CN102982202B - Based on the structural model modification method of defect mode - Google Patents

Abstract

The invention discloses a kind of structural model modification method based on defect mode.First primary data formats by the method, is obtained the primary data of required form by the adjustment of degree of freedom numbering and complex mode real number; The expansion of defect modal data, treats correction modal data and carries out Matrix QR Decomposition, finally obtain the matrix equation of defect modal data and solve by least square method after format; The determination of corrected parameter, utilizes the modal data after expansion set up the matrix equation of corrected parameter and utilize the iterative algorithm of structure-preserving to solve; Modifying model, the corrected parameter namely utilizing the iterative algorithm of structure-preserving to try to achieve and without spilling modified formulation calculate revised finite element model.The present invention can not only process the incomplete difficulty of modal data degree of freedom in actual measurement, thus avoid the process of modal expanding or model reduction, and revised model maintain original symmetrical structure and ensure that do not need revise mode updating before and after remain unchanged.

Description

Structural model correction method based on defect mode

Technical Field

The invention belongs to finite element model correction technology, in particular to a structural model correction method based on defect mode.

Background

In the technical fields of engineering such as aerospace, aviation, machinery, civil engineering, traffic and the like, a dynamic model of a structure must be established firstly to quantitatively and accurately analyze structure dynamics and solve the problem of structural vibration control commonly existing in engineering. The general modeling method includes theoretical modeling and experimental modeling.

Finite element method is commonly used in theoretical modeling engineering. The finite element method has the advantages of wide adaptability, high analysis speed, short design period, low cost compared with a structure dynamic test and the like, and is widely applied to engineering practice. However, in most cases the results obtained by finite element analysis do not match well with the results obtained experimentally. The phenomenon has two reasons, namely, errors exist in the finite element model due to improper modeling assumption, uncertainty of material characteristics, incorrect boundary conditions and the like during finite element modeling; the other is that the measured data is inaccurate due to the failure of experimental test equipment, the interference of experimental environment noise, improper sensor placement and the like. With the development of testing techniques, one generally considers the measured data to be reliable. When the deviation between the result obtained by the finite element analysis and the test result exceeds the range allowed by the engineering, the finite element model needs to be corrected, namely, the mass matrix, the damping matrix and the rigidity matrix are corrected, so that the dynamic analysis result of the corrected model is consistent with the corresponding test result.

In structural model correction, the following challenging practical problems are mainly faced: (ii) no overflow correction. The correction method not only fuses the measured modal data to the correction model, but also ensures that the residual modal of the correction model is consistent with the original model. ② the positive qualitative retention. The general correction method can only ensure that the corrected model has symmetry, and the positive or semi-positive nature can not be ensured, so that the model loses the physical significance. And incomplete measurement data freedom. The usual correction methods almost require that the measured mode degrees of freedom coincide with the analytical model degrees of freedom. However, in actual measurement, due to the limitation of the measurement equipment, the measurement points for obtaining feedback information are quite limited, and the measurement freedom degree is far smaller than that of the analysis model, i.e. the measured modal data is defective. To overcome this difficulty, two approaches are often employed, namely model reduction and modal expansion. The model reduction is to reduce the degree of freedom of the original analysis model, most typically Guyan's static polycondensation method and various improved and popularized polycondensation methods; the modal expansion is mainly realized by using an interpolation technology for each measured order of modes. Both model order reduction and modal expansion introduce computational errors.

The finite element model revision problem has been of widespread interest and research over the past forty years. Berman, Baruch, et al established the foundation work in this area in the first 70's of the last century. In the 90 s, Friswell and Motterhead (1.Friswell MI, Motterhead JE. FiniteElementmodel applying therapeutics, structures, Klume Acad academic publishers, 1995) summarized and reviewed the results of this field. The previous research mainly focuses on model correction of an undamped damping system, and the problem of model correction of a damping system is emphasized in recent years. Two damping system model correction methods for giving complete degree of freedom modal data are proposed by Kuo and Lin, etc., using the theory and method of secondary eigenvalue inverse problem (2.KuoYC, LinWWandXuSF. New methods for designing model updating schemes. AIAAjournal, 2006, 44 (6): 1310-13163. KuoYC, LinWWandXuSF. Anewmodel correcting methods for designing damping system model estimating arrangement. AIAAjournal, 2005, 43 (12): 2593-2598). Another type of method for damping system model correction is symmetric low rank correction. Zimmerman and widegren (4. zimmermann dc, widegren m. correctingfinitenestringmechaniste. aiaajournal, 1990, 28 (9): 1670-1676) developed methods of model correction using feature configuration. Carvalho et al (5.CarvalhoJ, DattaBN, LinWW, et al. symmetry typ reviviervationvaluebedinginfinite-elementmodeupdatin virostringstructures. journal of Soundand Vibration, 2006, 290 (3-5): 839-864) propose a feature value embedding technique, which not only ensures the symmetry of a modified model, but also keeps the residual modal data of the modified model consistent with the original model (no overflow), and has the disadvantage of not considering information for measuring feature vectors. Recently, Chu et al theoretically studied the overflow phenomenon of the damping system model correction and proposed a model correction method without overflow (6.ChuMT, LinWWand XuSF. Updatg quadrate model and switch on the top of the damping system model correction method. InversP Problers, 2007, 23 (1): 243-256), ChuDelin et al (7.ChuD, ChuMTand WW. Quadrature model up with the same precision, positivedefinition, and not-over. SIAMJohnalor model and analysis application, 2009, 31 (2): 546-564), and further discussed the model correction problem of maintaining the positive and no overflow of the damping system on the basis.

Since it is almost impossible to give modal data of complete degree of freedom in actual model correction, Carvalhoa et al (8.CarvalhoaJ, DattaBN, GuptaCA, et. effective. additional method for modeling with complete degrees of freedom and having no damping system model correction method is proposed, which uses modal data of incomplete degree of freedom measurement to perform model correction and can ensure that the correction is free from overflow, but for damping systems, no corresponding method has been proposed until no overflow model correction is performed using defective modal data of incomplete degree of freedom measurement.

Disclosure of Invention

The invention aims to provide a finite element model correction method for performing overflow-free correction on a damping vibration system based on defect modal data with incomplete measurement freedom.

The technical solution for realizing the purpose of the invention is as follows: a structural model correction method based on defect mode comprises the following steps:

step 1, formatting initial data, namely separating the measured freedom degree from the unmeasured freedom degree in modal data through freedom degree number adjustment, and performing real-valued transformation on complex modal data so as to ensure that all operations are executed in a real form;

step 2, expanding the defective modal data, namely performing matrix QR decomposition on the modal data to be corrected after formatting to obtain a matrix equation of the defective modal data, and solving the matrix equation through a least square algorithm;

step 3, determining correction parameters, establishing a matrix equation of the correction parameters by using the expanded modal data, and solving the correction parameters by a matrix equation iterative algorithm keeping a symmetrical structure;

and 4, correcting the original finite element model, namely determining the corrected structural finite element model by using the correction parameters obtained in the step 3 and the overflow-free correction format.

Compared with the prior art, the invention has the following remarkable advantages: 1) the method can process the defective modal data with incomplete measurement freedom without modal expansion or model reduction, and avoids introducing unnecessary errors, thereby improving the precision of the corrected model; 2) the correction method not only fuses the measured modal data with the corrected finite element model, but also keeps the modal data which does not participate in the correction unchanged before and after the correction, thereby ensuring that the corrected modal data in the corrected finite element model is consistent with the measured data, and the modal data which is not corrected is kept unchanged, namely the correction method is overflow-free.

The present invention is described in further detail below with reference to the attached drawing figures.

Drawings

Fig. 1 is a flowchart of a structural model modification method based on defect modalities.

Fig. 2 is a schematic diagram of an initial data formatting process.

FIG. 3 is a flowchart of a modified parameter matrix solution.

Detailed Description

With reference to fig. 1, a method for modifying a structural model based on a defect mode includes the following steps:

step 1, formatting initial data, namely separating the measured freedom degree from the unmeasured freedom degree in modal data through freedom degree number adjustment, and performing real-valued transformation on complex modal data so as to ensure that all operations are executed in a real form; with reference to fig. 2, the specific steps for formatting the initial data are:

step 1-1, setting all degrees of freedom as {1,2, … N }, { m₁，m₂，…m_lL measured degrees of freedom, the remaining unmeasured degrees of freedom are denoted as m_l+1，…m_N}；

Step 1-2, separating the measured degree of freedom from the unmeasured degree of freedom to obtain an arrangement matrix P (m)₁，m₂，…m_l，m_l+1，…m_N) Wherein P (m)₁，m₂，…m_l，m₁₊₁，…m_N) Is given by all column vectors of the identity matrix of order N according to (m)₁，m₂，…m_l，m_l+1，…m_N) The order of the two groups is rearranged to obtain;

step 1-3, passing the initial data through an array P (m)₁，m₂，…m_l，m_l+1，…m_N) And (3) performing transformation, namely performing the following transformation on the known modal data to be corrected:

$X_1=P(m_1,m_2,\·\·\·m_l,m_{l+1},\·\·\·m_N)X_1^{*};$

wherein,the modal data matrix to be corrected, X, for the initial measurement₁Is a transformed modal data matrix;

step 1-4, the complex modal data is real-valued, namely

$T_a{\mathrm{\Λ}}_1{T}_a^H={\mathrm{\Λ}}_{1R},$ $X_1{T}_a^H=X_{1R},$ $T_m{\mathrm{\Σ}}_1{T}_m^H={\mathrm{\Σ}}_{1R},$ $Y_1^{\left(m\right)}{T}_m^H=Y_{1R}^{\left(m\right)}.$

In the formula, Λ₁And X₁An eigenvalue matrix and an eigenvector matrix respectively composed of p pairs of features to be corrected, wherein the first 2k (2k ≤ p) are complex conjugate pairs ∑₁Andrespectively, an eigenvalue matrix consisting of p measured pairs of features and an eigenvector matrix of measured degrees of freedom, wherein the first 2n (2n ≦ p) are complex conjugate pairs, Λ_1R，X_1R，∑_1R，The real modal data matrix is obtained after the corresponding complex modal data matrix is transformed; the corresponding transformation matrix is

Step 2, expanding the defective modal data, namely performing matrix QR decomposition on the modal data to be corrected after formatting to obtain a matrix equation of the defective modal data, and solving the matrix equation through a least square algorithm; the specific steps for expanding the defect modal data are as follows:

step 2-1, the modal matrix X to be corrected_1RCarrying out matrix QR decomposition to obtain a matrix [ Q ]₁，Q₂]I.e. by

$X_{1R}=[Q_1,Q_2]\left[\begin{array}{c}R\\ 0\end{array}\right].$

Step 2-2, constructing defect modal dataEquation of matrix (2)

$Q_2^T{E}_r{Y}_{1R}^{\left(u\right)}=-Q_2^T{E}_l{Y}_{1R}^{\left(m\right)},$

Wherein E_l＝[e₁，...，e_k]，E_r＝[e_k+1，…，e_N]，e_iN is an N-order unit array I_NThe ith column;

step 2-3, solving least square solution of the matrix equation

$Y_{1R}^{\left(u\right)}=-{\left(Q_2^T{E}_r\right)}^{+}{Q}_2^T{E}_l{Y}_{1R}^{\left(m\right)}.$

WhereinIs a matrixMoore-Penrose generalized inverse of (1);

step 2-4, synthesizing the data determined in the step 2-3 to obtain an expanded real mode data matrix

$Y_{1R}=\left[\begin{array}{c}{Y}_{1R}^{\left(m\right)}\\ Y_{1R}^{\left(u\right)}\end{array}\right].$

Step 3, determining correction parameters, establishing a matrix equation of the correction parameters by using the expanded modal data, and solving the correction parameters by a matrix equation iterative algorithm keeping a symmetrical structure;

building modifications using extended modal dataPositive parameter phi_RSolving the correction parameters by a matrix equation iterative algorithm keeping a symmetrical structure, and combining the figure 3, the concrete steps of solving are as follows:

step 3-1, determining a matrix equation A phi_RB^T+EΦ_RF^TThe coefficient matrix for W is as follows:

A＝M_aX_1RΛ_1R，

$B={\mathrm{\Σ}}_{1R}^T({\mathrm{\Σ}}_{1R}^T{Y}_{1R}^T{M}_a{X}_{1R}{\mathrm{\Λ}}_{1R}-Y_{1R}^T{K}_a{X}_{1R}),$

E＝-K_aX_1R，

$F={\mathrm{\Σ}}_{1R}^T{Y}_{1R}^T{M}_a{X}_{1R}{\mathrm{\Λ}}_{1R}-Y_{1R}^T{K}_a{X}_{1R},$

$W=M_a{Y}_{1R}{\mathrm{\Σ}}_{1R}^2+C_a{Y}_{1R}{\mathrm{\Σ}}_{1R}+K_a{Y}_{1R}.$

wherein M is_a，C_a，K_aThe matrix of the mass, the damping and the rigidity of the original finite element model before correction;

step 3-2, determining a normal equation corresponding to the matrix equation, specifically:

A^TAΦ_RB^TB+A^TEΦ_RF^TB+E^TAΦ_RB^TF+E^TEΦ_RF^TF+B^TBΦ_RA^TA+B^TFΦ_RE^TA+F^TBΦ_RA^TE+F^TFΦ_RE^TE＝A^TWB+E^TWF+(A^TWB+E^TWF)^T.

step 3-3, arranging the normal equation into a matrix equation form related to X as follows:

$\underset{k=1}{\overset{4}{\mathrm{\Σ}}}{A}_{k}X{B}_k^T+\underset{k=1}{\overset{4}{\mathrm{\Σ}}}{B}_{k}X{A}_k^T=\mathrm{\Ω},$

in the formula

A₁＝A^TA，A₂＝A^TE，A₃＝E^TE，A₄＝E^TA，

B₁＝B^TB，B₂＝B^TF，B₃＝F^TF，B₄＝F^TB，

Ω＝A^TWB+E^TWF+(A^TWB+E^TWF)^T；

Step 3-4, solving the normal equation and setting an initial estimation X₀Is an identity matrix and a termination criterion Tol of 10^-8；

Step 3-5, making i equal to 0 and calculating

$R_0=\mathrm{\Ω}-\underset{k=1}{\overset{4}{\mathrm{\Σ}}}{A}_k{X}_0{B}_k^T-\underset{k=1}{\overset{4}{\mathrm{\Σ}}}{B}_k{X}_0{A}_k^T,$

$P_0=\underset{k=1}{\overset{4}{\mathrm{\Σ}}}{A}_k{R}_0{B}_k^T+\underset{k=1}{\overset{4}{\mathrm{\Σ}}}{B}_k{R}_0{A}_k^T.$

Step 3-6, judging the termination condition, if | | | R_i||_F< Tol, wherein | | | | noncircular vision_FIf the Frobenius norm is the Frobenius norm of the matrix, the calculation is terminated, otherwise, i is made to be i + 1;

step 3-7, calculating

$X_i=X_{i-1}+\frac{{\left|\right|R_{i-1}\left|\right|}_F^2}{{\left|\right|P_{i-1}\left|\right|}_F^2}{P}_{i-1},$

$R_i=\mathrm{\&Omega;}-\underset{k=1}{\overset{4}{\mathrm{\&Sigma;}}}{A}_k{X}_k{B}_k^T-\underset{k=1}{\overset{4}{\mathrm{\&Sigma;}}}{B}_k{X}_k{A}_k^T$

$=R_{i-1}-\frac{{\left|\right|R_{i-1}\left|\right|}_F^2}{{\left|\right|P_{i-1}\left|\right|}_F^2}(\underset{k=1}{\overset{4}{\mathrm{\&Sigma;}}}{A}_k{P}_{k-1}{B}_k^T+\underset{k=1}{\overset{4}{\mathrm{\&Sigma;}}}{B}_k{P}_{k-1}{A}_k^T),$

$P_i=\underset{k=1}{\overset{4}{\mathrm{\&Sigma;}}}{A}_k{R}_k{B}_k^T+\underset{k=1}{\overset{4}{\mathrm{\&Sigma;}}}{B}_k{R}_k{A}_k^T+\frac{{\left|\right|R_i\left|\right|}_F^2}{{\left|\right|R_{i-1}\left|\right|}_F^2}{P}_{i-1}.$

And then returning to the step 3-6.

And 4, correcting the original finite element model, namely determining the corrected structural finite element model by using the correction parameters obtained in the step 3 and the overflow-free correction format, and finishing the correction of the original finite element model. The modified finite element model can be obtained by the following non-overflow modification format:

$M=M_a-M_a{X}_{1R}{\mathrm{\&Lambda;}}_{1R}{\mathrm{\&Phi;}}_R{\mathrm{\&Lambda;}}_{1R}^T{X}_{1R}^T{M}_a,$

$C=C_a+M_a{X}_{1R}{\mathrm{\&Lambda;}}_{1R}{\mathrm{\&Phi;}}_R{X}_{1R}^T{K}_a+K_a{X}_{1R}{\mathrm{\&Phi;}}_R{\mathrm{\&Lambda;}}_{1R}^T{X}_{1R}^T{M}_a,$

$K=K_a-K_a{X}_{1R}{\mathrm{\&Phi;}}_R{X}_{1R}^T{K}_a.$

therefore, the method can process the defective modal data with incomplete measurement freedom degree without modal expansion or model order reduction, avoids introducing unnecessary errors and improves the precision of the corrected model; meanwhile, modal data which do not participate in the correction are kept unchanged before and after the correction, so that the correction is ensured to be free of overflow.

Claims (5)

1. A structural model correction method based on defect mode is characterized by comprising the following steps:

step 1, formatting modal data to be corrected, namely separating the measured freedom degree from the unmeasured freedom degree in the modal data through the adjustment of the number of the freedom degrees, and performing real-valued computation on complex modal data so as to ensure that all operations are executed in a real form;

step 2, expanding the defective modal data, namely performing matrix QR decomposition on the formatted to-be-corrected modal data to obtain a matrix equation of the defective modal data, and solving the matrix equation through a least square algorithm;

step 3, determining correction parameters, establishing a matrix equation of the correction parameters by using the expanded modal data, and solving the correction parameters by a matrix equation iterative algorithm keeping a symmetrical structure;

and 4, correcting the original finite element model, namely determining the corrected structural finite element model by using the correction parameters obtained in the step 3 and the overflow-free correction format.

2. The structural model modification method based on defect modality according to claim 1, wherein the specific steps of formatting the data of the modality to be modified in step 1 are as follows:

step 1-1, setting all degrees of freedom as {1,2, … N }, { m₁,m₂,…m_lL measured degrees of freedom, the remaining unmeasured degrees of freedom are denoted as m_l+1,…m_N}；

Step 1-2, separating the measured degree of freedom from the unmeasured degree of freedom to obtain an arrangement matrix P (m)₁,m₂,…m_l,m_l+1,…m_N) Wherein P (m)₁,m₂,…m_l,m_l+1,…m_N) Is given by all column vectors of the identity matrix of order N according to (m)₁,m₂,…m_l,m_l+1,…m_N) The order of the two groups is rearranged to obtain;

step 1-3, passing the modal data to be corrected through an arrangement matrix P (m)₁,m₂,…m_l,m_l+1,…m_N) And (3) performing transformation, namely performing the following transformation on the known modal data to be corrected:

wherein,for initially measured modal data to be corrected, X₁For transformed modal data；

Step 1-4, the complex modal data is real-valued, namely

In the formula, Λ₁And X₁Respectively forming an eigenvalue matrix and an eigenvector matrix which are formed by p feature pairs to be corrected, wherein the first 2k (2k is less than or equal to p) feature pairs are complex conjugate pairs; sigma₁Anda matrix of eigenvalues and measured degrees of freedom of eigenvectors, respectively, of p measured pairs of eigenvalues, the first 2n (2n ≦ p) of which are complex conjugate pairs Λ_1R,X_1R,Σ_1R，The real modal data matrix is obtained after the corresponding complex modal data matrix is transformed; the corresponding transformation matrix is

3. The method for modifying a structural model based on a defect modality according to claim 1, wherein the step 2 of expanding the data of the defect modality is specifically as follows:

step 2-1, the modal matrix X to be corrected_1RCarrying out matrix QR decomposition to obtain a matrix [ Q ]₁,Q₂]I.e. by

Step 2-2, constructing defect modal dataEquation of matrix (2)

Wherein E_l＝[e₁,...,e_k],E_r＝[e_k+1,...,e_N],e_iN is an N-order unit array I_NThe ith column;

step 2-3, solving least square solution of the matrix equation

WhereinIs a matrixMoore-Penrose generalized inverse of (1);

step 2-4, synthesizing the data determined in the step 2-3 to obtain an expanded real mode data matrix

4. The method for modifying a structural model based on a defect modality according to claim 1, wherein step 3 establishes a modification parameter Φ using the extended modality data_RSolving the correction parameters by a matrix equation iterative algorithm keeping a symmetrical structure, wherein the concrete steps of solving are as follows:

step 3-1, determining a matrix equation A phi_RB^T+EΦ_RF^TThe coefficient matrix for W is as follows:

A＝M_aX_1RΛ_1R,

E＝-K_aX_1R,

wherein M is_a,C_a,K_aThe matrix of the mass, the damping and the rigidity of the original finite element model before correction;

step 3-2, determining a normal equation corresponding to the matrix equation, specifically:

A^TAΦ_RB^TB+A^TEΦ_RF^TB+E^TAΦ_RB^TF+E^TEΦ_RF^TF+

B^TBΦ_RA^TA+B^TFΦ_RE^TA+F^TBΦ_RA^TE+F^TFΦ_RE^TE

＝A^TWB+E^TWF+(A^TWB+E^TWF)^T；

step 3-3, arranging the normal equation into a matrix equation form related to X as follows:

in the formula

A₁＝A^TA,A₂＝A^TE,A₃＝E^TE,A₄＝E^TA,

B₁＝B^TB,B₂＝B^TF,B₃＝F^TF,B₄＝F^TB,

Ω＝A^TWB+E^TWF+(A^TWB+E^TWF)^T；

Step 3-4, solving the normal equation and setting an initial estimation X₀Is an identity matrix and a termination criterion Tol of 10^-8；

Step 3-5, making i equal to 0 and calculating

Step 3-6, judging the termination condition, if | | | R_i||_F< Tol, where | | | R_i||_FIf the Frobenius norm is the Frobenius norm of the matrix, the calculation is terminated, otherwise, i is made to be i + 1;

step 3-7, calculating

And then returning to the step 3-6.

5. The defect mode based structural model modification method of claim 1, wherein the finite element model modified in step 4 is obtained by the following non-overflow modification format: