CN113111547A - Frequency domain finite element model correction method based on reduced basis - Google Patents
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Abstract
The embodiment of the invention discloses a frequency domain finite element model correction method based on a reduced 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 to 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 values and the measured values under the reduced coordinates and a sensitivity matrix of the residual errors and the sensitivity matrix related to the parameters to be corrected, solving a corresponding sensitivity equation to obtain updated parameters and a finite element model, and repeatedly iterating 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 precision, so that the dynamic finite element model of the bogie structure is efficiently corrected by utilizing actual test data.
Description
Technical Field
The invention belongs to the field of finite element model correction, and particularly relates to a frequency domain method for correcting a dynamic finite element model with a large number of degrees of freedom formed by a bogie structure.
Background
With the continuous improvement of the speed of modern trains, the requirements on the reliability, the precision and the like of the bogie structure are higher and higher, and the form, the parameters and the like of the bogie structure need to be analyzed from a plurality of angles such as dynamics, structural fatigue, reliability and the like in the stages of designing, using, improving and the like of the bogie structure, so that the establishment of a finite element model capable of reflecting the actual dynamic characteristics of the bogie is very important. However, due to the complexity of each part of the bogie, the modeling process inevitably needs to simplify the structure and use a plurality of mechanical units containing parameters for dispersion, and the dynamic characteristics of the bogie finite element model may not be consistent with the actual structure depending on the initial model parameters selected by experience; and the formed bogie finite element model is often large in scale, and the analysis and calculation efficiency is low. At present, model modification methods for large finite element models mainly include a method of condensing a finite element model modification formula to test degrees of freedom, a method of reducing degrees of freedom based on a substructure modal synthesis technology with modification modal parameters as a target, a modification method based on a proxy model, and the like. However, when the test freedom degree of the model polycondensation is less, a larger approximation error is easily introduced, and the calculation amount for forming the polycondensation matrix is also larger; although the correction method utilizing the substructure modal synthesis has high calculation precision and efficiency, the error and uncertainty of the modal identification process are introduced by taking modal parameters as the correction target; although the correction method based on the proxy model can quickly obtain the correction result after the response surface of the correction target about the parameter to be corrected is established, the process of forming the response surface needs to calculate enough data to avoid the distortion of the proxy model, and the calculation amount is undoubtedly huge. How to efficiently and accurately correct a large finite element model formed by a bogie structure based on frequency domain response data has important significance for numerical simulation of the structure in engineering.
Disclosure of Invention
In order to solve the problem of low efficiency when the existing model correction technology is applied to a bogie structure, the invention provides a finite element model frequency domain correction method based on a reduced basis, which can obviously improve the calculation efficiency on the premise of ensuring the precision.
In order to achieve the purpose, the implementation of the invention adopts the following technical scheme:
the embodiment of the invention provides a frequency domain finite element model correction method based on a reduced basis, which is used for correcting a finite element model by using frequency response function data, wherein the finite element model of a bogie to be corrected has low correction efficiency due to the huge number of degrees of freedom, and the method comprises the following steps:
the method comprises the following steps: determining the output freedom and the input freedom of a bogie structure for vibration test, obtaining a measurement frequency response function of the structure after testing, and converting the measurement frequency response function into a reduced coordinate through a base matrix;
step two: establishing a dynamic finite element model of the bogie structure, and calculating a frequency response function analysis value under a reduced coordinate according to the parameter value of the previous iteration step;
step three: calculating residual errors of frequency response function analysis values and measured values under the reduced coordinates and sensitivity matrixes of the residual errors and the sensitivity matrixes related to parameters to be corrected, and solving corresponding sensitivity equations to obtain updated parameters and finite element models after parameter increment;
step four: calculating the frequency response function amplitude correlation corresponding to the updated parameter, if the correlation meets the convergence criterion, outputting the parameter value and corresponding historical iteration data to finish correction, and otherwise, returning to the second step to iterate again until convergence;
the specific method for obtaining the test frequency response of the bogie structure under the reduced coordinate in the first step is as follows:
the input and output degrees of freedom in the bogie structure vibration test respectively refer to the degree of freedom for excitation in the hammer test and the degree of freedom for outputting structural response data. The specific determination method comprises the following steps: the arrangement of the output freedom degree clearly reflects the vibration mode of each order, and the node of the modal vibration mode of each order of the structure needs to be avoided when the input freedom degree is arranged. A test system consisting of a data acquisition system, a force hammer, a sensor and data analysis software can acquire a measurement frequency response function on the input and output freedom degrees through a multi-reference-point hammering test. Because the actual data inevitably contains measurement noise, the frequency response function of the resonance area is not only less polluted by the noise, but also more sensitive to the parameters of the structure, namely, has higher sensitivity, and the coefficient matrix of the formed sensitivity equation is more accurate and is not easy to be ill-conditioned, only the frequency response function corresponding to the frequency points near the resonance area is selected to participate in model modification. In the implementation, as shown in fig. 2, the measured frequency response function in each resonance region can be obtained by selecting the frequency point in the half-power bandwidth of the modal indicator function
Wherein the superscript i represents the ith resonance region, q represents the total number of resonance regions,
n is selected for the ith resonance region of the kth frequency pointiThe frequency point is set to be a frequency point,
is a frequency point
And measuring frequency response corresponding to the jth input degree of freedom.
In practical application, the complete degree of freedom of the finite element model is not matched with the test degree of freedom, the number of the complete degree of freedom of the finite element model is usually far larger than that of the test degree of freedom of the finite element model, and a test frequency response function of the complete degree of freedom is needed when the base matrix is used for coordinate conversion. Therefore, to convert the measured frequency response function to a reduced coordinate, it needs to be expanded first. Setting the measured frequency response function of the ith resonance area
Corresponding to a certain test mode of the structure (if the test mode is a repetition frequency mode, corresponding to two modes with the same motion form and different phases), the motion form of the structure in the resonance region is mainly determined by the mode of the certain test mode and the adjacent mode thereof, so that the test mode is selected and tested according to the mode correlation (MAC) and the frequency errorTesting the calculation mode of finite element model matched with mode, and adding several orders of adjacent calculation modes as expansion
Base phi ofF,iThe number of the added neighboring modes is determined by observing the complexity of the frequency domain displacement motion of the resonance region and the coupling degree between the corresponding test mode and the neighboring modes. Furthermore, for the case of free boundary conditions, the various order motion profiles of the structure will contain both elastic deformation and rigid body motion of greater magnitude, hence ΦF,iIt should also contain several orders of rigid body modes. Then is formed by
Test frequency response function after expansion to complete degree of freedom
Can be represented by the substrate as
Wherein q isiFor the corresponding fitting coefficients, the frequency response function expansion formula obtained by performing a series of matrix changes on the above formula is
Wherein
Is the component of the substrate in the test degree of freedom.
Finally, the expanded test frequency response function is taken
In a certain column
The structure test frequency response function under the reduced coordinate can be obtained after the inverse transformation of the base matrix
Wherein B issFor basis matrices, reducing the coordinate values means basis matrix BsThe corresponding component value. It can be divided into a mode shape basis matrix and a mode synthesis basis matrix according to different methods of reducing the degree of freedom. The former is directly taken as a plurality of low-order finite element model vibration modes covering an analysis frequency band; the latter is obtained by dividing the whole structure into a plurality of sub-structures, then obtaining a constraint mode set and a main mode set phi, and eliminating the dependent generalized coordinates through a transformation matrix T, S
Bs＝TΦS (5)
In practical application, because each of the two kinds of basis matrices has its advantages and disadvantages, it should be specifically selected according to the actual structure to be corrected: the mode shape is directly used as a base matrix, so that the method is simple and effective and is easy to understand, but when the number of degrees of freedom of the structure finite element model is large, the calculation of the mode needs to consume much time and calculation amount; the calculation amount required by the modal synthetic base calculated by the split substructure is greatly reduced along with the increase of the number of the substructures, but the process is more complicated, and when the number of the interface degrees of freedom between the substructures is larger, the reduced coordinates still have larger number of the degrees of freedom, and the calculation efficiency of the correction process is reduced. While both are similar for general structural vibrations, their ability to be fitted as a substrate to describe actual motion.
The concrete method for obtaining the analysis value of the bogie structure frequency response function under the reduced coordinates in the step two is as follows:
firstly, establishing a finite element model which is consistent with the dynamic characteristics of the bogie structure through any commercial finite element software, and if the finite element model is divided into a plurality of groups according to the types of materials and units, selecting a part which is sensitive to the structural response from the material parameters (such as elastic modulus, density and damping) of the groups as a parameter to be corrected. The dynamic matrix of each group of the finite element model is softAfter the element is derived, the function analysis value under the reduced coordinate can be obtained by the base matrix
Is parameterized as
Wherein
For the e-th subgroup of the finite element model in reduced coordinates, the mass and stiffness matrix, alphae、γeFor its respective correction factor;
then the proportional damping and the structural damping matrix, beta, under the reduced coordinateeIs its correction factor. Theta is alphae、βe、γeAnd (4) forming the parameter increment to be corrected. If some grouped material parameters are not used as correction objects in practice, the parameters are only required to be fixed as initial values of the parameters in a sensitivity equation and then solved.
The specific method for calculating the sensitivity matrix and identifying the parameter value increment to be corrected in the third step is as follows:
the above formula is a sensitivity equation derived by a Newton-Gauss method, and the mathematical meaning of the sensitivity equation is that the direction and the size of parameter increment which enables the frequency response function analysis value and the residual error of the measured value to be minimum under the current parameter are solved, wherein the above formula is the sensitivity equation derived by the Newton-Gauss method, and the mathematical meaning of the sensitivity equation is that the direction and the
Frequency response function measured value frequency points and analysis value frequency points in the optimal matching frequency pair sequence selected according to the amplitude correlation under the k-th group of frequency shift theory framework respectively, and k is 1, …, Nmatch，NmatchThen is the matched logarithm, S, delta f is the parameter increment theta to be corrected from the last iteration step or the initial iteration step at the pair of frequency pointsrThe determined sensitivity matrix and residual vector are calculated by the following specific formula
Wherein
For inputting the frequency response function analysis value under the reduced coordinate with the degree of freedom consistent with the test model,
the dynamic stiffness matrix of the finite element model under the reduced coordinate is obtained. And substituting the frequency response function analysis value and the measured value under the reduced coordinates calculated in the second step and the third step into the formula to calculate the sensitivity matrix of the residual error relative to the parameters, and solving the sensitivity equation to obtain the parameter increment of the next iteration step. However, in practical implementation, the coefficient matrix is often disturbed due to unavoidable measurement noise pollution and truncation error during the degree of freedom reduction, and in addition, the coefficient matrix often has a large condition number to further amplify the disturbance, so that accurate identification of parameters is difficult. To improve this problem, it is often necessary to perform more elaborate operations in the vibration testing and numerical modeling processes to reduce the main error sources, and to reasonably select the data involved in parameter identification and to reduce the ill-posed nature of the equation by using a certain numerical approach.
The specific method for judging whether the objective function value meets the convergence criterion in the fourth step is as follows:
the convergence criterion is also important in the actual model correction, and the reasonable criterion can obtain the most reasonable correction result with the least iteration steps, whereas the unreasonable criterion not only can increase the iteration steps, but also can cause divergence due to iteration to a parameter point losing physical significance. The method uses the residual v (omega) of the frequency response function measured value and the analysis value under the reduced coordinate with the structurekTheta), i.e. convergence is achieved when iteration is performed until the residual norm approaches to 0, which can also be understood geometrically as the approximation of the analysis frequency response function curve to the test frequency response function curve, so that if the frequency response amplitude correlation coefficient is used, the norm of theta) is considered to be the minimization target, i.e. the convergence is achieved when the iteration is performed until the residual norm approaches to 0
Describing the degree of coincidence of two curves, when their mean values at respective frequency points are greater than a threshold value sigmaminThen it is considered that convergence is reached, i.e.
In the specific implementation process, the criterion for judging the iterative convergence is not fixed, and different criteria can be set according to different actual correction targets so as to obtain a group of parameter values which make the dynamic characteristics of the corrected finite element model most accord with the expectation.
Compared with the prior art, the technical scheme of the invention has the following beneficial technical effects:
compared with the prior art, the frequency domain finite element model correction method based on the reduced basis can effectively reduce the matrix operation scale and improve the calculation efficiency of the correction algorithm for the finite element model with a huge number of degrees of freedom formed by the bogie structure by using the reduced basis method, so that the bogie finite element model which is consistent with the actual dynamic characteristics after correction can be efficiently obtained. The invention provides a beneficial scheme for solving the problem of low correction efficiency caused by large-scale finite element models in engineering.
Drawings
FIG. 1 is a block diagram of a process of a method for modifying a finite element model in a frequency domain based on a reduced basis according to the present invention;
FIG. 2 is a test frequency response function piecewise graph;
FIG. 3 is a pictorial illustration of a finite element model and a sub-structure partition of a bogie frame;
FIG. 4 is a graph comparing the initial analysis frequency response and the test frequency response before correction;
FIG. 5 is a graph comparing the corrected analysis frequency response with the test frequency response;
FIG. 6 is a comparison graph of the correlation between the analysis frequency response and the test frequency response amplitude before and after correction;
Detailed Description
The invention is described in detail below with reference to the figures and examples.
The invention provides a finite element model frequency domain correction method based on a reduced basis, which is characterized in that the displacement of a finite element model of a bogie structure is expressed as a modal comprehensive reduced basis or linear superposition of modal vibration type reduced basis, a sensitivity matrix of a frequency response function residual error under a reduced coordinate with respect to a selected parameter to be corrected is obtained through derivation, and a sensitivity equation of corresponding identification parameter increment is established according to a Newton-Gauss method, so that the correction process is carried out on the reduced coordinate, and the calculation efficiency is obviously improved on the premise of ensuring the precision. Meanwhile, model correction is carried out under the theoretical framework of the frequency shift technology, so that the accuracy of parameter identification is effectively improved, and the iterative convergence is increased; and a segmented frequency response expansion method with higher precision is deduced based on the modal expansion formula so as to reduce the influence of expansion errors on the correction process. The test example of the bogie frame proves that for a large finite element model with a large number of degrees of freedom, the method can efficiently obtain a correction model which is consistent with the dynamic characteristics of an actual structure.
The invention is illustrated by way of example of a finite element model modification of a bogie frame of the type shown in figure 3. The finite element model has 181356 degrees of freedom, if the traditional model correction method is adopted, the calculation efficiency is low and the noise resistance is poor, and the defects can be avoided by adopting the method of the invention.
A frequency domain finite element model modification method based on a reduced basis is characterized by comprising the following steps:
firstly, determining the output freedom and the input freedom of a structure for vibration test, obtaining a measurement frequency response function of the structure after testing, and converting the measurement frequency response function into a reduced coordinate through a base matrix;
secondly, establishing a dynamic finite element model of the structure, and calculating a frequency response function analysis value under a reduced coordinate according to the parameter value of the previous iteration step;
thirdly, calculating residual errors of frequency response function analysis values and measured values under the reduced coordinates and sensitivity matrixes of the residual errors and the sensitivity matrixes related to parameters to be corrected, and solving corresponding sensitivity equations to obtain updated parameters and finite element models after parameter increment;
and fourthly, calculating the frequency response function amplitude correlation corresponding to the updated parameter, if the correlation meets the convergence criterion, outputting the parameter value and corresponding historical iteration data to finish correction, and otherwise, returning to the second step to iterate again until convergence.
The specific method for obtaining the frequency response function of the structure under the reduced coordinate in the first step is as follows:
the input and output degrees of freedom in the structural vibration test respectively refer to the degree of freedom for excitation in the hammering test and the degree of freedom for outputting structural response data. The specific determination method comprises the following steps: the arrangement of the output freedom degree clearly reflects the vibration mode of each order, and the node of the modal vibration mode of each order of the structure needs to be avoided when the input freedom degree is arranged. A test system consisting of a data acquisition system, a force hammer, a sensor and data analysis software can acquire a measurement frequency response function on the input and output freedom degrees through a multi-reference-point hammering test. Because the actual data inevitably contains measurement noise, the frequency response function of the resonance area is not only less polluted by the noise, but also more sensitive to the parameters of the structure, namely, has higher sensitivity, and the coefficient matrix of the formed sensitivity equation is more accurate and is not easy to be ill-conditioned, only the frequency response function corresponding to the frequency points near the resonance area is selected to participate in model modification. In the implementation, as shown in fig. 2, the measured frequency response function in each resonance region can be obtained by selecting the frequency point in the half-power bandwidth of the modal indicator function
Wherein
For the k-th frequency point, the frequency of the first frequency point,
for the measured frequency response corresponding to the jth input degree of freedom at that frequency point,
the superscript i represents the ith resonance region, and q represents the total number of resonance regions.
In practical application, the complete degree of freedom of the finite element model is not matched with the test degree of freedom, the number of the complete degree of freedom of the finite element model is usually far larger than that of the test degree of freedom of the finite element model, and a test frequency response function of the complete degree of freedom is needed when the base matrix is used for coordinate conversion. Therefore, to convert the measured frequency response function to a reduced coordinate, it needs to be expanded first. Setting the measurement frequency response of the ith resonance area
Corresponding to a certain test mode of the structure (if the test mode is a repetition frequency mode, corresponding to two modes with the same motion form and different phases), the motion form of the structure in the resonance region is mainly determined by the mode and the adjacent modes thereof, so that the calculation mode of the finite element model matched with the test mode is selected through the vibration mode correlation (MAC) and the frequency error, and a plurality of adjacent calculation modes are added as the expansion mode
Substrate of (1), noted as ΦF,iThe number of the added neighboring modes is determined by observing the complexity of the frequency domain displacement motion of the resonance region and the coupling degree between the corresponding test mode and the neighboring modes. Further, for the free boundary condition case, the junctionThe motion state of each step of the structure will simultaneously include elastic deformation and rigid motion with larger amplitude, therefore phiF,iIt should also contain several orders of rigid body modes. Therefore it has the advantages of
Wherein
To expand the test frequency response to a complete degree of freedom, the above formula is subjected to a series of matrix changes to obtain
Wherein
Is the component of the substrate in the test degree of freedom.
Finally, the expanded one is taken
Frequency response of a certain row in
The structure test frequency response under the reduced coordinate can be obtained by the following formula
Wherein B issReduced coordinates are the basis matrix BsCorresponding to the component. It can be divided into a mode shape basis matrix and a mode synthesis basis matrix according to different methods of reducing the degree of freedom. The former is directly taken as a plurality of low-order finite element model vibration modes covering an analysis frequency band; the latter is obtained by dividing the whole structure into a plurality of sub-structures, then obtaining a constraint mode set and a main mode set phi, and eliminating the dependent generalized coordinates through a transformation matrix T, S
Bs＝TΦS (5)
In practical application, because each of the two kinds of basis matrices has its advantages and disadvantages, it should be specifically selected according to the actual structure to be corrected: the mode shape is directly used as a base matrix, so that the method is simple and effective and is easy to understand, but when the number of degrees of freedom of the structure finite element model is large, the calculation of the mode needs to consume much time and calculation amount; the calculation amount required by the modal synthetic base calculated by the split substructure is greatly reduced along with the increase of the number of the substructures, but the process is more complicated, and when the number of the interface degrees of freedom between the substructures is larger, the reduced coordinates still have larger number of the degrees of freedom, and the calculation efficiency of the correction process is reduced. While both are similar for general structural vibrations, their ability to be fitted as a substrate to describe actual motion.
The specific method for obtaining the structural frequency response function analysis value under the reduced coordinate in the second step is as follows:
firstly, establishing a finite element model which is consistent with the dynamic characteristics of an actual structure through any commercial finite element software, and if the finite element model is divided into a plurality of groups according to the types of materials and units, selecting a part which is sensitive to the response of the structure from the material parameters (such as elastic modulus, density and damping) of the groups as a parameter to be corrected. After the dynamic matrix of each group of the finite element model is derived from software, the function analysis value under the reduced coordinate can be obtained by the base matrix
Is parameterized as
Wherein
For the e-th subgroup of the finite element model in reduced coordinates, the mass and stiffness matrix, alphae、γeFor its respective correction factor;
then the proportional damping and the structural damping matrix, beta, under the reduced coordinateeIs its correction factor. Theta is alphae、βe、γeAnd (4) forming the parameter increment to be corrected. If some grouped material parameters are not used as correction objects in practice, the parameters are only required to be fixed as initial values of the parameters in a sensitivity equation and then solved.
In the third step, the specific method for calculating the sensitivity matrix and identifying the parameter value increment to be corrected is as follows:
the above equation is a sensitivity equation derived by a Newton-Gauss method, and the mathematical meaning of the equation is the parameter increment direction and size which enable the frequency response function analysis value and the residual error of the measured value to be minimum under the current parameter solving. Wherein
Frequency response function measured value frequency points and analysis value frequency points in the optimal matching frequency pair sequence selected according to the amplitude correlation under the k-th group of frequency shift theory framework respectively, and k is 1, …, Nmatch，NmatchThen the matched logarithm is obtained, S, delta f are the parameter increment theta to be corrected from the last iteration step at the pair of frequency pointsrThe determined sensitivity matrix and residual vector are calculated by the following specific formula
Wherein
For inputting the frequency response function analysis value under the reduced coordinate with the degree of freedom consistent with the test model,
the dynamic stiffness matrix of the finite element model under the reduced coordinate is obtained. And substituting the frequency response function analysis value and the measured value under the reduced coordinates calculated in the second step and the third step into the formula to calculate the sensitivity matrix of the residual error relative to the parameters, and solving the sensitivity equation to obtain the parameter increment of the next iteration step. However, in practical implementation, the coefficient matrix is often disturbed due to unavoidable measurement noise pollution and truncation error during the degree of freedom reduction, and in addition, the coefficient matrix often has a large condition number to further amplify the disturbance, so that accurate identification of parameters is difficult. To improve this problem, it is often necessary to perform more elaborate operations in the vibration testing and numerical modeling processes to reduce the main error sources, and to reasonably select the data involved in parameter identification and to reduce the ill-posed nature of the equation by using a certain numerical approach.
The third step is a specific method for judging whether the objective function value meets the convergence criterion, which is as follows:
the convergence criterion is also important in the actual model correction, and the reasonable criterion can obtain the most reasonable correction result with the least iteration steps, whereas the unreasonable criterion not only can increase the iteration steps, but also can cause divergence due to iteration to a parameter point losing physical significance. The method uses the residual v (omega) of the frequency response function measured value and the analysis value under the reduced coordinate with the structurekTheta), i.e. convergence is achieved when iteration is performed until the residual norm approaches to 0, which can also be understood geometrically as the approximation of the analysis frequency response function curve to the test frequency response function curve, so that if the frequency response amplitude correlation coefficient is used, the norm of theta) is considered to be the minimization target, i.e. the convergence is achieved when the iteration is performed until the residual norm approaches to 0
Describe the degree of coincidence of the two curves whenIts mean value at each frequency point is greater than a threshold value sigmaminThen it is considered that convergence is reached, i.e.
In the specific implementation process, the criterion for judging the iterative convergence is not fixed, and different criteria can be set according to different actual correction targets so as to obtain a group of parameter values which make the dynamic characteristics of the corrected finite element model most accord with the expectation.
Referring to fig. 1, the invention provides a frequency domain finite element model modification method based on a reduced basis, which comprises the following specific steps:
the method comprises the following steps: and acquiring a structure test frequency response under the reduced coordinate. A bogie frame supported by elasticity is adopted, 192 output degrees of freedom and 3 input degrees of freedom are selected in total, and then a hammering test is carried out to obtain a test frequency response function on the corresponding degree of freedom. Because the structure is a free boundary condition, the first 3-order rigid body mode of the integral 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 bases for expanding the test frequency response function, and the test frequency response function on the expanded complete degree of freedom is obtained. In addition, to obtain the basis matrix, as shown in fig. 3, the finite element framework model is divided into 3 substructures to perform modal synthesis to obtain a modal synthesis reduced basis matrix. The degree of freedom of the model under the natural coordinate is 181356, and the degree of freedom of the model under the reduced coordinate is 1830, so that the matrix operation scale in the correction process is greatly reduced. Finally, the expanded test frequency response can be converted to the reduced coordinate according to the reduced base matrix.
Step two: and calculating the frequency response function analysis value in the reduced coordinate by using the finite element model. The finite element model of the framework is divided into 10 groups as shown in FIG. 3. The modulus of elasticity, the density and the damping coefficient of each group are taken as parameters to be corrected, and the total number of the parameters is 30. After parameter initial values are set, commercial finite element software derives a mass matrix, a rigidity matrix, a damping matrix and the like of each group of the model, the mass matrix, the rigidity matrix, the damping matrix and the like are converted into a reduced coordinate through a reduced base matrix, and a frequency response function analysis value in the reduced coordinate is obtained through calculation. As shown in fig. 4, the fitting frequency response and the test frequency response pair before correction have a low frequency response function goodness of fit, that is, the dynamic characteristics of the initial finite element model are greatly different from the actual structure, and the parameters thereof need to be corrected.
Step three: a sensitivity matrix is calculated and the parameter value increment to be corrected is identified. And substituting the test frequency response and the analysis frequency response under the reduced coordinates obtained in the first step and the second step and the mass, rigidity and damping matrix of each group of the finite element model into a sensitivity formula to calculate and obtain a sensitivity matrix of the frequency response function residual under the reduced coordinates, and forming a sensitivity equation of each frequency point with the residual vector. And obtaining the increment of the parameter value to be corrected by solving an overdetermined equation set consisting of sensitivity equations at the selected frequency points.
Step four: and substituting the measured value and the analyzed value of the frequency response function of each frequency point into a formula to calculate the amplitude correlation of the frequency response function under the current parameter, outputting the corrected parameter value and corresponding iteration data when the amplitude correlation is greater than the set convergence threshold value of 0.85, otherwise, setting the current parameter as the initial value of the parameter and returning to the second step for iterating again until convergence. After 5 iteration steps, the correlation between the corrected analysis frequency response and the test frequency response is improved from 0.31 to 0.88, and the correlation is obviously improved. The comparison between the corrected finite element analysis frequency response and the test frequency response and the amplitude correlation thereof are shown in fig. 5 and 6. Therefore, the method can effectively improve the calculation efficiency on the premise of ensuring the correction precision for the correction problem of the bogie finite element model with huge number of degrees of freedom, and quickly and accurately obtain the dynamic finite element model which is consistent with the real vibration characteristic of the bogie structure.
While the present invention has been described in terms of a method for modifying a large finite element model formed from a bogie structure using frequency response function data, it should be understood that the present invention can be applied to the modification of a large finite element model formed from a complex vibration structure similar to a bogie without departing from the principles of the present invention, and such applications should be considered as the scope of the present invention.
Claims (6)
1. A frequency domain finite element model modification method based on a reduced basis is characterized by comprising the following steps:
firstly, determining the output freedom and the input freedom of a bogie structure for vibration test, obtaining a measurement frequency response function of the structure after test, and converting the measurement frequency response function into a reduced coordinate through a base matrix; setting the parameter value to be corrected of the initial iteration step;
secondly, establishing a dynamic finite element model of the bogie structure, and calculating a frequency response function analysis value under a reduced coordinate according to a to-be-corrected parameter value of the previous iteration step or the initial iteration step;
thirdly, calculating residual errors of frequency response function analysis values and measured values under the reduced coordinates and sensitivity matrixes of the residual errors and the sensitivity matrixes related to parameters to be corrected, and solving corresponding sensitivity equations to obtain updated parameters and finite element models after parameter increment;
and fourthly, calculating the frequency response function amplitude correlation corresponding to the updated parameter, if the correlation meets the convergence criterion, outputting the parameter value and corresponding historical iteration data to finish correction, and otherwise, returning to the second step to iterate again until convergence.
2. The method for modifying a finite element model in frequency domain based on reduced basis as claimed in claim 1, wherein the specific method for obtaining the measured frequency response function of the bogie structure in the first step is as follows:
determining the input and output freedom degrees of the structure to be tested, and carrying out a hammering test on the structure to be tested;
and obtaining the measurement frequency response function in each resonance area by selecting the frequency point in the half-power bandwidth of the modal indication function:
wherein the superscript i represents the ith resonance region, q represents the total number of resonance regions,
n is selected for the ith resonance region of the kth frequency pointiThe frequency point is set to be a frequency point,
is a frequency point
Measuring frequency response corresponding to the jth input degree of freedom;
setting the vibration form of the ith resonance area to be determined by a certain order of test mode of the structure, selecting a calculation mode of a finite element model matched with the order of test mode through vibration mode correlation MAC and frequency error, and adding a plurality of orders of adjacent calculation modes as an expansion test frequency response function
Base phi ofF,i，ΦF,iIncluding several orders of rigid body modes, is composed of
Test frequency response function after expansion to complete degree of freedom
Can be represented by the substrate as
Wherein q isiFor the corresponding fitting coefficients, the frequency response function expansion formula obtained by performing a series of matrix changes on the above formula is
Wherein
Is the component of the substrate in the test degree of freedom;
finally, the expanded test frequency response function is taken
In a certain column
The structure test frequency response function under the reduced coordinate can be obtained after the inverse transformation of the base matrix
Wherein B issIs a basis matrix.
3. The method for modifying a finite element model in a frequency domain based on a reduced basis as claimed in claim 2, wherein the input and output degrees of freedom of the structure to be tested refer to degrees of freedom for excitation during a hammering test and degrees of freedom for outputting response data of the structure, and the determining method comprises: the arrangement of the output freedom clearly reflects the vibration mode of each order, and the node of the modal shape of each order of the structure is avoided when the input freedom is arranged.
4. The method for frequency-domain finite element model modification based on reduced basis of claim 2, wherein the specific method for obtaining the analysis value of the bogie structure frequency response function under the reduced coordinates in the second step is as follows:
firstly, establishing a bogie finite element model which is in accordance with the dynamic characteristics of an actual structure, namely a structure to be detected, and selecting parameters sensitive to structural response as parameters to be corrected;
grouping the finite element models, and obtaining a frequency response function analysis value under a reduced coordinate after base matrix transformation
Is parameterized as
Wherein
For the e-th subgroup of the finite element model in reduced coordinates, the mass and stiffness matrix, alphae、γeFor its respective correction factor;
then the proportional damping and the structural damping matrix, beta, under the reduced coordinateeA correction factor for it; theta is alphae、βe、γeAnd (4) forming the parameter increment to be corrected.
5. The reduced radix based frequency domain finite element model modification method of claim 1, wherein the third step of calculating the sensitivity matrix and identifying the parameter value increment to be modified comprises the following steps:
the above formula is a sensitivity equation based on a Newton-Gauss optimization method, and the mathematical meaning of the sensitivity equation is that the parameter increment direction and the parameter increment size which enable the residual error between the frequency response function analysis value and the measured value to be minimum under the current parameter are solved; wherein
Frequency response function measured value frequency points and analysis value frequency points in the optimal matching frequency pair sequence selected according to the amplitude correlation under the k-th group of frequency shift theory framework respectively, and k is 1, …, Nmatch，NmatchThe matched logarithm is obtained; s and delta f are the pairIncrement theta of parameter to be corrected at frequency point by last iteration step or initial iteration steprThe determined sensitivity matrix and residual vector are calculated by the following specific formula
Wherein
For the reduced coordinate down-frequency response function analysis value corresponding to the jth input degree of freedom,
the dynamic stiffness matrix of the finite element model under the reduced coordinate is obtained;
and substituting the frequency response function analysis value and the measured value under the reduced coordinates calculated in the second step and the third step into a formula (10), calculating a sensitivity matrix of the residual error relative to the parameters, and solving a sensitivity equation to obtain the parameter increment of the next iteration step.
6. The method for modifying a finite element model in frequency domain based on reduced basis as claimed in claim 1, wherein the specific method for determining the convergence criterion of the objective function value in the third step is as follows:
residual v (ω) of the analysis valuekTheta), i.e. assuming convergence is reached when iteration is performed until the residual norm approaches 0, if frequency response amplitude correlation coefficients are used
Describing the degree of coincidence of two curves, when their mean values at respective frequency points are greater than a threshold value sigmaminThen it is considered that convergence is reached, i.e.
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