CN102982202A - Model correcting method based on defective modal structure - Google Patents

Abstract

The invention discloses a model correcting method based on a defective modal structure. The model correcting method first formats initial data, and the initial data of a required format is obtained through an adjustment of freedom degree numbers and a real number conversion of a complex mode. The extension of defective modal data is achieved by conducting matrix QR decomposition on model data to be corrected after the formatting of the initial data, obtaining a matrix equation of the defective modal data and utilizing least square method to solve the matrix equation. The confirming of corrected parameters is achieved by building a matrix equation of corrected parameters through extended modal data and utilizing a structure preserving iterative algorithm to solve the matrix equation. The corrected model is a finite element model calculated and corrected by the corrected parameters which are obtained through the structure preserving iterative algorithm and non-overflowing revising format. The model correcting method based on the defective modal structure can not only deal with the difficulty that the freedom degree of the modal data is not complete in actual measurement, thus avoiding a modal unfolding process or a model reduction process, but also keep the corrected model in a primary symmetrical structure and guarantee that the model which does not need correction is kept the same before and after the model correcting.

Description

Structural model modification method based on damaged mode

Technical field

The invention belongs to Finite Element Model Updating, particularly a kind of structural model modification method based on damaged mode.

Background technology

In field of engineering technology such as space flight, aviation, machinery, building, traffic, carry out quantitatively, exactly the structural dynamic Epidemiological Analysis, solve ubiquitous structural vibration control problem in the engineering, must set up first the kinetic model of structure.General modeling method has two kinds of theoretical modeling and Experimental modelings.

Finite Element Method commonly used on the theoretical modeling engineering.Finite Element Method because have wide adaptability, analysis speed is fast, the design cycle is short, compare the advantages such as expense is low with dynamic test of structure, in engineering practice, be widely applied.Yet, can not coincide well by the finite element analysis result who obtains and the result that experiment obtains as a rule.Cause this phenomenon that the reason of two aspects is arranged, the one, the uncertainty of inappropriate modeling assumption, material behavior, incorrect boundary condition etc. cause finite element model to have error during finite element modeling; To be that experiment test device fails, experimental situation noise, sensor placement location are incorrect etc. cause measurement data inaccurate for another.Along with the development of measuring technology, it is conventionally believed that measurement data is reliable.When the deviation of finite element analysis acquired results and test result exceeds the scope of engineering license, need to revise finite element model, be correction mass matrix, damping matrix and stiffness matrix, so that the dynamic analysis result of model is consistent with corresponding test findings after revising.

In the structural model correction, mainly face following challenging practical problems: 1. without overflowing correction.Modification method not only makes the modal data of measurement be melted into correction model, and the residue mode of correction model is consistent with master mould.2. orthotropicity keeps.Common modification method only can guarantee that revised model has symmetry, and orthotropicity or Positive generally can't guarantee, thereby make model lose physical significance.3. the measurement data degree of freedom is imperfect.It is consistent with the number of degrees of freedom, of analytical model that common modification method nearly all requires to survey the number of degrees of freedom, of mode.Yet in actual measurement, because the limitation of measuring equipment, the measuring point that obtains feedback information is very limited, measures the number of degrees of freedom, that number of degrees of freedom, is far smaller than analytical model, and the modal data of namely measuring is damaged.In order to overcome this difficulty, often adopt two kinds of methods, i.e. model reduction and Modal Expansion.Model reduction is the number of degrees of freedom, of the former analytical model of reduction, and most typical is the static condensation methods of Guyan and the condensation methods of various improvement and popularization; Modal Expansion mainly is to realize with interpolation technique by each the rank mode to actual measurement.But model reduction and Modal Expansion all can be introduced the error of calculation.

In four times more than ten years of past, finite element model correction problem has obtained paying close attention to widely and studying.As far back as the seventies in last century, the people such as Berman, Baruch have just established the element task in this field.The nineties, Friswell and Mottershead (1.Friswell M I, Mottershead J E.Finite Element model updating instructural dynamics.Klumer Academic Publishers, 1995) again the achievement in research in this field is summarized and summarized.Research before mainly concentrates on the model correction of undamped system, and the model correction problem of damping system was subject to the people's attention in recent years.Utilize theory and the method for quadratic character value indirect problem, two kinds of damping system model modification methods (2.Kuo Y C when Kuo and Lin etc. have proposed given complete degree of freedom modal data, Lin W W and XuS F.New methods for finite element model updating problems.AIAA Journal, 2006,44 (6): 1310～13163.Kuo Y C, Lin W W and Xu S F.A new model correcting method forquadratic eigenvalue problems using a symmetric eigenstructure assignment.AIAA Journal, 2005,43 (12): 2593～2598).The another kind of method of damping system model correction is symmetrical low-rank correction.Zimmerman and Widengren (4.Zimmerman D C, Widengren M.Correcting finite element models using asymmetric eigenstructure assignment technique.AIAA Journal, 1990,28 (9): 1670～1676) developed the method for utilizing Characteristic Structure Configuration to carry out the model correction.(the 5.Carvalho J such as Carvalho, Datta B N, LinW W, et al.Symmetry preserving eigenvalue embedding in finite-element model updating ofvibrating structures.Journal of Sound and Vibration, 2006,290 (3-5): 839～864) proposed the eigenwert embedded technology, not only guaranteed the symmetry of correction model, and so that residue modal data and the master mould of the rear model of correction are consistent (without overflowing), weak point is not consider to measure the information of proper vector.Recently, Chu etc. theoretically systematic study the spillover of damping system model correction, and a non-spill model modification method (6.Chu M T proposed, Lin W W and Xu S F.Updating quadratic models with no spillover effect onunmeasured spectral data.Inverse Problems, 2007,23 (1): 243～256), (the 7.Chu D such as Chu Del in, Chu M T and Lin W W.Quadratic model updating with symmetry, positive definiteness, andno spill-over.SIAM Journal on Matrix Analysis and Applications, 2009,31 (2): 546～564) maintenance orthotropicity and the non-spill model correction problem of damping system have been discussed further on this basis again.

Owing to may provide hardly the modal data of complete degree of freedom in the actual model correction, (the 8.Carvalhoa J such as Carvalhoa, Datta B N, Guptac A, et al.A direct method for model updating with incompletemeasured data and without spurious modes.Mechanical Systems and Signal Processing, 2007,21 (7): 2715～2731) proposed the model modification method of a undamped system, the modal data that the method utilizes degree of freedom not exclusively to measure carries out the model correction and can guarantee to revise is non-spill, but for damping system, utilize the incomplete damaged modal data of measurement degree of freedom to carry out going back so far the no one without the models on spillovers correction and propose corresponding method.

Summary of the invention

The object of the present invention is to provide and a kind ofly based on measuring the incomplete damaged modal data of degree of freedom vibration-damping system is carried out without overflowing the correction method for finite element model of correction.

The technical solution that realizes the object of the invention is: a kind of structural model modification method based on damaged mode, and step is as follows:

Step 1, to primary data format, namely adjust by the degree of freedom numbering degree of freedom that measures in the modal data separated with the degree of freedom that does not measure, and with complex mode data real number, thereby guarantee that all computings carry out under real form;

Step 2, damaged modal data is expanded, namely treated after the format and revise modal data and carry out Matrix QR Decomposition, obtain the matrix equation of damaged modal data, and find the solution by least-squares algorithm;

Step 3, corrected parameter is determined, utilized modal data after the expansion to set up the matrix equation of corrected parameter, and find the solution corrected parameter by the matrix equation iterative algorithm that keeps symmetrical structure;

Step 4, original finite element model is revised, namely utilized corrected parameter that step 3 obtains and revise form and determine revised structural finite element model without overflowing.

The present invention compared with prior art, its remarkable advantage is: 1) can process and measure the incomplete damaged modal data of degree of freedom and do not need to carry out mode and launch or model reduction, avoid introducing unnecessary error, thereby improve revise after the precision of model; 2) this modification method not only makes the modal data that measures be melted into revised finite element model, and the modal data that does not participate in revising remains unchanged before and after revising, thereby guaranteed that the modal data of this correction in the revised finite element model and measurement data are identical, the modal data that should not revise remains unchanged, and namely modification method is non-spill.

Below in conjunction with accompanying drawing the present invention is described in further detail.

Description of drawings

Fig. 1 is based on the structural model modification method process flow diagram of damaged mode.

Fig. 2 is primary data formatting procedure synoptic diagram.

Fig. 3 is corrected parameter Matrix Solving process flow diagram.

Embodiment

In conjunction with Fig. 1, a kind of structural model modification method based on damaged mode, step is as follows:

Step 1, to primary data format, namely adjust by the degree of freedom numbering degree of freedom that measures in the modal data separated with the degree of freedom that does not measure, and with complex mode data real number, thereby guarantee that all computings carry out under real form; In conjunction with Fig. 2, to primary data format concrete steps be:

Step 1-1, set whole degree of freedom be 1,2 ... N}, { m ₁, m ₂... m _lBeing l degree of freedom that measures, the remaining degree of freedom of not measuring is designated as { m _L+1... m _N;

Step 1-2, the degree of freedom that measures is separated with the degree of freedom that does not measure, obtain permutation matrix P (m ₁, m ₂... m _l, m _L+1... m _N), P (m wherein ₁, m ₂... m _l, m ₁₊₁... m _N) by all column vectors of N rank unit matrix according to (m ₁, m ₂... m _l, m _L+1... m _N) order rearrange and obtain;

Step 1-3, with primary data by ordered array P (m ₁, m ₂... m _l, m _L+1... m _N) carry out conversion, namely known modal data to be revised is carried out following conversion:

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

Wherein,

Be the positive-norm state data matrix to be repaired of initial measurement, X ₁Be the modal data matrix after the conversion;

Step 1-4, with complex mode data real number, 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, Λ ₁And X ₁Be respectively p feature to be revised eigenvalue matrix and the eigenvectors matrix to forming, wherein before 2k (2k≤p) is individual to be complex conjugate pair; ∑ ₁With

Be respectively p feature measuring to the eigenvalue matrix that forms and the eigenvectors matrix of having surveyed degree of freedom, wherein (2n≤p) is individual to be complex conjugate pair to front 2n; Λ _1R, X _1R, ∑ _1R,

Be the real modal data matrix that obtains after the conversion of corresponding complex mode data matrix; Corresponding transformation matrix is

Step 2, damaged modal data is expanded, namely treated after the format and revise modal data and carry out Matrix QR Decomposition, obtain the matrix equation of damaged modal data, and find the solution by least-squares algorithm; Damaged modal data expanded is specially:

Step 2-1, modal matrix X that will be to be revised _1RCarry out Matrix QR Decomposition, obtain matrix [Q ₁, Q ₂], namely

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

Step 2-2, construct damaged modal data

Matrix equation

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

E wherein _l=[e ₁..., e _k], E _r=[e _K+1..., e _N], e _i(i=1,2 ... N) be N rank unit matrix I _NI row;

Step 2-3, find the solution the least square solution of above-mentioned matrix equation

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

Wherein

It is matrix

The Moore-Penrose generalized inverse;

Step 2-4, step 2-3 established data is synthesized the real modal data matrix after being expanded

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

Step 3, corrected parameter is determined, utilized modal data after the expansion to set up the matrix equation of corrected parameter, and find the solution corrected parameter by the matrix equation iterative algorithm that keeps symmetrical structure;

Utilize the modal data after expanding to set up corrected parameter Φ _RMatrix equation, and find the solution corrected parameter by the matrix equation iterative algorithm that keeps symmetrical structure, in conjunction with Fig. 3, the concrete steps of finding the solution are:

Step 3-1, determine matrix equation A Φ _RB ^T+ E Φ _RF ^TThe matrix of coefficients of=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，

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

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

M wherein _a, C _a, K _aBe quality, damping and the stiffness matrix of revising front original finite element model;

Step 3-2, determine the normal equation that above-mentioned matrix equation is corresponding, be specially:

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, normal equation is put in order, is organized into about the matrix form equation of 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

A ₁＝A ^TA，A ₂＝A ^TE，A ₃＝E ^TE，A ₄＝E ^TA，

B ₁＝B ^TB，B ₂＝B ^TF，B ₃＝F ^TF，B ₄＝F ^TB，

Ω＝A ^TWB+E ^TWF+(A ^TWB+E ^TWF) ^T；

Step 3-4, find the solution above-mentioned normal equation, set initial estimation X ₀Be unit matrix and stop criterion Tol=10 ^-8

Step 3-5, make i=0 and calculate

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

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

Step 3-6, end condition is judged, if || R _i|| _F＜Tol, wherein || || _FBe the Frobenius norm of matrix, then stop calculating, otherwise make i=i+1;

Step 3-7, calculating

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

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

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

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

Return afterwards step 3-6.

Step 4, original finite element model is revised, namely utilized corrected parameter that step 3 obtains and revise form and determine revised structural finite element model without overflowing, finish the correction to original finite element model.Revised finite element model can overflow the correction form by following nothing and obtain:

$M=M_a-M_a{X}_{1R}{\mathrm{\Λ}}_{1R}{\mathrm{\Φ}}_R{\mathrm{\Λ}}_{1R}^T{X}_{1R}^T{M}_a,$

$C=C_a+M_a{X}_{1R}{\mathrm{\Λ}}_{1R}{\mathrm{\Φ}}_R{X}_{1R}^T{K}_a+K_a{X}_{1R}{\mathrm{\Φ}}_R{\mathrm{\Λ}}_{1R}^T{X}_{1R}^T{M}_a,$

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

As from the foregoing, method of the present invention can be processed the measurement incomplete damaged modal data of degree of freedom and not need to carry out mode expansion or model reduction, avoids introducing unnecessary error, thereby improves the precision of revising rear model; The modal data that does not participate in simultaneously revising remains unchanged before and after revising, and is non-spill thereby guaranteed to revise.

Claims (5)

1. structural model modification method based on damaged mode is characterized in that step is as follows:

Step 1, to primary data format, namely adjust by the degree of freedom numbering degree of freedom that measures in the modal data separated with the degree of freedom that does not measure, and with complex mode data real number, thereby guarantee that all computings carry out under real form;

Step 2, damaged modal data is expanded, namely treated after the format and revise modal data and carry out Matrix QR Decomposition, obtain the matrix equation of damaged modal data, and find the solution by least-squares algorithm;

Step 3, corrected parameter is determined, utilized modal data after the expansion to set up the matrix equation of corrected parameter, and find the solution corrected parameter by the matrix equation iterative algorithm that keeps symmetrical structure;

Step 4, original finite element model is revised, namely utilized corrected parameter that step 3 obtains and revise form and determine revised structural finite element model without overflowing.

2. the structural model modification method based on damaged mode according to claim 1 is characterized in that, in the step 1 to primary data format concrete steps is:

Step 1-1, set whole degree of freedom be 1,2 ... N}, { m ₁, m ₂... m _lBeing l degree of freedom that measures, the remaining degree of freedom of not measuring is designated as { m _L+1... m _N;

Step 1-2, the degree of freedom that measures is separated with the degree of freedom that does not measure, obtain permutation matrix P (m ₁, m ₂... m _l, m _L+1... m _N), P (m wherein ₁, m ₂... m _l, m _L+1... m _N) by all column vectors of N rank unit matrix according to (m ₁, m ₂... m _l, m _L+1... m _N) order rearrange and obtain;

Step 1-3, with primary data by ordered array P (m ₁, m ₂... m _l, m _L+1... m _N) carry out conversion, namely known modal data to be revised is carried out following conversion:

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

Wherein,

Be the positive-norm state data to be repaired of initial measurement, X ₁Be the modal data after the conversion;

Step 1-4, with complex mode data real number, 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, Λ ₁And X ₁Be respectively p feature to be revised eigenvalue matrix and the eigenvectors matrix to forming, wherein before 2k (2k≤p) is individual to be complex conjugate pair; ∑ ₁And Y ₁ ^(m)Be respectively p feature measuring to the eigenvalue matrix that forms and the survey freedom matrix of proper vector, wherein before 2n (2n≤p) is individual to be complex conjugate pair; Λ _1R, X _1R, ∑ _1R,

Be the real modal data matrix that obtains after the conversion of corresponding complex mode data matrix; Corresponding transformation matrix is

3. the structural model modification method based on damaged mode according to claim 1 is characterized in that, step 2 pair damaged modal data is expanded and is specially:

Step 2-1, modal matrix X that will be to be revised _1RCarry out Matrix QR Decomposition, obtain matrix [Q ₁, Q ₂], namely

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

Step 2-2, construct damaged modal data

Matrix equation

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

E wherein _l=[e ₁..., e _k], E _r=[e _K+1..., e _N], e _i(i=1,2 ... N) be N rank unit matrix I _NI row;

Step 2-3, find the solution the least square solution of above-mentioned matrix equation

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

Wherein

It is matrix

The Moore-Penrose generalized inverse;

Step 2-4, step 2-3 established data is synthesized the real modal data matrix after being expanded

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

4. the structural model modification method based on damaged mode according to claim 1 is characterized in that, step 3 utilizes the modal data after the expansion to set up corrected parameter Φ _RMatrix equation, and find the solution corrected parameter by the matrix equation iterative algorithm that keeps symmetrical structure, the concrete steps of finding the solution are:

Step 3-1, determine matrix equation A Φ _RB ^T+ E Φ _RF ^TThe matrix of coefficients of=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，

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

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

M wherein _a, C _a, K _aBe quality, damping and the stiffness matrix of revising front original finite element model;

Step 3-2, determine the normal equation that above-mentioned matrix equation is corresponding, be specially:

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, normal equation is put in order, is organized into about the matrix form equation of 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

A ₁＝A ^TA，A ₂＝A ^TE，A ₃＝E ^TE，A ₄＝E ^TA，

B ₁＝B ^TB，B ₂＝B ^TF，B ₃＝F ^TF，B ₄＝F ^TB，

Ω＝A ^TWB+E ^TWF+(A ^TWB+E ^TWF) ^T；

Step 3-4, find the solution above-mentioned normal equation, set initial estimation X ₀Be unit matrix and stop criterion Tol=10 ^-8

Step 3-5, make i=0 and calculate

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

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

Step 3-6, end condition is judged, if || R _i|| _F＜Tol, wherein || || _FBe the Frobenius norm of matrix, then stop calculating, otherwise make i=i+1;

Step 3-7, calculating

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

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

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

$P_i=\underset{k=1}{\overset{4}{\mathrm{\Σ}}}{A}_k{R}_k{B}_k^T+\underset{k=1}{\overset{4}{\mathrm{\Σ}}}{B}_k{R}_k{A}_k^T+\frac{{\left|\right|R_i\left|\right|}_F^2}{{\left|\right|R_{i-1}\left|\right|}_F^2}{P}_{i-1}.$ Return afterwards step 3-6.

5. the structural model modification method based on damaged mode according to claim 1 is characterized in that, revised finite element model overflows by following nothing and revises form and obtain in the step 4:

$M=M_a-M_a{X}_{1R}{\mathrm{\Λ}}_{1R}{\mathrm{\Φ}}_R{\mathrm{\Λ}}_{1R}^T{X}_{1R}^T{M}_a,$

$C=C_a+M_a{X}_{1R}{\mathrm{\Λ}}_{1R}{\mathrm{\Φ}}_R{X}_{1R}^T{K}_a+K_a{X}_{1R}{\mathrm{\Φ}}_R{\mathrm{\Λ}}_{1R}^T{X}_{1R}^T{M}_a,.$

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

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