Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 941689 9 pageshttpdxdoiorg1011552013941689
Research ArticleThe Use of Gramian Matrices for Aeroelastic Stability Analysis
Douglas Domingues Bueno1 Clayton Rodrigo Marqui1
Luiz Carlos Sandoval Goacutees1 and Paulo Joseacute Paupitz Gonccedilalves2
1 Technological Institute of Aeronautics (ITA) 12 228 900 Sao Jose dos Campos SP Brazil2 Universidade Estadual Paulista (UNESP) 17 033 360 Bauru SP Brazil
Correspondence should be addressed to Douglas Domingues Bueno ddbuenoitabr
Received 18 November 2012 Revised 1 March 2013 Accepted 1 March 2013
Academic Editor Cristian Toma
Copyright copy 2013 Douglas Domingues Bueno et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited
Most of the established procedures for analysis of aeroelastic flutter in the development of aircraft are based on frequency domainmethods Proposing new methodologies in this field is always a challenge because the new methods need to be validated by manyexperimental procedures With the interest for new flight control systems and nonlinear behavior of aeroelastic structures otherstrategies may be necessary to complete the analysis of such systems If the aeroelastic model can be written in time domain usingstate-space formulation for instance then many of the tools used in stability analysis of dynamic systems may be used to helpproviding an insight into the aeroelastic phenomenon In this respect this paper presents a discussion on the use of Gramianmatrices to determine conditions of aeroelastic flutterThemain goal of this work is to introduce how observability gramianmatrixcan be used to identify the system instability To explain the approach the theory is outlined and simulations are carried out ontwo benchmark problems Results are compared with classical methods to validate the approach and a reduction of computationaltime is obtained for the second example
1 Introduction
Between various physical phenomena involving fluid-struct-ure interaction flutter is probably the most representativetopic studied in engineering applications such as aircrafts andbridges The flutter phenomenon is an interaction betweenstructural dynamics and aerodynamics that results in diver-gent and destructive oscillations of motion [1]
In 1935 Theodorsen [2] proposed a method of flutteranalysis in a discrete system by including aerodynamic forcesin frequency domain and formulating the analysis as a com-plex eigenvalue problem Hassig [3] proposes the pk-methodwhere the unsteady aerodynamic matrix is represented bya function of the complex eigenvalues Using an iterativealgorithm the value of a reduced frequency converges to theimaginary part of a system eigenvalue Chen [4] also proposesa flutter method including a first-order damping term intothe equation of motion known as the g-method Accordingto the author this method generalizes the 119896-methods andpk-methods for reliable damping prediction and has proved
to be efficient in obtaining unlimited number of aerodynamiclag roots
These methodologies which are well established inthe research and engineering community were developeddecades ago and have been used in the development of almostall flying commercial and military aircraft
In this context this paper proposes an alternativeapproach for detecting flutter using observability Gramianmatrices The proposed methodology is developed in timedomain using state-space representation of the aeroelasticsystem The elements of a Gramian matrix are related to theenergy of vibration modes and can be seen as an improvedobservability matrix introduced by Kalman et al [5]
Gramian matrices have been used in the field of controlengineering Their fundamental concepts were proposed byMoore after introducing the balanced reduction for state-space models [6] and have been used for applications such asoptimal placement of sensors and actuators [7ndash9] After thiswork the approach was extended for unsteady aerodynamicand aeroelastic systems [10ndash13] According to [14] despite
2 Mathematical Problems in Engineering
these efforts Gramian matrices are still neglected by thescientific community
The principal objective of the paper is to show thatobservability Gramian matrices can be used to detect flutterThe rationale for using this approach is thatGramianmatricescontain in a single index information about the energy thatis transferred from the flux to the structure and then dissi-pated by any damping mechanism for each flight conditionconsidered in the analysis (flight envelope)
It is shown that theGramians are sensitive to flutter whereenergy transferred from the flux in a cycle is larger than theenergy dissipated by the damping mechanism
Part of the procedure to determine the Gramian matricesrequires a system defined in time domain including theaerodynamic forces If aerodynamic forces are written interms of reduced frequencies this can be done using one ofthemethods such as least square [15] minimum state [16] andthe mixed state [8]
The paper illustratersquos the process to obtain the Gramianmatrix from an aeroelastic system in time domain Twonumerical examples are used to compare the proposedmethod with the methods in the literature The first exampleis three degrees of freedom typical section airfoil and thesecond is the AGARD 4456 wing model developed usingfinite element method [17 18]
The paper shows that it is possible to obtain someadvantage in terms of computation time using the proposedmethodology
2 Aeroelastic Model for Stability Analysis
Assuming a general aeroelastic model that can be written inthe state-space form according to
x (119905) = Ax (119905) + B119888f119888(119905)
y (119905) = Cx (119905)
(1)
where x(119905) is the state vector A is the dynamic matrix B119888
is the input matrix f119888(119905) is a vector of external forces C is
the known as the output matrix and y(119905) is the output vectorUsing this equation the aeroelastic system can be described bydifferent aerodynamic theories and the system of equationscan be written in physical or modal coordinate systems(complementary information is presented in Appendix A)The time-invariant aeroelastic system defined by matrix A isstable for a range of velocities defined by increasing airspeedvalues 119881
1 119881
119895 119881
119899containing a flutter airspeed 119881
119865 if
119881 lt 119881119865
That stability could be verified by solving an eigenvalueproblem for each discrete airspeed point in the flight envelopeand checking if the real part of system eigenvalues is negativevalues This can be time consuming specifically for largedimension systems To overcome this process the observabil-ity Gramian method presented in this paper is based on thesolution of a set of linear equationsThe next section presentsthe bases to write the problem in appropriate format
3 Observability and Gramian Matrices
The concept of observability involves the dynamic matrix Aand the output matrix C A linear system or the pair (AC)is observable at instant of time 119905
0 if the state x(119905
0) can be
determined from the output y(119905) with 119905 isin [1199050 1199051] where
1199051gt 1199050is a finite instant of time If this is true for all initial time
1199050and all initial states x(119905
0) the system is said to be completely
observable [14]A linear time-invariant systemwith119898outputs is said to be
completely observable if and only if the observability matrixwith dimension 2119898
2
(2 + 119899lag) times 21198982
(2 + 119899lag) has hank119898(2 +
119899lag) With the observability matrix given by
O =
[[[[[[
[
CCACA2
CA119899minus1
]]]]]]
]
(2)
where 119899 is the dimension of matrix A This concept can beeasily implemented to verify system observability but maylead to numerical overflow for systems represented by largedimension matrices [14] Also only qualitative informationabout the system is provided (see [14 19 20])
An alternative approach that can be applied for large-order problems is to use the observability Gramian matrixThe observability Gramian matrix is defined to expressquantitative properties of the system considering it at time119905 lt infin written as
W119900(119905) = int
119905
0
[119890A119879119905C119879C119890
A119905] 119889119905 (3)
which according to [14] can be determinated as
W119900(119905) = A119879W
119900(119905) + W
119900(119905)A + C119879C (4)
where W119900(119905) indicates a time-variant property If a linear
time-invariant and stable system is considered then theobservability Gramian matrix can be computed using theLyapunov equation [14]
A119879W119900+ W119900A + C119879C = 0 (5)
An important property between observability andGramian matrices is that they share the same Kernel (ie theset of all vectors x for which Ax = 0)
Ker [W119900(119905)] = Ker [O (CA)] (6)
According to [21 22] one consequence of this property isthat the energy detected by an output state can be computedthrough the observability Gramian matrix This is done bywriting an expression for the energy detected (or observed)by the output y at time 119905
0caused by the systemrsquos initial state
x(0) such that
Energy [y (1199050)] = x119879 (0)W
119900(1199050) x (0) (7)
Mathematical Problems in Engineering 3
Equation (5) is only defined for a stable system [14]To include the flutter speed it is necessary to modifythe equation using the generalized ordinary cross-GramianmatrixW
119888119900 introduced by Zhou et al [23] defined for both
stable and unstable systems Then let X119892be the solution to
the Riccati equation
X119892F119892+ F119879119892X119892minus X119892G119892G119879119892X119892= 0 (8)
The observability Gramian matrix W119900is a submatrix of W
119866
which can be computed by solving the following equation
(F119892+ G119892M119892)W119866+ W119866(F119892+ G119892M119892)119879
+ G119892G119879119892
= 0 (9)
where
F119892= [
A 00 A119879] G
119892= [
[
B119888
C119879]
]
M119892= minusG119879119892X119892
W119866
= [W119888
W119888119900
W119888119900
W119900
]
(10)
and the modified output matrix C is used instead of C Thismodified output matrix C with dimension 119898 times 119898(2 + 119899lag) isdefined such that
y (119905) = Cx (119905)
y (119905) = 0 sdot sdot sdot 0 119906119898(119894)
(119905) 0 sdot sdot sdot 0119879
then C119894= [0 sdot sdot sdot 0
(119898+(119894minus1))1(119894)
0 sdot sdot sdot 0]
(11)
where the 119894th rowC119894satisfies the equation119910
119894(119905) = 119906
119898(119894)(119905) and
the other (119898minus1) rows are filled with zerosThus consideringC119894and the aeroelastic matrix A(119881
119895) defined at the airspeed
119881119895 the observability GramianmatrixW
119900(C119894 119881119895) = W
119900(119894 119895) is
computed by solving (9) for the pair (C119894 119881119895)The inputmatrix
B119888is written by [Mminus1
1198861198980119898(1+119899lag)times119898
]119879 to represent 119898 inputs
31 Complex Schur Decomposition Using a complex Schurdecomposition the Lyapunov equation can be reshaped as areal linear system of equation and solving itW
119900(119894119895)is obtained
[24]
[I otimes (F119892+ G119892M119892) + (F119879
119892+ M119879119892G119879119892) otimes I]vec(W
119866)
= vec(minusG119892G119879119892)
(12)
where I is an identity matrix with appropriate dimensionvec(sdot) makes a column vector out of a matrix by stacking itscolumns and otimes indicates a Kronecker product [25]
32 Gramian Paramater to Detect Flutter The hypothesisintroduced in this paper is that observability Gramianmatrixcontains information which can be used to indicate theamount of energy transferred from the air flow to thestructure In this case this amount of energy is maximum atthe airspeed at which the system becomes unstable
Stable Unstable
Mode 1
Mode 2
AirspeedFlutter speed
Gra
mm
ian
para
met
er
119881119865
Figure 1 Gramian parameter used to detect the flutter phe-nomenon
In order to prove this hypothesis a Gramian parameter120590119892(119894 119895) isin R+ is defined and obtained by computing a matrix
norm ofW119900 This parameter indicates two main features the
contribution of each aeroelastic mode absorbing energy fromthe air flow and the airspeed where flutter occurs
By computing 120590119892(119894 119895) for each pair119881
119895andC
119894it is possible
to build a matrix Σ (14) Assuming that aeroelastic instabilityoccurs in this speed range the flutter speed119881
119895is found for the
largest value of 120590119892(119894 119895)
max (Σ) = max119894119895
[120590119892] = 120590
max119892
such that if Δ119881 997888rarr 119889119881 997904rArr119889120590119892
119889119881= 0
(13)
where 119899V is the length of the airspeed vector
Σ =
[[[[[
[
120590119892(11)
sdot sdot sdot 120590119892(1119895)
sdot sdot sdot 120590119892(1119899V)
sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot
120590119892(1198941)
sdot sdot sdot 120590119892(119894119895)
sdot sdot sdot 120590119892(119894119899V)
sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot
120590119892(1198981)
sdot sdot sdot 120590119892(119898119895)
sdot sdot sdot 120590119892(119898119899V)
]]]]]
]
(14)
The 119894th Gramian parameter in each column of the matrixΣ containing 120590
max119892
is related to a measure of the energyabsorbed by the 119894th aeroelastic mode The values of 120590
119892in
each column can then be compared to determine the modecontribution On the other hand the row of Σwhich contains120590max119892
can be plotted to determine the airspeed of flutter (Thisis illustrated in Figure 1)
33Matrix Norm In practical implementations theGramianparameter 120590
119892(119894 119895) can be obtained by different matrix norms
Matrix norms are often used to provide quantitative infor-mation In this paper Frobenius norm is used (15) However
4 Mathematical Problems in Engineering
Table 1 Physical and geometric properties of the 2D airfoil
Aerodynamic semichord 119887 = 015mAirfoil mass 119872 = 50 kgPlunge frequency 120596
ℎ= 30Hz
Pitch frequency 120596120579= 45Hz
Control surface rotationfrequency 120596
120573= 12Hz
Air density 120588 = 12895 kgm3
Lag parameters (Rogerrsquosmethod) 120573
1= 02 120573
2= 12 120573
3= 16 120573
4= 18
Reduced frequencies 01 le 119896 le 20 Δ119896 = 01
Figure 2 119886 = minus040
Figure 2 119888 = 060
Distance between ce to cg 119909120579= 020
Distance between ce tocg (flap) 119909
120573= 00125
Radius of gyration of theflap referred to 119886
119903120573= (625 times 10
minus3
)12
Radius of gyration of theairfoil referred to 119886
119903120579= radic025
Elastic center ceCenter of gravity cg
ce
ca+120579119896120579
119896ℎ
+ℎ
119886
119887 119887
cg
119888
cgflap
minus120573
119903120573
119909120579
119903120579
119896120573
119909120573
Figure 2 Typical section 2D airfoil
similar results were obtained using other norms presented inAppendix B
120590(119865)
119892(119894 119895) = (sum
119894119903 119895119888
1003816100381610038161003816119908119900 (119894119903 119895119888)1003816100381610038161003816)
12
(15)
where119908119900(119894119903 119895119888) is the Gramianmatrix element of the 119894
119903th row
and 119895119888th column and 119894
119903 119895119888= 1 119898(2 + 119899lag)
4 Numerical Simulations
41 Typical Section Airfoil The effectiveness of the approachwas determined through simulations using two examplesThe first was a three-degree-of-freedom typical airfoil section(semichord 119887) The equations of motion describing the
Pitch mode
Control surface deflection
Plunge mode
0 5 10 15
16
14
12
10
8
6
4
2
0
Freq
uenc
y (H
z)
Airspeed (ms)
119881-119891 diagrammdashtypical section
Figure 3 Aeroelastic frequency as a function of airspeed (typicalsection airfoil)
015
01
005
0
minus005
minus01
minus015
minus02
minus025
minus03
minus035
Dam
ping
ratio
0 5 10 15
Pitch mode
127ms
Control surface deflection
Plunge mode
Airspeed (ms)
119881-119892 diagrammdashtypical section
Figure 4 Real part of the eigenvalues as a function of the airspeed(typical section airfoil)
aeroelastic response are presented in [17] The bidimen-sional system was formulated in modal coordinate systemconsidering the plunge and pitch modes and the controlsurface deflection to represent the third mode Its physicaland geometric properties are presented in Table 1 and anillustration of the airfoil is shown in Figure 2
The results of the proposed approach were comparedwith classical eigenvalues analysis Figures 3 and 4 presentaeroelastic frequency anddamping both plotted as a functionof the airspeed According to the literature although theplunge (or bending) mode is stable the system becomesunstable because the pitch (or torsion mode) is unstable Itis said that airfoil normally is undergoing a flutter oscillationcomposed of important contributions of these modes In
Mathematical Problems in Engineering 5
Grammian parametermdashtypical section airfoil
Plunge mode400
200120590119892
2 4 6 8 10 12 140
Airspeed (ms)
(a)
Grammian parametermdashtypical section airfoil
120590119892
2 4 6 8 10 12 14
40
20
0
Airspeed (ms)
Pitch mode
(b)
Grammian parametermdashtypical section airfoil
120590119892
2 4 6 8 10 12 14
Control surface deflection4
2
0
Airspeed (ms)
(c)
Figure 5 Gramian parameter computed by the Frobenius norm asa function of the airspeed (typical section airfoil)
Air density (kgm3)02 04 06 08 1 12
AGARD wing100
90
80
70
60
50
40
30
20
10
0
Freq
uenc
y (H
z)
Figure 6 Aeroelastic frequency as a function of the air density(AGARD wing)
this example flutter airspeed was computed equal to 119881 =
127msThe results of the proposed method are presented in
Figure 5 The flutter speed was correctly identified Similarresults were also obtained using the other norms and there-fore they are not shown herein
Air density (kgm3)02
02
04
04
06 08 1 1412
AGARD wing
Third mode
minus02
minus04
minus06
minus1
Dam
ping
ratio
Second mode
First mode
Fourth mode
0
0
minus08
Figure 7 Aeroelastic damping as a function of the air density(AGARD wing)
42 AGARD 4456 Wing The AGARD 4456 benchmarkwingwas also used to demonstrate themethodThe structuralmodel for the AGARD 4456 wing was developed usingfinite element method using the MSCNASTRAN The finiteelement model consisted of plate elements with single-layerorthotropic material The model has 231 nodes and 200elements Rotation of nodes was neglected allowing threedegrees of freedom per node See physical properties inAppendix C and complementary details in [18 26]
Aerodynamic and structuralmatrices were obtained fromMSCNASTRAN program (Aeroelastic Solution 145) for theMach number 050 and 120588REF = 1225 kgm3 The values ofreduced frequencies are presented in Appendix CThemodelhas a reference length of 2119887 = 05578m a sweep angle of45 degrees at the quarter chord line a semispan of 0762mand a taper ratio of 066 The flutter boundary is investigatedonly using the first four fundamental structural modes andtheir natural frequencies are respectively 119891
1= 945Hz 119891
2=
3969Hz 1198913
= 4945Hz and 1198914
= 9510Hz Details can befound in literature (eg see [18 26])
A state-space model was obtained through a MATLABimplementation for which the parameters of lag were chosenas 1205731
= 055 1205732
= 140 1205733
= 190 and 1205734
= 290 Theobservability Gramian matrix was computed for each pair(C119894 119881119895) and 120590
119892was computed using different matrix norms
Analysis was evaluated considering 61 pairs of (119881119895 120588119895) with
constantMach number Using the traditional pk-method theflutter speed was identified at 15802ms The classical 119881-119891and 119881-119892 diagrams are shown in Figures 6 and 7
With the proposedmethodology the Gramian parameter120590119892is showed in Figures 8 and 9 Based on Figure 8 it is
possible to note that the first and second aeroelastic modescontribute more to the flutter than the third and fourth
6 Mathematical Problems in Engineering
20
15
10
5
002 04 06 08 1 12
Air density (kgm3)
First mode Air density 06527
120590119892
Airspeed 15802
Grammian parametermdashAGARD wing
(a)
02 04 06 08 1 12
Air density (kgm3)
Second mode
120590119892
3
2
1
0
Grammian parametermdashAGARD wing
(b)
Figure 8 Gramian parameters computed by the Frobenius norm as a function of the air density (AGARD wing)
02 04 06 08 1 12
Air density (kgm3)
Third mode
120590119892
08
06
04
02
1
0
Grammian parametermdashAGARD wing
(a)
02 04 06 08 1 12
Air density (kgm3)
Fourth mode120590119892
08
06
04
02
1
0
Grammian parametermdashAGARD wing
(b)
Figure 9 Gramian parameters computed by the Frobenius norm as a function of the air density (AGARD wing)
06
05
04
03
02
01
010 20 30 40 50 60
Fligth envelope point
pk-methodGram EqCross-gram Eq
Tim
e sum
Computational cost comparison
Figure 10 Comparison of computational time
modes This information can be confirmed in Figure 7 thatshows small changes of aeroelastic damping for these modes
43 Comparison between pk-Method and the ProposedApproach This section shows that it is possible to reduce thecomputational cost to find the flutter speed using Gramian
matrices especially for industrial applications where thenumber of nominal and parametric cases of analysis can bevery large Based on the second example (AGARD wing) itis demonstrated that (5) and (9) can be conveniently usedto get small computational time in comparison with the pk-method
According to previous sections (5) is not valid forunstable systems that is the physics is represented if andonly if the system is stable Then if this equation is usedto compute the Gramian parameter the solution has nophysical meaning after the flutter velocity (including thatpoint) However the computational time to solve the linearsystem of equations (see Section 31) is smaller than that tosolve the iterative eigenvalue algorithm (pk-method) Thefollowing results were obtained using the personal computerwith operational system MS Windows 7 (64 bits) Intel(R)Core(TM) i5 CPUM460 253GHz and RAM 40GHz
Figure 10 shows that Gramian parameters computedfrom the cross-Gramian matrices have computational costlarger than classical pk-method However if they are com-puted from Gramian matrices (5) the computational timeis smaller In this context it is possible to use the followingprocedure
(1) Compute Gramian parameters using observabilityGramian matrices obtained from (5) for every points119875(120588119895 119881119895) into the flight envelope
(2) Identify the point 119875(120588119895 119881119895) = 119875(119895) at which Gramian
parameter is maximum according to (13)(3) Compute cross-Gramian parameters for points
around the maximum
Mathematical Problems in Engineering 7
10 20 30 40 50 60
Fligth envelope point
pk-methodGram EqCross-gram Eq
70 800
002
004
006
008
01
012
014
016
018
02
Com
puta
tiona
l tim
e
Timereduction48
Figure 11 Computational time pk-method versus Gramian param-eters (AGARD 4456 wing)
Using this procedure it is possible to obtain somereduction in the computational time to evaluate the systemstability This reduction is represented in Figure 11
5 Conclusions
Controllability and observability Gramian matrices havebeen used extensively in control design and for optimalplacement of sensors and actuators in smart structures Theyhave also been used for solving aeroelastic problems mainlyfor writing reduced-order models
This paper has investigated the applicability of observ-ability Gramian matrix to detect the flutter speed in twobenchmark structures It was shown that Gramian matricesrepresent the transfer of energy from the flow to the structureThat transfer of energy increases when the airspeed is close tothe flutter condition A classical method was used to compareand validate the results obtained by Gramian matrices
This approach does not compute frequency and dampingfor the aeroelastic modes However it allows to determinethe aeroelastic modes with largest contribution to the fluttermechanism and the airspeed where the system becomesunstable Additionally one advantage is related to the reduc-tion of computational time for analysis when compared toclassical pk-method
This work is a complementary effort to apply the conceptsfrom control theory in problems involving aeroelasticityin time domain It can also be used in flight flutter testsusing an identification method to obtain the state-spacerepresentation instead of identifying aeroelastic frequenciesand damping
Appendices
A State-Space Represenation ofan Aeroelastic System
An aeroelastic system consisting of structural parameters(mass damping and stiffness) subject to the forces fromthe fluid can be modeled in the Laplace domain using aphysical or modal coordinate system In general the modalcoordinates are used to truncate the systemof equations usingthe eigenvector extracted from structural mass and stiffnessmatrices Without loss of generality both numerical casestudies were performed using modal coordinates as shown inthe following equations
The matrices representing the structural parameters arethe mass M the viscous damping D and the stiffness K Inmost cases these matrices are obtained from finite elementmethod and are reduced to modal coordinates (representedby the subscript119898) the system displacement vector in modalcoordinates is u
119898 The aerodynamic influence matrix Q
depends on the parameters 119896 (reduced frequency) and 119898119872
(Mach Number) and 119902 is the dynamic pressure caused by theflowing fluid (Q can be obtained by methods such as DoubletLattice) Note that the proposed approach is not restricted tothis aerodynamic theory
1199042M119898u119898
(119904) + 119904D119898u119898
(119904) + K119898u119898
(119904) = 119902Q119898
(119898119872 119896) u119898
(119904)
(A1)
where 119904 is the Laplace variableIn this case the problem that arises from the conversion
of (A1) to time domain is the fact that Q has no Laplaceinverse This is overcome by writing a rational function torepresent the aerodynamic coefficients in time domain Inthis work Rogerrsquos approximation [15] is used containing apolynomial part representing the forces acting directly relatedto the displacements u(119905) and their first and second deriva-tives Also this equation has a rational part representing theinfluence of the wake acting on the structure with a timedelay
Q119898
(119904) = [
[
2
sum
119895=0
Q119898119895
119904119895
(119887
119881)
119895
+
119899lag
sum
119895=1
Q119898(119895+2)
(119904
119904 + (119887119881) 120573119895
)]
]
u119898
(119904)
(A2)
where 119899lag is the number of lag terms and 120573119895is the 119895th
lag parameter (119895 = 1 119899lag) The parameters 120573119895were
chosen arbitrarily to ensure equality of (A2) for each reducedfrequency and then their values are different for each systempresented herein The procedure to obtain the aerodynamicmatrices is discussed in [15]
Equation (A2) is substituted into (A1) allowing to writethe aeroelastic system in time domain using a state-spacerepresentation In this case the state vector presented in (1) isx(119905) = u
119898u119898
u119886119898
119879 where u
119886119898are states of lags required
for the approximation The dynamic matrix A is given by
8 Mathematical Problems in Engineering
A =
[[[[[[[[[[
[
minusMminus1119886119898D119886119898
minusMminus1119886119898K119886119898
119902Mminus1119886119898Q1198983
sdot sdot sdot 119902Mminus1119886119898Q119898(2+119899lag)
I 0 0 sdot sdot sdot 0I 0 (minus
119881
119887)1205731I 0 sdot sdot sdot
0 d sdot sdot sdot
I 0 sdot sdot sdot (minus
119881
119887)120573119899lag
I
]]]]]]]]]]
]
(A3)
where the matrices M119886119898 D119886119898 and K
119886119898are comprised of
terms containing structural and aerodynamic coefficientswhich are given by
M119886119898
= M119898
minus 119902(119887
119881)
2
Q1198982
D119886119898
= D119898
minus 119902(119887
119881)Q1198981
K119886119898
= K119898
minus 119902Q1198980
(A4)
B Norms for Computing theGramian Parameter
Thematrix norms for computing theGramian parameters arepresented as follows
B1infin-Norm Theinfin-normof W119900(119894119895) is themaximumof the
row sums that is
120590(infin)
119892(119894119895) = max
119894119903
10038161003816100381610038161003816100381610038161003816100381610038161003816
sum
119895119888
119908119900(119894119903 119895119888)
10038161003816100381610038161003816100381610038161003816100381610038161003816
(B1)
B2 2-Norm The 2-norm is computed based on the largestsingular value 120582
119894119903of [W119867
119900(119894119895)W
119900(119894119895)] 119894
119903= 1 119898(2 + 119899lag)
that is
120590(2119873)
119892(119894119895) = (max
119894119903
120582119894119903)
12
(B2)
whereW119867119900(119894119895) denotes a Hermitian matrix
B3 1-Norm The 1-norm is the maximum of the columnssums that is
120590(1119873)
119892(119894119895) = max
119895119888
10038161003816100381610038161003816100381610038161003816100381610038161003816
sum
119894119903
119908119900(119894119903 119895119888)
10038161003816100381610038161003816100381610038161003816100381610038161003816
(B3)
C AGARD 4456 Wind Structural Model
This appendix presents complementary information for theAGARD4456wind andmore details can be found in [18 26]
Figures 12 13 14 and 15 show the four structuralmodes considered in the finite element model obtained fromMSCNASTRAN program
Figure 12 AGARD 4456 wing-first structural mode
Figure 13 AGARD 4456 wing-second structural mode
Figure 14 AGARD 4456 wing-third structural mode
Figure 15 AGARD 4456 wing-fourth structural mode
The thickness distribution is governed by the airfoilshape The material properties used are 119864
1= 31511 119864
2=
04162GPa ] = 031 119866 = 04392GPa and 120588mat =
38198 kgm3 where 1198641and 119864
2are the moduli of elasticity in
the longitudinal and lateral directions ] is Poissonrsquos ratio 119866is the shear modulus in each plane and 120588 is the wing density
There are 10 elements in chordwise and 20 elementsin spanwise direction There are a total of 231 nodes and200 elements The boundary conditions of the wing areselected in accordance with the physical model The root iscantilevered except for the nodes at the leading and trailingedges Additionally the rotation around all axis degrees offreedom at all nodes is constrained to zero In this casethere are 666 degrees of freedom in the model in physicalcoordinates This work considered four modes to obtain themodel in modal coordinates system
Table 2 shows the reduced frequencies 119896 used to computethe aerodynamicmatrices for the second case study (AGARDwing 4456)
Mathematical Problems in Engineering 9
Table 2 Reduced frequencies for the case study AGARD wing4456
10minus3
210minus3
510minus3
10minus2
510minus2
01 02 03
05 06 08 10
15 20 30 40
Conflict of Interests
The authors declare that the research was conducted in theabsence of any commercial or financial relationships thatcould be construed as a potential conflict of interests
References
[1] H A Bisplingho L Raymond and R L Halfman Aeroelastic-ity Dover 1996
[2] T Theodorsen ldquoGeneral theory of aerodynamic instability andthe mechanism of flutterrdquo Tech Rep 496 NACA NationalAdvisory Committee for Aeronautics 1935
[3] H J Hassig ldquoAn approximate true damping solution of theflutter equation by determinant iterationrdquo Journal of Aircraftvol 8 no 11 pp 885ndash889 1971
[4] P C Chen ldquoDamping perturbation method for flutter solutionthe g-methodrdquo The American Institute of Aeronautics andAstronautics Journal vol 38 no 9 pp 1519ndash1524 2000
[5] R E Kalman Y C Ho and K S Narenda ldquoControllability oflinear dynamics systemrdquo Contribution to Dierential Equationsvol 1 no 2 pp 189ndash213 1962
[6] B C Moore ldquoPrincipal component analysis in linear systemscontrollability observability andmodel reductionrdquo IEEETrans-actions on Automatic Control vol 26 no 1 pp 17ndash32 1981
[7] A Arbel ldquoControllability measures and actuator placement inoscillatory systemsrdquo International Journal of Control vol 33 no3 pp 565ndash574 1981
[8] D Biskri R M Botez N Stathopoulos S Therien A Ratheand M Dickinson ldquoNew mixed method for unsteady aero-dynamic force approximations for aeroservoelasticity studiesrdquoJournal of Aircraft vol 43 no 5 pp 1538ndash1542 2006
[9] G Schulz and G Heimbold ldquoDislocated actuatorsensor posi-tioning and feedback design for flexible structuresrdquo Journal ofGuidance Control andDynamics vol 6 no 5 pp 361ndash367 1983
[10] M L Baker ldquoApproximate subspace iteration for construct-ing internally balanced reduced order models of unsteadyaerodynamic systemsrdquo in Proceedings of the 37th AIAAASMEASCEAHSASC Structures Structural Dynamics and MaterialsConference and Exhibit Salt Lake City Utah USA April 1996Technical Papers Part 2 (A96-26801 06-39)
[11] K Willcox and J Peraire ldquoBalanced model reduction via theproper orthogonal decompositionrdquo The American Institute ofAeronautics and Astronautics Journal vol 40 no 11 pp 2323ndash2330 2002
[12] D J Lucia P S Beran and W A Silva ldquoReduced-ordermodeling new approaches for computational physicsrdquo Progressin Aerospace Sciences vol 40 no 1-2 pp 51ndash117 2004
[13] M J Balas ldquoReduced order modeling for aero-elastic simula-tionrdquoAFOSRFinal ReportAFRL-SR-AR-TR06-0456Air ForceOffice of Scientiffic Research Arlington Va USA 2006
[14] W K Gawronski Dynamics and Control of Structures A ModalApproach Springer New York NY USA 1st edition 1998
[15] K Roger ldquoAirplane math modelling methods for active controldesignrdquo in Structural Aspectsof Control vol 9 of AGARDConference Proceeding pp 41ndash411 1977
[16] M Karpel ldquoDesign for active and passive flutter suppressionand gust alleviationrdquo Tech Rep 3482 National Aeronautics andSpace AdministrationmdashNASA 1981
[17] T Theodorsen and I E Garrick Mechanism of Flutter a The-oretical and Experimental Investigation of the Flutter ProblemNACA National Advisory Committee for Aeronautics 1940
[18] E C Yates AGARD Standard Aeroelastic Congurations forDynamic Response I-Wing 4456 Advisory Group for AerospaceResearch and Development 1988
[19] L A Aguirre ldquoControllability and observability of linearsystems some noninvariant aspectsrdquo IEEE Transactions onEducation vol 38 no 1 pp 33ndash39 1995
[20] L A Aguirre and C Letellier ldquoObservability of multivariatedifferential embeddingsrdquo Journal of Physics A vol 38 no 28pp 6311ndash6326 2005
[21] T Stykel ldquoGramian-based model reduction for descriptorsystemsrdquo Mathematics of Control Signals and Systems vol 16no 4 pp 297ndash319 2004
[22] Y Zhu and P R Pagilla ldquoBounds on the solution of the time-varying linear matrix differential equation (119905) = 119860
119867
(119905)119875(119905) +
119875(119905)119860(119905) + 119876(119905)rdquo IMA Journal of Mathematical Control andInformation vol 23 no 3 pp 269ndash277 2006
[23] K Zhou G Salomon and E Wu ldquoBalanced realization andmodel reduction for unstable systemsrdquo International Journal ofRobust and Nonlinear Control vol 9 no 3 pp 183ndash198 1999
[24] C D M Martin and C F V Loan ldquoSolving real linear systemswith the complex Schur decompositionrdquo SIAM Journal onMatrix Analysis andApplications vol 29 no 1 pp 177ndash183 2007
[25] R Bartels andG Stewart ldquoSolutions of thematrix equation 119886119909+
119909119887 = 119888rdquo Communications of ACM vol 15 no 9 pp 820ndash8261972
[26] P Pahlavanloo ldquoDynamic aeroelastic simulation of theAGARD4456 wing using edgerdquo Tech Rep FOI-R-2259-SE DefenceResearchAgencyDefence and Security SystemandTechnologyStockholm Swedish 2007
Submit your manuscripts athttpwwwhindawicom
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
these efforts Gramian matrices are still neglected by thescientific community
The principal objective of the paper is to show thatobservability Gramian matrices can be used to detect flutterThe rationale for using this approach is thatGramianmatricescontain in a single index information about the energy thatis transferred from the flux to the structure and then dissi-pated by any damping mechanism for each flight conditionconsidered in the analysis (flight envelope)
It is shown that theGramians are sensitive to flutter whereenergy transferred from the flux in a cycle is larger than theenergy dissipated by the damping mechanism
Part of the procedure to determine the Gramian matricesrequires a system defined in time domain including theaerodynamic forces If aerodynamic forces are written interms of reduced frequencies this can be done using one ofthemethods such as least square [15] minimum state [16] andthe mixed state [8]
The paper illustratersquos the process to obtain the Gramianmatrix from an aeroelastic system in time domain Twonumerical examples are used to compare the proposedmethod with the methods in the literature The first exampleis three degrees of freedom typical section airfoil and thesecond is the AGARD 4456 wing model developed usingfinite element method [17 18]
The paper shows that it is possible to obtain someadvantage in terms of computation time using the proposedmethodology
2 Aeroelastic Model for Stability Analysis
Assuming a general aeroelastic model that can be written inthe state-space form according to
x (119905) = Ax (119905) + B119888f119888(119905)
y (119905) = Cx (119905)
(1)
where x(119905) is the state vector A is the dynamic matrix B119888
is the input matrix f119888(119905) is a vector of external forces C is
the known as the output matrix and y(119905) is the output vectorUsing this equation the aeroelastic system can be described bydifferent aerodynamic theories and the system of equationscan be written in physical or modal coordinate systems(complementary information is presented in Appendix A)The time-invariant aeroelastic system defined by matrix A isstable for a range of velocities defined by increasing airspeedvalues 119881
1 119881
119895 119881
119899containing a flutter airspeed 119881
119865 if
119881 lt 119881119865
That stability could be verified by solving an eigenvalueproblem for each discrete airspeed point in the flight envelopeand checking if the real part of system eigenvalues is negativevalues This can be time consuming specifically for largedimension systems To overcome this process the observabil-ity Gramian method presented in this paper is based on thesolution of a set of linear equationsThe next section presentsthe bases to write the problem in appropriate format
3 Observability and Gramian Matrices
The concept of observability involves the dynamic matrix Aand the output matrix C A linear system or the pair (AC)is observable at instant of time 119905
0 if the state x(119905
0) can be
determined from the output y(119905) with 119905 isin [1199050 1199051] where
1199051gt 1199050is a finite instant of time If this is true for all initial time
1199050and all initial states x(119905
0) the system is said to be completely
observable [14]A linear time-invariant systemwith119898outputs is said to be
completely observable if and only if the observability matrixwith dimension 2119898
2
(2 + 119899lag) times 21198982
(2 + 119899lag) has hank119898(2 +
119899lag) With the observability matrix given by
O =
[[[[[[
[
CCACA2
CA119899minus1
]]]]]]
]
(2)
where 119899 is the dimension of matrix A This concept can beeasily implemented to verify system observability but maylead to numerical overflow for systems represented by largedimension matrices [14] Also only qualitative informationabout the system is provided (see [14 19 20])
An alternative approach that can be applied for large-order problems is to use the observability Gramian matrixThe observability Gramian matrix is defined to expressquantitative properties of the system considering it at time119905 lt infin written as
W119900(119905) = int
119905
0
[119890A119879119905C119879C119890
A119905] 119889119905 (3)
which according to [14] can be determinated as
W119900(119905) = A119879W
119900(119905) + W
119900(119905)A + C119879C (4)
where W119900(119905) indicates a time-variant property If a linear
time-invariant and stable system is considered then theobservability Gramian matrix can be computed using theLyapunov equation [14]
A119879W119900+ W119900A + C119879C = 0 (5)
An important property between observability andGramian matrices is that they share the same Kernel (ie theset of all vectors x for which Ax = 0)
Ker [W119900(119905)] = Ker [O (CA)] (6)
According to [21 22] one consequence of this property isthat the energy detected by an output state can be computedthrough the observability Gramian matrix This is done bywriting an expression for the energy detected (or observed)by the output y at time 119905
0caused by the systemrsquos initial state
x(0) such that
Energy [y (1199050)] = x119879 (0)W
119900(1199050) x (0) (7)
Mathematical Problems in Engineering 3
Equation (5) is only defined for a stable system [14]To include the flutter speed it is necessary to modifythe equation using the generalized ordinary cross-GramianmatrixW
119888119900 introduced by Zhou et al [23] defined for both
stable and unstable systems Then let X119892be the solution to
the Riccati equation
X119892F119892+ F119879119892X119892minus X119892G119892G119879119892X119892= 0 (8)
The observability Gramian matrix W119900is a submatrix of W
119866
which can be computed by solving the following equation
(F119892+ G119892M119892)W119866+ W119866(F119892+ G119892M119892)119879
+ G119892G119879119892
= 0 (9)
where
F119892= [
A 00 A119879] G
119892= [
[
B119888
C119879]
]
M119892= minusG119879119892X119892
W119866
= [W119888
W119888119900
W119888119900
W119900
]
(10)
and the modified output matrix C is used instead of C Thismodified output matrix C with dimension 119898 times 119898(2 + 119899lag) isdefined such that
y (119905) = Cx (119905)
y (119905) = 0 sdot sdot sdot 0 119906119898(119894)
(119905) 0 sdot sdot sdot 0119879
then C119894= [0 sdot sdot sdot 0
(119898+(119894minus1))1(119894)
0 sdot sdot sdot 0]
(11)
where the 119894th rowC119894satisfies the equation119910
119894(119905) = 119906
119898(119894)(119905) and
the other (119898minus1) rows are filled with zerosThus consideringC119894and the aeroelastic matrix A(119881
119895) defined at the airspeed
119881119895 the observability GramianmatrixW
119900(C119894 119881119895) = W
119900(119894 119895) is
computed by solving (9) for the pair (C119894 119881119895)The inputmatrix
B119888is written by [Mminus1
1198861198980119898(1+119899lag)times119898
]119879 to represent 119898 inputs
31 Complex Schur Decomposition Using a complex Schurdecomposition the Lyapunov equation can be reshaped as areal linear system of equation and solving itW
119900(119894119895)is obtained
[24]
[I otimes (F119892+ G119892M119892) + (F119879
119892+ M119879119892G119879119892) otimes I]vec(W
119866)
= vec(minusG119892G119879119892)
(12)
where I is an identity matrix with appropriate dimensionvec(sdot) makes a column vector out of a matrix by stacking itscolumns and otimes indicates a Kronecker product [25]
32 Gramian Paramater to Detect Flutter The hypothesisintroduced in this paper is that observability Gramianmatrixcontains information which can be used to indicate theamount of energy transferred from the air flow to thestructure In this case this amount of energy is maximum atthe airspeed at which the system becomes unstable
Stable Unstable
Mode 1
Mode 2
AirspeedFlutter speed
Gra
mm
ian
para
met
er
119881119865
Figure 1 Gramian parameter used to detect the flutter phe-nomenon
In order to prove this hypothesis a Gramian parameter120590119892(119894 119895) isin R+ is defined and obtained by computing a matrix
norm ofW119900 This parameter indicates two main features the
contribution of each aeroelastic mode absorbing energy fromthe air flow and the airspeed where flutter occurs
By computing 120590119892(119894 119895) for each pair119881
119895andC
119894it is possible
to build a matrix Σ (14) Assuming that aeroelastic instabilityoccurs in this speed range the flutter speed119881
119895is found for the
largest value of 120590119892(119894 119895)
max (Σ) = max119894119895
[120590119892] = 120590
max119892
such that if Δ119881 997888rarr 119889119881 997904rArr119889120590119892
119889119881= 0
(13)
where 119899V is the length of the airspeed vector
Σ =
[[[[[
[
120590119892(11)
sdot sdot sdot 120590119892(1119895)
sdot sdot sdot 120590119892(1119899V)
sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot
120590119892(1198941)
sdot sdot sdot 120590119892(119894119895)
sdot sdot sdot 120590119892(119894119899V)
sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot
120590119892(1198981)
sdot sdot sdot 120590119892(119898119895)
sdot sdot sdot 120590119892(119898119899V)
]]]]]
]
(14)
The 119894th Gramian parameter in each column of the matrixΣ containing 120590
max119892
is related to a measure of the energyabsorbed by the 119894th aeroelastic mode The values of 120590
119892in
each column can then be compared to determine the modecontribution On the other hand the row of Σwhich contains120590max119892
can be plotted to determine the airspeed of flutter (Thisis illustrated in Figure 1)
33Matrix Norm In practical implementations theGramianparameter 120590
119892(119894 119895) can be obtained by different matrix norms
Matrix norms are often used to provide quantitative infor-mation In this paper Frobenius norm is used (15) However
4 Mathematical Problems in Engineering
Table 1 Physical and geometric properties of the 2D airfoil
Aerodynamic semichord 119887 = 015mAirfoil mass 119872 = 50 kgPlunge frequency 120596
ℎ= 30Hz
Pitch frequency 120596120579= 45Hz
Control surface rotationfrequency 120596
120573= 12Hz
Air density 120588 = 12895 kgm3
Lag parameters (Rogerrsquosmethod) 120573
1= 02 120573
2= 12 120573
3= 16 120573
4= 18
Reduced frequencies 01 le 119896 le 20 Δ119896 = 01
Figure 2 119886 = minus040
Figure 2 119888 = 060
Distance between ce to cg 119909120579= 020
Distance between ce tocg (flap) 119909
120573= 00125
Radius of gyration of theflap referred to 119886
119903120573= (625 times 10
minus3
)12
Radius of gyration of theairfoil referred to 119886
119903120579= radic025
Elastic center ceCenter of gravity cg
ce
ca+120579119896120579
119896ℎ
+ℎ
119886
119887 119887
cg
119888
cgflap
minus120573
119903120573
119909120579
119903120579
119896120573
119909120573
Figure 2 Typical section 2D airfoil
similar results were obtained using other norms presented inAppendix B
120590(119865)
119892(119894 119895) = (sum
119894119903 119895119888
1003816100381610038161003816119908119900 (119894119903 119895119888)1003816100381610038161003816)
12
(15)
where119908119900(119894119903 119895119888) is the Gramianmatrix element of the 119894
119903th row
and 119895119888th column and 119894
119903 119895119888= 1 119898(2 + 119899lag)
4 Numerical Simulations
41 Typical Section Airfoil The effectiveness of the approachwas determined through simulations using two examplesThe first was a three-degree-of-freedom typical airfoil section(semichord 119887) The equations of motion describing the
Pitch mode
Control surface deflection
Plunge mode
0 5 10 15
16
14
12
10
8
6
4
2
0
Freq
uenc
y (H
z)
Airspeed (ms)
119881-119891 diagrammdashtypical section
Figure 3 Aeroelastic frequency as a function of airspeed (typicalsection airfoil)
015
01
005
0
minus005
minus01
minus015
minus02
minus025
minus03
minus035
Dam
ping
ratio
0 5 10 15
Pitch mode
127ms
Control surface deflection
Plunge mode
Airspeed (ms)
119881-119892 diagrammdashtypical section
Figure 4 Real part of the eigenvalues as a function of the airspeed(typical section airfoil)
aeroelastic response are presented in [17] The bidimen-sional system was formulated in modal coordinate systemconsidering the plunge and pitch modes and the controlsurface deflection to represent the third mode Its physicaland geometric properties are presented in Table 1 and anillustration of the airfoil is shown in Figure 2
The results of the proposed approach were comparedwith classical eigenvalues analysis Figures 3 and 4 presentaeroelastic frequency anddamping both plotted as a functionof the airspeed According to the literature although theplunge (or bending) mode is stable the system becomesunstable because the pitch (or torsion mode) is unstable Itis said that airfoil normally is undergoing a flutter oscillationcomposed of important contributions of these modes In
Mathematical Problems in Engineering 5
Grammian parametermdashtypical section airfoil
Plunge mode400
200120590119892
2 4 6 8 10 12 140
Airspeed (ms)
(a)
Grammian parametermdashtypical section airfoil
120590119892
2 4 6 8 10 12 14
40
20
0
Airspeed (ms)
Pitch mode
(b)
Grammian parametermdashtypical section airfoil
120590119892
2 4 6 8 10 12 14
Control surface deflection4
2
0
Airspeed (ms)
(c)
Figure 5 Gramian parameter computed by the Frobenius norm asa function of the airspeed (typical section airfoil)
Air density (kgm3)02 04 06 08 1 12
AGARD wing100
90
80
70
60
50
40
30
20
10
0
Freq
uenc
y (H
z)
Figure 6 Aeroelastic frequency as a function of the air density(AGARD wing)
this example flutter airspeed was computed equal to 119881 =
127msThe results of the proposed method are presented in
Figure 5 The flutter speed was correctly identified Similarresults were also obtained using the other norms and there-fore they are not shown herein
Air density (kgm3)02
02
04
04
06 08 1 1412
AGARD wing
Third mode
minus02
minus04
minus06
minus1
Dam
ping
ratio
Second mode
First mode
Fourth mode
0
0
minus08
Figure 7 Aeroelastic damping as a function of the air density(AGARD wing)
42 AGARD 4456 Wing The AGARD 4456 benchmarkwingwas also used to demonstrate themethodThe structuralmodel for the AGARD 4456 wing was developed usingfinite element method using the MSCNASTRAN The finiteelement model consisted of plate elements with single-layerorthotropic material The model has 231 nodes and 200elements Rotation of nodes was neglected allowing threedegrees of freedom per node See physical properties inAppendix C and complementary details in [18 26]
Aerodynamic and structuralmatrices were obtained fromMSCNASTRAN program (Aeroelastic Solution 145) for theMach number 050 and 120588REF = 1225 kgm3 The values ofreduced frequencies are presented in Appendix CThemodelhas a reference length of 2119887 = 05578m a sweep angle of45 degrees at the quarter chord line a semispan of 0762mand a taper ratio of 066 The flutter boundary is investigatedonly using the first four fundamental structural modes andtheir natural frequencies are respectively 119891
1= 945Hz 119891
2=
3969Hz 1198913
= 4945Hz and 1198914
= 9510Hz Details can befound in literature (eg see [18 26])
A state-space model was obtained through a MATLABimplementation for which the parameters of lag were chosenas 1205731
= 055 1205732
= 140 1205733
= 190 and 1205734
= 290 Theobservability Gramian matrix was computed for each pair(C119894 119881119895) and 120590
119892was computed using different matrix norms
Analysis was evaluated considering 61 pairs of (119881119895 120588119895) with
constantMach number Using the traditional pk-method theflutter speed was identified at 15802ms The classical 119881-119891and 119881-119892 diagrams are shown in Figures 6 and 7
With the proposedmethodology the Gramian parameter120590119892is showed in Figures 8 and 9 Based on Figure 8 it is
possible to note that the first and second aeroelastic modescontribute more to the flutter than the third and fourth
6 Mathematical Problems in Engineering
20
15
10
5
002 04 06 08 1 12
Air density (kgm3)
First mode Air density 06527
120590119892
Airspeed 15802
Grammian parametermdashAGARD wing
(a)
02 04 06 08 1 12
Air density (kgm3)
Second mode
120590119892
3
2
1
0
Grammian parametermdashAGARD wing
(b)
Figure 8 Gramian parameters computed by the Frobenius norm as a function of the air density (AGARD wing)
02 04 06 08 1 12
Air density (kgm3)
Third mode
120590119892
08
06
04
02
1
0
Grammian parametermdashAGARD wing
(a)
02 04 06 08 1 12
Air density (kgm3)
Fourth mode120590119892
08
06
04
02
1
0
Grammian parametermdashAGARD wing
(b)
Figure 9 Gramian parameters computed by the Frobenius norm as a function of the air density (AGARD wing)
06
05
04
03
02
01
010 20 30 40 50 60
Fligth envelope point
pk-methodGram EqCross-gram Eq
Tim
e sum
Computational cost comparison
Figure 10 Comparison of computational time
modes This information can be confirmed in Figure 7 thatshows small changes of aeroelastic damping for these modes
43 Comparison between pk-Method and the ProposedApproach This section shows that it is possible to reduce thecomputational cost to find the flutter speed using Gramian
matrices especially for industrial applications where thenumber of nominal and parametric cases of analysis can bevery large Based on the second example (AGARD wing) itis demonstrated that (5) and (9) can be conveniently usedto get small computational time in comparison with the pk-method
According to previous sections (5) is not valid forunstable systems that is the physics is represented if andonly if the system is stable Then if this equation is usedto compute the Gramian parameter the solution has nophysical meaning after the flutter velocity (including thatpoint) However the computational time to solve the linearsystem of equations (see Section 31) is smaller than that tosolve the iterative eigenvalue algorithm (pk-method) Thefollowing results were obtained using the personal computerwith operational system MS Windows 7 (64 bits) Intel(R)Core(TM) i5 CPUM460 253GHz and RAM 40GHz
Figure 10 shows that Gramian parameters computedfrom the cross-Gramian matrices have computational costlarger than classical pk-method However if they are com-puted from Gramian matrices (5) the computational timeis smaller In this context it is possible to use the followingprocedure
(1) Compute Gramian parameters using observabilityGramian matrices obtained from (5) for every points119875(120588119895 119881119895) into the flight envelope
(2) Identify the point 119875(120588119895 119881119895) = 119875(119895) at which Gramian
parameter is maximum according to (13)(3) Compute cross-Gramian parameters for points
around the maximum
Mathematical Problems in Engineering 7
10 20 30 40 50 60
Fligth envelope point
pk-methodGram EqCross-gram Eq
70 800
002
004
006
008
01
012
014
016
018
02
Com
puta
tiona
l tim
e
Timereduction48
Figure 11 Computational time pk-method versus Gramian param-eters (AGARD 4456 wing)
Using this procedure it is possible to obtain somereduction in the computational time to evaluate the systemstability This reduction is represented in Figure 11
5 Conclusions
Controllability and observability Gramian matrices havebeen used extensively in control design and for optimalplacement of sensors and actuators in smart structures Theyhave also been used for solving aeroelastic problems mainlyfor writing reduced-order models
This paper has investigated the applicability of observ-ability Gramian matrix to detect the flutter speed in twobenchmark structures It was shown that Gramian matricesrepresent the transfer of energy from the flow to the structureThat transfer of energy increases when the airspeed is close tothe flutter condition A classical method was used to compareand validate the results obtained by Gramian matrices
This approach does not compute frequency and dampingfor the aeroelastic modes However it allows to determinethe aeroelastic modes with largest contribution to the fluttermechanism and the airspeed where the system becomesunstable Additionally one advantage is related to the reduc-tion of computational time for analysis when compared toclassical pk-method
This work is a complementary effort to apply the conceptsfrom control theory in problems involving aeroelasticityin time domain It can also be used in flight flutter testsusing an identification method to obtain the state-spacerepresentation instead of identifying aeroelastic frequenciesand damping
Appendices
A State-Space Represenation ofan Aeroelastic System
An aeroelastic system consisting of structural parameters(mass damping and stiffness) subject to the forces fromthe fluid can be modeled in the Laplace domain using aphysical or modal coordinate system In general the modalcoordinates are used to truncate the systemof equations usingthe eigenvector extracted from structural mass and stiffnessmatrices Without loss of generality both numerical casestudies were performed using modal coordinates as shown inthe following equations
The matrices representing the structural parameters arethe mass M the viscous damping D and the stiffness K Inmost cases these matrices are obtained from finite elementmethod and are reduced to modal coordinates (representedby the subscript119898) the system displacement vector in modalcoordinates is u
119898 The aerodynamic influence matrix Q
depends on the parameters 119896 (reduced frequency) and 119898119872
(Mach Number) and 119902 is the dynamic pressure caused by theflowing fluid (Q can be obtained by methods such as DoubletLattice) Note that the proposed approach is not restricted tothis aerodynamic theory
1199042M119898u119898
(119904) + 119904D119898u119898
(119904) + K119898u119898
(119904) = 119902Q119898
(119898119872 119896) u119898
(119904)
(A1)
where 119904 is the Laplace variableIn this case the problem that arises from the conversion
of (A1) to time domain is the fact that Q has no Laplaceinverse This is overcome by writing a rational function torepresent the aerodynamic coefficients in time domain Inthis work Rogerrsquos approximation [15] is used containing apolynomial part representing the forces acting directly relatedto the displacements u(119905) and their first and second deriva-tives Also this equation has a rational part representing theinfluence of the wake acting on the structure with a timedelay
Q119898
(119904) = [
[
2
sum
119895=0
Q119898119895
119904119895
(119887
119881)
119895
+
119899lag
sum
119895=1
Q119898(119895+2)
(119904
119904 + (119887119881) 120573119895
)]
]
u119898
(119904)
(A2)
where 119899lag is the number of lag terms and 120573119895is the 119895th
lag parameter (119895 = 1 119899lag) The parameters 120573119895were
chosen arbitrarily to ensure equality of (A2) for each reducedfrequency and then their values are different for each systempresented herein The procedure to obtain the aerodynamicmatrices is discussed in [15]
Equation (A2) is substituted into (A1) allowing to writethe aeroelastic system in time domain using a state-spacerepresentation In this case the state vector presented in (1) isx(119905) = u
119898u119898
u119886119898
119879 where u
119886119898are states of lags required
for the approximation The dynamic matrix A is given by
8 Mathematical Problems in Engineering
A =
[[[[[[[[[[
[
minusMminus1119886119898D119886119898
minusMminus1119886119898K119886119898
119902Mminus1119886119898Q1198983
sdot sdot sdot 119902Mminus1119886119898Q119898(2+119899lag)
I 0 0 sdot sdot sdot 0I 0 (minus
119881
119887)1205731I 0 sdot sdot sdot
0 d sdot sdot sdot
I 0 sdot sdot sdot (minus
119881
119887)120573119899lag
I
]]]]]]]]]]
]
(A3)
where the matrices M119886119898 D119886119898 and K
119886119898are comprised of
terms containing structural and aerodynamic coefficientswhich are given by
M119886119898
= M119898
minus 119902(119887
119881)
2
Q1198982
D119886119898
= D119898
minus 119902(119887
119881)Q1198981
K119886119898
= K119898
minus 119902Q1198980
(A4)
B Norms for Computing theGramian Parameter
Thematrix norms for computing theGramian parameters arepresented as follows
B1infin-Norm Theinfin-normof W119900(119894119895) is themaximumof the
row sums that is
120590(infin)
119892(119894119895) = max
119894119903
10038161003816100381610038161003816100381610038161003816100381610038161003816
sum
119895119888
119908119900(119894119903 119895119888)
10038161003816100381610038161003816100381610038161003816100381610038161003816
(B1)
B2 2-Norm The 2-norm is computed based on the largestsingular value 120582
119894119903of [W119867
119900(119894119895)W
119900(119894119895)] 119894
119903= 1 119898(2 + 119899lag)
that is
120590(2119873)
119892(119894119895) = (max
119894119903
120582119894119903)
12
(B2)
whereW119867119900(119894119895) denotes a Hermitian matrix
B3 1-Norm The 1-norm is the maximum of the columnssums that is
120590(1119873)
119892(119894119895) = max
119895119888
10038161003816100381610038161003816100381610038161003816100381610038161003816
sum
119894119903
119908119900(119894119903 119895119888)
10038161003816100381610038161003816100381610038161003816100381610038161003816
(B3)
C AGARD 4456 Wind Structural Model
This appendix presents complementary information for theAGARD4456wind andmore details can be found in [18 26]
Figures 12 13 14 and 15 show the four structuralmodes considered in the finite element model obtained fromMSCNASTRAN program
Figure 12 AGARD 4456 wing-first structural mode
Figure 13 AGARD 4456 wing-second structural mode
Figure 14 AGARD 4456 wing-third structural mode
Figure 15 AGARD 4456 wing-fourth structural mode
The thickness distribution is governed by the airfoilshape The material properties used are 119864
1= 31511 119864
2=
04162GPa ] = 031 119866 = 04392GPa and 120588mat =
38198 kgm3 where 1198641and 119864
2are the moduli of elasticity in
the longitudinal and lateral directions ] is Poissonrsquos ratio 119866is the shear modulus in each plane and 120588 is the wing density
There are 10 elements in chordwise and 20 elementsin spanwise direction There are a total of 231 nodes and200 elements The boundary conditions of the wing areselected in accordance with the physical model The root iscantilevered except for the nodes at the leading and trailingedges Additionally the rotation around all axis degrees offreedom at all nodes is constrained to zero In this casethere are 666 degrees of freedom in the model in physicalcoordinates This work considered four modes to obtain themodel in modal coordinates system
Table 2 shows the reduced frequencies 119896 used to computethe aerodynamicmatrices for the second case study (AGARDwing 4456)
Mathematical Problems in Engineering 9
Table 2 Reduced frequencies for the case study AGARD wing4456
10minus3
210minus3
510minus3
10minus2
510minus2
01 02 03
05 06 08 10
15 20 30 40
Conflict of Interests
The authors declare that the research was conducted in theabsence of any commercial or financial relationships thatcould be construed as a potential conflict of interests
References
[1] H A Bisplingho L Raymond and R L Halfman Aeroelastic-ity Dover 1996
[2] T Theodorsen ldquoGeneral theory of aerodynamic instability andthe mechanism of flutterrdquo Tech Rep 496 NACA NationalAdvisory Committee for Aeronautics 1935
[3] H J Hassig ldquoAn approximate true damping solution of theflutter equation by determinant iterationrdquo Journal of Aircraftvol 8 no 11 pp 885ndash889 1971
[4] P C Chen ldquoDamping perturbation method for flutter solutionthe g-methodrdquo The American Institute of Aeronautics andAstronautics Journal vol 38 no 9 pp 1519ndash1524 2000
[5] R E Kalman Y C Ho and K S Narenda ldquoControllability oflinear dynamics systemrdquo Contribution to Dierential Equationsvol 1 no 2 pp 189ndash213 1962
[6] B C Moore ldquoPrincipal component analysis in linear systemscontrollability observability andmodel reductionrdquo IEEETrans-actions on Automatic Control vol 26 no 1 pp 17ndash32 1981
[7] A Arbel ldquoControllability measures and actuator placement inoscillatory systemsrdquo International Journal of Control vol 33 no3 pp 565ndash574 1981
[8] D Biskri R M Botez N Stathopoulos S Therien A Ratheand M Dickinson ldquoNew mixed method for unsteady aero-dynamic force approximations for aeroservoelasticity studiesrdquoJournal of Aircraft vol 43 no 5 pp 1538ndash1542 2006
[9] G Schulz and G Heimbold ldquoDislocated actuatorsensor posi-tioning and feedback design for flexible structuresrdquo Journal ofGuidance Control andDynamics vol 6 no 5 pp 361ndash367 1983
[10] M L Baker ldquoApproximate subspace iteration for construct-ing internally balanced reduced order models of unsteadyaerodynamic systemsrdquo in Proceedings of the 37th AIAAASMEASCEAHSASC Structures Structural Dynamics and MaterialsConference and Exhibit Salt Lake City Utah USA April 1996Technical Papers Part 2 (A96-26801 06-39)
[11] K Willcox and J Peraire ldquoBalanced model reduction via theproper orthogonal decompositionrdquo The American Institute ofAeronautics and Astronautics Journal vol 40 no 11 pp 2323ndash2330 2002
[12] D J Lucia P S Beran and W A Silva ldquoReduced-ordermodeling new approaches for computational physicsrdquo Progressin Aerospace Sciences vol 40 no 1-2 pp 51ndash117 2004
[13] M J Balas ldquoReduced order modeling for aero-elastic simula-tionrdquoAFOSRFinal ReportAFRL-SR-AR-TR06-0456Air ForceOffice of Scientiffic Research Arlington Va USA 2006
[14] W K Gawronski Dynamics and Control of Structures A ModalApproach Springer New York NY USA 1st edition 1998
[15] K Roger ldquoAirplane math modelling methods for active controldesignrdquo in Structural Aspectsof Control vol 9 of AGARDConference Proceeding pp 41ndash411 1977
[16] M Karpel ldquoDesign for active and passive flutter suppressionand gust alleviationrdquo Tech Rep 3482 National Aeronautics andSpace AdministrationmdashNASA 1981
[17] T Theodorsen and I E Garrick Mechanism of Flutter a The-oretical and Experimental Investigation of the Flutter ProblemNACA National Advisory Committee for Aeronautics 1940
[18] E C Yates AGARD Standard Aeroelastic Congurations forDynamic Response I-Wing 4456 Advisory Group for AerospaceResearch and Development 1988
[19] L A Aguirre ldquoControllability and observability of linearsystems some noninvariant aspectsrdquo IEEE Transactions onEducation vol 38 no 1 pp 33ndash39 1995
[20] L A Aguirre and C Letellier ldquoObservability of multivariatedifferential embeddingsrdquo Journal of Physics A vol 38 no 28pp 6311ndash6326 2005
[21] T Stykel ldquoGramian-based model reduction for descriptorsystemsrdquo Mathematics of Control Signals and Systems vol 16no 4 pp 297ndash319 2004
[22] Y Zhu and P R Pagilla ldquoBounds on the solution of the time-varying linear matrix differential equation (119905) = 119860
119867
(119905)119875(119905) +
119875(119905)119860(119905) + 119876(119905)rdquo IMA Journal of Mathematical Control andInformation vol 23 no 3 pp 269ndash277 2006
[23] K Zhou G Salomon and E Wu ldquoBalanced realization andmodel reduction for unstable systemsrdquo International Journal ofRobust and Nonlinear Control vol 9 no 3 pp 183ndash198 1999
[24] C D M Martin and C F V Loan ldquoSolving real linear systemswith the complex Schur decompositionrdquo SIAM Journal onMatrix Analysis andApplications vol 29 no 1 pp 177ndash183 2007
[25] R Bartels andG Stewart ldquoSolutions of thematrix equation 119886119909+
119909119887 = 119888rdquo Communications of ACM vol 15 no 9 pp 820ndash8261972
[26] P Pahlavanloo ldquoDynamic aeroelastic simulation of theAGARD4456 wing using edgerdquo Tech Rep FOI-R-2259-SE DefenceResearchAgencyDefence and Security SystemandTechnologyStockholm Swedish 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
Equation (5) is only defined for a stable system [14]To include the flutter speed it is necessary to modifythe equation using the generalized ordinary cross-GramianmatrixW
119888119900 introduced by Zhou et al [23] defined for both
stable and unstable systems Then let X119892be the solution to
the Riccati equation
X119892F119892+ F119879119892X119892minus X119892G119892G119879119892X119892= 0 (8)
The observability Gramian matrix W119900is a submatrix of W
119866
which can be computed by solving the following equation
(F119892+ G119892M119892)W119866+ W119866(F119892+ G119892M119892)119879
+ G119892G119879119892
= 0 (9)
where
F119892= [
A 00 A119879] G
119892= [
[
B119888
C119879]
]
M119892= minusG119879119892X119892
W119866
= [W119888
W119888119900
W119888119900
W119900
]
(10)
and the modified output matrix C is used instead of C Thismodified output matrix C with dimension 119898 times 119898(2 + 119899lag) isdefined such that
y (119905) = Cx (119905)
y (119905) = 0 sdot sdot sdot 0 119906119898(119894)
(119905) 0 sdot sdot sdot 0119879
then C119894= [0 sdot sdot sdot 0
(119898+(119894minus1))1(119894)
0 sdot sdot sdot 0]
(11)
where the 119894th rowC119894satisfies the equation119910
119894(119905) = 119906
119898(119894)(119905) and
the other (119898minus1) rows are filled with zerosThus consideringC119894and the aeroelastic matrix A(119881
119895) defined at the airspeed
119881119895 the observability GramianmatrixW
119900(C119894 119881119895) = W
119900(119894 119895) is
computed by solving (9) for the pair (C119894 119881119895)The inputmatrix
B119888is written by [Mminus1
1198861198980119898(1+119899lag)times119898
]119879 to represent 119898 inputs
31 Complex Schur Decomposition Using a complex Schurdecomposition the Lyapunov equation can be reshaped as areal linear system of equation and solving itW
119900(119894119895)is obtained
[24]
[I otimes (F119892+ G119892M119892) + (F119879
119892+ M119879119892G119879119892) otimes I]vec(W
119866)
= vec(minusG119892G119879119892)
(12)
where I is an identity matrix with appropriate dimensionvec(sdot) makes a column vector out of a matrix by stacking itscolumns and otimes indicates a Kronecker product [25]
32 Gramian Paramater to Detect Flutter The hypothesisintroduced in this paper is that observability Gramianmatrixcontains information which can be used to indicate theamount of energy transferred from the air flow to thestructure In this case this amount of energy is maximum atthe airspeed at which the system becomes unstable
Stable Unstable
Mode 1
Mode 2
AirspeedFlutter speed
Gra
mm
ian
para
met
er
119881119865
Figure 1 Gramian parameter used to detect the flutter phe-nomenon
In order to prove this hypothesis a Gramian parameter120590119892(119894 119895) isin R+ is defined and obtained by computing a matrix
norm ofW119900 This parameter indicates two main features the
contribution of each aeroelastic mode absorbing energy fromthe air flow and the airspeed where flutter occurs
By computing 120590119892(119894 119895) for each pair119881
119895andC
119894it is possible
to build a matrix Σ (14) Assuming that aeroelastic instabilityoccurs in this speed range the flutter speed119881
119895is found for the
largest value of 120590119892(119894 119895)
max (Σ) = max119894119895
[120590119892] = 120590
max119892
such that if Δ119881 997888rarr 119889119881 997904rArr119889120590119892
119889119881= 0
(13)
where 119899V is the length of the airspeed vector
Σ =
[[[[[
[
120590119892(11)
sdot sdot sdot 120590119892(1119895)
sdot sdot sdot 120590119892(1119899V)
sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot
120590119892(1198941)
sdot sdot sdot 120590119892(119894119895)
sdot sdot sdot 120590119892(119894119899V)
sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot
120590119892(1198981)
sdot sdot sdot 120590119892(119898119895)
sdot sdot sdot 120590119892(119898119899V)
]]]]]
]
(14)
The 119894th Gramian parameter in each column of the matrixΣ containing 120590
max119892
is related to a measure of the energyabsorbed by the 119894th aeroelastic mode The values of 120590
119892in
each column can then be compared to determine the modecontribution On the other hand the row of Σwhich contains120590max119892
can be plotted to determine the airspeed of flutter (Thisis illustrated in Figure 1)
33Matrix Norm In practical implementations theGramianparameter 120590
119892(119894 119895) can be obtained by different matrix norms
Matrix norms are often used to provide quantitative infor-mation In this paper Frobenius norm is used (15) However
4 Mathematical Problems in Engineering
Table 1 Physical and geometric properties of the 2D airfoil
Aerodynamic semichord 119887 = 015mAirfoil mass 119872 = 50 kgPlunge frequency 120596
ℎ= 30Hz
Pitch frequency 120596120579= 45Hz
Control surface rotationfrequency 120596
120573= 12Hz
Air density 120588 = 12895 kgm3
Lag parameters (Rogerrsquosmethod) 120573
1= 02 120573
2= 12 120573
3= 16 120573
4= 18
Reduced frequencies 01 le 119896 le 20 Δ119896 = 01
Figure 2 119886 = minus040
Figure 2 119888 = 060
Distance between ce to cg 119909120579= 020
Distance between ce tocg (flap) 119909
120573= 00125
Radius of gyration of theflap referred to 119886
119903120573= (625 times 10
minus3
)12
Radius of gyration of theairfoil referred to 119886
119903120579= radic025
Elastic center ceCenter of gravity cg
ce
ca+120579119896120579
119896ℎ
+ℎ
119886
119887 119887
cg
119888
cgflap
minus120573
119903120573
119909120579
119903120579
119896120573
119909120573
Figure 2 Typical section 2D airfoil
similar results were obtained using other norms presented inAppendix B
120590(119865)
119892(119894 119895) = (sum
119894119903 119895119888
1003816100381610038161003816119908119900 (119894119903 119895119888)1003816100381610038161003816)
12
(15)
where119908119900(119894119903 119895119888) is the Gramianmatrix element of the 119894
119903th row
and 119895119888th column and 119894
119903 119895119888= 1 119898(2 + 119899lag)
4 Numerical Simulations
41 Typical Section Airfoil The effectiveness of the approachwas determined through simulations using two examplesThe first was a three-degree-of-freedom typical airfoil section(semichord 119887) The equations of motion describing the
Pitch mode
Control surface deflection
Plunge mode
0 5 10 15
16
14
12
10
8
6
4
2
0
Freq
uenc
y (H
z)
Airspeed (ms)
119881-119891 diagrammdashtypical section
Figure 3 Aeroelastic frequency as a function of airspeed (typicalsection airfoil)
015
01
005
0
minus005
minus01
minus015
minus02
minus025
minus03
minus035
Dam
ping
ratio
0 5 10 15
Pitch mode
127ms
Control surface deflection
Plunge mode
Airspeed (ms)
119881-119892 diagrammdashtypical section
Figure 4 Real part of the eigenvalues as a function of the airspeed(typical section airfoil)
aeroelastic response are presented in [17] The bidimen-sional system was formulated in modal coordinate systemconsidering the plunge and pitch modes and the controlsurface deflection to represent the third mode Its physicaland geometric properties are presented in Table 1 and anillustration of the airfoil is shown in Figure 2
The results of the proposed approach were comparedwith classical eigenvalues analysis Figures 3 and 4 presentaeroelastic frequency anddamping both plotted as a functionof the airspeed According to the literature although theplunge (or bending) mode is stable the system becomesunstable because the pitch (or torsion mode) is unstable Itis said that airfoil normally is undergoing a flutter oscillationcomposed of important contributions of these modes In
Mathematical Problems in Engineering 5
Grammian parametermdashtypical section airfoil
Plunge mode400
200120590119892
2 4 6 8 10 12 140
Airspeed (ms)
(a)
Grammian parametermdashtypical section airfoil
120590119892
2 4 6 8 10 12 14
40
20
0
Airspeed (ms)
Pitch mode
(b)
Grammian parametermdashtypical section airfoil
120590119892
2 4 6 8 10 12 14
Control surface deflection4
2
0
Airspeed (ms)
(c)
Figure 5 Gramian parameter computed by the Frobenius norm asa function of the airspeed (typical section airfoil)
Air density (kgm3)02 04 06 08 1 12
AGARD wing100
90
80
70
60
50
40
30
20
10
0
Freq
uenc
y (H
z)
Figure 6 Aeroelastic frequency as a function of the air density(AGARD wing)
this example flutter airspeed was computed equal to 119881 =
127msThe results of the proposed method are presented in
Figure 5 The flutter speed was correctly identified Similarresults were also obtained using the other norms and there-fore they are not shown herein
Air density (kgm3)02
02
04
04
06 08 1 1412
AGARD wing
Third mode
minus02
minus04
minus06
minus1
Dam
ping
ratio
Second mode
First mode
Fourth mode
0
0
minus08
Figure 7 Aeroelastic damping as a function of the air density(AGARD wing)
42 AGARD 4456 Wing The AGARD 4456 benchmarkwingwas also used to demonstrate themethodThe structuralmodel for the AGARD 4456 wing was developed usingfinite element method using the MSCNASTRAN The finiteelement model consisted of plate elements with single-layerorthotropic material The model has 231 nodes and 200elements Rotation of nodes was neglected allowing threedegrees of freedom per node See physical properties inAppendix C and complementary details in [18 26]
Aerodynamic and structuralmatrices were obtained fromMSCNASTRAN program (Aeroelastic Solution 145) for theMach number 050 and 120588REF = 1225 kgm3 The values ofreduced frequencies are presented in Appendix CThemodelhas a reference length of 2119887 = 05578m a sweep angle of45 degrees at the quarter chord line a semispan of 0762mand a taper ratio of 066 The flutter boundary is investigatedonly using the first four fundamental structural modes andtheir natural frequencies are respectively 119891
1= 945Hz 119891
2=
3969Hz 1198913
= 4945Hz and 1198914
= 9510Hz Details can befound in literature (eg see [18 26])
A state-space model was obtained through a MATLABimplementation for which the parameters of lag were chosenas 1205731
= 055 1205732
= 140 1205733
= 190 and 1205734
= 290 Theobservability Gramian matrix was computed for each pair(C119894 119881119895) and 120590
119892was computed using different matrix norms
Analysis was evaluated considering 61 pairs of (119881119895 120588119895) with
constantMach number Using the traditional pk-method theflutter speed was identified at 15802ms The classical 119881-119891and 119881-119892 diagrams are shown in Figures 6 and 7
With the proposedmethodology the Gramian parameter120590119892is showed in Figures 8 and 9 Based on Figure 8 it is
possible to note that the first and second aeroelastic modescontribute more to the flutter than the third and fourth
6 Mathematical Problems in Engineering
20
15
10
5
002 04 06 08 1 12
Air density (kgm3)
First mode Air density 06527
120590119892
Airspeed 15802
Grammian parametermdashAGARD wing
(a)
02 04 06 08 1 12
Air density (kgm3)
Second mode
120590119892
3
2
1
0
Grammian parametermdashAGARD wing
(b)
Figure 8 Gramian parameters computed by the Frobenius norm as a function of the air density (AGARD wing)
02 04 06 08 1 12
Air density (kgm3)
Third mode
120590119892
08
06
04
02
1
0
Grammian parametermdashAGARD wing
(a)
02 04 06 08 1 12
Air density (kgm3)
Fourth mode120590119892
08
06
04
02
1
0
Grammian parametermdashAGARD wing
(b)
Figure 9 Gramian parameters computed by the Frobenius norm as a function of the air density (AGARD wing)
06
05
04
03
02
01
010 20 30 40 50 60
Fligth envelope point
pk-methodGram EqCross-gram Eq
Tim
e sum
Computational cost comparison
Figure 10 Comparison of computational time
modes This information can be confirmed in Figure 7 thatshows small changes of aeroelastic damping for these modes
43 Comparison between pk-Method and the ProposedApproach This section shows that it is possible to reduce thecomputational cost to find the flutter speed using Gramian
matrices especially for industrial applications where thenumber of nominal and parametric cases of analysis can bevery large Based on the second example (AGARD wing) itis demonstrated that (5) and (9) can be conveniently usedto get small computational time in comparison with the pk-method
According to previous sections (5) is not valid forunstable systems that is the physics is represented if andonly if the system is stable Then if this equation is usedto compute the Gramian parameter the solution has nophysical meaning after the flutter velocity (including thatpoint) However the computational time to solve the linearsystem of equations (see Section 31) is smaller than that tosolve the iterative eigenvalue algorithm (pk-method) Thefollowing results were obtained using the personal computerwith operational system MS Windows 7 (64 bits) Intel(R)Core(TM) i5 CPUM460 253GHz and RAM 40GHz
Figure 10 shows that Gramian parameters computedfrom the cross-Gramian matrices have computational costlarger than classical pk-method However if they are com-puted from Gramian matrices (5) the computational timeis smaller In this context it is possible to use the followingprocedure
(1) Compute Gramian parameters using observabilityGramian matrices obtained from (5) for every points119875(120588119895 119881119895) into the flight envelope
(2) Identify the point 119875(120588119895 119881119895) = 119875(119895) at which Gramian
parameter is maximum according to (13)(3) Compute cross-Gramian parameters for points
around the maximum
Mathematical Problems in Engineering 7
10 20 30 40 50 60
Fligth envelope point
pk-methodGram EqCross-gram Eq
70 800
002
004
006
008
01
012
014
016
018
02
Com
puta
tiona
l tim
e
Timereduction48
Figure 11 Computational time pk-method versus Gramian param-eters (AGARD 4456 wing)
Using this procedure it is possible to obtain somereduction in the computational time to evaluate the systemstability This reduction is represented in Figure 11
5 Conclusions
Controllability and observability Gramian matrices havebeen used extensively in control design and for optimalplacement of sensors and actuators in smart structures Theyhave also been used for solving aeroelastic problems mainlyfor writing reduced-order models
This paper has investigated the applicability of observ-ability Gramian matrix to detect the flutter speed in twobenchmark structures It was shown that Gramian matricesrepresent the transfer of energy from the flow to the structureThat transfer of energy increases when the airspeed is close tothe flutter condition A classical method was used to compareand validate the results obtained by Gramian matrices
This approach does not compute frequency and dampingfor the aeroelastic modes However it allows to determinethe aeroelastic modes with largest contribution to the fluttermechanism and the airspeed where the system becomesunstable Additionally one advantage is related to the reduc-tion of computational time for analysis when compared toclassical pk-method
This work is a complementary effort to apply the conceptsfrom control theory in problems involving aeroelasticityin time domain It can also be used in flight flutter testsusing an identification method to obtain the state-spacerepresentation instead of identifying aeroelastic frequenciesand damping
Appendices
A State-Space Represenation ofan Aeroelastic System
An aeroelastic system consisting of structural parameters(mass damping and stiffness) subject to the forces fromthe fluid can be modeled in the Laplace domain using aphysical or modal coordinate system In general the modalcoordinates are used to truncate the systemof equations usingthe eigenvector extracted from structural mass and stiffnessmatrices Without loss of generality both numerical casestudies were performed using modal coordinates as shown inthe following equations
The matrices representing the structural parameters arethe mass M the viscous damping D and the stiffness K Inmost cases these matrices are obtained from finite elementmethod and are reduced to modal coordinates (representedby the subscript119898) the system displacement vector in modalcoordinates is u
119898 The aerodynamic influence matrix Q
depends on the parameters 119896 (reduced frequency) and 119898119872
(Mach Number) and 119902 is the dynamic pressure caused by theflowing fluid (Q can be obtained by methods such as DoubletLattice) Note that the proposed approach is not restricted tothis aerodynamic theory
1199042M119898u119898
(119904) + 119904D119898u119898
(119904) + K119898u119898
(119904) = 119902Q119898
(119898119872 119896) u119898
(119904)
(A1)
where 119904 is the Laplace variableIn this case the problem that arises from the conversion
of (A1) to time domain is the fact that Q has no Laplaceinverse This is overcome by writing a rational function torepresent the aerodynamic coefficients in time domain Inthis work Rogerrsquos approximation [15] is used containing apolynomial part representing the forces acting directly relatedto the displacements u(119905) and their first and second deriva-tives Also this equation has a rational part representing theinfluence of the wake acting on the structure with a timedelay
Q119898
(119904) = [
[
2
sum
119895=0
Q119898119895
119904119895
(119887
119881)
119895
+
119899lag
sum
119895=1
Q119898(119895+2)
(119904
119904 + (119887119881) 120573119895
)]
]
u119898
(119904)
(A2)
where 119899lag is the number of lag terms and 120573119895is the 119895th
lag parameter (119895 = 1 119899lag) The parameters 120573119895were
chosen arbitrarily to ensure equality of (A2) for each reducedfrequency and then their values are different for each systempresented herein The procedure to obtain the aerodynamicmatrices is discussed in [15]
Equation (A2) is substituted into (A1) allowing to writethe aeroelastic system in time domain using a state-spacerepresentation In this case the state vector presented in (1) isx(119905) = u
119898u119898
u119886119898
119879 where u
119886119898are states of lags required
for the approximation The dynamic matrix A is given by
8 Mathematical Problems in Engineering
A =
[[[[[[[[[[
[
minusMminus1119886119898D119886119898
minusMminus1119886119898K119886119898
119902Mminus1119886119898Q1198983
sdot sdot sdot 119902Mminus1119886119898Q119898(2+119899lag)
I 0 0 sdot sdot sdot 0I 0 (minus
119881
119887)1205731I 0 sdot sdot sdot
0 d sdot sdot sdot
I 0 sdot sdot sdot (minus
119881
119887)120573119899lag
I
]]]]]]]]]]
]
(A3)
where the matrices M119886119898 D119886119898 and K
119886119898are comprised of
terms containing structural and aerodynamic coefficientswhich are given by
M119886119898
= M119898
minus 119902(119887
119881)
2
Q1198982
D119886119898
= D119898
minus 119902(119887
119881)Q1198981
K119886119898
= K119898
minus 119902Q1198980
(A4)
B Norms for Computing theGramian Parameter
Thematrix norms for computing theGramian parameters arepresented as follows
B1infin-Norm Theinfin-normof W119900(119894119895) is themaximumof the
row sums that is
120590(infin)
119892(119894119895) = max
119894119903
10038161003816100381610038161003816100381610038161003816100381610038161003816
sum
119895119888
119908119900(119894119903 119895119888)
10038161003816100381610038161003816100381610038161003816100381610038161003816
(B1)
B2 2-Norm The 2-norm is computed based on the largestsingular value 120582
119894119903of [W119867
119900(119894119895)W
119900(119894119895)] 119894
119903= 1 119898(2 + 119899lag)
that is
120590(2119873)
119892(119894119895) = (max
119894119903
120582119894119903)
12
(B2)
whereW119867119900(119894119895) denotes a Hermitian matrix
B3 1-Norm The 1-norm is the maximum of the columnssums that is
120590(1119873)
119892(119894119895) = max
119895119888
10038161003816100381610038161003816100381610038161003816100381610038161003816
sum
119894119903
119908119900(119894119903 119895119888)
10038161003816100381610038161003816100381610038161003816100381610038161003816
(B3)
C AGARD 4456 Wind Structural Model
This appendix presents complementary information for theAGARD4456wind andmore details can be found in [18 26]
Figures 12 13 14 and 15 show the four structuralmodes considered in the finite element model obtained fromMSCNASTRAN program
Figure 12 AGARD 4456 wing-first structural mode
Figure 13 AGARD 4456 wing-second structural mode
Figure 14 AGARD 4456 wing-third structural mode
Figure 15 AGARD 4456 wing-fourth structural mode
The thickness distribution is governed by the airfoilshape The material properties used are 119864
1= 31511 119864
2=
04162GPa ] = 031 119866 = 04392GPa and 120588mat =
38198 kgm3 where 1198641and 119864
2are the moduli of elasticity in
the longitudinal and lateral directions ] is Poissonrsquos ratio 119866is the shear modulus in each plane and 120588 is the wing density
There are 10 elements in chordwise and 20 elementsin spanwise direction There are a total of 231 nodes and200 elements The boundary conditions of the wing areselected in accordance with the physical model The root iscantilevered except for the nodes at the leading and trailingedges Additionally the rotation around all axis degrees offreedom at all nodes is constrained to zero In this casethere are 666 degrees of freedom in the model in physicalcoordinates This work considered four modes to obtain themodel in modal coordinates system
Table 2 shows the reduced frequencies 119896 used to computethe aerodynamicmatrices for the second case study (AGARDwing 4456)
Mathematical Problems in Engineering 9
Table 2 Reduced frequencies for the case study AGARD wing4456
10minus3
210minus3
510minus3
10minus2
510minus2
01 02 03
05 06 08 10
15 20 30 40
Conflict of Interests
The authors declare that the research was conducted in theabsence of any commercial or financial relationships thatcould be construed as a potential conflict of interests
References
[1] H A Bisplingho L Raymond and R L Halfman Aeroelastic-ity Dover 1996
[2] T Theodorsen ldquoGeneral theory of aerodynamic instability andthe mechanism of flutterrdquo Tech Rep 496 NACA NationalAdvisory Committee for Aeronautics 1935
[3] H J Hassig ldquoAn approximate true damping solution of theflutter equation by determinant iterationrdquo Journal of Aircraftvol 8 no 11 pp 885ndash889 1971
[4] P C Chen ldquoDamping perturbation method for flutter solutionthe g-methodrdquo The American Institute of Aeronautics andAstronautics Journal vol 38 no 9 pp 1519ndash1524 2000
[5] R E Kalman Y C Ho and K S Narenda ldquoControllability oflinear dynamics systemrdquo Contribution to Dierential Equationsvol 1 no 2 pp 189ndash213 1962
[6] B C Moore ldquoPrincipal component analysis in linear systemscontrollability observability andmodel reductionrdquo IEEETrans-actions on Automatic Control vol 26 no 1 pp 17ndash32 1981
[7] A Arbel ldquoControllability measures and actuator placement inoscillatory systemsrdquo International Journal of Control vol 33 no3 pp 565ndash574 1981
[8] D Biskri R M Botez N Stathopoulos S Therien A Ratheand M Dickinson ldquoNew mixed method for unsteady aero-dynamic force approximations for aeroservoelasticity studiesrdquoJournal of Aircraft vol 43 no 5 pp 1538ndash1542 2006
[9] G Schulz and G Heimbold ldquoDislocated actuatorsensor posi-tioning and feedback design for flexible structuresrdquo Journal ofGuidance Control andDynamics vol 6 no 5 pp 361ndash367 1983
[10] M L Baker ldquoApproximate subspace iteration for construct-ing internally balanced reduced order models of unsteadyaerodynamic systemsrdquo in Proceedings of the 37th AIAAASMEASCEAHSASC Structures Structural Dynamics and MaterialsConference and Exhibit Salt Lake City Utah USA April 1996Technical Papers Part 2 (A96-26801 06-39)
[11] K Willcox and J Peraire ldquoBalanced model reduction via theproper orthogonal decompositionrdquo The American Institute ofAeronautics and Astronautics Journal vol 40 no 11 pp 2323ndash2330 2002
[12] D J Lucia P S Beran and W A Silva ldquoReduced-ordermodeling new approaches for computational physicsrdquo Progressin Aerospace Sciences vol 40 no 1-2 pp 51ndash117 2004
[13] M J Balas ldquoReduced order modeling for aero-elastic simula-tionrdquoAFOSRFinal ReportAFRL-SR-AR-TR06-0456Air ForceOffice of Scientiffic Research Arlington Va USA 2006
[14] W K Gawronski Dynamics and Control of Structures A ModalApproach Springer New York NY USA 1st edition 1998
[15] K Roger ldquoAirplane math modelling methods for active controldesignrdquo in Structural Aspectsof Control vol 9 of AGARDConference Proceeding pp 41ndash411 1977
[16] M Karpel ldquoDesign for active and passive flutter suppressionand gust alleviationrdquo Tech Rep 3482 National Aeronautics andSpace AdministrationmdashNASA 1981
[17] T Theodorsen and I E Garrick Mechanism of Flutter a The-oretical and Experimental Investigation of the Flutter ProblemNACA National Advisory Committee for Aeronautics 1940
[18] E C Yates AGARD Standard Aeroelastic Congurations forDynamic Response I-Wing 4456 Advisory Group for AerospaceResearch and Development 1988
[19] L A Aguirre ldquoControllability and observability of linearsystems some noninvariant aspectsrdquo IEEE Transactions onEducation vol 38 no 1 pp 33ndash39 1995
[20] L A Aguirre and C Letellier ldquoObservability of multivariatedifferential embeddingsrdquo Journal of Physics A vol 38 no 28pp 6311ndash6326 2005
[21] T Stykel ldquoGramian-based model reduction for descriptorsystemsrdquo Mathematics of Control Signals and Systems vol 16no 4 pp 297ndash319 2004
[22] Y Zhu and P R Pagilla ldquoBounds on the solution of the time-varying linear matrix differential equation (119905) = 119860
119867
(119905)119875(119905) +
119875(119905)119860(119905) + 119876(119905)rdquo IMA Journal of Mathematical Control andInformation vol 23 no 3 pp 269ndash277 2006
[23] K Zhou G Salomon and E Wu ldquoBalanced realization andmodel reduction for unstable systemsrdquo International Journal ofRobust and Nonlinear Control vol 9 no 3 pp 183ndash198 1999
[24] C D M Martin and C F V Loan ldquoSolving real linear systemswith the complex Schur decompositionrdquo SIAM Journal onMatrix Analysis andApplications vol 29 no 1 pp 177ndash183 2007
[25] R Bartels andG Stewart ldquoSolutions of thematrix equation 119886119909+
119909119887 = 119888rdquo Communications of ACM vol 15 no 9 pp 820ndash8261972
[26] P Pahlavanloo ldquoDynamic aeroelastic simulation of theAGARD4456 wing using edgerdquo Tech Rep FOI-R-2259-SE DefenceResearchAgencyDefence and Security SystemandTechnologyStockholm Swedish 2007
Submit your manuscripts athttpwwwhindawicom
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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OptimizationJournal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
Table 1 Physical and geometric properties of the 2D airfoil
Aerodynamic semichord 119887 = 015mAirfoil mass 119872 = 50 kgPlunge frequency 120596
ℎ= 30Hz
Pitch frequency 120596120579= 45Hz
Control surface rotationfrequency 120596
120573= 12Hz
Air density 120588 = 12895 kgm3
Lag parameters (Rogerrsquosmethod) 120573
1= 02 120573
2= 12 120573
3= 16 120573
4= 18
Reduced frequencies 01 le 119896 le 20 Δ119896 = 01
Figure 2 119886 = minus040
Figure 2 119888 = 060
Distance between ce to cg 119909120579= 020
Distance between ce tocg (flap) 119909
120573= 00125
Radius of gyration of theflap referred to 119886
119903120573= (625 times 10
minus3
)12
Radius of gyration of theairfoil referred to 119886
119903120579= radic025
Elastic center ceCenter of gravity cg
ce
ca+120579119896120579
119896ℎ
+ℎ
119886
119887 119887
cg
119888
cgflap
minus120573
119903120573
119909120579
119903120579
119896120573
119909120573
Figure 2 Typical section 2D airfoil
similar results were obtained using other norms presented inAppendix B
120590(119865)
119892(119894 119895) = (sum
119894119903 119895119888
1003816100381610038161003816119908119900 (119894119903 119895119888)1003816100381610038161003816)
12
(15)
where119908119900(119894119903 119895119888) is the Gramianmatrix element of the 119894
119903th row
and 119895119888th column and 119894
119903 119895119888= 1 119898(2 + 119899lag)
4 Numerical Simulations
41 Typical Section Airfoil The effectiveness of the approachwas determined through simulations using two examplesThe first was a three-degree-of-freedom typical airfoil section(semichord 119887) The equations of motion describing the
Pitch mode
Control surface deflection
Plunge mode
0 5 10 15
16
14
12
10
8
6
4
2
0
Freq
uenc
y (H
z)
Airspeed (ms)
119881-119891 diagrammdashtypical section
Figure 3 Aeroelastic frequency as a function of airspeed (typicalsection airfoil)
015
01
005
0
minus005
minus01
minus015
minus02
minus025
minus03
minus035
Dam
ping
ratio
0 5 10 15
Pitch mode
127ms
Control surface deflection
Plunge mode
Airspeed (ms)
119881-119892 diagrammdashtypical section
Figure 4 Real part of the eigenvalues as a function of the airspeed(typical section airfoil)
aeroelastic response are presented in [17] The bidimen-sional system was formulated in modal coordinate systemconsidering the plunge and pitch modes and the controlsurface deflection to represent the third mode Its physicaland geometric properties are presented in Table 1 and anillustration of the airfoil is shown in Figure 2
The results of the proposed approach were comparedwith classical eigenvalues analysis Figures 3 and 4 presentaeroelastic frequency anddamping both plotted as a functionof the airspeed According to the literature although theplunge (or bending) mode is stable the system becomesunstable because the pitch (or torsion mode) is unstable Itis said that airfoil normally is undergoing a flutter oscillationcomposed of important contributions of these modes In
Mathematical Problems in Engineering 5
Grammian parametermdashtypical section airfoil
Plunge mode400
200120590119892
2 4 6 8 10 12 140
Airspeed (ms)
(a)
Grammian parametermdashtypical section airfoil
120590119892
2 4 6 8 10 12 14
40
20
0
Airspeed (ms)
Pitch mode
(b)
Grammian parametermdashtypical section airfoil
120590119892
2 4 6 8 10 12 14
Control surface deflection4
2
0
Airspeed (ms)
(c)
Figure 5 Gramian parameter computed by the Frobenius norm asa function of the airspeed (typical section airfoil)
Air density (kgm3)02 04 06 08 1 12
AGARD wing100
90
80
70
60
50
40
30
20
10
0
Freq
uenc
y (H
z)
Figure 6 Aeroelastic frequency as a function of the air density(AGARD wing)
this example flutter airspeed was computed equal to 119881 =
127msThe results of the proposed method are presented in
Figure 5 The flutter speed was correctly identified Similarresults were also obtained using the other norms and there-fore they are not shown herein
Air density (kgm3)02
02
04
04
06 08 1 1412
AGARD wing
Third mode
minus02
minus04
minus06
minus1
Dam
ping
ratio
Second mode
First mode
Fourth mode
0
0
minus08
Figure 7 Aeroelastic damping as a function of the air density(AGARD wing)
42 AGARD 4456 Wing The AGARD 4456 benchmarkwingwas also used to demonstrate themethodThe structuralmodel for the AGARD 4456 wing was developed usingfinite element method using the MSCNASTRAN The finiteelement model consisted of plate elements with single-layerorthotropic material The model has 231 nodes and 200elements Rotation of nodes was neglected allowing threedegrees of freedom per node See physical properties inAppendix C and complementary details in [18 26]
Aerodynamic and structuralmatrices were obtained fromMSCNASTRAN program (Aeroelastic Solution 145) for theMach number 050 and 120588REF = 1225 kgm3 The values ofreduced frequencies are presented in Appendix CThemodelhas a reference length of 2119887 = 05578m a sweep angle of45 degrees at the quarter chord line a semispan of 0762mand a taper ratio of 066 The flutter boundary is investigatedonly using the first four fundamental structural modes andtheir natural frequencies are respectively 119891
1= 945Hz 119891
2=
3969Hz 1198913
= 4945Hz and 1198914
= 9510Hz Details can befound in literature (eg see [18 26])
A state-space model was obtained through a MATLABimplementation for which the parameters of lag were chosenas 1205731
= 055 1205732
= 140 1205733
= 190 and 1205734
= 290 Theobservability Gramian matrix was computed for each pair(C119894 119881119895) and 120590
119892was computed using different matrix norms
Analysis was evaluated considering 61 pairs of (119881119895 120588119895) with
constantMach number Using the traditional pk-method theflutter speed was identified at 15802ms The classical 119881-119891and 119881-119892 diagrams are shown in Figures 6 and 7
With the proposedmethodology the Gramian parameter120590119892is showed in Figures 8 and 9 Based on Figure 8 it is
possible to note that the first and second aeroelastic modescontribute more to the flutter than the third and fourth
6 Mathematical Problems in Engineering
20
15
10
5
002 04 06 08 1 12
Air density (kgm3)
First mode Air density 06527
120590119892
Airspeed 15802
Grammian parametermdashAGARD wing
(a)
02 04 06 08 1 12
Air density (kgm3)
Second mode
120590119892
3
2
1
0
Grammian parametermdashAGARD wing
(b)
Figure 8 Gramian parameters computed by the Frobenius norm as a function of the air density (AGARD wing)
02 04 06 08 1 12
Air density (kgm3)
Third mode
120590119892
08
06
04
02
1
0
Grammian parametermdashAGARD wing
(a)
02 04 06 08 1 12
Air density (kgm3)
Fourth mode120590119892
08
06
04
02
1
0
Grammian parametermdashAGARD wing
(b)
Figure 9 Gramian parameters computed by the Frobenius norm as a function of the air density (AGARD wing)
06
05
04
03
02
01
010 20 30 40 50 60
Fligth envelope point
pk-methodGram EqCross-gram Eq
Tim
e sum
Computational cost comparison
Figure 10 Comparison of computational time
modes This information can be confirmed in Figure 7 thatshows small changes of aeroelastic damping for these modes
43 Comparison between pk-Method and the ProposedApproach This section shows that it is possible to reduce thecomputational cost to find the flutter speed using Gramian
matrices especially for industrial applications where thenumber of nominal and parametric cases of analysis can bevery large Based on the second example (AGARD wing) itis demonstrated that (5) and (9) can be conveniently usedto get small computational time in comparison with the pk-method
According to previous sections (5) is not valid forunstable systems that is the physics is represented if andonly if the system is stable Then if this equation is usedto compute the Gramian parameter the solution has nophysical meaning after the flutter velocity (including thatpoint) However the computational time to solve the linearsystem of equations (see Section 31) is smaller than that tosolve the iterative eigenvalue algorithm (pk-method) Thefollowing results were obtained using the personal computerwith operational system MS Windows 7 (64 bits) Intel(R)Core(TM) i5 CPUM460 253GHz and RAM 40GHz
Figure 10 shows that Gramian parameters computedfrom the cross-Gramian matrices have computational costlarger than classical pk-method However if they are com-puted from Gramian matrices (5) the computational timeis smaller In this context it is possible to use the followingprocedure
(1) Compute Gramian parameters using observabilityGramian matrices obtained from (5) for every points119875(120588119895 119881119895) into the flight envelope
(2) Identify the point 119875(120588119895 119881119895) = 119875(119895) at which Gramian
parameter is maximum according to (13)(3) Compute cross-Gramian parameters for points
around the maximum
Mathematical Problems in Engineering 7
10 20 30 40 50 60
Fligth envelope point
pk-methodGram EqCross-gram Eq
70 800
002
004
006
008
01
012
014
016
018
02
Com
puta
tiona
l tim
e
Timereduction48
Figure 11 Computational time pk-method versus Gramian param-eters (AGARD 4456 wing)
Using this procedure it is possible to obtain somereduction in the computational time to evaluate the systemstability This reduction is represented in Figure 11
5 Conclusions
Controllability and observability Gramian matrices havebeen used extensively in control design and for optimalplacement of sensors and actuators in smart structures Theyhave also been used for solving aeroelastic problems mainlyfor writing reduced-order models
This paper has investigated the applicability of observ-ability Gramian matrix to detect the flutter speed in twobenchmark structures It was shown that Gramian matricesrepresent the transfer of energy from the flow to the structureThat transfer of energy increases when the airspeed is close tothe flutter condition A classical method was used to compareand validate the results obtained by Gramian matrices
This approach does not compute frequency and dampingfor the aeroelastic modes However it allows to determinethe aeroelastic modes with largest contribution to the fluttermechanism and the airspeed where the system becomesunstable Additionally one advantage is related to the reduc-tion of computational time for analysis when compared toclassical pk-method
This work is a complementary effort to apply the conceptsfrom control theory in problems involving aeroelasticityin time domain It can also be used in flight flutter testsusing an identification method to obtain the state-spacerepresentation instead of identifying aeroelastic frequenciesand damping
Appendices
A State-Space Represenation ofan Aeroelastic System
An aeroelastic system consisting of structural parameters(mass damping and stiffness) subject to the forces fromthe fluid can be modeled in the Laplace domain using aphysical or modal coordinate system In general the modalcoordinates are used to truncate the systemof equations usingthe eigenvector extracted from structural mass and stiffnessmatrices Without loss of generality both numerical casestudies were performed using modal coordinates as shown inthe following equations
The matrices representing the structural parameters arethe mass M the viscous damping D and the stiffness K Inmost cases these matrices are obtained from finite elementmethod and are reduced to modal coordinates (representedby the subscript119898) the system displacement vector in modalcoordinates is u
119898 The aerodynamic influence matrix Q
depends on the parameters 119896 (reduced frequency) and 119898119872
(Mach Number) and 119902 is the dynamic pressure caused by theflowing fluid (Q can be obtained by methods such as DoubletLattice) Note that the proposed approach is not restricted tothis aerodynamic theory
1199042M119898u119898
(119904) + 119904D119898u119898
(119904) + K119898u119898
(119904) = 119902Q119898
(119898119872 119896) u119898
(119904)
(A1)
where 119904 is the Laplace variableIn this case the problem that arises from the conversion
of (A1) to time domain is the fact that Q has no Laplaceinverse This is overcome by writing a rational function torepresent the aerodynamic coefficients in time domain Inthis work Rogerrsquos approximation [15] is used containing apolynomial part representing the forces acting directly relatedto the displacements u(119905) and their first and second deriva-tives Also this equation has a rational part representing theinfluence of the wake acting on the structure with a timedelay
Q119898
(119904) = [
[
2
sum
119895=0
Q119898119895
119904119895
(119887
119881)
119895
+
119899lag
sum
119895=1
Q119898(119895+2)
(119904
119904 + (119887119881) 120573119895
)]
]
u119898
(119904)
(A2)
where 119899lag is the number of lag terms and 120573119895is the 119895th
lag parameter (119895 = 1 119899lag) The parameters 120573119895were
chosen arbitrarily to ensure equality of (A2) for each reducedfrequency and then their values are different for each systempresented herein The procedure to obtain the aerodynamicmatrices is discussed in [15]
Equation (A2) is substituted into (A1) allowing to writethe aeroelastic system in time domain using a state-spacerepresentation In this case the state vector presented in (1) isx(119905) = u
119898u119898
u119886119898
119879 where u
119886119898are states of lags required
for the approximation The dynamic matrix A is given by
8 Mathematical Problems in Engineering
A =
[[[[[[[[[[
[
minusMminus1119886119898D119886119898
minusMminus1119886119898K119886119898
119902Mminus1119886119898Q1198983
sdot sdot sdot 119902Mminus1119886119898Q119898(2+119899lag)
I 0 0 sdot sdot sdot 0I 0 (minus
119881
119887)1205731I 0 sdot sdot sdot
0 d sdot sdot sdot
I 0 sdot sdot sdot (minus
119881
119887)120573119899lag
I
]]]]]]]]]]
]
(A3)
where the matrices M119886119898 D119886119898 and K
119886119898are comprised of
terms containing structural and aerodynamic coefficientswhich are given by
M119886119898
= M119898
minus 119902(119887
119881)
2
Q1198982
D119886119898
= D119898
minus 119902(119887
119881)Q1198981
K119886119898
= K119898
minus 119902Q1198980
(A4)
B Norms for Computing theGramian Parameter
Thematrix norms for computing theGramian parameters arepresented as follows
B1infin-Norm Theinfin-normof W119900(119894119895) is themaximumof the
row sums that is
120590(infin)
119892(119894119895) = max
119894119903
10038161003816100381610038161003816100381610038161003816100381610038161003816
sum
119895119888
119908119900(119894119903 119895119888)
10038161003816100381610038161003816100381610038161003816100381610038161003816
(B1)
B2 2-Norm The 2-norm is computed based on the largestsingular value 120582
119894119903of [W119867
119900(119894119895)W
119900(119894119895)] 119894
119903= 1 119898(2 + 119899lag)
that is
120590(2119873)
119892(119894119895) = (max
119894119903
120582119894119903)
12
(B2)
whereW119867119900(119894119895) denotes a Hermitian matrix
B3 1-Norm The 1-norm is the maximum of the columnssums that is
120590(1119873)
119892(119894119895) = max
119895119888
10038161003816100381610038161003816100381610038161003816100381610038161003816
sum
119894119903
119908119900(119894119903 119895119888)
10038161003816100381610038161003816100381610038161003816100381610038161003816
(B3)
C AGARD 4456 Wind Structural Model
This appendix presents complementary information for theAGARD4456wind andmore details can be found in [18 26]
Figures 12 13 14 and 15 show the four structuralmodes considered in the finite element model obtained fromMSCNASTRAN program
Figure 12 AGARD 4456 wing-first structural mode
Figure 13 AGARD 4456 wing-second structural mode
Figure 14 AGARD 4456 wing-third structural mode
Figure 15 AGARD 4456 wing-fourth structural mode
The thickness distribution is governed by the airfoilshape The material properties used are 119864
1= 31511 119864
2=
04162GPa ] = 031 119866 = 04392GPa and 120588mat =
38198 kgm3 where 1198641and 119864
2are the moduli of elasticity in
the longitudinal and lateral directions ] is Poissonrsquos ratio 119866is the shear modulus in each plane and 120588 is the wing density
There are 10 elements in chordwise and 20 elementsin spanwise direction There are a total of 231 nodes and200 elements The boundary conditions of the wing areselected in accordance with the physical model The root iscantilevered except for the nodes at the leading and trailingedges Additionally the rotation around all axis degrees offreedom at all nodes is constrained to zero In this casethere are 666 degrees of freedom in the model in physicalcoordinates This work considered four modes to obtain themodel in modal coordinates system
Table 2 shows the reduced frequencies 119896 used to computethe aerodynamicmatrices for the second case study (AGARDwing 4456)
Mathematical Problems in Engineering 9
Table 2 Reduced frequencies for the case study AGARD wing4456
10minus3
210minus3
510minus3
10minus2
510minus2
01 02 03
05 06 08 10
15 20 30 40
Conflict of Interests
The authors declare that the research was conducted in theabsence of any commercial or financial relationships thatcould be construed as a potential conflict of interests
References
[1] H A Bisplingho L Raymond and R L Halfman Aeroelastic-ity Dover 1996
[2] T Theodorsen ldquoGeneral theory of aerodynamic instability andthe mechanism of flutterrdquo Tech Rep 496 NACA NationalAdvisory Committee for Aeronautics 1935
[3] H J Hassig ldquoAn approximate true damping solution of theflutter equation by determinant iterationrdquo Journal of Aircraftvol 8 no 11 pp 885ndash889 1971
[4] P C Chen ldquoDamping perturbation method for flutter solutionthe g-methodrdquo The American Institute of Aeronautics andAstronautics Journal vol 38 no 9 pp 1519ndash1524 2000
[5] R E Kalman Y C Ho and K S Narenda ldquoControllability oflinear dynamics systemrdquo Contribution to Dierential Equationsvol 1 no 2 pp 189ndash213 1962
[6] B C Moore ldquoPrincipal component analysis in linear systemscontrollability observability andmodel reductionrdquo IEEETrans-actions on Automatic Control vol 26 no 1 pp 17ndash32 1981
[7] A Arbel ldquoControllability measures and actuator placement inoscillatory systemsrdquo International Journal of Control vol 33 no3 pp 565ndash574 1981
[8] D Biskri R M Botez N Stathopoulos S Therien A Ratheand M Dickinson ldquoNew mixed method for unsteady aero-dynamic force approximations for aeroservoelasticity studiesrdquoJournal of Aircraft vol 43 no 5 pp 1538ndash1542 2006
[9] G Schulz and G Heimbold ldquoDislocated actuatorsensor posi-tioning and feedback design for flexible structuresrdquo Journal ofGuidance Control andDynamics vol 6 no 5 pp 361ndash367 1983
[10] M L Baker ldquoApproximate subspace iteration for construct-ing internally balanced reduced order models of unsteadyaerodynamic systemsrdquo in Proceedings of the 37th AIAAASMEASCEAHSASC Structures Structural Dynamics and MaterialsConference and Exhibit Salt Lake City Utah USA April 1996Technical Papers Part 2 (A96-26801 06-39)
[11] K Willcox and J Peraire ldquoBalanced model reduction via theproper orthogonal decompositionrdquo The American Institute ofAeronautics and Astronautics Journal vol 40 no 11 pp 2323ndash2330 2002
[12] D J Lucia P S Beran and W A Silva ldquoReduced-ordermodeling new approaches for computational physicsrdquo Progressin Aerospace Sciences vol 40 no 1-2 pp 51ndash117 2004
[13] M J Balas ldquoReduced order modeling for aero-elastic simula-tionrdquoAFOSRFinal ReportAFRL-SR-AR-TR06-0456Air ForceOffice of Scientiffic Research Arlington Va USA 2006
[14] W K Gawronski Dynamics and Control of Structures A ModalApproach Springer New York NY USA 1st edition 1998
[15] K Roger ldquoAirplane math modelling methods for active controldesignrdquo in Structural Aspectsof Control vol 9 of AGARDConference Proceeding pp 41ndash411 1977
[16] M Karpel ldquoDesign for active and passive flutter suppressionand gust alleviationrdquo Tech Rep 3482 National Aeronautics andSpace AdministrationmdashNASA 1981
[17] T Theodorsen and I E Garrick Mechanism of Flutter a The-oretical and Experimental Investigation of the Flutter ProblemNACA National Advisory Committee for Aeronautics 1940
[18] E C Yates AGARD Standard Aeroelastic Congurations forDynamic Response I-Wing 4456 Advisory Group for AerospaceResearch and Development 1988
[19] L A Aguirre ldquoControllability and observability of linearsystems some noninvariant aspectsrdquo IEEE Transactions onEducation vol 38 no 1 pp 33ndash39 1995
[20] L A Aguirre and C Letellier ldquoObservability of multivariatedifferential embeddingsrdquo Journal of Physics A vol 38 no 28pp 6311ndash6326 2005
[21] T Stykel ldquoGramian-based model reduction for descriptorsystemsrdquo Mathematics of Control Signals and Systems vol 16no 4 pp 297ndash319 2004
[22] Y Zhu and P R Pagilla ldquoBounds on the solution of the time-varying linear matrix differential equation (119905) = 119860
119867
(119905)119875(119905) +
119875(119905)119860(119905) + 119876(119905)rdquo IMA Journal of Mathematical Control andInformation vol 23 no 3 pp 269ndash277 2006
[23] K Zhou G Salomon and E Wu ldquoBalanced realization andmodel reduction for unstable systemsrdquo International Journal ofRobust and Nonlinear Control vol 9 no 3 pp 183ndash198 1999
[24] C D M Martin and C F V Loan ldquoSolving real linear systemswith the complex Schur decompositionrdquo SIAM Journal onMatrix Analysis andApplications vol 29 no 1 pp 177ndash183 2007
[25] R Bartels andG Stewart ldquoSolutions of thematrix equation 119886119909+
119909119887 = 119888rdquo Communications of ACM vol 15 no 9 pp 820ndash8261972
[26] P Pahlavanloo ldquoDynamic aeroelastic simulation of theAGARD4456 wing using edgerdquo Tech Rep FOI-R-2259-SE DefenceResearchAgencyDefence and Security SystemandTechnologyStockholm Swedish 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
Grammian parametermdashtypical section airfoil
Plunge mode400
200120590119892
2 4 6 8 10 12 140
Airspeed (ms)
(a)
Grammian parametermdashtypical section airfoil
120590119892
2 4 6 8 10 12 14
40
20
0
Airspeed (ms)
Pitch mode
(b)
Grammian parametermdashtypical section airfoil
120590119892
2 4 6 8 10 12 14
Control surface deflection4
2
0
Airspeed (ms)
(c)
Figure 5 Gramian parameter computed by the Frobenius norm asa function of the airspeed (typical section airfoil)
Air density (kgm3)02 04 06 08 1 12
AGARD wing100
90
80
70
60
50
40
30
20
10
0
Freq
uenc
y (H
z)
Figure 6 Aeroelastic frequency as a function of the air density(AGARD wing)
this example flutter airspeed was computed equal to 119881 =
127msThe results of the proposed method are presented in
Figure 5 The flutter speed was correctly identified Similarresults were also obtained using the other norms and there-fore they are not shown herein
Air density (kgm3)02
02
04
04
06 08 1 1412
AGARD wing
Third mode
minus02
minus04
minus06
minus1
Dam
ping
ratio
Second mode
First mode
Fourth mode
0
0
minus08
Figure 7 Aeroelastic damping as a function of the air density(AGARD wing)
42 AGARD 4456 Wing The AGARD 4456 benchmarkwingwas also used to demonstrate themethodThe structuralmodel for the AGARD 4456 wing was developed usingfinite element method using the MSCNASTRAN The finiteelement model consisted of plate elements with single-layerorthotropic material The model has 231 nodes and 200elements Rotation of nodes was neglected allowing threedegrees of freedom per node See physical properties inAppendix C and complementary details in [18 26]
Aerodynamic and structuralmatrices were obtained fromMSCNASTRAN program (Aeroelastic Solution 145) for theMach number 050 and 120588REF = 1225 kgm3 The values ofreduced frequencies are presented in Appendix CThemodelhas a reference length of 2119887 = 05578m a sweep angle of45 degrees at the quarter chord line a semispan of 0762mand a taper ratio of 066 The flutter boundary is investigatedonly using the first four fundamental structural modes andtheir natural frequencies are respectively 119891
1= 945Hz 119891
2=
3969Hz 1198913
= 4945Hz and 1198914
= 9510Hz Details can befound in literature (eg see [18 26])
A state-space model was obtained through a MATLABimplementation for which the parameters of lag were chosenas 1205731
= 055 1205732
= 140 1205733
= 190 and 1205734
= 290 Theobservability Gramian matrix was computed for each pair(C119894 119881119895) and 120590
119892was computed using different matrix norms
Analysis was evaluated considering 61 pairs of (119881119895 120588119895) with
constantMach number Using the traditional pk-method theflutter speed was identified at 15802ms The classical 119881-119891and 119881-119892 diagrams are shown in Figures 6 and 7
With the proposedmethodology the Gramian parameter120590119892is showed in Figures 8 and 9 Based on Figure 8 it is
possible to note that the first and second aeroelastic modescontribute more to the flutter than the third and fourth
6 Mathematical Problems in Engineering
20
15
10
5
002 04 06 08 1 12
Air density (kgm3)
First mode Air density 06527
120590119892
Airspeed 15802
Grammian parametermdashAGARD wing
(a)
02 04 06 08 1 12
Air density (kgm3)
Second mode
120590119892
3
2
1
0
Grammian parametermdashAGARD wing
(b)
Figure 8 Gramian parameters computed by the Frobenius norm as a function of the air density (AGARD wing)
02 04 06 08 1 12
Air density (kgm3)
Third mode
120590119892
08
06
04
02
1
0
Grammian parametermdashAGARD wing
(a)
02 04 06 08 1 12
Air density (kgm3)
Fourth mode120590119892
08
06
04
02
1
0
Grammian parametermdashAGARD wing
(b)
Figure 9 Gramian parameters computed by the Frobenius norm as a function of the air density (AGARD wing)
06
05
04
03
02
01
010 20 30 40 50 60
Fligth envelope point
pk-methodGram EqCross-gram Eq
Tim
e sum
Computational cost comparison
Figure 10 Comparison of computational time
modes This information can be confirmed in Figure 7 thatshows small changes of aeroelastic damping for these modes
43 Comparison between pk-Method and the ProposedApproach This section shows that it is possible to reduce thecomputational cost to find the flutter speed using Gramian
matrices especially for industrial applications where thenumber of nominal and parametric cases of analysis can bevery large Based on the second example (AGARD wing) itis demonstrated that (5) and (9) can be conveniently usedto get small computational time in comparison with the pk-method
According to previous sections (5) is not valid forunstable systems that is the physics is represented if andonly if the system is stable Then if this equation is usedto compute the Gramian parameter the solution has nophysical meaning after the flutter velocity (including thatpoint) However the computational time to solve the linearsystem of equations (see Section 31) is smaller than that tosolve the iterative eigenvalue algorithm (pk-method) Thefollowing results were obtained using the personal computerwith operational system MS Windows 7 (64 bits) Intel(R)Core(TM) i5 CPUM460 253GHz and RAM 40GHz
Figure 10 shows that Gramian parameters computedfrom the cross-Gramian matrices have computational costlarger than classical pk-method However if they are com-puted from Gramian matrices (5) the computational timeis smaller In this context it is possible to use the followingprocedure
(1) Compute Gramian parameters using observabilityGramian matrices obtained from (5) for every points119875(120588119895 119881119895) into the flight envelope
(2) Identify the point 119875(120588119895 119881119895) = 119875(119895) at which Gramian
parameter is maximum according to (13)(3) Compute cross-Gramian parameters for points
around the maximum
Mathematical Problems in Engineering 7
10 20 30 40 50 60
Fligth envelope point
pk-methodGram EqCross-gram Eq
70 800
002
004
006
008
01
012
014
016
018
02
Com
puta
tiona
l tim
e
Timereduction48
Figure 11 Computational time pk-method versus Gramian param-eters (AGARD 4456 wing)
Using this procedure it is possible to obtain somereduction in the computational time to evaluate the systemstability This reduction is represented in Figure 11
5 Conclusions
Controllability and observability Gramian matrices havebeen used extensively in control design and for optimalplacement of sensors and actuators in smart structures Theyhave also been used for solving aeroelastic problems mainlyfor writing reduced-order models
This paper has investigated the applicability of observ-ability Gramian matrix to detect the flutter speed in twobenchmark structures It was shown that Gramian matricesrepresent the transfer of energy from the flow to the structureThat transfer of energy increases when the airspeed is close tothe flutter condition A classical method was used to compareand validate the results obtained by Gramian matrices
This approach does not compute frequency and dampingfor the aeroelastic modes However it allows to determinethe aeroelastic modes with largest contribution to the fluttermechanism and the airspeed where the system becomesunstable Additionally one advantage is related to the reduc-tion of computational time for analysis when compared toclassical pk-method
This work is a complementary effort to apply the conceptsfrom control theory in problems involving aeroelasticityin time domain It can also be used in flight flutter testsusing an identification method to obtain the state-spacerepresentation instead of identifying aeroelastic frequenciesand damping
Appendices
A State-Space Represenation ofan Aeroelastic System
An aeroelastic system consisting of structural parameters(mass damping and stiffness) subject to the forces fromthe fluid can be modeled in the Laplace domain using aphysical or modal coordinate system In general the modalcoordinates are used to truncate the systemof equations usingthe eigenvector extracted from structural mass and stiffnessmatrices Without loss of generality both numerical casestudies were performed using modal coordinates as shown inthe following equations
The matrices representing the structural parameters arethe mass M the viscous damping D and the stiffness K Inmost cases these matrices are obtained from finite elementmethod and are reduced to modal coordinates (representedby the subscript119898) the system displacement vector in modalcoordinates is u
119898 The aerodynamic influence matrix Q
depends on the parameters 119896 (reduced frequency) and 119898119872
(Mach Number) and 119902 is the dynamic pressure caused by theflowing fluid (Q can be obtained by methods such as DoubletLattice) Note that the proposed approach is not restricted tothis aerodynamic theory
1199042M119898u119898
(119904) + 119904D119898u119898
(119904) + K119898u119898
(119904) = 119902Q119898
(119898119872 119896) u119898
(119904)
(A1)
where 119904 is the Laplace variableIn this case the problem that arises from the conversion
of (A1) to time domain is the fact that Q has no Laplaceinverse This is overcome by writing a rational function torepresent the aerodynamic coefficients in time domain Inthis work Rogerrsquos approximation [15] is used containing apolynomial part representing the forces acting directly relatedto the displacements u(119905) and their first and second deriva-tives Also this equation has a rational part representing theinfluence of the wake acting on the structure with a timedelay
Q119898
(119904) = [
[
2
sum
119895=0
Q119898119895
119904119895
(119887
119881)
119895
+
119899lag
sum
119895=1
Q119898(119895+2)
(119904
119904 + (119887119881) 120573119895
)]
]
u119898
(119904)
(A2)
where 119899lag is the number of lag terms and 120573119895is the 119895th
lag parameter (119895 = 1 119899lag) The parameters 120573119895were
chosen arbitrarily to ensure equality of (A2) for each reducedfrequency and then their values are different for each systempresented herein The procedure to obtain the aerodynamicmatrices is discussed in [15]
Equation (A2) is substituted into (A1) allowing to writethe aeroelastic system in time domain using a state-spacerepresentation In this case the state vector presented in (1) isx(119905) = u
119898u119898
u119886119898
119879 where u
119886119898are states of lags required
for the approximation The dynamic matrix A is given by
8 Mathematical Problems in Engineering
A =
[[[[[[[[[[
[
minusMminus1119886119898D119886119898
minusMminus1119886119898K119886119898
119902Mminus1119886119898Q1198983
sdot sdot sdot 119902Mminus1119886119898Q119898(2+119899lag)
I 0 0 sdot sdot sdot 0I 0 (minus
119881
119887)1205731I 0 sdot sdot sdot
0 d sdot sdot sdot
I 0 sdot sdot sdot (minus
119881
119887)120573119899lag
I
]]]]]]]]]]
]
(A3)
where the matrices M119886119898 D119886119898 and K
119886119898are comprised of
terms containing structural and aerodynamic coefficientswhich are given by
M119886119898
= M119898
minus 119902(119887
119881)
2
Q1198982
D119886119898
= D119898
minus 119902(119887
119881)Q1198981
K119886119898
= K119898
minus 119902Q1198980
(A4)
B Norms for Computing theGramian Parameter
Thematrix norms for computing theGramian parameters arepresented as follows
B1infin-Norm Theinfin-normof W119900(119894119895) is themaximumof the
row sums that is
120590(infin)
119892(119894119895) = max
119894119903
10038161003816100381610038161003816100381610038161003816100381610038161003816
sum
119895119888
119908119900(119894119903 119895119888)
10038161003816100381610038161003816100381610038161003816100381610038161003816
(B1)
B2 2-Norm The 2-norm is computed based on the largestsingular value 120582
119894119903of [W119867
119900(119894119895)W
119900(119894119895)] 119894
119903= 1 119898(2 + 119899lag)
that is
120590(2119873)
119892(119894119895) = (max
119894119903
120582119894119903)
12
(B2)
whereW119867119900(119894119895) denotes a Hermitian matrix
B3 1-Norm The 1-norm is the maximum of the columnssums that is
120590(1119873)
119892(119894119895) = max
119895119888
10038161003816100381610038161003816100381610038161003816100381610038161003816
sum
119894119903
119908119900(119894119903 119895119888)
10038161003816100381610038161003816100381610038161003816100381610038161003816
(B3)
C AGARD 4456 Wind Structural Model
This appendix presents complementary information for theAGARD4456wind andmore details can be found in [18 26]
Figures 12 13 14 and 15 show the four structuralmodes considered in the finite element model obtained fromMSCNASTRAN program
Figure 12 AGARD 4456 wing-first structural mode
Figure 13 AGARD 4456 wing-second structural mode
Figure 14 AGARD 4456 wing-third structural mode
Figure 15 AGARD 4456 wing-fourth structural mode
The thickness distribution is governed by the airfoilshape The material properties used are 119864
1= 31511 119864
2=
04162GPa ] = 031 119866 = 04392GPa and 120588mat =
38198 kgm3 where 1198641and 119864
2are the moduli of elasticity in
the longitudinal and lateral directions ] is Poissonrsquos ratio 119866is the shear modulus in each plane and 120588 is the wing density
There are 10 elements in chordwise and 20 elementsin spanwise direction There are a total of 231 nodes and200 elements The boundary conditions of the wing areselected in accordance with the physical model The root iscantilevered except for the nodes at the leading and trailingedges Additionally the rotation around all axis degrees offreedom at all nodes is constrained to zero In this casethere are 666 degrees of freedom in the model in physicalcoordinates This work considered four modes to obtain themodel in modal coordinates system
Table 2 shows the reduced frequencies 119896 used to computethe aerodynamicmatrices for the second case study (AGARDwing 4456)
Mathematical Problems in Engineering 9
Table 2 Reduced frequencies for the case study AGARD wing4456
10minus3
210minus3
510minus3
10minus2
510minus2
01 02 03
05 06 08 10
15 20 30 40
Conflict of Interests
The authors declare that the research was conducted in theabsence of any commercial or financial relationships thatcould be construed as a potential conflict of interests
References
[1] H A Bisplingho L Raymond and R L Halfman Aeroelastic-ity Dover 1996
[2] T Theodorsen ldquoGeneral theory of aerodynamic instability andthe mechanism of flutterrdquo Tech Rep 496 NACA NationalAdvisory Committee for Aeronautics 1935
[3] H J Hassig ldquoAn approximate true damping solution of theflutter equation by determinant iterationrdquo Journal of Aircraftvol 8 no 11 pp 885ndash889 1971
[4] P C Chen ldquoDamping perturbation method for flutter solutionthe g-methodrdquo The American Institute of Aeronautics andAstronautics Journal vol 38 no 9 pp 1519ndash1524 2000
[5] R E Kalman Y C Ho and K S Narenda ldquoControllability oflinear dynamics systemrdquo Contribution to Dierential Equationsvol 1 no 2 pp 189ndash213 1962
[6] B C Moore ldquoPrincipal component analysis in linear systemscontrollability observability andmodel reductionrdquo IEEETrans-actions on Automatic Control vol 26 no 1 pp 17ndash32 1981
[7] A Arbel ldquoControllability measures and actuator placement inoscillatory systemsrdquo International Journal of Control vol 33 no3 pp 565ndash574 1981
[8] D Biskri R M Botez N Stathopoulos S Therien A Ratheand M Dickinson ldquoNew mixed method for unsteady aero-dynamic force approximations for aeroservoelasticity studiesrdquoJournal of Aircraft vol 43 no 5 pp 1538ndash1542 2006
[9] G Schulz and G Heimbold ldquoDislocated actuatorsensor posi-tioning and feedback design for flexible structuresrdquo Journal ofGuidance Control andDynamics vol 6 no 5 pp 361ndash367 1983
[10] M L Baker ldquoApproximate subspace iteration for construct-ing internally balanced reduced order models of unsteadyaerodynamic systemsrdquo in Proceedings of the 37th AIAAASMEASCEAHSASC Structures Structural Dynamics and MaterialsConference and Exhibit Salt Lake City Utah USA April 1996Technical Papers Part 2 (A96-26801 06-39)
[11] K Willcox and J Peraire ldquoBalanced model reduction via theproper orthogonal decompositionrdquo The American Institute ofAeronautics and Astronautics Journal vol 40 no 11 pp 2323ndash2330 2002
[12] D J Lucia P S Beran and W A Silva ldquoReduced-ordermodeling new approaches for computational physicsrdquo Progressin Aerospace Sciences vol 40 no 1-2 pp 51ndash117 2004
[13] M J Balas ldquoReduced order modeling for aero-elastic simula-tionrdquoAFOSRFinal ReportAFRL-SR-AR-TR06-0456Air ForceOffice of Scientiffic Research Arlington Va USA 2006
[14] W K Gawronski Dynamics and Control of Structures A ModalApproach Springer New York NY USA 1st edition 1998
[15] K Roger ldquoAirplane math modelling methods for active controldesignrdquo in Structural Aspectsof Control vol 9 of AGARDConference Proceeding pp 41ndash411 1977
[16] M Karpel ldquoDesign for active and passive flutter suppressionand gust alleviationrdquo Tech Rep 3482 National Aeronautics andSpace AdministrationmdashNASA 1981
[17] T Theodorsen and I E Garrick Mechanism of Flutter a The-oretical and Experimental Investigation of the Flutter ProblemNACA National Advisory Committee for Aeronautics 1940
[18] E C Yates AGARD Standard Aeroelastic Congurations forDynamic Response I-Wing 4456 Advisory Group for AerospaceResearch and Development 1988
[19] L A Aguirre ldquoControllability and observability of linearsystems some noninvariant aspectsrdquo IEEE Transactions onEducation vol 38 no 1 pp 33ndash39 1995
[20] L A Aguirre and C Letellier ldquoObservability of multivariatedifferential embeddingsrdquo Journal of Physics A vol 38 no 28pp 6311ndash6326 2005
[21] T Stykel ldquoGramian-based model reduction for descriptorsystemsrdquo Mathematics of Control Signals and Systems vol 16no 4 pp 297ndash319 2004
[22] Y Zhu and P R Pagilla ldquoBounds on the solution of the time-varying linear matrix differential equation (119905) = 119860
119867
(119905)119875(119905) +
119875(119905)119860(119905) + 119876(119905)rdquo IMA Journal of Mathematical Control andInformation vol 23 no 3 pp 269ndash277 2006
[23] K Zhou G Salomon and E Wu ldquoBalanced realization andmodel reduction for unstable systemsrdquo International Journal ofRobust and Nonlinear Control vol 9 no 3 pp 183ndash198 1999
[24] C D M Martin and C F V Loan ldquoSolving real linear systemswith the complex Schur decompositionrdquo SIAM Journal onMatrix Analysis andApplications vol 29 no 1 pp 177ndash183 2007
[25] R Bartels andG Stewart ldquoSolutions of thematrix equation 119886119909+
119909119887 = 119888rdquo Communications of ACM vol 15 no 9 pp 820ndash8261972
[26] P Pahlavanloo ldquoDynamic aeroelastic simulation of theAGARD4456 wing using edgerdquo Tech Rep FOI-R-2259-SE DefenceResearchAgencyDefence and Security SystemandTechnologyStockholm Swedish 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
20
15
10
5
002 04 06 08 1 12
Air density (kgm3)
First mode Air density 06527
120590119892
Airspeed 15802
Grammian parametermdashAGARD wing
(a)
02 04 06 08 1 12
Air density (kgm3)
Second mode
120590119892
3
2
1
0
Grammian parametermdashAGARD wing
(b)
Figure 8 Gramian parameters computed by the Frobenius norm as a function of the air density (AGARD wing)
02 04 06 08 1 12
Air density (kgm3)
Third mode
120590119892
08
06
04
02
1
0
Grammian parametermdashAGARD wing
(a)
02 04 06 08 1 12
Air density (kgm3)
Fourth mode120590119892
08
06
04
02
1
0
Grammian parametermdashAGARD wing
(b)
Figure 9 Gramian parameters computed by the Frobenius norm as a function of the air density (AGARD wing)
06
05
04
03
02
01
010 20 30 40 50 60
Fligth envelope point
pk-methodGram EqCross-gram Eq
Tim
e sum
Computational cost comparison
Figure 10 Comparison of computational time
modes This information can be confirmed in Figure 7 thatshows small changes of aeroelastic damping for these modes
43 Comparison between pk-Method and the ProposedApproach This section shows that it is possible to reduce thecomputational cost to find the flutter speed using Gramian
matrices especially for industrial applications where thenumber of nominal and parametric cases of analysis can bevery large Based on the second example (AGARD wing) itis demonstrated that (5) and (9) can be conveniently usedto get small computational time in comparison with the pk-method
According to previous sections (5) is not valid forunstable systems that is the physics is represented if andonly if the system is stable Then if this equation is usedto compute the Gramian parameter the solution has nophysical meaning after the flutter velocity (including thatpoint) However the computational time to solve the linearsystem of equations (see Section 31) is smaller than that tosolve the iterative eigenvalue algorithm (pk-method) Thefollowing results were obtained using the personal computerwith operational system MS Windows 7 (64 bits) Intel(R)Core(TM) i5 CPUM460 253GHz and RAM 40GHz
Figure 10 shows that Gramian parameters computedfrom the cross-Gramian matrices have computational costlarger than classical pk-method However if they are com-puted from Gramian matrices (5) the computational timeis smaller In this context it is possible to use the followingprocedure
(1) Compute Gramian parameters using observabilityGramian matrices obtained from (5) for every points119875(120588119895 119881119895) into the flight envelope
(2) Identify the point 119875(120588119895 119881119895) = 119875(119895) at which Gramian
parameter is maximum according to (13)(3) Compute cross-Gramian parameters for points
around the maximum
Mathematical Problems in Engineering 7
10 20 30 40 50 60
Fligth envelope point
pk-methodGram EqCross-gram Eq
70 800
002
004
006
008
01
012
014
016
018
02
Com
puta
tiona
l tim
e
Timereduction48
Figure 11 Computational time pk-method versus Gramian param-eters (AGARD 4456 wing)
Using this procedure it is possible to obtain somereduction in the computational time to evaluate the systemstability This reduction is represented in Figure 11
5 Conclusions
Controllability and observability Gramian matrices havebeen used extensively in control design and for optimalplacement of sensors and actuators in smart structures Theyhave also been used for solving aeroelastic problems mainlyfor writing reduced-order models
This paper has investigated the applicability of observ-ability Gramian matrix to detect the flutter speed in twobenchmark structures It was shown that Gramian matricesrepresent the transfer of energy from the flow to the structureThat transfer of energy increases when the airspeed is close tothe flutter condition A classical method was used to compareand validate the results obtained by Gramian matrices
This approach does not compute frequency and dampingfor the aeroelastic modes However it allows to determinethe aeroelastic modes with largest contribution to the fluttermechanism and the airspeed where the system becomesunstable Additionally one advantage is related to the reduc-tion of computational time for analysis when compared toclassical pk-method
This work is a complementary effort to apply the conceptsfrom control theory in problems involving aeroelasticityin time domain It can also be used in flight flutter testsusing an identification method to obtain the state-spacerepresentation instead of identifying aeroelastic frequenciesand damping
Appendices
A State-Space Represenation ofan Aeroelastic System
An aeroelastic system consisting of structural parameters(mass damping and stiffness) subject to the forces fromthe fluid can be modeled in the Laplace domain using aphysical or modal coordinate system In general the modalcoordinates are used to truncate the systemof equations usingthe eigenvector extracted from structural mass and stiffnessmatrices Without loss of generality both numerical casestudies were performed using modal coordinates as shown inthe following equations
The matrices representing the structural parameters arethe mass M the viscous damping D and the stiffness K Inmost cases these matrices are obtained from finite elementmethod and are reduced to modal coordinates (representedby the subscript119898) the system displacement vector in modalcoordinates is u
119898 The aerodynamic influence matrix Q
depends on the parameters 119896 (reduced frequency) and 119898119872
(Mach Number) and 119902 is the dynamic pressure caused by theflowing fluid (Q can be obtained by methods such as DoubletLattice) Note that the proposed approach is not restricted tothis aerodynamic theory
1199042M119898u119898
(119904) + 119904D119898u119898
(119904) + K119898u119898
(119904) = 119902Q119898
(119898119872 119896) u119898
(119904)
(A1)
where 119904 is the Laplace variableIn this case the problem that arises from the conversion
of (A1) to time domain is the fact that Q has no Laplaceinverse This is overcome by writing a rational function torepresent the aerodynamic coefficients in time domain Inthis work Rogerrsquos approximation [15] is used containing apolynomial part representing the forces acting directly relatedto the displacements u(119905) and their first and second deriva-tives Also this equation has a rational part representing theinfluence of the wake acting on the structure with a timedelay
Q119898
(119904) = [
[
2
sum
119895=0
Q119898119895
119904119895
(119887
119881)
119895
+
119899lag
sum
119895=1
Q119898(119895+2)
(119904
119904 + (119887119881) 120573119895
)]
]
u119898
(119904)
(A2)
where 119899lag is the number of lag terms and 120573119895is the 119895th
lag parameter (119895 = 1 119899lag) The parameters 120573119895were
chosen arbitrarily to ensure equality of (A2) for each reducedfrequency and then their values are different for each systempresented herein The procedure to obtain the aerodynamicmatrices is discussed in [15]
Equation (A2) is substituted into (A1) allowing to writethe aeroelastic system in time domain using a state-spacerepresentation In this case the state vector presented in (1) isx(119905) = u
119898u119898
u119886119898
119879 where u
119886119898are states of lags required
for the approximation The dynamic matrix A is given by
8 Mathematical Problems in Engineering
A =
[[[[[[[[[[
[
minusMminus1119886119898D119886119898
minusMminus1119886119898K119886119898
119902Mminus1119886119898Q1198983
sdot sdot sdot 119902Mminus1119886119898Q119898(2+119899lag)
I 0 0 sdot sdot sdot 0I 0 (minus
119881
119887)1205731I 0 sdot sdot sdot
0 d sdot sdot sdot
I 0 sdot sdot sdot (minus
119881
119887)120573119899lag
I
]]]]]]]]]]
]
(A3)
where the matrices M119886119898 D119886119898 and K
119886119898are comprised of
terms containing structural and aerodynamic coefficientswhich are given by
M119886119898
= M119898
minus 119902(119887
119881)
2
Q1198982
D119886119898
= D119898
minus 119902(119887
119881)Q1198981
K119886119898
= K119898
minus 119902Q1198980
(A4)
B Norms for Computing theGramian Parameter
Thematrix norms for computing theGramian parameters arepresented as follows
B1infin-Norm Theinfin-normof W119900(119894119895) is themaximumof the
row sums that is
120590(infin)
119892(119894119895) = max
119894119903
10038161003816100381610038161003816100381610038161003816100381610038161003816
sum
119895119888
119908119900(119894119903 119895119888)
10038161003816100381610038161003816100381610038161003816100381610038161003816
(B1)
B2 2-Norm The 2-norm is computed based on the largestsingular value 120582
119894119903of [W119867
119900(119894119895)W
119900(119894119895)] 119894
119903= 1 119898(2 + 119899lag)
that is
120590(2119873)
119892(119894119895) = (max
119894119903
120582119894119903)
12
(B2)
whereW119867119900(119894119895) denotes a Hermitian matrix
B3 1-Norm The 1-norm is the maximum of the columnssums that is
120590(1119873)
119892(119894119895) = max
119895119888
10038161003816100381610038161003816100381610038161003816100381610038161003816
sum
119894119903
119908119900(119894119903 119895119888)
10038161003816100381610038161003816100381610038161003816100381610038161003816
(B3)
C AGARD 4456 Wind Structural Model
This appendix presents complementary information for theAGARD4456wind andmore details can be found in [18 26]
Figures 12 13 14 and 15 show the four structuralmodes considered in the finite element model obtained fromMSCNASTRAN program
Figure 12 AGARD 4456 wing-first structural mode
Figure 13 AGARD 4456 wing-second structural mode
Figure 14 AGARD 4456 wing-third structural mode
Figure 15 AGARD 4456 wing-fourth structural mode
The thickness distribution is governed by the airfoilshape The material properties used are 119864
1= 31511 119864
2=
04162GPa ] = 031 119866 = 04392GPa and 120588mat =
38198 kgm3 where 1198641and 119864
2are the moduli of elasticity in
the longitudinal and lateral directions ] is Poissonrsquos ratio 119866is the shear modulus in each plane and 120588 is the wing density
There are 10 elements in chordwise and 20 elementsin spanwise direction There are a total of 231 nodes and200 elements The boundary conditions of the wing areselected in accordance with the physical model The root iscantilevered except for the nodes at the leading and trailingedges Additionally the rotation around all axis degrees offreedom at all nodes is constrained to zero In this casethere are 666 degrees of freedom in the model in physicalcoordinates This work considered four modes to obtain themodel in modal coordinates system
Table 2 shows the reduced frequencies 119896 used to computethe aerodynamicmatrices for the second case study (AGARDwing 4456)
Mathematical Problems in Engineering 9
Table 2 Reduced frequencies for the case study AGARD wing4456
10minus3
210minus3
510minus3
10minus2
510minus2
01 02 03
05 06 08 10
15 20 30 40
Conflict of Interests
The authors declare that the research was conducted in theabsence of any commercial or financial relationships thatcould be construed as a potential conflict of interests
References
[1] H A Bisplingho L Raymond and R L Halfman Aeroelastic-ity Dover 1996
[2] T Theodorsen ldquoGeneral theory of aerodynamic instability andthe mechanism of flutterrdquo Tech Rep 496 NACA NationalAdvisory Committee for Aeronautics 1935
[3] H J Hassig ldquoAn approximate true damping solution of theflutter equation by determinant iterationrdquo Journal of Aircraftvol 8 no 11 pp 885ndash889 1971
[4] P C Chen ldquoDamping perturbation method for flutter solutionthe g-methodrdquo The American Institute of Aeronautics andAstronautics Journal vol 38 no 9 pp 1519ndash1524 2000
[5] R E Kalman Y C Ho and K S Narenda ldquoControllability oflinear dynamics systemrdquo Contribution to Dierential Equationsvol 1 no 2 pp 189ndash213 1962
[6] B C Moore ldquoPrincipal component analysis in linear systemscontrollability observability andmodel reductionrdquo IEEETrans-actions on Automatic Control vol 26 no 1 pp 17ndash32 1981
[7] A Arbel ldquoControllability measures and actuator placement inoscillatory systemsrdquo International Journal of Control vol 33 no3 pp 565ndash574 1981
[8] D Biskri R M Botez N Stathopoulos S Therien A Ratheand M Dickinson ldquoNew mixed method for unsteady aero-dynamic force approximations for aeroservoelasticity studiesrdquoJournal of Aircraft vol 43 no 5 pp 1538ndash1542 2006
[9] G Schulz and G Heimbold ldquoDislocated actuatorsensor posi-tioning and feedback design for flexible structuresrdquo Journal ofGuidance Control andDynamics vol 6 no 5 pp 361ndash367 1983
[10] M L Baker ldquoApproximate subspace iteration for construct-ing internally balanced reduced order models of unsteadyaerodynamic systemsrdquo in Proceedings of the 37th AIAAASMEASCEAHSASC Structures Structural Dynamics and MaterialsConference and Exhibit Salt Lake City Utah USA April 1996Technical Papers Part 2 (A96-26801 06-39)
[11] K Willcox and J Peraire ldquoBalanced model reduction via theproper orthogonal decompositionrdquo The American Institute ofAeronautics and Astronautics Journal vol 40 no 11 pp 2323ndash2330 2002
[12] D J Lucia P S Beran and W A Silva ldquoReduced-ordermodeling new approaches for computational physicsrdquo Progressin Aerospace Sciences vol 40 no 1-2 pp 51ndash117 2004
[13] M J Balas ldquoReduced order modeling for aero-elastic simula-tionrdquoAFOSRFinal ReportAFRL-SR-AR-TR06-0456Air ForceOffice of Scientiffic Research Arlington Va USA 2006
[14] W K Gawronski Dynamics and Control of Structures A ModalApproach Springer New York NY USA 1st edition 1998
[15] K Roger ldquoAirplane math modelling methods for active controldesignrdquo in Structural Aspectsof Control vol 9 of AGARDConference Proceeding pp 41ndash411 1977
[16] M Karpel ldquoDesign for active and passive flutter suppressionand gust alleviationrdquo Tech Rep 3482 National Aeronautics andSpace AdministrationmdashNASA 1981
[17] T Theodorsen and I E Garrick Mechanism of Flutter a The-oretical and Experimental Investigation of the Flutter ProblemNACA National Advisory Committee for Aeronautics 1940
[18] E C Yates AGARD Standard Aeroelastic Congurations forDynamic Response I-Wing 4456 Advisory Group for AerospaceResearch and Development 1988
[19] L A Aguirre ldquoControllability and observability of linearsystems some noninvariant aspectsrdquo IEEE Transactions onEducation vol 38 no 1 pp 33ndash39 1995
[20] L A Aguirre and C Letellier ldquoObservability of multivariatedifferential embeddingsrdquo Journal of Physics A vol 38 no 28pp 6311ndash6326 2005
[21] T Stykel ldquoGramian-based model reduction for descriptorsystemsrdquo Mathematics of Control Signals and Systems vol 16no 4 pp 297ndash319 2004
[22] Y Zhu and P R Pagilla ldquoBounds on the solution of the time-varying linear matrix differential equation (119905) = 119860
119867
(119905)119875(119905) +
119875(119905)119860(119905) + 119876(119905)rdquo IMA Journal of Mathematical Control andInformation vol 23 no 3 pp 269ndash277 2006
[23] K Zhou G Salomon and E Wu ldquoBalanced realization andmodel reduction for unstable systemsrdquo International Journal ofRobust and Nonlinear Control vol 9 no 3 pp 183ndash198 1999
[24] C D M Martin and C F V Loan ldquoSolving real linear systemswith the complex Schur decompositionrdquo SIAM Journal onMatrix Analysis andApplications vol 29 no 1 pp 177ndash183 2007
[25] R Bartels andG Stewart ldquoSolutions of thematrix equation 119886119909+
119909119887 = 119888rdquo Communications of ACM vol 15 no 9 pp 820ndash8261972
[26] P Pahlavanloo ldquoDynamic aeroelastic simulation of theAGARD4456 wing using edgerdquo Tech Rep FOI-R-2259-SE DefenceResearchAgencyDefence and Security SystemandTechnologyStockholm Swedish 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
10 20 30 40 50 60
Fligth envelope point
pk-methodGram EqCross-gram Eq
70 800
002
004
006
008
01
012
014
016
018
02
Com
puta
tiona
l tim
e
Timereduction48
Figure 11 Computational time pk-method versus Gramian param-eters (AGARD 4456 wing)
Using this procedure it is possible to obtain somereduction in the computational time to evaluate the systemstability This reduction is represented in Figure 11
5 Conclusions
Controllability and observability Gramian matrices havebeen used extensively in control design and for optimalplacement of sensors and actuators in smart structures Theyhave also been used for solving aeroelastic problems mainlyfor writing reduced-order models
This paper has investigated the applicability of observ-ability Gramian matrix to detect the flutter speed in twobenchmark structures It was shown that Gramian matricesrepresent the transfer of energy from the flow to the structureThat transfer of energy increases when the airspeed is close tothe flutter condition A classical method was used to compareand validate the results obtained by Gramian matrices
This approach does not compute frequency and dampingfor the aeroelastic modes However it allows to determinethe aeroelastic modes with largest contribution to the fluttermechanism and the airspeed where the system becomesunstable Additionally one advantage is related to the reduc-tion of computational time for analysis when compared toclassical pk-method
This work is a complementary effort to apply the conceptsfrom control theory in problems involving aeroelasticityin time domain It can also be used in flight flutter testsusing an identification method to obtain the state-spacerepresentation instead of identifying aeroelastic frequenciesand damping
Appendices
A State-Space Represenation ofan Aeroelastic System
An aeroelastic system consisting of structural parameters(mass damping and stiffness) subject to the forces fromthe fluid can be modeled in the Laplace domain using aphysical or modal coordinate system In general the modalcoordinates are used to truncate the systemof equations usingthe eigenvector extracted from structural mass and stiffnessmatrices Without loss of generality both numerical casestudies were performed using modal coordinates as shown inthe following equations
The matrices representing the structural parameters arethe mass M the viscous damping D and the stiffness K Inmost cases these matrices are obtained from finite elementmethod and are reduced to modal coordinates (representedby the subscript119898) the system displacement vector in modalcoordinates is u
119898 The aerodynamic influence matrix Q
depends on the parameters 119896 (reduced frequency) and 119898119872
(Mach Number) and 119902 is the dynamic pressure caused by theflowing fluid (Q can be obtained by methods such as DoubletLattice) Note that the proposed approach is not restricted tothis aerodynamic theory
1199042M119898u119898
(119904) + 119904D119898u119898
(119904) + K119898u119898
(119904) = 119902Q119898
(119898119872 119896) u119898
(119904)
(A1)
where 119904 is the Laplace variableIn this case the problem that arises from the conversion
of (A1) to time domain is the fact that Q has no Laplaceinverse This is overcome by writing a rational function torepresent the aerodynamic coefficients in time domain Inthis work Rogerrsquos approximation [15] is used containing apolynomial part representing the forces acting directly relatedto the displacements u(119905) and their first and second deriva-tives Also this equation has a rational part representing theinfluence of the wake acting on the structure with a timedelay
Q119898
(119904) = [
[
2
sum
119895=0
Q119898119895
119904119895
(119887
119881)
119895
+
119899lag
sum
119895=1
Q119898(119895+2)
(119904
119904 + (119887119881) 120573119895
)]
]
u119898
(119904)
(A2)
where 119899lag is the number of lag terms and 120573119895is the 119895th
lag parameter (119895 = 1 119899lag) The parameters 120573119895were
chosen arbitrarily to ensure equality of (A2) for each reducedfrequency and then their values are different for each systempresented herein The procedure to obtain the aerodynamicmatrices is discussed in [15]
Equation (A2) is substituted into (A1) allowing to writethe aeroelastic system in time domain using a state-spacerepresentation In this case the state vector presented in (1) isx(119905) = u
119898u119898
u119886119898
119879 where u
119886119898are states of lags required
for the approximation The dynamic matrix A is given by
8 Mathematical Problems in Engineering
A =
[[[[[[[[[[
[
minusMminus1119886119898D119886119898
minusMminus1119886119898K119886119898
119902Mminus1119886119898Q1198983
sdot sdot sdot 119902Mminus1119886119898Q119898(2+119899lag)
I 0 0 sdot sdot sdot 0I 0 (minus
119881
119887)1205731I 0 sdot sdot sdot
0 d sdot sdot sdot
I 0 sdot sdot sdot (minus
119881
119887)120573119899lag
I
]]]]]]]]]]
]
(A3)
where the matrices M119886119898 D119886119898 and K
119886119898are comprised of
terms containing structural and aerodynamic coefficientswhich are given by
M119886119898
= M119898
minus 119902(119887
119881)
2
Q1198982
D119886119898
= D119898
minus 119902(119887
119881)Q1198981
K119886119898
= K119898
minus 119902Q1198980
(A4)
B Norms for Computing theGramian Parameter
Thematrix norms for computing theGramian parameters arepresented as follows
B1infin-Norm Theinfin-normof W119900(119894119895) is themaximumof the
row sums that is
120590(infin)
119892(119894119895) = max
119894119903
10038161003816100381610038161003816100381610038161003816100381610038161003816
sum
119895119888
119908119900(119894119903 119895119888)
10038161003816100381610038161003816100381610038161003816100381610038161003816
(B1)
B2 2-Norm The 2-norm is computed based on the largestsingular value 120582
119894119903of [W119867
119900(119894119895)W
119900(119894119895)] 119894
119903= 1 119898(2 + 119899lag)
that is
120590(2119873)
119892(119894119895) = (max
119894119903
120582119894119903)
12
(B2)
whereW119867119900(119894119895) denotes a Hermitian matrix
B3 1-Norm The 1-norm is the maximum of the columnssums that is
120590(1119873)
119892(119894119895) = max
119895119888
10038161003816100381610038161003816100381610038161003816100381610038161003816
sum
119894119903
119908119900(119894119903 119895119888)
10038161003816100381610038161003816100381610038161003816100381610038161003816
(B3)
C AGARD 4456 Wind Structural Model
This appendix presents complementary information for theAGARD4456wind andmore details can be found in [18 26]
Figures 12 13 14 and 15 show the four structuralmodes considered in the finite element model obtained fromMSCNASTRAN program
Figure 12 AGARD 4456 wing-first structural mode
Figure 13 AGARD 4456 wing-second structural mode
Figure 14 AGARD 4456 wing-third structural mode
Figure 15 AGARD 4456 wing-fourth structural mode
The thickness distribution is governed by the airfoilshape The material properties used are 119864
1= 31511 119864
2=
04162GPa ] = 031 119866 = 04392GPa and 120588mat =
38198 kgm3 where 1198641and 119864
2are the moduli of elasticity in
the longitudinal and lateral directions ] is Poissonrsquos ratio 119866is the shear modulus in each plane and 120588 is the wing density
There are 10 elements in chordwise and 20 elementsin spanwise direction There are a total of 231 nodes and200 elements The boundary conditions of the wing areselected in accordance with the physical model The root iscantilevered except for the nodes at the leading and trailingedges Additionally the rotation around all axis degrees offreedom at all nodes is constrained to zero In this casethere are 666 degrees of freedom in the model in physicalcoordinates This work considered four modes to obtain themodel in modal coordinates system
Table 2 shows the reduced frequencies 119896 used to computethe aerodynamicmatrices for the second case study (AGARDwing 4456)
Mathematical Problems in Engineering 9
Table 2 Reduced frequencies for the case study AGARD wing4456
10minus3
210minus3
510minus3
10minus2
510minus2
01 02 03
05 06 08 10
15 20 30 40
Conflict of Interests
The authors declare that the research was conducted in theabsence of any commercial or financial relationships thatcould be construed as a potential conflict of interests
References
[1] H A Bisplingho L Raymond and R L Halfman Aeroelastic-ity Dover 1996
[2] T Theodorsen ldquoGeneral theory of aerodynamic instability andthe mechanism of flutterrdquo Tech Rep 496 NACA NationalAdvisory Committee for Aeronautics 1935
[3] H J Hassig ldquoAn approximate true damping solution of theflutter equation by determinant iterationrdquo Journal of Aircraftvol 8 no 11 pp 885ndash889 1971
[4] P C Chen ldquoDamping perturbation method for flutter solutionthe g-methodrdquo The American Institute of Aeronautics andAstronautics Journal vol 38 no 9 pp 1519ndash1524 2000
[5] R E Kalman Y C Ho and K S Narenda ldquoControllability oflinear dynamics systemrdquo Contribution to Dierential Equationsvol 1 no 2 pp 189ndash213 1962
[6] B C Moore ldquoPrincipal component analysis in linear systemscontrollability observability andmodel reductionrdquo IEEETrans-actions on Automatic Control vol 26 no 1 pp 17ndash32 1981
[7] A Arbel ldquoControllability measures and actuator placement inoscillatory systemsrdquo International Journal of Control vol 33 no3 pp 565ndash574 1981
[8] D Biskri R M Botez N Stathopoulos S Therien A Ratheand M Dickinson ldquoNew mixed method for unsteady aero-dynamic force approximations for aeroservoelasticity studiesrdquoJournal of Aircraft vol 43 no 5 pp 1538ndash1542 2006
[9] G Schulz and G Heimbold ldquoDislocated actuatorsensor posi-tioning and feedback design for flexible structuresrdquo Journal ofGuidance Control andDynamics vol 6 no 5 pp 361ndash367 1983
[10] M L Baker ldquoApproximate subspace iteration for construct-ing internally balanced reduced order models of unsteadyaerodynamic systemsrdquo in Proceedings of the 37th AIAAASMEASCEAHSASC Structures Structural Dynamics and MaterialsConference and Exhibit Salt Lake City Utah USA April 1996Technical Papers Part 2 (A96-26801 06-39)
[11] K Willcox and J Peraire ldquoBalanced model reduction via theproper orthogonal decompositionrdquo The American Institute ofAeronautics and Astronautics Journal vol 40 no 11 pp 2323ndash2330 2002
[12] D J Lucia P S Beran and W A Silva ldquoReduced-ordermodeling new approaches for computational physicsrdquo Progressin Aerospace Sciences vol 40 no 1-2 pp 51ndash117 2004
[13] M J Balas ldquoReduced order modeling for aero-elastic simula-tionrdquoAFOSRFinal ReportAFRL-SR-AR-TR06-0456Air ForceOffice of Scientiffic Research Arlington Va USA 2006
[14] W K Gawronski Dynamics and Control of Structures A ModalApproach Springer New York NY USA 1st edition 1998
[15] K Roger ldquoAirplane math modelling methods for active controldesignrdquo in Structural Aspectsof Control vol 9 of AGARDConference Proceeding pp 41ndash411 1977
[16] M Karpel ldquoDesign for active and passive flutter suppressionand gust alleviationrdquo Tech Rep 3482 National Aeronautics andSpace AdministrationmdashNASA 1981
[17] T Theodorsen and I E Garrick Mechanism of Flutter a The-oretical and Experimental Investigation of the Flutter ProblemNACA National Advisory Committee for Aeronautics 1940
[18] E C Yates AGARD Standard Aeroelastic Congurations forDynamic Response I-Wing 4456 Advisory Group for AerospaceResearch and Development 1988
[19] L A Aguirre ldquoControllability and observability of linearsystems some noninvariant aspectsrdquo IEEE Transactions onEducation vol 38 no 1 pp 33ndash39 1995
[20] L A Aguirre and C Letellier ldquoObservability of multivariatedifferential embeddingsrdquo Journal of Physics A vol 38 no 28pp 6311ndash6326 2005
[21] T Stykel ldquoGramian-based model reduction for descriptorsystemsrdquo Mathematics of Control Signals and Systems vol 16no 4 pp 297ndash319 2004
[22] Y Zhu and P R Pagilla ldquoBounds on the solution of the time-varying linear matrix differential equation (119905) = 119860
119867
(119905)119875(119905) +
119875(119905)119860(119905) + 119876(119905)rdquo IMA Journal of Mathematical Control andInformation vol 23 no 3 pp 269ndash277 2006
[23] K Zhou G Salomon and E Wu ldquoBalanced realization andmodel reduction for unstable systemsrdquo International Journal ofRobust and Nonlinear Control vol 9 no 3 pp 183ndash198 1999
[24] C D M Martin and C F V Loan ldquoSolving real linear systemswith the complex Schur decompositionrdquo SIAM Journal onMatrix Analysis andApplications vol 29 no 1 pp 177ndash183 2007
[25] R Bartels andG Stewart ldquoSolutions of thematrix equation 119886119909+
119909119887 = 119888rdquo Communications of ACM vol 15 no 9 pp 820ndash8261972
[26] P Pahlavanloo ldquoDynamic aeroelastic simulation of theAGARD4456 wing using edgerdquo Tech Rep FOI-R-2259-SE DefenceResearchAgencyDefence and Security SystemandTechnologyStockholm Swedish 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
A =
[[[[[[[[[[
[
minusMminus1119886119898D119886119898
minusMminus1119886119898K119886119898
119902Mminus1119886119898Q1198983
sdot sdot sdot 119902Mminus1119886119898Q119898(2+119899lag)
I 0 0 sdot sdot sdot 0I 0 (minus
119881
119887)1205731I 0 sdot sdot sdot
0 d sdot sdot sdot
I 0 sdot sdot sdot (minus
119881
119887)120573119899lag
I
]]]]]]]]]]
]
(A3)
where the matrices M119886119898 D119886119898 and K
119886119898are comprised of
terms containing structural and aerodynamic coefficientswhich are given by
M119886119898
= M119898
minus 119902(119887
119881)
2
Q1198982
D119886119898
= D119898
minus 119902(119887
119881)Q1198981
K119886119898
= K119898
minus 119902Q1198980
(A4)
B Norms for Computing theGramian Parameter
Thematrix norms for computing theGramian parameters arepresented as follows
B1infin-Norm Theinfin-normof W119900(119894119895) is themaximumof the
row sums that is
120590(infin)
119892(119894119895) = max
119894119903
10038161003816100381610038161003816100381610038161003816100381610038161003816
sum
119895119888
119908119900(119894119903 119895119888)
10038161003816100381610038161003816100381610038161003816100381610038161003816
(B1)
B2 2-Norm The 2-norm is computed based on the largestsingular value 120582
119894119903of [W119867
119900(119894119895)W
119900(119894119895)] 119894
119903= 1 119898(2 + 119899lag)
that is
120590(2119873)
119892(119894119895) = (max
119894119903
120582119894119903)
12
(B2)
whereW119867119900(119894119895) denotes a Hermitian matrix
B3 1-Norm The 1-norm is the maximum of the columnssums that is
120590(1119873)
119892(119894119895) = max
119895119888
10038161003816100381610038161003816100381610038161003816100381610038161003816
sum
119894119903
119908119900(119894119903 119895119888)
10038161003816100381610038161003816100381610038161003816100381610038161003816
(B3)
C AGARD 4456 Wind Structural Model
This appendix presents complementary information for theAGARD4456wind andmore details can be found in [18 26]
Figures 12 13 14 and 15 show the four structuralmodes considered in the finite element model obtained fromMSCNASTRAN program
Figure 12 AGARD 4456 wing-first structural mode
Figure 13 AGARD 4456 wing-second structural mode
Figure 14 AGARD 4456 wing-third structural mode
Figure 15 AGARD 4456 wing-fourth structural mode
The thickness distribution is governed by the airfoilshape The material properties used are 119864
1= 31511 119864
2=
04162GPa ] = 031 119866 = 04392GPa and 120588mat =
38198 kgm3 where 1198641and 119864
2are the moduli of elasticity in
the longitudinal and lateral directions ] is Poissonrsquos ratio 119866is the shear modulus in each plane and 120588 is the wing density
There are 10 elements in chordwise and 20 elementsin spanwise direction There are a total of 231 nodes and200 elements The boundary conditions of the wing areselected in accordance with the physical model The root iscantilevered except for the nodes at the leading and trailingedges Additionally the rotation around all axis degrees offreedom at all nodes is constrained to zero In this casethere are 666 degrees of freedom in the model in physicalcoordinates This work considered four modes to obtain themodel in modal coordinates system
Table 2 shows the reduced frequencies 119896 used to computethe aerodynamicmatrices for the second case study (AGARDwing 4456)
Mathematical Problems in Engineering 9
Table 2 Reduced frequencies for the case study AGARD wing4456
10minus3
210minus3
510minus3
10minus2
510minus2
01 02 03
05 06 08 10
15 20 30 40
Conflict of Interests
The authors declare that the research was conducted in theabsence of any commercial or financial relationships thatcould be construed as a potential conflict of interests
References
[1] H A Bisplingho L Raymond and R L Halfman Aeroelastic-ity Dover 1996
[2] T Theodorsen ldquoGeneral theory of aerodynamic instability andthe mechanism of flutterrdquo Tech Rep 496 NACA NationalAdvisory Committee for Aeronautics 1935
[3] H J Hassig ldquoAn approximate true damping solution of theflutter equation by determinant iterationrdquo Journal of Aircraftvol 8 no 11 pp 885ndash889 1971
[4] P C Chen ldquoDamping perturbation method for flutter solutionthe g-methodrdquo The American Institute of Aeronautics andAstronautics Journal vol 38 no 9 pp 1519ndash1524 2000
[5] R E Kalman Y C Ho and K S Narenda ldquoControllability oflinear dynamics systemrdquo Contribution to Dierential Equationsvol 1 no 2 pp 189ndash213 1962
[6] B C Moore ldquoPrincipal component analysis in linear systemscontrollability observability andmodel reductionrdquo IEEETrans-actions on Automatic Control vol 26 no 1 pp 17ndash32 1981
[7] A Arbel ldquoControllability measures and actuator placement inoscillatory systemsrdquo International Journal of Control vol 33 no3 pp 565ndash574 1981
[8] D Biskri R M Botez N Stathopoulos S Therien A Ratheand M Dickinson ldquoNew mixed method for unsteady aero-dynamic force approximations for aeroservoelasticity studiesrdquoJournal of Aircraft vol 43 no 5 pp 1538ndash1542 2006
[9] G Schulz and G Heimbold ldquoDislocated actuatorsensor posi-tioning and feedback design for flexible structuresrdquo Journal ofGuidance Control andDynamics vol 6 no 5 pp 361ndash367 1983
[10] M L Baker ldquoApproximate subspace iteration for construct-ing internally balanced reduced order models of unsteadyaerodynamic systemsrdquo in Proceedings of the 37th AIAAASMEASCEAHSASC Structures Structural Dynamics and MaterialsConference and Exhibit Salt Lake City Utah USA April 1996Technical Papers Part 2 (A96-26801 06-39)
[11] K Willcox and J Peraire ldquoBalanced model reduction via theproper orthogonal decompositionrdquo The American Institute ofAeronautics and Astronautics Journal vol 40 no 11 pp 2323ndash2330 2002
[12] D J Lucia P S Beran and W A Silva ldquoReduced-ordermodeling new approaches for computational physicsrdquo Progressin Aerospace Sciences vol 40 no 1-2 pp 51ndash117 2004
[13] M J Balas ldquoReduced order modeling for aero-elastic simula-tionrdquoAFOSRFinal ReportAFRL-SR-AR-TR06-0456Air ForceOffice of Scientiffic Research Arlington Va USA 2006
[14] W K Gawronski Dynamics and Control of Structures A ModalApproach Springer New York NY USA 1st edition 1998
[15] K Roger ldquoAirplane math modelling methods for active controldesignrdquo in Structural Aspectsof Control vol 9 of AGARDConference Proceeding pp 41ndash411 1977
[16] M Karpel ldquoDesign for active and passive flutter suppressionand gust alleviationrdquo Tech Rep 3482 National Aeronautics andSpace AdministrationmdashNASA 1981
[17] T Theodorsen and I E Garrick Mechanism of Flutter a The-oretical and Experimental Investigation of the Flutter ProblemNACA National Advisory Committee for Aeronautics 1940
[18] E C Yates AGARD Standard Aeroelastic Congurations forDynamic Response I-Wing 4456 Advisory Group for AerospaceResearch and Development 1988
[19] L A Aguirre ldquoControllability and observability of linearsystems some noninvariant aspectsrdquo IEEE Transactions onEducation vol 38 no 1 pp 33ndash39 1995
[20] L A Aguirre and C Letellier ldquoObservability of multivariatedifferential embeddingsrdquo Journal of Physics A vol 38 no 28pp 6311ndash6326 2005
[21] T Stykel ldquoGramian-based model reduction for descriptorsystemsrdquo Mathematics of Control Signals and Systems vol 16no 4 pp 297ndash319 2004
[22] Y Zhu and P R Pagilla ldquoBounds on the solution of the time-varying linear matrix differential equation (119905) = 119860
119867
(119905)119875(119905) +
119875(119905)119860(119905) + 119876(119905)rdquo IMA Journal of Mathematical Control andInformation vol 23 no 3 pp 269ndash277 2006
[23] K Zhou G Salomon and E Wu ldquoBalanced realization andmodel reduction for unstable systemsrdquo International Journal ofRobust and Nonlinear Control vol 9 no 3 pp 183ndash198 1999
[24] C D M Martin and C F V Loan ldquoSolving real linear systemswith the complex Schur decompositionrdquo SIAM Journal onMatrix Analysis andApplications vol 29 no 1 pp 177ndash183 2007
[25] R Bartels andG Stewart ldquoSolutions of thematrix equation 119886119909+
119909119887 = 119888rdquo Communications of ACM vol 15 no 9 pp 820ndash8261972
[26] P Pahlavanloo ldquoDynamic aeroelastic simulation of theAGARD4456 wing using edgerdquo Tech Rep FOI-R-2259-SE DefenceResearchAgencyDefence and Security SystemandTechnologyStockholm Swedish 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
Table 2 Reduced frequencies for the case study AGARD wing4456
10minus3
210minus3
510minus3
10minus2
510minus2
01 02 03
05 06 08 10
15 20 30 40
Conflict of Interests
The authors declare that the research was conducted in theabsence of any commercial or financial relationships thatcould be construed as a potential conflict of interests
References
[1] H A Bisplingho L Raymond and R L Halfman Aeroelastic-ity Dover 1996
[2] T Theodorsen ldquoGeneral theory of aerodynamic instability andthe mechanism of flutterrdquo Tech Rep 496 NACA NationalAdvisory Committee for Aeronautics 1935
[3] H J Hassig ldquoAn approximate true damping solution of theflutter equation by determinant iterationrdquo Journal of Aircraftvol 8 no 11 pp 885ndash889 1971
[4] P C Chen ldquoDamping perturbation method for flutter solutionthe g-methodrdquo The American Institute of Aeronautics andAstronautics Journal vol 38 no 9 pp 1519ndash1524 2000
[5] R E Kalman Y C Ho and K S Narenda ldquoControllability oflinear dynamics systemrdquo Contribution to Dierential Equationsvol 1 no 2 pp 189ndash213 1962
[6] B C Moore ldquoPrincipal component analysis in linear systemscontrollability observability andmodel reductionrdquo IEEETrans-actions on Automatic Control vol 26 no 1 pp 17ndash32 1981
[7] A Arbel ldquoControllability measures and actuator placement inoscillatory systemsrdquo International Journal of Control vol 33 no3 pp 565ndash574 1981
[8] D Biskri R M Botez N Stathopoulos S Therien A Ratheand M Dickinson ldquoNew mixed method for unsteady aero-dynamic force approximations for aeroservoelasticity studiesrdquoJournal of Aircraft vol 43 no 5 pp 1538ndash1542 2006
[9] G Schulz and G Heimbold ldquoDislocated actuatorsensor posi-tioning and feedback design for flexible structuresrdquo Journal ofGuidance Control andDynamics vol 6 no 5 pp 361ndash367 1983
[10] M L Baker ldquoApproximate subspace iteration for construct-ing internally balanced reduced order models of unsteadyaerodynamic systemsrdquo in Proceedings of the 37th AIAAASMEASCEAHSASC Structures Structural Dynamics and MaterialsConference and Exhibit Salt Lake City Utah USA April 1996Technical Papers Part 2 (A96-26801 06-39)
[11] K Willcox and J Peraire ldquoBalanced model reduction via theproper orthogonal decompositionrdquo The American Institute ofAeronautics and Astronautics Journal vol 40 no 11 pp 2323ndash2330 2002
[12] D J Lucia P S Beran and W A Silva ldquoReduced-ordermodeling new approaches for computational physicsrdquo Progressin Aerospace Sciences vol 40 no 1-2 pp 51ndash117 2004
[13] M J Balas ldquoReduced order modeling for aero-elastic simula-tionrdquoAFOSRFinal ReportAFRL-SR-AR-TR06-0456Air ForceOffice of Scientiffic Research Arlington Va USA 2006
[14] W K Gawronski Dynamics and Control of Structures A ModalApproach Springer New York NY USA 1st edition 1998
[15] K Roger ldquoAirplane math modelling methods for active controldesignrdquo in Structural Aspectsof Control vol 9 of AGARDConference Proceeding pp 41ndash411 1977
[16] M Karpel ldquoDesign for active and passive flutter suppressionand gust alleviationrdquo Tech Rep 3482 National Aeronautics andSpace AdministrationmdashNASA 1981
[17] T Theodorsen and I E Garrick Mechanism of Flutter a The-oretical and Experimental Investigation of the Flutter ProblemNACA National Advisory Committee for Aeronautics 1940
[18] E C Yates AGARD Standard Aeroelastic Congurations forDynamic Response I-Wing 4456 Advisory Group for AerospaceResearch and Development 1988
[19] L A Aguirre ldquoControllability and observability of linearsystems some noninvariant aspectsrdquo IEEE Transactions onEducation vol 38 no 1 pp 33ndash39 1995
[20] L A Aguirre and C Letellier ldquoObservability of multivariatedifferential embeddingsrdquo Journal of Physics A vol 38 no 28pp 6311ndash6326 2005
[21] T Stykel ldquoGramian-based model reduction for descriptorsystemsrdquo Mathematics of Control Signals and Systems vol 16no 4 pp 297ndash319 2004
[22] Y Zhu and P R Pagilla ldquoBounds on the solution of the time-varying linear matrix differential equation (119905) = 119860
119867
(119905)119875(119905) +
119875(119905)119860(119905) + 119876(119905)rdquo IMA Journal of Mathematical Control andInformation vol 23 no 3 pp 269ndash277 2006
[23] K Zhou G Salomon and E Wu ldquoBalanced realization andmodel reduction for unstable systemsrdquo International Journal ofRobust and Nonlinear Control vol 9 no 3 pp 183ndash198 1999
[24] C D M Martin and C F V Loan ldquoSolving real linear systemswith the complex Schur decompositionrdquo SIAM Journal onMatrix Analysis andApplications vol 29 no 1 pp 177ndash183 2007
[25] R Bartels andG Stewart ldquoSolutions of thematrix equation 119886119909+
119909119887 = 119888rdquo Communications of ACM vol 15 no 9 pp 820ndash8261972
[26] P Pahlavanloo ldquoDynamic aeroelastic simulation of theAGARD4456 wing using edgerdquo Tech Rep FOI-R-2259-SE DefenceResearchAgencyDefence and Security SystemandTechnologyStockholm Swedish 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of