Structural deflection recognition robustness improvement method based on Lagrange multiplier method

Abstract

According to the structure deflection recognition robustness improvement method based on the Lagrangian multiplier method, the robustness is the stability of inversion, and under the condition that measured data of a sensor has large noise, the method can invert the structure deflection in a high-fidelity mode. The method comprises the following steps: applying an impact load to a structure, collecting an acceleration signal when the structure is in a free vibration state, performing Fourier transform on the acceleration signal to obtain a spectrogram, and identifying the inherent frequency of the structure. Under the nth-order modal vibration of the structure, the strain deflection coefficient matrix of the order is fitted based on the least square method, and through the strain deflection coefficient matrix based on the Lagrange multiplier method and the strain vector quadratic construction method, the robustness influence of the strain sensing measuring point layout position difference and the pasting process quality degree on the deflection recognition result is reduced. The structure fits a multi-order modal coupling coefficient matrix under the action of front n-order inherent frequency simple harmonic force composite excitation.

Description

Technical Field

[0001] The 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 the operation of an aircraft, it will be deformed and damaged due to the effects of aerodynamics, gravity and inertia. The deformation of the wing structure is not only one of the important indicators to characterize the health of the wing structure, but also has an important impact on the performance of the wing-mounted equipment, such as airborne 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 annotation are difficult and the data has limitations. The mapping model has poor interpretability and is not suitable for the field of aircraft with extremely high requirements for safety and reliability.

[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 for the finite element model to obtain accurate modal parameters, resulting in errors between the actual calculation results and the actual results, which in turn affects the accuracy of deflection identification.

[0005] In addition, during the process of modal coefficient correction, 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 the strain vector, which will lead 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 calculation. Summary of the invention

[0006] Purpose of the invention: The technical problem to be solved by the present invention is to provide a method for improving the robustness of structural deflection identification based on the Lagrange multiplier method in view of the shortcomings of the prior art, comprising the following steps: Step 1: Install an acceleration sensor on the structure, apply an impact load to the structure, and make the structure vibrate freely. Under the free vibration state, collect acceleration data and perform Fourier transform to obtain a spectrum diagram, and 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 measuring points and the deflection information of the deflection measuring 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 deflection coefficient matrix is ​​constructed twice; Step 4, based on the Lagrange multiplier method and the conjugate gradient method, the strain vector is reconstructed; Step 5: Under the composite excitation of the simple harmonic force at the first n-order 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.

[0007] Step 1 includes: Step 1-1, applying initial excitation to the structure by knocking, allowing the structure to vibrate freely, and using 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. According to the amplitude of the spectrum, the natural frequency of the structure is identified.

[0008] 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 point 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 strains collected at M strain monitoring points of the structure; Step 2-2, fitting the strain deflection 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 the nth row and the Mth column 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 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 order mode;

[0009] make , is the deflection coefficient matrix of deflection measuring points 1 to N and strain deflection coefficient matrix of strain measuring points 1 to M under n-order modal vibration of the structure; Through calibration, the deflection of M deflection monitoring points of the structure is measured when the structure vibrates in the nth order mode. With strain , obtained by least squares fitting .

[0010] 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. n ) -1 Find the inverse matrix to reconstruct the matrix K as shown below: .

[0011] Step 4 includes: according to the boundary conditions of the deflection vector d and the numerical optimization method, the initial strain vector The strain vector is reconstructed by taking the boundary conditions as constraints and iterating the step size to minimize the difference between the iterative result and the fuzzy solution and to make the calculated deflection vector meet the boundary conditions. , the formula is: , Among them, st means restricted 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.

[0012] 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 d 1 to dM Column vector of ; Expand the square term of the objective function: , The Lagrangian function becomes: , right Find partial derivatives: , 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, while satisfying Under the condition of Minimum value of: , The conjugate gradient method gradually approaches 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; is the optimized reconstructed strain vector.

[0013] 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 the 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 frequency.

[0014] 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 , the vth element in the pth diagonal submatrix.

[0015] 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 described method.

[0016] The present invention also provides a storage medium storing a computer program or instruction. When the computer program or instruction is run on a computer, the steps of the method described are executed.

[0017] 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 by the measured strain and deflection data to avoid the strain / displacement modal matrix deviation problem caused by the inaccurate simulation mechanical model, thereby reducing the influence of the modal matrix deviation on the inversion accuracy of the structural deflection. Secondly, by combining the Lagrange multiplier method with the one-set method, the strain deflection coefficient matrix and strain vector in the process of secondary construction coefficient correction are used to reduce 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 results. BRIEF DESCRIPTION OF THE DRAWINGS

[0018] The present invention will be further described in detail below in conjunction with the accompanying drawings and specific embodiments, and the above and / or other advantages of the present invention will become more clear.

[0019] Figure 1 It is a flow chart of the method of the present invention.

[0020] Figure 2 Comparison between the reconstructed deflection value and the true value corresponding to the structural deflection measurement point under static load. DETAILED DESCRIPTION

[0021] 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, comprising: 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, and identify the first l-order natural frequencies of the structure; Apply initial excitation to the structure by knocking, so that the structure can 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 collected acceleration signal, x[r] is the discrete acceleration signal, X[P] is the frequency domain signal, j is the imaginary unit, and e is the natural constant. The spectrum is obtained and the natural frequency of the structure is identified by the amplitude of the spectrum.

[0022] Step 2: Distributed optical fiber 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; 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 point 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 strains collected at M strain monitoring points of the structure; Step 2-2, fitting the strain deflection coefficient matrix under the nth order modal vibration of the structure; Expand the overall structural strain-deflection conversion equation: , in, Represents the strain modal matrix The parameter at the nth row and the Mth column in ; When the structure vibrates in the nth order mode, we get: ,

[0023] in, is the deflection of M deflection monitoring points when the structure vibrates only in the nth 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.

[0024] make , is the deflection coefficient matrix of deflection measuring points 1~M and strain deflection coefficient matrix of strain measuring points 1~M under n-order modal vibration of the structure; Through calibration, the deflection of M deflection monitoring points of the structure is measured when the structure vibrates in the nth order mode. With strain , obtained by least squares fitting .

[0025] Step 3, constructing the strain deflection coefficient matrix based on the replace-one method; 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. n ) -1 Find the inverse matrix and reconstruct the matrix K as shown below: ; Step 4: According to the boundary conditions of the deflection vector d and the numerical optimization method, the initial strain vector The strain vector is reconstructed by taking the boundary conditions as constraints and iterating with a specific step length to minimize the difference between the iterative result and the fuzzy solution and to ensure that the calculated deflection vector meets the boundary conditions. , the formula is: , Among them, st means restricted 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.

[0026] Introduce constraints and combine the Lagrange multiplier method to construct 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 d 1 to d M Column vector of ; Expand the square term of the objective function , the Lagrangian function becomes: ,right Find partial derivatives: ,in, Express The partial derivative of , set the partial derivative to zero and we get ,right Taking the partial derivative and setting it to zero, we get .

[0027] 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 that Under the condition of Minimum value of: .

[0028] The conjugate gradient method gradually approaches 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: , , .

[0029] In each iteration, the calculation step size, solution vector, residual, conjugate coefficient and search direction are updated: , Iterate until the residual satisfies At this time, is the optimized reconstructed strain vector.

[0030] 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 fitted with the multi-order modal coupling coefficient matrix under the composite excitation of the simple harmonic force of 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 the 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 frequency.

[0031] After calculation, the multi-order modal coupling coefficient matrix is: .

[0032] like Figure 2 FIG. 1 is a schematic diagram of a specific embodiment provided by the present invention. 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.

[0033] 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 ways to implement the technical solution. The above is only a preferred embodiment of the present invention. It should be pointed out that for ordinary technicians in this technical field, several improvements and modifications can be made without departing from the principle of the present invention. These improvements and modifications should also be regarded as the protection scope of the present invention. All components not specified in this embodiment can 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, apply an impact load to the structure, and make the structure vibrate freely. Under the free vibration state, collect acceleration data and perform Fourier transform to obtain a spectrum diagram, and identify the first l-order natural frequencies of the structure. Step 2, arranging distributed optical fiber sensors and laser deflection sensors on the structure, collecting strain information of the structural strain measurement points and deflection information of the deflection measurement points under the n-th order modal vibration of the structure, and fitting the strain deflection coefficient matrix of the structure under the n-th order mode using the least square method; 1≤n≤l; Step 3, based on the replace-one method, the strain deflection coefficient matrix is ​​constructed twice; Step 4, based on the Lagrange multiplier method and the conjugate gradient method, the strain vector is reconstructed; Step 5: Under the composite excitation of the simple harmonic force at the first n-order 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, applying initial excitation to the structure by knocking, allowing the structure to vibrate freely, and using 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 diagram; and identify the structural natural frequency based on the amplitude of the spectrum diagram.

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 point 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 strains collected at M strain monitoring points of the structure; Step 2-2, fitting the strain deflection 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 the nth row and the Mth column 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 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 order mode; make , is the deflection coefficient matrix of deflection measuring points 1 to N and strain deflection coefficient matrix of strain measuring points 1 to M under n-order modal vibration of the structure; Through calibration, the deflection of M deflection monitoring points of the structure is measured when the structure vibrates in the nth order mode. With 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. 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 The strain vector is reconstructed by taking the boundary conditions as constraints and iterating the step size to minimize the difference between the iterative result and the fuzzy solution and to make the calculated deflection vector meet the boundary conditions. , the formula is: ， Among them, st means restricted 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, while satisfying Under the condition of Minimum value of: ， The conjugate gradient method gradually approaches 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 kth 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 the 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 frequency.

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 , 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 executes 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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