A robustness improvement method for structural deflection identification based on Lagrange multiplier method

Abstract

The present invention provides a method for improving the robustness of structural deflection identification based on the Lagrange multiplier method. Robustness is the stability of inversion. When the actual sensor data has large noise, this method can invert the structural deflection with high fidelity. The present invention includes: applying an impact load to the structure, collecting acceleration signals when it is in a free vibration state, performing Fourier transform on the acceleration signals to obtain a spectrum diagram, and identifying the natural frequency of the structure. Under the n-th order modal vibration, the structure fits the strain deflection coefficient matrix of this order based on the least squares method, and reduces the influence of the layout position difference of the strain sensor points and the quality of the pasting process on the robustness of the deflection identification result through the strain deflection coefficient matrix and strain vector quadratic construction method based on the Lagrange multiplier method. Under the composite excitation of the simple harmonic force of the first n-order natural frequencies, the structure fits a multi-order modal coupling coefficient matrix.

Description

Technical Field

[0001] The present invention relates to the field of methods for improving the robustness of structural deflection identification, and in particular to a method for improving the robustness of structural deflection identification based on a Lagrange multiplier method. Background Art

[0002] During operation, aircraft are subject to aerodynamic forces, gravity, and inertia, which can cause deformation and damage. Wing structural deformation is not only a key indicator of wing structural health, but also has a significant impact on the performance of wing-mounted equipment, such as onboard conformal antennas and weapons.

[0003] Structural deflection reconstruction methods based on strain information, such as the KO method, are susceptible to accumulated errors. Deep learning methods require a large number of learning samples, data acquisition and labeling are difficult and the data has limitations. The mapping model has poor interpretability and is not suitable for the field of aircraft where safety and reliability requirements are extremely high.

[0004] The modal superposition method has clear physical meaning and high computational efficiency. However, it usually requires the use of finite element analysis to obtain its modal parameters. For complex wing structures, it is difficult to obtain accurate modal parameters using finite element models, resulting in errors between the actual calculated results and the actual results, which in turn affects the accuracy of deflection identification.

[0005] In addition, during the modal coefficient correction process, differences in the layout positions of the strain sensor sensing points and the quality of the pasting process will also affect the calculation stability of the strain-displacement coefficient matrix and strain vector, which in turn leads to large fluctuations in the structural deflection identification results. Therefore, it is necessary to study a method to improve the robustness of the strain-displacement coefficient matrix and strain vector calculations. Summary of the Invention

[0006] Purpose of the invention: The technical problem to be solved by the present invention is to address the deficiencies of the existing technology and provide a method for improving the robustness of structural deflection identification based on the Lagrange multiplier method, comprising the following steps:

[0007] Step 1: Install an acceleration sensor on the structure and apply an impact load to the structure to make it vibrate freely. Under the free vibration state, collect acceleration data and perform Fourier transform to obtain a spectrum diagram to identify the first l-order natural frequencies of the structure.

[0008] Step 2: Distributed optical fiber sensors and laser deflection sensors are arranged on the structure. When the structure is in the nth order modal vibration, the strain information of the structural strain measurement points and the deflection information of the deflection measurement points are collected. The strain deflection coefficient matrix of the structure in the nth order mode is fitted using the least squares method; 1≤n≤ l ;

[0009] Step 3, based on the replace-one method, the strain-flexure coefficient matrix is ​​constructed twice;

[0010] Step 4: Based on the Lagrange multiplier method and the conjugate gradient method, the strain vector is constructed twice;

[0011] Step 5: Under the composite excitation of the simple harmonic force at the first n natural frequencies, the strain information of the structural strain measuring points and the deflection information of the deflection measuring points are collected, and the coupling coefficient matrix is ​​fitted using the least squares method.

[0012] Step 1 includes:

[0013] Step 1-1: Apply initial excitation to the structure by knocking, allowing the structure to vibrate freely, and use an accelerometer to record the acceleration response during the structural vibration attenuation process;

[0014] In step 1-2, the recorded vibration signal is Fourier transformed to obtain a spectrum. Based on the amplitude of the spectrum, the natural frequency of the structure is identified.

[0015] Step 2 includes:

[0016] Step 2-1, derive the overall strain-deflection conversion equation of the structure based on the modal superposition method;

[0017] According to the modal superposition method, the overall strain-deflection conversion equation of the structure is shown as follows. The structural deflection is calculated from the strain information of the strain measuring points and the modal parameters:

[0018] ,

[0019] Where T represents transpose, {d} M is the deflection of the M deflection monitoring points of the structure, is the deflection modal matrix, is the strain modal matrix, The strain collected from M strain monitoring points of the structure;

[0020] Step 2-2, fitting the strain-flexure coefficient matrix of the structure under the nth-order modal vibration;

[0021] Expand the overall structural strain-deflection conversion equation:

[0022] ,

[0023] in, Represents the strain modal matrix The parameter at row n and column M in ;

[0024] When the structure vibrates in the nth order mode, we get:

[0025] ,

[0026] in, is the deflection of M deflection monitoring points when the structure vibrates only in the nth order mode, is the deflection modal parameter corresponding to the Mth deflection measurement point when the structure vibrates only in the nth mode, is the strain modal parameter corresponding to the Mth strain measurement point when the structure vibrates only in the nth order mode, is the strain collected from M strain monitoring points when the structure vibrates only in the nth mode;

[0027] make , is the deflection coefficient matrix of the deflection measuring points 1 to N and the strain deflection coefficient matrix of the strain measuring points 1 to M under the n-th order modal vibration of the structure;

[0028] By means of calibration, the deflection of M deflection monitoring points of the structure is measured when the structure vibrates in the nth order mode. and strain , obtained by least squares fitting .

[0029] Step 3 includes:

[0030] According to the overall strain-deflection conversion equation and the substitution method, the boundary conditions of the deflection vector d are analyzed, and the inverse matrix (K n ) -1 The row vector r and column vector r are set to 0, and the elements corresponding to the rth row and rth column are set to 1, and the other values ​​remain unchanged. Then (K n ) -1 Find the inverse matrix to reconstruct the matrix K as shown below:

[0031] .

[0032] Step 4 includes: according to the boundary conditions of the deflection vector d and the numerical optimization method, the initial strain vector For fuzzy solution, with boundary conditions as constraints, through step iteration, the difference between the iterative result and the fuzzy solution is minimized and the calculated deflection vector meets the boundary conditions, and the strain vector is reconstructed. , the formula is:

[0033] ,

[0034] Among them, st means limited by, k MM represents the element in the Mth row and Mth column of matrix K, Represents the i-th strain component information of the reconstructed strain vector ε, represents the initial strain vector The i-th strain component information, d M Represents the Mth deflection component information of the deflection vector d.

[0035] Step 4 also includes: introducing constraints, combining the Lagrange multiplier method, and constructing the Lagrange function :

[0036] ,

[0037] in is the Lagrange multiplier, To reconstruct the strain vector With the initial strain vector The second norm of the difference, R represents the real number space, express belong dimensional real vector space, F is represented by d1 to d M Column vector of ;

[0038] Expand the square term of the objective function:

[0039] ,

[0040] The Lagrangian function becomes:

[0041] ,

[0042] right Find the partial derivative:

[0043] ,

[0044] in, Express The partial derivative of

[0045] Setting the partial derivative to zero yields:

[0046] ,

[0047] right Taking the partial derivative and setting it to zero, we get:

[0048] ,

[0049] The problem is finally transformed into minimizing under the constraint condition Kε=F and the initial strain vector The Euclidean distance between them defines the objective function for:

[0050] ,

[0051] Objective function The gradient of is:

[0052] ,

[0053] Solve the following optimization problem, satisfying Under the condition of Minimum value of:

[0054] ,

[0055] The conjugate gradient method gradually approximates the solution through iterative updates. Let r k is the residual, p k is the search direction, k is the number of iterations, α k To calculate the step size, β k is the conjugate coefficient, and its initial value is:

[0056] , , ,

[0057] in represents the initial value of the solution vector, represents the initial value of the residual, Indicates the initial value of the search direction;

[0058] In each iteration, the calculation step size, solution vector, residual, conjugate coefficient and search direction are updated:

[0059] ,

[0060] in represents the k-th iteration solution vector;

[0061] Iterate until the residual satisfies the following stopping condition:

[0062] ,

[0063] in, is the preset tolerance error; at this time is the optimized reconstructed strain vector.

[0064] Step 5 includes: repeating steps 2-2 to 4, setting n=1, 2, ..., c, and calculating K 1 , K 2 ,…,K c ; Fitting multi-order modal coupling coefficient matrix under the composite excitation of simple harmonic force at the first c-order natural frequency , the overall strain-deflection conversion equation of the structure is:

[0065] ;

[0066] in, It represents the deflection vector corresponding to the structure under the composite excitation of simple harmonic force at the first c-order natural frequency; It represents the strain vector of the structure under the composite excitation of simple harmonic force of the first c-order natural frequencies.

[0067] Step 5 also includes: after calculation, the multi-order modal coupling coefficient matrix is: ;

[0068] in is a constant, , , is the multi-order modal coupling coefficient In , the vth element in the pth diagonal submatrix.

[0069] The present invention also provides an electronic device, comprising a processor and a memory, wherein the memory stores program code, and when the program code is executed by the processor, the processor executes the steps of the method.

[0070] The present invention also provides a storage medium storing a computer program or instruction, which executes the steps of the method when the computer program or instruction is run on a computer.

[0071] Beneficial Effects: The present invention provides a method for improving the robustness of structural deflection identification based on the Lagrange multiplier method. First, the modal vibration coefficients are corrected using measured strain and deflection data to avoid the strain / displacement modal matrix deviation problem caused by inaccurate simulation mechanical models, thereby reducing the impact of modal matrix deviation on the accuracy of structural deflection inversion. Second, by combining the Lagrange multiplier method with the one-place method, the strain deflection coefficient matrix and strain vector are constructed during the coefficient correction process, reducing the impact of differences in the layout position of strain sensor points and the quality of the bonding process on the robustness of deflection identification results. BRIEF DESCRIPTION OF THE DRAWINGS

[0072] The present invention will be further described below in conjunction with the accompanying drawings and specific embodiments, and the above and / or other advantages of the present invention will become more apparent.

[0073] Figure 1 It is a flow chart of the method of the present invention.

[0074] Figure 2 Comparison between the reconstructed deflection value and the true value corresponding to the structural deflection measurement point under static load. DETAILED DESCRIPTION

[0075] like Figure 1 As shown, an embodiment of the present invention provides a method for improving the robustness of structural deflection identification based on the Lagrange multiplier method, including:

[0076] Step 1: Install an acceleration sensor on the structure, tap the structure to generate free vibration, collect acceleration data and perform Fourier transform to obtain a spectrum diagram to identify the first l-order natural frequencies of the structure;

[0077] Apply initial excitation to the structure by knocking, allowing the structure to vibrate freely. Use an accelerometer to record the acceleration vibration response during the structural vibration attenuation process; perform Fourier transform on the recorded vibration signal. , where P = 0, 1, …, L-1, L is the length of the acquired acceleration signal, x[r] is the discrete acceleration signal, X[P] is the frequency domain signal, j is the imaginary unit, and e is a natural constant. A spectrum is obtained, and the structural natural frequency is identified by the spectrum amplitude.

[0078] Step 2: Distributed fiber optic sensors and laser deflection sensors are arranged on the structure. The structure is at the nth order (1≤n≤ l ) Under modal vibration, the strain information of the structural strain measuring points and the deflection information of the deflection measuring points are collected, and the strain deflection coefficient matrix of the structure under the nth order mode is fitted using the least squares method;

[0079] Step 2-1, derive the overall strain-deflection conversion equation of the structure based on the modal superposition method;

[0080] According to the modal superposition method, the overall strain-deflection conversion equation of the structure is shown as follows. The structural deflection is calculated from the strain information of the strain measuring points and the modal parameters:

[0081] ,

[0082] Where T represents transpose, {d} M is the deflection of the M deflection monitoring points of the structure, is the deflection modal matrix, is the strain modal matrix, The strain collected from M strain monitoring points of the structure;

[0083] Step 2-2, fitting the strain-flexure coefficient matrix of the structure under the nth order modal vibration;

[0084] Expand the overall structural strain-deflection conversion equation:

[0085] ,

[0086] in, Represents the strain modal matrix The parameter at row n and column M in ;

[0087] When the structure vibrates in the nth order mode, we get:

[0088] ,

[0089] in, is the deflection of M deflection monitoring points when the structure vibrates only in the nth order mode, is the deflection modal parameter corresponding to the Mth deflection measurement point when the structure vibrates only in the nth mode, is the strain modal parameter corresponding to the Mth strain measurement point when the structure vibrates only in the nth order mode, is the strain collected from M strain monitoring points when the structure vibrates only in the nth mode.

[0090] make , is the deflection coefficient matrix of the deflection measuring points 1 to M and the strain deflection coefficient matrix of the strain measuring points 1 to M under the n-order modal vibration of the structure;

[0091] By means of calibration, the deflection of M deflection monitoring points of the structure is measured when the structure vibrates in the nth order mode. and strain , obtained by least squares fitting .

[0092] Step 3, construct the strain-flexure coefficient matrix based on the replace-one method;

[0093] According to the overall strain-deflection conversion equation and the substitution method, the boundary conditions of the deflection vector d are analyzed, and the inverse matrix (K n ) -1 The row vector r and column vector r are set to 0, and the elements corresponding to the rth row and rth column are set to 1, and the other values ​​remain unchanged. Then (K n ) -1 Find the inverse matrix and reconstruct the matrix K as shown below:

[0094] ;

[0095] Step 4: According to the boundary conditions of the deflection vector d and the numerical optimization method, the initial strain vector For fuzzy solution, with boundary conditions as constraints, through specific step iteration, the difference between the iterative result and the fuzzy solution is minimized and the calculated deflection vector meets the boundary conditions, and the strain vector is reconstructed. , the formula is:

[0096] ,

[0097] Among them, st means limited by, k MM represents the element in the Mth row and Mth column of matrix K, Represents the i-th strain component information of the reconstructed strain vector ε, represents the initial strain vector The i-th strain component information, dM Represents the Mth deflection component information of the deflection vector d.

[0098] Introduce constraints and combine the Lagrange multiplier method to construct the Lagrange function :

[0099] ,

[0100] in is the Lagrange multiplier, To reconstruct the strain vector With the initial strain vector The second norm of the difference, R represents the real number space, express belong dimensional real vector space, F is represented by d1 to d M Column vector of ;

[0101] Expand the square term of the objective function , the Lagrangian function becomes: ,right Find the partial derivative: ,in, Express The partial derivative of , set the partial derivative to zero and we get ,right Find the partial derivative and set the derivative to zero, and we get .

[0102] The problem is finally transformed into the constraint condition Next, minimize and the initial strain vector The Euclidean distance between them defines the objective function for: , the objective function The gradient of is: , solve the following optimization problem, under the condition Under the condition of Minimum value of: .

[0103] The conjugate gradient method gradually approximates the solution through iterative updates. Let r k is the residual, p k is the search direction, k is the number of iterations, α k To calculate the step size, β k is the conjugate coefficient, and its initial value is: , , .

[0104] In each iteration, the calculation step size, solution vector, residual, conjugate coefficient and search direction are updated:

[0105] ,

[0106] Iterate until the residual satisfies At this time, is the optimized reconstructed strain vector.

[0107] Step 5: Repeat steps 2-2 to 4 multiple times, let n = 1, 2, ..., c, and calculate K 1 , K 2 ,…,K c The structure is subjected to the composite excitation of the simple harmonic force at the first c-order natural frequency, and the multi-order modal coupling coefficient matrix is ​​fitted. , the overall strain-deflection conversion equation of the structure is:

[0108] ;

[0109] in, It represents the deflection vector corresponding to the structure under the composite excitation of simple harmonic force at the first c-order natural frequency; It represents the strain vector of the structure under the composite excitation of simple harmonic force of the first c-order natural frequencies.

[0110] After calculation, the multi-order modal coupling coefficient matrix is: .

[0111] like Figure 2 FIG. 1 is a schematic diagram of a specific embodiment provided by the present invention, wherein Figure 2 It can be seen that the deflection reconstruction value of the structural deflection measurement point under static load after optimization based on the Lagrange multiplier method is closer to the true value than before optimization. The robustness improvement method of structural deflection identification based on the Lagrange multiplier method can improve the accuracy of the deflection reconstruction value of the structural deflection measurement point under static / dynamic load.

[0112] The present invention provides a method for improving the robustness of structural deflection identification based on the Lagrange multiplier method. There are many methods and approaches for implementing this technical solution. The above is only a preferred embodiment of the present invention. It should be noted that those skilled in the art may make various improvements and modifications without departing from the principles of the present invention, and such improvements and modifications are also considered to be within the scope of protection of the present invention. Any components not specified in this embodiment may be implemented using existing technologies.

Claims

1. A method for improving the robustness of structural deflection identification based on the Lagrange multiplier method, characterized in that: The following steps are involved: Step 1: Install an acceleration sensor on the structure and apply an impact load to the structure to make it vibrate freely. Under the free vibration state, collect acceleration data and perform Fourier transform to obtain a spectrum diagram to identify the first l-order natural frequencies of the structure. Step 2: Distributed optical fiber sensors and laser deflection sensors are arranged on the structure. When the structure is in the nth order modal vibration, the strain information of the structural strain measurement points and the deflection information of the deflection measurement points are collected, and the strain deflection coefficient matrix of the structure in the nth order mode is fitted using the least squares method; 1≤n≤l; Step 3, based on the replace-one method, the strain-flexure coefficient matrix is ​​constructed twice; Step 4: Based on the Lagrange multiplier method and the conjugate gradient method, the strain vector is constructed twice; Step 5: Under the composite excitation of the simple harmonic force at the first n natural frequencies, the strain information of the structural strain measuring points and the deflection information of the deflection measuring points are collected, and the coupling coefficient matrix is ​​fitted using the least squares method.

2. The method according to claim 1, characterized in that Step 1 includes: Step 1-1: Apply initial excitation to the structure by knocking, allowing the structure to vibrate freely, and use an accelerometer to record the acceleration response during the structural vibration attenuation process; Step 1-2: Perform Fourier transform on the recorded vibration signal to obtain a spectrum; and identify the structural natural frequency based on the amplitude of the spectrum.

3. The method according to claim 2, characterized in that Step 2 includes: Step 2-1, derive the overall strain-deflection conversion equation of the structure based on the modal superposition method; According to the modal superposition method, the overall strain-deflection conversion equation of the structure is shown as follows. The structural deflection is calculated from the strain information of the strain measuring points and the modal parameters: ， Where T represents transpose, {d} M is the deflection of the M deflection monitoring points of the structure, is the deflection modal matrix, is the strain modal matrix, The strain collected from M strain monitoring points of the structure; Step 2-2, fitting the strain-flexure coefficient matrix of the structure under the nth-order modal vibration; Expand the overall structural strain-deflection conversion equation: ， in, Represents the strain modal matrix The parameter at row n and column M in ; When the structure vibrates in the nth order mode, we get: ， in, is the deflection of M deflection monitoring points when the structure vibrates only in the nth order mode, is the deflection modal parameter corresponding to the Mth deflection measurement point when the structure vibrates only in the nth mode, is the strain modal parameter corresponding to the Mth strain measurement point when the structure vibrates only in the nth order mode, is the strain collected from M strain monitoring points when the structure vibrates only in the nth mode; make , is the deflection coefficient matrix of the deflection measuring points 1 to N and the strain deflection coefficient matrix of the strain measuring points 1 to M under the n-th order modal vibration of the structure; By means of calibration, the deflection of M deflection monitoring points of the structure is measured when the structure vibrates in the nth order mode. and strain , obtained by least squares fitting .

4. The method according to claim 3, characterized in that Step 3 includes: According to the overall strain-deflection conversion equation and the substitution method, the boundary conditions of the deflection vector d are analyzed, and the inverse matrix (K n ) -1 The row vector r and column vector r are set to 0, and the elements corresponding to the rth row and rth column are set to 1, and the other values ​​remain unchanged. Then (K n ) -1 Find the inverse matrix to reconstruct the matrix K as shown below: 。 5. The method according to claim 4, characterized in that Step 4 includes: according to the boundary conditions of the deflection vector d and the numerical optimization method, the initial strain vector For fuzzy solution, with boundary conditions as constraints, through step iteration, the difference between the iterative result and the fuzzy solution is minimized and the calculated deflection vector meets the boundary conditions, and the strain vector is reconstructed. , the formula is: ， Among them, st means limited by, k MM represents the element in the Mth row and Mth column of matrix K, Represents the i-th strain component information of the reconstructed strain vector ε, represents the initial strain vector The i-th strain component information, d M Represents the Mth deflection component information of the deflection vector d.

6. The method according to claim 5, characterized in that Step 4 also includes: introducing constraints, combining the Lagrange multiplier method, and constructing the Lagrange function : ， in is the Lagrange multiplier, To reconstruct the strain vector With the initial strain vector The second norm of the difference, R represents the real number space, express belong dimensional real vector space, F is represented by d1 to d M Column vector of ; Expand the square term of the objective function: ， The Lagrangian function becomes: ， right Find the partial derivative: ， in, Express The partial derivative of Setting the partial derivative to zero yields: ， right Taking the partial derivative and setting it to zero, we get: ， The problem is finally transformed into minimizing under the constraint condition Kε=F and the initial strain vector The Euclidean distance between them defines the objective function for: ， Objective function The gradient of is: ， Solve the following optimization problem, satisfying Under the condition of Minimum value of: ， The conjugate gradient method gradually approximates the solution through iterative updates. Let r k is the residual, p k is the search direction, k is the number of iterations, α k To calculate the step size, β k is the conjugate coefficient, and its initial value is: ， ， ， in represents the initial value of the solution vector, represents the initial value of the residual, Indicates the initial value of the search direction; In each iteration, the calculation step size, solution vector, residual, conjugate coefficient and search direction are updated: ， in represents the k-th iteration solution vector; Iterate until the residual satisfies the following stopping condition: ， in, is the preset tolerance error; at this time is the optimized reconstructed strain vector.

7. The method according to claim 6, characterized in that Step 5 includes: repeating steps 2-2 to 4, setting n=1, 2, ..., c, and calculating K 1 , K 2 ,…,K c ; Fitting multi-order modal coupling coefficient matrix under the composite excitation of simple harmonic force at the first c-order natural frequency , the overall strain-deflection conversion equation of the structure is: ； in, It represents the deflection vector corresponding to the structure under the composite excitation of simple harmonic force at the first c-order natural frequency; It represents the strain vector of the structure under the composite excitation of simple harmonic force of the first c-order natural frequencies.

8. The method according to claim 7, characterized in that Step 5 also includes: after calculation, the multi-order modal coupling coefficient matrix is: ; in is a constant, , , is the multi-order modal coupling coefficient In , the vth element in the pth diagonal submatrix.

9. An electronic device, characterized in that: The method comprises a processor and a memory, wherein the memory stores program codes, and when the program codes are executed by the processor, the processor is caused to perform the steps of the method according to any one of claims 1 to 8.

10. A storage medium, characterized in that: A computer program or instruction is stored, and when the computer program or instruction is run on a computer, the steps of the method according to any one of claims 1 to 8 are executed.

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