Reduced-Order Models for the AeroelasticAnalysis of Ares Launch Vehicles

Walter A. Silva�

Veer N. Vatsay

Robert T. Biedronz

NASA Langley Research Center, Hampton, VA

This document presents the development and application of unsteady aerodynamic,structural dynamic, and aeroelastic reduced-order models (ROMs) for the ascent aeroe-lastic analysis of the Ares I-X ight test and Ares I crew launch vehicles using theunstructured-grid, aeroelastic FUN3D computational uid dynamics (CFD) code. Thepurpose of this work is to perform computationally-e�cient aeroelastic response cal-culations that would be prohibitively expensive via computation of multiple full-orderaeroelastic FUN3D solutions. These e�cient aeroelastic ROM solutions provide valuableinsight regarding the aeroelastic sensitivity of the vehicles to various parameters over arange of dynamic pressures.

Note To ReadersThe predicted performance and certain other fea-

tures and characteristics of the Ares I and Ares I-Xlaunch vehicles are de�ned by the U.S. Government tobe Sensitive but Unclassi�ed (SBU). Therefore, detailshave been removed from all plots and �gures.

Introduction

AT present, the development of CFD-basedreduced-order models (ROMs) is an area of active

research at several government, industry, and aca-demic institutions.1{7 Development of ROMs basedon the Volterra theory is one of several ROM methodscurrently under development.8{12

Silva and Bartels4 introduced the development oflinearized, unsteady aerodynamic state-space modelsfor prediction of utter and aeroelastic response usingthe parallelized, aeroelastic capability of the CFL3Dv6code. The results presented provided an importantvalidation of the various phases of the ROM devel-opment process. In Silva and Bartels,4 the Eigen-system Realization Algorithm (ERA),13 which trans-forms an impulse response (one form of ROM) intostate-space form (another form of ROM), was ap-plied for the development of the aerodynamic state-

�Senior Research Scientist, Aeroelasticity Branch, AIAA As-sociate Fellow.ySenior Research Scientist, Computational Aerosciences

Branch, AIAA Associate Fellow.zSenior Research Scientist, Computational Aerosciences

Branch, AIAA Senior Member.Copyright c by the American Institute of Aeronautics and

Astronautics, Inc. No copyright is asserted in the United Statesunder Title 17, U.S. Code. The U.S. Government has a royalty-free license to exercise all rights under the copyright claimed hereinfor Governmental Purposes. All other rights are reserved by thecopyright owner.

space models. The ERA is part of the SOCIT(System/Observer/Controller Identi�cation Toolbox).Flutter results for the AGARD 445.6 Aeroelastic Wingusing the CFL3Dv6 code were presented as well, in-cluding computational costs. Unsteady aerodynamicstate-space models were generated and coupled witha structural model within a MATLAB/SIMULINK14

environment for rapid calculation of aeroelastic re-sponses including the prediction of utter. Aeroelasticresponses computed directly using the CFL3Dv6 codeshowed excellent comparison with the aeroelastic re-sponses computed using the CFD-based ROM.

A primary purpose of this e�ort is to provide theAres I-X ight test and the Ares I crew launch vehi-cle projects with cost-e�ective analyses that provideinsight into the aeroelastic behavior of this class oflaunch vehicles. The present analysis develops and ap-plies aeroelastic ROMs that are based on CFD aeroe-lastic analyses that fully couple the ow �eld and thestructural exibility. These aeroelastic ROMs are gen-erated at each Mach number of interest along the ighttrajectory and are valid for a limited range of vehi-cle deformations and a range of dynamic pressuresat that Mach number. Due to the nature of launchvehicles, the mode shapes and frequencies of the ve-hicles are di�erent at each Mach number of interest.The application of ROMs for aeroelastic analyses canyield signi�cant computational e�ciency over the moretraditional method using full-order CFD aeroelasticsolutions.

The resultant aeroelastic ROMs are in a state-spaceformat suitable for use by other disciplines. There-fore, a secondary purpose of this e�ort is to providethe Guidance, Navigation, and Control group withROMs that may be suitable for inclusion into their full

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28th AIAA Applied Aerodynamics Conference28 June - 1 July 2010, Chicago, Illinois

AIAA 2010-4375

This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.

ascent vehicle simulation. The use of ROMs withinthis simulation would provide insight regarding thedynamic aeroelastic response of the vehicle along tra-jectory ight conditions.

Description of CFD and SystemIdenti�cation Methods

The following subsections brie y describe the aeroe-lastic version of the FUN3D code15 and the systemidenti�cation methods used in the ROM developmentprocess.

FUN3D Code and Grid

The development of ROMs is based on the use of theFUN3D code (version 10.5) to compute the ow�eld.FUN3D is a parallel, unstructured computational uiddynamics code that supports meshes containing anycombination of hexahedra, tetrahedra, prisms, andpyramids. The code is a node centered �nite vol-ume Euler and Navier-Stokes ow solver that includesthe Spalart-Allmaras (SA), the Menter shear stresstransport (SST), the Wilcox k-omega and the Abidk-epsilon turbulence models. In the present compu-tations the SA and the SST models have been used.The code has various options for computing the in-viscid ux quantities across volume faces, namely thevan Leer ux vector splitting, the Roe ux di�erencesplitting, the HLLC, the AUFS, LDFSS and centraldi�erencing. In the present computations the Roe uxdi�erence splitting was used. Several ux limiters arealso available such as the minmod Barth, Venkatakr-ishnan, van Leer, van Albada and the smooth limiter.For the present computations below Mach 0.9 no lim-iter was used and at Mach 0.9 and above the minmodlimiter was used. The present computations have beenperformed on the NASA Advanced Supercomputing(NAS) RTJones and Columbia systems and the NASALangley Research Center K cluster. The grids werepartitioned among 240 processors.

The baseline unstructured mesh had 23 millionnodes. This size baseline mesh was chosen based on ex-perience performing steady CFD for the Ares I projectwith FUN3D. Figure 1 is an image of the Ares I vehicleand Figure 2 is an image of the forward portion of theAres I with a sample surface grid.

The FUN3D code can be used to generate un-steady aerodynamic responses (generalized aerody-namic forces or GAFs) via the forcing of the modeshapes. This is the process that is used for the genera-tion of the unsteady aerodynamic ROM. The FUN3Dcode can also be used to generate aeroelastic responsesas iterations between the unsteady aerodynamics andthe modal structural model. These aeroelastic re-sponses from the FUN3D code are referred to as fullaeroelastic responses.

Fig. 1 Image of Ares I.

Fig. 2 Image of forward portion of Ares I withsample surface grid.

System Identi�cation Method

The development of discrete-time state-space mod-els that describe the modal dynamics of a structure hasbeen enabled by the development of algorithms suchas the Eigensystem Realization Algorithm (ERA)13

and the Observer Kalman Identi�cation (OKID)16 Al-gorithm. These algorithms perform state-space real-izations by using the Markov parameters (discrete-time impulse responses) of the systems of interest.These algorithms have been combined into one packageknown as the System/Observer/Controller Identi�ca-tion Toolbox (SOCIT)17 developed at NASA LangleyResearch Center.

There are several algorithms within the SOCIT thatare used for the development of unsteady aerody-namic discrete-time state-space models. The PULSEalgorithm is used to extract individual input/outputimpulse responses from simultaneous input/output re-sponses. For a four-input/four-output system, simul-taneous excitation of all four inputs1 yields four outputresponses. The PULSE algorithm is used to extractthe individual sixteen (four times four) impulse re-sponses that associate the response in one of the out-puts due to one of the inputs. Details of the PULSE

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algorithm are provided in the references. Once the in-dividual sixteen impulse responses are available, theyare then processed via the Eigensystem RealizationAlgorithm (ERA) in order to transform the sixteenindividual impulse responses into a four-input/four-output, discrete-time, state-space model. A brief sum-mary of the basis of this algorithm follows.

A �nite dimensional, discrete-time, linear, time-invariant dynamical system has the state-variableequations

x(k + 1) = Ax(k) + Bu(k) (1)

y(k) = Cx(k) + Du(k) (2)

where x is an n-dimensional state vector, u an m-dimensional control input, and y a p-dimensional out-put or measurement vector with k being the discretetime index. The transition matrix, A, characterizesthe dynamics of the system. The goal of system real-ization is to generate constant matrices (A, B, C) suchthat the output responses of a given system due to aparticular set of inputs is reproduced by the discrete-time state-space system described above.

For the system of Eqs. (1) and (2), the time-domainvalues of the systems discrete-time impulse responseare also known as the Markov parameters and are de-�ned as

Y (k) = CAk�1B (3)

with B an (n x m) matrix and C a (p x n) matrix.The ERA algorithm begins by de�ning the generalizedHankel matrix consisting of the discrete-time impulseresponses for all input/output combinations. The al-gorithm then uses the singular value decomposition(SVD) to compute the A, B, and C matrices.

In this fashion, the ERA is applied to unsteadyaerodynamic impulse responses to construct unsteadyaerodynamic state-space models.

ROM Development ProcessUnsteady Aerodynamic ROM

An outline of the improved simultaneous modal ex-citation ROM development process is as follows (seeFigure 3):

1. Generate the number of functions (from a se-lected family) that corresponds to the number of struc-tural mode shapes;

2. Apply the generated input functions simultane-ously via one FUN3D execution; these responses arecomputed directly from the restart of a steady rigidFUN3D solution (not at a particular dynamic pres-sure); for the present results, Walsh functions are usedas the input functions;

3. Using the simultaneous input/output responses,identify the individual impulse responses using thePULSE algorithm (within SOCIT);

4. Transform the individual impulse responses gen-erated in Step 3 into an unsteady aerodynamic state-space system using the ERA (within SOCIT);

Once the unsteady aerodynamic state-space ROM isgenerated (Step 4), the state-space model is validatedvia comparison with FUN3D results (i.e., ROM resultsvs. full FUN3D solution results). Additional ROMenhancements and capabilities are described in greaterdetail in the references.2

Fig. 3 Schematic of Unsteady Aerodynamic ROMProcess.

An important di�erence between the original ROMprocess and the improved ROM process is stated instep (2) of the outline above. For the original ROMprocess, if a static aeroelastic condition existed, then aROM was generated about a selected static aeroelasticcondition. So a static aeroelastic condition of interestwas de�ned (typically a dynamic pressure) and thatstatic aeroelastic condition was computed using theaeroelastic CFD code as a restart from a convergedsteady, rigid solution. Once a converged static aeroe-lastic solution was obtained, the ROM process wasapplied about that condition. This implies that theresultant ROM is, of course, limited in some sense tothe neighborhood of that static aeroelastic condition.Moving "too far away" from that condition could re-sult in loss of accuracy.

The reason for generating ROMs in this fashionwas because no method had been de�ned to enablethe computation of a static aeroelastic solution usinga ROM. Any ROMs generated in this fashion were,therefore, limited to the prediction of dynamic re-sponses about a static aeroelastic solution includingthe methods by Kim et al6 and by Raveh.7 The im-proved ROM method, however, includes a method forgenerating a ROM directly from a steady, rigid solu-tion. As a result, these improved ROMs can then beused to predict both static aeroelastic and dynamicsolutions for any dynamic pressure. In order to cap-ture a speci�c range of aeroelastic e�ects (previouslyobtained by selecting a particular dynamic pressure),the improved ROM method relies on the excitationamplitude to excite aeroelastic e�ects of interest. Thedetails of the method for using a ROM for computingboth static aeroelastic and dynamic solutions is pre-

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sented in another reference by the �rst author.2 Forthe present results, all responses were computed fromthe restart of a steady, rigid FUN3D solution, bypass-ing the need (and additional computational expense)to execute a static aeroelastic solution using FUN3D.

The Walsh functions that are applied simultane-ously to the aeroelastic FUN3D code in order to ex-cite all of the structural modes are generated basedon the time step de�ned for the aeroelastic analy-sis. Typically, the full FUN3D static and dynamicaeroelastic solutions (each one generated separately)are computed using di�erent time steps. The fullFUN3D dynamic aeroelastic solution typically has thelargest number of time steps so that su�cient cyclesof the aeroelastic response are generated to provideadequate frequency information when post-processed.The size of the time steps for both the static and dy-namic aeroelastic solutions are generally de�ned aslarge as possible for computationally e�cient solu-tions. For the generation of the unsteady aerodynamicROM, however, a time step is de�ned that is typi-cally smaller (by an order of magnitude or so) so thatrelevant frequencies and associated dynamics are ac-curately resolved by the SOCIT tools due to Walshinput functions. For this reason, the single FUN3Dsolution to obtain the Walsh responses may take aslong (CPU time) as a full FUN3D aeroelastic solution.However, the bene�t of generating the unsteady aero-dynamic ROM is that it can then be used to rapidlygenerate aeroelastic solutions at any other dynamicpressure and velocity (for a given Mach number) aswell as any variation in structural parameters (modaldamping, modal frequencies).

It should be mentioned that a full FUN3D aeroe-lastic solution for a given Mach number, dynamicpressure, and velocity consists of the computation of asteady rigid solution at that Mach number, followed bya static aeroelastic solution at that Mach number, dy-namic pressure, and velocity, which is then followed bya dynamic aeroelastic solution at the same conditions.The steady rigid solution is computed rapidly as thatsolution is independent of aeroelastic responses andtherefore independent of expensive mesh deformationiterations. The static aeroelastic response, restartedfrom the steady rigid solution, does require mesh defor-mations but an arti�cially high value of modal damp-ing is used to accelerate the solution to a convergedstatic aeroelastic response. The dynamic aeroelasticsolution, restarted from the converged static aeroelas-tic solution, requires the most computational time dueto the need for a certain number of cycles to accu-rately de�ne dominant frequencies in the aeroelasticresponse. The computational cost of a ROM FUN3Dsolution (solution with Walsh functions applied) is onthe order of a full FUN3D dynamic aeroelastic solu-tion. Additional details regarding full FUN3D aeroe-lastic solutions can be found in the reference by Bartels

et al.18

Structural Dynamic ROM

Normal modes analysis was performed withMSC.Nastran. Modal de ections were obtained bycreating nodes at stations along the centerline of thevehicle at which the average of the circumferentialouter mold line de ections at each axial station wascomputed using a wagon wheel interpolation connec-tion to the outer mold line of the vehicle. The averagede ection at the centerline node points was used tocreate the projection of the centerline modal de ec-tions onto the CFD surface. The interpolation fromthe FEM node points to the CFD surface mesh pointswas accomplished by a spline �t, except at CFD meshpoints beyond the �rst or last FEM node points wherea quadratic extrapolation of the FEM data was used.

A total of forty-four exible modes were used in thefull FUN3D aeroelastic analyses of the Ares I-X ve-hicle, thirty-seven exible modes were used in the fullFUN3D aeroelastic analyses of the Ares I vehicle, whileonly thirteen exible modes where used for the ROMaeroelastic analyses for both vehicles. The �rst thir-teen exible modes were used in the development ofROMs in order to be consistent with the number ofmodes being considered in related GNC analyses. Forthe majority of cases, no signi�cant di�erences wereidenti�ed either between the Ares I-X full FUN3D andROM aeroelastic analyses or between the Ares I fullFUN3D and ROM aeroelastic analyses.

Structural information from the Finite ElementAnalysis (MSC NASTRAN) such as generalized mass,generalized damping, and generalized frequencies(sti�ness) is used to generate a state-space model ofthe structural dynamic system using MATLAB (seeFigure 4). Once the ABCD matrices of the unsteadyaerodynamic system are generated for a given Machnumber, the unsteady aerodynamic system is com-bined with a set of ABCD matrices that de�ne thestructural dynamics. The combined use of these twosystems leads to the simulation of the aeroelastic sys-tem as the structural and aerodynamic systems inter-act.

Fig. 4 Schematic of Structural Dynamic ROMProcess.

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Aeroelastic ROM

The MATLAB/SIMULINK environment from theMathworks, Inc., is used to connect the state-spacemodel of the unsteady aerodynamics and the state-space model of the structural dynamics into an aeroe-lastic reduced order model as shown in Figure 5. Inaddition, the structural dynamic model can be easilyaltered to simulate various values of structural damp-ing and variations in modal frequencies. Simulation ofthe aeroelastic system with these variations providesa sensitivity study of the aeroelastic system to theseparametric variations. For the computation of theaeroelastic root loci, the state-space ROM of the un-steady aerodynamic system and the state-space ROMof the structural dynamic system are mathematicallycombined into a single state-space model.

Fig. 5 Schematic of combined structural/unsteadyaerodynamic ROM system, referred to as theAeroelastic ROM.

ResultsThe goal of this section is to present a sample of

results obtained in the process of generating and usingan unsteady aerodynamic ROM and, subsequently, anaeroelastic ROM. Using these methods, aeroelastic re-sults are then presented for the Ares I-X and the AresI vehicles at various Mach numbers. It is important toreiterate that results presented are for analyses usingthe �rst thirteen exible modes (no rigid-body modes).

When completed, the FUN3D Walsh solution is pro-cessed through MATLAB-based scripts that provideinformation regarding the error level of the subsequentunsteady aerodynamic ROM as compared to the fullFUN3D Walsh solution. Samples of the time-domainresponses due to the Walsh inputs for the full FUN3Dsolution and for the unsteady aerodynamic ROM arepresented in Figure 6 for the �rst exible mode and inFigure 7 for the second exible mode. For the sake of

Fig. 6 Comparison of time-domain responses inthe �rst exible mode due to Walsh input functionsfor full FUN3D solution (blue) and for unsteadyaerodynamic ROM (green).

clarity, discussions regarding the modes will focus oneach individual exible mode and not on the exiblemode pairs. That is, in some references, the �rst andsecond exible mode comprise the �rst bending modepair and are referred to as such.

Fig. 7 Comparison of time-domain responses inthe second exible mode due to Walsh input func-tions for full FUN3D solution (blue) and for un-steady aerodynamic ROM (green).

The mean error and maximum percent error for thetime-domain solutions for all thirteen exible modesare presented in Figure 8. These errors are computedper mode (generalized coordinate) in order to bet-ter understand the impact of the error on the overallaeroelastic solution.

An analogous comparison of these responses is alsoviewed in the frequency domain. Presented in the up-

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Fig. 8 The mean error and maximum percenterror for the time-domain solutions.

per plot of Figure 9 is a comparison of the frequencycontent of the responses presented in Figure 6. Pre-sented in the lower plot of Figure 9 is the frequencycontent of the input Walsh function for the �rst exiblemode.

Presented in Figure 10 are analogous plots corre-sponding to Figure 7 and the input Walsh function forthe second exible mode.

Fig. 9 (Upper plot): Comparison of frequency-domain responses in the �rst exible mode dueto Walsh input functions for full FUN3D solu-tion (blue) and for unsteady aerodynamic ROM(green). (Lower plot): Frequency content of Walshinput function applied to �rst exible mode.

Parameters within the PULSE and ERA algorithmsthat are used to generate the discrete-time state-spacemodel are varied accordlingly until an acceptable er-ror level is achieved. Upon achieving an acceptableerror level (on the order of 1 percent or less in maxi-mum percent error), the unsteady aerodynamic ROM

Fig. 10 (Upper plot): Comparison of frequency-domain responses in the second exible mode dueto Walsh input functions for full FUN3D solu-tion (blue) and for unsteady aerodynamic ROM(green). (Lower plot): Frequency content of Walshinput function applied to second exible mode.

is combined with the structural dynamic ROM to cre-ate the aeroelastic ROM. The aeroelastic ROM is thenused to predict the aeroelastic response at various dy-namic pressures, velocities, and structural dynamicparameter variations. These variations would requireindividual FUN3D solutions that could take on the or-der of days to compute while taking only seconds usingthe aeroelastic ROM.

Ares I-X

An important contribution of this e�ort is the abilityto rapidly generate root locus plots of the aeroelasticbehavior of the vehicle. Presented in Figure 11 is aroot locus plot in terms of frequency versus dampingratio for the �rst thirteen exible modes of the AresI-X vehicle at a Mach number (M) of 0.5 and severaldynamic pressures. Migration of the roots toward theright of the zero dynamic pressure value (Q=0 psi)indicates reduced damping (less stable); migration tothe left of the zero dynamic pressure value indicatesincreased damping (more stable). Due to the struc-tural symmetry of these launch vehicles, some of the exible modes are very similar to each other. Thatis, the �rst exible mode is a �rst-bending mode inthe longitudinal axis while the second exible mode isa �rst-bending mode in the lateral axis. As a result,some of the roots (symbols) in the root loci plots forsome of these modes may be very close together or farapart, depending on the nature of the aeroelastic re-sponse. As can be seen, at this condition, both the�rst and second exible modes exhibit a destabilizinge�ect as demonstrated by a migration of the roots tothe right of the Q=0 values. A comparison of the fullFUN3D aeroelastic response and the ROM aeroelastic

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response at this Mach number and nominal dynamicpressure is presented as Figure 12 where good compar-ison between the two responses is evident.

Fig. 11 Dynamic pressure root loci from aeroelas-tic ROM for nominal modal frequencies at M=0.5for the Ares I-X vehicle.

Via a simple modi�cation of the structural dynamicROM, the modal frequencies can be varied to accountfor a possible variation. Figure 13 presents the M=0.5root locus plot for the condition where the modal fre-quencies have been increased by 10 percent (�rst andsecond modes) and by 20 percent (remaining modes).As can be seen, the e�ect of the increased frequenciesis to provide a slight sti�ening of the overall aeroelasticresponse. This slight sti�ening is evident by comparingFigure 11 with Figure 13 showing a slightly decreasedrange of damping values for most of the modes wherethe scale is the same for both plots.

Figure 14 presents the root locus plot for the condi-tion where the modal frequencies have been decreasedby 10 percent (�rst and second modes) and by 20 per-cent (remaining modes). The decrease in the modalfrequencies yields a slight softening of the aeroelas-tic response resulting in a greater destabilizing e�ectfor the �rst two modes. This slight softening is evi-dent by comparing Figure 11 with Figure 13 showinga slightly increased range of damping values for mostof the modes where, again, the scale is the same forboth plots. There is clearly a tendency towards desta-bilizing behavior at this Mach number.

At M=0.95, a di�erent aeroelastic behavior is ob-served for the Ares I-X vehicle. As can be seen inFigure 15, there is an increase in stability with an in-crease in dynamic pressure at this Mach number forthe nominal modal frequencies. For an increase in themodal frequencies identical to that performed previ-ously, no detrimental e�ect is noticed at this Machnumber as seen in Figure 16. However, for the case ofdecreased modal frequencies, reduced stability is ob-served for the twelfth exible mode as indicated bythe migration of the roots of that mode to the right of

Fig. 12 Comparison of FUN3D and ROM aeroe-lastic responses at M=0.5 and nominal dynamicpressure for the �rst generalized coordinate (�rstelastic mode).

Fig. 13 Dynamic pressure root loci from aeroelas-tic ROM for increased modal frequencies at M=0.5for the Ares I-X vehicle.

the Q=0 values, as seen in Figure 17. This indicates apotential sensitivity of that mode to slight variationsin its modal frequency. Although not presented here,these aeroelastic ROM results compared very well withresults from full FUN3D analyses at this condition.

Analysis at a higher Mach number of 1.44 indicatesthat there is no e�ect on the stability of the vehicle dueto variations in the modal frequencies. The result fornominal modal frequencies is shown in Figure 18, whilethe result for increased modal frequencies is shown inFigure 19 and that for decreased modal frequenciesis shown in Figure 20. As can be seen, there is nomigration of any of the roots to the right of the Q=0values.

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Fig. 14 Dynamic pressure root loci from aeroelas-tic ROM for decreased modal frequencies at M=0.5for the Ares I-X vehicle.

Fig. 15 Dynamic pressure root loci from aeroelas-tic ROM for nominal modal frequencies at M=0.95for the Ares I-X vehicle.

Ares I

Unsteady aerodynamic and aeroelastic ROMs alsowere generated for the Ares I vehicle at several Machnumbers and for a speci�c variation in the structuraldynamics of the vehicle. This structural dynamic vari-ation is due in part to the inclusion of a dual-planeThrust Oscillation Isolator (TOI) modi�ed model andit results in a di�erent structural dynamic representa-tion than the baseline Ares I vehicle model. Althoughthe intent of the TOI system is to reduce vibrations as-sociated with the propulsion system, the TOI systemas modeled actually introduces additional exibilitythat has a detrimental e�ect on aeroelastic stability.In addition to the exibility introduced by the TOIsystem, the baseline Ares I vehicle is more exible thanthe Ares I-X vehicle. In this section, results are pre-sented for the M=1.00 condition.

Presented in Figure 21 is a comparison of the full

Fig. 16 Dynamic pressure root loci from aeroe-lastic ROM for increased modal frequencies atM=0.95 for the Ares I-X vehicle.

Fig. 17 Dynamic pressure root loci from aeroe-lastic ROM for decreased modal frequencies atM=0.95 for the Ares I-X vehicle.

Fig. 18 Dynamic pressure root loci from aeroelas-tic ROM for nominal modal frequencies at M=1.44for the Ares I-X vehicle.

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Fig. 19 Dynamic pressure root loci from aeroe-lastic ROM for increased modal frequencies atM=1.44 for the Ares I-X vehicle.

Fig. 20 Dynamic pressure root loci from aeroe-lastic ROM for decreased modal frequencies atM=1.44 for the Ares I-X vehicle.

FUN3D and the ROM aeroelastic responses at M=1.00for the �rst exible mode of the baseline Ares I vehicle.As can be seen, the comparison is very good with theROM result only slightly less stable (less damping)than the full FUN3D solution.

The root loci at M=1.00 for the baseline Ares I ve-hicle is presented in Figure 22. For this version ofthe vehicle, it can be seen that most of the aeroelasticroots migrate to the left of the Q=0 points, indicatingincreased damping, and, therefore, increased stability.However, for the �rst mode there appears to be a rootmigration to the right, indicating decrease dampingand, therefore, decreased stability.

Presented in Figure 23 is a comparison of the fullFUN3D and the ROM aeroelastic responses at M=1.00for the �rst exible mode of the TOI-modi�ed Ares Ivehicle. Once again, the comparison is very good withthe ROM result only slightly less stable (less damping)than the full FUN3D solution.

Finally, the root loci at M=1.00 for the TOI-

Fig. 21 Comparison of full FUN3D and ROMaeroelastic responses at M=1.00 for the �rst ex-ible mode of the baseline Ares-I vehicle (ROM-Green; FUN3D-Blue).

Fig. 22 Dynamic pressure root loci from aeroelas-tic ROM at M=1.00 for the baseline Ares I vehicle.

modi�ed Ares I vehicle is presented in Figure 24. Bycomparing this root loci with that for the baselineAres I vehicle, it can be seen that modi�cation ofthe Ares I vehicle to include the TOI system tendsto reduce the stability of the �rst exible mode asindicated by the migration of the roots of the �rst exible mode to the right of the Q=0 point beyondthe point seen in Figure 22. The rapid assessment ofthis reduced-stability condition was an important con-tribution of the ROM methodology as applied to thesevehicles. Subsequent full FUN3D aeroelastic solutionscon�rmed these ROM aeroelastic results.

Concluding RemarksMethods for generating unsteady aerodynamic

reduced-order models (ROMs) and aeroelastic ROMshave been presented. These methods were appliedtowards the development of unsteady aerodynamic,structural dynamic, and aeroelastic ROMs for the Ares

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Fig. 23 Comparison of full FUN3D and ROMaeroelastic responses at M=1.00 for the �rst ex-ible mode of the TOI-modi�ed Ares-I vehicle(ROM-Green; FUN3D-Blue).

Fig. 24 Dynamic pressure root loci from aeroelas-tic ROM at M=1.00 for the TOI-modi�ed Ares Ivehicle.

I-X and Ares I launch vehicles in order to quicklygenerate aeroelastic response information. A sampleof results presented included error minimization tech-niques, comparison of aeroelastic responses for the fullFUN3D solution versus the aeroelastic ROM solution,and root locus plots for parametric variations of modalfrequency. The ability to rapidly generate aeroelasticroot loci as a function of dynamic pressure at a givenMach number provided valuable insight regarding thenature of the aeroelastic response of these launch vehi-cles. Of particular signi�cance is the fact that aeroelas-tic interactions amongst the various modes for launchvehicles is unlike the typical aeroelastic interactions oflifting surfaces. Whereas the aeroelastic interactions ofa lifting surface may reveal a coalescence of two modesoriginally at di�erent frequencies, the aeroelastic inter-actions for these launch vehicles consist of a coupling ofmodes at very nearly the same frequency with minimal

variation in frequency for all modes as a function of dy-namic pressure. This should not be surprising as lauchvehicles are not lifting surfaces. Another importantcontribution of this ROM methodology was the iden-ti�cation of reduced aeroelastic stability of the Ares Ivehicle with the TOI system at a transonic Mach num-ber. The methods were shown to be a powerful toolto enable the rapid assessment of aeroelastic behaviorof this class of launch vehicles.

AcknowledgementThis work was performed in support of and funded

by the Ares I-X and the Ares I Program O�ces.

References1Silva, W. A., \Simultaneous Excitation of Multiple-

Input/Multiple-Output CFD-Based Unsteady AerodynamicSystems," Journal of Aircraft , Vol. 45, No. 4, July-August 2008,pp. 1267{1274.

2Silva, W. A., \Recent Enhancements to the Develop-ment of CFD-Based Aeroelastic Reduced Order Models,"48th AIAA/ASME/ASCE/AHS/ASC Structures, StructuralDynamics, and Materials Conference, No. AIAA Paper No.2007-2051, Honolulu, HI, April 23-26 2007.

3Silva, W. A., \Identi�cation of Nonlinear Aeroelastic Sys-tems Based on the Volterra Theory: Progress and Opportuni-ties," Journal of Nonlinear Dynamics, Vol. 39, Jan. 2005.

4Silva, W. A. and Bartels, R. E., \Development of Reduced-Order Models for Aeroelastic Analysis and Flutter PredictionUsing the CFL3Dv6.0 Code," Journal of Fluids and Structures,No. 19, 2004, pp. 729{745.

5Beran, P. S. and Silva, W. A., \Reduced-Order Modeling:New Approaches for Computational Physics," Presented at the39th AIAA Aerospace Sciences Meeting, 8-11 January 2001,Reno,NV , January 2001.

6Kim, T., Hong, M., Bhatia, K. G., and SenGupta,G., \Aeroelastic Model Reduction for A�ordable Computa-tional Fluid Dynamics-Based Flutter Analysis," AIAA Journal ,Vol. 43, 2005, pp. 2487{2495.

7Raveh, D. E., \Identi�cation of Computational-Fluid-Dynamic Based Unsteady Aerodynamic Models for AeroelasticAnalysis," Journal of Aircraft , Vol. 41, June 2004, pp. 620{632.

8Silva, W. A., Beran, P. S., Cesnik, C. E. S., Guendel,R. E., Kurdila, A., Prazenica, R. J., Librescu, L., Marzocca,P., and Raveh, D., \Reduced-Order Modeling: CooperativeResearch and Development at the NASA Langley ResearchCenter," CEAS/AIAA/ICASE/NASA International Forum onAeroelasticity and Structural Dynamics, June 2001.

9Silva, W. A., \Reduced-Order Models Based onLinear and Nonlinear Aerodynamic Impulse Responses,"CEAS/AIAA/ICASE/NASA International Forum on Aeroe-lasticity and Structural Dynamics, June 1999.

10Silva, W. A., Discrete-Time Linear and Nonlinear Aero-dynamic Impulse Responses for E�cient CFD Analyses, Ph.D.thesis, College of William & Mary, December 1997.

11Silva, W. A., \Application of Nonlinear Systems Theory toTransonic Unsteady Aerodynamic Responses," Journal of Air-craft , Vol. 30, 1993, pp. 660{668.

12Raveh, D. E., Levy, Y., and Karpel, M., \Aircraft Aeroe-lastic Analysis and Design Using CFD-Based Unsteady Loads,"41st Structures, Structural Dynamics, and Materials Confer-ence, No. 2000-1325, Atlanta, GA, April 2000.

13Juang, J.-N. and Pappa, R. S., \An Eigensystem Realiza-tion Algorithm for Modal Parameter Identi�cation and ModelReduction," Journal of Guidance, Control, and Dynamics,Vol. 8, 1985, pp. 620{627.

14\Registered Product of the MathWorks, Inc." .

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15Biedron, R. T. and Thomas, J., \Recent Enhancements tothe FUN3D Flow Solver for Moving-Mesh Applications," 47thAIAA Aerospace Sciences Meeting, No. 2009-1360, Orlando,FL, Jan. 5-8 2009.

16Juang, J.-N., Phan, M., Horta, L. G., and Longman, R. W.,\Identi�cation of Observer/Kalman Filter Markov Parameters:Theory and Experiments," Journal of Guidance, Control, andDynamics, Vol. 16, 1993, pp. 320{329.

17Juang, J.-N., Applied System Identi�cation, Prentice-HallPTR, 1994.

18Bartels, R., Chwalowski, P., Massey, S., Vatsa, V., Heeg,J., Mineck, R., and Wieseman, C., \Computational Aeroelas-tic Analysis of the Ares Crew Launch Vehicle During Ascent,"28th AIAA Applied Aerodynamics Conference, No. 2010-4374,Chicago, IL, June 28 - July 1 2010.

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