CN109839441A - A kind of bridge Modal Parameters Identification - Google Patents
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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
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 vibration mode, wherein the identification result can be used for evaluating the safety state of a bridge.
Background
The modal parameters (frequency, mode shape and damping) of a bridge are the main dynamic indexes reflecting the characteristics of the beam. Due to the fact that the damping action mechanism is complex, the frequency and the vibration mode play an extremely important role in detection and monitoring of the beam structure, and the damping device is widely applied to damage identification and state evaluation of the beam structure. Frequency and mode shape identification is therefore one of the important tasks in the field of bridge detection and monitoring.
The identification of the modal parameters of the bridge is essentially a process of processing dynamic response signals collected by the sensors. It is common practice to arrange sensors on the bridge to collect the dynamic response of the bridge, but the significant inherent drawbacks are not negligible. A great deal of manpower and material resources are needed for installing the sensor on the bridge; because the damage and the deterioration of the beam are usually tiny and local, in order to better and respectively realize the damage identification and the state evaluation of the bridge, the bridge vibration modes of dense measuring points are needed, and very dense sensors are required to be arranged on the bridge, so that the equipment quantity and the workload in the test process are increased, and the difficulty of data processing in modal parameter identification is increased. The accuracy of vibration pattern recognition directly restricts the effectiveness of the vibration pattern recognition in bridge detection and monitoring applications such as damage recognition and state evaluation, however, compared with the frequency, the vibration pattern recognition usually needs more complex and advanced data analysis means.
Disclosure of Invention
The invention provides a bridge modal parameter identification method for avoiding the defects in the prior art so as to avoid arranging sensors on a bridge and reduce the required number of sensors and the data processing difficulty.
The invention adopts the following technical scheme for solving the technical problems:
the method for identifying the modal parameters of the bridge is characterized in that: 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.
The method for identifying the modal parameters of the bridge is also characterized by comprising the following 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.
The method for identifying the modal parameters of the bridge is also characterized in that:
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, omegaiAnd phiiRespectively normalizing the vibration mode for the ith order angular frequency and the ith order mass of the bridge;
ith order angular frequency omega of bridgeiAnd ith order frequency f of bridgeiThe 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 isiAnd miI order mode stiffness and i order mode mass of the bridge,superscript T represents vector transposition; phi is ai,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＝ΔmuTu (6)
wherein u is a 1 XN row vector, characterized by equation (7)
u＝[u1… uj-1ujuj+1… uN]＝[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-systemSubstituting formula (4), ignoring the influence of other order modes except the ith order mode, to obtain 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 bridgeiAdditional mass Δ m, and bridge mode shape φ at the measurement point corresponding to the jth degree of freedomi,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 mARear axle mass mBAnd m isA≠mB；
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 isi,aAnd phii,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 omegaiSubstituting an equation (2) to finish the identification of the bridge frequency;
by using the ith order angular frequency omega of the bridgeiIth order angular frequency of two-axle vehicle-bridge systemAndand vehicle front axle mass mAAnd rear axle mass mBAnd 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 phii,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.
Compared with the prior art, the invention has the beneficial effects that:
1. the invention only adopts a single acceleration sensor arranged on the two-axis vehicle for testing, and obtains the bridge vibration mode by gradually placing the two-axis vehicle at different positions of the bridge. The problems that in the prior art, a plurality of sensors need to be arranged on a bridge to collect dynamic response in the test process, the number and the workload of equipment are large in the test process, and the difficulty of data processing in modal parameter identification is high are effectively solved.
2. The method well utilizes the physical relation of the frequency change of the two-axis vehicle-bridge system on the modal parameters of the bridge, namely the frequency and the vibration mode, the vibration mode is identified as frequency identification, the signal is processed only by adopting a Fourier transform means in the whole process, no complex and advanced data analysis means is needed, and the operation is simple and convenient.
Drawings
FIG. 1 is a schematic diagram of the vibration pattern recognition process of the present invention;
FIG. 2 is a numerical simulation three-span variable cross-section continuous beam bridge;
FIG. 3 is a typical dynamic response of a three-span variable cross-section continuous beam bridge under random excitation;
FIG. 4 is a frequency spectrum of a typical dynamic response of a three-span variable cross-section continuous beam bridge under random excitation;
FIG. 5 is a first-order vibration mode identification result of a three-span variable cross-section continuous beam bridge;
FIG. 6 is a second-order vibration pattern recognition result of the three-span variable cross-section continuous beam bridge;
FIG. 7 is a numerical simulation three-span truss bridge;
fig. 8 is a first-order vibration pattern recognition result of the three-span truss bridge;
FIG. 9 is a second-order vibration pattern recognition error of the three-span truss bridge;
table 1 shows the frequency identification result of the three-span variable cross-section continuous beam bridge;
table 2 shows the frequency identification result of the three-span truss bridge.
Detailed Description
In the method for identifying the modal parameters of the bridge in the embodiment, a single wireless acceleration sensor is mounted on a two-axis vehicle to form movable testing equipment, the two-axis vehicle is gradually placed at different positions of the bridge to be tested, the dynamic response of the two-axis vehicle-bridge system under environmental excitation is obtained, the dynamic response is subjected to frequency spectrum analysis through Fourier transform, the frequency of the two-axis vehicle-bridge system is obtained, and then the frequency and the vibration mode of the bridge are identified by utilizing the physical relation of the change of the angular frequency of the two-axis vehicle-bridge system on the modal parameters, namely the frequency and the vibration mode of the bridge.
In specific implementation, as shown in fig. 1, the method for identifying the bridge modal parameters is performed according to the following steps:
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 method comprises the following steps of placing movable testing equipment on a bridge, placing a rear shaft at a 1 st measuring point and placing a front shaft at a 2 nd measuring point, and obtaining dynamic response of a two-shaft vehicle-bridge system under environmental excitation by using an 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.
In specific implementation, the physical relation of the change of the frequency of the two-axis vehicle-bridge system with respect to the modal parameters of the bridge is determined 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, omegaiAnd phiiThe ith order angular frequency and the ith order vibration mode of the bridge are respectively.
Ith order angular frequency omega of bridgeiAnd ith order frequency f of bridgeiThe relationship is as in formula (2):
according to the analysis of vibration structure mode-theory, experiment and application, published by Tianjin university Press, Cao Shu et al, the frequency response function matrix H (omega) is as formula (3), H (omega) is a matrix of NxN:
H(ω)＝[K-ω2M]-1(3)
where 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 isiAnd miI order mode stiffness and i order mode mass of the bridge,superscript T represents vector transposition; phi is ai,jNormalizing the component of the vibration mode on the jth degree of freedom for the ith-order mass of the bridge;
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＝ΔmuTu (6)
wherein u is a 1 XN row vector, characterized by equation (7)
u＝[u1… uj-1ujuj+1… uN]＝[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-systemSubstituting formula (4), ignoring the influence of other order modes except the ith order mode, to obtain 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 bridgeiAdditional mass Δ m, and bridge mode shape φ at the measurement point corresponding to the jth degree of freedomi,jThe physical relationship of (a);
two-axle vehicleThe bridge system is a double-point additional mass based on which: for a two-axle vehicle with the wheelbase h, the front axle has the mass mARear axle mass mBAnd m isA≠mB；
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 isi,aAnd phii,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 omegaiAnd (5) substituting an equation (2) to complete the identification of the bridge frequency.
By using the ith order angular frequency omega of the bridgeiIth order angular frequency of two-axle vehicle-bridge systemAndand vehicle front axle mass mAAnd rear axle mass mBAnd 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 phii,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.
Example 1: FIG. 2 shows a three-span variable cross-section continuous bridge, the side span length is 18m, and the mid-span length is 24 m; the beam section is rectangular, the width is 0.5m, the beam height is linearly changed within the range of 6m from left to right near the middle support, the beam height is linearly changed from 0.6m to 1.0m, and the beam heights at other positions are all 0.6 m. The elastic modulus of the beam is 30Gpa, and the density is 2400kg/m3When the finite element method is adopted for simulation, the bridge is equally divided into 120 plane Euler beam units, random excitation is adopted, the dynamic response of the bridge is calculated by a Newmark- β method, the sampling frequency is 100Hz, and the time is 10 s.
An identification step:
step 1: selecting a two-axis vehicle V, and placing a single wireless acceleration sensor S on the two-axis vehicle V to form movable testing equipment; two-axle vehicle parameters are, front axle mass mA200kg rear axle massmB400kg, and 3 m.
Step 2: and arranging a measuring point C. Taking the support bearing point at the left end of the 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 3m, the number of the measuring points is sequentially increased from left to right, and the total number of the measuring points is 20;
and step 3: the movable testing device is arranged on a bridge B, a rear shaft is arranged at a 1 st measuring point, a front shaft is arranged at a 2 nd 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 the movable testing device; then, moving the vehicle to the right, placing the rear axle at the 2 nd measuring point and the front axle at the 3 rd measuring point, acquiring dynamic response of the system under environmental excitation by using an acceleration sensor on the movable testing equipment, and sequentially completing 19 tests from left to right, wherein a typical acceleration dynamic response is shown in fig. 3;
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 20 th measuring point, a front shaft is arranged at a 19 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 19 tests from right to left in sequence in the same way as the step 3;
and 5: performing spectrum analysis on the dynamic response obtained in the steps 4-5 through Fourier transform, wherein a typical spectrogram is shown in FIG. 4, the peak value is obvious, and the test frequency of each order is obtained through identification;
step 6: and (3) calculating the bridge circle frequency according to the formula (13) by using the test frequencies of each order obtained by the first test from left to right and the last test from right to left, and then calculating the bridge frequency according to the formula (2). The recognition results are compared with reference values in table 1.
And 7: and (4) according to the formula (12), 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. The comparison of the recognition results of the 1 st order and the 2 nd order with the reference values is shown in fig. 5 and 6, respectively. I represents the identification value, R represents the reference value, and E represents the error.
Example 2: the span of the three-span truss bridge shown in fig. 7 is 56m +80m +56m, the distance between adjacent vertical rods is 8m, the height of the vertical rods varies from 10m to 18m, the height of the vertical rod at the middle two supports is 18m, and the height of the vertical rod at the other supports is 10 m. The truss bridge comprises 175 members, each member has an elastic modulus of 200GPa and a density of 7850kg/m3The cross-sectional area of the upper chord and the lower chord is 0.760m2The cross-sectional area of the vertical rod is 0.280m2The cross-sectional area of other rod pieces is 0.360m2. And each rod piece is simulated by adopting a plane truss unit. Front axle weight m of two-axle vehicleA2000kg, rear axle weight mB4000kg, and 8 m. And 25 measuring points are arranged at equal intervals. The solving method and the identification process are the same as those of the example 1, and the identification results of the 1 st order and the 2 nd order frequencies and vibration modes are compared with the reference values as shown in table 2, fig. 8 and fig. 9, respectively. In fig. 8 and 9, I represents the identification value, R represents the reference value, and E represents the error.
TABLE 1
| Order of the scale | Identification value (Hz) | Reference value (Hz) | Error (%) |
| First order | 3.9143 | 3.9201 | -0.15 |
| Second order | 6.6692 | 6.6586 | 0.16 |
TABLE 2
| Order of the scale | Identification value (Hz) | Reference value (Hz) | Error (%) |
| First order | 4.3735 | 4.3687 | 0.11 |
| Second order | 5.7001 | 5.7073 | -0.13 |
Embodiments 1 and 2 fully illustrate that the method of the present invention can adopt a single acceleration sensor for testing, only needs a fourier transform technique to process signals, identifies the frequency and the vibration mode of the bridge, does not need to arrange sensors on the bridge, and reduces the number of the required sensors and the data processing difficulty.
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, omegaiAnd phiiRespectively normalizing the vibration mode for the ith order angular frequency and the ith order mass of the bridge;
ith order angular frequency omega of bridgeiAnd ith order frequency f of bridgeiThe 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 isiAnd miI order mode stiffness and i order mode mass of the bridge respectivelyi＝φi TKφi，mi＝φi TMφiThe superscript T represents the vector transposition; phi is ai,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＝ΔmuTu (6)
wherein u is a 1 XN row vector, characterized by equation (7)
u＝[u1…uj-1ujuj+1…uN]＝[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 bridgeiAdditional mass Δ m, and bridge mode shape φ at the measurement point corresponding to the jth degree of freedomi,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 mARear axle mass mBAnd m isA≠mB；
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 isi,aAnd phii,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 omegaiSubstituting an equation (2) to finish the identification of the bridge frequency;
by using the ith order angular frequency omega of the bridgeiIth order angular frequency of two-axle vehicle-bridge systemAndand vehicle front axle mass mAAnd rear axle mass mBAnd 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 phii,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.
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