Determination of Dynamically Equivalent

FE Models of Structures from Experimental Data

Taylan KARAAĞAÇLI1, Erdinç N. YILDIZ1 and H. Nevzat ÖZGÜVEN2

1TÜBİTAK-SAGE, The Scientific and Technological Research Council of Turkey-Defense Industries Research and Development Institute, 06261 Ankara, Turkey

2Middle East Technical University, Department of Mechanical Engineering

06531 Ankara, Turkey

ABSTRACT

In various applications it is important to determine dynamically equivalent spatial finite element (FE) model of complex structures. For instance, obtaining the FE model of an existing aerospace structure is a major requirement for reliable aeroelastic analysis. In such applications a reliable FE model may not be always available, and when this is so a dynamically equivalent FE model derived from modal test will be very useful. This paper presents a noble method to determine spatial FE model of a structure by using experimentally measured modal data along with the connectivity information of measurement points. The method is based on the mass and stiffness orthogonality equations written using experimentally determined mode shapes and natural frequencies. These equations are solved for geometric and material properties constituting global spatial mass and stiffness matrices of an initial FE model. Starting from this initial FE model, mass and stiffness orthogonality equations are updated iteratively employing experimentally obtained natural frequencies and corresponding eigenvectors from the FE model. Iterations are continued until eigensolution of the updated FE model closely correlates with experimentally measured modal data. A simulated case study on GARTEUR scaled aircraft model is presented in order to demonstrate the applicability of the method. NOMENCLATURE A Cross sectional area [ ]kA Coefficient matrix of structural identification equations derived from stiffness orthogonality

[ ]mA Coefficient matrix of structural identification equations derived from mass orthogonality

E Elastic modulus G Shear modulus [ ]I Identity matrix

1I Second moment of area about y axis in local coordinates of a beam element

2I Second moment of area about z axis in local coordinates of a beam element

12I Product moment of area J Polar moment of area

Proceedings of the IMAC-XXVIIIFebruary 1–4, 2010, Jacksonville, Florida USA

©2010 Society for Experimental Mechanics Inc.

[ ]K Stiffness matrix

[ ]ek Element stiffness matrix

L Length of a beam element [ ]M Mass matrix

[ ]em Element mass matrix m Number of degrees of freedom of the finite element model N Number of experimentally obtained normal modes n Total number of degrees of freedom of the FE model p Number of primary coordinates

[ ]T Transformation matrix used in Guyan’s Expansion [ ]eT Coordinate transformation matrix for a finite element matrix

[ ]U Left singular matrix [ ]V Right singular matrix

[ ]*U Conjugate transpose of left singular matrix

[ ]*V Conjugate transpose of right singular matrix

[ ]Σ Matrix whose diagonal elements are singular values [ ]Φ Mass normalized modal matrix

[ ]txΦ Mass normalized experimental modal matrix

[ ]xΦ Mass normalized experimental modal matrix expanded to the size of FE model

{ }φ Eigenvector

[ ]λ Diagonal matrix composed of the squares of undamped natural frequencies ρ Density 1. INTRODUCTION Dynamic characteristics of aerospace structures are to be known in order to predict their aeroelastic behavior. For instance, flutter characteristic of an aircraft structure can be studied by using its dynamically equivalent FE model. Flutter analysis is used to determine the safe flight envelop of an aircraft. Accuracy of the FE model has a major effect on the reliability of the flutter analysis and it can be achieved by correlating FE model of the aircraft structure with its experimentally measured modal data. There has been extensive research on correlation of dynamically equivalent FE model of a structure with experimental modal data. Usually, first a FE model is obtained from CAD data and then it is corrected/improved by using experimental data. Methods used to correct FE models of structures are generally called model updating procedures. Studies in the field of model updating can be classified in two groups: direct and indirect model updating techniques. One of the important examples of early direct updating procedures, namely the Method of Lagrange Multipliers, has been used by Baruch and Bar Itzhack [1] to correct stiffness matrices of structures, and also by Berman and Nagy [2] to correct mass and stiffness matrices of structures. Another important direct updating method has been proposed by Sidhu and Ewins [3] to determine the error of mass and stiffness matrices using analytical and experimental modal data. Those studies have been followed by various contributions [4-5] in the field of model updating during last twenty years. A recent contribution made by Carvalho et al. [6] in this area is distinguished among many others. The method has several advantages: Firstly, it does not need any model reduction or expansion, and secondly it is capable of preventing the appearance of spurious modes in the frequency range of interest. Although direct methods prove to be useful in exact matching of mathematically obtained and experimentally determined modal data, there exist some disadvantages that put the work on the secondarily

searched results side. One such disadvantage is that the original coordinate connectivity of the FE model is lost and resulting mass and stiffness matrices become fully populated. When connectivity in the stiffness matrix is lost, off-diagonal stiffness elements appear for the degrees of freedom among which no direct connection is present. Loss of connectivity information in model updating of an aircraft structure makes it almost impossible to adjust geometric and material properties of standard finite elements such as beams and shells to obtain spatial matrices dictated by direct updating methods. Even if changes may be applied by mass and spring elements, the response of the structure under static loading is degraded which makes aeroelastic studies such as divergence analysis impossible. Loss of connectivity may also degrade semi-definiteness of stiffness matrix and may cause rigid body modes of the aircraft structure to be lost. In case of applying indirect model updating procedures, adjustments in the FE model are made on individual elements in an iterative manner instead of adjustments in the whole system matrices as in the case of direct updating techniques. One of the most widely used indirect methods is the Inverse Eigen Sensitivity Method which makes use of sensitivies of experimentally measured modal data with respect to updating parameters of the FE model. Sensitivity matrix is determined by introducing perturbation to structural parameters of the FE model and analyzing the amount of change in the modal response. Although indirect methods such as Inverse Eigen Sensitivity Method are widely used in model updating of the FE model of structures [7-12] they have certain disadvantages. One such disadvantage is that FE counterparts of the experimentally measured normal modes must appear in the initial FE model to guarantee convergence. Actually this is a general requirement for any of the state-of-the art model updating techniques (direct or indirect) and dictates construction of a detailed initial FE model of the structure. Determination of an accurate initial FE model of an aircraft structure demands considerable time and effort if it is not accomplished by its original design team. Even if such an initial FE model is obtained, the application of the indirect model updating method requires considerable time and experience. The method presented in this paper is developed for structures such as aircraft structures that can be modeled by beam elements. However, the method can further be extended for structures that are modeled by using other type of elements as well. The method starts with an ‘empty’ FE model of which geometric and material properties are not assigned but only connectivity information is available. Connectivity is achieved by connecting measurement degree of freedoms (dofs) with beam elements of known lengths. Initial estimates of geometric and material properties required for the FE model are determined by mass and stiffness orthogonality equations derived from experimentally measured modal data. Sufficient number of modal data is required so that the number of equations obtained is more than the number of unknowns. Alternatively, the number of unknowns should be reduced below the number of equations, by grouping similar elements with the assumption that elements within the same group have the same geometric and material properties. By this way an initial FE model of the structure with eigenvectors corresponding to experimentally determined mode shapes is obtained. Then, the eigenvalues and eigenvectors of the initial FE model are calculated. Eigenvectors of the FE model corresponding to experimental mode shapes are used along with experimentally obtained natural frequencies in order to reconstruct mass and stiffness orthogonality equations. By using constrained least square solution of the orthogonality equations, updated geometric and material properties are calculated. In each iteration, solutions are constrained by lower and upper bounds to avoid divergence problem. Updated properties are used to construct the updated FE model whose eigenvectors are to be used in the next iteration. Iterations are continued until eigensolution of the updated FE model closely correlates with experimentally measured modal data. The application of the method developed is illustrated on a GARTEUR scaled aircraft structure and it proves to be a promising method that deserves to be tested on real aircraft structures. 2. THEORY The theory is presented for structures which can be modeled by Euler-Bernoulli beam elements. Although the extension of the method for structures which are modeled by using other types of elements is not trivial, the same approach can be used. 2.1. Construction of FE Model from Modal Data Starting with Empty Matrices and Connectivity Information The development of the method has started by asking the following questions: Starting only with the knowledge of experimentally measured modal data and of measurement points, is it possible to obtain dynamically equivalent spatial FE model of a real aircraft structure? And also, to what extend the complexity, i.e. number of dofs of the

FE model, may be reduced by still closely correlating the calculated modes with experimentally measured modal data. These questions are the manifestation of the desire of eliminating several difficulties arising in the application of currently available model updating techniques. In current methods, it is necessary to start with a pretty accurate initial FE model. In order to eliminate the time and effort required for the development of a detailed initial FE model, it is chosen to start with an empty FE model. This model is constructed by connecting measurement points with beam elements without assignment of any geometric or material properties, except beam lengths. The initial estimates of properties are then determined by using mass and stiffness orthogonality of experimentally obtained incomplete normal modes. In indirect model updating procedures, dofs of FE models are usually at least an order of magnitude larger than measurement dofs. In this method it is suggested to take measurement points as the nodal points of the FE model and connect them with beam elements. In case of an aircraft structure, the skeleton of the wings will consist of beam like structures such as ribs and spars. Adjustment of connectivity of the FE model so that its look will be similar to the skeleton of the aircraft of interest may help in more accurate and physically meaningful initial estimates of geometric and material properties. 2.2. Expansion of Experimentally Measured Normal Modes After the construction of a FE element mesh model, the next step is the determination of initial estimates of geometric and material properties. The necessary equations to solve for initial estimates of properties are derived from mass and stiffness orthogonality relations of experimentally determined normal modes. Consider the following mass and stiffness orthogonality equations:

[ ] [ ][ ] [ ]IMT =ΦΦ (1)

[ ] [ ][ ] [ ]λ=ΦΦ KT (2)

For a dof FE model, the mass normalized modal matrix will be a square matrix. However, experimental modal matrix corresponding to that FE model will be incomplete, since there will be no measurement for rotational dofs, and also measurement points will be less than the total nodes in the FE model. Moreover, experimental modal matrix will be truncated as we are only interested in modes within the frequency range of interest. Therefore the experimental modal matrix will be highly incomplete. However, as it will be clear in the following section, experimentally measured normal modes have to be expanded to the size of the eigenvectors of the FE model to be able to estimate the geometric and material parameters.

n nxn

Guyan’s Expansion [13] is a simple and reliable technique to be used for expansion of experimentally determined normal modes. In Guyan’s Expansion, the transformation matrix between primary and slave coordinates is determined from stiffness matrix of the FE model. Since for the initial FE model we have mesh information only, it seems there is no stiffness matrix to be used. However, the critical question is the following: to obtain transformation matrix, is it really necessary to use the stiffness matrix of the true FE model or a stiffness matrix of some other FE model with the same mesh can be used? Let us consider two different FE models of the same aircraft structure with the same connectivity, i.e. the same number of elements, number of nodes and lengths for each element, but different cross sectional and material properties for individual elements. Then the stiffness matrices will be different; but if translational dofs of nodal points of two structures are forced to have the same displacement pattern such as that given by an experimental mode shape, it is expected that unconstrained dofs such as rotational and slave dofs in one FE model will be close to the corresponding dofs in the other model. Based on this approximation, beam elements of empty FE model of the aircraft structure will be assigned arbitrary geometric and material properties to obtain a stiffness matrix, from which the transformation matrix required in Guyan’s Expansion Method will be calculated.

According to Guyan’s Expansion Method global stiffness matrix is partitioned to separate elements corresponding to primary and slave coordinates as follows:

[ ] ⎥⎦

⎤⎢⎣

⎡=

ppps

spss

KKKK

K (3)

Then the transformation matrix can be obtained as:

[ ] [ ] [ ]⎥⎦

⎤⎢⎣

⎡−=

−

][

1

IKKT spss (4)

Finally, experimentally measured mass normalized normal modes can be expanded into FE structure size by using the transformation matrix given in Eq. (4) as follows: [ ] [ ] [ ]pxNt

xnxpnxNx T Φ=Φ (5)

2.3. Derivation of Structural Identification Equations from Mass and Stiffness Orthogonality Relations Once experimental normal modes are expanded into FE model size, the next step is the derivation of structural identification equations from mass and stiffness orthogonality relations of normal modes. Consider a dof FE model with elements. Global mass and stiffness matrices can be expanded in terms of element mass and stiffness matrices as follows:

nk

[ ] ∑=

=k

enxnenxn kK

1][ (6)

[ ] ∑=

=k

enxnenxn mM

1

][ (7)

where and are element stiffness and mass matrices; [ ]ek [ em ] [ ]K and [ ]M are global stiffness and mass matrices, respectively. Actual size of element matrices is where m is the number of dofs of an element, but during assembly process they should be used in sparse forms (in the size of ) as can be seen in Eqs. (6) and (7).

mxmnxn

Substituting Eqs. (6) and (7) in Eqs. (1) and (2) yields

[ ] [ ][ ] [ ]Imk

ee

T =ΦΦ ∑=1

(8)

[ ] [ ][ ] [ ]λ=ΦΦ ∑=

k

ee

T k1

(9)

Eqs. (8) and (9) can be further expanded as follows:

{ } [ ] [ ] [ ]{ }{ }⎩⎨⎧

≠=

=+++srifsrif

mmm sk

Tr

01

21 φφ K (10)

{ } [ ] [ ] [ ]{ }{ }⎩⎨⎧

≠=

=+++srifsrif

kkk rsk

Tr

021

λφφ K (11)

from which it is possible to write

{ } [ ]{ } { } [ ]{ } { } [ ]{ }⎩⎨⎧

≠=

=++srifsrif

mmm sk

TrsTrsTr

01

21 φφφφφφ K (12)

{ } [ ]{ } { } [ ]{ } { } [ ]{ }⎩⎨⎧

≠=

=++srifsrif

kkk rsk

TrsTrsTr

021

λφφφφφφ K (13)

Since the individual element matrices in Eqs. (12) and (13) are sparse, they can be condensed as follows:

{ } [ ] { } { } [ ] { } 1111 mx

semxme

T

xm

renx

snxne

Txn

r mm φφφφ = (14)

{ } [ ] { } { } [ ] { } 1111 mx

semxme

T

xm

renx

snxne

Txn

r kk φφφφ = (15)

where [ ]mxmem and [ ]mxmek are local element matrices in global coordinates. { } 1mx

reφ represents the part of

the rth eigenvector { } 1nxrφ corresponding to the dofs of the element of interest.

[ ]mxmem and [ ]mxmek are obtained by transformation of element matrices in local coordinates to global coordinates as follows:

[ ] [ ] [ ][ ]ele

Temxme TmTm = (16)

[ ] [ ] [ ][ ]ele

Temxme TkTk = (17)

where [ ]mxmlem and [ ]mxml

ek are element matrices in local coordinates, and [ ]eT is the transformation matrix for the specific element, which transforms matrices from local (element) coordinates to global coordinates.

Replacing [ ]mxmlem and [ ]mxml

ek into right hand side of Eqs. (14) and (15) we have:

{ } [ ]{ } { } [ ] [ ][ ]( ){ }seele

Te

Tr

e

s

ee

Tr

e TmTm φφφφ = (18)

{ } [ ]{ } { } [ ] [ ][ ]( ){ }seele

Te

Tr

e

s

ee

Tr

e TkTk φφφφ = (19) Before substituting Eqs. (18) and (19) into Eqs. (12) and (13) one last arrangement is necessary as shown below:

{ } [ ] [ ][ ]( ){ } [ ]{ } [ ] [ ]{ }⎟⎠⎞⎜

⎝⎛⎟

⎠⎞⎜

⎝⎛=

s

eele

Tr

ee

s

eele

Te

Tr

e TmTTmT φφφφ (20)

{ } [ ] [ ][ ]( ){ } [ ]{ } [ ] [ ]{ }⎟⎠⎞⎜

⎝⎛⎟

⎠⎞⎜

⎝⎛=

s

eele

Tr

ee

s

eele

Te

Tr

e TmTTmT φφφφ (21)

Right hand sides of Eqs. (20) and (21) can be put in more compact forms with the following definition:

{ } [ ]{ }⎟⎠⎞⎜

⎝⎛=

r

ee

rl

e T φφ (22)

By using the above definition in Eqs. (20) and (21), and substituting them into Eqs. (12) and (13) we obtain:

{ } [ ]{ } { } [ ]{ } { } [ ]{ }⎩⎨⎧

≠=

=++srifsrif

mmmslk

lk

Trlk

rllTrlsll

Trl

01

222111 φφφφφφ K (23)

{ } [ ]{ } { } [ ]{ } { } [ ]{ }⎩⎨⎧

≠=

=++srifsrif

kkk rslk

lk

Trlk

sllTrlsll

Trl

0222111

λφφφφφφ K (24)

For 3-D Euler-Bernoulli beam elements, size of element matrices [ ]em and [ ]ek in local coordinates is 12x12 and parametric expressions for matrix elements are too lengthy to be presented in this paper. For that reason, element matrices of a simplified 2-D Euler-Bernoulli beam model without axial dof (see Figure 1) are given here just to explain the application of the method. However, the formulation is obtained for 3-D Euler-Bernoulli beam elements, and this formulation is used in the case study presented in this paper. Two-noded beam element shown in Figure 1 has 4 dofs and its element mass and stiffness matrices are given as follows:

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−−−

−−

=

22

22

3

46266126122646612612

LLLLLLLLLLLLL

LEIk le (25)

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−−−−−

=

22

22

422313221561354313422135422156

420LLLLLLLLLLLLL

ALmle

ρ (26)

On the other hand, { }rleφ vector given in Eqs. (23) and (24) is as follows:

{ } { }Trle 2211 θνθνφ = (27)

where ν and θ represent translational and rotational dofs at nodes, respectively.

Figure 1. 2-D Euler-Bernoulli beam model without axial dof

Substituting Eqs. (25), (26) and (27) into Eqs. (23) and (24), the following equations are obtained:

⎩⎨⎧

≠=

=+++srifsrif

AaAaAa krsk

rsrs

01

)()()( 2211 ρρρ K (28)

⎩⎨⎧

≠=

=+++srifsrif

EIbEIbEIb rk

rsk

rsrs

0)()()( 2211

λK (29)

where and are known constants calculated from mesh properties and experimental mode shapes. rs

earseb

It can easily be seen that coefficients and can be calculated by using experimental normal modes and matrices given in Eqs. (25) and (26). It is important to note that experimental normal modes should be mass normalized, which can be calculated by using experimentally obtained modal constant matrices [14].

rsea

rseb

When 3-D Euler Bernoulli beam elements are used, Eq. (29) takes the following form:

( )

⎩⎨⎧

≠=

=+++

++++++++

srifsrif

EAfEIb

EAfEIbEAfGJeEIdEIcEIb

r

kelement

krskk

rsk

element

rsrs

element

rsrsrsrsrs

0)()(

)(...)()()()()(

1

2

22212

1

11111121121111

λ4444 34444 21

KK

4444 34444 214444444444 34444444444 21

(30)

Equations (28) and (30) can be put into the following compact forms: [ ]{ } { }mmm bxA = (31) [ ]{ } { }kkk bxA = (32) where and are coefficient matrices that consist of and , … terms. { and ][ mA ][ kA

rsea

rseb

rsec

rsef }mx { }kx are

unknown vectors which consist of products of geometric and material properties as given in Eqs. (28) and (30). Eqs. (31) and (32) will be referred to as structural identification equations. The number of individual equations in matrix equations (31) and (32) is determined by the number of experimentally determined modes. If experimental modes have been extracted in the frequency range of interest, number of independent equations can be written for each of Eqs. (31) and (32).

N2/)1( +NN

If there are more unknowns than the number of equations because of the limited number of reliable mode shapes experimentally obtained in the frequency range of interest, the number of unknowns can be reduced by grouping similar beam elements with the assumption that elements within the same group have the same geometric and material properties. Then the initial estimates for the unknown system parameters can be calculated as follows:

{ } [ ]{ }mmm bAx += (33) { } [ ]{ }kkk bAx += (34) where [ ]+mA and [ ]+kA show pseudo inverses of [ ]mA and [ ]kA respectively. Studies have shown that experimental errors and errors introduced during expansion of experimental modes may give rise to ill conditioned coefficient matrices and . Another source of ill conditioning is the order of magnitude difference between the numerical values of the unknown parameters. Avoiding the problems due to ill conditioned coefficient matrices will be addressed in the next section.

][ mA ][ kA

2.4. Singular Value Decomposition (SVD) Analysis of Coefficient Matrices SVD analysis has shown that major singular values and corresponding left and right singular vectors of ideal coefficient matrices are still preserved in actual matrices. Nevertheless experimental errors, errors due to expansion of normal modes and unbalanced order of magnitude between unknowns introduce uncommon singular values and corresponding singular vectors in the actual matrices. Consider the following SVD of the actual coefficient matrices:

[ ] [ ][ ][ ]*mmmm VUA ∑= (35)

[ ] [ ][ ][ ]*kkkk VUA ∑= (36) Pseudo inverses of coefficient matrices turn out to be:

[ ] [ ][ ][ ]*mmmm UVA ++ ∑= (37)

[ ] [ ][ ][ ]*kkkk UVA ++ ∑= (38) where [ ]+∑m and [ ]+∑ k are pseudo inverses of [ ]m∑ and [ ]k∑ , respectively, and they are obtained by replacing every non-zero singular value by its reciprocal. Therefore, reciprocals of erroneous singular values of small magnitudes become dominant terms of the pseudo inverse of coefficient matrices. Consequently, Eqs. (33) and (34) may give rise to erroneous solutions. To avoid this, lower singular values of coefficient matrices have to be eliminated systematically. 2.5. Iterative Solution Procedure After determining initial estimates of products of geometric and material parameters by solving equations (33) and (34), mesh only FE model is completed by using elastic and inertial estimates obtained. Since initial estimates are least square solutions, eigensolution of the initial FE model will not perfectly correlate with experimental modes.

To improve correlation the following algorithm is applied: Step 1: Solve eigenvalues and eigenvectors of the initial FE model in order to obtain modal data corresponding to experimentally measured ones. Step 2: Reconstruct coefficient matrices of Eqs. (31) and (32) by using eigenvector counterparts of experimental normal modes to reduce ill conditioning. Step 3: Solve { and { from reconstructed versions of Eqs. (31) and (32), by using a non-linear least square solver with lower and upper bounds to avoid divergence problem.

}mx }kx

Step 4: After obtaining updated parameters in Step 3, update FE system matrices and go to Step 2. Use previous solution as initial guess for the non-linear least squares solver. Continue until eigenvalues and eigenvectors converge to their experimental counterparts. 3. CASE STUDY In this case study, the applicability and the accuracy of the method are demonstrated by using simulated experimental modal data derived from a FE model of SM-AG19 scaled aircraft structure of GARTEUR (Group for Aeronautical Research and Technology in Europe). The FE model shown in Figure 2, has 44 nodes, 40 elements and 264 dofs and its overall dimensions are: 2000 mm wing span and 1500 mm fuselage. Material properties of aluminum used in the FE model are: E = 70 GPa, ρ = 2800 kg/m3 ν = 0.3. The simulated experimental modal data are obtained as follows: 1. It is assumed that an experiment similar to the one used in the work of Kozak et al. [10] is carried out, and the

first 10 elastic modes are obtained. By using the FE model of the system with 264 dofs, the first 10 elastic modes are numerically calculated.

2. In order to simulate the experimental extraction of the modal vectors, only the 66 elements are retained in

each eigenvector by keeping the dofs corresponding to the measurements points in [10]. 3. Finally, the 10 incomplete eigenvectors are polluted with random multipliers changing between 0.95 and 1.05,

thus giving ± 5% error to the exact values.

Figure 2. FE Model of GARTEUR SM-AG19 test bed

The method presented in this paper is applied according to the following scenario: Only the locations of measurement points and the first 10 elastic experimental mass normalized mode shapes and corresponding natural frequencies are available for the structure. The objective is to derive a FE model of that structure which is dynamically equivalent to the actual FE model. Thus it is expected to have a good correlation between the pseudo-experimental modal data and the modal data obtained from the eigen solution of the FE model determined. The following procedure is applied in determining the FE model: 1. Measurement points are connected by beam elements to construct mesh only FE model of the structure

without any geometric or material properties. Accordingly, a mesh only FE model with 44 nodes, 40 elements and 264 dofs is obtained. Only available geometric properties are element lengths.

2. Arbitrary geometric and material properties are assigned to the beam elements to derive a stiffness matrix.

This stiffness matrix is used to calculate the transformation matrix between primary (measurement) and slave dofs in order to apply Guyan’s Expansion. Then, incomplete pseudo-experimental normal modes are expanded into the size of the FE model.

3. 110 structural identification equations are obtained by using the expanded normal modes. Half of these

equations are obtained from mass orthogonality and the other half are obtained from stiffness orthogonality. To reduce the number of unknowns (which is 240 in this example) below the number of equations, similar elements are collected into 4 groups with the assumption that elements within the same group have the same structural properties. Definition of each group is given in Table 1. Thus the total number of unknowns is reduced to 24.

4. By least square solution of 110 equations, unknowns are calculated. These are the initial estimates of the

structural parameters. By assigning these initial estimates of structural parameters to relevant beam elements stiffness and mass matrices (of sizes 264x264) of the whole system are obtained. Comparisons of the eigen solutions of the initial FE model with corresponding pseudo-experimental modal data are given in Table 2 and Figure 3.

5. In the last step, the iterative solution procedure presented in section 2.5 is applied and a converged FE model

whose eigen solutions are in very good correlation with pseudo-experimental modal data is obtained. Comparisons of eigen solutions of the converged FE model with corresponding pseudo-experimental modal data are given in Table 3 and Figure 4.

Table 1. Definition of element groups

Group No Group Definition 1 Wing

2 Fuselage

3 Vertical Stabilizer

4 Horizontal Stabilizer

Table 2. Correlation between pseudo-experimental modes and the modes of the initial FE model

Pseudo-Experimental

Modes

Pseudo-Experimental

Natural Frequencies (Hz)

Corresponding Modes of the

Initial FE Model

Natural Frequencies of

the Initial FE Model (Hz)

Difference of Natural

Frequencies (%)

1 5.62 1 4.96 -11.8 2 16.73 2 15.33 -8.4 3 37.37 3 35.09 -6.1 4 37.93 4 35.46 -6.5 5 38.02 5 35.47 -6.7 6 43.63 6 38.28 -12.3 7 46.27 7 62.01 34.0 8 54.83 8 68.20 24.4 9 67.57 10 74.85 10.8

10 73.13 9 71.74 -1.9

Figure 3. MAC matrix between pseudo-experimental modes and the modes of the initial FE model

Table 3. Correlation between pseudo-experimental modes and the modes of the converged FE model

Pseudo-Experimental

Modes

Pseudo-Experimental

Natural Frequencies (Hz)

Corresponding Modes of the

Converged FE Model

Natural Frequencies of the Converged FE Model (Hz)

Difference of Natural

Frequencies (%)

1 5.62 1 5.65 0.5 2 16.73 2 17.02 1.7 3 37.37 3 37.30 -0.2 4 37.93 4 37.94 0.0 5 38.02 5 38.03 0.0 6 43.63 6 43.63 0.0 7 46.27 7 46.28 0.0 8 54.83 8 54.63 -0.4 9 67.57 9 67.57 0.0

10 73.13 10 73.16 0.0

Figure 4. MAC matrix between pseudo-experimental modes and the modes of the converged FE model The excellent agreement observed in Table 3 and Figure 4 shows the accuracy of the method developed. 4. CONCLUSIONS In this paper, a noble method is presented to determine spatial FE models of structures by using experimentally measured modal data and connectivity information. The method aims to eliminate considerable time and effort required to obtain a complex and accurate initial FE model which is a must in every state-of-the art model updating scheme. The theory of the method is presented for structures that can be dynamically modeled by beam elements, although the general concept can be applied to structures that can be modeled by using other types of elements as well. In this method, first of all, a mesh only stick FE model is constructed by connecting measurement points with beam elements. Secondly, structural identification equations are derived from mass and

stiffness orthogonality of experimentally obtained incomplete normal modes. Then, from the solutions of structural identification equations initial estimates for geometric and material properties of beam elements are obtained, and thus spatial stiffness and mass matrices of the whole structure are constructed. Finally, starting from this initial FE model, an iterative procedure is applied to obtain an ultimate FE model whose eigensolutions correlating well with experimental modal data. The applicability of the method is demonstrated on a GARTEUR scaled aircraft structure. Based on the performance of the identified spatial FE model in predicting modal characteristics of the structure, it is concluded that the method presented deserves to be tested on real aircraft structures, which will be the next step in the ongoing research. 5. REFERENCES

[1] M. Baruch, Y. Bar Itzhack, Optimal Weighted Orthogonalization of Measured Modes, AIAA Journal, 16(4), 346-351, 1978 [2] A. Berman, E.J. Nagy, Improvement of a Large Analytical Model Using Test Data, AIAA Journal, 21(8), 1168- 1173, 1983 [3] J. Sidhu, D.J. Ewins, Correlation of Finite Element and Modal Test Studies of a Practical Structure, Proc of 2nd IMAC, 756-762, Orlando, Florida, 1984 [4] B. Caesar, Update and Identification of Dynamic Mathematical Models, Proc. of 4th IMAC, 394-401, Los Angeles, California 1986. [5] H. P. Gypsin, Critical Application of the Error Matrix Method for Localization of Finite Element Modeling Inaccuracies, Proceedings of 4th IMAC, 1339-1351, K.U. Leuven, 1986. [6] J. Carvalho, B. N. Datta, A. Gupta and M. Lagadapati, A Direct Method for Model Updating with Incomplete

Measured Data and without Spurious Modes, Mechanical Systems and Signal Processing, 21, 2715-2731, 2007

[7] D. Göge, 2003 Automatic Updating of Large Aircraft Models Using Experimental Data from Ground Vibration Testing, Aerospace Science and Technology, 7, 33-45, 2003 [8] J. M. W. Lee, G.R. Parker, Application of Design Sensitivity Analysis to Improve Correlation Between Analytical and Test Mode, MSC 1989 Wold Users’ Conference Proceedings, Los Angeles, California, USA, 1989 [9] M. T. Kozak, M.D. Cömert, H. N. Özgüven, A Model Updating Routine Based on the Minimization of a New Frequency Response Based Index for Error Localization, 25th IMAC, Orlando, Florida, 2007 [10] M. T. Kozak, M. Öztürk, H. N. Özgüven, A Method In Model Updating Using Miscorrelation Index Sensitivity, Mechanical Systems and Signal Processing, 23(6), 1747-1758, 2008 [11] R. M. Lin, D. J. Ewins, Model Updating Using FRF Data, 15 International Seminar on Modal Analysis, 141-162, Leuven-Belgium, 1990 [12] P.O. Larsson, P. Sas, Model Updating Based on Forced Vibration Testing Using Numerically Stable Formulations, 10th IMAC, San Diego, USA, 966-974, 1992 [13] R. J. Guyan, Reduction of Stiffness and Mass Matrices, AIAA Journal, 3(2), 380, 1965 [14] J. He, Z. Fu, Modal Analysis, Butterworth Heinemann, 2001