research article the use of gramian matrices for

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)


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


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


Plunge mode

Airspeed (ms)


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


Grammian parametermdashtypical section airfoil

Plunge mode400

200120590119892

2 4 6 8 10 12 140

Airspeed (ms)

(a)


120590119892

2 4 6 8 10 12 14

40

20

0

Airspeed (ms)

Pitch mode

(b)


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


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


(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


(a)

02 04 06 08 1 12

Air density (kgm3)

Fourth mode120590119892

08

06

04

02

1

0


(b)


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


10 20 30 40 50 60



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


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)


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


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


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)






X119892F119892+ F119879119892X119892minus X119892G119892G119879119892X119892= 0 (8)


119866


(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)


y (119905) = Cx (119905)

y (119905) = 0 sdot sdot sdot 0 119906119898(119894)

(119905) 0 sdot sdot sdot 0119879


(119898+(119894minus1))1(119894)

0 sdot sdot sdot 0]

(11)


119894(119905) = 119906

119898(119894)(119905) and




119900(C119894 119881119895) = W

119900(119894 119895) is



1198861198980119898(1+119899lag)times119898



119900(119894119895)is obtained

[24]

[I otimes (F119892+ G119892M119892) + (F119879

119892+ M119879119892G119879119892) otimes I]vec(W

119866)

= vec(minusG119892G119879119892)

(12)



Stable Unstable

Mode 1

Mode 2


Gra

mm

ian

para

met

er

119881119865






119895andC




largest value of 120590119892(119894 119895)

max (Σ) = max119894119895

[120590119892] = 120590

max119892


119889119881= 0

(13)


Σ =

[[[[[

[

120590119892(11)

sdot sdot sdot 120590119892(1119895)

sdot sdot sdot 120590119892(1119899V)


120590119892(1198941)

sdot sdot sdot 120590119892(119894119895)

sdot sdot sdot 120590119892(119894119899V)


120590119892(1198981)

sdot sdot sdot 120590119892(119898119895)

sdot sdot sdot 120590119892(119898119899V)

]]]]]

]

(14)


max119892


119892in









ℎ= 30Hz



120573= 12Hz



1= 02 120573

2= 12 120573

3= 16 120573

4= 18



Figure 2 119888 = 060



120573= 00125


119903120573= (625 times 10

minus3

)12


119903120579= radic025


ce

ca+120579119896120579

119896ℎ

+ℎ

119886

119887 119887

cg

119888

cgflap

minus120573

119903120573

119909120579

119903120579

119896120573

119909120573



120590(119865)

119892(119894 119895) = (sum

119894119903 119895119888

1003816100381610038161003816119908119900 (119894119903 119895119888)1003816100381610038161003816)

12

(15)


119903th row

and 119895119888th column and 119894

119903 119895119888= 1 119898(2 + 119899lag)



Pitch mode


Plunge mode

0 5 10 15

16

14

12

10

8

6

4

2

0

Freq

uenc

y (H

z)

Airspeed (ms)



015

01

005

0

minus005

minus01

minus015

minus02

minus025

minus03

minus035

Dam

ping

ratio

0 5 10 15

Pitch mode

127ms


Plunge mode

Airspeed (ms)







Plunge mode400

200120590119892

2 4 6 8 10 12 140

Airspeed (ms)

(a)


120590119892

2 4 6 8 10 12 14

40

20

0

Airspeed (ms)

Pitch mode

(b)


120590119892

2 4 6 8 10 12 14


2

0

Airspeed (ms)

(c)


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)






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




1= 945Hz 119891

2=

3969Hz 1198913

= 4945Hz and 1198914



= 055 1205732

= 140 1205733

= 190 and 1205734








20

15

10

5

002 04 06 08 1 12

Air density (kgm3)


120590119892

Airspeed 15802


(a)

02 04 06 08 1 12

Air density (kgm3)

Second mode

120590119892

3

2

1

0


(b)


02 04 06 08 1 12

Air density (kgm3)

Third mode

120590119892

08

06

04

02

1

0


(a)

02 04 06 08 1 12

Air density (kgm3)


08

06

04

02

1

0


(b)


06

05

04

03

02

01

010 20 30 40 50 60



Tim

e sum











around the maximum


10 20 30 40 50 60



70 800

002

004

006

008

01

012

014

016

018

02

Com

puta

tiona

l tim

e

Timereduction48



5 Conclusions





Appendices







1199042M119898u119898

(119904) + 119904D119898u119898

(119904) + K119898u119898

(119904) = 119902Q119898

(119898119872 119896) u119898

(119904)

(A1)



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)





119898u119898

u119886119898

119879 where u




A =

[[[[[[[[[[

[

minusMminus1119886119898D119886119898

minusMminus1119886119898K119886119898

119902Mminus1119886119898Q1198983



119881


0 d sdot sdot sdot


119881

119887)120573119899lag

I

]]]]]]]]]]

]

(A3)




M119886119898

= M119898

minus 119902(119887

119881)

2

Q1198982

D119886119898

= D119898

minus 119902(119887

119881)Q1198981

K119886119898

= K119898

minus 119902Q1198980

(A4)




row sums that is

120590(infin)

119892(119894119895) = max

119894119903

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119895119888

119908119900(119894119903 119895119888)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(B1)


119894119903of [W119867

119900(119894119895)W

119900(119894119895)] 119894

119903= 1 119898(2 + 119899lag)

that is

120590(2119873)

119892(119894119895) = (max

119894119903

120582119894119903)

12

(B2)



120590(1119873)

119892(119894119895) = max

119895119888

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119894119903

119908119900(119894119903 119895119888)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(B3)









1= 31511 119864

2=

04162GPa ] = 031 119866 = 04392GPa and 120588mat =

38198 kgm3 where 1198641and 119864







10minus3

210minus3

510minus3

10minus2

510minus2

01 02 03

05 06 08 10

15 20 30 40



References























119867

(119905)119875(119905) +














Volume 2014




Journal of











Journal of


Function Spaces






Algebra














X119892F119892+ F119879119892X119892minus X119892G119892G119879119892X119892= 0 (8)


119866


(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)


y (119905) = Cx (119905)

y (119905) = 0 sdot sdot sdot 0 119906119898(119894)

(119905) 0 sdot sdot sdot 0119879


(119898+(119894minus1))1(119894)

0 sdot sdot sdot 0]

(11)


119894(119905) = 119906

119898(119894)(119905) and




119900(C119894 119881119895) = W

119900(119894 119895) is



1198861198980119898(1+119899lag)times119898



119900(119894119895)is obtained

[24]

[I otimes (F119892+ G119892M119892) + (F119879

119892+ M119879119892G119879119892) otimes I]vec(W

119866)

= vec(minusG119892G119879119892)

(12)



Stable Unstable

Mode 1

Mode 2


Gra

mm

ian

para

met

er

119881119865






119895andC




largest value of 120590119892(119894 119895)

max (Σ) = max119894119895

[120590119892] = 120590

max119892


119889119881= 0

(13)


Σ =

[[[[[

[

120590119892(11)

sdot sdot sdot 120590119892(1119895)

sdot sdot sdot 120590119892(1119899V)


120590119892(1198941)

sdot sdot sdot 120590119892(119894119895)

sdot sdot sdot 120590119892(119894119899V)


120590119892(1198981)

sdot sdot sdot 120590119892(119898119895)

sdot sdot sdot 120590119892(119898119899V)

]]]]]

]

(14)


max119892


119892in









ℎ= 30Hz



120573= 12Hz



1= 02 120573

2= 12 120573

3= 16 120573

4= 18



Figure 2 119888 = 060



120573= 00125


119903120573= (625 times 10

minus3

)12


119903120579= radic025


ce

ca+120579119896120579

119896ℎ

+ℎ

119886

119887 119887

cg

119888

cgflap

minus120573

119903120573

119909120579

119903120579

119896120573

119909120573



120590(119865)

119892(119894 119895) = (sum

119894119903 119895119888

1003816100381610038161003816119908119900 (119894119903 119895119888)1003816100381610038161003816)

12

(15)


119903th row

and 119895119888th column and 119894

119903 119895119888= 1 119898(2 + 119899lag)



Pitch mode


Plunge mode

0 5 10 15

16

14

12

10

8

6

4

2

0

Freq

uenc

y (H

z)

Airspeed (ms)



015

01

005

0

minus005

minus01

minus015

minus02

minus025

minus03

minus035

Dam

ping

ratio

0 5 10 15

Pitch mode

127ms


Plunge mode

Airspeed (ms)







Plunge mode400

200120590119892

2 4 6 8 10 12 140

Airspeed (ms)

(a)


120590119892

2 4 6 8 10 12 14

40

20

0

Airspeed (ms)

Pitch mode

(b)


120590119892

2 4 6 8 10 12 14


2

0

Airspeed (ms)

(c)


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)






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




1= 945Hz 119891

2=

3969Hz 1198913

= 4945Hz and 1198914



= 055 1205732

= 140 1205733

= 190 and 1205734








20

15

10

5

002 04 06 08 1 12

Air density (kgm3)


120590119892

Airspeed 15802


(a)

02 04 06 08 1 12

Air density (kgm3)

Second mode

120590119892

3

2

1

0


(b)


02 04 06 08 1 12

Air density (kgm3)

Third mode

120590119892

08

06

04

02

1

0


(a)

02 04 06 08 1 12

Air density (kgm3)


08

06

04

02

1

0


(b)


06

05

04

03

02

01

010 20 30 40 50 60



Tim

e sum











around the maximum


10 20 30 40 50 60



70 800

002

004

006

008

01

012

014

016

018

02

Com

puta

tiona

l tim

e

Timereduction48



5 Conclusions





Appendices







1199042M119898u119898

(119904) + 119904D119898u119898

(119904) + K119898u119898

(119904) = 119902Q119898

(119898119872 119896) u119898

(119904)

(A1)



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)





119898u119898

u119886119898

119879 where u




A =

[[[[[[[[[[

[

minusMminus1119886119898D119886119898

minusMminus1119886119898K119886119898

119902Mminus1119886119898Q1198983



119881


0 d sdot sdot sdot


119881

119887)120573119899lag

I

]]]]]]]]]]

]

(A3)




M119886119898

= M119898

minus 119902(119887

119881)

2

Q1198982

D119886119898

= D119898

minus 119902(119887

119881)Q1198981

K119886119898

= K119898

minus 119902Q1198980

(A4)




row sums that is

120590(infin)

119892(119894119895) = max

119894119903

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119895119888

119908119900(119894119903 119895119888)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(B1)


119894119903of [W119867

119900(119894119895)W

119900(119894119895)] 119894

119903= 1 119898(2 + 119899lag)

that is

120590(2119873)

119892(119894119895) = (max

119894119903

120582119894119903)

12

(B2)



120590(1119873)

119892(119894119895) = max

119895119888

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119894119903

119908119900(119894119903 119895119888)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(B3)









1= 31511 119864

2=

04162GPa ] = 031 119866 = 04392GPa and 120588mat =

38198 kgm3 where 1198641and 119864







10minus3

210minus3

510minus3

10minus2

510minus2

01 02 03

05 06 08 10

15 20 30 40



References























119867

(119905)119875(119905) +














Volume 2014




Journal of











Journal of


Function Spaces






Algebra












ℎ= 30Hz



120573= 12Hz



1= 02 120573

2= 12 120573

3= 16 120573

4= 18



Figure 2 119888 = 060



120573= 00125


119903120573= (625 times 10

minus3

)12


119903120579= radic025


ce

ca+120579119896120579

119896ℎ

+ℎ

119886

119887 119887

cg

119888

cgflap

minus120573

119903120573

119909120579

119903120579

119896120573

119909120573



120590(119865)

119892(119894 119895) = (sum

119894119903 119895119888

1003816100381610038161003816119908119900 (119894119903 119895119888)1003816100381610038161003816)

12

(15)


119903th row

and 119895119888th column and 119894

119903 119895119888= 1 119898(2 + 119899lag)



Pitch mode


Plunge mode

0 5 10 15

16

14

12

10

8

6

4

2

0

Freq

uenc

y (H

z)

Airspeed (ms)



015

01

005

0

minus005

minus01

minus015

minus02

minus025

minus03

minus035

Dam

ping

ratio

0 5 10 15

Pitch mode

127ms


Plunge mode

Airspeed (ms)







Plunge mode400

200120590119892

2 4 6 8 10 12 140

Airspeed (ms)

(a)


120590119892

2 4 6 8 10 12 14

40

20

0

Airspeed (ms)

Pitch mode

(b)


120590119892

2 4 6 8 10 12 14


2

0

Airspeed (ms)

(c)


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)






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




1= 945Hz 119891

2=

3969Hz 1198913

= 4945Hz and 1198914



= 055 1205732

= 140 1205733

= 190 and 1205734








20

15

10

5

002 04 06 08 1 12

Air density (kgm3)


120590119892

Airspeed 15802


(a)

02 04 06 08 1 12

Air density (kgm3)

Second mode

120590119892

3

2

1

0


(b)


02 04 06 08 1 12

Air density (kgm3)

Third mode

120590119892

08

06

04

02

1

0


(a)

02 04 06 08 1 12

Air density (kgm3)


08

06

04

02

1

0


(b)


06

05

04

03

02

01

010 20 30 40 50 60



Tim

e sum











around the maximum


10 20 30 40 50 60



70 800

002

004

006

008

01

012

014

016

018

02

Com

puta

tiona

l tim

e

Timereduction48



5 Conclusions





Appendices







1199042M119898u119898

(119904) + 119904D119898u119898

(119904) + K119898u119898

(119904) = 119902Q119898

(119898119872 119896) u119898

(119904)

(A1)



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)





119898u119898

u119886119898

119879 where u




A =

[[[[[[[[[[

[

minusMminus1119886119898D119886119898

minusMminus1119886119898K119886119898

119902Mminus1119886119898Q1198983



119881


0 d sdot sdot sdot


119881

119887)120573119899lag

I

]]]]]]]]]]

]

(A3)




M119886119898

= M119898

minus 119902(119887

119881)

2

Q1198982

D119886119898

= D119898

minus 119902(119887

119881)Q1198981

K119886119898

= K119898

minus 119902Q1198980

(A4)




row sums that is

120590(infin)

119892(119894119895) = max

119894119903

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119895119888

119908119900(119894119903 119895119888)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(B1)


119894119903of [W119867

119900(119894119895)W

119900(119894119895)] 119894

119903= 1 119898(2 + 119899lag)

that is

120590(2119873)

119892(119894119895) = (max

119894119903

120582119894119903)

12

(B2)



120590(1119873)

119892(119894119895) = max

119895119888

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119894119903

119908119900(119894119903 119895119888)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(B3)









1= 31511 119864

2=

04162GPa ] = 031 119866 = 04392GPa and 120588mat =

38198 kgm3 where 1198641and 119864







10minus3

210minus3

510minus3

10minus2

510minus2

01 02 03

05 06 08 10

15 20 30 40



References























119867

(119905)119875(119905) +














Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Plunge mode400

200120590119892

2 4 6 8 10 12 140

Airspeed (ms)

(a)


120590119892

2 4 6 8 10 12 14

40

20

0

Airspeed (ms)

Pitch mode

(b)


120590119892

2 4 6 8 10 12 14


2

0

Airspeed (ms)

(c)


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)






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




1= 945Hz 119891

2=

3969Hz 1198913

= 4945Hz and 1198914



= 055 1205732

= 140 1205733

= 190 and 1205734








20

15

10

5

002 04 06 08 1 12

Air density (kgm3)


120590119892

Airspeed 15802


(a)

02 04 06 08 1 12

Air density (kgm3)

Second mode

120590119892

3

2

1

0


(b)


02 04 06 08 1 12

Air density (kgm3)

Third mode

120590119892

08

06

04

02

1

0


(a)

02 04 06 08 1 12

Air density (kgm3)


08

06

04

02

1

0


(b)


06

05

04

03

02

01

010 20 30 40 50 60



Tim

e sum











around the maximum


10 20 30 40 50 60



70 800

002

004

006

008

01

012

014

016

018

02

Com

puta

tiona

l tim

e

Timereduction48



5 Conclusions





Appendices







1199042M119898u119898

(119904) + 119904D119898u119898

(119904) + K119898u119898

(119904) = 119902Q119898

(119898119872 119896) u119898

(119904)

(A1)



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)





119898u119898

u119886119898

119879 where u




A =

[[[[[[[[[[

[

minusMminus1119886119898D119886119898

minusMminus1119886119898K119886119898

119902Mminus1119886119898Q1198983



119881


0 d sdot sdot sdot


119881

119887)120573119899lag

I

]]]]]]]]]]

]

(A3)




M119886119898

= M119898

minus 119902(119887

119881)

2

Q1198982

D119886119898

= D119898

minus 119902(119887

119881)Q1198981

K119886119898

= K119898

minus 119902Q1198980

(A4)




row sums that is

120590(infin)

119892(119894119895) = max

119894119903

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119895119888

119908119900(119894119903 119895119888)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(B1)


119894119903of [W119867

119900(119894119895)W

119900(119894119895)] 119894

119903= 1 119898(2 + 119899lag)

that is

120590(2119873)

119892(119894119895) = (max

119894119903

120582119894119903)

12

(B2)



120590(1119873)

119892(119894119895) = max

119895119888

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119894119903

119908119900(119894119903 119895119888)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(B3)









1= 31511 119864

2=

04162GPa ] = 031 119866 = 04392GPa and 120588mat =

38198 kgm3 where 1198641and 119864







10minus3

210minus3

510minus3

10minus2

510minus2

01 02 03

05 06 08 10

15 20 30 40



References























119867

(119905)119875(119905) +














Volume 2014




Journal of











Journal of


Function Spaces






Algebra










20

15

10

5

002 04 06 08 1 12

Air density (kgm3)


120590119892

Airspeed 15802


(a)

02 04 06 08 1 12

Air density (kgm3)

Second mode

120590119892

3

2

1

0


(b)


02 04 06 08 1 12

Air density (kgm3)

Third mode

120590119892

08

06

04

02

1

0


(a)

02 04 06 08 1 12

Air density (kgm3)


08

06

04

02

1

0


(b)


06

05

04

03

02

01

010 20 30 40 50 60



Tim

e sum











around the maximum


10 20 30 40 50 60



70 800

002

004

006

008

01

012

014

016

018

02

Com

puta

tiona

l tim

e

Timereduction48



5 Conclusions





Appendices







1199042M119898u119898

(119904) + 119904D119898u119898

(119904) + K119898u119898

(119904) = 119902Q119898

(119898119872 119896) u119898

(119904)

(A1)



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)





119898u119898

u119886119898

119879 where u




A =

[[[[[[[[[[

[

minusMminus1119886119898D119886119898

minusMminus1119886119898K119886119898

119902Mminus1119886119898Q1198983



119881


0 d sdot sdot sdot


119881

119887)120573119899lag

I

]]]]]]]]]]

]

(A3)




M119886119898

= M119898

minus 119902(119887

119881)

2

Q1198982

D119886119898

= D119898

minus 119902(119887

119881)Q1198981

K119886119898

= K119898

minus 119902Q1198980

(A4)




row sums that is

120590(infin)

119892(119894119895) = max

119894119903

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119895119888

119908119900(119894119903 119895119888)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(B1)


119894119903of [W119867

119900(119894119895)W

119900(119894119895)] 119894

119903= 1 119898(2 + 119899lag)

that is

120590(2119873)

119892(119894119895) = (max

119894119903

120582119894119903)

12

(B2)



120590(1119873)

119892(119894119895) = max

119895119888

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119894119903

119908119900(119894119903 119895119888)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(B3)









1= 31511 119864

2=

04162GPa ] = 031 119866 = 04392GPa and 120588mat =

38198 kgm3 where 1198641and 119864







10minus3

210minus3

510minus3

10minus2

510minus2

01 02 03

05 06 08 10

15 20 30 40



References























119867

(119905)119875(119905) +














Volume 2014




Journal of











Journal of


Function Spaces






Algebra










10 20 30 40 50 60



70 800

002

004

006

008

01

012

014

016

018

02

Com

puta

tiona

l tim

e

Timereduction48



5 Conclusions





Appendices







1199042M119898u119898

(119904) + 119904D119898u119898

(119904) + K119898u119898

(119904) = 119902Q119898

(119898119872 119896) u119898

(119904)

(A1)



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)





119898u119898

u119886119898

119879 where u




A =

[[[[[[[[[[

[

minusMminus1119886119898D119886119898

minusMminus1119886119898K119886119898

119902Mminus1119886119898Q1198983



119881


0 d sdot sdot sdot


119881

119887)120573119899lag

I

]]]]]]]]]]

]

(A3)




M119886119898

= M119898

minus 119902(119887

119881)

2

Q1198982

D119886119898

= D119898

minus 119902(119887

119881)Q1198981

K119886119898

= K119898

minus 119902Q1198980

(A4)




row sums that is

120590(infin)

119892(119894119895) = max

119894119903

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119895119888

119908119900(119894119903 119895119888)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(B1)


119894119903of [W119867

119900(119894119895)W

119900(119894119895)] 119894

119903= 1 119898(2 + 119899lag)

that is

120590(2119873)

119892(119894119895) = (max

119894119903

120582119894119903)

12

(B2)



120590(1119873)

119892(119894119895) = max

119895119888

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119894119903

119908119900(119894119903 119895119888)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(B3)









1= 31511 119864

2=

04162GPa ] = 031 119866 = 04392GPa and 120588mat =

38198 kgm3 where 1198641and 119864







10minus3

210minus3

510minus3

10minus2

510minus2

01 02 03

05 06 08 10

15 20 30 40



References























119867

(119905)119875(119905) +














Volume 2014




Journal of











Journal of


Function Spaces






Algebra










A =

[[[[[[[[[[

[

minusMminus1119886119898D119886119898

minusMminus1119886119898K119886119898

119902Mminus1119886119898Q1198983



119881


0 d sdot sdot sdot


119881

119887)120573119899lag

I

]]]]]]]]]]

]

(A3)




M119886119898

= M119898

minus 119902(119887

119881)

2

Q1198982

D119886119898

= D119898

minus 119902(119887

119881)Q1198981

K119886119898

= K119898

minus 119902Q1198980

(A4)




row sums that is

120590(infin)

119892(119894119895) = max

119894119903

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119895119888

119908119900(119894119903 119895119888)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(B1)


119894119903of [W119867

119900(119894119895)W

119900(119894119895)] 119894

119903= 1 119898(2 + 119899lag)

that is

120590(2119873)

119892(119894119895) = (max

119894119903

120582119894119903)

12

(B2)



120590(1119873)

119892(119894119895) = max

119895119888

10038161003816100381610038161003816100381610038161003816100381610038161003816

sum

119894119903

119908119900(119894119903 119895119888)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(B3)









1= 31511 119864

2=

04162GPa ] = 031 119866 = 04392GPa and 120588mat =

38198 kgm3 where 1198641and 119864







10minus3

210minus3

510minus3

10minus2

510minus2

01 02 03

05 06 08 10

15 20 30 40



References























119867

(119905)119875(119905) +














Volume 2014




Journal of











Journal of


Function Spaces






Algebra











10minus3

210minus3

510minus3

10minus2

510minus2

01 02 03

05 06 08 10

15 20 30 40



References























119867

(119905)119875(119905) +














Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article the use of gramian matrices for

Documents