are satisfied: However, due to the exponential decay term Monotonically decreasing on the time axis, such that: A n (t)≠A n (L / v-t); The theoretical symmetry of the signal is destroyed by the exponential decay.

6. The damping and modal recovery method of an arbitrary supported bridge based on left and right traveling vehicles according to claim 5 is characterized in that: The specific steps of step S4 are as follows: In order to restore the symmetry of the signal, the penalty term e is introduced αt ,make: Among them, α is a parameter that needs to be determined through optimization, when the following formula is satisfied: a = x n oh n ; The signal restores complete symmetry, and α is now recorded as α * , solve for the optimal parameter α * The steps are as follows: Step S41: Evaluate the signal symmetry by DTW and calculate the DTW distance between the signal and the symmetrical signal. The specific steps are as follows: Step S411, local distance calculation: First, calculate at each time step t i and t j The local distance d(i,j) between them is: Step S412, construct a distance matrix: Then, fill a distance matrix D(i,j) through dynamic programming to represent the cumulative minimum distance from the starting point to the point (i,j): D(i,j)=d(i,j)+min(D(i-1,j),D(i,j-1),D(i-1,j-1)); Among them, 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 line; Each element D(i,j) of the matrix is ​​calculated by selecting the smallest value of the previous element and then adding the distance of the current point. The first element of the matrix is ​​initialized to D(0,0) = 0; Step S413: Calculate the final DTW distance: The lower right 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: D final =D(N,N); Step S42: Use the binary search method to optimize within a given interval to minimize the DTW distance. The specific steps are as follows: Step S421, initialization: set the initial interval [α min ,α max ] and calculate the midpoint Step S422, calculate DTW distance: for the current α mid , calculate the corresponding DTW distance D(α mid ); Step S423, adjust the interval: according to D(α mid ) to narrow the search range: If D(α mid ) is smaller than the target value or the previous distance, indicating that the current α mid Closer to the optimal solution, update α min is α mid ; If D(α mid ) is greater than the target value, update α max is α mid ; Step S43: Set the convergence criterion. When the interval length is less than the preset accuracy threshold, stop the optimization and output the optimal parameter α * , the expression is as follows: |α max -α min |≤∈ threshold ； In the formula, ∈ threshold represents the preset accuracy threshold, |α max -α min | indicates the length of the interval.

7. The damping and modal recovery method of an arbitrary supported bridge based on left and right traveling vehicles according to claim 6 is characterized in that: The calculation formula of the bridge damping ratio in step S5 is as follows: In the formula, ω n Represents the bridge frequency, generally taken as ω n =ω Dn .

8. The damping and modal recovery method of an arbitrary supported bridge based on left and right traveling vehicles according to claim 7 is characterized in that: The signal compensated and normalized in step S5 is as follows: By introducing the time-space transformation t = x / v and normalizing it, we can get and Respectively represent the restored mode shapes of the bridge structure from left to right and from right to left. In order to obtain the mode shapes that meet the conventional definition, A mirror flip operation along the x-axis is performed to achieve a normalized representation of the mode shapes.

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