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

Based on the structural model modification method of defect mode

Technical field

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

Background technology

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

Conventional Finite Element Method in theoretical modeling engineering.Finite Element Method owing to having wide adaptability, the advantage such as analysis speed is fast, the design cycle is short, expense is low compared with dynamic test of structure, be widely applied in engineering practice.But the result obtained by finite element analysis as a rule can not be coincide well with the result that experiment obtains.Cause this phenomenon to have the reason of two aspects, one when being finite element modeling inappropriate modeling assumption, the uncertainty of material behavior, incorrect boundary condition etc. cause finite element model to there is error; Another is experiment test device fails, experimental situation noise, sensor placement location are incorrect etc. causes measurement data inaccurate.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 that engineering permits, need to revise finite element model, i.e. correction mass matrix, damping matrix and stiffness matrix, makes the dynamic analysis result revising rear model be consistent with corresponding test findings.

In structural model correction, mainly face following challenging practical problems: 1. revise without spilling.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 ensure that revised model has symmetry, and orthotropicity or Positive generally cannot ensure, thus make model lose physical significance.3. measurement data degree of freedom is imperfect.Common modification method nearly all requires that the number of degrees of freedom, of surveying mode is consistent with the number of degrees of freedom, of analytical model.But in actual measurement, due to the limitation of measuring equipment, the measuring point obtaining feedback information is very limited, measure the number of degrees of freedom, that number of degrees of freedom, is far smaller than analytical model, the modal data namely measured is defect.In order to overcome this difficulty, normal employing two kinds of methods, i.e. model reduction and Modal Expansion.Model reduction is the number of degrees of freedom, reducing former analytical model, and most typical is the static condensation method of Guyan and the condensation methods of various improvement and popularization; Modal Expansion is mainly by using interpolation technique to realize to each rank mode of actual measurement.But model reduction and Modal Expansion all can introduce the error of calculation.

In time more than ten years past four, FEM updating problem has obtained to be paid close attention to widely and studies.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.FriswellMI, MottersheadJE.FiniteElementmodelupdatinginstructuraldyna mics.KlumerAcademicPublishers, 1995) again the achievement in research in this field summarized and summarize.Research before mainly concentrates on the Modifying model of undamped system, and the Modifying model problem of damping system was subject to the people's attention in recent years.Utilize the Theories and methods of quadratic eigenvalue indirect problem, Kuo and Lin etc. propose two kinds of damping system model modification method (2.KuoYC during given complete degree of freedom modal data, LinWWandXuSF.Newmethodsforfiniteelementmodelupdatingprob lems.AIAAJournal, 2006, 44 (6): 1310 ~ 13163.KuoYC, LinWWandXuSF.Anewmodelcorrectingmethodforquadraticeigenv alueproblemsusingasymmetriceigenstructureassignment.AIAA Journal, 2005, 43 (12): 2593 ~ 2598).The another kind of method of damping system Modifying model is symmetrical low-rank correction.Zimmerman and Widengren (4.ZimmermanDC, WidengrenM.Correctingfiniteelementmodelsusingasymmetrice igenstructureassignmenttechnique.AIAAJournal, 1990,28 (9): 1670 ~ 1676) method utilizing Characteristic Structure Configuration to carry out Modifying model has been developed.(the 5.CarvalhoJ such as Carvalho, DattaBN, LinWW, etal.Symmetrypreservingeigenvalueembeddinginfinite-eleme ntmodelupdatingofvibratingstructures.JournalofSoundandVi bration, 2006,290 (3-5): 839 ~ 864) propose eigenwert embedded technology, not only ensure that the symmetry of correction model, and making to revise the residue modal data of rear model and master mould is consistent (without overflowing), weak point does not consider the information of measurement proper vector.Recently, Chu etc. are the systematic study spillover of damping system Modifying model theoretically, and propose a non-spill model modification method (6.ChuMT, LinWWandXuSF.Updatingquadraticmodelswithnospillovereffec tonunmeasuredspectraldata.InverseProblems, 2007, 23 (1): 243 ~ 256), (the 7.ChuD such as ChuDelin, ChuMTandLinWW.Quadraticmodelupdatingwithsymmetry, positivedefiniteness, andnospill-over.SIAMJournalonMatrixAnalysisandApplicatio ns, 2009, 31 (2): 546 ~ 564) the maintenance orthotropicity of damping system is discussed again on this basis further and non-spill Modifying model problem.

Due to the modal data of complete degree of freedom may be provided in the Modifying model of reality hardly, (the 8.CarvalhoaJ such as Carvalhoa, DattaBN, GuptacA, etal.Adirectmethodformodelupdatingwithincompletemeasured dataandwithoutspuriousmodes.MechanicalSystemsandSignalPr ocessing, 2007, 21 (7): 2715 ~ 2731) model modification method of a undamped system is proposed, the modal data that the method utilizes degree of freedom not exclusively to measure carries out Modifying model and can guarantee that correction is non-spill, but for damping system, utilize measurement degree of freedom incomplete defect modal data to carry out going back no one so far without models on spillovers correction and propose corresponding method.

Summary of the invention

The object of the present invention is to provide a kind of correction method for finite element model based on measurement degree of freedom incomplete defect modal data, vibration-damping system being carried out to revise without spilling.

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

Step 1, to primary data format, namely by the adjustment of degree of freedom numbering by the degree of freedom measured in modal data with unmeasured to degree of freedom be separated, and by complex mode data real number, thus guarantee that all computings perform under real form;

Step 2, defect modal data to be expanded, namely treat after format and revise modal data and carry out Matrix QR Decomposition, obtain the matrix equation of defect modal data, and solved by least-squares algorithm;

Step 3, corrected parameter to be determined, utilize the modal data after expansion to set up the matrix equation of corrected parameter, and solve corrected parameter by the matrix equation iterative algorithm of maintenance symmetrical structure;

Step 4, original finite element model to be revised, the corrected parameter namely utilizing step 3 to obtain and determine revised structural finite element model without spilling modified formulation.

Compared with prior art, its remarkable advantage is in the present invention: 1) can process and measure the incomplete defect modal data of degree of freedom and do not need to carry out modal expanding or model reduction, avoids introducing unnecessary error, thus improves the precision revising rear model; 2) this modification method not only makes the modal data measured be melted into revised finite element model, and the modal data not participating in revising remains unchanged before and after correction, thus ensure that the modal data of this correction in revised finite element model and measurement data are coincide, 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.

Accompanying drawing explanation

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

Fig. 2 is primary data formatting procedure schematic diagram.

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

Embodiment

Composition graphs 1, a kind of structural model modification method based on defect mode, step is as follows:

Step 1, to primary data format, namely by the adjustment of degree of freedom numbering by the degree of freedom measured in modal data with unmeasured to degree of freedom be separated, and by complex mode data real number, thus guarantee that all computings perform under real form; Composition graphs 2, to primary data format concrete steps is:

Step 1-1, set whole degree of freedom as 1,2 ... N}, { m ₁, m ₂... m _lbe l the degree of freedom measured, remaining unmeasured degree of freedom is designated as { m _l+1... m _n;

Step 1-2, by the degree of freedom measured with unmeasured to degree of freedom be separated, obtain permutation matrix P (m ₁, m ₂... m _l, m _l+1... m _n), wherein P (m 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, by primary data by ordered array P (m ₁, m ₂... m _l, m _l+1... m _n) convert, namely to the conversion that known modal data to be revised carries out below:

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

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

Step 1-4, by 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 formula, Λ ₁and X ₁be respectively p feature to be revised to the eigenvalue matrix of composition and eigenvectors matrix, wherein, 2k (2k≤p) is individual is complex conjugate pair; ∑ ₁with be respectively p feature measuring to the eigenvalue matrix of composition and the eigenvectors matrix having surveyed degree of freedom, wherein front 2n (2n≤p) is individual is complex conjugate pair; Λ _1R, X _1R, ∑ _1R, for the real modal data matrix obtained after corresponding complex mode data matrix conversion; Corresponding transformation matrix is

Step 2, defect modal data to be expanded, namely treat after format and revise modal data and carry out Matrix QR Decomposition, obtain the matrix equation of defect modal data, and solved by least-squares algorithm; Carry out expansion to defect modal data to be specially:

Step 2-1, by modal matrix X 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, structure defect modal data matrix equation

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

Wherein E _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-th row;

Step 2-3, solve 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 moore-Penrose generalized inverse;

Step 2-4, step 2-3 established data to be 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 to be determined, utilize the modal data after expansion to set up the matrix equation of corrected parameter, and solve corrected parameter by the matrix equation iterative algorithm of maintenance symmetrical structure;

The modal data after expansion is utilized to set up corrected parameter Φ _rmatrix equation, and by keeping the matrix equation iterative algorithm of symmetrical structure to solve corrected parameter, composition graphs 3, the concrete steps solved 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，

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

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

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

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

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 arranged, the matrix form equation be organized into about X is 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 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, solve above-mentioned normal equation, setting initial estimation X ₀for 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 to be judged, if || R _i|| _f< Tol, wherein || || _ffor the Frobenius norm of matrix, then stop calculating, otherwise make i=i+1;

Step 3-7, calculating

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

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

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

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

Return step 3-6 afterwards.

Step 4, original finite element model to be revised, the corrected parameter namely utilizing step 3 to obtain and determine revised structural finite element model without spilling modified formulation, complete the correction to original finite element model.Revised finite element model overflows modified formulation by nothing below and obtains:

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

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

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

As from the foregoing, method of the present invention can process to be measured the incomplete defect modal data of degree of freedom and not to need to carry out modal expanding or model reduction, avoids introducing unnecessary error, thus improves the precision revising rear model; The modal data simultaneously not participating in revising remains unchanged before and after correction, thus ensure that correction is non-spill.

Claims (5)

1., based on a structural model modification method for defect mode, it is characterized in that, step is as follows:

Step 1, treat and revise modal data format, namely by the adjustment of degree of freedom numbering by the degree of freedom measured in modal data with unmeasured to degree of freedom be separated, and by complex mode data real number, thus guarantee that all computings perform under real form;

Step 2, defect modal data to be expanded, namely Matrix QR Decomposition is carried out to the positive-norm state data to be repaired after format, obtain the matrix equation of defect modal data, and solved by least-squares algorithm;

Step 3, corrected parameter to be determined, utilize the modal data after expansion to set up the matrix equation of corrected parameter, and solve corrected parameter by the matrix equation iterative algorithm of maintenance symmetrical structure;

Step 4, original finite element model to be revised, the corrected parameter namely utilizing step 3 to obtain and determine revised structural finite element model without spilling modified formulation.

2. the structural model modification method based on defect mode according to claim 1, is characterized in that, treats correction modal data format concrete steps to be in step 1:

Step 1-1, set whole degree of freedom as 1,2 ... N}, { m ₁, m ₂... m _lbe l the degree of freedom measured, remaining unmeasured degree of freedom is designated as { m _l+1... m _n;

Step 1-2, by the degree of freedom measured with unmeasured to degree of freedom be separated, obtain permutation matrix P (m ₁, m ₂... m _l, m _l+1... m _n), wherein P (m 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, by positive-norm state data to be repaired by permutation matrix P (m ₁, m ₂... m _l, m _l+1... m _n) convert, namely to the conversion that known modal data to be revised carries out below:

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

Step 1-4, by complex mode data real number, namely

In formula, Λ ₁and X ₁be respectively p feature to be revised to the eigenvalue matrix of composition and eigenvectors matrix, wherein, 2k (2k≤p) is individual is complex conjugate pair; Σ ₁with be respectively the eigenvalue matrix of p feature to composition and the freedom matrix of survey of proper vector that measure, wherein, 2n (2n≤p) is individual is complex conjugate pair; Λ _1R, X _1R, Σ _1R, for the real modal data matrix obtained after corresponding complex mode data matrix conversion; Corresponding transformation matrix is

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

Step 2-1, by modal matrix X to be revised _1Rcarry out Matrix QR Decomposition, obtain matrix [Q ₁, Q ₂], namely

Step 2-2, structure defect modal data matrix equation

Wherein E _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-th row;

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

Wherein it is matrix moore-Penrose generalized inverse;

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

4. the structural model modification method based on defect mode according to claim 1, is characterized in that, the modal data after step 3 utilizes expansion sets up corrected parameter Φ _rmatrix equation, and by keeping the matrix equation iterative algorithm of symmetrical structure to solve corrected parameter, the concrete steps solved 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,

E＝-K _aX _1R,

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

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

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 arranged, the matrix form equation be organized into about X is as follows:

In 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, solve above-mentioned normal equation, setting initial estimation X ₀for unit matrix and stop criterion Tol=10 ^-8;

Step 3-5, make i=0 and calculate

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

Step 3-7, calculating

Return step 3-6 afterwards.

5. the structural model modification method based on defect mode according to claim 1, is characterized in that, in step 4, revised finite element model is obtained by nothing spilling modified formulation below: