NASA Technical Reports Server (NTRS) 20150006860: In-Flight Aeroelastic Stability of the Thermal Protection System on the NASA HIAD, Part II: Nonlinear Theory and Extended Aerodynamics

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In-Flight Aeroelastic Stability of the Thermal Protection
System on the NASA HIAD, Part II: Nonlinear Theory and
Extended Aerodynamics
∗ †
Benjamin D. Goldman and Earl H. Dowell
DukeUniversity,Durham,NC,27708,USA
‡
Robert C. Scott
NASALangleyResearchCenter,Hampton,VA,23681,USA
Conical shell theory and a supersonic potential flow aerodynamic theory are used to study the
nonlinear pressure buckling and aeroelastic limit cycle behavior of the thermal protection system
for NASA’s Hypersonic Inflatable Aerodynamic Decelerator. The structural model of the thermal
protection system consists of an orthotropic conical shell of the Donnell type, resting on several cir-
cumferentialelasticsupports. ClassicalPistonTheoryisusedinitiallyfortheaerodynamicpressure,
but was found to be insufficient at low supersonic Mach numbers. Transform methods are applied
to the convected wave equation for potential flow, and a time-dependent aerodynamic pressure cor-
rectionfactor is obtained. The Lagrangianof theshell systemis formulatedin termsof thegeneral-
izedcoordinatesforalldisplacementsandtheRayleigh-Ritzmethodisusedtoderivethegoverning
differential-algebraicequationsofmotion. Aeroelasticlimitcycleoscillationsandbucklingdeforma-
tionsarecalculatedinthetimedomainusingaRunge-KuttamethodinMATLAB.
◦
Threeconicalshellgeometrieswereconsideredinthepresentanalysis:a3-meterdiameter70 cone,
◦ ◦
a3.7-meter70 cone,anda6-meterdiameter70 cone. The6-meterconfigurationwasloadedstatically
andtheresultswerecomparedwithanexperimentalloadtestofa6-meterHIAD.Thoughagreement
between theoretical and experimental strains was poor, the circumferential wrinkling phenomena
observedduringtheexperimentswascapturedbythetheoryandaxialdeformationswerequalitatively
similarinshape. WithPistonTheoryaerodynamics,thenonlinearflutterdynamicpressuresofthe3-
meter configuration were in agreement with the values calculated using linear theory, and the limit
cycle amplitudes were generally on the order of the shell thickness. The effect of axial tension was
studiedforthisconfiguration,andincreasingtensionwasfoundtodecreasethelimitcycleamplitudes
whenthecircumferentialelasticsupportswereneglected,butresultedinmorecomplexbehaviorwhen
thesupportswereincluded. Thenominalflutterdynamicpressureofthe3.7-meterconfigurationwas
significantlylowerthanthatofthe3-meter,anditwasfoundthattwosetsofnaturalmodescoalesce
tofluttermodesnearthesamedynamicpressure. Thisresultedinasignificantdropinthelimitcycle
frequencies at higher dynamic pressures, where the flutter mode with the lower frequency becomes
morecritical. Pre-bucklingpressureloadsandtheaerodynamicpressurecorrectionfactorwerestudied
forallgeometries, andtheseeffectsresultedinsignificantlylowerflutterboundariescomparedwith
Piston Theory alone. The maximum dynamic pressure predicted by aerodynamic simulations of a
proposed3.7-meterHIADvehiclewasstilllowerthananyofthecalculatedflutterdynamicpressures,
suggestingthataeroelasticeffectsforthisvehicleareoflittleconcern.
∗
PhDcandidate,DepartmentofMechanicalEngineeringandMaterialsScience,Box90300HudsonHall,StudentMemberAIAA
†
WilliamHollandProfessor, DepartmentofMechanicalEngineeringandMaterialsScience, Box90300HudsonHall, Honorary
FellowAIAA
‡
Sr.AerospaceEngineer,AeroelasticityBranch,AssociateFellowAIAA
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Nomenclature
a∞ Speedofsound, m/s
a ,b ,c Modalcoordinatesforthethreeshelldisplacements
mn mn mn
(cid:2) (cid:2) (cid:2)
a,b,c Modalcoordinatesforthethreeshelldisplacements,invectorform
Dy,Dθ ShellBendingstiffnessinthe y and θ directions,respectively, Pa m3
Dyθ Shellin-planetwistingstiffness, Pa m3
Ey,Eθ ShellYoung’sModulusinthe y and θ directions,respectively, Pa
f Frequency, Hz
f Criticalfrequency,attheflutterordivergenceboundary, Hz
crit
Gyθ Shearmodulus, Pa
h Shellthickness, m
I ModifiedBesselfunctionofthefirstkind
n
k Elasticsupportstiffness, N/m
s
K ModifiedBesselfunctionofthesecondkind
n
m Shellmassperarea, kg/m2,ortheaxialmodenumber
m Totalnumberofaxialmodes
tot
M LocalMachnumber
L
nˆ Circumferentialmodenumber
n nˆ/sinα
n Totalnumberofcircumferentialmodes
tot
n Criticalcircumferentialmodenumber,attheflutterordivergenceboundary
crit
Na Tensionforcesinthey-direction, N/m
y
N∗ −tanβp /2
y s
Na Totalin-planeforceiny-direction(duetopressureandtension), N/m
y,tot
Nθ∗ −tanβps
Nθa,tot Totalappliedin-planeforcein θ-direction, N/m
p TemporalLaplacetransformvariable
p Staticpressuredifferential, Pa
s
q Localdynamicpressure, Pa
A,L
q Criticallocaldynamicpressure,attheflutterordivergenceboundary, Pa
A,Lcrit
r Cylindricalshellradialcoordinate, m
R Averageradiusofthecylindricalorconicalshell, m
r˜ rδ
s SpatialLaplacetransformvariable
u,v,w Shelldisplacements, m
∗∗
w DoubleLaplacetransformofthedeflectionwithrespecttospaceandtime
U∞ Free-streamflowvelocity, m/s
x Cylindricalshellaxialcoordinate
y Coordinatealongtheshellmeridian
y ,y Locationsoftheshellminorandmajorends,respectively
1 2
( − ) ( − )
y¯ Dimensionlessshellcoordinate, y y / y y
1 2 1
α Fouriertransformvariable(s =iα)
β Shellhalf-coneangle
αm, βm S(cid:2)hellin-planemodalfunctions
δ (M2(s+p/U∞)2−s2)
Δp Unsteadyaerodynamicpressure
Δp∗α SpatialLaplacetransformofthepressure
Δp∗∗ DoubleLaplacetransformofthepressurewithrespecttospaceandtime
(cid:6)y,(cid:6)θ,(cid:6)yθ Middlesurfacestrainsinthe y, θ, yθ directions,respectively
χy, χθ, χyθ Middlesurfacecurvaturesinthe y, θ, yθ directions,respectively
θ Shellcircumferentialcoordinate
φ θsinα,orthecircumferentiallydependentvelocitypotential
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Φ Circumferentiallyindependentvelocitypotential
Φ∗∗ DoubleLaplacetransformofthevelocitypotential,withrespecttospaceandtime
ψ Shellaxialmodalfunction
m
γ Rδ
νy, νθ Poisson’sratiointhe y and θ directions,respectively
τ Temporalintegrationvariable
I. Introduction
Thepurposeofthisreportseries,consistingofthecurrentworkandreference[1],istocharacterizethe
in-flight aeroelastic behavior of a proposed thermal protection system2 for the NASA Hypersonic Inflat-
ableAerodynamicDecelerator(HIAD).3 TheHIADisaninflatableaeroshellwithadiametersignificantly
larger than that of current rigid aeroshells. Its size is not limited by the launch vehicle shroud, allowing
heavier payloads to be delivered to higher elevations of planets like Mars. The HIAD is composed of
twoseparatebutinteractingstructuralcomponents: astackedinflatabletoroidsubstructureandathermal
protection system (TPS). An illustration of these components is shown in Fig. 1, for the 3-meter diameter
HIADusedintheInflatableRe-EntryVehicleExperiment.4 TheTPSisflexibleenoughtobepackedintoa
smallvolumefortransportandexpandswiththeinflationofthetoroidsubstructureinspace. Asaresult
of this flexibility, the TPS may be susceptible to static or dynamic aeroelastic instabilities such as "diver-
gence" (buckling) or "flutter," which may be induced by fluid-structure interaction during atmospheric
re-entry.
Figure1: IllustrationoftheHIADusedintheInflatableRe-EntryVehicleExperiment.4
As a first attempt at characterizing the aeroelastic behavior of this system, experimental aerothermal
performancetests2,5 wereconductedonsmallrectangularTPScouponsintheNASA8’HighTemperature
Tunnel. Observations of oscillatory motion and failure of these samples during testing suggested that
aeroelastic effects needed additional study, and theoretical model development was required. Goldman,
Dowell,andScott6,7 developedseveraltheoreticalaeroelasticmodelsfortheTPScoupons,andagreement
between theory and experiment for select cases was achieved. A rough prediction of in-flight stability
can be made from these plate-like models; however, more precise predictions require structural models
consistentwiththein-flightTPSgeometry,i.e. thatofoneormoreconicalshells.
In Part I1 of this report series, linearized conical shell theory and Piston Theory aerodynamics were
usedtodeterminetheaeroelasticstabilityboundariesfortheTPSonthe3-meterHIAD.Severalaeroelastic
modelsweredeveloped,andinmostcases,flutterwasfoundtobetheprimaryinstability. Thelimitations
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of linear theory did not allow the amplitude of flutter oscillations to be determined, and the nonlinear
effect of static pressure was ignored. Pressure-induced buckling was not studied since the buckling pres-
sures predicted by linear theory were orders of magnitude lower than the pressures seen in flight. In the
current analysis, nonlinear conical shell theory is used to study the buckling and aeroelastic limit cycle
oscillations over a range of dynamic pressures above the linear flutter boundary. The initial deformation
causedbythestaticpressureloadingisalsoincluded. Anextendedaerodynamictheoryisalsodeveloped
to study more accurately the aeroelastic behavior in Mach number regimes where Piston Theory is less
accurate.
II. Background and Previous Work
Details on thethermal protection system modeling can befound in reference [1]. A brief summary of
the structural models and TPS will be provided here, since the current analysis is based on the previous
work. Thegeneration1TPSconfiguration2consistsofonelayerofAluminizedKaptonKevlar(AKK),four
layers of Pyrogel 2250, and two layers of Nextel 440-BF20. The AKK is composed of Kevlar woven fabric,
coated with thin layers of Aluminum and Kapton. The Pyrogel is a soft sponge-like insulator, which
ensures heating of the AKK is low enough to prevent structural failure. The Nextel is a woven fabric
thermal barrier that takes most of the heat load. This system of interacting layers has been simplified for
computational efficiency and can be approximated as a shell-like structure. Three structural models have
beendeveloped,andengineeringillustrationsaregiveninFig.2.
The single shell model consists of a single truncated conical shell. The separate layers of the thermal
protection system are approximated as a single layer of AKK with added mass to account for the Nextel
and Pyrogel. Tension in the axial direction can also be applied at the shell boundaries. The boundaries
aresimply-supportedinallthreeshelldisplacementdirections.
The elastically supported single shell model consists of a truncated conical shell with six middle sur-
face circumferential elastic supports. These supports take into account the presence of the HIAD’s toroid
substructure, in a linear sense. As in the previous case, the three separate layers of the TPS are approxi-
mated as a single layer of AKK with added mass for the Nextel and Pyrogel, tension can be applied, and
theboundaryconditionsremainthesame.
The elastically supported three-layer shell model consists of one AKK shell and one Nextel shell sep-
arated by a Pyrogel linear elastic layer. The Nextel shell is one layer of Nextel material with twice the
nominalmass,astwolayersofNextelarestitchedtogetherintheTPS.SincethePyrogelisasoft,sponge-
likematerial,itisapproximatedasalinearelasticlayer. Thislayerisassumedtobebondedtothetopand
bottomoftheAKKandNextelsurfaces,respectively. Theboundaryconditionsarethesameforalllayers,
andareidenticaltotheprevioustwocases.
The linear aeroelastic equations of motion corresponding to each of the structural models above were
derivedusingtheRayleigh-RitzmethodwithaLagrangianintermsofthegeneralizedcoordinatesforall
shelldisplacements. TheaerodynamicpressureisgivenbycylindricalshellPistonTheory,8 andincluded
in the equations as generalized forces. Aeroelastic solutions were obtained by writing the equations as
an eigenvalue problem, and observing the behavior of the eigenvalues and eigenvectors with increasing
flowdynamicpressure. Sincealinearstructuraltheoryisemployed,onlythestabilityboundariescouldbe
determinedusingthismethod. Thecriticalmodesanddynamicpressuresobtainedfromtheeigenanalysis
willbenecessarytosimplifygreatlythenonlinearcalculationsandreducecomputationtime.
III. Nonlinear Methods
The goal of the current work is to study the nonlinear pressure-buckling and aeroelastic response
of the TPS structural models developed in Part I1 of this report series. Since the nonlinear problem is
substantiallymorecomplex,onlythesingleshell(A)andelastically-supportedsingleshell(B)modelswill
be considered here. Additionally, for the purposes of determining the worst-case scenario for flutter and
LCO,furtheranalysisofmodelCisunnecessarysincelineartheorypredictstheflutterboundariesforthis
casetobenearlyanorderofmagnitudelargerthanmodelsAandB.
ThenonlinearequationsofmotionarederivedfromthefullDonnellstrain-displacementrelationsand
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ModelA:Singleshell.
ModelB:Elasticallysupportedsingleshell.
ModelC:Elasticallysupportedthree-layershell
(notconsideredinthenonlinearanalysis).
Figure2: ConicalshellstructuralmodelsoftheTPS.
the Rayleigh-Ritz is employed. A time-integration procedure has been developed to solve the resulting
set of coupled differential-algebraic equations. Typical results for the full nonlinear aeroelastic system
include limit cycle oscillations (LCOs), in which the flutter amplitude grows initially but is limited by
the nonlinearities in the structure. Frequency spectra and time-dependent deflection shapes can also be
calculatedusingthismethod. Theproposednonlinearanalysisisadvantageousbecausethemagnitudeof
flutter oscillations can be determined, which may be necessary for fatigue analyses of the TPS materials.
Thestaticshapeandstraindistributionoftheshellsatpressureshigherthanthelinearbucklingpressure
canalsobedeterminedandcomparedwithexperimentalstaticloadtests.
III.A. NonlinearAeroelasticEquationsofMotion
The first step in applying the Rayleigh-Ritz method is to derive the total strain and elastic potential
energy, kinetic energy, and non-conservative virtual work for the nonlinear shell system. To form the
strainenergies,thestrain-displacement,stress-strain,andmoment-curvaturerelationsareneeded.
ThenonlinearDonnellstrain-displacementrelationsgivenbySeide9 areusedhere:
(cid:3) (cid:4)
ε = ∂u + 1 ∂w 2
y ∂y 2 ∂y (cid:3) (cid:4)
ε = u−wcotβ + 1∂v + 1 ∂w 2 (1)
θ y y∂φ 2y2 ∂φ
ε = ∂v − v + 1∂u + 1∂w∂w
yθ ∂y y y∂φ y ∂y ∂φ
Thecurvaturesarelinearfunctionsofdeflection(anditsderivatives):
χ = ∂2w
y ∂y2
χθ = 1y∂∂wy + y12∂∂2φw2 (2)
χ = 1 ∂2w − 1 ∂w
yθ y∂y∂φ y2 ∂φ
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Theforceandmomentrelationsare:
(cid:5) (cid:6)
Ny = 1−Eνyyhνθ (cid:5)εy+νθεθ(cid:6)
Nθ = 1−Eνθθhνy εθ+νyεy (3)
Nyθ = Gyθhεyθ
(cid:5) (cid:6)
My = −Dy(cid:5)χy+νθχθ(cid:6)
Mθ = −Dθ χθ+νyχy (4)
Myθ = −Dyθχyθ
The three shell displacements are expanded using modal series with summations over both the axial
andcircumferentialmodes. Adoublesummationisnecessarysincethestructuralnonlinearitiesintroduce
couplingamongthecircumferentialmodes,whichwerepreviouslyuncoupledinthelinearanalysis. Note
that these expansions only allow for standing waves along the shell circumference. The axial modes may
be any functions that satisfy the geometric boundary conditions. Sine functions are used for simply-
supported boundaries, where all three displacements vanish at the shell edges. The modal expansions
are:
u(y,θ,t) = ∑∑a (t)cosnφψ (y)
mn m
m n
v(y,θ,t) = ∑∑bmn(t)sinnφψm(y) (5)
m n
w(y,θ,t) = ∑∑c (t)cosnφψ (y)
mn m
m n
Thestrainenergyisconstructedbyintegratingovertheentireshellsurfacetheforcestimesstrainsand
momentstimescurvatures:
(cid:7) (cid:7)
US = 1 y2 2π(cid:5)Nyεy+Nθεθ+Nyθεyθ−Myχy−Mθχθ−2Myθχyθ(cid:6)ydφdy (6)
2 y1 0
ThemembraneenergyassociatedwithstaticaxialorpressureloadingisgivenbyMcNeal:10
(cid:7) (cid:7) (cid:3) (cid:8) (cid:9) (cid:4)
UIS = 12 y1y2 02π Nya,totΘ2θ+Nθa,totΘ2y+ Nya,tot+Nθa,tot Θ2n ydφdy (7)
wherethestrainrotationvectorsare:
Θ = −∂w
θ ∂y
Θy = ytavnβ − 1y∂∂wφ (8)
Θ = v + 1∂v − 1 ∂u
n 2y y∂y 2y∂φ
Thetotalappliedin-planeforcesarerelatedtothestaticpressureandtensionbytherelation:
Na = −ytanβps −Na
y,tot 2 y (9)
Nθa,tot = −ytanβps
A derivation of the relationship between in-plane forces and the externally-applied static pressure can be
foundintheappendix.
ThetoroidfoundationontheHIADisapproximatedbyseverallinearcircumferentialelasticsupports.
Theelasticpotentialenergyforthesesupportsis:
(cid:7)
U = ks 2πsinβ∑w2 y| dφ (10)
ES
2 0 i y=yi
where y referstotheelasticsupportlocationalongtheshellmiddlesurface.
i
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Thetotalkineticenergy,neglectingrotatoryinertia,is:
(cid:7) (cid:7) (cid:10)(cid:11) (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:13)
T = m y2 2π ∂u 2+ ∂v 2+ ∂w 2 ydφdy (11)
2 y1 0 ∂t ∂t ∂t
Thevirtualworkandgeneralizedforcesare:
(cid:7) (cid:7)
δW = − 2πsinα y2(Δp+p )δwydφdy (12)
s
0 y1
(cid:7) (cid:7)
Q = 2πsinα y2((Δp) +p )cosnφψ ydφdy (13)
mnsk sk s m
0 y1
where (Δp) isthemodalformoftheaerodynamicpressure.
sk
These generalized forces include the unsteady aerodynamic pressure as well as static pressure due to
themeanflow,whichisknownfromstaticCFDcomputations11 oftheHIADvehicle. Inmostcases,Piston
Theory will be used for the aerodynamic pressure. In the next section, an extended aerodynamic theory
willbedevelopedthatwillmodifysignificantlytheformoftheaerodynamicgeneralizedforces.
After substituting Eq. 5 into Eqs. 6 - 12, constructing the Lagrangian, and applying the Lagrange
equation for each modal coordinate, a large system of ordinary differential equations is obtained for all
three shell displacements. This system of equations is difficult to solve because the in-plane frequencies
for shells are often orders of magnitude higher than out-of-plane deflection frequencies, requiring a very
small time step in the computational scheme. However, since the in-plane inertia is small relative to the
out-of-plane inertia, it may be neglected. This requires that the following linear system involving the
in-planecoordinatesbesolvedateverytimestep:
⎡ (cid:3) (cid:4)(cid:5) (cid:6) (cid:3) (cid:4) ⎤(cid:18) (cid:19)
⎣ A+AT+ N(cid:3)y∗+Nθ∗ (cid:4)N +NT (cid:3) B(cid:4)+(cid:5) Ny∗+Nθ∗ O(cid:6) T (cid:5) (cid:6) ⎦ (cid:2)a
BT+ Ny∗+Nθ∗ O C+(cid:18)CT+ Ny∗+Nθ∗ PP +(cid:19)PPT +Nθ∗ QQ+QQT (cid:2)b
= − D(cid:2)c+C(cid:2)CJ
E(cid:2)c+C(cid:2)CL+Nθ∗R(cid:2)c
(14)
(cid:2)
AlongwiththeODEfor c:
(cid:20) (cid:21) (cid:3) (cid:20) (cid:21) (cid:4)
I+IT (cid:2)¨c = − DT(cid:2)a+E(cid:2)b+ F +FT (cid:2)c +Q(cid:2)A+ES(cid:2)c
(cid:3) (cid:3) (cid:4) (cid:4) (cid:3) (cid:4)
(15)
+Nθ∗ RT(cid:2)b+ S+ST (cid:2)c +Ny∗ T +TT (cid:2)c+CC(cid:2) C
The full expressions for the vectors and coefficient matrices in the equations above are provided in the
(cid:2)
appendix. Note that the right-hand-side of Eq. 14 is a nonlinear function of the deflection coordinate c,
(cid:2) (cid:2)
but the system is linear with respect to the in-plane coordinates a and b. This method, in which Eq. 14
is solved at every time-step, was found to be nearly one order of magnitude faster than solving the full
systemincludingin-planeinertia. Ananalysiswith12axialand2circumferentialmodeswasperformed,
anditwasfoundthatin-planeinertiahadlittleeffectonboththeflutterboundariesandLCOamplitudes.
Nonlinear aeroelastic solutions are calculated by first performing the linear analysis described in Part
I.1 Linear theory predicts a single critical circumferential mode as well as coalescence of two axial modes
at the flutter dynamic pressure. A nonlinear time integration simulation is then performed with initial
conditions (displacements) in the coordinates for these critical modes only. It was found that the system
is generally not sensitive to the magnitude of the initial conditions, and giving initial conditions in non-
critical modes also produces the same results, but additional computation time is then required to reach
a steady state. Additionally, at least two circumferential modes are needed to introduce coupling in the
nonlineartermssuchthattheyarenonzero(theonlyexceptiontothisisif n =0.)
crit
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III.B. ExtendedPotentialFlowAerodynamicTheory
Since the local flow solution behind the HIAD’s bow shock is known from static CFD simulations,11
Piston Theory can be used as an initial approximation for the aerodynamic pressure. This approach
has been used in several previous works on conical shell flutter, including Dixon and Hudson12 and
Mahmoudkhani et al.13 Krumhaar8 derived a Piston Theory curvature correction term for cylindrical
shells, which can be easily modified for the conical shell by letting R = ysinβ. The "corrected" piston
theoryusedformostcomputationsinthispaperisgivenby:
(cid:22) (cid:23)
2q ∂w 1 ∂w 1 w
Δp = A,L + − (16)
ML ∂y U∞ ∂t 2ML ysinβ
Piston Theory is a convenient tool for aeroelastic computations, since it can be included directly into
the structural equations of motion as generalized forces, independent of temporal or spacial memory.
However, as the Mach number approaches 1 supersonically, this approximation may no longer be suf-
ficient. Generally M2 >> 1 is the necessary condition for validity.14 Since the maximum local Mach
number on the HIAD surface is approximately 1.5 - 2, investigation into a more elaborate aerodynamic
theoryisneeded. Fullpotentialflowtheoryisthenextlevelofapproximation,butadirectsolutionusing
transform methods is cumbersome and difficult, especially when performing the inversion of transform
integrals. Following the method of Dowell and Bliss for 2-D airfoils,15 Potential Flow theory can be used
to extend conical shell Piston Theory to a lower Mach number range by expanding the transform of the
aerodynamic pressure in powers of inverse Mach number. Under the local flow assumption, we start by
transforming the convected wave equation for potential flow in cylindrical coordinates with respect to
bothtimeandthestreamwisecoordinatex:
(cid:22) (cid:11) (cid:12)(cid:23)
1∂Φ ∂2Φ n2 ∂2Φ 1 ∂2Φ ∂2Φ ∂2Φ
LsLp r ∂r + ∂r2 − r2Φ+ ∂x2 − a2∞ ∂t2 +2U∞∂x∂t +U∞2 ∂x2 (17)
where φ(x,r,θ,t) = Φ(x,r,t)cosnφ. ThetransformsresultinthefollowingmodifiedBesselequation:
∂2Φ∗∗ ∂Φ∗∗ (cid:20) (cid:21)
r˜2 +r˜ − n2+r˜2 Φ∗∗ =0 (18)
∂r˜2 ∂r˜
(cid:3) (cid:4)
2
wherer˜ =rδ, δ2 = M2 s+ p −s2,andthedoubleLaplacetransformofthepotentialis:
U∞
(cid:7) (cid:7)
∞ ∞
Φ∗∗ = Φ(x,r,t)e−sx−ptdxdt (19)
0 0
ThegeneralsolutiontoEq.18isalinearcombinationofmodifiedBesselfunctions:
Φ∗∗ = c I (r˜)+c K (r˜) (20)
1 n 2 n
andafterapplyingthefinitenessconditionasrapproachesinfinity,
Φ∗∗ = c K (r˜) (21)
2 n
The boundary condition at all points along the shell axis requires that the local velocity on the shell
surfacebeequaltothedownwash: (cid:24)
∂φ(cid:24)(cid:24) = ∂w + ∂w
∂r(cid:24)r=R U∞∂x ∂t (22)
andthetransformofthisboundaryconditionis:
(cid:24)
∂Φ∗∗ (cid:24)
∂r˜ δ(cid:24)(cid:24)r˜=Rδ = (U∞s+p)w∗∗ (23)
After applying the transform of the boundary condition to the transform of the potential, the unknown
coefficient c canbedetermined. Thecompleteexpressionforthetransformofthepotentialis:
2
( ) ∗∗
Φ∗∗ = KKn(cid:4)(r˜Rwδ)δ (U∞s+p) (24)
n
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ThepressureateverypointalongtheshellaxisisgivenbytheBernoulliequation:
(cid:11) (cid:12)(cid:24)
∂φ ∂φ (cid:24)
Δp = −ρ ∂t +U∞∂x (cid:24)(cid:24)r=R (25)
After taking the transform of Eq. 25 and using Eq. 24, the transform of the aerodynamic pressure is
obtained: (cid:11) (cid:12)
Δp∗∗ = −ρR(p+U∞s)2w∗∗ γKKn((cid:4)γ(γ)) (26)
n
where γ = Rδ. Following Krumhaar8 and Bateman et al.,16 the asymptotic expansion of the quantity in
parentheses(forlarge γ)is:
(cid:11) (cid:12)
γKKn((cid:4)γ(γ)) = −γ−1+ 12γ−2+ n22 − 38 γ−3+... (27)
n
This expansion can be further simplified by applying the Dowell and Bliss method,15 whereby algebraic
manipulationisusedtore-writeeachterminEq.27,andapplyingthebinomialtheorem. Thisprocedure
isoutlinedintheappendix. Theresultingtransformofthepressure,includingtermsuptoO(1/M5) is:
(cid:25) (cid:3) (cid:4)
Δp∗∗ = ρU∞2W∗∗ M1 s+ Up∞ − 2R1M2(cid:3)+ 2M(cid:4)3(ss+2 (cid:19)Up∞) − R2Mn322(−s+38Up∞)
− s2 − 3 n22−38 s2 (28)
2RM4(s+Up∞)2 2R2M5(s+Up∞)3
ThefirsttwotermsinEq.28correspondtoclassicalPistonTheoryandKrumhaar’scurvaturecorrection
factor. To form the aerodynamic generalized forces, the transform of the pressure must be inverted back
to physical space and time. It is convenient; however, to bypass the spatial inversion of the pressure and
insteadinvertthetransformofthegeneralizedforcesinthefollowingmanner:
1. InvertEq.28withrespecttotime. Requiresaconvolutionintegralofthespatialtransformofdeflec-
tionandthetemporalexponentialintegrals.
2. ApplytheBromwichintegralinFourierspace,usings =iαandx = (y−y1)cosβ(changeofvariable
forconicalshellreferenceframe):
(cid:7)
cosβ ∞ Δp∗αeiα(y−y1)cosβdα (29)
2π −∞
Notethatthisintegralisnotcalculatedexplicitlyuntilstep4.
3. Perform all spatial integrations, including the transform of the deflection and the axial and circum-
ferentialintegralsfromEq.13.
4. Integratewithrespectto α overtherealinfinitedomain.
Iftheorderoftheaboveprocedureisfollowed,onlyintegrationwithrespecttoαandτ (temporalintegra-
tionvariable)mustbecomputednumerically. Theinfinitelimitsfortheintegralwithrespectto α maybe
truncated based on how the integrand decays, which depends on the axial mode number, Mach number,
speedofsound,andshellgeometry. AplotoftheintegrandoftheKernelfunctionvs. α isgiveninFig.3
for the 3.7 meter shell at Mach 2, with a∞=340 m/s. For the first three modes shown, integration may be
truncated at α = 30, but limits must be extended for higher modes. The integrand of the Kernel function
isalwayssymmetricabout α =0.
Theresultingcorrectionfactorfortheaerodynamicgeneralizedforcesis:
(cid:7)
QA,corr =2q cosβ∑∑Ξ1 tc (t)A (t−τ)dτ (30)
sk A nk mn mnsk
n m 0
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1.5
Integrand of A
1010
Integrand of A
2010
1 Integrand of A
3010
A
f
o
d
n 0.5
a
r
g
e
t
n
I
0
−0.5
−30 −20 −10 0 10 20 30
α
Figure(cid:1)3:(cid:1)Integrand(cid:1)of(cid:1)the(cid:1)Aerodynamic(cid:1)kernel(cid:1)function(cid:1)vs.(cid:1)the(cid:1)Fourier(cid:1)variable(cid:1)α,(cid:1)3.7-meter(cid:1)shell,(cid:1)M =(cid:1)2,(cid:1)
L(cid:1)
a∞(cid:1)=(cid:1)340(cid:1)m/s.
( )
ThecompleteexpressionfortheKernel A t isgivenintheappendix. Notethatsincethisexpression
mnsk
is a function of the circumferential mode number, it is effectively a three-dimensional extension of Piston
Theory.
PlotsoftheKernelasafunctionoftimearegiveninFig.4,showingtheinfluenceofthefirstthreeaxial
modes on the first axial mode. The Kernel decays rapidly to zero at a characteristic time that depends
on the aerodynamic and structural parameters, but is mostly influenced by the Mach number. The case
shown in Fig. 4 has the same parameters as Fig. 3. Here, the characteristic time is approximately 0.00068
seconds. This is the amount of deflection time history required for numerical bookkeeping in order to
computetheconvolutionintegralinEq.30.
IV. Results
ResultsarecomputedforaspecificsetoftestcasesdevelopedincoordinationwithHIADinvestigators
at the NASA Langley Research Center. The first set of results are from validation studies, ensuring that
the nonlinear buckling and aeroelastic simulations are consistent with the linear theory in Part I.1 Next,
deflection shapes and strains for a static load simulation are presented, corresponding to experimental
tests performed on a 6-meter HIAD article. Aeroelastic limit cycle oscillations are then calculated for
various cases, and the effects of axially-applied tension are examined for the 3-meter geometry. Finally,
pre-buckling pressure loads and the extended potential flow aerodynamic model are included in the
aeroelasticsimulationsandresultsarecalculatedfora3.7-metershell.
A summary of the numerical test cases studied in this paper (in the order presented) are given in
−
Table 1. Simply-supported modal functions are used in all computations, as indicated by the “ 1” iden-
tifier in the model description (see Part I1). Note that almost all of the cases presented are for model B,
which is the elastically-supported conical shell. Results for model A, the single shell, are only included
forcomparisonintheparameterstudyexaminingtheeffectsofaxially-appliedtension.
IV.A. Verification,ConvergenceStudiesandLinear/NonlinearModelComparison
To verify that the time-integration scheme has been implemented correctly, various comparisons were
madewiththelineartheoryusedinPartI.1 Notethatthislinearanalysiswasfoundtobeconsistentwith
previouslypublishedresultsonthefreevibrationsandflutterofconicalshells.
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