A double-hanger system parameter identification method based on frequency trajectory intersection characteristics

Abstract

The application discloses a double-hanging cable system parameter identification method based on frequency trajectory intersection characteristics, belongs to the field of bridge engineering structure health monitoring, and is suitable for a double-hanging cable system of a suspension bridge and a tied arch bridge. The method first collects hanging cable structure parameters and measured modal frequencies, builds a dynamic model, generates an overall dynamic stiffness matrix and a frequency characteristic equation by means of a dynamic stiffness method, and establishes a mapping relationship between to-be-identified parameters and modal frequencies; frequency response functions are constructed relying on each order of measured frequencies, and frequency trajectories conforming to frequency constraints are extracted; a set of intersection points is obtained by intersecting multiple orders of trajectories with each other, and then an intersection density analysis is adopted to determine an identification result; when there is only one common intersection point of the trajectories, the parameter of the point is taken; and when there is no unique intersection point, a parameter corresponding to a global maximum value of intersection density is selected. The application does not need to preset initial values of parameters, does not need to perform iterative calculation, supports separate cable force identification and cable force-bending stiffness joint identification, can compensate errors caused by model mismatch, and effectively improves parameter identification precision and stability.

Description

Technical Field

[0001] This invention belongs to the field of bridge engineering structural health monitoring technology, and in particular relates to a method for identifying cable force and bending stiffness parameters applicable to double-cable systems of suspension bridges and tied arch bridges. Background Technology

[0002] Double-cable systems are widely used in major engineering structures such as suspension bridges and tied-arch bridges. A double-cable system typically consists of two parallel cables and a rigid vibration damper, causing the two cables to couple during vibration. Therefore, its dynamic characteristics differ significantly from those of a single cable, making traditional cable force identification theories based on single-cable models difficult to apply directly to double-cable systems. Cable force is a crucial parameter for evaluating the stress state and safety performance of double-cable structures, and its identification accuracy directly affects bridge operational safety monitoring and condition assessment. However, the dynamic characteristics of a double-cable system are influenced not only by cable force but also by factors such as bending stiffness, boundary conditions, and material properties. Since a cable is composed of multiple steel strands, its bending stiffness is not a constant value but depends on the cable-forming process and cable force, making accurate results difficult to obtain. When using a fixed bending stiffness for cable force identification, model parameter mismatch can easily lead to increased identification errors. Therefore, it is necessary to introduce bending stiffness parameters for joint identification during the cable force identification process to reduce the impact of model parameter mismatch on the cable force identification results and improve the accuracy and engineering applicability of cable force identification.

[0003] Existing technologies for identifying parameters in dual-span cable systems have the following drawbacks. First, traditional frequency-based methods rely on the vibration theory of single-cable strings to establish an explicit frequency-force correspondence, which cannot be directly applied to dual-span cable systems with significant coupling effects. Second, optimization problems with the deviation between measured and theoretical frequencies as the objective function require preset initial parameter values. The convergence result depends on the selection of these initial values, and iteration is prone to convergence to local optima, failing to obtain a global optimum. Third, in practical engineering, measured frequencies are affected by multiple factors such as test errors, environmental noise, temperature changes, and finite element model simplification errors. Iterative methods based on minimizing a single objective function lack robust mechanisms for handling multi-source errors.

[0004] Therefore, there is an urgent need for a parameter identification method for a double-sling system that does not require defining an optimization objective function, iterative optimization, or presetting initial values, and is adaptable to engineering errors. Summary of the Invention

[0005] To address the above shortcomings, this invention proposes a parameter identification method for dual-cable systems based on frequency trajectory intersection characteristics. This method unifies cable force identification and cable force-bending stiffness joint identification into a single parameter expression framework. It constructs frequency response functions based on the measured modal frequencies of each order and extracts their continuous peak sets to form corresponding peak trajectories. The parameters to be identified are determined by the intersection characteristics of different peak trajectories. When frequency deviations caused by testing errors prevent the trajectories from strictly intersecting at a unique common intersection point, an intersection density analysis method is introduced to determine the optimal identification result for the parameters to be identified.

[0006] The technical solution provided by this invention includes the following steps:

[0007] Step 1: Obtain the structural parameters and measured modal frequencies of the double-sling system.

[0008] The structural parameters include the length of each sling. Mass per unit length Vibration damper position parameters Elastic modulus Initial moment of inertia Initial cable force The elastic modulus and initial moment of inertia are known structural parameters in the cable force identification mode, and reference values ​​in the cable force-bending stiffness joint identification mode; the initial cable force is a reference value for the parameter search range. The measured modal frequencies are obtained by performing vibration tests on the double-cable system, collecting acceleration signals, and extracting the natural frequencies of each order through spectrum analysis. The measured modal frequencies are denoted as follows: , The frequency order for parameter identification is usually taken as... .

[0009] Step 2: Establish the frequency characteristic equation of the double-suspension system.

[0010] A dynamic model of the double-cable system is established based on the dynamic stiffness method, yielding the overall dynamic stiffness matrix and frequency characteristic equations. The specific modeling process is as follows:

[0011] The double-sling system consists of two parallel slings and a vibration damper between them. The vibration damper is located between the two slings and divides each sling into a lower segment and an upper segment. (Regarding the first...) root sling ( ) cable segment ( Indicates the lower cable segment. (Representing the upper cable segment), establish the differential equation of free vibration motion in the plane:

[0012] (1)

[0013] in For the first root suspension cable Each cable segment is located at... place, time The in-plane lateral displacement, For the first The bending stiffness of the sling. For the first The tension of the sling. For the first Mass per unit length of the sling.

[0014] Using the method of separation of variables, let ,in, For the mode shape function of the cable segment, , Let be the angular frequency of the vibration of the double-sling structure. Substituting into equation (1), we get:

[0015] (2)

[0016] make , , Transformed into:

[0017] (3)

[0018] in, This is the ratio of axial force to bending stiffness of the sling. .

[0019] Equation (3) is a fourth-order homogeneous differential equation with constant coefficients. Its general solution consists of hyperbolic and trigonometric function terms, which can be expressed as a combination of four linearly independent fundamental solutions, containing four undetermined integral constants. In this model, the vibration damper is considered a massless rigid body, and its own mass and moment of inertia are not considered. The undetermined constants of the general solution of each cable segment are determined by the following three sets of conditions: (a) boundary conditions of zero displacement and zero rotation at both ends of the cable; (b) displacement compatibility condition of equal lateral displacement of the two cables at the vibration damper position; (c) nodal force equilibrium condition of zero resultant shear force and bending moment of the cable segments on both sides of the vibration damper. All constraints are organized into a homogeneous linear equation system about the undetermined constants. The coefficient matrix of this equation system is the overall dynamic stiffness matrix. .in, Given a set of known structural parameters, the cable force identification mode includes... The cable-bending stiffness joint identification mode includes ; The parameter vector to be identified. is the structural modal angular frequency.

[0020] To unify different recognition modes, this invention defines the parameter vector to be recognized as follows: Depending on the target being identified, the parameter vector has the following two forms: (1) The cable force identification mode is It is suitable for working conditions where the bending stiffness of the sling can be accurately determined. The cable forces of the two slings are respectively; (2) The cable force-bending stiffness joint identification mode is as follows: This is suitable for situations where the two suspension cables have identical structural parameters and similar stress. The equivalent cable force in a double-sling system represents the equivalent cable force of a single sling. It represents the equivalent bending stiffness of a double-sling system, representing the equivalent bending stiffness of a single sling, and reflects the overall bending dynamic characteristics of the system.

[0021] The condition for a double-sling system to have a non-zero vibration solution is that the homogeneous equations have a non-trivial solution, i.e.:

[0022] (4)

[0023] Equation (4) is the frequency characteristic equation of the double-suspension system, and establishes the mapping relationship between the parameter vector to be identified and the structural modal frequencies.

[0024] Step 3: Calculate the frequency response function and extract the frequency trajectories of each order.

[0025] For each measured modal frequency Within the parameter space formed by the parameter vectors to be identified, a corresponding frequency response function is constructed based on the frequency characteristic equation. A continuous set of peak values ​​satisfying the constraints of the frequency characteristic equation is extracted, and the curve connecting the continuous set of peak values ​​is defined as the frequency trajectory corresponding to the measured modal frequency of that order. The specific steps are as follows:

[0026] First, determine the parameters to be identified. The search scope, of which The search scope is , The search scope is .

[0027] For cable force parameters The cable tension is primarily determined by the structure's self-weight, vehicle load, temperature effects, construction errors, and operational status. Since the cable tension in actual service conditions may vary in both directions relative to the design cable tension, the design cable tension or previously tested cable tension is usually used as the reference. As a reference value, the search range is determined based on the variation range of cable forces in the engineering structure. Preferably, the search range is set to... .

[0028] For bending stiffness parameters Its moment of inertia is determined by both the elastic modulus and the moment of inertia of the cable material. For a cable described using an equivalent circular cross section, its moment of inertia can be expressed as: ,in, This refers to the equivalent cross-sectional diameter of the sling. In actual engineering service, due to factors such as material performance degradation, wire damage, corrosion, and changes in connection conditions, the effective bending stiffness of the sling is usually reduced to varying degrees compared to the theoretically calculated value. Therefore, when identifying actual operating conditions, the theoretically calculated bending stiffness should be used. When used as a reference value, the search range for the bending stiffness parameter is determined based on the structural service condition, and preferentially covers the parameter range below the theoretically calculated value, i.e. It should be noted that the search range for bending stiffness can be adjusted according to the verification conditions to cover the region of parameters to be analyzed. When the structure lacks reliable prior parameters or there is a special service environment, the search range can be appropriately expanded according to the structure type, the degree of material property degradation, and engineering experience.

[0029] Then, the parameter space is discretized. Let the parameters be... , Divided into and For each discrete node, the corresponding discrete grid step size is as follows:

[0030] , (5)

[0031] in, , These represent the number of discrete nodes in the two parameter directions, respectively. The number of discrete nodes is determined based on a combination of recognition accuracy requirements and computational efficiency. A smaller discrete grid step size can improve the parameter space search resolution and parameter recognition accuracy, but it will increase the computational load. When the grid step size is too large, it may lead to the omission of peak regions or trajectory position shifts, thereby reducing parameter recognition accuracy. Therefore, the grid division should be determined comprehensively based on target recognition accuracy and computational efficiency.

[0032] Finally, a peak response function is established, and the frequency trajectory formed by the set of continuous peaks is extracted. For each grid point... Convert the measured frequency to the angular frequency. Substitute the dynamic stiffness matrix into the frequency equation residual function:

[0033] (6)

[0034] when When the frequency equation is exactly satisfied Since the determinant of the dynamic stiffness matrix changes drastically near its zero value, directly searching for the zero-value region of the residuals is highly sensitive to deviations and numerical errors. Therefore, the peak response function is defined as:

[0035] (7)

[0036] The closer the residual is to 0, the larger the value of the frequency response function becomes, forming a local peak; after traversing all grid points, the first value is obtained in the parameter space formed by the parameter vectors to be identified. Response surface of first frequency .

[0037] Peak points are extracted using a local maximum search algorithm. Connecting all consecutive peak points forms a continuous curve, thus obtaining the frequency trajectory corresponding to that frequency order. This trajectory represents a continuous set of parameters that potentially satisfy the frequency equation constraints within the parameter space formed by the parameter vectors to be identified, representing a certain order of measured modal frequency. The set of frequency trajectories corresponding to all measured modal frequencies is as follows: Different frequency trajectories together constrain the position of the true parameters.

[0038] Step 4: Find the intersection points between trajectories of different frequencies to obtain the set of intersection points.

[0039] The specific steps are as follows:

[0040] First, construct the trajectory difference function. For the... Rank and first The frequency trajectories of each order are represented as curves on the parameter plane formed by the parameter vectors to be identified. and Suppose that both trajectories can be represented as .in, The first coordinate in the corresponding coordinate plane, The second coordinate corresponding to the coordinate plane. Within the common parameter range of the two trajectories, the parameters... Uniform discretization yields a discrete point set. The trajectories will be... and The second parameter coordinates corresponding to each discrete point are represented as follows: and Construct the difference function between the two trajectories:

[0041] (8)

[0042] Next, the intersection points between each trajectory are determined. Along the parameters... The process iterates through all discrete points, detecting changes in the sign of the difference function. When adjacent discrete points exist... and satisfy At this point, it is determined whether the two frequency trajectories intersect within the interval. For intervals with sign changes, linear interpolation, bisection method, or other numerical interpolation methods are used to solve for the intersection point coordinates. ,in, To meet The interpolation solution, .

[0043] Finally, a set of intersection points is formed. For all... Repeat the above operation for frequency trajectory combinations. Each trajectory pair may generate zero, one, or more intersections. Collect all valid intersections to form an intersection set. Ideally, when there are no measurement or model errors, all trajectory intersections coincide at the same location, which is the true value of the parameter to be identified. However, when affected by measurement errors, environmental noise, and model errors, the trajectory intersections will cluster around the true parameter, requiring the intersection density analysis method in step five to determine the identification result.

[0044] Step 5: Determine the identification results using the intersection density analysis method.

[0045] With the set of intersections Based on this, the intersection density analysis method is used to determine the final identification result. The intersection density function is defined as follows:

[0046] (9)

[0047] Where M is the total number of intersection points in the set of trajectory intersection points. Set of intersection points The Middle The coordinates of the intersection points For the search radius, For kernel function, This represents the Euclidean distance in the plane containing the parameter to be identified.

[0048] The search radius Based on a comprehensive analysis of the measured frequency signal-to-noise ratio and the dispersion of the intersection points, the following parameters are determined within the normalized parameter space: The kernel value is set to 0.3 to 0.5 times the standard deviation of the Euclidean distance between the intersection points. A smaller value is used when the measured frequency signal-to-noise ratio is high to improve the positioning accuracy of the intersection area; a larger value is used when the signal-to-noise ratio is low to reduce the impact of test errors on trajectory dispersion. The kernel function can be any one of the Gaussian kernel function, the Epanechnikov kernel function, or the uniform kernel function. Preferably, a Gaussian kernel function is used.

[0049] Within the parameter space formed by the parameter vectors to be identified, the intersection density function value is calculated by traversing it according to a preset step size. The optimal identification parameters satisfy: The parameter corresponding to the global maximum value of the intersection density function is taken as the final recognition result.

[0050] When frequency trajectories cannot strictly intersect due to test errors, model errors, or environmental factors, the intersection density function can reflect the degree of clustering of the intersection points, and its peak position corresponds to the optimal estimate of the parameter to be identified. When the frequency trajectories strictly or approximately strictly intersect at the same point, the intersection density function forms a unique peak near the common intersection point.

[0051] The method of this invention employs a non-iterative identification approach based on frequency trajectory intersection analysis. It establishes the frequency response function corresponding to the measured modal frequencies by performing deterministic traversal calculations on each combination of parameters to be identified, and determines the position of the parameters to be identified based on the intersection relationships of parameter constraint sets of different orders. When the parameter constraint set cannot form a unique intersection point due to test errors or model errors, the optimal identification result is determined through intersection density analysis.

[0052] This invention has the following advantages: First, it redefines parameter identification from an optimization problem to a trajectory intersection problem, eliminating the need to construct an optimization objective function or iteratively seek optimization, thus overcoming the shortcomings of traditional iterative methods that rely on initial values ​​and are prone to getting trapped in local optima. Second, the intersection density analysis mechanism handles multi-source engineering errors through kernel density estimation, resulting in robust identification results. Third, it supports two modes: separate identification of cable force and joint identification of cable force and bending stiffness, applicable to various working conditions such as equal cable force and non-equal cable force. Fourth, the entire identification process of this invention adopts deterministic parameter space traversal and trajectory intersection analysis, eliminating the need for convergence judgment and iterative iteration, resulting in a simple calculation process, high degree of automation, and good repeatability of the identification results. Attached Figure Description

[0053] Figure 1 Flowchart of the parameter identification method for the dual-sling system of the present invention

[0054] Figure 2 Schematic diagram of the dynamic model of the double-sling system

[0055] Figure 3 3D schematic diagram of frequency trajectory of 20 m double suspension cable numerical model

[0056] Figure 4 20 m double suspension cable force identification frequency trajectory intersection diagram

[0057] Figure 5 Intersection diagram of frequency trajectories for joint identification of cable force and bending stiffness of 60.1 m double suspension cables (intersection density method) Detailed Implementation

[0058] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0059] Example 1: Cable Force Identification

[0060] A 20 m double-cable numerical model with equal cable force and equal bending stiffness was adopted as the benchmark model, with basic parameters E1=E2=1.95×10¹¹Pa and I1=I2=2.26×10⁻ 6 m 4 , m1=m2=45.7kg / m, H1=H2=1.5×10 6 N, L1=L2=20m, u=0.5.

[0061] Using the frequency characteristic equation of the double-suspension system established in this invention, the first eight in-plane modal frequencies of the double-suspension system were calculated using the bisection method, which are 4.8041 Hz, 9.7168 Hz, 14.7729 Hz, 24.2404 Hz, 25.7731 Hz, 32.2577 Hz, 38.3609 Hz, and 46.2839 Hz, respectively.

[0062] Step 1 Implementation: Select the first 8 modal frequencies mentioned above as the measured input frequencies; known structural parameters include cable length, mass per unit length, vibration damper position parameters, elastic modulus, and moment of inertia of the cross section.

[0063] Step Two: Establish a dynamic model of the double-cable system based on the structural parameters for each working condition. Establish the in-plane free vibration control equations for each segment of the two cables, and construct the overall dynamic stiffness matrix using the dynamic stiffness method. A parameter vector to be identified is established by combining the boundary conditions of the suspension cable, the displacement compatibility conditions at the vibration damper, and the nodal force balance conditions. The correspondence between the frequency and the structural modal frequencies. This is established through the frequency characteristic equation. Construct a joint identification model.

[0064] Step 3: Calculate the frequency response function and extract the frequency trajectories of each order. Set the parameter search range to H1, H2 ∈ [1.3 × 10⁻⁶]. 6 1.7×10 6 N is divided into 401 discrete nodes in each parameter direction, with ΔH1=ΔH2=1×10 3 Discretization is performed using N as the step size. The frequency response function is calculated for each measured modal frequency. . Figure 3 The three-dimensional distribution plot of the frequency response function corresponding to the first-order frequency is given. The peak points are extracted using the neighborhood local maxima algorithm, and all consecutive peak points are connected to form a continuous curve, thus obtaining the frequency trajectory corresponding to this order frequency. .

[0065] Step four implementation: Using the difference function method combined with linear interpolation, the pairwise intersections of all 8th order frequency trajectories are calculated to form a total of For each combination of trajectories, iterate through all combinations to obtain the set of intersection points Ω. Figure 4 It can be seen that under the 20 m constant cable force condition, the intersection points of each pair of trajectories are concentrated at H1=H2=1.5×10. 6 At point N, it is indicated that frequency trajectory intersection analysis can effectively constrain the location of the parameter to be identified.

[0066] Step 5: The intersection density analysis method is used to determine the final identification result. A Gaussian kernel function is used to construct the intersection density function, and the search radius h is taken as 0.3 times the standard deviation of the Euclidean distance between the intersection points. Under this condition, the trajectories have a unique common intersection point, and the final identified cable forces of the two suspension cables are H1=H2=1.5×10⁻⁶. 6 N, with good recognition accuracy.

[0067] Example 2: Joint Identification of Cable Stress and Bending Stiffness

[0068] An engineering verification was conducted using a double-span suspension bridge over the Yellow River in Lanzhou as an example. Its basic parameters are: L1=L2=60.1m, E1=E2=1.95×10¹¹Pa, I1=I2=7.73×10⁻⁻⁻⁴. 7 m 4 m1=m2=11.10kg / m, design cable force H1=H2=1.15×10 5 N, vibration damper position parameter u=0.5

[0069] Step 1 Implementation: Accelerometers were used to collect the vibration response of the suspension cables on-site. The first five in-plane measured modal frequencies of the dual-sling system were obtained through environmental excitation and spectrum analysis, namely: 0.889 Hz, 2.701 Hz, 3.632 Hz, 4.603 Hz, and 5.642 Hz. These first five modal frequencies were selected as the measured input frequencies. Known structural parameters included cable length, mass per unit length, and vibration damper position parameters.

[0070] Step Two: Establish a dynamic model of the double-cable system based on the structural parameters of the actual bridge. Establish the in-plane free vibration control equations for each segment of the two cables, and construct the overall dynamic stiffness matrix using the dynamic stiffness method. A parameter vector to be identified is established by combining the boundary conditions of the suspension cable, the displacement compatibility conditions at the vibration damper, and the nodal force balance conditions. The correspondence between the frequency and the structural modal frequencies. This is established through the frequency characteristic equation. Construct a joint identification model.

[0071] Step 3: Calculate the frequency response function and extract the frequency trajectories of each order. Set the parameter search range to H∈[0.92×10]. 5 1.38×10 5 N, EI∈[0.904×105 1.51×10 5 N·m 2 Each parameter direction is divided into 401 discrete nodes, with ΔH = 1.15 × 10 2 N, ΔEI = 1.515 × 10 2 N·m 2 Discretize the frequency response function for each measured modal frequency, using a set step size. The peak points are extracted using the neighborhood local maxima algorithm. Connecting all consecutive peak points forms a continuous curve, thus obtaining the frequency trajectory corresponding to that frequency order. .

[0072] Step four implementation: Using the difference function method combined with linear interpolation, the pairwise intersections of all 5th order frequency trajectories are calculated to form a total of For each trajectory combination, the set of trajectory intersection points Ω is obtained by traversing all trajectory combinations. Due to factors such as degradation of the actual bridge material properties, differences in boundary conditions, test errors, and model simplification, the trajectories of each frequency order do not form a unique common intersection point, but instead exhibit clustered distribution characteristics near the actual parameters.

[0073] Step 5: The final identification result is determined using intersection density analysis. A Gaussian kernel function is used to construct the intersection density function, with the search radius h set to 0.5 times the standard deviation of the Euclidean distance between the intersection points. The parameter corresponding to the global maximum value of the intersection density function is used as the final identification result. The peak trajectory corresponding to the measured modal frequency of the 4th order does not form a stable intersection region with the peak trajectories of other orders. The dispersion of this order trajectory is significantly higher than that of the other orders, presumably due to interference from test noise and modal identification bias. Therefore, it is judged as an abnormal trajectory and removed before further identification calculations. The remaining peak trajectories are as follows: Figure 5 As shown. Based on the residual peak trajectory, the cable force-bending stiffness joint identification was performed, and the equivalent cable force of the cable was found to be 1.1799 × 10⁻⁶. 5 N, relative to the design cable force error of 2.6%, and the equivalent bending stiffness of the cable is obtained as 1.233x10. 5 The bending stiffness is reduced by approximately 22.3% (N·m) compared to the theoretically calculated value, which is consistent with the engineering principle that the effective bending stiffness of slings in service is generally reduced. This result can effectively compensate for model parameter mismatch and improve the accuracy and stability of cable force identification.

Claims

1. A method for parameter identification of a dual-sling system based on frequency trajectory intersection characteristics, characterized in that, Includes the following steps: Step 1: Obtain the structural parameters and measured modal frequencies of the double-sling system. The structural parameters include the length, mass per unit length, damper position parameters, elastic modulus, initial moment of inertia, and initial cable force of each sling; the elastic modulus... The initial moment of inertia is a known structural parameter in the cable force identification mode and a reference value in the cable force-bending stiffness joint identification mode; the initial cable force is the reference benchmark value for the parameter search interval; the measured modal frequencies are the first N measured modal frequencies participating in parameter identification. Step 2: Establish a dynamic model of the double-sling system, construct the overall dynamic stiffness matrix using the dynamic stiffness method, and establish the correspondence between the parameter vectors in the parameter space to be identified and the structural modal frequencies based on the frequency characteristic equation; Step 3: For each measured modal frequency, construct the corresponding frequency response function in the parameter space to be identified, extract the continuous peak region that satisfies the frequency equation constraint, and form the corresponding frequency trajectory; Step 4: Find the intersection of each pair of frequency trajectories corresponding to different frequencies, determine the intersection points, and traverse all combinations of frequency trajectories to obtain the set of intersection points. Step 5: Construct an intersection density function based on the set of trajectory intersection points. When all trajectories have a unique common intersection point, use the parameter corresponding to the common intersection point as the recognition result; when there is no unique common intersection point, use the parameter to be identified corresponding to the global maximum value of the intersection density function as the final recognition result.

2. The method according to claim 1, characterized in that, The dual-sling system consists of two parallel slings and a rigid damping frame. The damping frame divides each sling into a lower sling segment and an upper sling segment along the length of the sling. The position parameter of the damping frame is defined as the ratio of the length along the sling from the damping frame to the lower end of the sling to the total length of the sling.

3. The method according to claim 1, characterized in that, The parameters to be identified in the parameter space adopt a unified parameter vector. It is indicated that, depending on the target being identified, the parameter vector takes any of the following forms: (a) under cable force identification mode ,in (a) The cable forces of the two slings; (b) Cable force-bending stiffness joint identification mode Where H is the equivalent cable force of the double cable system, and EI is the equivalent bending stiffness of the double cable system.

4. The method according to claim 3, characterized in that, The cable force-bending stiffness joint identification mode is suitable for working conditions in actual bridges where bending stiffness parameters are difficult to obtain accurately. By synchronously identifying equivalent bending stiffness parameters, the influence of model parameter mismatch on cable force identification results is reduced.

5. The method according to claim 1, characterized in that, In step two, the dynamic model of the double cable system is established based on the control equation of free vibration in the cable plane. The overall dynamic stiffness matrix is ​​assembled by combining the consolidation boundary conditions at both ends of the cable, the displacement coordination conditions at the vibration damping frame, and the nodal force balance conditions.

6. The method according to claim 1, characterized in that, In step two, the overall dynamic stiffness matrix is ​​established based on the dynamic model of the double-sling system. ,in Given a set of known structural parameters, The structural modal circular frequency; the overall dynamic stiffness matrix is ​​assembled from the dynamic equations of each cable segment, boundary conditions, displacement compatibility conditions at the vibration damping frame, and nodal force balance conditions, and is based on the frequency characteristic equation. Establish the correspondence between the parameter vector to be identified and the structural modal frequencies.

7. The method according to claim 1, characterized in that, In step three, the frequency response function is constructed based on the residual of the frequency equation. ,in For the first The residual function of the frequency equation corresponding to the measured modal frequencies of the first order. , For the first The measured modal frequencies are obtained; the parameter combinations that satisfy the constraints of the frequency characteristic equation form a continuous set of peak values, and the corresponding frequency trajectory is obtained by extracting the peak set.

8. The method according to claim 1, characterized in that, In step four, the intersection point of the frequency trajectory is determined by the trajectory intersection algorithm; the trajectory intersection algorithm adopts one or a combination of the difference function method, interpolation method, and geometric distance method.

9. The method according to claim 1, characterized in that, In step five, the intersection density function is constructed based on kernel density estimation and is defined as follows: Where M is the total number of intersection points in the set of trajectory intersection points. The first in the set of intersection points The coordinates of the intersection points For the search radius, For kernel function, The distance is the Euclidean distance in the plane of the parameter to be identified; the kernel function is any one of the Gaussian kernel function, Epanechnikov kernel function, or uniform kernel function.

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