Cable force inversion identification method based on linear least square and full-field vibration mode information
By acquiring full-field vibration mode information of the cable using a monocular camera and image processing technology, and combining linear least squares method and optimization equations, the problem of insufficient cable force identification accuracy of frequency method in short cable, large sag and complex boundary scenarios is solved, and high-precision cable force inversion is achieved.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- CHONGQING UNIVERSITY OF SCIENCE AND TECHNOLOGY
- Filing Date
- 2026-03-25
- Publication Date
- 2026-06-30
AI Technical Summary
In existing technologies, the frequency method is not accurate enough for cable force identification in short cables, large sag and complex boundary scenarios, and traditional acceleration sensors lack vibration mode information, making it difficult to meet the requirements of high-precision testing.
A monocular camera was used to capture video of cable vibration. Phase information was extracted by Gabor convolution, and natural frequencies were identified by combining power spectrum peak picking and parabolic interpolation. Principal component analysis and CP unmixing matrix were used to obtain full-field mode shape information, the spatial partial derivatives of the mode shape function were estimated, optimization equations were constructed, and the cable force was solved by linear least squares method.
It achieves high-precision cable force identification in scenarios with short cables, large sag, and complex boundaries, overcomes the accuracy bottleneck of traditional frequency methods, and provides high-precision time-domain inversion of cable force.
Smart Images
Figure CN122306287A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of cable force testing technology, and in particular to a cable force inversion and identification method based on linear least squares and full-field mode information. Background Technology
[0002] Cables are the core load-bearing components of long-span bridges, and accurate measurement of cable forces is crucial for understanding the true stress state of the bridge structure and ensuring its service safety and long-term performance stability. Existing cable force measurement methods are mainly divided into two categories: frequency methods and optimization algorithms. Among them, the frequency method, with its mature principles, simple operation, strong field adaptability, and compatibility with long-term online monitoring, has become the most widely used cable force measurement technology in the engineering field. Its core is to establish an analytical mapping relationship between the cable's natural frequency and cable force; the test accuracy highly depends on the effectiveness of this mapping relationship. However, while the frequency method has high accuracy in identifying the cable forces of slender cables, its accuracy is limited for short cables, where bending stiffness is significantly affected.
[0003] Based on this, some scholars have applied optimization algorithms to the vibration equations or benchmark finite element models of cables to perform inversion identification of cable forces. Currently, the cable state data used for inversion identification typically consists of the cable's frequency and mode shape, which can be obtained by combining the cable's vibration response signal with modal analysis algorithms. Traditional vibration response signals are often tested using accelerometers. However, due to the large deployment conditions and wiring workload, the number of measurement points on a single cable is usually small, resulting in a significant lack of mode shape information. In addition, the inherent shortcomings, such as low testing efficiency, high-altitude operation risks, and the ease with which added mass can introduce systematic errors, make it difficult to meet the increasingly stringent ultra-high precision testing requirements.
[0004] In recent years, non-contact structural mode shape measurement technology based on image phase information has enabled continuous mode shape measurement across the entire length of cables, effectively compensating for the lack of mode shape information in traditional accelerometer measurements. However, how to establish a high-precision cable force inversion method based on this type of full-field mode shape measurement results, which can accurately account for the effects of bending stiffness and damping, and overcome the technical bottleneck of low accuracy of the frequency method in short cables, large sag, and complex boundary scenarios, remains an urgent technical problem to be solved. Summary of the Invention
[0005] The purpose of this invention is to provide a cable force inversion and identification method based on linear least squares and full-field mode information, which solves the problem of insufficient accuracy of existing frequency methods in cable force identification under short cable, large sag and complex boundary scenarios.
[0006] To achieve the above objectives, this invention provides a cable force inversion and identification method based on linear least squares and full-field mode shape information, comprising the following steps: Obtain the full-field vibration mode information and corresponding modal response of the cable; Based on the full-field vibration mode information, estimate the spatial partial derivatives of the mode function at different locations on the cable under the specified modal order; Select several representative measurement points and extract the spatial partial derivatives of the mode function of each representative measurement point at the specified modal order; Based on the vibration differential equation of the cable, an optimization equation for cable force inversion is constructed using the modal response and the spatial partial derivative. Substitute the data from multiple representative measuring points into the optimization equation to construct an overdetermined system of equations, and solve it using the linear least squares method to obtain the cable force.
[0007] This includes obtaining the full-field vibration mode information and corresponding modal response of the cable, specifically including: A monocular camera was used to capture video of cable vibration, and image information from each frame was extracted. Extracting phase information from an image using Gabor convolution; Based on the phase information, the natural frequencies of the cable are identified by a combination of power spectrum peak picking and parabolic interpolation. By combining principal component analysis with CP unmixing matrix, the mode shapes of the cable are identified and the modal responses of each order are extracted.
[0008] The method employed is power spectrum peak picking combined with parabolic interpolation to identify the natural frequencies of the cable, specifically including: The power spectrum calculation is based on frequency refinement of the full-segment signal of the enhanced filtered modal coordinates. By screening the effective peaks of free decaying vibrations to eliminate spurious peaks caused by noise, and using the parabolic interpolation formula to calculate the offset of the spectral peak position, the refined modal frequencies are obtained.
[0009] Specifically, a combination of principal component analysis and CP unmixing matrix was used to identify the mode shapes of the cable and extract the modal responses of each order, including: The high-dimensional displacement response is projected onto a low-dimensional principal component subspace. The orthogonal unmixing matrix is obtained by optimization in the principal component subspace using the CP algorithm. The mixed principal components are linearly transformed to obtain the decoupled modal coordinates. Finally, the physical modal basis is obtained by combining them.
[0010] Specifically, estimating the spatial partial derivatives of the mode functions at different locations on the cable under a specified modal order includes: The spatial partial derivatives of the mode shape function are estimated using the central difference formula, where the second and fourth spatial partial derivatives, representing the measurement point j, are calculated according to the following formulas:
[0011] In the formula, To represent the mode amplitude at measurement point j, Let (j-1) represent the mode amplitude of the first node (j-1) to the left of measuring point j. Let (j+1) represent the mode amplitude of the first node (j+1) to the right of measuring point j. Let (j+2) represent the mode amplitude of the second node to the right of measuring point j. Let represent the mode amplitude of the second node (j-2) to the left of the measuring point j, and h is the equidistant step size of the cable space discreteness.
[0012] Among them, several representative measurement points were selected, specifically including: The representative measuring points are evenly distributed in the region from L / 6 to 5L / 6 along the length of the cable, where L is the length of the cable, to avoid the boundary effect areas at both ends.
[0013] Specifically, based on the vibration differential equation of the cable, an optimization equation for cable force inversion is constructed using the modal response and the spatial partial derivatives, including: The optimization equation is shown below:
[0014] In the formula, , EI is the cross-sectional stiffness of the cable, T is the cable force, m is the mass per unit length of the cable, c is the viscous damping coefficient per unit length of the cable, A(t) is the fourth-order differential displacement term at the two measuring points, B(t) is the second-order differential displacement term at the two measuring points, G(t) is the acceleration term at the two measuring points, and D(t) is the velocity term at the two measuring points. For modal coordinates, To represent the mode amplitude of measuring point 2, Let represent the mode amplitude of measuring point 1.
[0015] The construction of an overdetermined system of equations specifically includes: Multiple representative measurement points are paired to obtain multiple sets of measurement point combinations. The second-order spatial partial derivatives, fourth-order spatial partial derivatives, and modal response data of each set of measurement points at the specified modal order are substituted into the optimization equation to form an overdetermined set of equations.
[0016] The solution is obtained using the linear least squares method, specifically including: By setting the mass per unit length of the cable to a known measured value, the overdetermined equations are solved using linear least squares, yielding the optimal and unique solutions for cable force, bending stiffness, and damping coefficient.
[0017] This invention presents a cable force inversion and identification method based on linear least squares and full-field mode shape information. First, it captures cable vibration video using a monocular camera and extracts phase information using Gabor convolution. Power spectrum peak picking and parabolic interpolation are then used to identify natural frequencies. Principal component analysis and CP unmixing matrix combination are employed to obtain the full-field mode shape information and corresponding modal responses of the cable. Next, the central difference formula is used to estimate the second and fourth spatial partial derivatives at representative measurement points under different modal orders. An optimization equation incorporating cable force, bending stiffness, and damping coefficients is constructed based on the cable vibration differential equation. Finally, by pairwise combining multiple representative measurement points and substituting the data into the optimization equation to form an overdetermined set of equations, the linear least squares method is used to solve for the optimal and unique solutions for cable force, bending stiffness, and damping coefficients. This method overcomes the insufficient accuracy of traditional frequency methods in identifying cable force in short cables, large sag, and complex boundary scenarios, achieving high-precision time-domain inversion of cable force. Attached Figure Description
[0018] To more clearly illustrate the technical solutions in the embodiments of this application or the prior art, the accompanying drawings used in the description of the embodiments or the prior art will be briefly introduced below.
[0019] Figure 1 This is a schematic diagram of the representative measuring point of the present invention.
[0020] Figure 2 This is a flowchart of the cable force inversion identification method based on linear least squares and full-field mode information of the present invention.
[0021] Figure 3 This is a diagram showing the first mode shape and measurement point arrangement of Embodiment 1 of the present invention.
[0022] Figure 4 This is a comparison chart of the inverted values and the true values of Embodiment 1 of the present invention.
[0023] Figure 5 This is a flowchart of the cable force inversion identification method based on linear least squares and full-field vibration mode information of the present invention. Detailed Implementation
[0024] The embodiments of the present invention are described in detail below. Examples of the embodiments are shown in the accompanying drawings. The embodiments described below with reference to the accompanying drawings are exemplary and intended to explain the present invention, but should not be construed as limiting the present invention.
[0025] Please refer to Figures 1 to 5 This invention provides a cable force inversion and identification method based on linear least squares and full-field mode shape information, comprising the following steps: S101: Obtain the full-field vibration mode information and corresponding modal response of the cable; Specifically, firstly, a monocular camera is used to capture video of the cable vibration and extract image information from each frame. The frame rate of a general camera can be up to 60 frames per second. In order to better identify the vibration mode, the frame rate is usually set to 60 frames per second. Then, phase information containing local vertical structure response, pixel relative position, specific spatial frequency texture and inter-frame subpixel displacement encoding is extracted from the image through Gabor convolution; Next, power spectrum peak picking combined with parabolic interpolation was used to identify the natural frequencies of the cable. Power spectrum calculation was performed based on frequency refinement of the entire modal coordinate signal after enhancement filtering. This was achieved by setting the minimum peak height to 1% of the maximum absolute value of the signal and the minimum peak protrusion to 0.01, thus filtering out effective peaks of free decaying vibrations and eliminating spurious peaks caused by noise. The parabolic interpolation process was as follows: after obtaining the power spectral density curve using the Pwelch method, the index of the maximum spectral peak was first located. If this index met the boundary conditions, five adjacent power spectral density values were selected. Using the three middle points (the peak point and one point to its left and right), the offset of the peak position was calculated using the parabolic interpolation formula. Finally, the offset was multiplied by the frequency resolution and superimposed onto the initial frequency corresponding to the peak point to obtain the refined modal frequencies.
[0026] Then, PCA (Principal Component Analysis) principal component vectors and CP unmixing matrices are combined to identify the mode shapes of the cable at different orders. This allows for the extraction of mode shape values at any point on the cable under different modal orders, as well as the extraction of the modal responses of the cable at different orders. Modal response refers to the projected component of the total response of the structure under excitation along a certain natural mode shape direction, exhibiting a single-degree-of-freedom vibration at that natural frequency and spatially distributed according to the corresponding mode shape.
[0027] The displacement response of a linear structure satisfies the modal expansion formula. ,in The matrix represents the true modal shape, and q(t) represents the coordinates of a completely decoupled single-degree-of-freedom mode. In the specific identification process, the high-dimensional (12-dimensional) displacement response is projected onto a low-dimensional (r-dimensional) principal component subspace to eliminate noise and redundancy, ensuring that the true modal information is completely contained within this subspace. Since PCA only achieves spatial decorrelation, the resulting principal component sequence... The signal remains a mixture of multiple modes, with coupling between components. Therefore, within the r-dimensional principal component subspace, the orthogonal unmixing matrix W is obtained through optimization using the CP algorithm. A linear transformation is then performed on the mixed principal components to obtain the decoupled modal coordinates. At this point, each component of q is an independent single-frequency, single-degree-of-freedom response.
[0028] Substituting the above transformation into the response expression, we get Where Ur is the principal component basis matrix obtained by PCA truncation, this expression perfectly matches the principle of modal superposition. The combined result is... It is a physical modal basis, whose column vectors satisfy mass orthogonality, ensuring that the structural response under this basis can be completely decoupled into independent modal coordinates of each order, ultimately achieving modal decoupling.
[0029] S102: Based on the full-field vibration mode information, estimate the spatial partial derivatives of the mode function at different positions on the cable under the specified modal order; Specifically, for different modal orders, the second and fourth spatial partial derivatives of the mode shape function at any point on the cable are estimated using the central difference formula. The calculation formula is as follows:
[0030] In the formula, To represent the mode amplitude at measurement point j, Let (j-1) represent the mode amplitude of the first node (j-1) to the left of measuring point j. Let (j+1) represent the mode amplitude of the first node (j+1) to the right of measuring point j. Let (j+2) represent the mode amplitude of the second node to the right of measuring point j. Let represent the mode amplitude of the second node (j-2) to the left of the measuring point j, and h is the equidistant step size of the cable space discreteness.
[0031] The spatial distance h is determined as follows: first, calculate the average distance between adjacent representative measuring points, and then set h to one-fifth of the average distance. This is to ensure that there is enough space around each representative measuring point to arrange measuring points of ±2h and ±h, and to avoid overlap.
[0032] S103: Select several representative measurement points and extract the spatial partial derivatives of the mode shape function of each representative measurement point at the specified modal order; Specifically, representative measurement points are selected for inversion identification, and the second and fourth spatial partial derivatives of the mode functions at specified modal orders of these representative measurement points are extracted. In practice, s representative measurement points are evenly distributed on each mode of the cable. The more measurement points, the more accurate the measurement. Generally, 5 points are sufficient. These s measurement points are set in the L / 6 to 5L / 6 region of the cable. The L / 6 to 5L / 6 region is selected to avoid the boundary effect region at both ends.
[0033] S104: Based on the vibration differential equation of the cable, construct an optimization equation for cable force inversion using the modal response and the spatial partial derivative; Specifically, based on the second and fourth spatial partial derivatives and modal coordinates of these representative measurement points at specified modal orders, optimization equations for cable force inversion are constructed. These optimization equations are derived from the partial differential equations of cable vibration, and their specific expressions are as follows:
[0034] In the formula, , EI is the cross-sectional stiffness of the cable, T is the cable force, m is the mass per unit length of the cable, c is the viscous damping coefficient per unit length of the cable, A(t) is the fourth-order differential displacement term at the two measuring points, B(t) is the second-order differential displacement term at the two measuring points, G(t) is the acceleration term at the two measuring points, and D(t) is the velocity term at the two measuring points. For modal coordinates, To represent the mode amplitude of measuring point 2, Let represent the mode amplitude of measuring point 1.
[0035] S105: Substitute the data from multiple representative measuring points into the optimization equation to construct an overdetermined system of equations, and solve it using the linear least squares method to obtain the cable force.
[0036] Specifically, the s representative measurement points are paired (each pair contains two measurement points and their corresponding modal coordinates and mode amplitudes), resulting in a total of The measurement point group is a combination, where k is the nth representative measurement point. The second and fourth spatial partial derivatives and modal response data of each measurement point at the specified modal order are substituted into the above optimization equation, and the cable force T is inverted and identified by combining linear least squares.
[0037] Since simultaneously inverting EI, cable force T, mass per unit length m, and damping coefficient c will result in infinitely many solutions, and the mass per unit length of the cable is relatively easy to obtain, the value of m is taken as the actual measured value. Based on this, the overdetermined equation system composed of multiple sets of measurement point data is solved using the linear least squares method, minimizing the sum of squared errors between the observed data and the theoretical model's predicted values, thus obtaining the optimal and unique solution for bending stiffness EI, cable force T, and damping coefficient c. Here, EI, T, and C are obtained through... Inverse this formula, where m is known, and multiple measurement points are combined to form a system of equations to solve the equation.
[0038] It avoids the strong dependence of traditional frequency methods on boundary conditions and cable length parameters, and can fully account for the effects of damping and inertia terms. Combined with the structural mode shape extraction method based on image phase, it can achieve high-precision time-domain inversion of cable force, with better theoretical rigor and engineering applicability.
[0039] Example 1: The effectiveness of this invention was demonstrated through numerical simulation. Parameters of a real bridge short cable were selected: length 12.8m, diameter 0.24m, weight per unit length 97.5kg / m, and elastic modulus E of 2.05*10⁻⁶. 5 MPa, designed as a dead load cable force of 4200KN for the bridge. The first-order mode shape function of the cable was obtained through finite element analysis (specifically, the parameters of the actual short cables of the bridge mentioned above), and the cable force was inverted and identified. Simultaneously, to simulate the testing errors present in actual mode shape tests, 5% noise was added to the theoretical mode shape values calculated by the finite element analysis, introducing a certain amplitude variation to the mode shape.
[0040] Five representative measuring points were selected. These points are located in the L / 6 to 5L / 6 region of the first-order vibration mode. Two points were also placed on either side of each of these five representative measuring points to calculate the difference values of the second- and fourth-order vibration mode functions at those points. In engineering, the weight per unit length of the cable is relatively easy to obtain; therefore, in this embodiment, the value of m is set to a fixed value to allow for the use of linear least squares to find the optimal and unique solution for the value of T.
[0041] Depend on Figures 3 to 4 As shown in Table 1, the inversion T-value error of the method proposed in this invention is 0.7073%.
[0042]
[0043] Table 1 Summary of Inversion Results Figure 3 As can be seen in this embodiment, a total of 5 representative measurement points are set, and two measurement points are set on both sides of each representative measurement point so that the representative measurement points can perform second-order and fourth-order differences. Figure 4 As can be seen from Table 1, the error between the inverted T value and the true value is 0.7073%, the error of EI is 0.2258%, and the error of c is 0.1313%.
[0044] To compare the results with those obtained using the frequency method. Based on the frequency method formula... Cable force identification was performed. The frequency f was taken as 9.8569 Hz, obtained from the first mode shape calculated by numerical simulation; the m value was taken as 97.5 kg / m; and the l value was taken as 12.8 m. The calculated cable force was 6208.193 kN. A comparison was made between the frequency method calculation and the designed dead load cable force of the completed bridge (4200 kN). It can be seen that the error of the frequency method is... The method proposed in this invention has an inversion T-value error of 0.7073%, which effectively solves the accuracy bottleneck of the frequency method in scenarios with short cables, large sag, and complex boundaries.
[0045] The above-disclosed embodiments are merely one or more preferred embodiments of this application and should not be construed as limiting the scope of this application. Those skilled in the art can understand that all or part of the processes for implementing the above embodiments and equivalent changes made in accordance with the claims of this application still fall within the scope of this application.
Claims
1. A cable force inversion and identification method based on linear least squares and full-field mode shape information, characterized in that, Includes the following steps: Obtain the full-field vibration mode information and corresponding modal response of the cable; Based on the full-field vibration mode information, estimate the spatial partial derivatives of the mode function at different locations on the cable under the specified modal order; Select several representative measurement points and extract the spatial partial derivatives of the mode function of each representative measurement point at the specified modal order; Based on the vibration differential equation of the cable, an optimization equation for cable force inversion is constructed using the modal response and the spatial partial derivative. Substitute the data from multiple representative measuring points into the optimization equation to construct an overdetermined system of equations, and solve it using the linear least squares method to obtain the cable force.
2. The cable force inversion and identification method based on linear least squares and full-field mode information as described in claim 1, characterized in that, Obtain the full-field vibration mode information and corresponding modal response of the cable, specifically including: A monocular camera was used to capture video of cable vibration, and image information from each frame was extracted. Extracting phase information from an image using Gabor convolution; Based on the phase information, the natural frequencies of the cable are identified by a combination of power spectrum peak picking and parabolic interpolation. By combining principal component analysis with CP unmixing matrix, the mode shapes of the cable are identified and the modal responses of each order are extracted.
3. The cable force inversion and identification method based on linear least squares and full-field mode information as described in claim 2, characterized in that, The natural frequencies of the cable are identified using a combination of power spectrum peak picking and parabolic interpolation, specifically including: The power spectrum calculation is based on frequency refinement of the full-segment signal of the enhanced filtered modal coordinates. By screening the effective peaks of free decaying vibrations to eliminate spurious peaks caused by noise, and using the parabolic interpolation formula to calculate the offset of the spectral peak position, the refined modal frequencies are obtained.
4. The cable force inversion and identification method based on linear least squares and full-field mode information as described in claim 2, characterized in that, Using a combination of principal component analysis and CP unmixing matrix, the modal shapes of the cable are identified and the modal responses of each order are extracted, specifically including: The high-dimensional displacement response is projected onto a low-dimensional principal component subspace. The orthogonal unmixing matrix is obtained by optimization in the principal component subspace using the CP algorithm. The mixed principal components are linearly transformed to obtain the decoupled modal coordinates. Finally, the physical modal basis is obtained by combining them.
5. The cable force inversion and identification method based on linear least squares and full-field mode information as described in claim 1, characterized in that, Estimate the spatial partial derivatives of the mode functions at different locations on the cable under a specified modal order, specifically including: The spatial partial derivatives of the mode shape function are estimated using the central difference formula, where the second and fourth spatial partial derivatives, representing the measurement point j, are calculated according to the following formulas: ， In the formula, To represent the mode amplitude at measurement point j, Let (j-1) represent the mode amplitude of the first node (j-1) to the left of measuring point j. Let (j+1) represent the mode amplitude of the first node (j+1) to the right of measuring point j. Let (j+2) represent the mode amplitude of the second node to the right of measuring point j. Let represent the mode amplitude of the second node (j-2) to the left of the measuring point j, and h is the equidistant step size of the cable space discreteness.
6. The cable force inversion and identification method based on linear least squares and full-field mode information as described in claim 1, characterized in that, Several representative measurement points were selected, specifically including: The representative measuring points are evenly distributed in the region from L / 6 to 5L / 6 along the length of the cable, where L is the length of the cable, to avoid the boundary effect areas at both ends.
7. The cable force inversion and identification method based on linear least squares and full-field mode information as described in claim 1, characterized in that, Based on the vibration differential equation of the cable, an optimization equation for cable force inversion is constructed using the modal response and the spatial partial derivative, specifically including: The optimization equation is shown below: ； In the formula, , EI is the cross-sectional stiffness of the cable, T is the cable force, m is the mass per unit length of the cable, c is the viscous damping coefficient per unit length of the cable, A(t) is the fourth-order differential displacement term at the two measuring points, B(t) is the second-order differential displacement term at the two measuring points, G(t) is the acceleration term at the two measuring points, and D(t) is the velocity term at the two measuring points. For modal coordinates, To represent the mode amplitude of measuring point 2, Let represent the mode amplitude of measuring point 1.
8. The cable force inversion and identification method based on linear least squares and full-field mode information as described in claim 7, characterized in that, Constructing an overdetermined system of equations, specifically including: Multiple representative measurement points are paired to obtain multiple sets of measurement point combinations. The second-order spatial partial derivatives, fourth-order spatial partial derivatives, and modal response data of each set of measurement points at the specified modal order are substituted into the optimization equation to form an overdetermined set of equations.
9. The cable force inversion and identification method based on linear least squares and full-field mode information as described in claim 8, characterized in that, The solution is obtained using the linear least squares method, specifically including: By setting the mass per unit length of the cable to a known measured value, the overdetermined equations are solved using linear least squares, yielding the optimal and unique solutions for cable force, bending stiffness, and damping coefficient.