Bridge Modal Vibration Mode Extraction Method Based on Combined Vehicle Test System

Abstract

The present invention relates to a method for extracting bridge modal vibration modes based on a combined vehicle test system, belonging to the technical field of bridge health monitoring. The method includes: establishing a combined vehicle test system and performing finite element modeling on the combined vehicle test system; establishing a vehicle model, a bridge model, and a vehicle-bridge coupling model according to the combined vehicle test system; using the excitation vehicle in the combined vehicle test system to excite the bridge vibration and recording the vehicle-bridge coupling response through the test vehicle; extracting the bridge modes from the vehicle or contact point response of the test vehicle by the time domain decomposition method. Compared with the vehicle response, the contact point response of the present invention shows better performance in identifying the bridge modal vibration modes.

Description

Technical Field

[0001] The present invention belongs to the technical field of bridge health monitoring, and relates to a method for extracting bridge modal shapes based on a combined vehicle test system. Background Art

[0002] Bridge modes are important indicators for evaluating the structural health status. Therefore, the extraction of bridge modes has attracted extensive attention. So far, forced vibration testing and ambient vibration testing are two main measurement methods. The first method is to excite the vibration of the bridge structure by an artificial impact device or a drop hammer, but normal traffic needs to be closed during the test. The second method does not require traffic closure and excites the bridge vibration through natural or environmental excitations such as traffic or wind. Based on this, ambient vibration testing has been widely used in modal shape extraction. Recording the dynamic response of the bridge under external excitations is the first step in identifying modal shapes. The next key step is to extract modal shapes from these recorded responses. Input-output methods such as frequency response functions (FRFs), and output-only methods such as frequency domain decomposition (FDD) and stochastic subspace identification (SSI) have been widely used to extract modal shapes. Even though an artificial impact device can be used to control the magnitude of the excitation force, it is difficult to precisely control the magnitude of the excitation force in actual measurements, especially in the case of long-span bridge structures. Therefore, the use of input-output methods to identify modal parameters is mainly applied to laboratory environments or small and medium-sized bridges. For this reason, the identification of modal parameters using only output dynamic responses has received more attention and applications.

[0003] As one of the output-only methods, the vehicle scanning method (VSM) has been successfully used to extract bridge properties such as frequencies, modal shapes, damping ratios, and damage identification. Compared with the need to install hundreds of sensors on the bridge, VSM has the characteristics of high efficiency, low cost, and strong mobility. Bridge modal properties can be directly identified from the responses of moving vehicles. The moving vehicle can serve as both an excitation source and a recorder, so VSM has attracted extensive attention worldwide.

[0004] The application of the vehicle scanning method to extract the modal shapes of bridges has been realized in many studies. For example, Zhang et al. first identified the modal curvatures of beam and plate structures from the dynamic responses of moving vehicles excited by self-excitation devices. Subsequently, Zhang et al. further proposed a method to identify modal shapes by extracting instantaneous frequencies (IFs) from the responses of moving concentrated masses. Yang et al. used the Hilbert transform to extract the modal shapes of simply supported bridges from the responses of sensor-equipped vehicles. To mitigate the negative impact of road surface roughness, Kong et al. used a tractor with two trailers, and by subtracting the response of one trailer from that of the other, the modal characteristics of the bridge were extracted from the residual response. Malekjafarian and O'Brien identified modal shapes from the responses of two sensor-equipped vehicles, using short-time FDD (frequency domain decomposition), where the bridge was first divided into multiple parts and a multi-stage process was given to obtain modal shapes through FDD. After that, they successively used multiple laser devices to extract the modal shapes of bridges from passing vehicles and proposed an algorithm for estimating modal shapes using a truck-trailer test system. Qi and Au constructed the modal shapes of bridges through moving vehicles under impact excitation. Similarly, Li et al. proposed a method to estimate the modal parameters of bridges from the responses of two sensor-equipped vehicles through the SSI (subspace identification) method, in which one vehicle served as a fixed reference sensor and the other as a moving sensor. From existing studies, road surface roughness is the largest external interference affecting the identification accuracy of the vehicle scanning method. Although existing studies can greatly weaken its negative impact, it is still impossible to eliminate its interference in a real sense. Summary of the Invention

[0005] In view of this, the purpose of the present invention is to provide a method for extracting the modal shapes of bridges based on a combined vehicle test system. The method consists of an excitation vehicle and two detection vehicles. When the excitation vehicle travels across the bridge at a certain speed accuracy, the two detection vehicles are stationary at preset bridge measurement points to record the vibration responses of the bridge; subsequently, by extracting the responses at the contact points of the detection vehicles and combining with the time-domain decomposition method, the modal shapes of the bridge are identified.

[0006] To achieve the above purpose, the present invention provides the following technical solutions:

[0007] A method for extracting the modal shapes of bridges based on a combined vehicle test system, which includes the following steps:

[0008] S1. Establish a combined vehicle test system and perform finite element modeling on the combined vehicle test system;

[0009] S2. Establish a vehicle model, a bridge model, and a vehicle-bridge coupling model according to the combined vehicle test system;

[0010] S3. Use the excitation vehicle in the combined vehicle test system to stimulate the bridge vibration, and record the vehicle-bridge coupling response through the test vehicle;

[0011] S4. Extract the bridge mode from the vehicle or contact point response of the test vehicle by the time-domain decomposition method.

[0012] Furthermore, in step S1, the combined vehicle test system includes one or more excitation vehicles, one or more test vehicles, and at least one bridge, wherein the excitation vehicle is used to stimulate the bridge vibration, and the test vehicle is used to record the vehicle-bridge coupling response;

[0013] The excitation vehicle has two degrees of freedom, namely the vertical motion y e and the pitching motion whose properties include: vehicle mass m e , moment of inertia J e , spring stiffness coefficient k e , damping coefficient c e ;

[0014] The test vehicle is a single degree of freedom, and its properties include: body mass m t , spring stiffness k t and damping coefficient c t of the oscillator; the test vehicle measures its own vertical motion through an accelerometer;

[0015] The bridge adopts the Bernoulli-Euler type, with a span length of L, a mass per unit length of m, and a stiffness of EI, and its vertical displacement is expressed as y b ;

[0016] During operation, the excitation vehicle drives across the bridge at a constant speed v, while the test vehicle stops at a pre-selected position.

[0017] Furthermore, in step S1, using the numerical finite element method, the bridge structure is divided into N two-dimensional beam elements and combined with the vehicle system to form a vehicle-bridge coupling element, and the finite element modeling of the combined vehicle test system is obtained;

[0018] Ensure that both the excitation vehicle and the test vehicle are in rigid contact with the bridge, and there is no sliding or climbing phenomenon of the vehicle on the bridge surface, then the motion equation of the combined vehicle test system is expressed as:

[0019]

[0020] where the subscripts 'v' and 'b' represent the vehicle and the bridge respectively; and y respectively represent the column vectors of acceleration, velocity, and displacement of the CVTS; M i , C i , K i, F i (i = v, b) represent the mass matrix, damping matrix, stiffness matrix, and external force vector respectively.

[0021] Furthermore, in step S2, the mass matrix of the vehicle model is expressed according to the energy principle as:

[0022]

[0023] The stiffness matrix of the vehicle model is expressed as:

[0024]

[0025] The damping matrix of the vehicle model is expressed as:

[0026]

[0027] Among them, at least one excitation vehicle is set, and x test vehicles are set and numbered sequentially.

[0028] Furthermore, in step S2, the bridge structure is divided into N two-dimensional beam elements, and each element has four degrees of freedom; the mass matrix M of the bridge b is only composed of its own components, and the mass matrix is expressed as:

[0029]

[0030] where m represents the mass per unit length of the bridge; [N] represents the row vector of the shape function, and it is expressed as:

[0031] [N] = [N1 N2 N3 N4]

[0032] N1 = 1 - 3(ξ / l) 2 + 2(ξ / l) 3

[0033]

[0034] N3 = 3(ξ / l) 2 - 2(ξ / l) 3

[0035]

[0036] where ξ represents the local coordinate starting from the left side of the node; l represents the length of the contact unit;

[0037] The stiffness matrix K of the bridge b includes the components of the bridge itself and the components caused by the vehicle and is expressed as:

[0038]

[0039] wherein obtained by assembling its element stiffness matrix where the symbol d represents the second derivative of the row vector of shape functions with respect to the local coordinate ξ;

[0040] At the nth excitation, the test vehicles numbered x1 and x2 are respectively deployed on the ith and jth bridge elements. The bridge structure is evenly divided into N elements, and the length of each element is denoted as l; for each test vehicle, their positions in the local coordinate system are respectively and For the excitation vehicle, at the t τ th time step, the front and rear wheels are respectively located on the pth and qth bridge elements; at this time step, the positions of the front and rear wheels are l f and l r , and Le represents the half distance between the two wheels; then the component of the stiffness matrix caused by the vehicle is calculated as follows:

[0041]

[0042] where the subscript τ represents the number of the vehicle-bridge coupling element; the row vector of shape functions at the current time step is obtained by substituting the local coordinate of the element into the motion equation of the CVTS, and [N′] τ represents the first derivative of the row vector of shape functions with respect to the local coordinate ξ.

[0043] The damping matrix C b of the bridge is composed of its own component and the component caused by the vehicle assembled, and is expressed as:

[0044]

[0045] Viscous damping is simulated using Rayleigh damping, and the calculation method is as follows:

[0046]

[0047] where α and β are damping coefficients, expressed as α = 2ζ g ω1ω2 / (ω1 + ω2) and β = 2ζ b / (ω1 + ω2), ω1 and ω2 respectively represent the first and second natural frequencies of the bridge, and ζ b is the damping ratio of the bridge.

[0048] Furthermore, in step S2, at the t τ th time step, the vehicle-bridge coupling stiffness matrix Kvb Expressed as:

[0049]

[0050] where h = 1, 2 represent the front wheel and the rear wheel respectively;

[0051] Coupling stiffness matrix K bv Expressed as:

[0052]

[0053] Vehicle - bridge coupling damping matrix C vb and C bv Expressed as:

[0054]

[0055] Furthermore, in step S3, during the process of exciting the bridge vibration, the loads borne by the bridge include external loads and vehicle loads, where the external load vector is expressed as:

[0056]

[0057] where r represents the roughness of the road surface;

[0058] The vehicle load vector is expressed as:

[0059]

[0060] The time - varying Newmark - β integration method is used to solve the coupled dynamic equation.

[0061] Furthermore, in step S3, the motion equation of the x - th test vehicle is expressed as:

[0062]

[0063] where, represents the column vector of the acceleration of the x - th test vehicle, represents the column vector of the velocity of the x - th test vehicle, y tx represents the column vector of the displacement of the x - th test vehicle, y c represents the contact point response, and the contact point response is calculated by back - calculation according to the above formula.

[0064] Furthermore, in step S4, the acceleration response is analyzed by the time - domain decomposition method to extract the modal information of the bridge;

[0065] where, the modal decomposition process of the acceleration response is:

[0066] The acceleration of the bridge under any load excitation at time t is expressed as:

[0067]

[0068] Among them, is the output acceleration column vector, and p represents the number of sensors; Φ i = [φ 1i … φ 1p T is the i-th order bridge vibration mode; λ i (t) is the i-th order modal contribution factor, indicating the acceleration contribution at time t.

[0069] Retrieve the first n order modes from the discrete acceleration signal. At the sampling time k, the acceleration of the bridge under any load excitation is expressed as:

[0070]

[0071] Among them, ε t (k) represents a p×1 truncation error column vector, which is expressed as

[0072] Use a digital band-pass filter to extract the i-th order single-mode response, then the i-th filtered acceleration response is expressed as:

[0073]

[0074] Among them, ε f (k) represents the noise vector, which is the sample sum of the noise and the residual at time k.

[0075] The noise vector is decomposed into a p - 1 dimensional orthogonal noise space:

[0076]

[0077] Among them, Ψ j = [ψ 1j … ψ pj T is the j-th orthogonal noise basis mode, and d j (k) is the contribution of the j-th noise mode to the total noise vector.

[0078] Substitute the p - 1 dimensional noise vector into the calculation formula of the i-th filtered acceleration response, then the i-th filtered acceleration response is further described as:

[0079]

[0080] If N samples are recorded in the on-site test, then the N filtered acceleration responses are extended to:

[0081] ​​

[0082] Simplified to:

[0083]

[0084] Among them, Y i represents the measured acceleration response containing only the i-th order bridge mode; λ i = [λ i (1) … λ i (N)] T is the contribution column vector of the i-th order acceleration response; the second term represents the j-th orthogonal noise basis mode and its contribution column vector.

[0085] Considering the cross-effect between each sensor, calculate the cross-correlation matrix of the i-th order single-mode response:

[0086]

[0087] Considering the principle of orthogonal basis, further transform it into:

[0088]

[0089] Among them,

[0090] Rewrite as:

[0091] E i = UΩU T

[0092] Among them, represents the singular vector matrix of matrix Y i ; represents the singular value matrix of matrix Y i , and the order of the singular values is q i > σ1 > σ2 > … > σ p-1 .

[0093] By performing singular value decomposition on matrix E i and extracting the first singular vector, it is the i-th order mode shape

[0094] The beneficial effects of the present invention are as follows:

[0095] The present invention provides a modal shape method based on time-domain identification and its process. This method is established on a combined vehicle test system (CVTS). The CVTS includes two test vehicles and one excitation vehicle. Among them, the test vehicles are used to record the responses of the vehicle-bridge coupling system, and the excitation vehicle is used to excite the vibration of the bridge. Accelerometer sensors are installed on the test vehicles to measure their vertical vibrations. Then, a time-domain decomposition (TDD) analysis tool is used to extract the modal shapes from the responses of the test vehicles or the contact points. Compared with the vehicle responses, the contact point responses show better performance in identifying the bridge modal shapes.

[0096] Other advantages, objectives, and features of the present invention will be described to some extent in the subsequent specification, and to some extent, will be obvious to those skilled in the art based on the study of the following text, or can be taught from the practice of the present invention. The objectives and other advantages of the present invention can be achieved and obtained through the following specification. Brief Description of the Drawings

[0097] In order to make the objectives, technical solutions, and advantages of the present invention clearer, the present invention will be described in preferred detail below in conjunction with the drawings, where:

[0098] Figure 1 is a simplified model schematic diagram of the combined vehicle test system;

[0099] Figure 2 is a finite element modeling schematic diagram of the combined vehicle test system;

[0100] Figure 3 is a schematic diagram of the No. 1 test vehicle model under the nth excitation;

[0101] Figure 4 is a schematic diagram of the No. 2 test vehicle model under the nth excitation;

[0102] Figure 5 is the excitation vehicle model schematic diagram at time step t τ ;

[0103] Figure 6 is a schematic diagram of the contact point between the test vehicle and the bridge;

[0104] Figure 7 is a schematic diagram of the layout of the bridge test points in the embodiment;

[0105] Figure 8 is a schematic diagram of the simulated result of the road roughness in the embodiment;

[0106] Figure 9 is the contact point acceleration response obtained by inverse calculation in the time domain and frequency domain of the No. 1 test vehicle in the embodiment, where Figure 9(a) is the time-domain response, Figure 9 (b) is the frequency-domain response;

[0107] Figure 10 are the contact point acceleration responses obtained by time-domain and frequency-domain back-calculation of the No. 2 test vehicle under the embodiment, where Figure 10 (a) is the time-domain response, Figure 10 (b) is the frequency-domain response;

[0108] Figure 11 are the comparisons of the modal vibration modes identified from the contact point response and vehicle response of different orders with the theoretical modal vibration modes, where Figure 11 (a) is the schematic diagram of the vibration mode comparison of the first order, Figure 11 (b) is the schematic diagram of the vibration mode comparison of the second order, Figure 11 (c) is the schematic diagram of the vibration mode comparison of the third order, Figure 11 (d) is the schematic diagram of the vibration mode comparison of the fourth order;

[0109] Figure 12 is the schematic diagram of the MAC values of the contact point response and vehicle response, where Figure 12 (a) is the MAC value of the contact point response, Figure 12 (b) is the MAC value of the vehicle response. Detailed implementation manners

[0110] The following uses specific specific examples to illustrate the implementation manners of the present invention. Those skilled in the art can easily understand other advantages and effects of the present invention from the content disclosed in this specification. The present invention can also be implemented or applied through other different specific implementation manners. Various details in this specification can also be modified or changed based on different viewpoints and applications without departing from the spirit of the present invention. It should be noted that the drawings provided in the following embodiments only illustrate the basic concept of the present invention in a schematic manner. Without conflict, the following embodiments and the features in the embodiments can be combined with each other.

[0111] Among them, the drawings are only used for exemplary illustration, showing only schematic diagrams, rather than physical diagrams, and cannot be understood as a limitation to the present invention; in order to better illustrate the embodiments of the present invention, some components in the drawings will be omitted, enlarged or reduced, which do not represent the sizes of actual products; for those skilled in the art, it is understandable that some well-known structures and their descriptions in the drawings may be omitted.

[0112] In the drawings of the embodiments of the present invention, the same or similar reference numerals correspond to the same or similar components; in the description of the present invention, it should be understood that if there are terms such as "upper", "lower", "left", "right", "front", "rear", etc. indicating the orientation or positional relationship, they are based on the orientation or positional relationship shown in the drawings. This is only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation. Therefore, the terms describing the positional relationship in the drawings are only for illustrative purposes and should not be construed as a limitation of the present invention. For those of ordinary skill in the art, the specific meanings of the above terms can be understood according to specific circumstances.

[0113] Please refer to Figures 1 to 12 , a method for extracting the bridge modal vibration mode based on a combined vehicle test system, which uses the combined vehicle test system (CVTS) to extract the bridge mode. The CVTS includes two test vehicles and one excitation vehicle. The test vehicle is used to record the response of the vehicle-bridge interaction (VBI) system, while the excitation vehicle is used to excite the bridge vibration. Subsequently, a time-domain decomposition (TDD) analysis tool is applied to extract the mode from the vehicle or contact point (CP) response of the test vehicle.

[0114] Embodiment

[0115] A method for extracting the bridge modal vibration mode based on a combined vehicle test system of the present invention includes the following steps:

[0116] S1. Establish a combined vehicle test system and perform finite element modeling on the combined vehicle test system;

[0117] S2. Establish a vehicle model, a bridge model, and a vehicle-bridge coupling model according to the combined vehicle test system;

[0118] S3. Use the excitation vehicle in the combined vehicle test system to excite the bridge vibration and record the vehicle-bridge coupling response through the test vehicle;

[0119] S4. Extract the bridge mode from the vehicle or contact point response of the test vehicle by the time-domain decomposition method.

[0120] In step S1, the present embodiment proposes a continuous vehicle test system (CVTS). The CVTS includes test vehicles and excitation vehicles, where the test vehicles are used to record the response of the vehicle-bridge coupling system, and the excitation vehicles are used to excite the bridge vibration. Strictly speaking, the number of test vehicles and excitation vehicles can be multiple. However, for the sake of simplicity, the present embodiment selects one excitation vehicle and two test vehicles to demonstrate the feasibility and accuracy of the proposed modal identification method.

[0121] Consider as Figure 1A simple beam as shown is subjected to an excitation vehicle and two test vehicles. The excitation vehicle is modeled as a system with two degrees of freedom (DOF), including vertical motion y e and pitching motion The properties of the excitation vehicle are as follows: vehicle mass m e , moment of inertia J e , spring stiffness coefficient k e , damping coefficient c e . The test vehicles are assumed and designed as single-degree-of-freedom systems and modeled as oscillators with body mass m t , spring stiffness k t and damping coefficient c t . An accelerometer is installed on the body to measure its vertical motion. The properties of all test vehicles are considered consistent in this study. The beam is of Bernoulli-Euler type, with a span length of L, mass per unit length of m, and stiffness of EI. The vertical displacement of the beam is denoted as y b . During operation, the excitation vehicle travels across the bridge at a constant speed v, while the test vehicles are parked at pre-selected positions. Depending on the number of test vehicles, the test vehicles will move to pre-selected positions and get ready for the arrival of the next excitation vehicle.

[0122] Due to the complex measurement system, it is difficult to obtain a closed-form solution for the response of the test vehicles. Based on this, in this embodiment, the numerical finite element method is adopted. The bridge structure is divided into N two-dimensional beam elements and combined with the vehicle system to form some typical vehicle-bridge interaction (VBI) elements, resulting in the finite element modeling of the combined vehicle test system. The finite element modeling of the CVTS is as shown in Figure 2 .

[0123] It is ensured that both the excitation vehicle and the test vehicles maintain rigid contact with the bridge, and there is no sliding or climbing of the vehicles on the bridge surface. In this case, according to the energy principle, the motion equations of the CVTS can be written as:

[0124]

[0125] where the subscripts 'v' and 'b' represent the vehicle and the bridge respectively; and y represent the column vectors of the acceleration, velocity, and displacement of the CVTS respectively; M i , C i , K i , F i (i = v, b) represent the mass matrix, damping matrix, stiffness matrix, and external force vector respectively. In Figure 2 , y t1 and y t1 are the vertical displacements of the 1st and 2nd test vehicles. Other parameters are the same asFigure 1 The parameters in

[0126] In step S2, for the excitation vehicle and the test vehicle in the combined vehicle test system, their motion states are different within a certain test time period. For example, generally, the test vehicle is in a stationary state, while the excitation vehicle is driving on the bridge deck. In this embodiment, as described above, one excitation vehicle and two test vehicles are set, denoted as the No. 1 test vehicle and the No. 2 test vehicle respectively. For the excitation vehicle and the test vehicle, the mass matrix of the vehicle model can be expressed according to the energy principle as:

[0127]

[0128] The stiffness matrix of the vehicle model can be expressed as:

[0129]

[0130] The damping matrix of the vehicle model can be expressed as:

[0131]

[0132] The bridge structure is divided into N two-dimensional beam elements, and each element has four degrees of freedom. Since the influence of the wheels is ignored, the mass matrix M b is only composed of its own components. First, the element mass and stiffness matrices are established, and then they are assembled into the global matrix. The mass matrix can be expressed as:

[0133]

[0134] where m represents the mass per unit length of the bridge; [N] represents the row vector of the shape function. Here, the cubic Hermite function is used and expressed as:

[0135] [N] = [N1 N2 N3 N4]

[0136] N1 = 1 - 3(ξ / l) 2 + 2(ξ / l) 3

[0137]

[0138] N3 = 3(ξ / l) 2 - 2(ξ / l) 3

[0139]

[0140] where ξ represents the local coordinate starting from the left side of the node; l represents the length of the contact element.

[0141] Due to the influence of vehicle-bridge coupling, the stiffness matrix K of the bridge b includes the components of the bridge itself and the components caused by the vehicle can be expressed as:

[0142]

[0143] where is obtained by assembling its element stiffness matrix . The symbol d represents the second derivative of the shape function row vector with respect to the local coordinate ξ.

[0144] Assume that at the nth excitation, the 1st and 2nd test vehicles are respectively deployed on the ith and jth bridge elements, as Figure 3 and Figure 4 shown. The bridge structure is evenly divided into N elements, and the length of each element is denoted as l. For each test vehicle, their positions in the local coordinate system are respectively and For the excitation vehicle, at the t τ th time step, the front and rear wheels are respectively located on the pth and qth bridge elements, as Figure 5 shown. At this time step, the positions of the front and rear wheels are respectively l f and l r . Le represents the half distance between the two wheels. Based on these assumptions, the component of the stiffness matrix caused by the vehicle is calculated as follows:

[0145]

[0146] where the subscript τ represents the number of the VBI (Vehicle-Bridge Interaction) element. The shape function row vector at the current time step can be obtained by substituting the local coordinate of the element into the motion equation of the CVTS. [N′] τ represents the first derivative of the shape function row vector with respect to the local coordinate ξ.

[0147] The damping matrix C of the bridge b is also assembled from its own components and the components caused by the vehicle and can be expressed as:

[0148]

[0149] Here, in order to facilitate the solution of the dynamic equation, Rayleigh damping is used in this embodiment to simulate viscous damping, and the calculation method is as follows:

[0150]

[0151] where α and β are damping coefficients, expressed as α = 2ζ g ω1ω2 / (ω1 + ω2) and β = 2ζ b / (ω1 + ω2), where ω1 and ω2 represent the first and second natural frequencies of the bridge respectively. ζ b is the damping ratio of the bridge.

[0152] For the second term it can be expressed as:

[0153]

[0154] Based on this, at the t τ th time step, the vehicle-bridge coupling stiffness matrix K vb can be expressed as:

[0155]

[0156] where h = 1, 2 represent the front wheel and the rear wheel respectively. It can be seen that the first term represents the coupling stiffness matrix from the test vehicle, while the other two terms are the coupling stiffness matrices from the excitation vehicle.

[0157] For the coupling stiffness matrix K bv , it can be expressed as:

[0158]

[0159] Similarly, the vehicle-bridge coupling damping matrix C vb and C bv can be expressed as:

[0160]

[0161] In step S3, this embodiment uses an excitation vehicle to induce bridge vibration. However, at the same time, the bridge itself is also subjected to external loads such as wind, earthquake, and impact loads. Among them, the external loads mainly include static loads and loads caused by road surface roughness. Then, the external load vector can be expressed as:

[0162]

[0163] where r represents the roughness of the road surface.

[0164] Additionally, the bridge also needs to bear the load of the vehicle. According to the established motion equation, the vehicle load vector can be expressed as:

[0165]

[0166] So far, the equations of motion of the combined vehicle test system have been obtained using the above sub-matrices and sub-load vectors. Then, the time-varying Newmark-β integration method, where β = 0.25 and γ = 0.5, is used to solve the coupled dynamics equations.

[0167] During the field test, the contact point response between the vehicle and the bridge is obtained through the test vehicle. The contact point represents the point where the wheel contacts the road surface at a certain time step. It has been verified that it is a better input signal for identifying the bridge modal parameters. Usually, the bridge frequency is masked because the amplitude of the vehicle frequency is more significant than that of the bridge frequency. Compared with the vehicle response, the contact point response is not affected by the vehicle frequency, so the negative impact brought by the vehicle frequency can be eliminated. Therefore, the contact point response is a better choice for identifying the characteristics of the bridge model. Unfortunately, the contact point response cannot be directly obtained during the test. To solve this problem, the present invention adopts a method of numerically back-calculating the contact point response from the measured vehicle response. Taking test vehicle 1 as an example, Figure 6 Schematic diagram of the contact point between the test vehicle and the bridge.

[0168] The equation of motion of the x-th test vehicle can be expressed as:

[0169]

[0170] Where, The column vector representing the acceleration of the x-th test vehicle, The column vector representing the velocity of the x-th test vehicle, y tx The column vector representing the displacement of the x-th test vehicle, y c Represents the contact point response, and the contact point response can be back-calculated according to the above formula.

[0171] Taking the equation of motion of the 1st test vehicle as an example, the specific back-calculation process is: that is, taking x = 1, it is expressed as:

[0172]

[0173] Moving the variables related to the contact point response in the above formula to the left side of the equation and other variables to the right side, the above formula can be converted to:

[0174]

[0175] Where, ξ t Is the damping ratio of the test vehicle, and the calculation method is ω t Is the natural frequency of the vehicle, and the calculation method is Represents the contact point response between the tire and the bridge.

[0176] Take the second derivative of the transformed formula and further transform it into:

[0177]

[0178] Where, represents the acceleration response of the contact point; Q(t) represents the generalized load input force of the contact point.

[0179] By solving the differential equation, the contact point response is obtained:

[0180]

[0181] Therefore, the discrete contact point response can be expressed as:

[0182]

[0183] Where Δt represents the sampling time interval.

[0184] In the field test, the acceleration response of the vehicle can be measured However, the term cannot be solved directly. To solve this problem, they can be obtained by the central difference method as follows:

[0185]

[0186] Where, and are the acceleration responses of the test vehicle at discrete time steps i+1, i, and i-1 respectively.

[0187] In step S4, the time domain decomposition (TDD) method is used to decompose the contact point acceleration response obtained in step S3 to extract the modal information of the bridge. Specifically, its modal decomposition process is as follows:

[0188] The acceleration of the bridge under any load excitation at time t is expressed as:

[0189]

[0190] Where, is the output acceleration column vector, p represents the number of sensors; Φ i =[φ 1i …φ 1p T is the i-th order bridge vibration mode; λ i (t) is the i-th order modal contribution factor, representing the acceleration contribution at time t.

[0191] ​Due to the limited number of sensors, only the first n modal orders can be retrieved from the discrete acceleration signals. At the sampling time k, the acceleration of the bridge under any load excitation can also be expressed as:

[0192]

[0193] where ε t (k) represents a p×1 column vector of truncation errors, usually taken as the zero vector, which is expressed as

[0194] Using a digital band - pass filter to extract the i - th single - modal response, then the i - th filtered acceleration response can be expressed as:

[0195]

[0196] where ε f (k) represents the noise vector, which is the sum of samples of noise and residuals at time k. The noise vector can be decomposed into a p - 1 - dimensional orthogonal noise space, such as:

[0197]

[0198] where Ψ j =[ψ 1j … ψ pj T is the j - th orthogonal noise basis mode, and d j (k) is the contribution of the j - th noise mode to the total noise vector.

[0199] Substituting the p - 1 - dimensional noise vector into the calculation formula of the i - th filtered acceleration response, then the i - th filtered acceleration response can be further described as:

[0200]

[0201] If N samples are recorded in the field test, then the N filtered acceleration responses can be expanded as:

[0202]

[0203] Simplified to:

[0204]

[0205] where Y i represents the measured acceleration response containing only the i - th bridge mode; λ i =[λ i (1) … λ i (N)] T ​is the contribution column vector of the i-th order acceleration response. The second term represents the j-th orthogonal noise basis mode and its contribution column vector.

[0206] Considering the cross-effect between each sensor, calculate the cross-correlation matrix of the i-th order single-mode response:

[0207]

[0208] Considering the principle of orthogonal basis, further transform it into:

[0209]

[0210] where

[0211] It can be seen that the above formula can be rewritten as:

[0212] E i =UΩU T

[0213] where represents the singular vector matrix of matrix Y i ; represents the singular value matrix of matrix Y i , and the order of singular values is q i >σ1>σ2>…>σ p-1 .

[0214] Obviously, the required i-th order modal shape can be obtained by extracting the first singular vector after performing singular value decomposition (SVD) on matrix E i .

[0215] In summary, the main steps for identifying the modal shape include:

[0216] Step 1: Determine the number of preselected test points and the number of runs of the excitation vehicle;

[0217] Step 2: Move the test vehicle to the predefined position and let the excitation vehicle drive across the bridge deck;

[0218] Step 3: Record the acceleration responses of all preselected test points;

[0219] Step 4: Determine the upper and lower filtering frequencies and use a band-pass filter to extract the first n order single-mode responses;

[0220] Step 5: Use the TDD method to gradually identify the corresponding modal shape through the equation E i =UΩU T .

[0221] This embodiment also conducts a numerical simulation verification for the method of the present invention.

[0222] To verify the accuracy and feasibility of the proposed method for extracting bridge modal shapes using vehicle or control point (CP) responses, a typical simply supported bridge will be used. The properties of the test vehicle, excitation vehicle, and bridge are shown in Table 1. The driving speed of the excitation vehicle is kept constant at 10 m / s. According to the properties of the vehicle and the bridge, the first two natural frequencies of the bridge are 2.36 Hz and 9.45 Hz, respectively, and the frequency of the test vehicle is 5.13 Hz. Due to the symmetry of the excitation vehicle, the vehicle's heave and pitch vibrations are decoupled, and their vibration frequencies are 3.18 Hz and 7.64 Hz, respectively. As Figure 7 shown, 12 test points are evenly distributed on the bridge, and two points are measured simultaneously during each round of excitation. The bridge is divided into 30 beam elements, and the time step used is 0.001 s.

[0223] Table 1

[0224]

[0225] The road surface roughness is regarded as the main external excitation, and its influence cannot be ignored during the test. Therefore, the trigonometric function synthesis (TFS) method is used to simulate the road surface profile, and the expression is:

[0226]

[0227] where n i is the spatial frequency, taken as Δn = (n u - n l ) / N, where N represents the total number of discrete frequencies; n l and n u represent the lower and upper limits of the spatial frequency, respectively; θ i represents a random phase angle in the range of 0 to 2π; d i represents the amplitude of the road surface roughness; G d (n) is the power spectral density (PSD) function determined by the roughness level; G d (n0) represents the PSD value at n0 = 0.1 cycle / m, which is determined by the road surface roughness level.

[0228] In this embodiment, grade A is adopted, and G d (n0) = 0.001 × 10 -6 m 3 . The spatial frequency range is within [1 - 100], which is evenly divided into 2500 frequency points. The length of the section considered is 100 m. Figure 8 Fig. is a schematic diagram of the simulation result of road roughness.

[0229] The 11th and 12th test points are selected to calculate the acceleration response at the contact points. Figure 9The contact point acceleration response obtained by time-domain and frequency-domain back-calculation for the No. 1 test vehicle under the embodiment, where, Figure 9 (a) is the time-domain response, Figure 9 (b) is the frequency-domain response; Figure 10 The contact point acceleration response obtained by time-domain and frequency-domain back-calculation for the No. 2 test vehicle under the embodiment, where, Figure 10 (a) is the time-domain response, Figure 10 (b) is the frequency-domain response. The results show that the results of the back-calculation method in the time domain and frequency domain are consistent with the numerical solutions. The accuracy of the proposed algorithm for back-calculating the contact point response using the vehicle acceleration response is verified, providing a practical method for obtaining the contact point response in field tests.

[0230] Theoretically, the modal shapes of the bridge can be identified from the contact point or vehicle responses. Since the contact point response is not affected by the vehicle frequency, its performance is better than that of the vehicle response. Therefore, this embodiment also studies the comparison of identifying modal shapes using the contact point response and vehicle response respectively. To quantify the identification accuracy of the contact point and vehicle responses, the Modal Assurance Criterion (MAC) is used here, and its expression is:

[0231]

[0232] where, {φ e} and {φ t} are the retrieved and theoretical modal shapes respectively.

[0233] Figure 11 Shows the comparison of modal shapes identified from the contact point response and vehicle response for different orders, where, Figure 11 (a) is the schematic diagram of the mode shape comparison for the first order, Figure 11 (b) is the schematic diagram of the mode shape comparison for the second order, Figure 11 (c) is the schematic diagram of the mode shape comparison for the third order, Figure 11 (d) is the schematic diagram of the mode shape comparison for the fourth order. It can be observed that the first four-order modal shapes can be successfully extracted from the contact point response, while only the first two-order modal shapes can be well identified from the test vehicle response. The modal shapes identified using the contact point response are in good agreement with the theoretical modal shapes, which proves the accuracy of the proposed method. To further quantify its accuracy and compare the identified modal accuracy with the vehicle response, Figure 12 Shows the schematic diagram of the MAC values using the contact point response and vehicle response, where, Figure 12 (a) is the MAC value of the contact point response, Figure 12(b) is the MAC value of the vehicle response. From these values, it can be seen that even when identifying high-order modes, the contact point response can maintain high accuracy. However, it is difficult to extract high-order modes from the test vehicle response, and the MAC values of the third and fourth-order modes identified are 0.73024 and 0.19662 respectively. Therefore, the contact point response is a better alternative input signal for extracting the bridge modal vibration mode.

[0234] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not 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 the technical solutions of the present invention can be modified or equivalently replaced without departing from the spirit and scope of the present technical solution, and they should all be covered within the scope of the claims of the present invention.

Claims

1. A method for extracting bridge modal vibration modes based on a combined vehicle test system, characterized in that: It includes the following steps: S1. Establish a combined vehicle test system and perform finite element modeling on the combined vehicle test system; S2. Establish a vehicle model, a bridge model, and a vehicle-bridge coupling model according to the combined vehicle test system; S3. Use the excitation vehicle in the combined vehicle test system to excite the bridge vibration, and record the vehicle-bridge coupling response through the test vehicle; S4. Extract the bridge mode from the vehicle or contact point response of the test vehicle by the time-domain decomposition method; In step S2, the bridge structure is divided into N two-dimensional beam elements, and each element has four degrees of freedom; Stiffness matrix of the bridge including the components of the bridge itself and the components caused by the vehicle , expressed as: Among them obtained by assembling its element stiffness matrix where EI is the bridge stiffness; At the nd excitation, the th and th test vehicles are respectively deployed on the th and th bridge units. The bridge structure is evenly divided into units, and the length of each unit is denoted as ; for each test vehicle, their positions in the local coordinate system are respectively and ; For actuating the vehicle, at the -th time step, the front wheel and the rear wheel are respectively located on the -th and the -th bridge elements; at this time step, the positions of the front wheel and the rear wheel are respectively and , L where \(e\) represents the half distance between the two wheels; then the component of the stiffness matrix caused by the vehicle is calculated as follows: where the subscript represents the number of the axle coupling unit; the row vector of the shape function at the current time step is obtained by substituting the local coordinates of the unit into the motion equation of the combined vehicle test system, represents the row vector of the shape function, represents the first derivative of the row vector of the shape function with respect to the local coordinate ; represents the spring stiffness of the test vehicle, represents the spring stiffness of the excitation vehicle, represents the damping coefficient of the excitation vehicle; Damping matrix of the bridge Composed of its own components And components caused by vehicles Assembled, expressed as: Use Rayleigh damping to simulate viscous damping, and the calculation method is as follows: Wherein and are damping coefficients, expressed as and , and respectively represent the first and second natural frequencies of the bridge, is the damping ratio of the bridge; represents the damping coefficient of the test vehicle.

2. The method for extracting the bridge modal vibration mode based on the combined vehicle test system according to claim 1, wherein: In step S1, the combined vehicle test system includes one or more excitation vehicles, one or more test vehicles, and at least one bridge. Among them, the excitation vehicle is used to excite the bridge vibration, and the test vehicle is used to record the vehicle-bridge coupling response; The actuated vehicle has two degrees of freedom, namely vertical motion and pitching motion , and its properties include: vehicle mass , moment of inertia , spring stiffness coefficient , damping coefficient ; The test vehicle has a single degree of freedom, and its properties include: body mass , spring stiffness and damping coefficient ; The test vehicle measures its own vertical motion through an accelerometer; The bridge is of the Bernoulli-Euler type, with a span length of L, a mass per unit length of m, and a stiffness of EI. Its vertical displacement is expressed as ; During operation, the excitation vehicle travels across the bridge at a constant speed v, while the test vehicle stops at a pre-selected position.

3. The method for extracting bridge modal vibration modes based on a combined vehicle test system according to claim 2, wherein: In step S1, adopt the numerical finite element method, divide the bridge structure into N two-dimensional beam elements, and combine with the vehicle system to form a vehicle-bridge coupling element, and obtain the finite element modeling of the combined vehicle test system; Ensure that both the excitation vehicle and the test vehicle are in rigid contact with the bridge, and there is no sliding or climbing phenomenon of the vehicle on the bridge surface, then the motion equation of the combined vehicle test system is expressed as: Among them, the subscripts 'v' and 'b' represent the vehicle and the bridge respectively; 、 and represent the column vectors of acceleration, velocity, and displacement of the combined vehicle test system respectively; 、 、 、 represent the mass matrix, damping matrix, stiffness matrix, and external force vector respectively.

4. The method for extracting the bridge modal vibration mode based on the combined vehicle test system according to claim 3, characterized in that: In step S2, the mass matrix of the vehicle model is expressed according to the energy principle as: The stiffness matrix of the vehicle model is expressed as: The damping matrix of the vehicle model is expressed as: Among them, at least one incentive vehicle is set, and the test vehicles are set to be vehicles, and they are numbered in sequence.

5. The method for extracting bridge modal vibration modes based on a combined vehicle test system according to claim 4, characterized in that: In step S2, the mass matrix of the bridge is composed only of its own components, and the mass matrix is expressed as: Among them represents the mass per unit length of the bridge; represents the row vector of the shape function, which is expressed as: Among them, represents the local coordinate starting from the left side of the node; l represents the length of the contact unit.

6. The method for extracting the bridge modal vibration mode based on the combined vehicle test system according to claim 5, wherein: In step S2, at the th time step, the vehicle-bridge coupling stiffness matrix is expressed as: Among them respectively represent the front wheel and the rear wheel; Coupling stiffness matrix It is expressed as: Vehicle-bridge coupling damping matrix and are expressed as: 。 7. The method for extracting the bridge modal vibration mode based on the combined vehicle test system according to claim 6, characterized in that: In step S3, during the process of exciting the bridge vibration, the loads borne by the bridge include external loads and vehicle loads. Among them, the external load vector is expressed as: Where r represents the roughness of the road surface; The vehicle load vector is expressed as: Adopt the time-varying Newmark-β integration method to solve the coupled dynamics equation.

8. The method for extracting the bridge modal vibration mode based on the combined vehicle test system according to claim 7, characterized in that: In step S3, the motion equation of the x-th test vehicle is expressed as: Among them, denotes the column vector of the acceleration of the x-th test vehicle, denotes the column vector of the speed of the x-th test vehicle, denotes the column vector of the displacement of the x-th test vehicle, denotes the contact point response, denotes the contact point response between the tire and the bridge, and the contact point response is calculated by inverse calculation according to the above formula.

9. The method for extracting the bridge modal vibration mode based on the combined vehicle test system according to claim 8, characterized in that: In step S4, use the time-domain decomposition method to analyze the acceleration response and extract the modal information of the bridge; Among them, the modal decomposition process of the acceleration response is: The acceleration of the bridge under any load excitation at time t is expressed as: Among them, is the output acceleration column vector, p represents the number of sensors; is the i-th order bridge vibration mode; is the i-th order modal contribution factor, indicating the acceleration contribution at time t; Retrieve the first n modes from the discrete acceleration signal. At the sampling time k, the acceleration of the bridge under any load excitation is expressed as: Among them, represents a p×1 truncated error column vector, which is expressed as ; The i-th order single-mode response is extracted using a digital band-pass filter, and the i-th filtered acceleration response is expressed as: wherein, represents a noise vector, which is the sample sum of the noise and the residual at time k; The noise vector is decomposed into a p-1 dimensional orthogonal noise space: Among them, is the j-th orthogonal noise basis mode, is the contribution of the j-th noise mode to the total noise vector; Substitute the p-1 dimensional noise vector into the calculation formula of the i-th filtered acceleration response, then the i-th filtered acceleration response is further described as: If N samples are recorded in the field test, the N filtered acceleration responses are extended as: Simplified to: Among them, represents the measured acceleration response containing only the i-th order bridge mode; is the contribution column vector of the i-th order acceleration response; the second term represents the j-th orthogonal noise basis mode and its contribution column vector; Considering the cross-effect between each sensor, calculate the cross-correlation matrix of the i-th single-mode response; Considering the principle of orthogonal basis, further transform it into: Among them, , ; Rewritten as: Among them, represents the singular vector matrix of the matrix ; represents the singular value matrix of the matrix , and the order of the singular values is ; By performing singular value decomposition on the matrix and extracting the first singular vector, the i-th order modal shape is obtained .

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