Bridge modal parameter identification method based on random subspace and biaxial vehicle response

Abstract

The invention discloses a bridge modal parameter identification method based on random subspace and biaxial vehicle response, belongs to the technical field of bridge health detection, and solves the problem of precision misalignment when a traditional indirect detection method synchronously identifies the inherent frequency, vibration mode and damping ratio of a bridge. And the technical bottleneck that the modal shape is easily interfered by bridge damping is overcome. The method comprises the following steps: eliminating vehicle frequency interference based on a double-axle vehicle-bridge contact point response algorithm, identifying the frequency and damping ratio of a bridge of detection data by using a random subspace method, and identifying a bridge modal shape by using a spatial relationship between front and rear contact point responses; contact point response signals are obtained through a double-axle vehicle-bridge contact point response algorithm, interference of vehicle frequency can be effectively removed, front and back contact point response signals are processed through a random subspace method, the accurate frequency and damping ratio of a bridge can be obtained, and when the vibration mode of the bridge is recognized, the vibration mode of the bridge is recognized. And the influence of bridge damping on bridge vibration mode identification can be effectively eliminated.

Description

Technical Field

[0001] The present invention belongs to the technical field of bridge health detection, and particularly relates to a method for identifying bridge modal parameters based on random subspace and biaxial vehicle response. Background Art

[0002] Bridges are an important part of the transportation network and play a crucial role in economic development and social activities. As of the end of 2023, there were 1,079,300 highway bridges in the country with a total length of 95,288.2 kilometers, an increase of 46,100 bridges and 9,523.3 kilometers respectively compared to the end of 2022. As the service time of bridges increases, they are damaged by natural and human factors, resulting in a continuous decline in the structural performance of bridges. Therefore, it is particularly necessary to monitor the health status of bridges and timely detect safety problems existing in bridges.

[0003] When a vehicle travels on a bridge, due to the coupling effect of the vehicle-bridge system, the vibration data of the vehicle body contains the vibration information of the bridge. Therefore, signal processing technology can be used to extract the modal information of the bridge from the moving vehicle response, thereby realizing the indirect identification of bridge modal parameters. This method is also known as the vehicle scanning method. Currently, due to its advantages such as strong mobility and high economy, this method has been widely studied.

[0004] Chinese Patent CN109839441B discloses a method for identifying bridge modal parameters. The method includes: installing a single wireless acceleration sensor on a two-axle vehicle to form a mobile test device, gradually placing the two-axle vehicle at different positions on the bridge for testing, obtaining the dynamic response of the two-axle vehicle-bridge system under environmental excitation, performing spectral analysis on the dynamic response through Fourier transform to obtain the frequency of the two-axle vehicle-bridge system, and then using the physical relationship between the change of the angular frequency of the two-axle vehicle-bridge system and the bridge modal parameters, namely frequency and mode shape, to identify the bridge frequency and mode shape; however, the traditional indirect detection method has problems of inaccurate accuracy in synchronously identifying the natural frequency, mode shape, and damping ratio of the bridge, and the technical bottleneck that the modal mode shape is easily interfered by the bridge damping. In view of the above problems, we propose a method for identifying bridge modal parameters based on random subspace and biaxial vehicle response. Summary of the Invention

[0005] The purpose of the present invention is to provide a method for identifying bridge modal parameters based on random subspace and biaxial vehicle response in view of the deficiencies of the prior art, which solves the problems of inaccurate accuracy in synchronously identifying the natural frequency, mode shape, and damping ratio of the bridge by the traditional indirect detection method, and the technical bottleneck that the modal mode shape is easily interfered by the bridge damping.

[0006] The present invention is implemented as follows. A method for identifying bridge modal parameters based on random subspace and biaxial vehicle response includes:

[0007] Use a dual-axis measurement vehicle equipped with a suspension system as a mobile inspection vehicle, and collect inspection data using vertical acceleration sensors installed inside the dual-axis measurement vehicle;

[0008] Load the inspection data, eliminate frequency interference based on the dual-axis vehicle-bridge contact point response algorithm, and perform data partitioning and processing on the inspection data;

[0009] Use the stochastic subspace method to identify the frequency and damping ratio of the bridge from the inspection data;

[0010] By recursively calculating the modal amplitudes at the midpoints of each section of the bridge, establish the modal vibration shape of the nth order of the bridge, identify the modal vibration shape of the bridge, load the frequency, damping ratio, and modal vibration shape of the bridge, and integrate them into a bridge modal parameter set to complete the identification of bridge modal parameters.

[0011] Preferably, the method of collecting inspection data using vertical acceleration sensors installed inside the dual-axis measurement vehicle includes:

[0012] Install vertical acceleration sensors (S r and S f ) at the centers of the front (f) and rear (r) axles of the dual-axis measurement vehicle respectively;

[0013] According to the vehicle-bridge coupling principle, when the dual-axis measurement vehicle passes over the bridge, the bridge generates vertical vibrations, and then the vibrations are transmitted to the test vehicle body through the wheels, causing the vehicle body to generate vertical vibrations. The vertical vibration response of the vehicle body is collected in real time by the acceleration sensors installed at the axle centers.

[0014] Preferably, the method of performing data partitioning and processing on the inspection data includes:

[0015] Use the dual-axis vehicle-bridge contact point response algorithm to eliminate the interference of the vehicle vertical frequency f v,D and the vehicle rotation frequency f r,D , and calculate the dual-axis vehicle-bridge contact point response;

[0016] Divide the bridge along the driving path into k (k = L / d) sections according to the vehicle body length, where L is the length of the bridge and d is the length of the vehicle's front and rear axles. The dual-axis measurement vehicle moves along one side of the bridge deck at a constant speed, and divides the collected front and rear contact point response signals into k sections.

[0017] Preferably, the method of identifying the modal vibration shape of the bridge includes:

[0018] Load the collected front and rear contact point response signals, and process the contact point response signals of each section using the random subspace method. The sections passed by the front and rear wheels are the k-th and (k - 1)-th sections respectively, and the mode factors of the midpoints of the bridge sections passed by the front and rear wheels are obtained as A j,k and A j,k-1 (j = f, r);

[0019] Taking the rear contact point as the reference point and the front contact point as the response point, compare the front and rear mode factors A k,j (j = f, r). The ratio is the recurrence factor Its calculation formula is as follows:

[0020]

[0021] In the above formula, is the n-th recurrence factor of the vehicle when it travels to the midpoint of the k-th section of the bridge. When the double-axle vehicle moves forward a distance d each time, the value is updated accordingly;

[0022] By recursively calculating the modal amplitude Φ n,k at the midpoint of each section of the bridge, the modal shape of the n-th order of the bridge is established. Its calculation formula is as follows:

[0023]

[0024] Φ n = {Φ n,1 , Φ n,2 , …, Φ n,k , …, Φ n,K}.

[0025] In the formula, Φ n,k is the n-th order modal amplitude at the midpoint of the k-th section of the bridge, Φ n is the n-th order modal shape of the bridge. The initial state of the vehicle is staying on the first section of the bridge. Assume that the modal amplitude of the first section of the bridge is 1, that is, (Φ n,1 = 1). Use the above method to extract the modal shape and eliminate the influence of bridge damping.

[0026] Preferably, when calculating the contact point response of the double-axle vehicle-bridge, the motion equations of the vehicle body in the vertical and rotational directions are as follows:

[0027]

[0028]

[0029] In the above formula, y v (t) and θ v (t) are respectively the vertical displacement and pitch angle of the vehicle body center of gravity, y wj(t) is the vertical displacement of the front and rear wheels. The vertical motion equations of the front and rear wheels are as follows:

[0030]

[0031] The second derivative with respect to time t is taken for the vertical and rotational motion equations of the vehicle body, the vibration signal is converted into a quantity related to acceleration, and the wheel response is solved, which is expressed as:

[0032]

[0033] In the formula:

[0034]

[0035] In the above formula, and are the vertical acceleration response and rotational acceleration response of the double-axle vehicle, and their calculation formulas are:

[0036]

[0037] Since the actually obtained acceleration response data is discrete, the wheel response formula is sorted out as:

[0038]

[0039] Based on the wheel response, the double-axle vehicle-bridge contact point response is calculated. The second derivative is taken for the vertical motion equations of the front and rear wheels, the vibration signal is converted into a quantity related to acceleration, and the contact point response is sorted out into the following formula:

[0040]

[0041] In the formula:

[0042]

[0043] For discrete data, the contact point response is converted into the following formula:

[0044]

[0045] Preferably, when the detection data is partitioned and processed, a state space expression of the double-axle vehicle-bridge coupling system is established;

[0046] Among them, the bridge is modeled as a linear system with N degrees of freedom, and the motion equation of the bridge is expressed as follows:

[0047]

[0048] In the formula, where M B , C B , KB ∈R N×N are the mass matrix, damping matrix, and stiffness matrix of the bridge respectively. U(t) ∈ R N ×1 is the displacement vector, F C ∈R N×1 is the contact force between the bridge and the vehicle, and its expression is:

[0049]

[0050] In the formula, U Bj (t) is the vertical displacement of the bridge under the action of the front and rear wheels, is the position function of the front and rear contact points when the double - axle vehicle is moving;

[0051] According to the state - space theory, the motion equation of the bridge is transformed into the following form:

[0052]

[0053] In the formula, I N is the N - order identity matrix, and the above formula is rewritten as:

[0054]

[0055] Among them:

[0056]

[0057] The contact - point response is converted into the state - space form, and the contact - point response is expressed in the following form:

[0058]

[0059] Combining the motion equation of the bridge and the expression of the contact - point response, the observation equation of the contact - point acceleration response is obtained:

[0060]

[0061] In the formula:

[0062]

[0063] Based on the fact that the acceleration signal collected by the sensor is discrete, the motion equation of the bridge and the observation equation of the contact - point acceleration response are converted into discrete - state - space equations:

[0064] X k+1 = A d X k + B d F k ,

[0065] Y k = CXk +DF k ,.

[0066] In the formula:

[0067] A d = exp(AΔt), B d = A -1 (A - I N )B,

[0068] C = C(t)| t＝kΔt , D = D(t)| t＝kΔt ,

[0069] X k = X(kΔt),

[0070] F k = F C | t＝kΔt .

[0071] In the above formula, k represents the time step k, and Δt is the sampling interval;

[0072] Convert the discrete - type state - space equation into the state - space expression of the two - axis vehicle - bridge coupling system:

[0073] X k+1 = A d X k + w k ,

[0074] Y k = CX k + v k ,.

[0075] In the above formula, w k is the system noise term, and v k is the measurement noise term.

[0076] Preferably, the method for identifying the frequency and damping ratio of the bridge from the detection data by using the stochastic subspace method includes:

[0077] Identify the response signals of the front and rear contact points in the detection data, and form the following Hankel matrix with the obtained response signals of the front and rear contact points:

[0078]

[0079] The above Hankel matrix is divided into two parts, where Y p ∈R mi×j represents "past output", and Y f ∈R mi×jDenoted as "future output", where m is the number of sensors, the calculated output covariance forms a Toeplitz matrix as shown below:

[0080]

[0081] The Toeplitz matrix decomposition property is expressed as:

[0082]

[0083] In the formula O i is the observation matrix, C i is the control matrix. Using singular value decomposition (SVD), the Toeplitz matrix can be decomposed to obtain the following results:

[0084]

[0085] Combining the Toeplitz matrix decomposition property and the Toeplitz matrix decomposition results, the observation matrix is obtained:

[0086]

[0087] By using two sub - matrices of O i the bridge state matrix A is obtained, and the bridge state matrix A is expressed as:

[0088]

[0089] In the above formula represents the pseudo - inverse matrix, and the specific values are as follows:

[0090]

[0091] Using logarithmic matrix operations to continuousize the A matrix, the following results are obtained:

[0092]

[0093] In the formula, logm represents matrix operation, and the obtained A c matrix is subjected to eigenvalue decomposition:

[0094]

[0095] In the above formula, is a diagonal matrix composed of complex eigenvalues, ψ c is the eigenvector, λ n is the n - th eigenvalue in the eigenvalue matrix, (*) represents complex conjugate. The eigenvalues in the eigenvalue matrix Λ c appear in pairs and are conjugate to each other, where the eigenvalue λ nRelated to the natural frequency f of the system n and the damping ratio ξ n ;

[0096]

[0097] In the formula, j represents the imaginary unit, and the frequency and damping ratio of the system are calculated according to the following formula:

[0098]

[0099] In the above formula, the eigenvalues of Re and Im are the real part and the imaginary part, respectively, expressed as:

[0100] Re(λ n ) = -ξ n ω n ,

[0101] Compared with the prior art, the embodiments of the present application mainly have the following beneficial effects:

[0102] In the embodiments of the present invention, the contact point response signal is obtained by using the biaxial vehicle-bridge contact point response algorithm. This method can effectively remove the interference of the vehicle frequency, making the recognition effect better. The random subspace method is used to process the front and rear contact point response signals to obtain the accurate frequency and damping ratio of the bridge. At the same time, the state space expression of the biaxial vehicle-bridge coupling system is established. Based on the state space theory, the bridge motion equation and the biaxial vehicle contact point response are respectively converted into the state equation and the output equation, providing theoretical support for the subsequent identification of bridge modal parameters based on the biaxial vehicle contact point response and the random subspace method. And in order to eliminate the influence of bridge damping on the identification of bridge vibration modes, a method for identifying bridge vibration modes based on the random subspace method is creatively proposed. This method can effectively eliminate the influence of bridge damping on the identification of bridge modal vibration modes. BRIEF DESCRIPTION OF THE DRAWINGS

[0103] Figure 1 is a schematic implementation flowchart of the bridge modal parameter identification method based on the random subspace and the biaxial vehicle response provided by the present invention.

[0104] Figure 2 Shows a simplified diagram of a simply supported bridge model with damping for a biaxial measurement vehicle.

[0105] Figure 3 Shows the time domain diagram of the biaxial vehicle-bridge contact point response obtained based on the method of the present invention.

[0106] Figure 4 Shows the Fourier transform frequency domain diagram of the biaxial vehicle-bridge contact point response obtained based on the method of the present invention.

[0107] Figure 5Shows the stable diagram of bridge frequencies calculated by the random subspace algorithm.

[0108] Figure 6 Shows the first-order modal vibration mode of the bridge restored based on the method of the present invention.

[0109] Figure 7 Shows the second-order modal vibration mode of the bridge restored based on the method of the present invention. Detailed implementation manners

[0110] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by those of ordinary skill in the technical field to which this application belongs; the terms used in the specification of this application are only for the purpose of describing specific embodiments and are not intended to limit this application; the terms "including" and "having" and any variations thereof in the specification and claims of this application and the above drawings are intended to cover non-exclusive inclusion. The terms "first", "second", etc. in the specification and claims of this application or the above drawings are used to distinguish different objects and not to describe a specific order.

[0111] The traditional indirect detection method has problems of inaccurate accuracy when synchronously identifying the natural frequency, vibration mode and damping ratio of a bridge, and the technical bottleneck that the modal vibration mode is easily interfered by the bridge damping. In view of the above problems, we propose a method for identifying bridge modal parameters based on random subspace and biaxial vehicle response. Briefly, when implementing the method, first use a biaxial measurement vehicle equipped with a suspension system as a moving detection vehicle, and use vertical acceleration sensors installed in the biaxial measurement vehicle to collect detection data. Then, based on the biaxial vehicle-bridge contact point response algorithm, eliminate frequency interference, perform data partitioning and processing on the detection data, use the random subspace method to identify the frequency and damping ratio of the bridge in the detection data, and finally establish the modal amplitude of the midpoint of each section of the bridge through recursive calculation, establish the nth-order modal vibration mode of the bridge, and identify the bridge modal vibration mode. In the embodiment of the present invention, the contact point response signal is obtained by using the biaxial vehicle-bridge contact point response algorithm. This method can effectively remove the interference of vehicle frequencies, making the identification effect better. The random subspace method is used to process the front and rear contact point response signals to obtain accurate bridge frequencies and damping ratios. At the same time, a state space expression of the biaxial vehicle-bridge coupling system is established. Based on state space theory, the bridge motion equation and the biaxial vehicle contact point response are respectively converted into a state equation and an output equation, providing theoretical support for subsequent identification of bridge modal parameters based on the biaxial vehicle contact point response and the random subspace method. And in order to eliminate the influence of bridge damping on the identification of the bridge vibration mode, a method for identifying the bridge vibration mode based on the random subspace method is creatively proposed. This method can effectively eliminate the influence of bridge damping on the identification of the bridge modal vibration mode.

[0112] It should be noted that as one of the time-domain methods, the random subspace method does not require the conversion of signals in the time-frequency domain. It has high recognition accuracy, and at the same time has the advantages of robustness and stability, and has received extensive attention in the research of modal parameter identification. At present, in the vehicle scanning method, the frequency-domain method and the time-frequency domain method are mainly used to identify the bridge modal parameters, while the application research of the random subspace method is relatively less, and in these researches, mainly individual bridge modal parameters are identified, and comprehensive identification is not carried out. The modal parameters of a bridge reflect the dynamic characteristics of the structure. Bridge structure damage will cause changes in parameters such as frequency, damping ratio, and vibration mode. Therefore, in the vehicle scanning method, the comprehensive identification of bridge modal parameters (frequency, damping ratio, vibration mode) using the random subspace method is of great significance for bridge health detection. In this application, the random subspace method is used to identify bridge modal parameters (frequency, damping ratio, modal vibration mode) in the field of vehicle scanning method, and the identification formulas and technical processes for bridge frequency, damping ratio, and modal vibration mode are established using the biaxial vehicle contact point response. At the same time, this method can effectively eliminate the influence of bridge damping when extracting the bridge modal vibration mode. It is of great significance for bridge health detection.

[0113] The embodiment of the present invention provides a method for identifying bridge modal parameters based on random subspace and biaxial vehicle response, as Figure 1 shown, the method for identifying bridge modal parameters based on random subspace and biaxial vehicle response includes:

[0114] Data acquisition: Use a biaxial measurement vehicle equipped with a suspension system as a moving detection vehicle, and use vertical acceleration sensors installed in the biaxial measurement vehicle to collect detection data;

[0115] It should be noted that during data acquisition, the specific steps are as follows:

[0116] S10, install vertical acceleration sensors (S r and S f ) at the centers of the front (f) and rear (r) axles of the biaxial measurement vehicle respectively;

[0117] S20, according to the vehicle-bridge coupling principle, when the biaxial measurement vehicle passes over the bridge, the bridge generates vertical vibration, and then the vibration is transmitted to the test vehicle body through the wheels, causing the vehicle body to generate vertical vibration. The vertical vibration response generated by the vehicle body is collected in real time by the acceleration sensors installed at the axle centers;

[0118] Data division and processing: Load the detection data, eliminate frequency interference based on the biaxial vehicle-bridge contact point response algorithm, and perform data division and processing on the detection data;

[0119] Among them, during data division and processing:

[0120] S30, Use the biaxial vehicle-bridge contact point response algorithm to eliminate the interference of the vehicle's vertical frequency f v,D and the vehicle's rotational frequency f r,D , calculate the biaxial vehicle-bridge contact point response, so as to enhance the identification effect of bridge modal parameters;

[0121] S40, Divide the bridge along the driving path into k (k = L / d) segments according to the vehicle body length, where L is the length of the bridge and d is the length of the vehicle's front and rear axles. The biaxial measurement vehicle is located on one side of the bridge deck (the rear wheel is at the initial position of the bridge deck) and moves along the bridge deck at a constant speed. Divide the collected front and rear contact point response signals into k segments;

[0122] Bridge modal parameter identification: Use the stochastic subspace method to identify the frequency and damping ratio of the bridge from the detection data. By recursively calculating the modal amplitude at the midpoint of each segment of the bridge, establish the modal shape of the nth order of the bridge, identify the bridge modal shape, load the frequency, damping ratio, and modal shape of the bridge, and integrate them into the bridge modal parameter set to complete the identification of bridge modal parameters.

[0123] Among them, when identifying bridge modal parameters, the specific steps are as follows:

[0124] S50, Use the stochastic subspace method to identify the frequency and damping ratio of the bridge from the detection data;

[0125] S60, Load the collected front and rear contact point response signals (the duration of each segment is, is the driving speed of the vehicle), and use the stochastic subspace method to process the contact point response signals of each segment. Among them, the sections passed by the front and rear wheels are the kth and (k - 1)th segments, and the mode shape factors of the midpoints of the bridge sections passed by the front and rear wheels are obtained as A j,k and A j,k-1 (j = f, r);

[0126] S70, Taking the rear contact point as the reference point and the front contact point as the response point, compare the front and rear mode shape factors A k,j (j = f, r), and obtain the recursion factor This ratio is the recursion factor Its calculation formula is as follows:

[0127]

[0128] In the above formula, is the nth order recursion factor when the vehicle travels to the midpoint of the kth segment of the bridge. When the biaxial vehicle travels forward a distance d each time, value is updated accordingly.

[0129] S80, Establish the modal shape of the nth order of the bridge by recursively calculating the modal amplitude Φ n,k of the midpoint of each segment of the bridge. Its calculation formula is as follows:

[0130]

[0131] Φ n ={Φ n,1 ,Φ n,2 ,…,Φ n,k ,…,Φ n,K}.

[0132] In the formula, Φ n,k is the nth-order modal amplitude at the midpoint of the kth segment of the bridge, Φ n is the nth-order vibration mode of the bridge. The initial state of the vehicle is staying on the first segment of the bridge. Assume the modal amplitude of the first segment of the bridge is 1, that is, (Φ n,1 =1). Use the above method to extract the modal vibration mode and eliminate the influence of bridge damping;

[0133] In the embodiments of the present invention, the contact point response signal of the dual-axis vehicle-bridge is obtained by using the dual-axis vehicle-bridge contact point response algorithm. This method can effectively remove the interference of the vehicle frequency, making the recognition effect better. Use the stochastic subspace method to process the front and rear contact point response signals to obtain the accurate frequency and damping ratio of the bridge. At the same time, the state space expression of the dual-axis vehicle-bridge coupling system is established. Based on the state space theory, the bridge motion equation and the dual-axis vehicle contact point response are respectively converted into the state equation and the output equation, providing theoretical support for the subsequent identification of bridge modal parameters based on the dual-axis vehicle contact point response and the stochastic subspace method. And in order to eliminate the influence of bridge damping on the identification of the bridge vibration mode, a method for identifying the bridge vibration mode based on the stochastic subspace method is creatively proposed. This method can effectively eliminate the influence of bridge damping on the identification of the bridge modal vibration mode.

[0134] It should be noted that the dual-axis measurement vehicle travels on the simply supported bridge with damping at a speed of v. The dual-axis vehicle is modeled as a four-degree-of-freedom system. Figure 2 Shows the simplified diagram of the dual-axis measurement vehicle on the simply supported bridge model with damping. Among them, the vehicle body is regarded as a rigid beam with a length of d and a mass of M V , and the moment of inertia is J V . The front and rear axles are connected to the vehicle body through the suspension system. The vehicle body includes two degrees of freedom: vertical and pitching, and each of the two wheels has one degree of freedom. The suspension system is modeled as two groups of spring and damper units with stiffness (k sf , k sr ) and damping coefficients (c sf , c sr ). The subscripts (f, r) represent the front and rear axles. The masses of the wheels are m wf and m wr respectively. Each tire is modeled as a spring with a stiffness of k wj and a damping coefficient of c wjThe spring shock absorber unit. The distances from the front and rear axles to the center of gravity of the vehicle body are d j (j = f, r), where d = d f +d r 。The bridge is simulated as an Euler-Bernoulli beam, with a bridge span of L, a unit mass of m, an elastic modulus of E, and a sectional moment of inertia of I. u cj and u cr are the vertical displacements of the front and rear contact points of the double-axle vehicle-bridge response respectively.

[0135] When the vehicle is moving on the bridge, due to the coupling effect of the vehicle-bridge system, the vibration data of the vehicle body contains the vibration information of the bridge. Therefore, signal processing techniques can be used to extract the modal information of the bridge from the response of the moving vehicle, so as to indirectly identify the modal parameters of the bridge.

[0136] When the double-axle vehicle passes through the bridge, taking the Figure 2 appearance of the bridge object in, the vertical motion equation of the bridge is as follows:

[0137]

[0138] In the above formula, u(x, t) is the vertical displacement of the bridge, where F(t) is the contact force generated by the front and rear axle loads p j (j = f, r):

[0139]

[0140] In the above formula, δ() and H() are the Dirac constant and the unit step function respectively, and p j is the load of the front and rear axles:

[0141]

[0142] In the formula, g is the acceleration due to gravity, and the time t j when the front and rear axles enter the bridge and the time T when the vehicle passes through the bridge are respectively:

[0143]

[0144] Based on the above control equation, the analytical solution of the vertical displacement response u(x, t) of the bridge can be derived as follows:

[0145]

[0146] In the above formula, ω b,n is the nth natural frequency of the bridge, and Δ stn,j is defined as the nth static deflection deformation caused by the front and rear axle loads p j , and ω bD,n is the nth damping frequency of the bridge, ωd,n is the driving frequency of the vehicle, ξ b,n is the damping ratio of the nth order of the bridge, and the specific parameters are as follows:

[0147]

[0148]

[0149] A bD.n =-A d,n ,

[0150] where S n is the speed parameter, and the expression is as follows:

[0151]

[0152] By moving the contact relationship between the double - axle vehicle and the bridge and setting x = vt in the vertical motion equation of the bridge, the expressions for the responses of the front and rear contact points of the double - axle vehicle - bridge are obtained:

[0153]

[0154] It can be seen from the formula that the expression of the contact - point response contains the modal information of the bridge. Based on this, the present invention will use the stochastic subspace method to identify the modal parameters of the bridge. In addition, it is worth noting that the formula does not contain the vehicle's own frequency, which proves that the contact - point response can effectively filter out the vehicle frequency and achieve the effect of enhancing the identification.

[0155] However, in reality, the contact - point response is difficult to measure directly and needs to be calculated using the vertical response of the vehicle body. Therefore, when calculating the double - axle vehicle - bridge contact - point response in the embodiments of the present invention,

[0156] such as Figure 2 shown, when the double - axle measurement vehicle passes through the bridge, the vertical and rotational motion equations of the vehicle body are as follows:

[0157]

[0158] In the above formula, y v (t) and θ v (t) are respectively the vertical displacement and pitch angle of the vehicle body's center of gravity, and y wj (t) is the vertical displacement of the front and rear wheels. The vertical motion equations of the front and rear wheels are as follows:

[0159]

[0160] The second - order derivative with respect to time t of the vertical and rotational motion equations of the vehicle body is taken to convert the vibration signal into a quantity related to acceleration, and the wheel response is solved, which is expressed as:

[0161]

[0162] In the formula:

[0163]

[0164] In the above formula, and are the vertical acceleration response and rotational acceleration response of the biaxial vehicle, and their calculation formulas are:

[0165]

[0166] Since the actually obtained acceleration response data is discrete, thus, the wheel response formula is organized as:

[0167]

[0168] Based on the wheel response to calculate the biaxial vehicle-bridge contact point response, the vertical motion equations of the front and rear wheels are differentiated twice, and the vibration signal is converted into a quantity related to acceleration. The contact point response is organized into the following formula:

[0169]

[0170] In the formula:

[0171]

[0172] For discrete data, the contact point response is converted into the following formula:

[0173]

[0174] In the embodiment of the present invention, when performing data partitioning and processing on the detection data, a state space expression of the biaxial vehicle-bridge coupling system is established;

[0175] Among them, the bridge is modeled as a linear system with N degrees of freedom, and the motion equation of the bridge is expressed as follows:

[0176]

[0177] In the formula, where M B , C B , K B ∈R N×N are respectively the mass matrix, damping matrix, and stiffness matrix of the bridge. U(t)∈R N ×1 is the displacement vector, F C ∈R N×1 is the contact force between the bridge and the vehicle, and its expression is:

[0178]

[0179] Where U Bj (t) is the vertical displacement of the bridge under the action of the front and rear wheels, is the position function of the front and rear contact points when the double-axle vehicle is traveling;

[0180] According to the state space theory, the motion equation of the bridge is transformed into the following form:

[0181]

[0182] Where I N is the N-order identity matrix, and the above formula is rewritten as:

[0183]

[0184] Among them:

[0185]

[0186] The data used in the present invention is the contact point response derived above, so it is necessary to convert the contact point response into the state space form, where the contact point response is expressed in the following form:

[0187]

[0188] Combining the motion equation of the bridge and the expression of the contact point response, the observation equation of the contact point acceleration response is obtained:

[0189]

[0190] Where:

[0191]

[0192] Based on the fact that the acceleration signal collected by the sensor is discrete, the motion equation of the bridge and the observation equation of the contact point acceleration response are converted into discrete state space equations:

[0193] X k+1 =A d X k +B d F k ,

[0194] Y k =CX k +DF k ,.

[0195] Where:

[0196] A d =exp(AΔt), B d =A-1 (A - I N )B,

[0197] C = C(t)| t＝kΔt , D = D(t)| t＝kΔt ,

[0198] X k = X(kΔt),

[0199] F k = F C | t＝kΔt .

[0200] In the above formula, k represents the time step k, and Δt is the sampling interval;

[0201] Convert the discrete - state - space equation into the state - space expression of the two - axis vehicle - bridge coupling system:

[0202] X k+1 = A d X k + w k ,

[0203] Y k = CX k + v k ,.

[0204] In the above formula, w k is the system noise term, and v k is the measurement noise term.

[0205] It should be noted that the method for identifying the frequency and damping ratio of the bridge using the stochastic subspace method for the detected data includes:

[0206] Identify the response signals of the front and rear contact points in the detected data, and form the following Hankel matrix with the obtained response signals of the front and rear contact points:

[0207]

[0208] The above Hankel matrix is divided into two parts. Y p ∈R mi×j represents "past output", and Y f ∈R mi×j represents "future output", m is the number of sensors, and the calculated output covariance forms the following Toeplitz matrix:

[0209]

[0210] The decomposition property of the Toeplitz matrix is expressed as:

[0211]

[0212] In the formula O i is the observation matrix, and C i is the control matrix. By using singular value decomposition (SVD), the Toeplitz matrix can be decomposed to obtain the following results:

[0213]

[0214] Combining the Toeplitz matrix decomposition characteristics and the Toeplitz matrix decomposition results, the observation matrix is obtained:

[0215]

[0216] By using two sub-matrices of O i the bridge state matrix A is obtained, and the bridge state matrix A is expressed as:

[0217]

[0218] In the above formula represents the pseudo-inverse matrix, and the specific values are as follows:

[0219]

[0220] Using logarithmic matrix operations, the A matrix is made continuous to obtain the following results:

[0221]

[0222] In the formula, logm represents matrix operation. The obtained A c matrix is subjected to eigenvalue decomposition:

[0223]

[0224] In the above formula, is a diagonal matrix composed of complex eigenvalues, ψ c is the eigenvector, λ n is the nth eigenvalue in the eigenvalue matrix, (*) represents complex conjugate, and the eigenvalues in the eigenvalue matrix Λ c appear in pairs and are conjugate to each other. Among them, the eigenvalue λ n is related to the natural frequency f n and damping ratio ξ n of the system;

[0225]

[0226] In the above formula, j represents the imaginary unit, and the frequency and damping ratio of the system are calculated according to the following formula:

[0227]

[0228] In the above formula, the eigenvalues of Re and Im are the real part and the imaginary part, respectively, and are expressed as:

[0229] Re(λ n ) = -ξ n ω n ,

[0230] It should be noted that when the stochastic subspace algorithm is used to identify the bridge vibration mode, sufficient spatial information is required to identify the bridge vibration mode. Measuring the dynamic response of the bridge deck using a moving dual-axis vehicle beam equipped with two sensors can simulate the effect of arranging a large number of sensors along the bridge deck. The bridge deck is divided into k (k = L / d) segments according to the vehicle body length d. The dual-axis vehicle is located on one side of the bridge deck (the rear wheels are at the initial position of the bridge deck) and moves along the bridge deck at a constant speed. The response signals of the front and rear contact points collected are divided into k segments, as shown in the following formula.

[0231] Y = {Y1, Y2, …, Y k , …, Y K},

[0232]

[0233] In the above formula, Y f,k and Y r,k-1 respectively represent the k-th segment of the front contact response and the (k - 1)-th segment of the rear contact response;

[0234] For the k-th segment of the bridge, it can be calculated that:

[0235]

[0236] In the formula, C k is the first m rows of the observation matrix O i , ψ n,k is the n-th order eigenvector of the k-th segment obtained by matrix calculation of A c , A r,k-1 is the vibration mode factor of the rear contact point response, i.e., the (k - 1)-th segment, and A f,k is the vibration mode factor of the front contact point response, i.e., the k-th segment.

[0237] Specifically, when identifying the bridge modal parameters based on the stochastic subspace and the dual-axis vehicle response, the bridge span L = 32 m, the density of the curved bridge ρ = 2,400 kg / m 3 , the elastic modulus E = 27.5 GPa, the cross-sectional moment of inertia I = 0.15 m 4 , and the bridge damping ratio ξ bA bridge with a value of 0.01 was tested. A double-axle measurement vehicle with a suspension system was used, and vertical acceleration sensors were installed at the centers of the front and rear axles to collect the vertical acceleration response of the vehicle. Due to the vehicle-bridge coupling effect, the acceleration response collected by the acceleration sensors contains the vibration information of the bridge. The length of the double-axle vehicle body is 2m, and the mass is M v = 1,000 kg and the moment of inertia J V = 500 kg·m 2 , the distances from the front and rear axles to the midline of the vehicle body are 0.9m and 1.1m, and the damping and stiffness of the suspension are c sf = c sr = 1 kN·s / m, k sf = k sr = 500 kN / m, and the masses of the front and rear wheels are m wf = m wr = 100 kg. The damping and stiffness of the front and rear wheels are c wf = c wr = 1 kN·s / m, k wf = k wr = 1,000 kN / m. The driving speed of the double-axle vehicle is v = 2 m / s.

[0238] In order to verify that the bridge modal parameter identification method based on random subspace and the response of a double-axle vehicle can effectively identify the bridge frequency, damping ratio, and modal vibration shape by using the double-axle vehicle-bridge contact response and the random subspace method, numerical simulations were carried out. Figure 3 is the time-domain diagram of the double-axle vehicle-bridge contact point response obtained based on the method of the present invention, Figure 4 is the frequency-domain diagram of the Fourier transform of the double-axle vehicle-bridge contact point response obtained based on the method of the present invention. Figure 5 is the stable diagram of the bridge frequency calculated by the random subspace algorithm. Figure 6 is the first-order modal vibration shape of the bridge restored based on the method of the present invention, Figure 7 is the second-order modal vibration shape of the bridge restored based on the method of the present invention. Table 1 shows the frequency identification results, and Table 2 shows the damping ratio identification results.

[0239] Table 1 Frequency Identification Results

[0240]

[0241] Table 2 Damping Ratio Identification Results

[0242]

[0243] From Table 1 - Table 2, Figures 3 - 6It can be seen that the present invention can efficiently and accurately identify the frequencies of bridges. As can be seen from Table 1 (frequency identification results), the identified frequencies are highly consistent with the natural frequencies of the actual bridges, indicating that the present invention has high precision in frequency identification. Moreover, through the method of the present invention, the damping ratio of bridges can be accurately measured. Table 2 (damping ratio identification results) shows that the identified damping ratio is very close to the actual value, verifying the accuracy of the method. The present invention creatively establishes a new formula for constructing the modal vibration mode of bridges by using the spatial relationship between the front and rear wheels of a double-axle vehicle. From Figure 6 and Figure 7 (the first and second order vibration mode modes of the bridge recovered), it can be seen that the recovered vibration mode modes are clear and accurate, and can effectively eliminate the influence of bridge damping. The numerical simulation results show that the method of the present invention is not only theoretically feasible, but also has high accuracy and reliability in practical applications, providing an effective tool for bridge modal parameter identification.

[0244] In summary, the present invention provides a method for identifying bridge modal parameters based on random subspace and double-axle vehicle response. In the embodiments of the present invention, the contact point response signal is obtained by using the double-axle vehicle-bridge contact point response algorithm. This method can effectively remove the interference of vehicle frequencies, making the identification effect better. The random subspace method is used to process the front and rear contact point response signals to obtain the accurate frequencies and damping ratios of the bridge. At the same time, the state space expression of the double-axle vehicle-bridge coupling system is established. Based on state space theory, the bridge motion equation and the double-axle vehicle contact point response are respectively converted into state equations and output equations, providing theoretical support for subsequent bridge modal parameter identification based on double-axle vehicle contact point response and random subspace method. And in order to eliminate the influence of bridge damping on the identification of bridge vibration modes, a method for identifying bridge vibration modes based on random subspace method is creatively proposed, which can effectively eliminate the influence of bridge damping on the identification of bridge modal vibration modes.

[0245] In the embodiments of the present invention, the contact point response is calculated by using the double-axle vehicle-contact point response algorithm, and the state space expression of the double-axle vehicle-bridge coupling system is established, providing theoretical support for subsequent bridge modal parameter identification based on double-axle vehicle contact point response and random subspace method. Subsequently, the random subspace method is used to process the contact point response, and the frequencies and damping ratios of the bridge can be obtained. In order to eliminate the influence of bridge damping on the identification of modal vibration modes, a modal vibration mode identification formula and technical process are creatively established.

[0246] It should be noted that, for the foregoing embodiments, for the sake of simple description, they are all expressed as a series of action combinations. However, those skilled in the art should be aware that the present invention is not limited by the described action sequence, because according to the present invention, certain steps may be performed in other sequences or simultaneously. Secondly, those skilled in the art should also be aware that the embodiments described in the specification are all preferred embodiments, and the actions and modules involved are not necessarily essential to the present invention.

[0247] The above embodiments are only used to illustrate the technical solutions of the present invention, rather than limiting the protection scope of the invention. Obviously, the described embodiments are only some embodiments of the present invention, rather than all embodiments. Based on these embodiments, all other embodiments obtained by those of ordinary skill in the art without making creative efforts belong to the scope of protection of the present invention. Although the present invention has been described in detail with reference to the above embodiments, those of ordinary skill in the art can still, without conflict, make combinations, additions, deletions or other adjustments to the features in the embodiments of the present invention according to the circumstances without making creative efforts, so as to obtain different technical solutions that do not essentially deviate from the concept of the present invention, and these technical solutions also belong to the scope of protection of the present invention.

Claims

1. A method for identifying bridge modal parameters based on random subspace and biaxial vehicle response, characterized in that Including: Taking the dual-axis measurement vehicle equipped with a suspension system as a mobile detection vehicle, and collecting detection data using a vertical acceleration sensor installed inside the dual-axis measurement vehicle; Loading the detection data, eliminating frequency interference based on the dual-axis vehicle-bridge contact point response algorithm, and performing data partitioning and processing on the detection data; Identifying the frequency and damping ratio of the bridge from the detection data using the stochastic subspace method; By recursively calculating the modal amplitude at the midpoint of each section of the bridge, establishing the modal shape of the nth order of the bridge, identifying the modal shape of the bridge, loading the frequency, damping ratio, and modal shape of the bridge, and integrating them into a bridge modal parameter set to complete the identification of bridge modal parameters.

2. The method for identifying bridge modal parameters based on random subspace and biaxial vehicle response according to claim 1, wherein: The method of collecting detection data using a vertical acceleration sensor installed inside the dual-axis measurement vehicle includes: Install the vertical acceleration sensors (S r and S f ) at the centers of the front (f) and rear (r) axles of the biaxial measurement vehicle respectively; According to the vehicle-bridge coupling principle, when the dual-axis measurement vehicle passes over the bridge, the bridge generates vertical vibrations, and then the vibrations are transmitted to the vehicle body of the test vehicle through the wheels, causing the vehicle body to generate vertical vibrations. The vertical vibration response generated by the vehicle body is collected in real time.

3. The method for identifying bridge modal parameters based on random subspace and biaxial vehicle response according to claim 1, wherein: The method of performing data partitioning and processing on the detection data includes: Eliminating the interference of the vehicle vertical frequency f v,D and the vehicle rotational frequency f r,D to calculate the response of the two-axle vehicle-bridge contact point; Dividing the bridge along the driving path into k (k = L / d) sections according to the vehicle body length, where L is the length of the bridge and d is the length between the front and rear axles of the vehicle. The dual-axis measurement vehicle moves along one side of the bridge deck at a constant speed, and divides the collected front and rear contact point response signals into k sections.

4. The method for identifying bridge modal parameters based on random subspace and biaxial vehicle response according to claim 3, wherein: The method of identifying the modal shape of the bridge includes: Load the collected front and rear contact point response signals, and use the stochastic subspace method to process the contact point response signals of each segment. The sections passed by the front and rear wheels are the k-th and (k - 1)-th segments respectively, and the mode shape factors of the midpoints of the bridge sections passed by the front and rear wheels are obtained as A j,k and A j,k-1 (j = f, r); Taking the subsequent contact point as the reference point and the prior contact point as the response point, the prior and subsequent mode shape factors A k,j (j = f, r) are compared, and this ratio is the recurrence factor The calculation formula thereof is as follows: In the above formula, is the nth recursive factor when the vehicle travels to the midpoint of the kth section of the bridge. When the double-axle vehicle travels forward a distance d each time, the value is updated accordingly; By recursively calculating the modal amplitude Φ of the midpoint of each section of the bridge n,k , the modal shape of the nth order of the bridge is established, and its calculation formula is as follows: where Φ n,k is the modal amplitude of the nth order at the midpoint of the kth segment of the bridge, and Φ n is the nth order vibration mode of the bridge. The initial state of the vehicle is staying on the first segment of the bridge. Assume that the modal amplitude of the first segment of the bridge is 1, i.e., (Φ n,1 = 1). Use the above method to extract the modal vibration mode and eliminate the influence of bridge damping.

5. The method for identifying bridge modal parameters based on random subspace and biaxial vehicle response according to claim 3, wherein: When calculating the dual-axis vehicle-bridge contact point response, the motion equations of the vehicle body in the vertical and rotational directions are as follows: In the above formula, y v (t) and θ v (t) are respectively the vertical displacement and pitch angle of the vehicle body center of gravity, and y wj (t) is the vertical displacement of the front and rear wheels. The vertical motion equations of the front and rear wheels are as follows: Performing a second derivative with respect to time t on the motion equations of the vehicle body in the vertical and rotational directions, converting the vibration signal into a quantity related to acceleration, and solving for the wheel response, which is expressed as: Where: In the above formula, and are the vertical acceleration response and rotational acceleration response of the biaxial vehicle, and their calculation formulas are as follows: Since the actually obtained acceleration response data is discrete, the wheel response formula is sorted out as: Calculating the dual-axis vehicle-bridge contact point response based on the wheel response, performing a second derivative on the vertical motion equations of the front and rear wheels, converting the vibration signal into a quantity related to acceleration, and sorting out the contact point response into the following formula: Where: For discrete data, the contact point response is converted into the following formula:

6. The method for identifying bridge modal parameters based on random subspace and biaxial vehicle response according to claim 5, wherein: When performing data partitioning and processing on the detection data, establishing the state space expression of the dual-axis vehicle-bridge coupling system; Among them, the bridge is modeled as a linear system with N degrees of freedom, and the motion equation of the bridge is expressed as follows: where M B , C B , K B ∈R N×N are respectively the mass matrix, damping matrix, and stiffness matrix of the bridge, U(t) ∈ R N×1 is the displacement vector, F C ∈R N×1 is the contact force between the bridge and the vehicle, and its expression is: where U Bj (t) is the vertical displacement of the bridge under the action of the front and rear wheels, is the position function of the front and rear contact points when the double-axle vehicle is traveling; According to the state space theory, the motion equation of the bridge is converted into the following form: where I N is an N-order identity matrix, and the above equation is rewritten as: Where: Converting the contact point response into the state space form, where the contact point response is expressed in the following form: Combining the motion equation of the bridge and the contact point response expression, obtaining the observation equation of the contact point acceleration response: Where: Based on the fact that the acceleration signal collected by the sensor is discrete, converting the motion equation of the bridge and the observation equation of the contact point acceleration response into discrete state space equations: X k+1 = A d X k + B d F k , Y k = CX k + DF k ,. Where: A d = exp(AΔt), B d = A -1 (A - I N )B, C = C(t)| t＝kΔt , D = D(t)| t＝kΔt , F k = F C | t＝kΔt . In the above formulas, k represents the time step k, and Δt is the sampling interval; Converting the discrete state space equations into the state space expression of the dual-axis vehicle-bridge coupling system: X k+1 = A d X k + w k , Y k = CX k + v k ,. In the above formula, w k is the system noise term, and v k is the measurement noise term.

7. The method for identifying bridge modal parameters based on random subspace and biaxial vehicle response according to claim 6, wherein: The method of identifying the frequency and damping ratio of the bridge from the detection data using the stochastic subspace method includes: Identifying the front and rear contact point response signals in the detection data, and forming the following Hankel matrix with the obtained front and rear contact point response signals: The above Hankel matrix is divided into two parts, namely Y p ∈R mi×j representing "past output", Y f ∈R mi×j representing "future output", m is the number of sensors, and the calculated output covariance forms a Toeplitz matrix as shown below: The Toeplitz matrix decomposition property is expressed as: In the formula O i is the observation matrix, and C i is the control matrix. The Toeplitz matrix can be decomposed by using singular value decomposition (SVD) to obtain the following results: Combining the Toeplitz matrix decomposition property and the Toeplitz matrix decomposition result, obtaining the observation matrix: By utilizing two sub-matrices of O i the bridge state matrix A is obtained, and the bridge state matrix A is expressed as: In the above formula represents the pseudo-inverse matrix, and the specific values are as follows: The continuous transformation of matrix A is achieved using logarithmic matrix operations, resulting in the following: where logm represents performing matrix operations to obtain matrix A c and performing eigenvalue decomposition on the matrix: In the above formula, a diagonal matrix composed of complex eigenvalues, ψ c is the eigenvector, λ n is the nth eigenvalue in the eigenvalue matrix, (*) represents the complex conjugate, and the eigenvalues in the eigenvalue matrix Λ c appear in pairs and are conjugate to each other, where the eigenvalue λ n is related to the natural frequency f n and the damping ratio ξ n ; where j represents the imaginary unit, and the frequency and damping ratio of the system are calculated as follows: In the above equation, the eigenvalues of Re and Im are the real and imaginary parts, respectively, and are expressed as:

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