46 AIAA/SDM, 18–21/April/2005, Austin, TX, USA

Output-Only Approach for Finite Element Model

Updating of AB-204 Helicopter Blade

L. Balis Crema∗ and G. Coppotelli†

University of Rome “La Sapienza”, Rome, 00184, ITALY

The recent developments in the operative modal analysis made possible a new approachin the estimate of the modal parameters. In fact, they could be evaluated consideringonly the responses of the structure when subjected to the “natural” excitation duringits operative life, output–only analysis. Therefore, this approach could reduce the costsneeded for the experimental investigations, since no input measurements are required.Moreover, it takes into consideration the actual loadings and boundary conditions actingon the structure leading to a more accurate identification of the modal parameters. Theexperimental model so far identified could be considered as reference for a further updatingof the structural model. The developed updating procedure, is based on the minimizationof an error function, representing the differences between the numerical and experimentalmodel, by means of design variables associated to the finite element model. The errorfunction considered is built from the evaluation of the sensitivity of correlation functionsof the Frequency Response Functions, FRFs, to design parameters. In this paper theeffectiveness of output-only experimental analysis in both the estimate of the frequencyresponse functions and, then, in the structural updating of the finite element model of anAB-204 helicopter blade is investigated. Specifically, a comparison between the updatedfinite element models obtained both from the FRFs, gained from the well establishedexperimental modal analysis, and from the output only analysis will be performed.

Nomenclature

G Power Spectral Density MatrixH Frequency Response Function Matrixd Excitation LevelU Matrix of the Singular VectorsΣ Diagonal Matrix of the Singular Valuesωk, ωn Frequency point, Eigenvalueψ, φ Eigenvector, Mass Normalized EigenvectorS, p Sensitivity matrix, Vector of the Design ParametersP Design ParameterCR, CP Weighting Matrices

SubscriptA, X Analytical, Experimentalx, f Response, Loading

Superscriptk Mode Index

∗Full Professor, Dipartimento di Ingegneria Aerospaziale e Astronautica, Via Eudossiana, 16.†Assistant Professor, Dipartimento di Ingegneria Aerospaziale e Astronautica, Via Eudossiana, 16, AIAA Member.Copyright c© 2005 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

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American Institute of Aeronautics and Astronautics Paper 2005-2249

46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference18 - 21 April 2005, Austin, Texas

AIAA 2005-2249

Copyright © 2005 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

I. Introduction

Structural model validation is generally performed with the aid of experimental tests in order to assesthe behavior of the Finite Element model with respect to both static and dynamic analyses. Most of theexperimental techniques, developed for the identification of the dynamic parameters, require the measurementof both the excitation and the response signals. Then, the frequency response functions evaluated at theexperimental degrees of freedom are disposable: natural frequencies, damping ratios, and modes shapescould be estimated through curve-fitting-based algorithms.1 These dynamic characteristics could be alsoidentified from techniques that require the measurements of responses only, assuming the structure excitedby a broad band loading. Therefore, these output only techniques not only consider the “natural” excitationacting on the structure during its operative life, and take into account the actual boundary conditions,2,3 butalso could reduce the overall costs associated to the experimental activities since the test setup would notinclude any input measurement devices. Moreover, they represent the preferred approaches for the dynamiccharacterization of the structures when the input load is impossible to measure such as the turbulenceexcitation acting on aircrafts and civil structures, the wave excitation on cruising ships. In this framework,this paper presents a first development of a numerical/experimental procedure aimed to identify the actualdynamic signature of a rotating helicopter blade to be used to validate the corresponding numerical model,and control strategies for the reduction of the vibration levels.

Several output only techniques have been developed and reported in literature. it is possible to groupthem in time and frequency domain approaches. In the first approach, the modal model of the structure isidentified from a state space formulation, solved in time domain, by means of the well known orthogonal-projection technique, also known as Stochastic Subspace Identification or SSI.4,5 On the other side, the modalparameters could be estimated from the singular value decomposition of the power spectral density matrixevaluated for each frequency line in the available spectrum, Frequency Domain Decomposition achievedperforming the Fast Fourier Transform of the output responses(FDD).6 Recently, a further output onlytechniques, based on the Hilbert transform of the power spectral density matrix, has been developed for thedirect estimate of the biased Frequency Response Functions.7,8 These output only approaches present alsoan important drawback related to unavailability of the generalized masses that could not be evaluated sincethe input level is unknown. The problem has been solved by researchers taking advantages of the sensitivityof the modal parameters to structural modifications.9 Performing different experimental tests with differentmass loadings the generalized masses could be then evaluated from the measure of the shifts in the naturalfrequencies of the vibrating structure.10–12

Therefore, the dynamic model of the structure could be expressed using both the modal parameters, andthe frequency response functions, FRFs. This last dynamic model representation has been considered as thereference model for the structural updating of the corresponding F.E. model. The developed methodology,based on the Predictor-Corrector approach,13 belongs to the sensitivity based approaches that require theminimization of an error function through an iterative procedure that allows one to identify the designvariables to which correspond the region of the numerical model scarcely modeled.13,14 The error functionis given by the differences between the numerical and the experimental correlation function of the FRFs,being the last estimates the result of the output–only analysis.

In this paper, the behavior of both the output only procedure, used for estimating the modal parameters,and the developed updating procedure, considered for the validation of the F.E. model, has been investigatedapplying them to the AB-204 helicopter blade. Indeed, not only the dynamic response model of the nonrotating blade will be compared with the one estimated through the input/output analysis, but it will bealso considered as reference model in the further updating procedure. Finally, the updated F.E. model willbe compared with the one achieved when considering the experimental response model obtained using theFRFs.

II. Theoretical Background

A. Modal Parameters Estimates

Recalling the relationship between the input, subscript f , and the output, subscript x, the relationshipbetween the power spectral density matrix, G, and the frequency response functions matrix, H, could beexpressed as:15

G(ω)xx = H(ω)∗G(ω)ffH(ω)T (1)

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American Institute of Aeronautics and Astronautics Paper 2005-2249

where superscripts “ ∗ ” and “ T ” denote complex conjugate and transpose respectively. If the input signalis stationary with zero mean white noise, then Gff exhibits uncorrelated and frequency independent values.Moreover, when a lightly damped structure is considered, the frequency response function H(ω)ij , evaluatedbetween the i-th output degree of freedom and the j-th input load, could be expressed in terms of poles andresidues,6 therefore, Eq. (1) becomes:

G(ω)xx =M∑

k=1

dkψ(k)ψ(k)T

jω − λk+

d∗kψ(k)∗ψ(k)H

jω − λ∗k(2)

where M is the number of modes in the frequency band of interest, dk is a scalar quantity depending of theunknown excitation level, whereas ψk and λk are the k − th mode shape and pole respectively (superscript“H” denotes an hermitian transformation). Now, if a Singular Value Decomposition, SVD, of the powerspectral density matrix of the response is performed, and recalling that Gxx = Gxx

H , it is possible to obtain:

G(ω)xx = U(ω)Σ(ω)U(ω)H (3)

where U(ω) is the matrix containing the singular vectors and Σ(ω) is the diagonal matrix of the singularvalues. If the above SVD is performed at ω = ωnk

only one singular value σ(p)k is non zero if the mode

shapes are well separated: this is the case where the structure almost vibrates as a SDOF system nearthe resonant frequency. The k − th mode is then estimated through the corresponding singular vector u

(p)k

of the aforementioned singular value σ(p)k , as it could be easily obtained by comparing Eqs. (2) and (3).

The SDOF hypothesis could be overcome. Since the singular value decomposition will identify as many“important” singular values (with respect to the others) as the number of such close modes, it is possible toobtain the auto power spectral density associated to only one mode by filtering it with a narrow band-passfilter centered on the natural frequency of the mode of interest. Finally, performing an Inverse Fast FourierTransform of the auto PSD matrix, it is possible to estimate the SDOF auto correlation function, and thenthe damping ratio could be achieved by means of the logarithmic decrement method.16

Unfortunately, with this technique it is not possible to estimate the generalized parameters, such as thegeneralized masses, because the excitation force is unknown. As a result, the mode shapes obtained sofar, called “operational mode shapes”, remain unscaled, restricting then the applicability of the operationalmodal models. This limitation could be overcame recalling the effects a structural modification produces onthe modal parameters,9 as briefly reported in the following subsection.

B. Mode Shape Scaling

The undamped k − th eigenfrequency, ωnk, and (not normalized) mode, ψ(k), could be obtained from the

knowledge of the mass, M, and the stiffness, K, matrices by the well known relationship:(K − ω2

nkM

)ψ(k) = 0 (4)

If the structure is perturbed by a change of both the stiffness, and the mass distribution, denoted with ∆Kand ∆M respectively, then the previous Eq. 4 become:9,11

(K + ∆K)(ψ(k) + ∆ψ(k)

)= (M + ∆M)

(ψ(k) + ∆ψ(k)

) (ω2

k + ∆ω2k

)(5)

where ∆ω2k and ∆ψ(k) are the corresponding changes in the k − th natural frequency and mode shapes.

Assuming that the perturbation in the properties of the structure does not affect the mode shapes, pre-multiplying the above Eq. 5 by ψ(k)T

, and neglecting higher order terms, the k− th generalized mass, givenby definition as mk := ψ(k)T

Mψ(k), is:

mk = ψ(k)T

Mψ(k) =ψ(k)T

∆Kψ(k) − ω2kψ

(k)T

∆Mψ(k)

∆ω2k

(6)

Therefore, a first order approximation for the sensitivity of the natural frequencies for light damped struc-tures, allows one the estimate of the k − th mass normalized eigenvector, φ(k) as:

φ(k) =√

1mk

ψ(k) (7)

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American Institute of Aeronautics and Astronautics Paper 2005-2249

This scaling procedure has been applied to the k − th operational mode shape of the AB-204 helicopterblade, adding small known masses at selected DOFs so that the changes in the poles of the system, i.e.,∆ω2

k, could be experimentally evaluated. As reported in Ref.12 the added masses should be small enoughand located so that to comply with the first order approximation hypothesis, and should not be placed nearthe node of the mode shapes of interest in order to reduce the uncertainties in the estimate of the shifts ofthe poles of the system.

Once the operational mode shapes are correctly scaled with respect to the modal masses, the corre-sponding frequency response functions, when the modal damping ratio, ζk, considered, are finally givenby:

Hij(ω) =M∑

k=1

φ(k)i φ

(k)j

ω2nk

− ω2 + j2ωωnkζk

(8)

C. Structural Updating

The updating procedure considered in this paper is a sensitivity–based method that iteratively solves a set oflinear equations with respect to changes in the P design variables, chosen for the updating by the structuralanalyst.13,14,17,18 These design parameters, casted in a vector p having P components are related, in general,to both the mass, and the stiffness elemental matrices. The differences in the dynamic behavior between theactual experimental model, and the numerical predictions could be defined introducing the following twocorrelation functions of the FRFs:

χs(ωk) =|HH

ijX(ωk) HijA

(ωk)|2

[HHijX

(ωk) HijX(ωk)] [HH

ijA(ωk) HijA

(ωk)]

(9)

χa(ωk) =2|HH

ijX(ωk) HijA

(ωk)|[HH

ijX(ωk) HijX

(ωk)] + [HHijA

(ωk) HijA(ωk)]

where the subscript X and A refer to experimental and analytical FRFs, Hij(ωk) is defined over a discretenumber of frequency lines, Nf , disposable from the test, with the corresponding values of angular frequencywk (k=1,..,Nf ). The previous correlation functions, for each angular frequency, are real quantities defined in[0 1] range depending on the correlation between the numerical and experimental FRFs. Specifically, χs(ωk)represents the correlation between the “shapes” of the corresponding FRFs, whereas χa(ωk) is associated totheir amplitudes. Therefore, the actual differences in the dynamic behavior could be arranged in the vectorε, with 2Nf components, defined as:

ε =

1 − χs(ω1)...

1 − χs(ωNf)

1 − χa(ω1)...

1 − χa(ωNf)

Moreover, the effects of the changes in the design parameters, could be caught, in a first order approxi-

mation, by a sensitivity matrix, Sp, defined as:

Sp =

∂χs(ω1)∂p1

∂χs(ω1)∂p2

· · · ∂χs(ω1)∂pP

· · · · · · · · · · · ·∂χs(ωNf

)

∂p1

∂χs(ωNf)

∂p2· · · ∂χs(ωNf

)

∂pP∂χa(ω1)

∂p1

∂χa(ω1)∂p2

· · · ∂χa(ω1)∂pP

· · · · · · · · · · · ·∂χa(ωNf

)

∂p1

∂χa(ωNf)

∂p2· · · ∂χa(ωNf

)

∂pP

(10)

The sensitivity matrix could be obtained from the derivatives of the correlation functions, with respect tounit variation of the design parameters, that, in turn, could be achieved from the derivatives of the dynamic

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American Institute of Aeronautics and Astronautics Paper 2005-2249

stiffness. Finally, the updating algorithm solves, iteratively, the following linear system of equations withrespect to the changes of the design parameters ∆p

Sp∆p = ε (11)

Since Nf is in general greater than the number of design parameters considered for the updating, thenthe solution of the Eq. 11 is achieved via a least square technique based on an error weighting function,Bayesan Parameter Estimator, that involves two weighting matrices, i.e., CR and CP , in order to give moreimportance to those frequencies points where the FRFs are better.13 Therefore, minimizing the followingfunctional:

F = εT CRε+ ∆pT CP ∆p (12)

the solution vector ∆p, for each iteration step, is then obtained by performing:

∆p =(ST

p CRSp + CP

)−1ST

p CRε (13)

Considering two consecutive iteration steps, the new design parameters are obtained from the old ones as:

pnew = pold + ∆p (14)

III. Results

A. Identification of the Blade Structure

The span of the considered helicopter blade is 6100 [mm] with a chord of 530 [mm], whereas the total weightis 88 [Kg]. The skin, the sandwich core, and the main spar are both in aluminum alloys. The reference finiteelement model of the helicopter blade is formed by twenty beam elements whose elastic and inertia propertieshave been derived from previous experimental analysis.18 As an example, the out-of-plane bending stiffness,and torsional stiffness, as function the blade span are reported in Figs. 1, 2, whereas the position of the elasticaxis is reported in Fig. 3 in which the effect of the changes of the thickness of the skin located at a distance ofabout 3 m from the blade root is evident. The output-only experimental analysis of the helicopter blade has

Figure 1. AB 204 Out of plane Stiffness

been performed in a free-free boundary condition obtained using elastic suspensions: 3 measurements, in theout-of-plane direction, for each of the 11 blade sections - corresponding to the leading edge, blade main spar,and trailing edge positions - have been considered for a total of 33 experimental degrees of freedom. Theresponses of the structure, have been recorded using 214 sampling points with a sampling time equal to 5 [ms],whereas the corresponding power spectral density matrix has been averaged using 8 data blocks of length2048. From this analysis it was possible to perform the singular value decomposition of the power spectraldensity matrix for each frequency line available from the experimental tests, as described in Eq. 3. Reportingthe obtained singular values as function of the frequency, i.e., the frequency domain decomposition diagramof Fig. 4, it was possible to identify, from their maximum values, 6 natural frequencies, 3 of which are veryclosed probably due to in/out-of-plane/torsion mode coupling. In the following, only the out-of-plane modes

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American Institute of Aeronautics and Astronautics Paper 2005-2249

Figure 2. AB 204 Torsional Stiffness

Figure 3. AB 204 shear center position (from the leading edge)

will be considered reducing to 5 the number of considered modal deformations. Furthermore, applying thelogarithmic decrement technique to the SDOF free decay, corresponding to each identified natural frequency,it was possible to estimate the damping ratios. These modal parameters, denoted as OO, have been found

0 10 20 30 40 50 60 70 80 90 100−100

−80

−60

−40

−20

0

20

40

60

80Singular Values of Averaged Spectral Density Matrix−Selected Natural Frequencies−Frequency Domain Decomposition

Frequency [Hz]

[dB

| (m

/s2 )2 /H

z]

Figure 4. Natural frequency estimates from FDD technique

to be in a very good agreement with those achieved from the input/output analysis, IO, as reported in Tabs.1, and 2 where the differences in the natural frequencies and in the damping ratios estimates have beenreported. In fact, the output-only analysis, based on the FDD approach, led to practically the same estimate

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American Institute of Aeronautics and Astronautics Paper 2005-2249

of the natural frequencies as the more common FRF-based identification algorithms, here performed usingthe LMS CADA-PC software. The developed output-only technique was also capable to identify almost

Table 1. Comparison between the natural frequency estimates

Mode # f IOn [Hz ] fOO

n [Hz ] ε %1 7.49 7.43 0.862 21.67 21.60 0.363 43.70 43.55 0.284 44.98 44.85 0.305 72.79 72.69 0.14

Table 2. Comparison between the damping ratios estimates

Mode # ζIOn % ζOO

n %1 0.17 0.372 0.14 0.193 0.39 0.384 0.14 0.205 0.02 0.09

the same mode shapes as the LMS CADA-PC software, as depicted in Fig. 5 in which the well knownMAC index has been graphically represented. From this figure, the coupling between the out-of-plane andtorsional modes, corresponding to the third and fourth mode is apparent. Nevertheless, an overestimate of

1

23

4

5

0

0.2

0.4

0.6

0.8

1

FDDLMS CADA−PC

Figure 5. Correlation matrix between output-only (FDD), and input/output (LMS CADA-PC) eigenvectors

the damping ratios associated to both the first and the fifth mode are achieved by the output-only approachprobably due to leakage effects when estimating the power spectral density matrix of the structural responses.A further experimental investigation has been performed in order to estimate the modal masses, as describedin subsection B of the theoretical background section. With the aid of a preliminary F.E. analysis, assessedto identify the best positions for the added mass with respect to both the type of the mode shapes, and theshifts of the natural frequencies (first order approximation) it was chosen to place five additional lumpedmasses, with the weight of 1 [Kg] each, in correspondence of the experimental points 7, 12, 15, 23, as depictedin Fig. 6. As a consequence, an average shift of the natural frequencies of about 5% has been reported.

Once the modal masses have been evaluated, and, in turn, the normalized mode shapes are disposable,then the frequency response function could be synthesized using Eq. 8. As one can see from Fig. 7 - wherethe amplitudes, in db, of both the measured and synthesized FRFs of the driving point relative to the blade

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American Institute of Aeronautics and Astronautics Paper 2005-2249

1

2

3

4

5

6

7

8

9

10

11

12

13

14 15

16

17 18

19

20 21

22

23 24

25

26 27

28

29 30

31

32 33

Blade Root

Blade Tip

Leading Edge

Trailing Edge

1 [Kg]

1 [Kg]

1 [Kg]

1 [Kg]

1 [Kg]

Figure 6. Added Mass Positioning on the Blade Structure

tip in the out-of-plane direction, i.e., H11 are reported - a good correlation near the peak of resonanceshas been achieved except for the closed modes where a better modal identification will probably reduce theuncertainties in the estimate of the corresponding modal mass.

10 20 30 40 50 60 70

10−9

10−8

10−7

10−6

10−5

10−4

10−3

f [Hz]

|H11

|

Figure 7. Comparison between the measured (blue line) and synthesized (red line), from output-only data,FRFs

B. Structural Updating

The previous dynamic identification of the AB-204 helicopter blade, pointed out a strong coupling of theout-of-plane bending/torsion behavior. Therefore, in order to reduce the computational time, the updatingprocedure has been split into two steps. Assuming the mass distribution correctly identified in the numericalmodel, and taking into account the linearity of the system, first the updating of the out-of plane behaviorof the blade has been performed, then, starting from these partial results, the torsional stiffness distributionwill updated. For both the updating processes, the H11(ω) will be considered, whereas a frequency bandof [5 − 40] Hz, that includes the first two out-of-plane modes, has been used when updating the bendingstiffness, whereas the [30 − 60] Hz frequency band, required to consider the first “global” torsional mode,i.e., mode number 4, has been used for the torsional stiffness instead. The design parameters used for theupdating procedure were associated to elemental stiffness: since the finite element model is built with 20beam elements (numbered from the blade root to the tip), 40 updating parameters, 20 for the bending,and 20 for the torsion, were introduced. Using the FRF from the input/output analysis, the 8 − th, andthe 10 − th finite element have been found to be updated in the bending behavior, as shown in Fig. 8where a comparison between the FRFs is reported. From this result, the bending stiffness of the 8th finiteelement should be increased of about 75%, whereas the 10 − th element should reduce it of about 30%.Next, the torsional distribution seemed to be updated by the stiffness associated to the 2 − nd, 4 − th, and

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American Institute of Aeronautics and Astronautics Paper 2005-2249

10− th element. In this case, the corresponding elemental torsional stiffness should be all increased of about230%, 10%, and 250% respectively. Combining these two information, the updated finite element model

5 10 15 20 25 30 35

10−7

10−6

10−5

10−4

10−3

f [Hz]|H

11|

ExperimentalNumerical InitialNumerical Updated

Figure 8. AB-204 Helicopter Blade - Out-of-Plane Stiffness Updating with FRF

exhibited a reduction in the differences between the natural frequencies considered for the updating, namelymode number 1, 2, and 4, as reported in Fig. 9 where the eigenfrequency shifts, for each mode and foreach updating phase, are reported. Nevertheless, the updating procedure seemed to be not able to increasethe correlation between the mode number 3, i.e., the one presenting the major coupling in the bending andtorsion behavior, since the error in the natural frequencies rose from about 2.5% to almost 4.5%, as it could beseen from Fig. 9. The same updating procedure has been performed using the synthesized H11(ω) achieved

1 2 3 4 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Mode #

Nat

ural

Fre

quen

cy S

hift

(%)

InitialBending UpdatingTorsional Updating

Figure 9. AB-204 Helicopter Blade - Natural Frequency Updating with FRF

form the output-only analysis. The updating results regarding the out-of-plane bending stiffness distributionsuggested a reduction in the value of both the 11− th, and the 14− th elemental stiffness of about 20% and15% respectively, whereas an increase of the stiffness linked to 15 − th finite element of almost 80%. Doingso, the errors in the natural frequencies relative to the out-of-plane bending modes, are almost reduced bya factor 2 with an increase of the correlation of the FRFs, as reported in Fig. 10, and 11. Moreover, theupdating of the torsional mode has been considered. After few iteration steps, it was possible to identify anupdated model by changing the torsional elemental stiffness of the 2 − nd, 4 − th, 5 − th, 8 − th, and the10 − th finite elements by increasing the stiffness of about 160%, 100%, 50%, 200%, and 25% respectively.Considering these suggestions, a good dynamic model for representing the torsion of the helicopter blade hasbeen achieved, as reported in Fig. 12 where the comparison among the FRFs is depicted. Combining theresults obtained in the updating procedures for both the bending, and the torsional stiffness distribution, abetter correlated dynamic finite element model has been achieved. Indeed, the resulting eigenfrequency shiftsnot only have been greatly reduced for the modes considered in the updating procedure, i.e., mode number1, 2, and 4, but also a slight increase in the correlation with the experimental data of the mode with anevident coupling of the bending and torsion behavior (mode number 3) has been achieved. On the contrary,

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American Institute of Aeronautics and Astronautics Paper 2005-2249

1 20

0.5

1

1.5

2

Mode #

Nat

ural

Fre

quen

cy S

hift

(%)

InitialFinal

Figure 10. AB-204 Helicopter Blade Out-Of-Plane Bending Modes - Natural Frequency Updating with FDD

5 10 15 20 25 30 35

10−7

10−6

10−5

10−4

10−3

f [Hz]

|H11

|

ExperimentalNumerical InitialNumerical Updated

Figure 11. AB-204 Helicopter Blade - Out-of-Plane Stiffness Updating with FDD

40 42 44 46 48 50 5210

−11

10−10

10−9

f [Hz]

|H11

|

ExperimentalNumerical InitialNumerical Updated

Figure 12. AB-204 Helicopter Blade - Torsional Stiffness Updating with FDD

the out of band mode, i.e., mode number 5, exhibited a reduction in the correlation of the correspondingnatural frequency, Fig. 13. The previous updating analyses, performed using two different methodologiesin estimating the frequency response functions, allowed one the identification of some differences in thecorresponding resulting finite element models. Indeed, both the FDD, and FRF based updating procedureconverged to the identification of discrepancies in the torsional distribution used for finite element model nearthe root section, as reported in Fig. 14, probably due to presence of the stiffening ribs, and to the difficultin the estimate of the torsional constant, GJ , from static tests.18 Finally, updating the out-of-plane bending

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American Institute of Aeronautics and Astronautics Paper 2005-2249

1 2 3 4 50

0.5

1

1.5

2

2.5

3

3.5

Mode #

Nat

ural

Fre

quen

cy S

hift

(%)

InitialBending UpdatingTorsional Updating

Figure 13. AB-204 Helicopter Blade - Natural Frequency Updating with FDD

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 60000

2

4

6

8

10

12x 10

13

Blade Span [mm]

GJ

[Kg(

mm

2 /s2 )]

Stiffness Increase

Figure 14. AB-204 Helicopter Blade - Updating of the torsional stiffness

stiffness distribution, with the experimental data from the FDD, and FRF analysis, practically identifiedthe same increase in the stiffness of the central blade sections, Fig. 15. This is equivalent to consider themost critical design parameters those near the elastic suspensions and near the discontinuity in the thicknessof the skin.18 Moreover, the different experimental data led to the identification of other critical regionsof the blade. Although the resulting dynamic numerical models showed practically the same behavior, the

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 60000.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2x 10

14

Blade Span [mm]

EI [

Kg(

m3 /s2 )]

Stiffness Increase

Stiffness Reduction

Stiffness Reduction

Stiffness Increase

FRF Updating

FDD Updating

Figure 15. AB-204 Helicopter Blade - Updating of the out-of-plane stiffness

one achieved using the frequency response function from the output only analysis has to be preferred since

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American Institute of Aeronautics and Astronautics Paper 2005-2249

the capability to update all the modes in the considered frequency range, at least for this application. Thiscould be viewed as a consequence of the property of the FDD approach to isolate a “cleaner” mode shape,since the use of the SVD decomposition of the power spectral density matrix of the responses.

IV. Concluding Remarks

An output-only approach has been developed and employed in order to update a beam-like finite elementmodel of the AB-204 helicopter blade. The first step of this approach requires the estimate of the modalparameters from the responses of the structure only without having any measurements of the input loads.Therefore, from the good results obtained by the developed procedure, this technique could be consideredas a privileged candidate for the dynamic identification of systems where the loading is difficult, or evenimpossible, to achieve such as in the rotating helicopter blades in actual operating conditions. The output-only analysis gives the natural frequencies, damping ratios, and mode shapes, but a further experimentalinvestigation is needed to evaluate the modal masses. The problem has been solved by a sensitivity analysisof the natural frequencies to structural modifications. Then, the experimental response model has beensynthesized and used for the updating procedure. The results have been compared with the ones gainedby the well established input/output approach. The output-only procedure not only presents some benefitsrelated to the reduction of both the costs and the time needed for the experimental survey, but also theupdated finite element model appears to be able to increase the correlation of all the mode shapes in theconsidered frequency band, with respect to the one achievable by the standard input/output technique.

Acknowledgments

This research has been supported by “Progetto Giovani Ricercatori: Dinamica ed Aeroelasticicta deiRotori di Elicottero”, University of Rome “La Sapienza”, 2001.

References

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