Studiesonflutterprediction
Gabriela STROE1 Irina-Carmen Andrei2
Abstract
The purpose of this paper is to study the instability of the dynamic flutter. The justification is expressed by the fact that the phenomenon of flutter in the flight envelope of an aircraft results in irreversible structural deformation and consequently to serious damage, and therefore the mathematical modeling and its validation are very important. The instability of the dynamic flutter is characterized by critical speed and critical pulsation of oscillatory movements.
In this paper were analyzed the quasi-stationary model and the Theodorsen model for calculating the aerodynamic forces and torques, and a comparison was carried on.
The fluid-structure couplingisdonebyrewritingthe equations,considering thatthe forcesare givenbyclosed formulas. Theodorsenmodelandthe quasi-stationary model are applicabletomathematical modeling of and p-k and V-gflutter.
Tomodeling offreevorticesaerodynamicforcesandmomentsrequires that theequationswhich describe both the motion ofthe structureandthe fluid flowmust be integratedsimultaneouslyintime. The couplingfluid/structureis considered as a combination of two systems that describe the aeroelastic behaviour of the structure.
Keywords:typical section, quasi-stationary model, flutter prediction.
I. INTRODUCTION
Flutter's dynamicinstabilityischaracterizedbycriticalspeedandcriticalpulsationofoscillatorymovements.
The phenomenonofflutterintheflight envelopeof anaircraftresults inirreversiblestructuraldeformityand consequentlytoseriousdamage. This requiresverycarefulvalidation ofthe computationalmodelused. WithincreasingMach numberandincidenceofflight, theflowbecomescriticalextradosaprofileforMach numbersbetween0.4and 0.7, the firstshock waveformingataboutMach0.1higher.Dynamic responseisa transientresponseormovementof aircraftstructuralcomponentsproducedasa resultofgustsof air, suddenorders, shocks, etc. Forflexiblestructures, aeroelasticresponseofthe structureinteractswith theflow, resulting incomplex situations. For example, structuralvibrationscausealternatinglift offandreattachmentof thelayer.Nonstationaryloadsgreaterinteractionwithinthe structurecausingunusualaeroelasticphenomenathatcansignificantlychange theflightenvelope [1], [2], [3].
1PhD Student, MSc AE, Lecturer, University POLITEHNICA of Bucharest, Romania
2 PhD, MSc AE, Lecturer, University POLITEHNICA of Bucharest, Romania
To describethesetwophenomenaintroducethe conceptoftypicalsection. Thisisachievedsectionedwingwithaplane parallel to theplaneof symmetryofthedistancey.The winghasabendingmovementandtorsionalsobecausewehavetwotypesofmovements(Figure 1) [1], [2].
Fig.1.Twotypesofmovements
Commonmathematicalmodeldescribingthesetwophenomenaisobtained fromthe Lagrangeformalismwhichconsistsofthe followingequation:
(1)
WhereTiskinetic energyexpressed in terms ofgeneralizedcoordinateandgeneralizedspeed, thepotential energyUexpressed in function of and, terms ofelasticdeformationandthe correspondinggeneralizedcoordinatesandgeneralizedforcesQi, fromthe workofexternalforceson the nature ofaerodynamics, mass, etc.
Fig.2. Typicalsection
Typicalsectionalview ofthemodelin Figure2, the kinetic energypotential energythatare givenbyrelations [1]:
(2)
(3)
For modelwithtwodegreesoffreedom
(4)
(5)
For the modelwiththreedegreesoffreedom bysubstitutionwe obtainLagrange equationsof the form:
(6)
whereMismassmatrix, Cthedampingmatrix, stiffness matrixKandvectoron the rightisfromthevirtualmechanical workandrepresentsaerodynamicforces.Startingfromthis systemwe canformulateappropriatemathematicalmodelsflutteranddynamic response.
IntheTheodorsenmodelandthe modelforcalculating thequasi-aerodynamic forcesand torques, fluid-structure couplingisdonebyrewritingthe equations,considering thatthe forcesare givenbyclosed formulas. Theodorsenmodeland quasi-calculus models areapplicabletop-kand V-gflutter [1], [2].
Tomodelthecalculation offreevorticesaerodynamicforcesandmomentsrequires that theequationsthatdescribe the motion ofthe structureandequationsdescribingthe fluid flowisintegratedsimultaneouslyintime. Solvingnumericallythe couplingfluid/structureraisessomeproblemsbecausethe equationsthatdescribe thebehavior ofthe structurearetreatedinaLagrangeanreference system, whilethe equationsthatdescribefluid flowaredescribedinanEuleriancoordinate systemtype. On theother hand, the deformationstructureinevitablyleadstoa change(partial or total) of the borderbetweenfluidandstructure, whichinvolvesintegratingthe equations offluidflowoncellvolumecontrolso,amobile computingnetwork.
The couplingfluid/structureareviewedasacombination of twosystemsthatdescribe theaeroelasticbehaviorof the structure. The problemmaybesupplementedby the equationsof motionofthe network,apseudo-structural systemwithits owndynamic [3].
If aerodynamic forces are calculated with a model of free vortices, efficient method tocalculate the flutter speed is the ‘root locus design’ [1], [3].
Since aerodynamic forces are those which introduce energy into the system and their value depends on the speed for a given configuration (characteristic mass, elastic and geometric structure) will be able to calculate the critical flutter speed, speed which if exceeded, the system becomes unstable dynamic and virtually destroyed.
Consequently, the critical wave speed is defined as the speed at which the motion is harmonic structure and oscillation damping (structural and aerodynamic) is zero.
Determination of wave conditions (wave speed and frequency associated) is significantly dependent wind model adopted, a harmonic oscillator system (proposed by Theodorsen) approximating reality better than a quasi-stationary. In what follows, in terms of aerodynamics will be waving this study both for simplified cases based on the study of the quasi-stationary aerodynamic forces as well as periodic nonstationary.
II. Dynamic response
Dynamic response is a transient response or movement of aircraft structural components produced as a result of the forces burst data, sharp commands, different shocks, etc.. It will present three methods for calculating the dynamic response (time integration methods of the equations of motion) applied to an aeroelastic model.
The first method called the Newmark method is a method based on implicit discretization of the equations. The time constant is chosen and the periods of oscillation are known. A second method called HHT method (Hilbert, Hughes, Taylor) are working on the systems in second order differential equations and uses the physical meaning of terms such as displacements, velocities and accelerations. A third method, Runge-Kutta requires transforming the system initially in a first-order differential system, with unknown but not both movements and speeds [1].
Runge-Kutta method
Whether first-order differential equations of the form:
/ (7)wherefunctionsdescribeingeneralnonlinearformsinindependent variablexandthe dependent variable.
Note: If thesystemof differential equationsishigher orderthanweproceed tochangethe order.
IfEulermethodfor solvingordinarydifferentialequationssystemmovingbetweentwosuccessivecalculation pointsxkand xk+1,is done byapplyingacorrectionvalue yk, determinedbyproductintegrationstep handderivativesolution y(x), calculatedthe leftmostintervalyk,using thefunctionfthatdefinesthe differential equation [1].
One canwritetherefore:
/ (8)ForCauchyversion, improvingaccuracyisobtainedby calculating thecorrectionto be appliedyk tothederivativefunctiony(x) at the middlerangexk+1/2=xk+h/2.Calculatingthe derivativeat this pointrequiresanapproximationforyk+1/2, obtained usingaclassicalEulerstep, i.e.:
/ (9)Runge-Kutta methodsgeneralizethisprinciple. From apointofknowncoordinates(xkyk), Runge-Kutta methodsare advancingto the next pointvalueby applyingvalueykone linear combination ofcorrectionsKj, weightedby anumberofcoefficientsrjto bedetermined :
/ (10)Inthisrelationrindicatesthe orderof the method. Runge-Kutta typemethodshaveseveraladvantagesamong whichtherecall:
(i) direct methods, sononeedto useauxiliarypowermethods;
(ii)Taylorseriesareidenticaltothe termofrankr, soit is possibleto estimatethetruncationerror;
(iii) require evaluationofpartialderivativesof thefunctionf(x, y), only thefunctionfitself.
CorrectionsKjis determinedas theproductof theintegrationstephwithspecific valuesofthe functionf,calculatedinpointsnear thepoint(xk, yk):
In all thecasesare considered:
/ (12)Therefore
/ (13)/ (14)
It remainstospecifyvaluescoefficients, rj, jandji. Thesecoefficientsaredeterminedby imposingthe conditionthat theRunge-Kutta formula
/ (15)tocoincidewith theTaylorseries expansionup to orderr.It is notedthatthisconditionleadstoasystemoflinear equationswithunknownsrj, jandji, which generallycontainsfewerequationsthanunknowns. Solvingthissystemrequiresapriorispecificationof someunknown, which makes theformulatogenerateavirtuallyunlimitednumberofRunge-Kutta methods [1].
Inpractice,weknownonlyafewsuchmethods.Further, itwillpresentsomeparticularcasesofRunge-Kutta methods.Caser=1(formula first-orderRunge-Kutta).Inthis caserelationsaccount, togetherwiththe restriction11=1,leadsto:
/ (16)Fig. 3.
That Eulermethodisjustthe usualformula. Figure3isoutlinedhowthe advancingsolutionmethodorder IRunge-Kutta. The TruncationErrorat the first stepisof the orderh2.
Caser= 2(Runge-Kutta formulaofordertwo)
Equationsforcalculatingleadin thiscase, successively, to [1]:
/ (17)To determine theunknowncoefficientsthisexpressiondevelops afterthe powersof habout the point (xk, yk)andretainonlylinearterms:
/ (18)Ontheotherhand, theTaylorseries expansionleads to:
/ (19)Imposingthe conditionthathislasttwoexpressionscoincide, it follows:
/ (20)Ifitrequiresapriori21=0,resulting: 22=1and2=21=1/2,thata firstversionofRunge-Kutta formula, thesecond order:
/ (21)If yourequire22=1/2, resulting21=1/2and 2= 21=1,respectively:
/ (22)Fig. 4.
Figure4isoutlinedhowthemethodadvancesthe solutionofsecond orderRunge-Kutta, theintermediatestepsarewhiteheadsandblackheadsarethe pointscalculated. Applyingastepusingthe above formulasgenerateatruncationerroroforderh3.
Caser= 3(Runge-Kutta formulaoforderthree).Usingsimilarreasoningheld for
r=2, we reach asystemofsixequationswith eightunknowns. Runge-Kutta formulaoforderthreemostcommonare [1]:
/ (23)Beingathird ordermethodisassociatedtruncationerrorof orderh4.
Caser= 4(Runge-Kutta formulaoforderfour). ThisRunge-Kutta methoditismostcommonlyused inpracticeandusesa systemofcoefficientsthatleadstothe formula:
/ (24)Runge-Kutta methodoforderfourhasatruncationerroroforderh5.
Whereappropriatesectiontypicalsystemwith twodegreesoffreedom(extension methodfor themodelwith threedegrees offreedomisimmediate) systemwe havesolved [1]:
Where
.
RungeKuttamethodinvolvesthe followingprocessingscheme:
The vector ofunknowns is, where.
/ (26)/ (27)
/ (28)
We notefurther
/ (29)Giventhe systemandinitial conditions [1], [2]:
/ (30)III. Numerical results
Inthoseconditionsare obtainedfor theequilibrium position (see Table 1)
Table 1. Conditions for equilibrium position
The numerical resultsare plottedandthe calculationwasperformed usingtwo models ofaerodynamic forces(Theodorsen modelandquasi-stationary model).
Theharmonicoscillator(Theodorsen model) / Thequasi-stationary model
Fig.5.
Theharmonicoscillator(Theodorsen model) / Thequasi-stationary model
Fig.6.
Theharmonicoscillator(Theodorsen model) / Thequasi-stationary model
Fig.7.
IV. Conclusions and remarks
In the Theodorsen model and the quasi-stationary modelfor calculating the aerodynamic forces and torques, fluid-structure coupling is done by rewriting the equations, considering that the forces are given by closed formulas. Theodorsen model and the quasi-stationarymodel areapplicable to calculus models p-K andV-g flutter.
To model the calculation of free vortices aerodynamic forces and moments requires that the equations that describe the motion of the structure and equations describing the fluid flow is integrated simultaneously in time.
Solving numerically the coupling fluid / structure raises some problems because the equations that describe the behavior of the structure are treated in a Lagrangean reference system, while the equations that describe fluid flow are described in an Eulerian coordinate system type. On the other hand, the deformation structure inevitably leads to a change (partial or total) of the border between fluid and structure, which involves integrating the equations of fluidflow on cell volume
control so, a mobile computing network.
The coupling fluid / structure are viewed as a combination of two systems that
describe the aeroelastic behavior of the structure.
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