CN109740216B - Modal separation degree judging method for structural model correction - Google Patents
Modal separation degree judging method for structural model correction Download PDFInfo
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Abstract
The invention discloses a mode separation degree judging method for correcting a structural model, which comprises the steps of firstly calculating an MAC matrix of an original structural model; then, the state of the structure is changed by selecting points one by one, and the MAC matrix of the new structure model relative to the original model is calculated; and finally subtracting the maximum value in each MAC matrix obtained by changing the structural state from the maximum value in the MAC matrix of the original model, wherein the obtained result is used as the mode confidence coefficient criterion sensitivity of each condition. And then comparing the sensitivity of the mode confidence coefficient criterion one by one, wherein points with low sensitivity indicate that the mode of a certain order obtained after the model is changed is similar to the mode of a certain order of the original model, and the mode separation degree is poor, so that an approximate equation can be obtained, and the superposition solution of the equation is not facilitated. The invention is a prediction of the suitability of the additional mass point or the rigidity changing point in the model correction, can be realized through computer simulation, omits the process of actually testing the points one by one, and improves the efficiency.
Description
Technical Field
The invention belongs to the technical field of structural dynamics modeling, and particularly relates to a mode separation degree judging method for structural model correction.
Background
Model correction is a key technology in the technical field of structural dynamics modeling, and an accurate model after correction provides great convenience for dynamic characteristic analysis of a structure. The existing methods for correcting various models have the following same points: and substituting test data of the model into a correction equation by taking the test data of the model as a correction basis of the finite element model, and solving parameters to be corrected of the model. However, the number of test modes available through experiments is often limited, so that there may be cases where the unknowns to be solved are more than the correction equations. This is a problem of uncertainty in the solution, and its calculation accuracy is poor. The usual methods for solving the above problems are: and carrying out modal tests for a plurality of times by adding mass or changing local rigidity for a plurality of times, and adding the new Cheng Die obtained by the modal test after changing the structure into the original equation to change the static uncertain equation set into an overdetermined equation set so as to improve the correction accuracy degree.
However, not all positions are suitable for adding mass or changing stiffness, and when determining whether a position is suitable, it is considered that the new structure obtained after adding mass or changing stiffness at that position should have a greater degree of modal separation from the original structure. The resulting modified equations may also be approximated if the resulting modality approximates, and the effect of the approximated equations for superposition solutions may not be ideal. If the mass is added or the rigidity is changed from point to point, the test is carried out, the correction equation is obtained and then is compared, the whole process is very complicated, and the workload is greatly increased.
Therefore, for solving the static and unstable situation of insufficient equation number during model correction, a simple mode separation degree judging method is needed to judge the mode approximation degree of the new model and the original model after the additional mass of the model or the rigidity is changed, so that a proper additional mass or rigidity change position is selected structurally for mode test.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provides a mode separation degree judging method for correcting a structural model. The method is suitable for selecting the position of the additional mass or the position of the changed rigidity when the model is corrected, and avoids the phenomenon that the mode is approximate due to low mode separation degree after the additional mass or the rigidity is changed, thereby affecting the accuracy of solving the unknown number after the equation is overlapped. The method can be carried out on a finite element model, the comparison is carried out through the simulation result, the comparison result is used for prediction, the process of testing on test pieces one by one is omitted, the workload is reduced, and the working efficiency is improved.
In order to achieve the above purpose, the present invention adopts the following technical scheme.
The invention discloses a mode separation degree judging method for correcting a structural model, which is characterized by comprising the following steps of:
step 1, according to the data of each order of vibration modes of the finite element model to be corrected
Substituting the model confidence coefficient into a model confidence coefficient criterion formula to solve a model confidence coefficient matrix, namely MAC, between the mode shapes of each order of the finite element model to be corrected A A matrix;
step 2, selecting a plurality of positions with additional mass or rigidity being changeable on the finite element model to be corrected, and calculating the finite element mode after the additional mass or rigidity being changed at each position to obtain the mode shape of each order of the new finite element model;
step 3, regarding each time of adding mass or changing rigidity in the step 2 to obtain a new finite element model as a new condition, wherein each new condition corresponds to a new finite element model, and each new finite element model corresponds to each order of mode shape; substituting the mode shapes of each order of the new finite element model and each order of the finite element model to be corrected into a mode confidence coefficient criterion formula according to the mode shapes of each order of the new finite element model and each order of the finite element model to be corrected, and calculating a mode confidence coefficient matrix MAC of the new finite element model and the finite element model to be corrected, which are obtained under each condition AB ；
Step 4, firstly, the MAC obtained in the step 1 A Maximum value MAC of selected element in matrix Amax The MAC obtained after calculation in each case in the step 3 is respectively selected AB Maximum value MAC of element in matrix ABmax With the MAC Amax Subtracting the MAC under each of the conditions ABmax The mode confidence criterion sensitivity S in each case is obtained MAC ；
Step 5, comparing the mode confidence coefficient criterion sensitivity S calculated under each condition MAC Size of S MAC The smaller the model, the closer the mode shapes of the two finite element models are; then after adding mass or changing model rigidity at the position, the obtained new finite element model is close to the finite element model to be corrected, solving is not facilitated, and the position is abandoned in the test.
The step 1 comprises the following steps:
let the ith order mode of finite element model to be corrected be
The j-th order mode is->
The correlation between the modes of the finite element model can be described by using a mode confidence coefficient criterion, and the formula is as follows:
wherein, MAC Aij Representing the ith order mode of vibration of the finite element model to be corrected
And the j-th order mode->
A modality confidence between; />
Represents the transpose of the ith order mode of vibration, +.>
Taking a mode after representing the product of the transpose of the ith order mode and the jth order mode,
representing the square after taking the modulus; the denominator is the product of two modes;
MAC Aij the value of (2) is between 0 and 1, reflects the correlation between the two-order vibration modes, and the closer to 1, the closer the two-order vibration modes are; if the estimated order is m-order, MAC A The MAC matrix is an m multiplied by m dimensional matrix, and according to a formula, the MAC matrix obtained by comparing the mode shapes of each order of the finite element model is 1.
The step 2 comprises the following steps:
2-1) initial selection of positions: selecting a plurality of positions which are easy to attach mass or change rigidity on the test piece as initial selection positions;
2-2) modifying the finite element model: according to the initial selection result, carrying out structural modification of additional mass or rigidity change in the finite element model to be modified one by one, thereby obtaining a new finite element model;
2-3) modal analysis: and carrying out modal analysis on the new finite element models with the modified structures in sequence to obtain the mode shape of each order of each new finite element model.
The step 3 comprises the following steps:
setting finite element model to be correctedThe ith order mode is
The j-th order vibration mode of the new finite element model obtained by adding mass or changing rigidity on a certain position of the finite element model to be corrected is +.>
The relevance of the two models can be described by a modal confidence criterion, the formula being:
MAC ABij representing the ith order mode of vibration of the finite element model to be corrected
J-th order mode of vibration of new finite element model after adding mass or changing rigidity ∈>
A modality confidence between; />
Representing the transposition of the ith order mode of the finite element model to be corrected, < >>
Taking a model after representing the product of the transpose of the ith order vibration mode of the finite element model to be corrected and the jth order vibration mode of the finite element model after adding mass or changing rigidity;
MAC ABij the value of (2) is between 0 and 1, reflecting the correlation between the two modes; the closer this value is to 1, the closer the two-order mode shape is; if the estimated order is m-order, MAC AB Is an m x m dimensional matrix;
according to formula (2), performing modal confidence calculation between each new finite element model obtained in step (2) and the finite element model to be corrected to obtain matrix MAC respectively AB 。
The step 4 comprises the following steps:
4-1) the MAC obtained in step 1 A The maximum value of the selected element in the matrix is denoted as MAC A max . The mode confidence coefficient matrix between the new finite element model and the model to be corrected obtained after adding mass or changing rigidity at each position in the step 3 is MAC AB The mode confidence matrix obtained by adding mass or changing rigidity at the nth position is recorded as MAC ABn Finding out the MAC ABn Maximum MAC of elements in matrix ABn max ；
4-2) recording the variation of the maximum value of the element in the MAC matrix before and after the added mass or the rigidity change of the finite element model to be corrected as the mode confidence coefficient criterion sensitivity, and then the mode confidence coefficient criterion sensitivity formula before and after the added mass or the rigidity change at the nth position is as follows:
S nMAC ＝MAC A max -MAC ABn max (3)
from step 1, the MAC in the formula A max Is a matrix MAC A The maximum value element of (3) is 1, so that the formula (3) is simplified as:
S nMAC ＝1-MAC ABn max (4)。
the step 5 comprises the following steps:
and (3) comparing the mode confidence coefficient sensitivity calculated in the step (4) one by one, when the sensitivity value of some positions is obviously lower than other sensitivity values, indicating that after the mass is added or the rigidity is changed at the positions, the obtained vibration mode is similar to a model of the finite element model to be corrected, and then the equation introduced during model correction can be similar to the previous equation, so that the solution is not facilitated, and therefore, the positions are discarded when the positions are selected on the test piece.
Compared with the prior art, the invention has the following advantages and beneficial effects:
1. the invention is suitable for preparation and screening before the correction of the structural model, in particular to any structural model correction method which needs to utilize additional mass or change rigidity for multiple tests, and has wide application range.
2. The method can point out the modal separation degree between the new finite element model and the finite element model to be corrected, which is obtained after the finite element model of the test piece is added with mass or rigidity is changed, and provides a reference for the position selection of the added mass or rigidity on the test piece. The problem of correction equation approximation caused by vibration mode approximation can be effectively avoided, and the accuracy of solving during model correction can be improved.
3. The invention can operate on the finite element model, predicts the actual result by using the finite element result, avoids the complicated operation of adding mass point by point or changing the rigidity test on the real structure, and improves the working efficiency.
Drawings
FIG. 1 is a flow chart of a method of one embodiment of the present invention.
Fig. 2 is a system architecture of a finite element model of an embodiment of the present invention.
Detailed Description
In the structural model correction, the condition that the parameters to be corrected are more than the number of the equations is usually encountered, and the number of the equations needs to be increased to solve the problem of uncertainty of the solution. The conventional method for increasing the number of processes is to add concentrated mass to the structure or change the local rigidity of the structure, and the principle of selecting the points is to obtain a new structure with larger distinction from the original mode after changing the state of the structure as much as possible. According to the mode separation degree judging method for structural model correction, based on mode confidence coefficient criterion sensitivity, the mode separation degree of a structure with a previous structure is judged after the mass is added or the rigidity is changed at certain positions through the change of a MAC (mode confidence coefficient criterion, modal Assurance Criteria) matrix of the mode correlation from the mode shape angle of a finite element model. Firstly, calculating an MAC matrix of a finite element model to be corrected (namely an original structure model); then, the state of the structure is changed by selecting points one by one, and the MAC matrix of a new finite element model (namely, a new structure model after the added mass or the curvature is changed) relative to the original model is calculated; and finally subtracting the maximum value in each MAC matrix obtained by changing the structural state from the maximum value in the MAC matrix of the original model, wherein the obtained result is used as the mode confidence coefficient criterion sensitivity of each condition. And then comparing the calculated mode confidence coefficient criterion sensitivities one by one, wherein points with low sensitivity indicate that a certain order of vibration mode obtained after the model is changed is similar to a certain order of vibration mode of the original model, and the mode separation degree is poor, so that an approximate equation can be obtained, and the superposition solution of the equation is not facilitated. The method is a prediction of whether the additional mass point or the rigidity change point is suitable or not in the model correction, and can be realized through computer simulation, so that the process of actually measuring the points one by one for trying is omitted, and the efficiency is improved.
The mode confidence coefficient criterion sensitivity refers to a mode confidence coefficient Matrix (MAC) between the obtained new structure model and the original structure model after adding mass or changing rigidity at a certain position of the original structure AB Maximum value (MAC) of element in matrix ABmax ) Modal confidence Matrix (MAC) A Maximum value (MAC) of element in matrix Amax ) Is a change in the amount of change in (a). The change degree of the vibration mode of the new structure model relative to the original structure model after the original structure is added with mass or rigidity is changed is reflected.
The invention is described in further detail below with reference to the accompanying drawings.
Fig. 1 is a flowchart of an embodiment of a mode separation degree determination method for structural model correction according to the present invention. As shown in fig. 1, finite element model to be corrected and vibration mode data of model to be corrected in block diagram
Model self-modal confidence matrix MAC to be corrected A Corresponding to step 1. New finite element model and vibration mode data of new finite element model in block diagram +.>
Corresponding to step 2. Modal confidence matrix MAC (media access control) of model to be modified and new model in block diagram AB Corresponding to step 3. Matrix maximum MAC in block diagram A max Matrix maximum value MAC AB max Modality confidence criterion sensitivity S nMAC Corresponding to step 4. Step 5 is to make the confidence criterion sensitivity S for the last block diagram content mode nMAC Is a comparison with the screening of the above.
The method of the embodiment of the invention comprises the following steps:
step 1, according to the data of each order of vibration modes of the finite element model to be corrected
Substituting the model confidence coefficient into a model confidence coefficient criterion formula to solve a model confidence coefficient matrix, namely MAC, between the mode shapes of each order of the finite element model to be corrected A A matrix.
The step 1 comprises the following steps:
let the ith order mode of finite element model to be corrected be
The j-th order mode is->
The correlation between the modes of the finite element model can be described by using a mode confidence coefficient criterion, and the formula is as follows:
wherein, MAC Aij Representing the ith order mode of vibration of the finite element model to be corrected
And the j-th order mode->
A modality confidence between; />
Represents the transpose of the ith order mode of vibration, +.>
Taking a mode after representing the product of the transpose of the ith order mode and the jth order mode,
representing the square after taking the modulus; the denominator is the product of two modes;
MAC Aij the value of (2) is between 0 and 1, reflects the correlation between the two-order vibration modes, and the closer to 1, the closer the two-order vibration modes are; if the estimated order is m-order, MAC A The MAC matrix is an m multiplied by m dimensional matrix, and according to a formula, the MAC matrix obtained by comparing the mode shapes of each order of the finite element model is 1.
And 2, selecting a plurality of positions with additional mass or rigidity being changeable on the finite element model to be corrected, and calculating the finite element mode after the additional mass or rigidity being changed at each position to obtain the mode shape of each order of the new finite element model.
The step 2 comprises the following steps:
2-1) initial selection of positions: selecting a plurality of positions which are easy to attach mass or change rigidity on the test piece as initial selection positions;
2-2) modifying the finite element model: according to the initial selection result, carrying out structural modification of additional mass or rigidity change in the finite element model to be modified one by one, thereby obtaining a new finite element model;
2-3) modal analysis: and carrying out modal analysis on the new finite element models with the modified structures in sequence to obtain the mode shape of each order of each new finite element model.
Step 3, regarding each time of adding mass or changing rigidity in the step 2 to obtain a new finite element model as a new condition, wherein each new condition corresponds to a new finite element model, and each new finite element model corresponds to each order of mode shape; substituting the mode shapes of each order of the new finite element model and each order of the finite element model to be corrected into a mode confidence coefficient criterion formula according to the mode shapes of each order of the new finite element model and each order of the finite element model to be corrected, and calculating a mode confidence coefficient matrix MAC of the new finite element model and the finite element model to be corrected, which are obtained under each condition AB 。
The step 3 comprises the following steps:
let the ith order mode of finite element model to be corrected be
The j-th order vibration mode of the new finite element model obtained by adding mass or changing rigidity on a certain position of the finite element model to be corrected is +.>
The relevance of the two models can be described by a modal confidence criterion, the formula being:
MAC ABij representing the ith order mode of vibration of the finite element model to be corrected
J-th order mode of vibration of new finite element model after adding mass or changing rigidity ∈>
A modality confidence between; />
Representing the transpose of the ith order mode of vibration of the finite element model to be corrected,
taking a model after representing the product of the transpose of the ith order vibration mode of the finite element model to be corrected and the jth order vibration mode of the finite element model after adding mass or changing rigidity;
MAC ABij the value of (2) is between 0 and 1, reflecting the correlation between the two modes; the closer this value is to 1, the closer the two-order mode shape is; if the estimated order is m-order, MAC AB Is an m x m dimensional matrix;
according to the formula (2), each new finite element model obtained in the step (2) is carried out with the finite element model to be correctedMode confidence coefficient calculation is carried out to respectively obtain matrix MAC AB 。
Step 4, firstly, the MAC obtained in the step 1 A Maximum value MAC of selected element in matrix Amax The MAC obtained after calculation in each case in the step 3 is respectively selected AB Maximum value MAC of element in matrix ABmax With the MAC Amax Subtracting the MAC under each of the conditions ABmax The mode confidence criterion sensitivity S in each case is obtained MAC 。
The step 4 comprises the following steps:
4-1) the MAC obtained in step 1 A The maximum value of the selected element in the matrix is denoted as MAC A max . The mode confidence coefficient matrix between the new finite element model and the model to be corrected obtained after adding mass or changing rigidity at each position in the step 3 is MAC AB The mode confidence matrix obtained by adding mass or changing rigidity at the nth position is recorded as MAC ABn Finding out the MAC ABn Maximum MAC of elements in matrix ABn max ；
4-2) recording the variation of the maximum value of the element in the MAC matrix before and after the added mass or the rigidity change of the finite element model to be corrected as the mode confidence coefficient criterion sensitivity, and then the mode confidence coefficient criterion sensitivity formula before and after the added mass or the rigidity change at the nth position is as follows:
S nMAC ＝MAC A max -MAC ABn max (3)
from step 1, the MAC in the formula A max Is a matrix MAC A The maximum value element of (3) is 1, so that the formula (3) is simplified as:
S nMAC ＝1-MAC ABn max (4)。
step 5, comparing the mode confidence coefficient criterion sensitivity S calculated under each condition MAC Size of S MAC The smaller the model, the closer the mode shapes of the two finite element models are; then after adding mass or changing model rigidity at the position, the obtained new finite element model is close to the finite element model to be corrected, which is unfavorable for solvingThis position should be discarded during the test.
The step 5 comprises the following steps:
and (3) comparing the mode confidence coefficient sensitivity calculated in the step (4) one by one, when the sensitivity value of some positions is obviously lower than other sensitivity values, indicating that after the mass is added or the rigidity is changed at the positions, the obtained vibration mode is similar to a model of the finite element model to be corrected, and then the equation introduced during model correction can be similar to the previous equation, so that the solution is not facilitated, and therefore, the positions are discarded when the positions are selected on the test piece.
In a word, when the finite element model is corrected, if the solution equation number is increased by the method of adding mass or changing rigidity, the problem of uncertainty in solution is solved, and the invention can point out the change degree of the mode shape of the test piece after adding mass or changing rigidity, and provide a reference for the position selection of adding mass or changing rigidity on the test piece. The problem of equation approximation caused by vibration mode approximation is effectively avoided, and the accuracy of solving during model correction can be improved. In addition, the method of the invention predicts the actual result by using the finite element result through operating on the finite element model, avoids the complicated operation of point-by-point test on the real structure and improves the working efficiency.
The application effect of the mode separation degree judging method in model correction is verified through a general viscous damping system model. Assume a system architecture of a finite element model as shown in fig. 2.
The stiffness, mass and damping arrays K, M, C in the system structure of the finite element model shown in fig. 2 are respectively:
the parameters provided with the finite element model are as follows;
k 1 ＝k 2 ＝k 3 ＝k 4 ＝k 6 ＝k 7 ＝10 6 N/m，k 5 ＝k 8 ＝k 9 ＝2×10 6 N/m，m 1 ＝m 2 ＝m 5 ＝2kg，m 3 ＝m 4 ＝m 6 ＝1kg，c 1 ＝c 2 ＝c 3 ＝10 3 ns/m. The mode frequencies of each order obtained by the model structure analysis are-2.99+/-94.40 i, -114.64 +/-136.40 i, -40.11+/-151.66 i, -112.80 +/-230.41 i, -4.06+/-277.89 i and-3.93+/-392.21 i (unit: hz).
Whereas the real stiffness matrix, mass matrix and damping matrix K * 、M * 、C * The method comprises the following steps of:
the complex frequencies calculated from the real model structure are-123.68 + -86.78 i, -2.59+ -94.67 i, -35.42+ -151.22 i, -152.29 + -210.9 i, -1.35+ -278.31 i, -5.23+ -399.7 i (unit: hz).
Calculating the frequency error between the finite element model and the real model by the following formula:
wherein omega a Representation is limitedFrequency, omega obtained by meta-simulation t Indicating the frequency of the experimental test. Wherein the numerator is the absolute value of the difference between the finite element simulation frequency and the test frequency, and the denominator is the absolute value of the test frequency. The calculation results are shown in the following table. Wherein, the complex frequencies are arranged in order of the imaginary parts from small to large, corresponding to the order of the mode orders from low to high.
TABLE 1 frequency error between finite element model and real model before correction
As can be seen from the data in the table, the 1 st, 2 nd and 4 th order frequency errors of the model are larger, and the finite element model is corrected by adopting an orthogonal model orthogonal mode improvement method containing damping. In the correction, due to insufficient number of equations, the problem of uncertainty in solving can be generated, and the number of equations needs to be increased by an additional mass method. The mass blocks m1, m2, m3, m4, m5 and m6 are sequentially used as the positions to be selected of 6 additional concentrated masses, the mode confidence coefficient criterion sensitivity after the concentrated masses of 1kg are respectively added to the 6 positions is sequentially calculated by adopting the judging method of the mode separation degree, and the calculation results are shown in the following table.
TABLE 2 Modal confidence criterion sensitivity with added mass for each position
From the data in the table, after adding 1kg of concentrated mass at position 1, the sensitivity of the mode confidence criterion is obviously lower than that of the results obtained at other positions, which indicates that the mode separation degree is not high. The correction equation calculated after adding the mass at the position 1 is superimposed to the original correction equation, and the correction effect may not be ideal. The correction effect obtained by adding mass to each position is calculated in order, and the prediction is verified.
And respectively carrying out superposition solution on the obtained correction equation after the concentrated mass is added at each position and the correction equation obtained before the concentrated mass is not added, wherein the corrected result is as follows:
TABLE 3 correction results for adding concentrated masses to position 1
Table 4 correction results of the additional concentrated mass at position 2
TABLE 5 correction of the additional concentrated mass at position 3
TABLE 6 correction results for adding concentrated quality at position 4
TABLE 7 correction results of the additional concentrated mass at position 5
Table 8 correction results of the additional concentrated mass at position 6
As can be seen by comparing the data in the above table, the correction effect is poor after the correction equation obtained by adding mass to the position 1 is superimposed on the original correction equation. After the mass is added to the position 1, the mode separation degree of the obtained new structure and the original structure is insufficient, the obtained correction equation is similar to the original equation, and the problem of unstable equation number is not solved.
The modal confidence criterion sensitivities listed in table 2 accurately predict the correction effect of an example after an additional mass. The value of the modal confidence criterion sensitivity at position 1 is significantly lower than the values of the other positions, so selecting this position should be avoided when the mass is added to the selected position. The mode separation degree judging method can only calculate on the finite element model, so that the position of the additional mass or rigidity change can be guided to the actual structure, blindness in position selection is avoided, and the correction efficiency is improved.
Claims (1)
1. The mode separation degree judging method for correcting the structural model is characterized by comprising the following steps of:
step 1, according to the data of each order of vibration modes of the finite element model to be corrected
Substituting the model confidence coefficient into a model confidence coefficient criterion formula to solve a model confidence coefficient matrix, namely MAC, between the mode shapes of each order of the finite element model to be corrected A A matrix;
step 2, selecting a plurality of positions with additional mass or rigidity being changeable on the finite element model to be corrected, and calculating the finite element mode after the additional mass or rigidity being changed at each position to obtain the mode shape of each order of the new finite element model;
step 3, regarding each time of adding mass or changing rigidity in the step 2 to obtain a new finite element model as a new condition, wherein each new condition corresponds to a new finite element model, and each new finite element model corresponds to each order of mode shape; substituting the mode shapes of each order of the new finite element model and each order of the finite element model to be corrected into a mode confidence coefficient criterion formula according to the mode shapes of each order of the new finite element model and each order of the finite element model to be corrected, and calculating a mode confidence coefficient matrix MAC of the new finite element model and the finite element model to be corrected, which are obtained under each condition AB ；
Step 4, firstly, the MAC obtained in the step 1 A Selecting the value M of the maximum element in the matrixAC Amax The MAC obtained after calculation in each case in the step 3 is respectively selected AB Maximum value MAC of element in matrix ABmax With the MAC Amax Subtracting the MAC under each condition respectively ABmax The mode confidence criterion sensitivity S in each case is obtained MAC ；
Step 5, comparing the mode confidence coefficient criterion sensitivity S calculated under each condition MAC Size of S MAC The smaller the model is, the more the mass is added or the rigidity is changed at the position, the obtained vibration mode of a certain order is similar to the model of a certain order of the finite element model to be corrected, and the equation introduced during model correction is similar to the previous equation, so that the solution is not facilitated; therefore, when selecting the locations of the additional mass or the change in stiffness on the test piece, these locations are discarded;
the step 1 comprises the following steps:
let the ith order mode of finite element model to be corrected be
The j-th order mode is->
The correlation between the modes of the finite element model can be described by using a mode confidence coefficient criterion, and the formula is as follows:
wherein, MAC Aij Representing the ith order mode of vibration of the finite element model to be corrected
And the j-th order mode->
A modality confidence between; />
Representing the ith order mode of vibrationTranspose of->
Taking a mode after representing the product of the transpose of the ith order mode and the jth order mode,
representing the square after taking the modulus; the denominator is the product of two modes;
MAC Aij the value of (2) is between 0 and 1, reflects the correlation between the two-order vibration modes, and the closer to 1, the closer the two-order vibration modes are; if the estimated order is m-order, MAC A The MAC matrix is an m multiplied by m dimensional matrix, and according to a formula, the MAC matrix is obtained by comparing the mode shapes of each order of the finite element model, and diagonal elements of the MAC matrix are all 1;
the step 2 comprises the following steps:
2-1) initial selection of positions: selecting a plurality of positions which are easy to attach mass or change rigidity on the test piece as initial selection positions;
2-2) modifying the finite element model: according to the initial selection result, carrying out structural modification of additional mass or rigidity change in the finite element model to be modified one by one, thereby obtaining a new finite element model;
2-3) modal analysis: carrying out modal analysis on the new finite element models after the structure modification in sequence to obtain each order of modal shape of each new finite element model;
the step 3 comprises the following steps:
let the ith order mode of finite element model to be corrected be
The j-th order vibration mode of the new finite element model obtained by adding mass or changing rigidity on a certain position of the finite element model to be corrected is +.>
The correlation of the finite element model to be corrected and the new finite element model obtained after the additional mass or the rigidity is changed can be described by using a mode confidence coefficient criterion, and the formula is as follows:
MAC ABij representing the ith order mode of vibration of the finite element model to be corrected
J-th order mode of vibration of new finite element model after adding mass or changing rigidity ∈>
A modality confidence between; />
Representing the transposition of the ith order mode of the finite element model to be corrected, < >>
Taking a model after representing the product of the transpose of the ith order vibration mode of the finite element model to be corrected and the jth order vibration mode of the finite element model after adding mass or changing rigidity;
MAC ABij the value of (2) is between 0 and 1, reflecting the correlation between the two modes; the closer this value is to 1, the closer the two-order mode shape is; if the estimated order is m-order, MAC AB Is an m x m dimensional matrix;
according to formula (2), performing modal confidence calculation between each new finite element model obtained in step (2) and the finite element model to be corrected to obtain matrix MAC respectively AB ；
The step 4 comprises the following steps:
4-1) the MAC obtained in step 1 A The maximum value of the selected element in the matrix is denoted as MAC Amax The method comprises the steps of carrying out a first treatment on the surface of the The mode confidence coefficient matrix between the new finite element model and the model to be corrected obtained after adding mass or changing rigidity at each position in the step 3 is MAC AB The mode confidence matrix obtained by adding mass or changing rigidity at the nth position is recorded as MAC ABn Finding out the MAC ABn Maximum MAC of elements in matrix ABnmax ；
4-2) recording the variation of the maximum value of the element in the MAC matrix before and after the added mass or the rigidity change of the finite element model to be corrected as the mode confidence coefficient criterion sensitivity, and then the mode confidence coefficient criterion sensitivity formula before and after the added mass or the rigidity change at the nth position is as follows:
S nMAC ＝MAC Amax -MAC ABnmax (3)
from step 1, the MAC in the formula Amax Is a matrix MAC A The maximum value element of (3) is 1, so that the formula (3) is simplified as:
S nMAC ＝1-MAC ABnmax (4)。
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