CN113111547B - Frequency domain finite element model correction method based on reduced basis - Google Patents

Abstract

The embodiment of the invention discloses a frequency domain finite element model correction method based on a reduction basis, which relates to the field of dynamic finite element model correction, and comprises the following steps: carrying out vibration test on the bogie structure to obtain a measurement frequency response function of the bogie structure, and converting the bogie structure into a reduced coordinate through a base matrix; establishing an initial finite element model of the structure, and calculating a frequency response function analysis value under a reduced coordinate; and calculating residual errors of the frequency response function analysis value and the measured value under the reduced coordinates and a sensitivity matrix related to the parameter to be corrected, solving a corresponding sensitivity equation to obtain updated parameters and a finite element model, and repeating iteration by taking the updated parameters and the finite element model as an initial finite element model until the residual errors are minimum. The method is suitable for the correction problem of the bogie finite element model with huge number of degrees of freedom, and can obviously improve the calculation efficiency on the premise of ensuring the accuracy, so that the dynamic finite element model of the bogie structure is efficiently corrected by using actual test data.

Description

Frequency domain finite element model updating method based on reduced basis

Technical Field

The invention belongs to the field of finite element model correction, and in particular relates to a frequency domain method for correcting a dynamic finite element model with a huge number of degrees of freedom formed by a bogie structure.

Background technique

With the continuous increase in the speed of modern trains, the requirements for the reliability and precision of bogie structures are becoming higher and higher. In the stages of bogie structure design, use, and improvement, it is necessary to analyze its form and parameters from multiple perspectives such as dynamics, structural fatigue, and reliability. Therefore, it is crucial to establish a finite element model that can reflect the actual dynamic characteristics of the bogie. However, due to the complexity of the various components of the bogie, the modeling process inevitably requires the simplification of the structure and the use of a variety of mechanical units with parameters for discretization. The initial model parameters selected based on experience may make the dynamic characteristics of the bogie finite element model inconsistent with the actual structure; and the resulting bogie finite element model is often large in scale and has low efficiency in analysis and calculation. At present, the model correction methods for large finite element models mainly include the method of reducing the finite element model correction formula to the test degree of freedom, the method of reducing the degree of freedom based on the substructure modal synthesis technology with the goal of correcting the modal parameters, and the correction method based on the proxy model. However, when the test freedom degree is small, the model shrinkage is easy to introduce large approximation errors, and the calculation amount of the shrinkage matrix itself is also large; although the correction method using substructure modal synthesis has high calculation accuracy and efficiency, it introduces errors and uncertainties in the modal identification process by taking modal parameters as the correction target; and the correction method based on the proxy model can quickly obtain the correction result after establishing the response surface of the correction target with respect to the parameter to be corrected, but the process of forming the response surface requires calculating enough data to avoid the distortion of the proxy model, and its calculation amount is undoubtedly huge. How to efficiently and accurately correct the large finite element model formed by the bogie structure based on the frequency domain response data is of great significance for the numerical simulation of the structure in engineering.

Summary of the invention

In order to overcome the problem of low efficiency of existing model correction technology when applied to bogie structure, the present invention proposes a finite element model frequency domain correction method based on a reduced basis, which significantly improves the calculation efficiency while ensuring accuracy. By using this invention, a large finite element model formed by the bogie can be corrected quickly and accurately so that its dynamic characteristics are consistent with the actual test results.

In order to achieve the above object, the present invention is implemented by adopting the following technical solutions:

An embodiment of the present invention provides a frequency domain finite element model correction method based on a reduced basis, the method is used for finite element model correction using frequency response function data, the bogie finite element model to be corrected has a large number of degrees of freedom, resulting in low correction efficiency, the method comprises the following steps:

Step 1: Determine the output degrees of freedom and input degrees of freedom for vibration testing of the bogie structure, obtain the measured frequency response function of the structure after the test, and transform it to the reduced coordinates through the basis matrix;

Step 2: Establish a dynamic finite element model of the bogie structure, and calculate the frequency response function analysis value under the reduced coordinate according to the parameter value of the previous iteration step;

Step 3: Calculate the residuals between the frequency response function analysis value and the measured value under the reduced coordinates and their sensitivity matrix with respect to the parameters to be corrected, and solve the corresponding sensitivity equation to obtain the updated parameters and finite element model after the parameter increment;

Step 4: Calculate the amplitude correlation of the frequency response function corresponding to the updated parameter. If it meets the convergence criterion, output the parameter value and the corresponding historical iteration data to complete the correction. Otherwise, return to step 2 and iterate again until convergence.

The specific method for obtaining the bogie structure test frequency response under reduced coordinates in step 1 is as follows:

The input and output degrees of freedom during the vibration test of the bogie structure refer to the degrees of freedom for excitation during the hammer test and the degrees of freedom for outputting the structural response data, respectively. The specific determination method is as follows: the layout of the output degrees of freedom should clearly reflect the vibration forms of each order, and the layout of the input degrees of freedom needs to avoid the nodes of the modal vibration modes of each order of the structure. The test system composed of a data acquisition system, a force hammer, a sensor and a data analysis software can obtain the measured frequency response functions on the input and output degrees of freedom through a multi-reference point hammer test. Since the actual data inevitably contains measurement noise, the frequency response function of the resonance zone is not only less polluted by noise, but also more sensitive to the parameters of the structure, that is, the sensitivity is greater, and the coefficient matrix of the sensitivity equation formed by it is also more accurate and less prone to pathological conditions. Therefore, only the frequency response functions corresponding to the frequency points near the resonance zone are selected to participate in the model correction. In the specific implementation process, as shown in Figure 2, the measured frequency response functions in each resonance zone can be obtained by selecting the frequency points within the half-power bandwidth of the modal indicator function.

The superscript i represents the i-th resonance zone, and q is the total number of resonance zones. For the kth frequency point, a total of N _i frequency points are selected in the i-th resonance zone. Frequency point The measured frequency response corresponding to the j-th input degree of freedom at .

In practical applications, the complete degrees of freedom of the finite element model and the test degrees of freedom are generally not matched, and the number of the former is often much larger than the latter. When using the basis matrix for coordinate transformation, the test frequency response function of the complete degrees of freedom is required. Therefore, in order to transform the measured frequency response function into the reduced coordinate, it is necessary to expand it first. Suppose the measured frequency response function of the i-th resonance zone is Corresponding to a certain test mode of the structure (if it is a repeated frequency mode, it corresponds to two modes with the same motion form but different phases), the motion form of the structure in the resonance area is mainly determined by this mode and its adjacent modes. Therefore, the calculation mode of the finite element model that matches this test mode is selected through vibration mode correlation (MAC) and frequency error, and several adjacent calculation modes are added as expansion. The basis Φ _{F,i of the resonance region is Φ F,i} , and the number of adjacent modes to be added should be determined by observing the complexity of the frequency domain displacement motion of the resonance region and the degree of coupling between the corresponding test mode and the adjacent mode. In addition, for the free boundary condition, the motion of each order of the structure will include both elastic deformation and large-scale rigid body motion, so Φ _F,i should also include several rigid body modes. Then Test frequency response function after expansion to full degrees of freedom This basis can be expressed as

Where q _i is the corresponding fitting coefficient. By performing a series of matrix changes on the above formula, the frequency response function expansion formula can be obtained as follows:

in is the component of the basis on the test degree of freedom.

Finally, take the expanded test frequency response function In a column After the inverse transformation of the basis matrix, the structural test frequency response function under the reduced coordinates can be obtained:

Where _Bs is the basis matrix, and the reduced coordinate value means the component value corresponding to the basis matrix _Bs . According to different methods of reducing degrees of freedom, it can be divided into modal vibration mode basis matrix and modal comprehensive basis matrix. The former is directly taken as a number of low-order finite element model vibration modes covering the analysis frequency band; the latter is obtained by dividing the overall structure into a number of substructures, the constrained mode set and the main mode set Φ, and eliminating the non-independent generalized coordinates through the transformation matrices T and S.

_Bs ＝TΦS (5)

In actual application, since the two basis matrices have their own advantages and disadvantages, they should be selected according to the actual structure to be corrected: directly using the modal vibration shape as the basis matrix is simple, effective and easy to understand, but when the number of degrees of freedom of the structural finite element model is large, it takes more time and calculation to calculate its mode; the amount of calculation required for the modal synthesis basis calculated by the split substructures decreases significantly with the increase of the number of substructures, but the process is more complicated, and when the number of degrees of freedom of the interface between the substructures is large, the reduced coordinates will still have a large number of degrees of freedom, reducing the calculation efficiency of the correction process. For general structural vibrations, the two have similar fitting capabilities as a basis to describe actual motion.

The specific method for obtaining the bogie structure frequency response function analysis value under reduced coordinates in step 2 is as follows:

First, a finite element model that is consistent with the dynamic characteristics of the bogie structure is established through any commercial finite element software. If it is divided into several groups according to material and unit type, a part of the material parameters (such as elastic modulus, density, damping) of these groups that are more sensitive to the structural response is selected as the parameters to be corrected. After the dynamic matrix of each group of the finite element model is exported from the software, the function analysis value under the reduced coordinate can be obtained from the basis matrix The parameterized expression is

in To reduce the mass and stiffness matrix of the e-th group of the finite element model under the coordinate, α _e and γ _e are their corresponding correction coefficients; is the proportional damping and structural damping matrix under the reduced coordinates, and _βe is its correction coefficient. θ is the parameter increment to be corrected composed of _αe , _βe , and _γe . If some grouped material parameters are not used as correction objects in practice, they only need to be fixed as the initial values of the parameters in the sensitivity equation and then solved.

The specific method for calculating the sensitivity matrix and identifying the increment of the parameter value to be corrected in step 3 is as follows:

The above formula is the sensitivity equation derived from the Newton-Gauss method. Its mathematical meaning is to solve the parameter increment direction and size that minimizes the residual between the frequency response function analysis value and the measured value under the current parameters. are the frequency response function measurement value frequency point and analysis value frequency point in the best matching frequency pair sequence selected according to the amplitude correlation under the framework of the kth group of frequency shift theory, and k＝1,…, _Nmatch , _Nmatch is the number of matched pairs, S, Δf are the sensitivity matrix and residual vector determined by the parameter increment ^θr to be corrected in the previous iteration step or the initial iteration step at this pair of frequency points, and the specific calculation formula is:

in The frequency response function analysis value under the reduced coordinates with the input degrees of freedom consistent with the test model, is the dynamic stiffness matrix of the finite element model under the reduced coordinates. At this point, the frequency response function analysis value and the measured value under the reduced coordinates calculated in steps 2 and 3 are substituted into the above formula to calculate the sensitivity matrix of the residual with respect to the parameters, and the parameter increment of the next iteration step is obtained after solving the sensitivity equation. However, in the actual implementation process, the coefficient matrix is often disturbed due to the inevitable measurement noise pollution and the truncation error when the degrees of freedom are reduced. In addition, it often has a large condition number, which further amplifies the disturbance, making it difficult to accurately identify the parameters. In order to improve this problem, it is often necessary to operate more finely in the vibration test and numerical modeling process to reduce the main sources of error, and at the same time reasonably select the data involved in parameter identification, and take certain numerical means to reduce the morbidity of the equation.

The specific method for determining whether the objective function value meets the convergence criterion in step 4 is as follows:

Convergence criteria are also very important in actual model correction. Reasonable criteria can obtain the most reasonable correction results with the least number of iterations. On the contrary, unreasonable criteria may not only increase the number of iterations, but also lead to divergence due to iteration to a parameter point that loses physical meaning. The proposed method takes the norm of the residual v(ω _k ,θ) between the measured value and the analyzed value of the frequency response function of the structure in the reduced coordinate as the minimization target, that is, it is considered that convergence is achieved when the residual norm approaches 0. Geometrically, this can also be understood as the analytical frequency response function curve is close to the test frequency response function curve. Therefore, if the frequency response amplitude correlation coefficient is used Describes the degree of overlap of the two curves. When the mean value at each frequency point is greater than the threshold _σmin, it is considered to have reached convergence, that is,

In the specific implementation process, the criterion for judging iterative convergence is not fixed, and different criteria can be set according to different actual correction targets to obtain a set of parameter values that make the dynamic characteristics of the corrected finite element model most consistent with expectations.

Compared with the prior art, the technical solution of the present invention has the following beneficial technical effects:

The present invention provides a frequency domain finite element model correction method based on a reduced basis. Compared with the prior art, the present invention can effectively reduce the scale of matrix operations and improve the calculation efficiency of the correction algorithm for the finite element model with a large number of degrees of freedom formed by the bogie structure by using the reduced basis method, thereby efficiently obtaining a bogie finite element model that is corrected and consistent with the actual dynamic characteristics. The invention provides a beneficial solution to the common problem of low correction efficiency due to the large scale of the finite element model in engineering.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG1 is a flowchart of a frequency domain finite element model correction method based on a reduced basis according to the present invention;

Figure 2 is a segmented diagram of the test frequency response function;

FIG3 is a finite element model of the bogie frame and a substructure division diagram;

Figure 4 is a comparison of the initial analysis frequency response and the test frequency response before correction;

Figure 5 is a comparison of the corrected analysis frequency response and the test frequency response;

Figure 6 is a comparison of the correlation between the analysis frequency response and the test frequency response amplitude before and after correction;

Detailed ways

The present invention is described in detail below with reference to the accompanying drawings and examples.

The present invention proposes a frequency domain correction method for finite element models based on a reduced basis, which expresses the displacement of the finite element model of the bogie structure as a linear superposition of a modal comprehensive reduced basis or a modal vibration type reduced basis, and derives the sensitivity matrix of the frequency response function residual in the reduced coordinates with respect to the selected parameters to be corrected, and establishes the sensitivity equation of the corresponding identification parameter increment based on the Newton-Gauss method, so that the correction process is carried out in the reduced coordinates, and the calculation efficiency is significantly improved under the premise of ensuring accuracy. At the same time, the model is corrected under the theoretical framework of the frequency shifting technology, which effectively improves the accuracy of parameter identification and increases the convergence of iterations; and a high-precision segmented frequency response expansion method is derived based on the modal expansion formula to reduce the influence of the expansion error on the correction process. The experimental example of the bogie frame proves that for large finite element models with a large number of degrees of freedom, the proposed method can efficiently obtain a corrected model that is consistent with the actual structural dynamic characteristics.

The present invention is described by taking the finite element model correction of a certain type of bogie frame shown in Figure 3 as an example. The finite element model has a total of 181356 degrees of freedom. If the traditional model correction method is used, the calculation efficiency is low and the noise resistance is poor. However, the method of the present invention can avoid these defects.

The frequency domain finite element model correction method based on reduced basis is characterized in that the method comprises the following steps:

The first step is to determine the output degrees of freedom and input degrees of freedom for the vibration test of the structure. After the test, the measured frequency response function of the structure is obtained and converted to the reduced coordinates through the basis matrix;

The second step is to establish a dynamic finite element model of the structure and calculate the frequency response function analysis value in the reduced coordinate according to the parameter values of the previous iteration step;

The third step is to calculate the residuals of the frequency response function analysis value and the measured value under the reduced coordinates and its sensitivity matrix with respect to the parameters to be corrected, and solve the corresponding sensitivity equation to obtain the updated parameters and finite element model after the parameter increment;

The fourth step is to calculate the amplitude correlation of the frequency response function corresponding to the updated parameter. If it meets the convergence criterion, the parameter value and the corresponding historical iteration data are output to complete the correction. Otherwise, return to step 2 and iterate again until convergence.

The specific method for obtaining the measured frequency response function of the structure under reduced coordinates in the first step is as follows:

The input and output degrees of freedom during structural vibration testing refer to the degrees of freedom for excitation during hammer tests and the degrees of freedom for outputting structural response data, respectively. The specific determination method is as follows: the layout of the output degrees of freedom should clearly reflect the vibration forms of each order, and the layout of the input degrees of freedom needs to avoid the nodes of each modal vibration mode of the structure. The test system composed of a data acquisition system, a force hammer, a sensor and a data analysis software can obtain the measured frequency response functions on the input and output degrees of freedom through a multi-reference point hammer test. Since the actual data inevitably contains measurement noise, the frequency response function of the resonance zone is not only less polluted by noise, but also more sensitive to the parameters of the structure, that is, the sensitivity is greater, and the coefficient matrix of the sensitivity equation formed by it is also more accurate and less prone to pathological conditions. Therefore, only the frequency response functions corresponding to the frequency points near the resonance zone are selected to participate in the model correction. In the specific implementation process, as shown in Figure 2, the measured frequency response functions in each resonance zone can be obtained by selecting the frequency points within the half-power bandwidth of the modal indicator function.

in is the kth frequency point, is the measured frequency response corresponding to the jth input degree of freedom at this frequency point, The superscript i represents the i-th resonance zone, and q is the total number of resonance zones.

In practical applications, the complete degrees of freedom of the finite element model and the test degrees of freedom are generally not matched, and the number of the former is often much larger than the latter. When using the basis matrix for coordinate transformation, the test frequency response function of the complete degrees of freedom is required. Therefore, in order to transform the measured frequency response function into the reduced coordinate, it is necessary to expand it first. Suppose the measured frequency response of the i-th resonance zone is Corresponding to a certain test mode of the structure (if it is a repeated frequency mode, it corresponds to two modes with the same motion form but different phases), the motion form of the structure in the resonance area is mainly determined by this mode and its adjacent modes. Therefore, the calculation mode of the finite element model that matches this test mode is selected through vibration mode correlation (MAC) and frequency error, and several adjacent calculation modes are added as expansion. The basis of is denoted as Φ _F,i , and the number of adjacent modes added should be determined by observing the complexity of the frequency domain displacement motion form of the resonance zone and the degree of coupling between the corresponding test mode and the adjacent mode. In addition, for the free boundary condition, the motion forms of each order of the structure will include both elastic deformation and large-scale rigid body motion, so Φ _F,i should also include several rigid body modes. Therefore,

in To expand the test frequency response to full freedom, a series of matrix changes can be performed on the above formula to obtain

in is the component of the basis on the test degree of freedom.

Finally, take the expanded Frequency response of a column The structural test frequency response under reduced coordinates can be obtained from the following formula:

Where _Bs is the basis matrix, and the reduced coordinates refer to the corresponding components of the basis matrix _Bs . According to different methods of reducing degrees of freedom, it can be divided into modal vibration basis matrix and modal comprehensive basis matrix. The former is directly taken as a number of low-order finite element model vibration modes covering the analysis frequency band; the latter is obtained by dividing the overall structure into a number of substructures, the constrained mode set and the main mode set Φ, and eliminating the non-independent generalized coordinates through the transformation matrices T and S.

_Bs ＝TΦS (5)

In actual application, since the two basis matrices have their own advantages and disadvantages, they should be selected according to the actual structure to be corrected: directly using the modal vibration shape as the basis matrix is simple, effective and easy to understand, but when the number of degrees of freedom of the structural finite element model is large, it takes more time and calculation to calculate its mode; the amount of calculation required for the modal synthesis basis calculated by the split substructures decreases significantly with the increase of the number of substructures, but the process is more complicated, and when the number of degrees of freedom of the interface between the substructures is large, the reduced coordinates will still have a large number of degrees of freedom, reducing the calculation efficiency of the correction process. For general structural vibrations, the two have similar fitting capabilities as a basis to describe actual motion.

The specific method for obtaining the structural frequency response function analysis value under reduced coordinates in the second step is as follows:

First, a finite element model that is consistent with the actual structural dynamic characteristics is established through any commercial finite element software. If it is divided into several groups according to material and unit type, a part of the material parameters (such as elastic modulus, density, damping) that is more sensitive to the structural response is selected from these groups as the parameters to be corrected. After the dynamic matrix of each group of the finite element model is exported from the software, the function analysis value under the reduced coordinate can be obtained from the basis matrix The parameterized expression is

in To reduce the mass and stiffness matrix of the e-th group of the finite element model under the coordinate, α _e and γ _e are their corresponding correction coefficients; is the proportional damping and structural damping matrix under the reduced coordinates, and _βe is its correction coefficient. θ is the parameter increment to be corrected composed of _αe , _βe , and _γe . If some grouped material parameters are not used as correction objects in practice, they only need to be fixed as the initial values of the parameters in the sensitivity equation and then solved.

The specific method for calculating the sensitivity matrix and identifying the increment of the parameter value to be corrected in the third step is as follows:

The above formula is the sensitivity equation derived from the Newton-Gauss method, and its mathematical meaning is to solve the parameter increment direction and size that minimizes the residual between the frequency response function analysis value and the measured value under the current parameters. are the frequency response function measurement value frequency point and analysis value frequency point in the best matching frequency pair sequence selected according to the amplitude correlation under the framework of the kth group of frequency shift theory, and k＝1,…, _Nmatch , _Nmatch is the number of matched pairs, S and Δf are the sensitivity matrix and residual vector determined by the parameter increment ^θr to be corrected in the previous iteration step at this pair of frequency points, and the specific calculation formula is:

in The frequency response function analysis value under the reduced coordinates with the input degrees of freedom consistent with the test model, is the dynamic stiffness matrix of the finite element model under the reduced coordinates. At this point, the frequency response function analysis value and the measured value under the reduced coordinates calculated in steps 2 and 3 are substituted into the above formula to calculate the sensitivity matrix of the residual with respect to the parameters, and the parameter increment of the next iteration step is obtained after solving the sensitivity equation. However, in the actual implementation process, the coefficient matrix is often disturbed due to the inevitable measurement noise pollution and the truncation error when the degrees of freedom are reduced. In addition, it often has a large condition number, which further amplifies the disturbance, making it difficult to accurately identify the parameters. In order to improve this problem, it is often necessary to operate more finely in the vibration test and numerical modeling process to reduce the main sources of error, and at the same time reasonably select the data involved in parameter identification, and take certain numerical means to reduce the morbidity of the equation.

The specific method for judging whether the objective function value satisfies the convergence criterion in the third step is as follows:

Convergence criteria are also very important in actual model correction. Reasonable criteria can obtain the most reasonable correction results with the least number of iterations. On the contrary, unreasonable criteria may not only increase the number of iterations, but also lead to divergence due to iteration to a parameter point that loses physical meaning. The proposed method takes the norm of the residual v(ω _k ,θ) between the measured value and the analyzed value of the frequency response function of the structure in the reduced coordinate as the minimization target, that is, it is considered that convergence is achieved when the residual norm approaches 0. Geometrically, this can also be understood as the analytical frequency response function curve is close to the test frequency response function curve. Therefore, if the frequency response amplitude correlation coefficient is used Describes the degree of overlap of the two curves. When the mean value at each frequency point is greater than the threshold _σmin, it is considered to have reached convergence, that is,

In the specific implementation process, the criterion for judging iterative convergence is not fixed, and different criteria can be set according to different actual correction targets to obtain a set of parameter values that make the dynamic characteristics of the corrected finite element model most consistent with expectations.

As shown in FIG1 , the present invention provides a frequency domain finite element model correction method based on a reduced basis, and the specific steps of the method are as follows:

Step 1: Obtain the structural test frequency response under reduced coordinates. A bogie frame with elastic support was adopted, and a total of 192 output degrees of freedom and 3 input degrees of freedom were selected, and then a hammer test was performed to obtain the test frequency response function on the corresponding degrees of freedom. Since the structure is a free boundary condition, the first three rigid body modes of the overall finite element model, the modes corresponding to all orders in the analysis frequency band, and the two-order modes outside the analysis frequency band are used as the basis of the expanded test frequency response function to obtain the expanded test frequency response function on the complete degrees of freedom. In addition, in order to obtain the basis matrix, as shown in Figure 3, the frame finite element model is divided into three substructures for modal synthesis to obtain the modal synthesis reduced basis matrix. The degrees of freedom of the model under natural coordinates are 181356, and the degrees of freedom of the model under reduced coordinates are only 1830, which greatly reduces the matrix calculation scale of the correction process. Finally, the expanded test frequency response can be converted to the reduced coordinates according to the reduced basis matrix.

Step 2: Calculate the frequency response function analysis value under reduced coordinates from the finite element model. The frame finite element model is divided into 10 groups, as shown in Figure 3. The elastic modulus, density and damping coefficient of each group are used as parameters to be corrected, totaling 30 parameters. After setting the initial values of the parameters, the mass matrix, stiffness matrix, damping matrix, etc. of each group of the model are exported by the commercial finite element software, and converted to the reduced coordinates by reducing the basis matrix, and the frequency response function analysis value under the reduced coordinates is calculated. The comparison between the fitting frequency response and the test frequency response before correction is shown in Figure 4. It can be seen that the frequency response function has a low degree of consistency, that is, there is a large difference between the dynamic characteristics of the initial finite element model and the actual structure, and its parameters need to be corrected.

Step 3: Calculate the sensitivity matrix and identify the increment of the parameter value to be corrected. Substitute the test frequency response, analysis frequency response and mass, stiffness and damping matrix of each group of the finite element model under the reduced coordinates obtained in steps 1 and 2 into the sensitivity formula to calculate the sensitivity matrix of the residual of the frequency response function under the reduced coordinates, and form the sensitivity equation of each frequency point with the residual vector. The increment of the parameter value to be corrected is obtained by solving the overdetermined system of equations composed of the sensitivity equations at the selected frequency points.

Step 4: Substitute the measured value and the analyzed value of the frequency response function at each frequency point into the formula to calculate the amplitude correlation of the frequency response function under the current parameters. When it is greater than the set convergence threshold of 0.85, the corrected parameter value and the corresponding iterative data are output, otherwise the current parameter is set to the initial value of the parameter and return to step 2 to iterate again until convergence. After 5 iterative steps, the correlation between the corrected analysis frequency response and the test frequency response is increased from 0.31 to 0.88, which is significantly improved. The comparison between the corrected finite element analysis frequency response and the test frequency response and their amplitude correlation are shown in Figures 5 and 6. Therefore, the method of the present invention can effectively improve the calculation efficiency while ensuring the correction accuracy for the correction problem of the bogie finite element model with a large number of degrees of freedom, and quickly and accurately obtain a dynamic finite element model that is consistent with the actual vibration characteristics of the bogie structure.

The present invention is generally applicable to the problem of correcting a large finite element model formed by a bogie structure using frequency response function data. The above is only a preferred embodiment of the present invention. It should be pointed out that for ordinary technicians in this technical field, without departing from the principle of the present invention, the present invention can also be applied to the correction of a large finite element model formed by a complex vibration structure similar to a bogie, and these applications should also be regarded as the scope of protection of the present invention.

Claims (4)

1. A frequency domain finite element model correction method based on a reduced basis, characterized in that the method comprises the following steps: The first step is to determine the output degrees of freedom and input degrees of freedom for the vibration test of the bogie structure, obtain the measured frequency response function of the structure after the test, and transform it into the reduced coordinates through the basis matrix; at the same time, set the parameter values to be corrected in the initial iteration step; The specific method for obtaining the measured frequency response function of the bogie structure in the first step is as follows: Determine the input and output degrees of freedom of the structure to be tested, and perform a hammer test on the structure to be tested; The measured frequency response function in each resonance region is obtained by selecting the frequency points within the half-power bandwidth of the mode indicator function: The superscript i represents the i-th resonance zone, and q is the total number of resonance zones. For the kth frequency point, a total of N _i frequency points are selected in the i-th resonance zone. Frequency point The measured frequency response corresponding to the j-th input degree of freedom at ; Assume that the vibration form of the i-th resonance zone is determined by a certain order test mode of the structure, select the calculation mode of the finite element model that matches this order test mode through the vibration mode correlation MAC and frequency error, and add several adjacent calculation modes as the expanded test frequency response function The basis Φ _F,i of Φ _F,i contains several rigid body modes, then Test frequency response function after expansion to full degrees of freedom This basis can be expressed as Where q _i is the corresponding fitting coefficient. By performing a series of matrix changes on the above formula, the frequency response function expansion formula can be obtained as follows: in is the component of the base on the test degree of freedom; Finally, take the expanded test frequency response function In a column After the inverse transformation of the basis matrix, the structural test frequency response function under the reduced coordinates can be obtained: Where Bs _is the basis matrix; The second step is to establish a dynamic finite element model of the bogie structure, and calculate the frequency response function analysis value under the reduced coordinate according to the parameter value to be corrected in the previous iteration step or the initial iteration step; The third step is to calculate the residuals of the frequency response function analysis value and the measured value under the reduced coordinates and its sensitivity matrix with respect to the parameters to be corrected, and solve the corresponding sensitivity equation to obtain the updated parameters and finite element model after the parameter increment; The fourth step is to calculate the amplitude correlation of the frequency response function corresponding to the updated parameter. If it meets the convergence criterion, the parameter value and the corresponding historical iteration data are output to complete the correction. Otherwise, return to step 2 and iterate again until convergence. The specific method for judging whether the objective function value meets the convergence criterion in the fourth step is as follows: The norm of the residual v(ω _k ,θ) of the analysis value is minimized, that is, it is considered that convergence is achieved when the residual norm approaches 0. If the frequency response amplitude correlation coefficient is used Describes the degree of overlap of the two curves. When the mean value at each frequency point is greater than the threshold _σmin, it is considered to have reached convergence, that is, 2. According to the frequency domain finite element model correction method based on reduced basis in claim 1, it is characterized in that the input and output degrees of freedom of the structure to be tested refer to the degrees of freedom for excitation during the hammer test and the degrees of freedom for outputting structural response data, and the determination method is: the arrangement of the output degrees of freedom clearly reflects the vibration forms of each order, and the input degrees of freedom are arranged to avoid the nodes of each order of modal vibration type of the structure. 3. The frequency domain finite element model correction method based on reduced basis according to claim 1 is characterized in that the specific method of obtaining the frequency response function analysis value of the bogie structure under reduced coordinates in the second step is as follows: Firstly, a bogie finite element model is established which is consistent with the actual structure, i.e., the dynamic characteristics of the structure to be measured, and the parameters sensitive to the structural response are selected as the parameters to be corrected; The finite element model is grouped and the frequency response function analysis value under the reduced coordinate is obtained after basis matrix transformation The parameterized expression is in To reduce the mass and stiffness matrix of the e-th group of the finite element model under the coordinate, α _e and γ _e are their corresponding correction coefficients; is the proportional damping and structural damping matrix under the reduced coordinate, _βe is its correction coefficient; θ is the parameter increment to be corrected composed of _αe , _αe , and _γe . 4. The frequency domain finite element model correction method based on reduced basis according to claim 1 is characterized in that the specific method of calculating the sensitivity matrix and identifying the increment of the parameter value to be corrected in the third step is as follows: The above formula is a sensitivity equation based on the Newton-Gauss optimization method. Its mathematical meaning is to solve the parameter increment direction and size that minimizes the residual between the frequency response function analysis value and the measured value under the current parameters; are the frequency response function measurement value frequency point and analysis value frequency point in the best matching frequency pair sequence selected according to the amplitude correlation under the framework of the kth group of frequency shift theory, and k＝1,…,N _match , N _match is the number of matched pairs; S and Δf are the sensitivity matrix and residual vector determined by the parameter increment θ ^r to be corrected in the previous iteration step or the initial iteration step at this pair of frequency points, and the specific calculation formula is: in is the frequency response function analysis value under reduced coordinates corresponding to the jth input degree of freedom, is the dynamic stiffness matrix of the finite element model in reduced coordinates; Substituting the frequency response function analysis value and the measured value under the reduced coordinates calculated in steps 2 and 3 into formula (10) can calculate the sensitivity matrix of the residual with respect to the parameters, and the parameter increment of the next iteration step can be obtained after solving the sensitivity equation.