CN109839441A - A kind of bridge Modal Parameters Identification - Google Patents

Abstract

The invention discloses a kind of bridge Modal Parameters Identifications, it is characterized in that: forming movable detection equipment in the upper single Wireless Acceleration Sensor of installation of two-axle car, the two-axle car is gradually placed at the different location of bridge and is tested, obtain dynamic response of the two-axle car-Modular Bridge System under environmental excitation, spectrum analysis is carried out by dynamic response described in Fourier transform pairs, obtain two-axle car-Modular Bridge System frequency, recycle the variation of two-axle car-Modular Bridge System angular frequency about bridge modal parameter, that is, frequency and vibration shape physical relation, identify bridge frequency and the vibration shape.

Description

A method for identifying modal parameters of bridges

technical field

The invention relates to the field of bridge monitoring and detection, in particular to a method for identifying bridge modal parameters including frequency and mode shape, and the identification result can be used to evaluate the safety state of the bridge.

Background technique

The modal parameters (frequency, mode shape and damping) of the bridge are the main dynamic indicators reflecting the characteristics of the beam. Due to the complex damping mechanism, frequency and mode shape play an extremely important role in the detection and monitoring of beam structures, and are widely used in beam structure damage identification and condition assessment. Therefore, frequency and mode shape identification has become one of the important tasks in the field of bridge detection and monitoring.

The identification of bridge modal parameters is essentially a process of processing the dynamic response signals collected by sensors. The usual practice is to arrange the sensors on the bridge to collect the dynamic response of the bridge, but its obvious inherent defects cannot be ignored. For example, installing sensors on bridges requires a lot of manpower and material resources; since beam damage and deterioration are usually small and local, in order to better realize bridge damage identification and condition assessment, bridge mode shapes with dense measurement points are required, which is Therefore, very dense sensors must be arranged on the bridge, which not only increases the number of equipment and workload during the testing process, but also increases the difficulty of data processing in the identification of modal parameters. The accuracy of mode shape identification directly restricts its effectiveness in bridge detection and monitoring applications such as damage identification and condition assessment. However, compared with frequency, mode shape identification usually requires more complex and advanced data analysis methods.

SUMMARY OF THE INVENTION

The present invention provides a bridge modal parameter identification method in order to avoid the above-mentioned shortcomings of the prior art, so as to avoid arranging sensors on the bridge and reduce the required number of sensors and the difficulty of data processing.

The present invention adopts the following technical scheme for solving the technical problem:

The characteristics of the bridge modal parameter identification method of the present invention are: a single wireless acceleration sensor is installed on a two-axle vehicle to form a movable test equipment, the two-axle vehicle is gradually placed at different positions of the bridge for testing, and the two-axle vehicle- The dynamic response of the bridge system under the environmental excitation, the frequency of the two-axle vehicle-bridge system is obtained by performing spectrum analysis on the dynamic response through Fourier transform, and then the change of the frequency of the two-axle vehicle-bridge system is used for the bridge modal parameters, namely Physical relationship between frequencies and mode shapes, identifying bridge frequencies and mode shapes.

The characteristics of the bridge modal parameter identification method of the present invention also include the following steps:

Step 1: Determine the parameters of the two-axle vehicle, including: the ratio of the mass of the two-axle vehicle to the mass of the bridge is 0.02-0.05, the mass ratio of the front axle to the rear axle of the two-axle vehicle is 1.5-2.5, the wheelbase of the two-axle vehicle not less than 1m;

Step 2: Select the bridge mode shape measurement point, take the support point of the left end of the bridge as the first measurement point, select the measurement points from left to right, the distance between the two adjacent measurement points is the vehicle wheelbase, and the measurement point The numbers increase sequentially from left to right, and the total number of measuring points is T;

Step 3: Place the movable test equipment on the bridge, the rear axle is placed at the first measuring point, the front axle is placed at the second measuring point, and the acceleration sensor on the movable test equipment is used to obtain the two-axle vehicle-bridge The dynamic response of the system under environmental excitation; then, move the vehicle to the right, the rear axle is placed at the second measuring point, and the front axle is placed at the third measuring point, and the acceleration sensor on the movable test equipment is used to obtain the system's environmental conditions. The dynamic response under excitation, complete the T-1 test from left to right in sequence;

Step 4: Reverse the direction of the vehicle and start the test from the right end of the bridge; first place the rear axle at the T-th measuring point and the front axle at the T-1-th measuring point, and use the acceleration sensor on the movable test equipment to obtain the two-axle vehicle - Dynamic response of bridge system under environmental excitation; complete T-1 tests from right to left sequentially in the same manner as in step 3;

Step 5: Perform spectrum analysis on the dynamic responses obtained in steps 3 and 4 through Fourier transform, and obtain test frequencies of each order after identification;

Step 6: Use the test frequencies of each order obtained from the first test from left to right and the last test from right to left to obtain the frequencies of each order of the bridge after identification;

Step 7: Identify the bridge mode shape using the physical relationship between the change in the angular frequency of the vehicle-bridge system and the bridge mode shape at the corresponding measurement location.

The characteristics of the bridge modal parameter identification method of the present invention are also as follows:

Determine the physical relationship of the change in frequency of the two-axle vehicle-bridge system with respect to the bridge modal parameters as follows:

The finite element model with the total number of degrees of freedom N is used for simulation, and the dynamic equation of the undamped bridge is shown in formula (1):

Among them, K and M are the stiffness matrix and mass matrix of the bridge, respectively, the size of the matrix is N×N, ω _i and φ _i are the i-th order angular frequency and the i-th order mass normalized mode shape of the bridge, respectively;

The relationship between the i-th order angular frequency ω _i of the bridge and the i-th order frequency f _i of the bridge is shown in formula (2):

The frequency response function matrix H(ω) is as in formula (3), and H(ω) is an N×N matrix:

H(ω)=[K-ω ² M] ^-1 (3)

Among them, ω is the frequency variable;

The element H _j,j (ω) of the jth row and jth column in the frequency response function matrix H(ω) is represented by equation (4):

where k _i and m _i are the i-th mode stiffness and the i-th mode mass of the bridge, respectively, The superscript T represents the vector transposition; φ _i,j is the component on the jth degree of freedom of the i-th mode shape of the bridge;

Based on single-point additional mass: when an additional mass Δm acts on the jth degree of freedom of the bridge finite element model, a single-point additional mass-bridge system is formed, and the dynamic equation of the single-point additional mass-bridge system is shown in Eq. (5 ):

in, and are the i-th angular frequency and the i-th mode shape of the single-point additional mass-bridge system, respectively, and ΔM is represented by equation (6):

ΔM= ^Δmu Tu (6)

where u is a 1×N row vector, represented by Eq. (7)

u=[u ₁ … u _j-1 u _j u _j+1 … u _N ]=[0 … 0 1 0 … 0] (7)

Attach a single point of mass - the i-th angular frequency of the system Substitute into equation (3), and use equation (5) to obtain equation (8):

Expand and transform Equation (8) to obtain Equation (9):

Attach a single point of mass - the i-th angular frequency of the system Substitute into Equation (4), except for the i-th order mode, ignoring the influence of other order modes, Equation (10) is obtained:

Using Equation (9), Equation (10) and the characteristics of the mass-normalized mode shape, Equation (11) is obtained:

Equation (11) characterizes the additional mass at a single point - the i-th angular frequency of the bridge system The physical relationship between the bridge's i-th angular frequency ω _i , the additional mass Δm, and the bridge mode shape φ _i,j at the measurement point corresponding to the j-th degree of freedom;

The two-axle vehicle-bridge system is a two-point additional mass based on the two-point additional mass: for a two-axle vehicle with a wheelbase h, the front axle mass is m _A , the rear axle mass is m _B , and m _A ≠ m _B ;

The first measurement position is defined as: the front axle is located at measurement point a, 2≤a≤T, the rear axle is at measurement point b, 1≤b≤T-1, and the i-th order of the two-axle vehicle-bridge system in the first measurement position is defined. The angular frequency is recorded as

The second measurement position is defined as: the front axle is located at the measuring point b, the rear axle is at the measuring point a, and the i-th frequency of the two-axle vehicle-bridge system in the second measurement position is recorded as According to formula (11), the relational formula is established as formula (12):

Among them, φ _i,a and φ _i,b are the components on the measuring point a and the measuring point b of the i-th mode shape of the bridge, respectively;

The i-th angular frequency of the two-axle vehicle-bridge system obtained from the first test from left to right is denoted as

The i-th angular frequency of the two-axle vehicle-bridge system obtained from the last test from right to left is denoted as Substitute into equation (12) to obtain the i-th angular frequency of the bridge represented by equation (13):

in,

Substitute the i-th angular frequency ω _i of the bridge obtained from equation (13) into equation (2) to complete the identification of the bridge frequency;

Using the i-th angular frequency of the bridge ω _i , the i-th angular frequency of the two-axle vehicle-bridge system and As well as the vehicle front axle mass m _A and the rear axle mass m _B , the components φ _i,a and φ _i,b on the measuring point a and the measuring point b of the i-th vibration mode of the bridge are calculated according to formula (12). By analogy, the components of the i-th mode shape of the bridge at all measuring points are obtained, and the identification of the bridge mode shape is completed.

Compared with the prior art, the beneficial effects of the present invention are reflected in:

1. The present invention only uses a single acceleration sensor installed on a two-axle vehicle for testing, and the bridge mode shape is obtained by gradually placing the two-axle vehicle at different positions of the bridge. The method effectively avoids the problems in the prior art that multiple sensors need to be arranged on the bridge to collect dynamic responses during the testing process, which leads to the huge number of equipment and workload in the testing process and the high difficulty of data processing in the identification of modal parameters.

2. The present invention makes good use of the physical relationship between the bridge modal parameters, that is, the frequency and the mode shape, from the change of the frequency of the two-axle vehicle-bridge system. The identification of the mode shape is frequency identification, and the whole process is only processed by means of Fourier transform. Signal, does not require complex and advanced data analysis means, easy to operate.

Description of drawings

Fig. 1 is the process schematic diagram that the method of the present invention carries out mode shape identification;

Figure 2 shows the numerical simulation of a three-span variable-section continuous girder bridge;

Figure 3 shows the typical dynamic response of a three-span variable-section continuous girder bridge under random excitation;

Figure 4 is a spectrum diagram of a typical dynamic response of a three-span variable-section continuous girder bridge under random excitation;

Figure 5 shows the identification results of the first vibration mode of a three-span variable-section continuous girder bridge;

Figure 6 shows the identification results of the second vibration mode of the three-span variable-section continuous girder bridge;

Figure 7 shows the numerical simulation of a three-span truss bridge;

Figure 8 shows the identification results of the first vibration mode of the three-span truss bridge;

Figure 9 shows the identification error of the second vibration mode of the three-span truss bridge;

Table 1 shows the frequency identification results of three-span variable-section continuous girder bridges;

Table 2 shows the frequency identification results of three-span truss bridges.

Detailed ways

The bridge modal parameter identification method in this embodiment is to install a single wireless acceleration sensor on the two-axle vehicle to form a movable test equipment, and gradually place the two-axle vehicle at different positions of the bridge for testing, and obtain the two-axle vehicle-bridge system. For the dynamic response under environmental excitation, the frequency of the two-axle vehicle-bridge system is obtained by performing spectrum analysis on the dynamic response through Fourier transform, and then the change of the angular frequency of the two-axle vehicle-bridge system is used for the bridge modal parameters, namely frequency and mode shape. physical relationships, identifying bridge frequencies and mode shapes.

In the specific implementation, as shown in Figure 1, the bridge modal parameter identification method is carried out according to the following steps:

Step 1: Determine the parameters of the two-axle vehicle, including: the ratio of the mass of the two-axle vehicle to the mass of the bridge is 0.02-0.05, the mass ratio of the front axle to the rear axle of the two-axle vehicle is 1.5-2.5, the wheelbase of the two-axle vehicle not less than 1m;

Step 2: Select the bridge mode shape measurement point, take the support point of the left end of the bridge as the first measurement point, select the measurement points from left to right, the distance between the two adjacent measurement points is the vehicle wheelbase, and the measurement point The numbers increase sequentially from left to right, and the total number of measuring points is T;

Step 3: Place the movable test equipment on the bridge, the rear axle is placed at the first measuring point, and the front axle is placed at the second measuring point, and the acceleration sensor on the movable test equipment is used to obtain the two-axle vehicle-bridge system. Dynamic response under environmental excitation; then, move the vehicle to the right, the rear axle is placed at the second measuring point, the front axle is placed at the third measuring point, and the acceleration sensor on the movable test equipment is used to obtain the system under environmental excitation. , complete the T-1 tests from left to right in sequence;

Step 4: Reverse the direction of the vehicle and start the test from the right end of the bridge; first place the rear axle at the T-th measuring point and the front axle at the T-1-th measuring point, and use the acceleration sensor on the movable test equipment to obtain the two-axle vehicle - Dynamic response of bridge system under environmental excitation; complete T-1 tests from right to left sequentially in the same manner as in step 3;

Step 5: Perform spectrum analysis on the dynamic responses obtained in steps 3 and 4 through Fourier transform, and obtain test frequencies of each order after identification;

Step 6: Use the test frequencies of each order obtained from the first test from left to right and the last test from right to left to obtain the frequencies of each order of the bridge after identification;

Step 7: Identify the bridge mode shape using the physical relationship between the change in the angular frequency of the vehicle-bridge system and the bridge mode shape at the corresponding measurement location.

In the specific implementation, the physical relationship between the frequency change of the two-axle vehicle-bridge system and the bridge modal parameters is determined as follows:

The finite element model with the total number of degrees of freedom N is used for simulation, and the dynamic equation of the undamped bridge is shown in formula (1):

Among them, K and M are the stiffness matrix and mass matrix of the bridge, respectively, the size of the matrix is N×N, and ω _i and φ _i are the i-th angular frequency and the i-th mode shape of the bridge, respectively.

The relationship between the i-th order angular frequency ω _i of the bridge and the i-th order frequency f _i of the bridge is shown in formula (2):

According to "Modal Analysis of Vibration Structures - Theory, Test and Application" by Cao Shuqian, published by Tianjin University Press, the frequency response function matrix H(ω) is as in formula (3), and H(ω) is a matrix of N×N :

H(ω)=[K-ω ² M] ^-1 (3)

where ω is the frequency variable.

The element H _j,j (ω) of the jth row and jth column in the frequency response function matrix H(ω) is represented by equation (4):

where k _i and m _i are the i-th mode stiffness and the i-th mode mass of the bridge, respectively, The superscript T represents the vector transposition; φ _i,j is the component on the jth degree of freedom of the normalized mode shape of the i-th order mass of the bridge;

Based on single-point additional mass: when an additional mass Δm acts on the jth degree of freedom of the bridge finite element model, a single-point additional mass-bridge system is formed. The dynamic equation of the single-point additional mass-bridge system is shown in Equation (5):

in, and are the i-th angular frequency and the i-th mode shape of the single-point additional mass-bridge system, respectively, and ΔM is represented by equation (6):

ΔM= ^Δmu Tu (6)

where u is a 1×N row vector, represented by Eq. (7)

u=[u ₁ … u _j-1 u _j u _j+1 … u _N ]=[0 … 0 1 0 … 0] (7)

Attach a single point of mass - the i-th angular frequency of the system Substitute into equation (3), and use equation (5) to obtain equation (8):

Expand and transform Equation (8) to obtain Equation (9):

Attach a single point of mass - the i-th angular frequency of the system Substitute into Equation (4), except for the i-th order mode, ignoring the influence of other order modes, Equation (10) is obtained:

Using Equation (9), Equation (10) and the characteristics of the mass-normalized mode shape, Equation (11) is obtained:

Equation (11) characterizes the additional mass at a single point - the i-th angular frequency of the bridge system The physical relationship between the bridge's i-th angular frequency ω _i , the additional mass Δm, and the bridge mode shape φ _i,j at the measurement point corresponding to the j-th degree of freedom;

The two-axle vehicle-bridge system is a two-point additional mass based on the two-point additional mass: for a two-axle vehicle with a wheelbase h, the front axle mass is m _A , the rear axle mass is m _B , and m _A ≠ m _B ;

The first measurement position is defined as: the front axle is located at measurement point a, 2≤a≤T, the rear axle is at measurement point b, 1≤b≤T-1, and the i-th order of the two-axle vehicle-bridge system in the first measurement position is defined. The angular frequency is recorded as

The second measurement position is defined as: the front axle is located at the measuring point b, the rear axle is at the measuring point a, and the i-th frequency of the two-axle vehicle-bridge system in the second measurement position is recorded as According to formula (11), the relational formula is established as formula (12):

Among them, φ _i,a and φ _i,b are the components on the measuring point a and the measuring point b of the i-th mode shape of the bridge, respectively;

The i-th angular frequency of the two-axle vehicle-bridge system obtained from the first test from left to right is denoted as

The i-th angular frequency of the two-axle vehicle-bridge system obtained from the last test from right to left is denoted as Substitute into equation (12) to obtain the i-th angular frequency of the bridge represented by equation (13):

in,

Substitute the i-th angular frequency ω _i of the bridge obtained from equation (13) into equation (2) to complete the identification of the bridge frequency.

Using the i-th angular frequency of the bridge ω _i , the i-th angular frequency of the two-axle vehicle-bridge system and As well as the vehicle front axle mass m _A and the rear axle mass m _B , the components φ _i,a and φ _i,b on the measuring point a and the measuring point b of the i-th vibration mode of the bridge are calculated according to formula (12). By analogy, the components of the i-th mode shape of the bridge at all measuring points are obtained, and the identification of the bridge mode shape is completed.

Example 1: Figure 2 shows a three-span variable cross-section continuous girder bridge, the side span length is 18m, and the middle span length is 24m; the beam section is rectangular, the width is 0.5m, and the beam The height changes linearly, from 0.6m to 1.0m, and the beam height at other positions is 0.6m. The elastic modulus of the beam is 30Gpa, and the density is 2400kg/m ³ . When simulated by the finite element method, the bridge is divided into 120 plane Euler beam elements equidistantly. Using random excitation, the dynamic response of the bridge is calculated by the Newmark-β method. The sampling frequency is 100Hz and the time is 10s.

Identification steps:

Step 1: Select a two-axle vehicle V, and place a single wireless acceleration sensor S on the two-axle vehicle V to form a movable test equipment; the two-axle vehicle parameters are, the front axle mass m _A = 200kg, the rear axle mass m _B = 400kg , the wheelbase h=3m.

Step 2: Arrange measuring point C. Take the support point at the left end of the bridge as the first measuring point, select the measuring points in turn from left to right, the distance between two adjacent measuring points is 3m, the number of measuring points increases from left to right, and the total number of measuring points is 20;

Step 3: Place the movable test device on the bridge B, place the rear axle at the first measuring point and the front axle at the second measuring point, and use the acceleration sensor on the movable test device to obtain the two-axle vehicle-bridge system Dynamic response under environmental excitation; then, move the vehicle to the right, the rear axle is placed at the second measuring point, the front axle is placed at the third measuring point, and the acceleration sensor on the movable test equipment is used to obtain the system under environmental excitation. 19 tests from left to right are completed in sequence, and the typical acceleration dynamic response is shown in Figure 3;

Step 4: Reverse the direction of the vehicle and start the test from the right end of the bridge; first place the rear axle at the 20th measuring point and the front axle at the 19th measuring point, and use the acceleration sensor on the movable test equipment to obtain the two-axle vehicle-bridge Dynamic response of the system under environmental excitation; 19 tests from right to left are completed sequentially in the same manner as in step 3;

Step 5: Perform spectrum analysis on the dynamic responses obtained in steps 4 to 5 through Fourier transform. The typical spectrogram is shown in Figure 4. The peaks are obvious, and the test frequencies of each order are obtained after identification;

Step 6: Using the test frequencies of each order obtained from the first test from left to right and the last test from right to left, calculate the bridge circular frequency according to formula (13), and then calculate the bridge frequency according to formula (2). The comparison between the identification results and the reference value is shown in Table 1.

Step 7: According to equation (12), identify the bridge mode shape by using the physical relationship between the change of the angular frequency of the vehicle-bridge system and the bridge mode shape at the corresponding measurement position. The comparison of the recognition results and reference values of the first and second orders are shown in Figure 5 and Figure 6, respectively. I represents the identification value, R represents the reference value, and E represents the error.

Example 2: The three-span truss bridge shown in Figure 7, the span arrangement is 56m+80m+56m, the distance between adjacent vertical rods is 8m, and the height of the vertical rods varies from 10m to 18m, of which the middle two supports The height of the vertical rod is 18m, and the height of the vertical rod at other supports is 10m. ^The truss bridge consists of 175 members, the elastic modulus of each member is ^200Gpa , and the density is 7850kg/m ³ . is 0.360m ² . Each member is simulated by plane truss element. The front axle weight of the two-axle vehicle is m _A = 2000kg, the rear axle weight m _B = 4000kg, and the wheelbase is h = 8m. 25 measuring points are arranged at equal distances. The solution method and identification process are consistent with those of Example 1. The comparison of the identification results of the first and second order frequencies and mode shapes with the reference values are shown in Table 2, Figure 8 and Figure 9, respectively. In Figs. 8 and 9, I represents an identification value, R represents a reference value, and E represents an error.

Table 1

Order Identification value (Hz) Reference value (Hz) error(%) level one 3.9143 3.9201 -0.15 second order 6.6692 6.6586 0.16

Table 2

Order Identification value (Hz) Reference value (Hz) error(%) level one 4.3735 4.3687 0.11 second order 5.7001 5.7073 -0.13

Embodiments 1 and 2 fully demonstrate that the method of the present invention can be tested by a single acceleration sensor, only Fourier transform technology is needed to process the signal, and the frequency and mode shape of the bridge can be identified. the number of sensors and the difficulty of data processing.

Claims (3)

1. A method for identifying modal parameters of a bridge is characterized by comprising the following steps: a single wireless acceleration sensor is installed on a two-axis vehicle to form movable testing equipment, the two-axis vehicle is gradually placed at different positions of a bridge to be tested, dynamic response of a two-axis vehicle-bridge system under environmental excitation is obtained, frequency spectrum analysis is carried out on the dynamic response through Fourier transform, frequency of the two-axis vehicle-bridge system is obtained, and then the bridge frequency and the vibration mode are identified by utilizing the physical relation of the change of the frequency of the two-axis vehicle-bridge system on bridge modal parameters, namely the frequency and the vibration mode.

2. The method for identifying the modal parameters of the bridge according to claim 1, comprising the steps of:

step 1: determining two-axis vehicle parameters, comprising: the mass ratio of the two-axis vehicle to the bridge is 0.02-0.05, the mass ratio of the front axle to the rear axle of the two-axis vehicle is 1.5-2.5, and the wheelbase of the two-axis vehicle is not less than 1 m;

step 2: selecting a bridge vibration mode measuring point, taking a support bearing point at the left end of a bridge as a 1 st measuring point, sequentially selecting measuring points from left to right, wherein the distance between every two adjacent measuring points is the vehicle wheelbase, the number of the measuring points is sequentially increased from left to right, and the total number of the measuring points is T;

and step 3: the movable testing equipment is arranged on a bridge, the rear shaft is arranged at the 1 st measuring point, the front shaft is arranged at the 2 nd measuring point, and the dynamic response of the two-shaft vehicle-bridge system under the environmental excitation is obtained by utilizing the acceleration sensor on the movable testing equipment; then, moving the vehicle to the right, placing the rear shaft at the 2 nd measuring point and the front shaft at the 3 rd measuring point, and obtaining the dynamic response of the system under the environmental excitation by utilizing the acceleration sensor on the movable testing equipment to sequentially complete the T-1 tests from left to right;

and 4, step 4: reversing the vehicle direction and starting the test from the right end of the bridge; firstly, a rear shaft is arranged at a T-th measuring point, a front shaft is arranged at a T-1-th measuring point, and the dynamic response of the two-shaft vehicle-bridge system under the environmental excitation is obtained by utilizing an acceleration sensor on movable testing equipment; completing the T-1 tests from right to left in sequence in the same way as the step 3;

and 5: carrying out spectrum analysis on the dynamic response obtained in the step 3 and the step 4 through Fourier transform, and obtaining test frequencies of various orders through identification;

step 6: identifying and obtaining each order frequency of the bridge by using each order test frequency obtained by the first test from left to right and the last test from right to left;

and 7: and identifying the bridge vibration mode by utilizing the physical relation between the change of the angular frequency of the vehicle-bridge system and the bridge vibration mode at the corresponding measurement position.

3. The method for identifying the modal parameters of the bridge according to claim 2, wherein:

determining a physical relationship of changes in frequency of the two-axis vehicle-bridge system with respect to a modal parameter of the bridge as follows:

a finite element model with the total degree of freedom N is adopted for simulation, and the dynamic equation of the undamped bridge is as shown in the formula (1):

wherein, K and M are respectively a rigidity matrix and a quality matrix of the bridge, and the matrix size is NXN, omega_iAnd phi_iRespectively normalizing the vibration mode for the ith order angular frequency and the ith order mass of the bridge;

ith order angular frequency omega of bridge_iAnd ith order frequency f of bridge_iThe relationship is as in formula (2):

the frequency response function matrix H (ω) is as in equation (3), H (ω) being a matrix of N × N:

H(ω)＝[K-ω²M]^-1(3)

wherein ω is a frequency variable;

element H of j row and j column of j in frequency response function matrix H (omega)_j,j(ω) is characterized by formula (4):

wherein k is_iAnd m_iI order mode stiffness and i order mode mass of the bridge respectively_i＝φ_i ^TKφ_i，m_i＝φ_i ^TMφ_iThe superscript T represents the vector transposition; phi is a_i,jIs the component of the ith order vibration mode of the bridge in the jth degree of freedom;

based on the single point of additional mass: when an additional mass deltam acts on the jth degree of freedom of the bridge finite element model, a single-point additional mass-bridge system is formed, and the dynamic equation of the single-point additional mass-bridge system is as shown in the formula (5):

wherein,andthe ith order angular frequency and ith order mode of the single point additional mass-bridge system, respectively, Δ M is characterized by equation (6):

ΔM＝Δmu^Tu (6)

wherein u is a 1 XN row vector, characterized by equation (7)

u＝[u₁…u_j-1u_ju_j+1…u_N]＝[0…0 1 0…0](7)

Ith order angular frequency of a single point additional mass-systemSubstituting formula (3), and obtaining formula (8) using formula (5):

and (3) unfolding and transforming the formula (8) to obtain a formula (9):

ith order angular frequency of a single point additional mass-systemSubstitution of formula (4) ignoring other orders except the i-th orderInfluence, yielding formula (10):

using the characteristics of equation (9), equation (10), and the mass normalized mode shape, equation (11) is obtained:

equation (11) represents the ith order angular frequency of the single-point additional mass-bridge systemIth order angular frequency omega of bridge_iAdditional mass Δ m, and bridge mode shape φ at the measurement point corresponding to the jth degree of freedom_i,jThe physical relationship of (a);

the two-axis vehicle-bridge system is a double-point additional mass based on the following steps: for a two-axle vehicle with the wheelbase h, the front axle has the mass m_ARear axle mass m_BAnd m is_A≠m_B；

Defining the first measurement bit as: the front axis is positioned at a measuring point a, a is more than or equal to 2 and less than or equal to T, the rear axis is positioned at a measuring point b, b is more than or equal to 1 and less than or equal to T-1, and the ith order angular frequency of the two-axis vehicle-bridge system in the first measuring position is recorded as

Defining the second measurement bit as: the front axis is positioned at a measuring point b, the rear axis is positioned at a measuring point a, and the ith order frequency of the two-axis vehicle-bridge system in the second measuring position is recorded asEstablishing a relation according to the formula (11) as shown in the formula (12):

wherein phi is_i,aAnd phi_i,bComponents of the ith order vibration mode of the bridge on a measuring point a and a measuring point b are respectively;

the ith order angular frequency of the two-axis vehicle-bridge system obtained from the first test from left to right is recorded as

The ith order angular frequency of the two-axis vehicle-bridge system obtained from the last test from right to left is recorded asObtaining a bridge ith order angular frequency characterized by equation (13) in place of equation (12):

wherein,

subjecting the bridge obtained in the formula (13) to an i-th order angular frequency omega_iSubstituting an equation (2) to finish the identification of the bridge frequency;

by using the ith order angular frequency omega of the bridge_iIth order angular frequency of two-axle vehicle-bridge systemAndand vehicle front axle mass m_AAnd rear axle mass m_BAnd calculating components phi on a measuring point a and a measuring point b of the ith order vibration mode of the bridge according to the formula (12)_i,aAnd phi_i,b. And by analogy, obtaining the ith order vibration mode component of the bridge at all the measuring points, and completing the identification of the bridge vibration mode.