51st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Orlando, FL, 12-15 April 2010

Aeroelastic Identification of a Flying UAVs by OutputOnly Data with Applications on Vibration Passive

Control

Franco Mastroddi∗ Giuliano Coppotelli† Andrea Cantella‡

Università degli Studi di Roma “La Sapienza", Via Eudossiana, 16, 00184, Rome, ITALY

This paper shows the capabilities of a system identification approach, based on theexperimental measurements of Output-Only, O-O, data, to monitor the aeroelastic char-acteristics of fixed-wing Unmanned Aerial Vehicles, UAVs, during their actual operativeconditions. Traditional Input/Output identification techniques are not easily carried outfor aeroelastic systems in operative flight conditions because of the intrinsic difficulty onmeasuring actual input loads. Therefore, only the response output level should be desirablyused for the identification of aeroelastic systems. Then, how the use of the O-O approachallowed to passively reduce the operative aeroelastic vibrations, via piezoelectric-patch de-vices (PZTs,) mounted aboard the UAV is presented. In order to validate the proposedapproach, a preliminary flight test campaign has been carried out. Data recorded aboardthe Unmanned Aerial vehicle demonstrated the effectiveness of the PZT patches whosedesign has been based on the O-O system identification performed in the first part of thepresent work.

I. Introduction

Several approaches, have been proposed by researchers for the identification of the dynamic propertiesof systems by output-only data, and some of them could be regarded as candidate methodologies for an“on-line” identification of modal parameters. In an attempt to categorize the methodologies developed bydifferent researchers, two main approaches could be identified: both of them are based on the hypothesisthat the structure is randomly excited by a “broad band” loading. To the first group belong those methodsbased on the analysis of the output responses in the time domain, also known as Stochastic SubspaceIdentification (SSI) based methods, Refs. [6,7], whereas the second group refers to those methods devoted tothe identification of the modal parameters in the frequency domain, i.e., Frequency Domain Decomposition(FDD), Refs. [8]. Approaching the problem of the dynamic identification of system in time domain, thestate-space formulation is solved by a so-called orthogonal-projection technique in order to achieve the modalmodel of the structure.9–11 On the other hand, it is possible to achieve the modal parameters by consideringthe frequency domain properties of the output signals.8,13,14 In the FDD approach, by evaluating thesingular values of the power spectral density matrix for each frequency line available from the experimentalsetup, natural frequencies, mode shapes, and damping ratios can be derived, Refs. [8,12]. Nevertheless, bothSSI, and FDD approaches could not identify the modal masses or the modal scale factors because the inputloading is not measured. This drawback has been overcome, Refs. [15–17], considering the sensitivity of themodal parameters to structural changes, as reported in the pioneer work due to Dr. De Vries, Ref. [18].

In this paper, the capability to experimentally identify both the aeroelastic frequency-response functionsand the aeroelastic stability scenario corresponding to different operative flight conditions of a flying Un-manned Aerial Vehicle (namely, a YAK 112 Airworld model) from output-only data will be investigated.Assuming the aircraft randomly excited by broad band loading or practically by the actual aerodynamic-load noise, the corresponding time-domain responses measured in flight by means of a set of accelerometers

∗Professor, Department of Aerospace and Astronautical Engineering.†Professor, Department of Aerospace and Astronautical Engineering.‡Graduate EngineerCopyright c© 2010 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

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51st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference<BR> 18th12 - 15 April 2010, Orlando, Florida

AIAA 2010-2556

Copyright © 2010 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

connected to a dedicated acquisition system, are used to build a Power Spectral Density matrix, Refs. [1,2].Then, using the Frequency Domain Decomposition technique, FDD, the poles and the complex eigenvectorsof the aeroelastic system can be evaluated for different flight conditions. As a consequence, not only astability diagram devoted to the flutter identification can be assessed and compared with the predictionscoming from the standard numerical investigation, but the aeroelastic frequency response matrix can beexplicitly identified too. Moreover, the approach allows also to identify, via the same direct experimentalflight test data, the generalized unsteady aerodynamic operators associated to the aircraft at its differentflight phases. The considered flight regimes range from low to high speed in the flight envelope. Moreover,the approach allows also to identify, via the same direct experimental flight test data, the generalized un-steady aerodynamic operators associated to the aircraft at its different flight phases. This issue is possibleby utilizing also the UAV natural frequencies and modes of vibrations that can be evaluated by the sametechnique but by means of an output-only vibration test. The experimental-based estimates for the unsteadyaerodynamic operators can be compared with the numerical ones typically obtained by commercial codes inthe aeroelastic modeling. In the second part of the paper, the O-O experimental approach has been used toexperimentally investigate the capability of a vibration suppression approach to reduce the vibration levelsaboard the fixed-wing UAV. The developed approach belongs to the so called passive methods (Refs. [32–34])and uses the electro-mechanical characteristics of piezoelectric patches to transform the mechanical energyinto electrical energy.

II. System identification using Output-Only data.

In this section a general procedure for the identification of poles and natural modes (both structuraland aeroelastic) of a system, will be presented. The correlation matrix, Rx(τ), between the responses of avibrating structure evaluated at N degrees of freedom, arranged in a vector x(t), supposed to be excited bya random stationary loading, is defined at a generic time shift, τ , as:

Rx(τ) = E[x(t),x(t+ τ)T ] (1)

where the E represents the “expected” value operator. Moreover, if M normal modes, ψ(j), j = 1,Mare used to describe the dynamic behavior of the structure, then the modal transformation matrix, Ψ =[ψ(1),ψ(2), . . . ,ψ(M)], could be used to express the correlation of the output responses in term of the Mmodal amplitudes, q(t). Indeed, substituting x(t) = Ψq(t) into the previous Eq. (1), the N ×N correlationmatrix is given by:

Rx(τ) = E[Ψq(t), (Ψq(t+ τ))T ] = ΨE[q(t),q(t+ τ)T ]ΨT = ΨRq(τ)ΨT

where Rq(τ) is the correlation matrix defined over the modal coordinates, which is diagonal if the load actingon the structure is uncorrelated. Since each element of correlation matrix satisfy the Wiener–Kintchine condi-tion, then the power spectral density matrix of the output responses, Sx(ω), is given by Fourier transformingRx(τ), that is:

Sx(ω) = ΨSq(ω)ΨT (2)

where Sq(ω) is a diagonal matrix of non-negative real values, whose generic element is indicated as Sql

containing the auto power spectral density terms, of the modal amplitudes20,21 . As a consequence, thecross power spectral density related to the i-th and the j-th response signals evaluated at a frequency lineωr, is given by:

Sx,ij(ωr) = ψ(r)i Sqr

(ωr)ψ(r)j +

M∑l=1,l 6=r

ψ(l)i Sql

(ωr)ψ(l)j (3)

If the spectral properties of Sx(ω) are investigated through the Singular Value Decomposition, svd, recallingthat this decomposition is a linear transformation, it follows that:

svd [Sx(ωr)]ij = svd[ψ

(r)i Sqr (ωr)ψ

(r)j

]+

M∑l=1,l 6=r

svd[ψ

(l)i Sql

(ωr)ψ(l)j

](4)

2

Generally, the Singular Value Decomposition of a generic square and symmetric matrix A with dimensionN is given by:

svd[A]ij = [UΣUT ]ij =N∑

p=1

N∑q=1

UiqΣpqUjp (5)

in which U is an orthogonal matrix, i.e., UT U = I, whose columns are called singular vectors, and Σ is adiagonal matrix of singular values, then the previous Eq. (4) reduces to:

svd [Sx(ωr)]ij = ψ(r)i Sqr

(ωr)ψ(r)j +

M∑l=1,l 6=r

ψ(l)i Sql

(ωr)ψ(l)i (6)

The properties of the singular value decomposition could be therefore used as a powerful mean for themodal parameter estimate of a structure: indeed, if the system behaves as a single degree of freedom, SDOF,in the neighbourhood of the resonant frequency, and if ωr and ψr, considered in Eq. (6), refer to the r− thnatural frequency and the corresponding eigenvector respectively, then the r− th term in Eq. (3), and thenin Eq. (6), will dominate. Therefore, comparing Eq. (5) and Eq. (6), if one evaluates the dependence of thesingular values, of the power spectral density matrix, as a function of the frequency line, a good estimate of thenatural frequency and mode shape is given by the frequency line to which corresponds the (local) maximumof the singular value together with the corresponding singular vector. If the structure presents more thanone orthogonal mode, vibrating at the same natural frequency, then the singular value decomposition willgive more then one dominant singular value: an estimate of such modes is, again, given by the correspondingsingular vectors.8 Once the natural frequency has been estimated, the corresponding damping ratio can bealso evaluated by the Inverse Fourier Transform of the power spectral density functions across the frequencyline corresponding to the r − th natural frequency identifying the free decay of the system for that naturalfrequency, Refs. [8]. Therefore, the damped natural frequencies and damping ratio are disposable evaluating,for example, the crossing times and applying logarithm decrement technique, respectively.8,16,17 Finally,the frequency response function, between the i-th and j-th degree of freedom of the whole system, could beestimated using the relation:

Hij(ω) =M∑

k=1

φ(k)i φ

(k)j

ω2nk − ω2 + j2ωωnkζ2

k

(7)

in which φ(k)i is the mass normalized eigenvector associated to the k−th mode and the i-th degree of freedom.

III. Aeroelastic and unsteady-aerodynamic identification using O-O data

In the present section the above concept on the O-O approach are applied to the output of a fixedwing aeroelastic system in order to identify the aeroelastic modal characteristics together with the unsteadyaerodynamic operator. A linear aeroelastic system is described in the Laplace domain as[

s2M + K− qDE(s;U∞,M∞)]q = fg (8)

where˜denotes the Laplace-transform operator, M and K the (diagonal) modal mass and stiffness matrices,E is the generalized-aerodynamic-force matrix, qD = 1

2%∞U2∞ is the dynamic pressure, U∞ the flight speed,

and M∞ the flight Mach number, whereas fg is the Laplace transform of the modal external load vector,for example, a gust load. Note that the vector q collects the modal co-ordinates. Thus, one has the modaltransformation in the Laplace domain

x =N∑

n=1

ψ(n)qn (9)

where N is the number of the structural modes ψ(n) assumed for the aeroelastic analysis.The aeroelastic system can be re-written in first order form in the Laplace domain as

sq = vsv = −M−1Kq + qDM−1E(s)q + fg

(10)

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Considering that

x = q, vT (11)

and reducing the system reported in Eq. (10) in matricial form as

sx =

[0 I

−M−1K + qDM−1E(s) 0

]x +

0

M−1fg

(12)

one can obtainx = [sI−A(s)]−1fg = H(s)fg (13)

where

A(s) =

[0 I

−M−1K + qDM−1E(s) 0

](14)

The aeroelastic stability analysis is performed on the homogeneous form of Eq. (8) considering the eigen-problem [

s2nM + K− qDE(sn;U∞;M∞)]wn = 0 (15)

which generally gives, because of the typical trascendental nature of the aerodynamic matrix E with respectto the Laplace variable s, a not-finite number of eigenvalues sn and eigenvectors wn. Therefore, this not-standard eigenproblem is typically solved by means of an iterative algorithms (Refs. [23,24]). If one defines

vn :=

wn

snwn

(16)

the relation written in Eq. (15) gives the same not-standard eigenproblem in a first order form

sn

wn

snwn

=

[0 I

−M−1K + qDM−1E(sn) 0

]vn (17)

orA(sn)vn = snvn (18)

The Equation (17) can be formally written as a generalized eigenvalue problem as

A(sn)V = VS (19)

where V is a [2N× 2N] matrix that collects the complex N eigenvectors vn together with their complexconjugated

V := [v(1)...v(2)

... . . ....v(N)

...v∗(1) ...v∗

(2) ... . . ....v∗

(N)] (20)

and S is a (diagonal) eigenvalues matrix with eigenvalues ordered as the corresponding eigenvectors in Eqn.20. Indeed, we have supposed that the aeroelastic poles and eigenvectors are all complex and recast in orderto have first the N distinct and then the N corresponding complex conjugated. The left hand side of Eq.(19) means that for each product row × column the matrix A(sn) changes with n. Considering the followingpartitions for the matrices V and S

V =

[V11 V12

V21 V22

]S =

[S1 00 S2

](21)

where

S1 =

s1 0

. . .0 sn

S2 =

s∗1 0

. . .0 s∗n

(22)

4

and remembering that

A(sn) =

[0 I

−M−1(K− qDE(sn)) 0

]=:

[0 I

B(sn) 0

](23)

the generalized eigenvalue problem reduced in Eq. (19) becomes[V21 V22

B(sn)V11 B(sn)V12

]=

[V11S1 V12S2

V21S1 V22S2

](24)

that shows four matrix equalities, whereas the aeroelastic eigenproblem written in second order form in Eq.(15) can be reduced in compact form as

WS2 = B(sn)W (25)

where W is the [N×N ] eigenvector matrix collecting theN eigenvectors wn on its column. Now, a coordinatetransformation has been introduced to deal with the first order forced aeroelastic problem in the Laplacedomain reported in Eq. (10)

x = Vη (26)

or using Eq. (11) and (21), qv

=

[V11 V12

V21 V22

]η1

η2

(27)

Then, using Eq. (27), the aeroelastic system written in first order form in the Laplace domain is

s(V21η1 + V22η2) + M−1K(V11η1 + V12η2)− qDM−1E(s)(V11η1 + V12η2) = M−1fg

which gives:

s(V21η1 + V22η2) + (M−1[K− qDM−1E(s)

])(V11η1 + V12η2) = M−1fg (28)

At this point of the theoretical formulation, the following piece-wise approximation is introduced: indeed,the aerodynamic matrix E(s) is approximated with the values that it assumes in correspondence of theaeroelastic poles E(sn), then

E(s) u E(sn) (29)

when s is in a specific neighbourood of sn. The assumption reported in Eq. (29) implies that (see also Eqn.14):

−M−1[K− qDM−1E(s)

]u B(sn) (30)

With such an approximation, the response aeroelastic problem indicated in Eq. (28) within the coordinatetransformation becomes:

s(V21η1 + V22η2)−B(sn)[V11η1 + V12η2] = M−1fg (31)

Using equalities shown in Eq. (24) for B(sn)V11 and B(sn)V12 the previous becomes

V21(sI− S1)η1 + V22(sI− S2)η2 = M−1fg (32)

Moreover, substituting the coordinate transformation assumed in Eq. (27) into:

sq = v (33)

one hass(V11η1 + V12η2) = V21η1 + V22η2 (34)

or, using the first row of Eq. (24)

s(V11η1 + V12η2) = V11S1η1 + V12S2η2 (35)

orV11(sI− S1)η1 + V12(sI− S2)η2 = 0 (36)

5

Finally, the final Eq. (36) and (32)V11(sI− S1)η1 + V12(sI− S2)η2 = 0

V21(sI− S1)η1 + V22(sI− S2)η2 = M−1fg(37)

can be written in a matricial form as[V11 V12

V21 V22

][Is− S1 0

0 Is− S2

]η1

η2

=

0

M−1fg

(38)

Then, the state-space variable ηT := ηT1 , η

T2 is given by

η =

. . . 01/(s− sn)

. . .1/(s− s∗n)

0. . .

V−1

0

M−1fg

(39)

or, using Eq. (26),

qv

= V

. . . 01/(s− sn)

. . .1/(s− s∗n)

0. . .

V−1

0

M−1

fg (40)

which are the spectral approximated representation of the Aeroelastic transfer function H(s) based onaeroelastic poles sn and eigenvectors v(n) or w(n) (Eqs. (13) and (15)). This transfer function matrix canbe partitioned such that

q = Hq(s)fgv = Hv(s)fg

(41)

where [Hq(s)Hv(s)

]= V

[1/(s− sn) 0

0 1/(s− s∗n)

]V−1

0

M−1

(42)

Under the approximation introduced in Eq. (29) and with the relation written in Eq. (40) it is possibleto have an estimate of the aeroelastic transfer function matrix Hq(s) using the aeroelastic eigenvalues sn

and the aeroelastic eigenvectors wn obtained both via numerical and/or experimental data. The modalmass matrix M can be also evaluated with numerical data as well. Thus, one could compare the aeroelastictranfer function matrix HExp

q (s) as obtained via experimental data (using Eq. (42) with experimental polessn and eigenvectors w(n)) with the aeroelastic tranfer function matrix HNum

q (s) obtained via numericaldata (using Eq. (42) with numerical poles sn and eigenvectors w(n)). However, the aeroelastic modes canbe experimentally determined, for example using Output-Only analysis, not in modal coordinates (as thevectors w(n) are) but in phisical coordinates. This drawback has been overcome by means of the knowledgeof structural modes. Indeed, the m - th aeroelastic mode with physical components, φAer(m)

j , can be writtensa a function of the structural modes, φStru(n)

j , evaluated via numerical data as:

φAer(m)

j =Nmode∑

n

φStru(n)

j w(m)n (43)

6

or, in vector form,φAer(m)

=[φStru(1)

|φStru(2)| . . . |φStru(Nmodes)

]wm (44)

This equation can be used to determine the m − th complex vector w(m) in a least-squared sense once thecomplex vector φAer(m) has been experimentally determined.

As a further by-product of the theory, one can also estimate an average aerodynamic operator via experi-mental data written in Fourier domain. Indeed, once the (experimental) aeroelastic transfer function matrixHExp

q (ω) , has been identified with Eq. (42) one can use Eq. (8) as follows:

H(Exp)−1

q = −ω2M + K− qDE(ω,U∞,M∞) (45)

From such a last relation one can obtain an experimental estimate of the aerodynamic operator as:

qDE(ω,U∞,M∞) = −ω2M + K−HExp−1

q (ω) (46)

where the modal mass and the modal stiffness matrices could be evaluated via experimental modal testanalysis.

IV. Finite Element modelling of PZT devices as passive dampers.

Piezoelectric materials may be integrated into a host structure to increase damping and then to reducethe amplitude of vibrations.31,32 This effect can be achieved either by an active control system or by passivetechniques. Two approaches are possible for a purely passive employment of piezo devices: the first oneconsists of loading the external electrical circuit of the devices with resistors, whilst the second one impliesthe use of resistors and inductors, so as to get resonant circuits with the inherent capacity of the piezosystem.31 The device with a resistor and an inductor, forms a resonant circuit with the inherent capacitorof the piezo device. This system, as shown in Refs. [31, 32], is similar, although not equal, to a mechanicalvibration absorber.29,30 Since the stiffness is a frequency function, it can be introduced into a commercialFinite Element code for linear structural analysis (the MSC.NASTRAN has been considered in the presentwork) as the device would perform a viscoelastic behavior. This procedure not only permits to use differentdevices with different electric loads, but it also allows a continuous change of the optimal values for resistorsand inductors in order to face the different dynamic conditions of the host structure namely, the differentflight conditions for an aeroelastic system.Let us consider a laminar or piezo patch with e3 indicating the normal direction to the middle plane ofthe element, and e1 and e2 two orthogonal directions in the plane. Supposing a linear electromechanicalbehavior for the piezo material, considering for it a small finite Cartesian element, the local constitutiveelectro-mechanical equations in the Laplace domain are Ref. [31]:(

I

S

)=

[sCT

p sAd

d′L−1 ssc

](V

T

)(47)

where I represents the vector of electrical current flowing along the three directions, S the vector of internaldeformations, V the vector of electrical potential applied on the element through the electrodes, and Tthe vector of internal stresses. From Eq.47 it’s possible to obtain the mathematical modellization of apiezoelectric element (an exhaustive description of this method can be find in Ref. [33]).In this work the resonant circuits have been used in order to dissipate the electrical energy, where someresistors (R) and inductors (L) are used to shunt the piezoelectric electrodes. In this case the complexstiffness of the shunted passive element in the Fourier domain is:

K¯

RL

sh= K

¯ sc

1 + k2

[1− 1− ω2LCS

P − jωRCSP

(1− ω2LCSP )2 + (ωRCS

P )2

](48)

The piezo material with its electrical circuit, when integrated with the host structure, behaves like avibration absorber. Considering the analogy with the dynamical absorbers, the optimal value of inductance

7

and resistance as functions of the natural frequencies of the system in open and short circuit conditions are:

Lopt =1

CSp (ωnoc)2

(49)

Ropt =

√2 [(ωnoc)2 − (ωnsc

)2]CS

p (ωnoc)2(50)

Once the piezo local frequency dependent matrix K¯ sh

(ω) is assembled with the overall host structure, aunique frequency dependent stiffness matrix K

¯(ω) has been built by a Finite Element code and employed in

the analysis. The final aeroelastic modelling in a state-space form Ref. [33] introduced in Appendix A bymeans of Finite-State rappresentation for the unsteady aerodynamic.24

V. Numerical and experimental results

In this section the numerical/experimental results aeroelastic identification and PZT passive vibrationcontrols have been presented. The approach for the identification of aeroelastic system, described in theprevious sections, has been applied to an Unmanned Aerial Vehicle, UAV, during its actual flight condition.The aircraft structure that has been considered in this work is a YAK 112 airworld model, Fig. 1(a), havinga wing span of 2.75 [m], a length of 2.00 [m], and with a weight of 13.7 [Kg] including the equipment and fuelweights. The fuselage and fin form a unique fiberglass shell with a thickness of 0.003 [m]. The wing structurehas a spar in poliuretan covered by a very thin layer of fiberglass. The wing and tail skins (thickness of0.0022 [m]) form a ply with a central layer of obeche wood and two fiberglass thin layers on the ends. Thegeometry of the numerical model has been designed as shown in Fig. 1(b). In a furter step, an F.E. modelhas been carried out: the corresponding F.E. characteristics are summarized in Tab. 1.

(a) Yak 112 Airworld (b) CAD Model

Figure 1. Yak 112 Airworld

Table 1. Summary of the FE model

Element Type Number PositionCBAR 20 Struts

CONM2 1 EngineCQUAD4 3367 Fuselage, Wing, TailCTRIA3 92 Fuselage, Wing, TailRBE2 20 Constrain

A. Structural dynamic system identification - Numerical vs Experimental results

In this section the results have been showed for the structural identification analysis. First, the characteristicsand differences about the first four evaluated structural modes of vibration (in term of natural frequenciesand corresponding modal shapes) between numerical and experimental models have been presented. The

8

numerical analysis has been carried out using MSC.NASTRAN solver, whereas the experimental one has beenperformed using the Input/Output technique. Figure 2 show a comparison between the experimental andnumerical structural modes whereas, Table 2 reports a comparison between the numerical and experimentalresults for the natural frequencies of vibration. It is worthwhile to point out that the experimental modal

Table 2. Correlated structural modes

fn Num fn I-O Error %12.84 12.89 0.4316.54 16.80 0.3931.80 34.41 4.5838.38 40.01 4.29

X

Y

Z

4.17-001

4.91-005

4.17-001

3.89-001

3.61-001

3.33-001

3.05-001

2.78-001

2.50-001

2.22-001

1.94-001

1.67-001

1.39-001

1.11-001

8.33-002

5.56-002

2.78-0024.91-005

default_Fringe :

Max 4.17-001 @Nd 104453

Min 4.91-005 @Nd 900004

default_Deformation :

Max 4.17-001 @Nd 104453

Patran 2007 r2 08-Jan-10 14:11:40

Fringe: Untitled.SC1, A1:Mode 6 : Freq. = 12.839, Eigenvectors, Translational, Magnitude, (NON-LAYERED)

Deform: Untitled.SC1, A1:Mode 6 : Freq. = 12.839, Eigenvectors, Translational,

X

Y

Z

(a) 1st num. mode

X

Y

Z

2.12-001

2.12-001

1.98-001

1.84-001

1.70-001

1.56-001

1.42-001

1.28-001

1.14-001

1.00-001

8.61-002

7.22-002

5.82-002

4.42-002

3.02-002

1.62-0022.26-003

default_Fringe :

Max 2.12-001 @Nd 83016

Min 2.26-003 @Nd 82349

default_Deformation :

Max 2.12-001 @Nd 83016

Patran 2007 r2 08-Jan-10 14:12:18

Fringe: Untitled.SC1, A1:Mode 8 : Freq. = 16.863, Eigenvectors, Translational, Magnitude, (NON-LAYERED)

Deform: Untitled.SC1, A1:Mode 8 : Freq. = 16.863, Eigenvectors, Translational,

X

Y

Z

(b) 2nd num. mode

X

Y

Z

1.16-001

7.67-005

1.16-001

1.09-001

1.01-001

9.31-002

8.53-002

7.76-002

6.98-002

6.21-002

5.43-002

4.66-002

3.88-002

3.11-002

2.33-002

1.56-002

7.83-0037.67-005

default_Fringe :

Max 1.16-001 @Nd 104378

Min 7.67-005 @Nd 83050

default_Deformation :

Max 1.16-001 @Nd 104378

Patran 2007 r2 13-Jan-10 10:32:27

Fringe: Untitled.SC1, A1:Mode 10 : Freq. = 31.802, Eigenvectors, Translational, Magnitude, (NON-LAYERED)

Deform: Untitled.SC1, A1:Mode 10 : Freq. = 31.802, Eigenvectors, Translational,

X

Y

Z

(c) 3rd num. mode

X

Y

Z

2.19-001

2.19-001

2.05-001

1.90-001

1.75-001

1.61-001

1.46-001

1.32-001

1.17-001

1.02-001

8.79-002

7.32-002

5.86-002

4.40-002

2.94-002

1.48-0022.15-004

default_Fringe :

Max 2.19-001 @Nd 104378

Min 2.15-004 @Nd 82341

default_Deformation :

Max 2.19-001 @Nd 104378

Patran 2007 r2 08-Jan-10 14:13:27

Fringe: Untitled.SC1, A1:Mode 12 : Freq. = 38.383, Eigenvectors, Translational, Magnitude, (NON-LAYERED)

Deform: Untitled.SC1, A1:Mode 12 : Freq. = 38.383, Eigenvectors, Translational,

X

Y

Z

(d) 4th num. mode

(e) 1st O-O mode (f) 2nd 0-0 mode (g) 3rd 0-0 mode (h) 4th 0-0 mode

Figure 2. First four structural modes investigated

parameters are practically the same of those estimated in a standard Input/Output ground vibration test.28The slight differences are due to the fact that the tests have been carried out considering little changes in thesystem configuration. The Modal Assurance Criterion (MAC), able to evaluate the parallelism level of twovectors, has been used to compare the structural modal shapes coming from the numerical and experimentalanalysis. In Table 3 the MAC correlations between the numerical and experimental eigenvectors representing

Table 3. MAC Correlation

Structural mode # MAC1 0.242 0.693 0.474 0.90

the modal shape of vibration are showed. It’s important to emphasize that the first MAC value evaluatedbetween the corresponding eigenvectors is very low and this shows a lower correlation in term of modalshapes with respect to frequency.

B. Aeroelastic eigenspectrum identification - Experimental results

In this section the aeroelastic stability scenario that has been identified by determining experimentallyvia O-O analysis the aeroelastic poles is presented. Three different operative flight conditions have beenidentified as "Headwind" (R1), "Turning" (V2), and "Tailwind" (R2) phases respectively (Tabs. 4 and 5).

9

During these phases the aeroelastic poles and eigenvectors have been evaluated by using the Output-Onlytechnique. The aeroelastic frequecies (namely, the poles immaginary part divided 2π) and the dampingcoefficients (namely, the ratio between the pole real and immaginary part with a changed sign) obtainedby carrying out the Output-Only analysis during each chosen reference conditions are reported in Tab. 4and in Tab. 5. Again, the experimentally estimated aeroelastic parameters are in agreement with thoseestimated in ground vibration test and therefore they could be considered as a good representation of theaeroelasticity of the flying UAV. In Tabs. 4 and 5 the last columns are referred respectively to the values offrequencies and damping coefficients obtain with an aeroelastic stability analysis on the FEM model usingMSC.NASTRAN solver (namely, SOL 145) with air speed U∞ = 20 m/s, Mach number M∞ = 0 and airdensity ρ∞ = 1.2Kg/m3 (close to Tailwind condition).

Table 4. Aeroelastic Frequency changing with Flight Conditions

Mode Headwind (R1) Turning (V2) Tailwind (R2) Numerical Tailwind

N fn[Hz] fn[Hz] fn[Hz] fn[Hz]I 12.00 12.00 12.50 12.54II 18.00 16.00 16.50 16.50III 33.00 34.00 33.75 32.65IV 42.00 — 41.00 39.08

Table 5. Damping Coefficients changing with Flight Conditions

Mode Headwind (R1) Turning (V2) Tailwind (R2) Numerical Tailwind

N ζ[%] ζ[%] ζ[%] ζ[%]I 2.98 5.34 1.34 8.94II 1.92 4.59 0.57 5.32III 1.40 3.56 0.21 10.18IV 0.78 — 0.15 2.68

In Figure 3, the experimental roots locus obtained by using experimental aeroelastic poles, with theflight velocity as parameter, is shown. Each of the four experimental aeroelastic modes investigated duringthe flight test presents three points in the diagram that correspond to the three different flight conditionsrespectively moving into the diagram from right to left. It is noticed that, as the air speed is increased, theexperimental aeroelastic poles move away from the imaginary axis increasing, in absolute value, their realparts . This result shows that the aeroelastic system gets more stable by increasing the flight velocity. In thisway the flutter instability is conjured and then the aeroelastic system keep stable within the manouveringrange considered in the analysis.

C. Aeroelastic Frequency Response Function experimental-based syntesis and AerodynamicsIdentification

In this section the characteristics of the Aeroelastic Frequency Response H(f) and of the Unsteady Aero-dynamic Operator E(k) evaluated both via numerical data, using MSC.NASTRAN solver, and via flighttest through the described mathematical approach are presented. In order to check the proposed syntesismethod for the aeroelastic FRF matrix (Eq. 42) in Figure 4(a) the differences between the numerical directsolution (MSC.NASTRAN SOL 146 solver) and the numerical syntetized solution (Eq. 42) are presented.The differences arising in Fig. 4(a) between the direct solution and the eigen-based solution of Eq. 42 aredue to piecewise approximation in frequency domain for the aerodinamic matrix as in Eq. 29. Moreover, thedifference between Aeroelastic Frequency Response Matrix H syntetized from experimental data assumingthe same piecewise approximation for the aerodinamic matrix and that given by numerical syntetic analysisare shown in Fig. 4(b) that report the second diagonal component referred to the second aeroelastic mode.The experimental H22 is characterized by a greater coupling with the other three aeroelastic modes, whereasthe numerical one has only one peak corresponding to the frequency of the second aeroelastic mode. Next,

10

Figure 3. Roots Locus

(a) H22 Numerical sytesis vs direct (b) H22 Numerical syntesis vs experimental syntesis

Figure 4. Aeroelastic Frequency Response: H22

Figure 5. Effect of PZT devices on the aeroelastic FRF (component H22). Numerical simulations.

11

Figure 5 shows the actual capability of the PZT devices using a numerical simulation: indeed, the Figuredepicts that, using only two PZT patches (one for each half-wing on the main spar) a weak attenuation ofabout the 1dB in the first tuned peak can be reached. This result would justify the further experimentalresults obtained in ground conditions (see later Figure 9).

The Generalized Aerodinamic Force (GAF) matrix has been also experimentally identified using Eq. 46.The difference between the experimental and numerical GAF Matrix E, in terms of variation of the real andimaginary parts of each its components, with respect of the reduced frequency k is shown in Fig. 6. TheFigure shows the second diagonal component referred to the second aeroelastic mode. The real part of the

(a) E22 Real Part (b) E22 Imaginary Part

Figure 6. Aerodynamic Operator Components vs Reduced Frequency: E22

second experimental components of the unsteady aerodynamic operator, E22, shown in Fig. 6(a) with redmarks, is quite an average value of the real component obtained carring out the numerical simulation. Ideed,the numerical curve cross the experimental one in correspondence of a reduced frequency value as confirmedby Eq. 29. The corresponding experimental imaginary parts of the E22 component, Fig. 6(b), have the samenegative slope in the neighbourhood of zero frequency as the numerical ones.

D. Damping passive-controls using PZT devices

1. Test on a cantilever beam

In order to check the method presented, an experimental test on a simple structure has been carried out.An aluminum cantilever beam - dimensions: 0.230 × 0.027 × 0.003 (m) - with one piezoceramic patch ACXQP16N bonded at the clamped end, has been tested. The impressed force was given by an ElettromagneticalShaker, and the response was measured by a PCB accelerometer, positioned at the tip end of the beam.A shunt circuit composed by a resistance and a syntetic inductance (as indicated in Eq. 48-49) has connectedto the PZT device Fig.7; a more exhaustive description of the syntetic inductance can be find in Ref. [35].The objective is to damp the first bending mode of the cantilever beam that is around 24.55 Hz. In orderto reach this goal, the circuit is syntonized to the correct frequency using the following values of optimalresistance and inductance: R = 3618Ω L = 186.81H. The experimental FRFs are reported in Fig.8 in threedifferent condition of PZT device: OC (Open Circuit), SC (Short Circuit) and Tuned (Shunted circuit).

2. Test on the UAV

In the previous section it has been tested the system composed by structure-PZT device-shunt circuit. Inthe present section the objective is to apply this system on the UAV wing in order to damp the first bendingmode of the principal flying surfaces. Three PZT devices have glued in the joint zone between the wingand the fuselage. This is the zone characterized by the maximum deformation of the bending mode. Inorder to check the damping effect of PZT devices on this complex structure, it has been carried out a simpleexperimental test evaluating the FRF with and without the presence of the passive tuned circuit. Afterthis test it has been obtained the FRFs (Fig.9) in the three different condition of PZT device: OC, SC andTuned. It important to point out that no pratical damping effect in the Tuned condition have been obtained.This is probably due to low value of the ratio between the area of the PZT devices and area to dampingthat decreases the PZT effectiveness.

12

Figure 7. Shunt circuit used

23 24 25 26 27 285

10

15

20

25

30

35

Frequenza (Hz)

|FR

F| (g

/N) d

b

OCSCTuned

Figure 8. Experimental FRFs with resonant load and PZT in OC, SC and Tuned condition on the cantilever.

Figure 9. Experimental FRFs with resonant load and PZT in OC, SC and Tuned condition on the UAV.

Indeed, the experimental test on the cantilever beam the ratio is 1/5, whereas in the experimental test onthe UAV it is of about 1/100. For this reason no flight test have been performed to validate the effectiveness

13

of the piezo devices in operative conditions.

VI. CONCLUDING REMARKS

In this paper an Output-Only-based procedure for the identification of the aeroelsatic modal charac-teristics, the aeroelastic frequency-response functions, the aeroelastic stability scenario, and the unsteadyaerodynamic operator of a typical aeronautical configurations is proposed. The procedure has been appliedusing actual experimental data obtained in a flight test performed in a UAV. Moreover, the same procedurehas been applied to check the capability of PZT patch applied on the UAV wing surface to passively dampthe aeroelastic vibration level.

The procedure is based on on the FDD method that allowed the representation of the aeroelastic system interms of eigenvalue and eigenvector estimates. Then, introducing some identified model approximation, theaeroelastic frequency-response-function matrix for the proposed configuration has been also estimated withthe aid of a well correlated Finite Element model. Finally, the GAF matrix for the proposed configurationhas been evaluated coherently with the assumed approximation for the estimated frequency-response matrix.

Using the flight test data and applying the proposed procedure, the aeroelastic stability scenario hasbeen identified in operative conditions in terms of aeroelastic poles. Moreover, the unsteady aerodynamicoperator in terms of modal coordinates has been also evaluated by the flight test data. Finally, the simulationincluding the behavior of passive PZT patch have have not showed practical capability of such a devicesin practical operative conditions. However, the proposed procedure has showed its actual and potentialcapabilities for a direct identification of the structural, aeroelastic, and aerodynamic characteristics of agiven aircraft configuration directly in operative conditions.

A. PZT modeling in linear Aeroelastic Analysis

In order to integrate in a monolitic formulation the PZT modeling (section 4) with the aeroelasticmodeling (section 3), a finite-state modeling for the aerodinamic operator is here introduced.24,33 In thelinear modeling of the unsteady aerodynamic, the vector eA in can be expressed as a linear combination ofthe lagrangean variables q, i.e., , in the Laplace domain as Eq. (8):

eA = qDE(s;U∞,M∞)q (51)

where qD = 12%∞U

2∞ is the dynamic pressure and E is the so-called generalized aerodynamic force (GAF)

matrix. Let us consider the following approximated representation of the aerodynamic operator by meansof a finite–state technique: this consists on using the aerodynamic data, given by the aerodynamic panelcodes (i.e., the complex evaluation of the matrix E for prescribed values of si = jωi), to determine a matrixpolynomial approximation of such a matrix with respect to the Laplace variable s. The structure of suchapproximation is given by:

Eapp(s) = A0 + A1

(s c

2 U∞

)+ A2

(s c

2 U∞

)2

+ C

(Is c

2 U∞− A

)−1

B (52)

where A0,A1,A2,A,B,C are matrices of interpolation in a specific range of flight speed depending on theMach number. The unknown matrices are obtained with a minimization process of the global error betweenthe sampled aerodynamic data in the frequency domain and the analytical function given by Eq. (52). It isalso worth pointing out that the process shown before, basically consisting on substituting the aerodynamicmatrix E (which originally is a transcendental function of s) with a rational polynomial approximation in s,corresponds in the time domain in approximating the aerodynamic time integral-differential operator with apurely differential one, but with more (aerodynamic) states included. Indeed, the first three matrices havethe same modal model dimensions, while A has dimensions (Npoles, Npoles), B (Npoles,M), C (M,Npoles),where Npoles is the number of the mentioned aerodynamic poles added in the finite-state approximationprocess.Referring to a condition with only one piezo (Np = 1) and without any external gust force, the Lagrange

14

equation of the linear aeroelastic problem with complex stiffness of the shunted passive element yields:[s2I + Ω2 +

[k2K(p,sc) −

k2

α14π2 (s− s1)(s− s2)

K(p,sc)

]]q =

qD

[s2c2

4U2∞

A2 +sc

2U∞A1 + A0 + C

(sc

2U∞I− A

)−1

B

]q (53)

Then, using the following positions:

v = sq (54)

g =k2

α14π2 (s− s1)(s− s2)

K(p,sc)q (55)

r =(

sc

2U∞I− A

)−1

Bq (56)

h = sg (57)

(where v, g, h have the same dimensions of the modal modelM , while r isNpoles dimensional) the problem canbe written in a first order form by introducing the following state-space matrix (4M+Npoles)×(4M+Npoles):

A =

0 I 0 0 0V1 V3 V4 V2 0

2U∞c B 0 2U∞

c A 0 00 0 0 0 I

k2 4π2

α1K(p,sc) 0 0 −s1s2I (s1 + s2)I

(58)

where

V1 = −[I− qD

c2

4U2∞

A2

]−1 (Ω2 + k2K(p,sc) − qDA0

)(59)

V2 =[I− qD

c2

4U2∞

A2

]−1

(60)

V3 = −[I− qD

c2

4U2∞

A2

]−1(−qD

c

2U∞A1

)(61)

V4 =[I− qD

c2

4U2∞

A2

]−1

CqD (62)

It is worth noting that the dynamics of the mechanical system takes essentially place in q and v subspaces,the aerodynamics in r, and the electro-dynamics of piezoelectric element in g and h.Thus, as a matter of fact, the problem can be written in the Laplace domain:

sx = Ax (63)

where x = q, v, r, g, hT is the state-space vector, A is an explicit function of flight speed (U∞) and airdensity (%∞), and an implicit function of Mach number (M∞) through the matrices A2,A1,A0,A,B,C;furthermore, A is a function of the tuning parameters (α0, α1, and α2) of the piezoelectrics devices, and ofthe electromechanical coupling coefficient (k31). In order to study the system aeroelastic stability, problemcan be associated to a standard eigenproblem:[

snI− A(U∞, %∞,M∞, α1, α2, k31)]u(n) = 0 (64)

whose solutions, e.g., at various U∞, allow one to draw the locus of the roots sn and then to make a stabilityanalysis at fixed altitude (%∞) and Mach number (M∞).

15

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