Rotary Wing Aeroelasticity-A Historical Perspective

JOURNAL OF AIRCRAFT

Vol. 40, No. 6, November–December 2003

Rotary Wing Aeroelasticity—A Historical Perspective

Peretz P. FriedmannThe University of Michigan, Ann Arbor, Michigan 48109-2140

andDewey H. Hodges

The Georgia Institute of Technology, Atlanta, Georgia 30332-0150

This paper provides a historical perspective of the fundamental developments that have played a centralrole in rotary-wing dynamics and aeroelasticity and have had a major impact on the design of rotary-wingaircraft. The paper describes a historical progression starting with the classical � ap-pitch problem that em-ulated � xed-wing behavior and describes the evolution of the dynamic and aeroelastic problems into thosethat are unique to rotorcraft, such as the � ap-lag problem, the lag-pitch problem, and the coupled � ap-lag-torsional problem. Subsequently, the coupled rotor/fuselage aeromechanical problems such as ground and airresonance are considered. A description of the evolution of the methodology used in the formulation and solu-tion of these types of problems is also provided, emphasizing the structural and aerodynamic models requiredfor their effective formulation and solution. The mathematical techniques used for solving the rotary-wing aeroe-lastic problems in hover and forward � ight are also described. The primary emphasis of the paper is on aeroe-lastic stability, and aeroelastic response is only treated brie� y. The paper focuses on contributions that havehistorical value because they represent landmark treatments. Because of the large amount of material avail-able, an all-inclusive treatment of the research done in this � eld is impractical, and the paper has unavoidableomissions.

Nomenclaturea = acceleration vector[B.Ã/] = tranformation matrix for multiblade

coordinatesb = semichordCd0 = pro� le drag coef� cientC.k/ = Theodorsen’s lift de� ciency functionC 0.k; m; Nhw/ = Loewy’s lift de� ciency function

Peretz P. Friedmann is Francois-Xavier Bagnoud Professor of Aerospace Engineering in the Aerospace Engi-neering Department of the University of Michigan, Ann Arbor. He received his B.S. and M.S. degrees in aero-nautical engineering from the Technion-Israel Institute of Technology, and his Sc.D. (1972) in aeronautics andastronautics from M.I.T. Prior to entering the academic world, Dr. Friedmann worked in Israel Aircraft In-dustries, and was Research Assistant at the Aeroelastic and Structures Laboratory at MIT. He has been withthe University of Michigan since January 1999. Between 1972 and 1998 he was a Professor in the Mechan-ical and Aerospace Engineering Department of the University of California, Los Angeles. Between 1988 and1991 he served as the Chairman of the Department. Dr. Friedmann has been engaged in research on rotary-wing and � xed-wing aeroelasticity, active control of vibrations, hypersonic aeroelasticity, � utter suppression,structural dynamics and structural optimization with aeroelastic constraints and he has published extensively(over 235 journal and conference papers). He was the recipient of the 1984 American Society of Mechani-cal Engineers Structures and Materials Award, and he is a Fellow of AIAA (since 1991). He was the recipi-ent of the AIAA Structures, Structural Dynamics and Materials (SDM) Award for 1996, and the AIAA SDMLecture Award at the 38th SDM Conference, in 1997. He is the recipient of the Spirit of St. Louis Medalfor 2003.

Dewey H. Hodges, Professor of Aerospace Engineering, Georgia Institute of Technology, obtained his B.S. inaerospace engineering in 1969 from the University of Tennessee, and his M.S. in 1970 and Ph.D. in 1973, bothin aeronautics and astronautics from Stanford University. Prior to joining Georgia Tech in 1986, Prof. Hodgeswas Research Scientist at the U.S. Army Aero� ightdynamics Directorate at Ames Research Center for 16 years.He has published over 230 technical papers in journals and conference proceedings in the � elds of rotorcraftaeroelasticity, structural mechanics, dynamics, � nite element analysis, and computational optimal control. Prof.Hodges is a Fellow of the AIAA and a member of the American Helicopter Society, the American Academy ofMechanics, and the American Society of Mechanical Engineers. He is presently a member of the Editorial Boardsof the International Journal of Solids and Structures and the Journal of Engineering Mechanics. He has served asan Associate Editor of AIAA Journal and of Vertica.

Received 30 April 2003; revision received 30 April 2003; accepted for publication 15 July 2003. Copyright c° 2003 by Peretz P. Friedmann and Dewey H.Hodges. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. Copies of this paper may be made for personal or internaluse, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include thecode 0021-8669/03 $10.00 in correspondence with the CCC.

CW = weight coef� cient[C .Ã/] = symbolic matrix, representing linear

damping effectsOex ; Oey; Oez = unit vectors in the directions of the

coordinates, x0 , y0 , z0, respectively beforedeformation

Oe0x ; Oe0

y; Oe0z = triad Oex ; Oey; Oez after deformation

e1 = offset of blade root from axis of rotation

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1020 FRIEDMANN AND HODGES

fFN L .Ã; q; Pqg = complete nonlinear state vector loadingh = plunging motion, used in unsteady

aerodynamicsNhw = .hw =b/ nondimensionalwake spacingI³ = blade inertia about lag hingek; (!b=U ) = reduced frequencyL = unsteady lift, per unit length based on

Greenberg’s theory[L.Ã/] = linear coef� cient matrixl = length of elastic part of the bladem = .!=Ä/ frequency ratioN or Nb = number of bladesfN .q; Ã/g = nonlinear vectorq = unknown state vectorR = blade radiusR = position vector of a mass point of blade

cross section, in blade-� xed,rotating reference frame

[S] = transformationmatrix between triads(Oex ; Oey ; Oez) and (Oe0

x ; Oe0y ; Oe0

z)u; v; w = components of the displacementof a point

on the elastic axis of the blade in directions,Oex ; Oey , and Oez , respectively,subscript kimplies kth blade

V = pulsating � ow velocity in Greenberg’s theory1V = varying part of VV0 = constant part of Vvi = mean induced velocity at the rotor discfXg = generalized coordinate vectorxA = blade cross-sectionalaerodynamic

center (A.C.) offset from elastic axis(E.A.), positive for A.C. before E.A.

fz.Ã/g = known periodic forcing®0 = constant part of pitch, or angle of attack¯ = � ap angle¯p = preconing, inclination of the feathering axis

with respect to the hub plane measured in avertical plane

¯0 = steady � ap angle¯1; ¯2 = rigid-body � apping angle for teetering rotor° = Lock number1® = change in angle of attack for dynamic stall±¸ = perturbation in steady in� ow ratio² = basis for order of magnitude, associated with

typical elastic blade slopes³ = lag angle´SLi = viscous structural damping coef� cients

in percent of critical damping, for thelag modes

µ = total pitch angleµ0 = steady pitch angleµ1s; µ1c = cyclic pitch componentsN = constant part of the in� ow ratio¸1s; ¸1c = cyclic components of in� ow ratio¹ = V cos®R=ÄR advance ratio½A = density of air¾ = blade solidity ratio: blade area/disk areaÁ = rotation of a cross section of the blade around

the elastic axisÃ = azimuth angle of blade (Ã D Ät ) measured

from straight aft position = angular velocity vectorN!F1; N!L1; N!T 1 = First rotating natural frequencies in � ap, lag,

and torsion, respectively, nondimensionalizedwith respect to Ä

!µ = torsional frequency

I. Introduction and Background

T HE 100th anniversary of the Wright brothers’ historic � ightis being celebrated by a variety of events, and several survey

papers dealing with various aspects of aeroelasticity are also beingwritten for this occasion. The present paper focuses on rotary-wingaeroelasticity. Its objective is to provide a historical perspective onthis fascinating � eld.

When reviewing research in rotary-wing aeroelasticity (RWA),it is important to note a few historical facts. The Wright broth-ers � ew in 1903, and Sikorsky built and started � ying the � rstoperational helicopter, the R-4 or (VS-316), in 1942. The R-4was a three-bladed helicopter with a rotor diameter of 11.6 mand was powered by a 185-hp engine. Thus, there is an initialgap of 39 years between � xed-wing and rotary-wing technolo-gies. Therefore, it is not surprising that certain rotary-wing prob-lems, particularly those pertaining to unsteady aerodynamics, arestill not well understood. The situation is further compoundedby the complexity of the vehicle when compared to � xed-wingaircraft.

The � eld of rotary-wing aeroelasticity has been a very activearea of research during the last 40 years. This research activity hasgenerateda largenumberof papers,whichcombinedwith the papersin this area published between 1945–1963, constitutes a large bodyof literature that is impossible to review in a single survey paper.Fortunately,a considerablenumberof reviewpapersand bookshavealso been published.

These review papers, when considered in chronological order,provide a historical perspective on the evolution of the � eld.1¡14

One of the � rst signi� cant reviews of rotary-wing dynamic andaeroelasticproblems was providedby Loewy,12 where a wide rangeof dynamic problems was reviewed in considerabledetail. A morelimited survey emphasizing the role of unsteady aerodynamicsandvibration problems in forward � ight was presented by Dat.2 Twocomprehensivereviewsof rotary-wingaeroelasticitywerepresentedby Friedmann.3;4 In Ref. 3 a detailedchronologicaldiscussionof the� ap-lagand coupled� ap-lag-torsionproblems in hover and forward� ight was presented, emphasizing the inherently nonlinear natureof the hingeless blade aeroelasticstability problem. The nonlineari-ties consideredwere geometricalnonlinearitiescaused by moderateblade de� ections. In Ref. 4, the role of unsteady aerodynamics, in-cluding dynamic stall, was examined, together with the treatmentof nonlinear aeroelastic problems in forward � ight. Finite elementsolutions to RWA problems were also considered, together withthe treatment of coupled rotor-fuselageproblems. Another detailedsurvey by Ormiston13 discussed the aeroelasticityof hingeless andbearingless rotors, in hover, from an experimental and theoreticalpoint of view.

Althoughaeroelasticstabilityplaysan importantrole in thedesignof rotor systems, the aeroelasticresponseproblemas representedbythe rotorcraft vibration and dynamic loads prediction plays an evenmore critical role. Thus, two other surveys have dealt exclusivelywith vibration and its control in rotorcraft.15;16 These papers focuson the vibrations caused by the aeroelastic response of the rotor,and the study of various passive, semiactive, and active devices forcontrolling such vibrations.

Johnson10;11 has published a comprehensive review paper ad-dressing both aeroelastic stability and vibration problems for ad-vanced rotor systems. In a sequel5 to his previous review papers,Friedmann discussed the principal developments that have takenplace between 1983–1987, emphasizingnew methods for formulat-ing aeroelastic problems, advances in treatment of the aeroelasticproblem in forward � ight, coupled rotor-fuselage analyses, struc-tural blade modeling, structural optimization, and the use of activecontrol for vibration reduction and stability augmentation.

A comprehensive report,14 which contains a detailed review ofthe theoretical and experimentaldevelopment in the aeroelasticandaeromechanical stability of helicopters and tilt-rotor aircraft, car-ried out under U.S. Army/NASA sponsorship during the period1967–1987 was prepared by Ormiston et al. Somewhat later, keyideas and developmentsin four speci� c areas— 1) role of geometricnonlinearitiesin RWA, 2) structural modeling of composite blades,3) coupled rotor-fuselage aeromechanical problems and their ac-tive control, and 4) higher harmonic control for vibration reductionin rotorcraft—were considered by Friedmann.6 At the same time

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FRIEDMANN AND HODGES 1021

Chopra1 surveyed the state of the art in aeromechanical stability ofhelicopters, including pitch � ap, � ap lag, coupled � ap lag torsion,air and ground resonance. Advances in aeromechanical analysis ofbearingless, circulation-controlled, and composite rotors were alsotreated in this detailed paper. Perhaps the most comprehensive pa-per on RWA was written by Friedmann and Hodges.9 This papercontains close to 350 references and dwells on all of the importantaspects of rotary-wing aeroelastic stability and response problems.The treatment is broadand comprehensiveand is currentup to 1991.A partial review of some recent developmentscan also be found inRef. 7.

In additionto the numerouspapersdealingwith the subjectof thisreview, this topic is also described in a number of books. Amongthese, the most notable one is Johnson’s17 monumental treatise onhelicoptertheory,which containsextensive,detailed,and usefulma-terialon aerodynamic,dynamic,andmathematicalaspectsof rotary-wing aerodynamics, dynamics, and aeroelasticity. A more recentbook18 treats several aeroelastic and structural dynamic problemsin rotorcraft. Quite recently, Leishman19 has written an excellentbook on helicopter aerodynamics, which contains good treatmentsof unsteady aerodynamics, rotor wake models, and dynamic stall.

The principal objectives of this paper are as follows:1) Present the historical evolution of modern rotary-wing aeroe-

lasticity, starting with the isolated blade aeroelastic problem andprogressing to the coupled rotor fuselage aeromechanical problem.

2) Present the evolution of the methodology for formulation andsolution of rotary-wing aeroelastic problems. The principal focuswill be on aeroelastic stability; therefore, the aeroelastic responseproblem will be mentioned only brie� y.

3) Describe some current trends so as to illustrate considerabledifferences between current and past endeavors.

The paper will not attempt to provide a comprehensive literaturereview of all of the papers published in the � eld. Instead, it willfocus on particular studies that have a historical value because theyrepresent an important contribution to the � eld of RWA.

To understandthe historicaldevelopmentof RWA, it is importantto recognize that the mathematical models capable of simulatingrotary-wingaeroelasticbehaviorwere intimately linked to the typesof helicopter rotors used. The evolutionof the various types of mainrotor systems was the principal driver that provided the impetus forthe developmentof the mathematicalmodeling tools. The � rst gen-eration of helicopters used articulated blades. A typical articulatedrotor hub togetherwith an idealizedrepresentationfor mathematicalmodeling are shown in Fig. 1. For this class of rotors, the dynamicsof the bladeare characterizedby the � ap ¯ , lag ³ , and pitch µ angles,which allow the blade to move as a rigid body.Flexible bending andtorsionaldisplacementcan be added to the displacementsas a resultof the rigid-body motion.

A few years later teeteringrotors, shownin Fig. 2, weredevelopedand used extensively on helicopters manufactured by Bell as well

Fig. 1 Typical articulated hub (top) and typical articulated blademodel (bottom).

Fig. 2 Typical teetering blade model.

a)

b)

Fig. 3 Typical hingeless rotor hub (top) and two views of a typicalhingeless blade used in mathematical modeling (bottom).

as other companies.These blades also have a � apping hinge, exceptthat now the rigid-body� ap angle on the � rst blade is equal and op-posite to that on the second blade, that is, ¯1 D ¡¯2; elastic � ap, lag,and torsional deformation can be superimposed on the rigid-body� apping motion. Teetering rotors were suitable primarily for lighterhelicoptersbecause the size of the blades for heavy helicopters cre-ates almost insurmountabledynamic problems.

The next step in the evolution of rotor systems was the develop-ment of the hingeless rotors shown in Fig. 3. Hingeless rotor con� g-urationsstartedappearingin theearly1960sandbecameoperationalin the late 1960s and early 1970s. Figure 3 depicts a typical exam-ple of a hingeless hub together with a typical model for a hingelessblade. These blades have no � ap or lag hinges. The pitch bearingis still needed to introduce the collective and cyclic components ofpitch.

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Fig. 4 Typical bearingless rotor hub.

The � nal step in the evolution of main rotor systems is the bear-ingless rotor depicted in Fig. 4. Bearingless rotor con� gurationsstarted appearing in the late 1960s and early 1970s. However, theywere incorporated in helicopters that went into production only inthe late 1990s. This rotor has no hinges; both the � ap and lag de-grees of freedom are cantilevered. The pitch bearing is replacedby a � exbeam, and the pitch inputs to the blade are provided byelastically twisting the blade using the pitch horn.

With this background it is now possible to review some of themost important developments in RWA. For convenience, the timeperiod from the mid 1940s to the present is divided into three prin-cipal periods: 1) the early years, 1945–1970, when engineers andresearchers were struggling to accommodate new developments inrotor hardware; 2) the golden age, 1970–2000, when many impor-tant contributionswere made leadingto a much betterunderstandingof the methods for formulating and solving the RWA problem; and3) the 21st century or period of re� nement, 2000–present, when thelarge computing power currently available is utilized to re� ne theaccuracy and reliability of the methods for formulating and solvingaeroelastic problems, by introducing computational aeroelasticityand combining it with control,acoustics,and optimizationin a moregeneral aeromechanical framework.

Each of these periods is considered in detail in the followingsections.

II. Early Years (1945–1970)A. Isolated Blade Stability

The state of the art emerges when reading all of the papers pub-lished during this time period. However, an excellent descriptionofthis periodcan be found in Loewy’s outstandingsurveypaper.12 Theinsight provided by Loewy is augmented by several other surveysthat partiallycover this time period.2;3;20 This was an interestingpe-riod characterizedby rapid hardware developments combined witha lack of sophisticatedmodels capable of replicating the aeroelasticbehavior.The appropriatemethodologyfor formulatingand solvingthe rotary-wingbehaviorwas not well understood,and the � eld wasstrongly in� uenced by the desire to adapt the most successful toolsthat have proven themselves for the � xed-wing static and dynamicaeroelastic problems to the rotary-wing case. Since the majorityof the rotor systems were articulated, the analyses developed wereaimed at modeling the blade con� guration shown in Fig. 1.

A landmark contribution in this area was a paper by Miller andEllis.21 The formulation of the problem was carried out by usingthe direct Newtonian approach and writing the equations of mo-ment equilibrium about the hinge. An important facet of this paper,which was somewhat typical also of other papers generated in thisperiod, was the fact that the individuals associated with the workhad industrial experience and outstanding intuitive understandingof the physics of the problem. Thus, even without achieving a com-pletely accurate formulation(i.e., some terms in the equationscouldbe missing, but they were usually quite small) the conclusions andthe insight provided were usually quite accurate.

The basic problem treated was the coupled � ap-pitch problem,with ¯ and µ degrees of freedom shown in Fig. 1, augmented byblade elastic bending. For aeroelastic stability the emphasis wason hover, using unsteady aerodynamics that representedessentially

Fig. 5 Typical � ap-pitch stability boundaries, showing divergence and� utter as a function of blade c.g. offset from feathering axis xI /c, ° = 12,!¯ = 1, and c/R = 0.05.

a quasi-steady version of Theodorsen theory.22 This resembled theclassicalbending-torsion� utter analysisof a � xed wing, augmentedby the aerodynamic and inertia terms caused by rotation. Bladestability was determined from linear constant coef� cient equations,which resembled the small perturbation equations commonly usedin � xed-wing aeroelasticity.

A typical stability boundary associatedwith this type of analysisis shown in Fig. 5. The stability boundary is plotted by providingthe torsional stiffness (!µ =Ä) in per rev, plotted against the off-set of the cross-sectional center of gravity behind the featheringaxis. Several interesting aspects are noteworthy. Both divergenceand � utter boundariesare evident.Divergencedependson the offsetbetween feathering axis and cross-sectionalc.g. offset. This differsfrom � xed-wing divergence, which depends strictly on offset be-tween elastic axis and aerodynamic center. It can be shown thatTheodorsen-typeunsteadyaerodynamicshas only a minor effect onthe � utter boundary because the reduced frequency ke is low.21 Twoother importanteffects identi� ed in Ref. 21 were the effect of steadyconing and steady in-plane bending. It was noted21 that steady elas-tic � apping de� ections have a minor effect on blade stability for anarticulated rotor. On the other hand, it was emphasized that steadyelastic in-planede� ection can have a major effect on blade stability,particularly for nonuniform spanwise mass distribution.21 There-fore, this was one of the � rst studies to pinpoint the signi� cance ofthe lag degree of freedom in rotary-wing aeroelastic stability.

The type of stability boundary shown in Fig. 5 can be modi� edsigni� cantly by kinematic coupling K p between the � ap and feath-ering degrees of freedom as shown in Refs. 23 and 24. Pitch-� apcoupling can be introduced by a skewed � ap-hinge geometry rel-ative to the radial axis of the blade as shown in Fig. 6a, or by anappropriatepositioningof the pitch link relative to the � ap hinge asshown in Fig. 6b. The pitch-� ap coupling is represented by

1µ D ¡K p1¯ (1)

for the geometry shown in Fig. 6a, K p D tan ±3 , K p > 0, � ap updecreases the blade pitch. Positive pitch-� ap coupling acts as anaerodynamicspringonthe� apmotionandhasa signi� cantin� uenceon � ap-pitch stability.

As mentioned, the importance of large steady in-plane de� ectionon � ap-pitch instabilitieswas identi� ed in Ref. 21. Thus, it was onlynatural that the next type of instability to receive attention was thepitch-lag instability.

The � rst comprehensive study of the pitch-lag instability wascarried out by P. C. Chou.25;26 This instabilitywas encountereddur-ing the whirl tower testing of a very light rotor blade designed bythe Prewitt Aircraft Company for the Vertol H-21 helicopter.High-amplitude oscillation occurred at low Ä, high collective and max-imum power, primarily in lead lag, at a frequency of 0:318=rev(close to lag frequency) and lag amplitude of 30 deg. No coupling

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Fig. 6a Pitch-� ap couplingcaused by skewed � ap hinge.

Fig. 6b Pitch-� ap coupling.

between rotor and tower dynamics was found, and despite the largeoscillations the blades sustained no damage.

A comprehensiveanalysis of this instability for hover was devel-oped by Chou.25;26 The analysis was linear and restricted to fullyarticulated rotors with inelastic blades. A lag damper assumed tohave constant viscous damping C³ was included in the analysis. Itwas found that the instability was caused by pitch-lag coupling

1µ D ¡KL 1³ (2)

introduced by skewed lag hinges located outboard of the � appinghinge. An elegantapproximatestability criterionwas obtained fromthe analytical model

C³ C2KL ¯2

0 ÄI³

[1 ¡ .¯0=µ0/K p]µ0> 0 (3)

which facilitated the design of stable blades.

During the mid-1960s, two new types of rotor systems, tilt rotorsand hingeless rotors, emerged. The modeling of this class of rotorsystems started a lengthy preoccupationwith one of the most inter-esting and vexing dynamic problems, the coupled � ap-lag aeroelas-tic problem. The � rst paper attempting to develop a model for the� ap-lag instability for hingeless and teetering rotors was presentedby Young.27 The equations of motion for hover and forward � ightwere derived in an ad hoc manner. The author recognized that tocapture the mechanism of instability the coupling between the � apand lag degrees of freedom, caused by aerodynamic and corioliseffects, is required.Because these two types of terms are nonlinear,they were included, but not in a consistent manner, that is, whereasterms having a certain order of magnitude were included, othershaving a similar order of magnitude were missing in the equationsof motion. The effects of elastic modes and advance ratio were alsoincorporated in an approximate manner. Inspired by the stabilitycriterion shown in Eq. (3), the author derived a fairly complicatedstability criterion for the � ap-lag case for both hover and forward� ight. Using this stability criterion, a number of sweeping conclu-sions were reached, some of these were incorrect, some partiallycorrect, and a few were correct. The paper correctly identi� ed thelag degree of freedom as the trigger for the � ap-lag instability, andit also identi� ed the aerodynamicand inertial coupling terms as im-portant. However, the stability criterion was false; and, therefore,the conclusion that “: : :all current rotor types are susceptible [to in-stability] in the speed range of 125–150 knots, or at lower speeds athigh altitude: : :” was also incorrect.27 Subsequently, Hohenemserand Heaton28 treated the same problem using a different formula-tion, which suffered from inaccuracies similar to Young’s causedby a variety of approximations.Instead of a stability criterion, theytried to determine blade stability by using a somewhat unconven-tional numerical integration scheme. The results presented in thepaper were mainly of a qualitative nature.

Both studies failed to clearly identify the natureof the � ap-lag in-stabilityproblembecause they did not accountfor the critical role ofthe elastic or structuralcouplingbetween the � ap and lag degreesoffreedom. In retrospect,this is somewhat surprisingbecausea monu-mental NASA technicalreportwritten by Houbolt and Brooks29 wasavailableat that time and it containedthe correct structuralcouplingterms which were required for the proper treatment of this problem.It was important to mention that Ref. 29 was overlooked by manystudies on RWA conducted during this time period, and its valuewas only recognized belatedly, in the late 1960s and early 1970s.

It is remarkable that while treatments of � ap pitch, pitch lag, andeven � ap lag were presented for the case of hover a comprehensiveanalysis of coupled � ap-lag-torsionalblade stability in hover failedto materialize. Although a set of suitable equations were derived inRef. 30, numerical results illustrating blade aeroelastic behavior inhover were not computed.

Up to this point, the aeroelasticproblems discussed were mainlythose associated with the hovering � ight condition, which is gov-erned by differential equations with constant coef� cients. One ofthe earliest papers to recognize the effect of periodic coef� cientscaused by forward � ight on � apping motion was Horvay.31 Theperiodic equations were solved using Hill’s method of in� nite de-terminants. Clearly, because only the � apping degree of freedomwas considered the level of parametric excitation that is necessaryto cause an instabilityhad to be quite large.This in turn leads to veryhigh advance ratios that do not occur during normal operating con-ditions of rotors in forward � ight unless one slows the rotor down.This approach to dealing with the effect of periodic coef� cients wasused in the coupled � ap-lag-torsional analysis of rotor blades pre-sented in Bielawa’s dissertation.30 However, the numerical resultsobtained were inconclusive.

The studies considered up to now were based on quasi-steadyorunsteady aerodynamic models that were developed essentially forthe � xed-wing aeroelastic problem. However, there was a growingawarenessthatRWA requiresunsteadyaerodynamicmodelscapableof representing the complicated aerodynamic environment presenton a helicopter. The � rst important rotary-wing unsteady aerody-namic theory developed for hover is the work of Loewy.32 This

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Fig. 7 Idealized wake geometry for Loewy’s incompressible unsteadyaerodynamic model.

theory is a generalization of Theodorsen’s theory, and it providesa useful approximation to the unsteady wake beneath the hover-ing rotor. The geometry for Loewy’s model is illustrated by Fig. 7.In this theory the effect of the spiral returning wake beneath therotor is taken into account approximately. The wakes, in� nite innumber, lie in planes parallel to the disc of the rotor and are asso-ciated with both previous blades (for an N -bladed rotor) and pre-vious revolutions. The nondimensional wake spacing Nhw D .2¼vi =ÄNb/ D 4 N=¾ .

The airfoil dynamics in this theory are identical to the simpleharmonic pitch-and-plungemotion postulated in Theodorsen’s the-ory. Loewy has shown that for this case the unsteady aerodynamiclift and moment can be written in a form identical to Theodorsen’stheory, except that Theodorsen’s lift de� ciency function C.k/ isreplaced by a more complicated lift de� ciency function given byC 0.k; m; Nhw/.

Loewy’s theory is restricted to low in� ow ratios, which impliesa lightly loaded disc. This theory was used for the � rst time tostudy “wake � utter” in Refs. 23 and 24, and the classical � ap-pitch stability boundary shown in Fig. 5 is modi� ed by several nar-row instability regions present above the � utter boundary shown inFig. 5.

Another useful aerodynamictheory developedin this time periodwas Greenberg’s theory.33 The theory recognizes that in addition toconstant velocity of oncoming � ow the blade can also experiencea time-dependent, pulsating velocity variation caused by in-planemotion (lead lag). Furthermore, in addition to harmonic variationin angle of pitch a constant pitch angle is also imposed on the air-foil. Greenberg’s theory is a modi� cation to Theodorsen’s theory toaccount for these effects. Thus, the unsteady lift on the blade crosssection is given by

L D1

2½Aab2 d2h

dt 2C .V0 C 1V /

d1®

dt

C .®0 C 1®/d.1V /

dt¡ xA ¡

b

2d21®

dt 2

C½AaV bC.k/dh

dtC ®01V C V01® C .b ¡ xA/

d1®

dt

C½AaV b[V0®0 C 1¾ V C.2k/] (4)

where V D V0 C 1V ; 1V D ¾V V0ei!t ; and ®0 D constant pitchsetting.

The last two terms in this theory represent, respectively,the staticlift (underlined)and a nonlinear term in the perturbationquantities(underbraced), which is usually neglected in rotary-wing applica-tions of this theory. Greenberg’s theory is approximate because itneglects the effect of fore and aft excursions of the blade or theeffect of the pulsating � ow velocity relative to the mean velocity on

the wake. Reference 33 also provides an appropriateexpression forthe moment. Although Loewy’s theory was applied to RWA aeroe-lastic problems shortly after its initial development, Greenberg’stheory was not used until the mid-1970s, when its value was � nallyrecognized.

Another concern associated with aerodynamic loading that ma-terialized during this period was stall � utter. An important earlyinvestigation of stall � utter was conducted by Ham.34 Retreatingblade stall on a model rotor in forward � ight was considered, andlarge torsional motion with a frequency close to the blade torsionalnatural frequencywas found after the blade entered the stall region.The sensitivity of the blade torsional amplitude to several parame-ters was studied. Increases in speed and rearward shift of the bladecross-sectionalcenterof gravitycaused increasesin theamplitudeoftorsional oscillation. However, increases in torsional damping andtorsionalstiffness reduced the amplitudes.The physicalmechanismcausing the vibration was associatedwith reduction in aerodynamicpitch damping caused by stall, which led to large-amplitude tor-sional loads and high blade loads.

In an important sequel to this study, Ham and Young35 conducteda study of stall � utter using a model rotor in hover. A single-degree-of-freedom limit-cycle torsional oscillation,with a frequency closeto thenaturaltorsionallag frequencyof theblade,was found to occurat high collective pitch settings. The origin of this torsional motionwas indicatedby experimentalstudyof chordwisepressurevariationon the model rotor during the stable limit-cycle oscillation.Using asimple analysis, the relationshipsbetween the torsional motion andthe effective damping in pitch in presence of stall are determined.Also the effect of reduced frequencyon limit-cycle amplitudes wasexperimentally measured. The implication of the results obtainedfor the case of forward � ight were also discussed, and a simplenumerical method for approximating the boundary of stable pitch-torsional oscillation in forward � ight was described and shown toproduce good correlation with � ight-test results.

B. Coupled Rotor-Fuselage ProblemsIn addition to isolated blade stability and response problems just

discussed, one also encounters coupled rotor-fuselageproblems asdepicted in Fig. 8. Two types of problems were encountered.Whenthe helicopter is on the ground, a mechanical instability couples in-plane blade motion with displacementof the axis of rotation causedby roll or pitch; this is usually denoted by the term “ground res-onance. The second instability is in � ight, and again it is causedby coupling between blade in-plane (lag) motion and body roll orpitch. This aeromechanical problem is usually denoted by the termair resonance. This terminologyis unfortunatebecauseneitherphe-nomenon has anything to do with resonance.

Early in the development of rotorcraft, ground resonance and itsavoidance were identi� ed as major design issues. The � rst de� ni-tive study of ground resonance was carried out by Coleman andFeingold.36 This report is a collection of the work done earlier bythose two authors on two bladed rotors on isotropic and anisotropicsupports, as well as rotors having three or more blades. The groundresonance represents coupling between a low-frequency lag mode(in the nonrotating frame) and a natural frequency of the structure

Fig. 8 Coupled rotor/fuselage system.

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supporting the hub. This coupling produces lateral and longitudi-nal displacement of the rotor center of gravity from the center ofrotation. Articulated rotors and hingeless rotors with lag frequencybelow 1/rev are susceptible to this instability. Ground resonance isvery destructive. Although ground resonance was well understoodin this time period, only a limited understanding of air resonanceexisted.

A valuable study conduced in the late 1960s37 examined the air-and ground-resonance characteristics of a soft in-plane hingelessrotor system used on an experimental XH-51A helicopter built byLockheed. The rotating fundamental lag frequency of soft-in-planerotors is below 1/rev. The particular rotor considered in this studyhad a “matched stiffness” con� guration, which eliminated part ofthe elastic coupling between the � ap and lag degrees of freedomand causes the rotor to be more susceptible to the � ap-lag type ofinstability. The paper has an excellent graphical description of themechanism of ground/air resonance for soft-in-plane hingeless ro-tors, which occurs when the rotor rpm is is such that Ä ¡ !i p isclose to a body natural frequency. In this case the center of grav-ity of the rotor disk is whirling about the center of rotation at anangular velocity !i p , as shown in Fig. 9. The Ä ¡ !i p curves relatebody frequencies, as shown in Fig. 10. For the articulated rotorhelicopter a critical body frequency for ground resonance coin-cides with the driving frequency when the rotor speed is at a valuebelow the operating speed, which corresponds to the leftmost cir-cle on the � gure. The coincidence between the inclined dash–dotline with the double line marked (on the ground) indicates groundresonance.The soft-in-planerotor tested and described in the papercan encounter ground resonance when it is at an rpm above the op-erating speed as shown by the intersection of the solid line and thedouble line denoted (on the ground); however, it can encounter air

Fig. 9 Rotor-in-plane mode in the nonrotating system.

Fig. 10 Driving and body frequency relationships.

resonance if the rotor is slowed in � ight, as indicated by the inter-section of the solid line and the double line (in the air). As a result ofthe special constructionof the rotor, both ground and air resonancewere demonstrated experimentally, and analytical results were cor-related with experimental data. However, the conclusions reachedin this study were not de� nitive, mainly because of incompleteunderstanding of the appropriate structural dynamic modeling ofhingeless rotor systems.

C. Summary of the State of the ArtTo set the stage for a discussionof thenext time period,a summary

of the state of the art for the early time period is useful:1) The pitch-� ap and pitch-lag instabilities of articulated rotors

were reasonablywell understood,particularly for the case of hover.However, therewas considerableconfusionaboutthe� ap-lagtypeofinstability.The unsteady aerodynamicswas approximated by usingTheodorsen- and Loewy-type unsteady aerodynamics.

2) For the case of forward � ight, there was some understandingofthe role of equationswith periodic coef� cients and its mathematicalimplications. However, there was little appreciation for effectivenumerical methods for dealing with such equations.There was alsogrowing appreciation for the important role of retreating blade stalland stall � utter.

3) There was a goodunderstandingof the ground-resonanceprob-lem, particularly for articulated rotors. The important role of lagdampers for preventing this problem was also appreciated.

However, despite the remarkable progress made and the success-ful design, engineering analysis, and production of a large numberof successful helicopters, the state of the art had major de� cien-cies that needed to be overcomebefore additionalprogress could bemade. These de� ciencies are summarized here:

1) The � ap-lag instability problem was not well understood.2) There was only limited appreciationof systematic approaches

to formulating and solving RWA problems.3) Hingeless rotor aeroelastic behavior and air resonance were

not understood.4) There was no appreciation for the important role of structural

dynamic models capable of representing coupled � ap-lag-torsionaldynamics in formulating RWA problems.

5) The role of geometric nonlinearities in RWA was not wellunderstood.

6) Unsteady aerodynamic models, wake models, and dynamicstall models were not available.

7) Treatments of the true RWA problem, as represented by thecoupled� ap-lag-torsionalproblemin hover and forward � ight, werenot available.

III. Golden Age (1970–2000)A. Overview of Principal Developments

This period was characterized by rising to the challenges posedby the unsolved problems summarized at the end of the precedingsection. The accomplishments of this period were summarized inthe various survey papers mentioned in the introductoryportion ofthis paper. Before discussing the most important accomplishmentsin detail, it is useful to distinguish between two types: 1) accom-plishments in modeling the aeroelastic behavior of rotor blade andcoupled rotor-fuselagesystems and 2) developmentof modern rotorsystems, such as hingeless and bearingless, used on various rotor-craft being producedworldwide. Clearly, these two types of accom-plishments are intertwined because modern rotor systems cannotbe developedwithout certain aeroelasticmodeling capability.Also,new modeling capabilities are being developed to meet the chal-lenges of the hardware designer. Emphasis in this paper is on themost important developments in aeroelastic modeling techniques.

Some key developments in modeling of aeroelastic behaviorthat have occurred during this period are listed here: 1) recog-nition of the fundamental role of structural modeling and asso-ciated kinematic assumptions in the proper formulation of theRWA problem; 2) unsteady aerodynamics for attached and sep-arated � ow; 3) development of systematic tools for formulating

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and solving RWA problems; 4) understanding of the basic cou-pled � ap-lag aeroelastic problem in hover and forward � ight; 5)understanding of the coupled � ap-lag-torsional problem in hoverand forward � ight; 6) understanding of air and ground resonance;7) modeling of composite rotor blades; 8) modeling of hingeless,bearingless, and swept tip rotor blades; and 9) development ofcomprehensive analysis codes capable of modeling several RWAproblems.

A detailed descriptionof all of these items within the frameworkof a single paper is quite dif� cult, and therefore one has to be selec-tive so as to limit the paper to a reasonable length.

B. Role of Structural ModelingInitially, structural models for isotropic rotor blades were

linear,29;38 and thus no distinctionwas made between the deformedand undeformed blade con� gurations.The aeroelastic formulationsdevelopedin the late1960swere allbasedon theHouboltandBrooksequations.29 In the late 1960sand early 1970s it was recognizedthatgeometrical nonlinearities caused by moderate de� ections neededto be incorporated in the aeroelastic operators associated with therotary-wing aeroelastic problem. The distinction between the un-deformed and deformed blade geometries also produces nonlinearterms that have to be included in the inertia and aerodynamicopera-tors. Moderate-de� ection beam theories capableof representingthecoupled � ap-lag-torsionaldynamics of rotor blades were developedprimarily between 1970–1980, and during the next decade large de-� ection theorieswere derived.The inceptionof moderate de� ectiontheories can be found in two dissertations that were published inthe same year.39;40 An integral part of moderate de� ection theorieswas ordering schemes, which allowed one to neglect higher-orderterms in the structural, aerodynamic, and inertia operators asso-ciated with the aeroelastic problem. Subsequently, the equationsevolved, and more careful derivation of the structural part resultedin equations that have formed the basis of numerous aeroelasticstudies.41;42

The source and structure of the geometrically nonlinear termsassociated with structural rotations are conveniently illustrated bya transformation between the triad of unit vectors describing thedeformed and undeformed state of a hingeless blade, as shown inFig. 11. Only four independent functions (three displacement vari-ables and one rotation)are neededfor the exact formof this transfor-mation becauseof a constraint that the plane in which the vectors Oey

and Oez lie remainsnormal to the deformed beam elastic axis. If thesevectors are, in turn, assumed to lie in the deformed beam cross sec-tion, then this constraint becomes analogous to the Euler–Bernoullihypothesis for a large-deformation theory. Such a transformation,based on the assumption of small strains and � nite rotations (asso-ciated with the twist angle and bending slopes), has the followingmathematical form:

Oe0x

Oe0y

Oe0z

D [S]

Oex

Oey

Oez

(5)

where the elements of the transformation matrix [S] determine theaccuracy or order of the theory. A typical transformation whereterms up to the third order are accounted for is given here:

S11 D 1 ¡ 12 v2

;x C w2;x ; S12 D v;x ; S13 D w;x

S21 D ¡ v;x ¡ Áw;x C 12v;x w2

;x ; S22 D 1 ¡ 12v2

;x ¡ Áv;x w;x

S23 D Á ¡ 12 w2

;x Á; S31 D ¡ w;x ¡ Áv2;x ¡ 1

2 v2;x w;x

S32 D ¡ Á C v;x w;x ¡ 12 v2

;x Á ; S33 D 1 ¡ 12 w2

;x (6)

Such a transformation can be assumed to imply the existence ofan ordering scheme in which third-order terms, in terms of blade

Fig. 11 Geometry of thebladeelastic axisbefore andafter deformation(top) and blade cross-sectional geometry before and after deformation(bottom).

slopes, are neglected.Such an ordering scheme implies

O.1/ C O.²3/ »D O.1/ (7)

where blade slopes are assumed to be moderate and of magnitude² , that is, 0:10 · ² · 0:20. Use of a less accurate ordering scheme

O.1/ C O.²2/ »D O.1/ (8)

will lead to the neglectof the third-orderterms in Eqs. (6). A word ofcaution is in order at this point. To allow for the treatment of appliedmoments, the virtual rotation must be obtained as a function of thedeformation variables. The variation must be taken prior to the ne-glect of the third-order terms; otherwise, the expressions for virtualrotationwill be incorrect(see Refs. 43 and 44 for more detail on thispoint). Transformationsof the form of Eq. (5) have been used as thebasis for moderate-de� ection beam theories, which are suitable forthe aeroelastic stability and response analysisof isotropichingelessand bearingless rotor blades. Once a transformation represented byEq. (5) is available, it is used to derive the inertia and aerodynamicloadsactingon theblade.Thus, these termspermeatethroughthe en-tire set of equationsof motion describingthe dynamics of the blade.

Consider as an example the treatment of the coupled � ap-lag-torsional dynamics of an isolated blade in forward � ight. For thiscase the ordering scheme would be based on the order of magnitude

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assumptions given here:

w;x D v;x D Á D O.²/

e1

RD

b

RD ¯p D N D ¸1s D ¸1c D w

RD v

RD O.²/

µ D µ1c D µ1s D O.1/

u D x I =R D xA=R D O.²2/; Cd0=a D O ²32

x= l D @

@xD @

@ÃD ¹ D O.1/ (9)

Application of such an ordering scheme leads to the neglect of nu-merous higher-order terms. Furthermore, modern computer pack-ages capable of algebraicmanipulation, such as Mathematica®, canbe used togetherwith an orderingscheme to generateequationswitha desired level of accuracy.45

Finally, such a scheme is based on common sense and experiencewith practical blade con� gurations.Thus, it should be applied witha certain degree of � exibility.

Structural models for moderate, as well as large de� ection beam(or blade) theories, have been often validated by correlating themwith experimentaldata obtained in a static experimentconductedatPrinceton.46

The development of moderate de� ection beam theories was fol-lowed by structural models that use only the smallness of the ex-tensional strain; otherwise, the analysis allows for arbitrarily largede� ections and rotations. This approach completely eliminates theneed for an ordering scheme. This type of model is more consis-tent and mathematically more elegant than blade models based onordering schemes. References 47–50 are representative of the � rststudies that have established this more accurate approach.

When the strain is assumed to be small, two developments arefeasible, depending on the representationof the cross-sectionalde-formation. Consider the rotation of the deformed beam sectionalframe, which is assumed to be arbitrarily large; this is denoted asglobal rotation. Furthermore, consider the rotation of a material el-ement at some point in the cross section caused by cross-sectionaldeformation.This so-called local rotationis relative to the deformedbeam sectional frame. The simpler development assumes that localrotation is of the order of the strain, whereas the more general oneassumes that the local rotation is of the order of the square root ofthe strain. In either case the beam deformation can be expressedin terms of six generalized strain measures: the extension of thereference axis, two shear strains at the reference axis, the elastictwist, and two elastic bendingmeasures. Because of the presenceofshear-strainmeasures, three independentorientationvariables mustbe allowed, as in Ref. 47. That is, it is not possible to express twoof the orientation variables in terms of the derivatives of the threedisplacement variables. Also, although not necessary for static andlow-frequency analysis of composite rotor blades, the presence ofshear strain in these developmentsimproves their accuracy in appli-cations to composite rotor blade analysis when transverse bendingmodes higher than the � rst are involved.

C. Unsteady Aerodynamics for Attached and Separated FlowAccurate modeling of the unsteady aerodynamic loads required

for aeroelasticstability and responsecalculationcontinuesto be oneof the major challenges facing both the analyst and the designer.The combination of the blade advancing and rotational speed is aformidablesourceof complexityin the � ow� eld surroundingthe ro-tor. At large values of the advance ratio, the aerodynamic � ow� eldaround the blade undergoes such variations that there are problemsof transonic � ow, with the shock waves on the advancing blade tip,problems of � ow reversal (reversed� ow region) and low-speed,un-steady stall on the retreating blade, and problems caused by highblade-sweep angle for various azimuthal locations. Modern sweptand curved-tip blade geometries further complicate this problem.Furthermore, the time-varying geometry of the wake, which is an

important source of unsteady loads, vibration and noise, is an ex-cruciatingly complex problem that is an order of magnitude morecomplicated that the wake geometry of � xed wings.

When dealing with the unsteady aerodynamic problem, one canmake a wide array of assumptions, which lead to diverse models,starting with simple and computationally ef� cient models and cul-minating in models, which are capable of simulating the more in-tricate details of the unsteady � ow. A detailed description of un-steady aerodynamic models for rotary-wing applications has beenpresented in books,17;19 as well as a couple of review papers.51;52

1. Attached-Flow Unsteady AerodynamicsFrom the � rst partof this paper, it is evidentthat theunsteadyaero-

dynamic models available were limited to two-dimensional incom-pressible theories such as Theodorsen, Greenberg,33 and Loewy.32

Because of the low reduced frequency associated with RWA prob-lems, unsteady aerodynamic effects have been found to be of lessthan critical importance. Furthermore, because of its wake struc-ture Theodorsen’s theory is not suitable for rotary-wingapplication,whereas Loewy’s theory is limited to lightly loaded rotors.

It is also important to recognize that both are frequency domaintheories, which are not suitable for forward � ight, where the equa-tions of dynamic equilibrium have periodic coef� cients. For con-venient mathematical treatment of equations with periodic coef� -cients, time-domain theories are required. Therefore, Greenberg’stheory with appropriate modi� cations53¡56 has been often used inRWA, with the assumption that the aerodynamics are quasisteady,C.k/ D 1. For this case the theory was also used in forward � ight.

Loewy’s theory has been extended to include compressibilityef-fects. However, these theories have been rarely used in coupled� ap-lag-torsionalanalysis in hover.53

Frequency-domain theories have a signi� cant de� ciency whenbeing applied to aeroelastic stability calculations because the as-sumption of simple harmonic motion upon which they are basedimplies that they are strictly only valid at the stability boundary.Thus, they provide no information on system damping before orafter the � utter condition is reached, and standard stability anal-yses based on conventional eigenanalysis, such as the root locusmethod, cannot be used. Furthermore, as indicated before, these arenot suitable for rotary-wing aeroelasticanalyses in forward � ight orapplications where the transient response of the aeroelastic systemis required. Thus, there is a need for unsteadyaerodynamictheoriesthat are capableof modelingunsteadyaerodynamicloads in the timedomain for � nite-time arbitrary motion of an airfoil, representingthe cross section of an oscillating rotor blade. The term “arbitrarymotion” is usedhere to denotegrowingor decayingoscillationswitha certain frequency.A number of such theorieswere developed,andRefs. 5 and 57 contain a uni� ed description of such theories.

Time-domain airfoil theories are extensions of previousfrequency-domain theories, using an approach developed byEdwards58 to extendTheodorsen’s theory to the time domain.Time-domain versions of Greenberg’s theory can be found in Ref. 59, anda time-domain version of Loewy’s theory was presented in Ref. 60.

A particularly useful time-domain theory, which has been usedfrequently in rotary-wing aeromechanical applications, is the dy-namic in� ow model, which was developed and used � rst at the be-ginning of the 1980s.61¡63 The mathematical form of the dynamicin� ow model in both hover and forward � ight clearly indicates thatit is an arbitrary motion, time-domain theory. The most widely usedversion of dynamic in� ow is that developed by Pit and Peters,63

which is suitableforboth hoverand forward � ight.The model repre-sents unsteadyglobalwake effects in a simple form and is applicableto the entire rotor. The assumption in this theory is that, for rela-tively low frequencies, actuator disk theory is valid for both steadyand unsteady conditions.Therefore, dynamic in� ow is essentially alow-frequency approximation to the unsteady aerodynamics of therotor. The total induced velocity on the rotor disk is assumed toconsistof a steady in� ow ¸0 (for trim loadings)and a perturbationalin� ow, denoted ±¸, as a result of transient loadings.The total in� owis expressed as:

¸ D ¸0 C ±¸ (10)

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where ±¸ is assumed to be given by

±¸ D ¸1 C ¸1cr

RcosÃ C ¸1s

r

Rsin Ã (11)

in which the in� ow variables ¸1 , ¸1c, and ¸1s are related to theperturbationalthrust coef� cient, roll and pitch-momentcoef� cientsacting on the rotor through the following relation:

[M]

P1

P1c

P1s

C [L]¡1

¸1

¸1c

¸1s

DCT

¡CM y

CM x P:A:

(12)

P.A. stands for perturbational aerodynamics. The elements of [M]and [L]¡1 can be obtained either theoretically, by using momen-tum theory,63 or experimentally.Dynamic in� ow models have beenparticularlyuseful for coupled rotor-fuselageaeromechanicalprob-lems in both hover and forward � ight, and they have been used forisolated rotor stability analyses.64

Subsequently, the concept of dynamic in� ow has led to the de-velopment of a complete unsteady aerodynamic model applicableto RWA.65;66 In this theory the induced � ow on the rotor disk is ex-pandedin Fouriercoef� cients(azimuthally)andspatialpolynomials(radially). The coef� cients of these expansion terms are shown toobey a closed-form set of ordinary differentialequationswith bladeloading (from any source) as the forcing functions. The obviousadvantage of such an approach is that the resultant equations canbe used for arbitrary motions in the time domain (time-marchingor Floquet), in the frequency domain (harmonic balance), or in theeigenvalue domain (conventional stability analysis) to any degreeof resolution as dictated by the application.

This theory is derived from the linear potential equations with askewed cylindrical wake. Wake contraction can also be modeled.For hover the results of this theory agree with Loewy’s model. Aconvenient feature of this theory is that it can be easily coupledwith Floquet solution of the equations of motion in forward � ight.A shortcoming of the theory is that it cannot model the importanteffect of blade-vortex interaction, which can be captured only byfree wake models.

2. Separated-Flow Unsteady Aerodynamics—Dynamic StallDynamic stall is a strong nonlinear unsteady aerodynamiceffect

associated with � ow separation and reattachment, which plays amajor role in aeroelastic stability and response calculations. Gooddescriptions of dynamic stall can be found in Refs. 17 and 19. Inthe early years dynamic stall was not well understood, and modelsthat would allow one to incorporate dynamic stall in an aeroelasticanalysis were not available.

Dynamic stall is associatedwith the retreating blade and borderson the reversed � ow region, as shown in Fig. 12. For such condi-tions the angle of attack of the blade cross section can be very large.Although the torsional response of the blade is relatively low undernormal conditions, at the � ight envelope boundary,where dynamicstall effects are pronounced, large transient-torsionalexcursion canbe excited, accompaniedby low negative damping in pitch. This, inturn, generates excessive control and blade vibratory loads, whichimpose speed and load limitations on the rotor system as a whole. Itcan also cause stall � utter. Because of its importance,dynamic stallhas been the subject of a large number of studies, which have led toa good physical understandingof this complex aerodynamiceffect.Some of the earlier work on this topic was done by Ham,67 and sub-sequent experimentaland analyticalwork of Carr68 and McCroskeyand his associates69 has led to improved physical understandingof this phenomenon. Attempts at simulation of dynamic stall us-ing computational � uid dynamics have not been successful in re-producing the quantitative characteristics at operational Reynoldsnumbers. The need to incorporate the important dynamic-stall ef-fects in rotary-wing aeroelastic stability and response calculationshas led to the development of semi-empirical dynamic-stall mod-els that capture the most important features of dynamic stall with

Fig. 12 Schematic illustration of reversed � ow region and dynamic-stall region.

reasonableaccuracy.Semi-empiricalmodels can reproduce the hys-teretic lift, moment and drag curves for a given airfoil quite accu-rately. These models have a number of common features. They areintended to incorporate two-dimensional airfoil unsteady aerody-namic effects in analytical studies in the time domain, and they aresuitable for stepwise numerical integration in time. All models areempirical, and various free parameters in the model are determinedby � tting the theory to experimental data obtained from oscillatingairfoil tests.

Several dynamic-stall models have been developed. However,only two have withstood the test of time and are in widespreaduse currently. These are the ONERA and the Leishman–Beddoesdynamic-stallmodels. Both distinguishbetween two principal � owregions: the attached, and the separated-� ow regions.

The ONERA model developed by Dat,51 Dat et al.,70 and Tranand Petot71 is based on the time-domain representationof the airfoilsection operating before, during, and in the poststall regime whileit performs essentially arbitrary motions. The model utilizes theproperties of differential equations to simulate the different effectsthat can be identi� ed on an oscillatingairfoil, such as pseudoelastic,viscous and inertial effects, and the effect of the � ow time history.The theoryalso recognizesthat, in the linearrangeof airfoilmotions,Theodorsen’s lift-de� ciency function represents the aerodynamictransfer function for the airfoil, relating the downwash velocity atthe three-quarter chord to circulatory lift. Furthermore, the theoryis based on approximating the aerodynamic transfer function byrational functions. In the nonlinear range the model consists of asystem of differential equations containing unsteady linear termswhose coef� cients are functions of the angle of attack and steady-� ow nonlinear terms.

The ONERA model has beenmodi� ed and improved by Rogers72

and Peters.73 These changeshaveproduceda modi� ed theory,whichin the attached-� ow region is consistent with classical unsteadyaerodynamics and in which circulation has been introduced as anew dependent variable. The ONERA model contains an approxi-mate correction for compressibility and no correction for the effectof sweep. The most recent version of this model was documentedby Petot.74 The coef� cients in the equations of this model are deter-mined by parameteridenti� cation from experimentalmeasurementson oscillatingairfoils.The model requires22 empirical coef� cients.Figure 13 shows typicalhystereticlift and moment coef� cients com-puted with the ONERA dynamic-stall model for a NACA 0012

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Fig. 13 Typical hysteretic lift and moment coef� cients computed withthe ONERA dynamic-stall model.

airfoil at M D 0:379, k D 0:075, and a time-varying angle of attack® D 10:3 deg C8:1 sin !t .

The Leishman–Beddoes model was developed originally byBeddoes in the mid-1970s.75;76 Subsequently, it was extended byLeishman,77¡79 and it has become a comprehensive and maturemodel. The model is capableof representingthe unsteady lift, pitch-ing moment, and drag characteristics of an airfoil undergoing dy-namic stall. This model consists of three distinct components: 1) anattached-� ow model for the unsteady linear airloads,2) a separated-� ow model for the nonlinear airloads, and 3) a dynamic-stallmodelfor the leading-edgevortex-inducedairloads. The model contains arigorous representationof compressibility in the attached-� ow partof the model, using compressible indicial response functions. Thetreatment of nonlinear aerodynamic effects associated with sepa-rated � ows are derived from the Kirchoff–Helmholtz model to de-� ne an effectiveseparationpoint that can begeneralizedempirically.The model uses relatively few empirical constants,with all but fourderived from static airfoil data.

3. Wake ModelsThe description of aerodynamic loading is incomplete without

mentioning wake models. A detailed description of wake modelsand their historical development is outside the scope of this paper,but can be found in Chapter 10 of Ref. 19. Accurate modeling ofthe wake and, in particular, free-wakemodels plays a critical role inaeroelasticresponseand bladevibratoryload calculations.However,it appears that accurate modeling of the wake is less important foraeroelastic stability analyses.

D. Development of Systematic Methods for Formulating and SolvingRotary-Wing Aeroelastic Problems1. Formulation of Equations of Motion

Formulationof completeaeroelasticequationsof motion requiresa combination of structural, aerodynamic, and inertia terms. It is inthis area that very signi� cant advances have been made comparedto the early period when equationsof motion were formulatedusingad hoc methods augmentedby good physical insight.The structural

and aerodynamic ingredients have been described in the precedingsections.The inertia loads are obtained in a straightforwardmannerby using D’Alambert’s principle and combining it with Newton’sSecond Law. The transformation between the undeformed and de-formed states representedby Eq. (5) determines the position vectorof a mass point in the deforming blade, and the acceleration vectoris given by using vector mechanics:

a D RR C 2 £ R C P £ R C £ . £ R/ (13)

Although the derivation of the inertia terms is conceptually sim-ple, the practical implementation can be tedious from an algebraicpoint of view, particularly when one is interested in coupled ro-tor fuselage aeromechnical problems. In the 1970s and early 1980sequations of motion used to be derived manually leading to longand algebraically cumbersome expressions. Examples illustratingthe complexity of the equations for isolated blade aeroelastic prob-lems in forward � ight80;81 or coupled rotor-fuselage problems82¡84

clearly indicate the tediousnature of such tasks, even when orderingschemes are used. Furthermore,when advancedaerodynamicssuchas dynamic-stallor wake models are used82 it is impossibleto obtainexplicit equations of motion.

The equations are partial differential equations, and their solu-tion requiresdiscretizationto eliminate the spatial dependence.Dis-cretizationand the solutionof the equationrequiresfurtheralgebraiceffort. Finally, the � nite element method, which was � rst used fora rotary-wing aeroelastic problem in 1980 (Ref. 85) and has be-come the most effective spatial discretization and solution methodcurrently used in RWA, tends to obscure the precise boundariesbetween problem formulation and solution.

Since the early 1970s, two distinct approaches for formulatingisolated blade or coupled rotor-fuselage equations of motion haveemerged.The � rst approach is usually denoted the explicit approachbecauseit leads to a set of detailedaeroelasticequationsofmotion inwhich all of the terms (aerodynamic, structural,and inertial) appearin explicit form. Explicit equations are usually derived using order-ing schemes to neglect higher-order terms in a systematic manner.The outcome of this process consistsof a set of nonlinearpartialdif-ferential equations in the space and time domain. These equationscan also contain integralexpressionscausedby centrifugaland otherterms. An alternative approach can be denoted as the implicit ap-proach. In this approach the detailed expressions of the aeroelasticequations of motion are avoided; instead, the aerodynamic, iner-tia, and structural operators are usually generated in matrix forminside the computer. When this approach is used, the boundariesbetween the formulation and solution process, particularly in spa-tial discretization, tend to be blurred. When the implicit approachis used, ordering schemes are no longer required. Furthermore, theimplicit approach frequently mandates iterative solutions. For con-venience and clarity the implementation of these two approacheswill be discussed by describing their application to two separateclasses of problems, namely, isolated-blade problems and coupledrotor-fuselageproblems.

Isolated-bladecase. A good example of explicit formulation ofequations of motion for the case of hover can be found in Refs. 41and 54. Explicit formulations for forward � ight can be found inRefs. 80 and 81. The algebraic task for deriving such equationswas too cumbersome, and, therefore, with increases in computerpower these tasks have been relegated to the computer. One of the� rst derivations of a set of coupled � ap-lag-torsional equations ofmotion for a hingeless rotor blade in forward � ight, using a specialpurpose symbolic processor written in FORTRAN, was presentedby Reddy and Warmbrodt.86

In the mid-1980s LISP workstations utilizing the MACSYMAsymbolicmanipulativepackagebecamecommerciallyavailableandwere used to derivecoupled � ap-lag-torsionalequationsfor a hinge-less blade in hover, including terms up to the third order.45;87;88 Bythe early 1990s regular workstations could be used in conjunctionwith MACSYMA to obtain explicit equations for hingeless rotorblades in forward � ight in a routine manner.89;90

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Another approach for avoiding the algebraic derivation associ-ated with the formulation of the RWA problem is to use the � niteelement approach. For this approach the equations of motion aregenerated in a numerical form, as part of the solution process. Theapproach can be used to obtain either explicit or implicit formu-lations and solutions. This approach will be described later in thispaper because the � nite element formulation is strongly linked tothe solution methodology.

Coupled rotor-fuselage case. Formulation of coupled rotor-fuselage equations for a typical con� guration, like that shown inFig. 8, has similarities to the isolated-blade problem, although anumberof substantialdifferencesdo exist: 1) the equationsare morecomplicated because of numerous additional terms associated withthe fuselage rigid-body degrees of freedom, which contribute tothe complexity of the inertia and aerodynamic loads; 2) if orderingschemes are used, combined with an explicit formulation, a mod-i� ed form of the ordering scheme has to be used to restrict theequations to a manageable size; 3) rotor-fuselage coupling has tobe performed in a careful and systematic manner; and 4) when thefuselage itself is also consideredas a � exiblebody, a furthercompli-cation in problem formulation emerges. Again, like in the isolatedblade case, both explicit and implicit formulations of the coupledrotor-fuselageproblem are available.

The early derivations, developed in the late 1970s, for this classof problems were explicit, usually done manually and resulted inlengthyequations.References 91 and 92 good examples of coupledrotor-fuselageequations in hover, whereas Ref. 84 is representativeof typical equationsfor forward � ight. In the same period Johnson93

has developed a coupled rotor-fuselage model suitable for forward� ight using an implicit approach. In the studies just mentioned, thefuselage was assumed to be rigid. In the mid-1980s there was aneed to obtain coupled rotor-fuselage equations capable of mod-eling single- and twin-rotor con� gurations in hover and forward� ight, and thus these were derived and solved in Refs. 83 and 94.The fuselage was modeled as a � exible beam with bending andtorsional degrees of freedom. This study represents one of the mostcomplicatedexplicit sets of equationsderivedmanually.Three yearslater they were rederived using MACSYMA and found to be errorfree.

Obviously, this class of problems is an ideal candidate for sym-bolic derivationof equations motion. The � rst explicit formulationsbased on symbolic manipulation were carried out in Ref. 95.

An interestingimplicit approach,based on a hybrid � nite elementcombined with a primitive multibody formulation, was used in theGRASP program.96 The required matrix elements were generatedby numerical evaluationof hierarchicalexpressions in the code. Asone of the � rst multi-� exible-body approaches applicable to rotor-craft,GRASP possesseda lot of modeling � exibility.Unfortunately,however, the only version released until the time development washalted was limited to the hovering � ight condition.

2. Methods of SolutionThe solution of rotary-wing aeroelastic stability and response

problems is usually carried out in two stages.The � rst stage consistsof the spatial discretizationof the equations of motion followed bya solution in the time domain. In the second stage, namely, thetime-domain solution, two different approaches are possible; onecan solve the equation in a blade-� xed, rotating coordinate systemor in a hub-� xed, nonrotatingcoordinate system. One also needs todistinguishbetween solutions for hover and those for forward � ight.

Spatial discretization. The � rst step in the solution of rotary-wing aeroelasticstabilityor responseproblems is eliminationof thespatial dependencein the nonlinearpartial differentialequations(orappropriateenergy expressions),which describe the system. Appli-cation of suitable discretizationmethods will yield a set of coupled,nonlinear,ordinary differentialequations in the time domain. Threeapproaches for spatial discretizationhave been used: 1) spatial dis-cretization based on global methods, 2) spatial discretizationbased

on the � nite element method, and 3) spatial discretizationbased onmatrix method. The third approach is mentioned only for historicalreasons. This approach has not withstood the test of time and willnotbe discussedhere, but informationon this approachcan be foundin Ref. 9.

During the 1970s, the preferred discretization methods wereglobal, such as the well-knownRayleigh–Ritz or Galerkinmethods,as shown in Refs. 3, 54, 86, and 97 based on free-vibration modesof the rotating blade. For hover, both coupled and uncoupledmodeshavebeenused,where thecouplingofmodes is causedby the collec-tive pitch setting. For forward � ight the use of uncoupled modes ismore convenient because cyclic pitch introduces time-varyingcou-pling. Discretization based on global modes is cumbersome and isbesthandledby symboliccomputationor numericalimplementationusing Gaussian quadrature.

Since 1980, the � nite element method has emerged as the mostversatile spatial discretization method. In addition to eliminatingthe algebraic manipulative labor required for the solution of theproblem, it also serves as the basis for the implicit formulationsdiscussed in the preceding section. Furthermore, the � nite elementmethod is ideally suited for modeling composite rotor blades andcomplicated redundant structural systems such as encountered inbearingless rotors. For rotary-wing aeroelastic problems two ap-proaches have been used: 1) weighted residual Galerkin-type � niteelement methods and 2) local Rayleigh–Ritz � nite element methodusing conventional as well as higher-order elements. Recognizingthat the rotary-wing aeroelasticproblem is geometricallynonlinear,it shouldbe emphasized that � nite element formulationfor this classof problems is more intricate than its � xed-wing counterpart.

The � rst � nite element treatment of the rotary-wing aeroelasticproblemin hover and forward � ight, usinga Galerkin-typeweightedresidual � nite-element method, can be found in Refs. 85, 98, and99. First, the coupled � ap-lag problem was treated,85;98 and subse-quently the coupled � ap-lag-torsional problem was formulated.99

The geometry for this problem is shown in Fig. 14. The bending de-grees of freedom were interpolated using cubic (or Hermite) inter-polation,whereas quadratic interpolationwas used for the torsionaldegree of freedom (not shown). In Refs. 85, 98, and 99 an explicitformulation was used; however, later the same method was used inan implicit formulation to solve the coupled � ap-lag-torsionalprob-lem of straight and swept-tip hingeless rotor blades in hover andforward � ight.100;101

The localRayleigh–Ritz type � nite elementmethodwas � rst usedby Sivaneri and Chopra102;103 to study the behavior of hingeless102

and bearingless103 rotors. Again, the bending degrees of freedomwere treated using Hermite interpolation,and torsion was treated by

Fig. 14 Geometry of the elastic axis of the hingeless deformed bladeand schematic representation of the � nite element model.

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linear interpolation.102 Ritz-type higher-order � nite elements werecombined with an implicit formulation and used in the GRASPprogram to solve aeroelastic problems in hover.96

Time-domain solutionof the equations. After spatial discretiza-tion the equations of motion are reduced to nonlinear ordinary dif-ferential form. In forward � ight these equations have periodic coef-� cients. The mathematical structure of these general equations canbe written in the following symbolic form4:

[M]fRqg C [C.Ã/]fPqg C [K .Ã/]fqg D fFNL.Ã; q; Pq/g (14)

where it is understoodthat thematrices[M ] and [C.Ã/] containbothaerodynamicand inertial contributions,whereas the matrix [K .Ã/]contains aerodynamic, inertial, as well as structural contributions.All nonlinear effects are combined in a vector fFN L .Ã; q; Pq/g.

When time-domain unsteady aerodynamics such as Eqs. (12) areused, these equations have to be appended to Eqs. (14) and solvedjointly.61 When discussing methods of solution for Eq. (14), it isconvenient to consider hover and forward � ight separately. It isalso useful to distinguishbetween isolated-bladeand coupled-rotor-fuselage analyses.

Consider the � rst isolated blade case in hover. For this caseEq. (14) has constant coef� cients. Linearizing the equationsabout anonlinearequilibriumpositiongives a goodapproximationto aeroe-lastic stability boundaries.3;4;39;54;85 Thus, it is common practice towrite perturbation equations that are linearized about the nonlinearstatic equilibrium position. This equilibrium position is obtainedfrom the solution of a system of a system of nonlinear algebraicequations.These equations are usually solved by Newton–Raphsoniteration.3;4;104 Lack of convergencecan be indicativeof divergence.Stability boundaries are obtained from solving the standard eigen-value problem for the linearized system. The real part of the eigen-values determines the aeroelastic stability boundaries of the blade.This approachalso predicts reliable stabilityboundariesin the pres-ence of static stall.105 This basic approach has evolved during the1970s and has been used ever since then without any signi� cantchanges.

Consider next the isolated blade in forward � ight, and assumethat dynamic-stall effects are neglected. The aeroelastic stabilityand response problem is based again upon Eq. (14). Reliable solu-tions for stability can be obtained by linearizingthe nonlinear equa-tions of motion about an appropriateequilibriumposition.82;86;97;106

In forward � ight the appropriate equilibrium position is a periodicsolution of Eq. (14). Calculation of the time-dependent periodicequilibrium position, representing the blade response solution, isinherently coupled with the trim state, or � ight mechanics, of thecomplete helicopter in forward � ight. The degree of sophisticationwith which this coupling is accomplished can affect the accuracyof the aeroelastic analysis. The trim state of a typical helicopter isdepicted in Fig. 15.

Two different trim procedures4;97;106 have been used: 1) propul-sive trim, which simulates actual forward � ight conditions where

Fig. 15 Geometry showing trimmed aircraft in propulsive trim.

horizontal and vertical force equilibrium is maintained and zeropitching and rolling moments are enforced,and 2) moment or windtunnel trim, which simulatesconditionsunderwhich the rotorwouldbe tested in a wind tunnel.Horizontal and vertical force equilibriumis not enforced because the helicopter is mounted on a supportingstructure. Therefore, only the requirement of zero moments on therotor is imposed.

Initially, the trim proceduresused in the mid 1970s to mid 1980scoupled trim only partially to the aeroelastic analysis, and the � rsttime trim requirements on all three � ap, lag, and torsional degreesof freedom were imposed was in Ref. 86. However, from the late1980s onward coupled trim/aeroelastic analyses have been used ina routine manner.

The most widely used method for solvingthe forward � ight prob-lem is based on the direct solutionof Eq. (14) in the rotating system,by using the theory of differential equations with periodic coef� -cients. This approach is facilitated by rewriting Eq. (14) in � rst-order, state-variable form

f Py.Ã/g D fz.Ã/g C [L.Ã/]fy.Ã/g C fN .Ã; y; Py/g (15)

where fz.Ã/g and [L.Ã/] areperiodicmatriceswith common period2¼ , and fN .Ã; y; Py/g represents the nonlinear contributions, and

fy.Ã/g DPq.Ã/

¢ ¢ ¢ ¢ ¢ ¢q.Ã/

(16)

It is evident from Eq. (15) that the aeroelasticstability and responseproblem are coupled. Therefore, solutions have to be obtained intwo stages.4;97 First the nonlinear time-dependentequilibriumposi-tion of the blade is obtained, and next the equations are linearizedabout the time-dependentequilibrium position by writing perturba-tion equation about the periodic nonlinear equilibrium.This secondstage yields a linear periodic system for which blade stability can beobtained using Floquet theory. In the early years there was consid-erable confusion regarding the treatment of equationswith periodiccoef� cients. One of the � rst papers to point out that calculation ofthe Floquet transition matrix at the end of one period, using numer-ical integration, is an effective way for dealing with � apping dy-namics of a rotor in forward � ight was published in the early 1970s(Ref. 107).A couple of years later it was shown108 that the transitionmatrix at the end of a period is a key ingredient for examiningaeroe-lastic stability in forward � ight. An effective numerical techniquefor calculating the transition matrix at the end of one period wasalso presented.108 This approach was generalized in a later paper109

that presented very ef� cient numerical techniques for dealing withperiodic systems by using Floquet theory. Later, the methods devel-oped for the stability of linear periodic systems were extended toobtain the response for the linear case, as well as the nonlinear caseusing quasilinearization.97 The importanceof the treatmentof equa-tions with periodic systems has played an important role in RWAin forward � ight and numerous papers on this topic have been writ-ten since the 1970s, including several survey papers.110;111 Threedistinct methods for calculating the nonlinear equilibrium positionassociated with Eq. (15), about which the perturbation equationsare linearized, have emerged. These are quasilinearization,97 peri-odic shooting,112 and the � nite element method applied in the timedomain.113

An alternative approach to the solution of Eq. (14) is based onthe introductionof the multibladecoordinate transformation,whichtransforms the variables fqg from the rotating,blade-� xed referenceframe, to a nonrotating, hub-� xed reference frame.66;110;114;115 Fol-lowing Hohenemser and Yin,115 these coordinates are frequentlyused in rotor dynamics. Use of such a transformation implies thatthe blade degrees of freedom in the blade-� xed system are re-placed by a set of nonrotating coordinates that describe the mo-tions of the hub plane and its deformation in a hub-� xed system.In a symbolic form the multiblade coordinate transformation iswritten as

fqg D [B.Ã/]fXg (17)

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where fqg are the original coordinates in Eq. (14) and fXg are themultiblade coordinates. A convenient stage to apply the multibladecoordinates is after the process of linearization. In such cases forrotors that have three blades or more, the multiblade coordinatetransformationenables one to replace the periodic system by a con-stant coef� cient approximation, which can be treated by conven-tional eigenvalue calculation methods. Such an approximation isusually reliable61;62 for advance ratios ¹ < 0:25, as indicated byresearch conducted in the early 1980s. Multiblade coordinates areconvenient to use with dynamic in� ow.

The use of multiblade coordinate transformation is particularlyuseful for coupled rotor-fuselageproblems in hover. In this case useof these coordinates eliminates the periodic coef� cients from theequations of motion, for rotors having three blades or more.83;94;115

Next, solution of the coupled rotor-fuselage problem in forward� ight is brie� y discussed. The equations of motion for this prob-lem have again the mathematical form representedby Eq. (14). Forlevel � ightconditionswith constantadvanceratio¹, bladenonlinearequilibrium, in forward � ight, is described by Eq. (14). Helicoptertrim and the equilibrium solution are extracted simultaneously us-ing a numerical harmonic balance solution.116;117 The equations arelinearized by writing appropriate perturbation equations. After lin-earizationa multibladecoordinatetransformationis introduced.Theoriginal equation, Eq. (14), has periodic coef� cients with a fun-damental nondimensional frequency of unity; however, the trans-formed system has periodic coef� cients with a higher fundamentalfrequency. These higher-frequency periodic terms have a reducedin� uence on the behavior of the system and can be ignored in someanalyses at low advance ratios. Once the transformation is carriedout, the system is rewritten in � rst-order form

f Pxg D [A.Ã/]fxg (18)

where fxg contains fX g and f PX g.The fundamental frequency of the coef� cient matrices depends

on the number of rotor blades Nb . For an odd-bladed system thefundamental frequency is Nb per revolution, whereas for an even-bladed system the fundamental frequency is Nb=2 per revolution.Stability can now be determinedusing either an eigenvalueanalysis(for hover) or Floquet theory for the periodic problem in forward� ight. An approximate stability analysis in forward � ight is alsopossible by performing an eigenanalysison the constant coef� cientportion of the system matrices.

When strong nonlinearities,such as dynamic stall, are present inthe equations, direct numerical integration is used to obtain bladeresponse. Representative examples of the solution of blade dynam-ics in the presence of aerodynamic nonlinearities can be found inRefs. 118 and 119. Finally, it should be mentioned that when us-ing dynamic-stall models the procedure described in this section,consisting of linearizing the perturbation equations about a non-linear time-dependent equilibrium position and extracting stabilityinformation using Floquet theory, might not be reliable.

E. Flap-Lag Problem in Hover and Forward FlightOne of the important contributions of the research conducted in

the early 1970s was the fundamental understanding of the impor-tance of the lead-lag degrees of freedom in rotary-wingaeroelastic-ity. It was shown that the � ap-lag type of instability is a result ofdestabilizing inertia and aerodynamic coupling effects associatedwith the two bendingmotions present in this problem. It stems fromthe low aerodynamic damping in the lead-lag motion.

The best treatment of the � ap-lag problem in hover was byOrmiston and Hodges,120 who used a simple centrally hinged,spring-restrainedmodel of a hingeless blade, similar to that shownin Fig. 16, except that the hinge offset was zero and the blade wastorsionally rigid. Using a linear analysis, they have shown that thisinstability is eliminated by elastic coupling, caused by the pitchsetting on the blade, which couples blade bending in and out ofthe plane of rotation. There was very little awareness of the im-portance of this effect on hingeless blade stability. The treatmentof the � ap-lag problem, based on a � exible hingeless blade model,

Fig. 16 Offset-hinged spring restrained model of a hingeless blade.

Fig. 17 Typical � ap-lag stability boundary in hover without elasticcoupling and zero structural damping.

using one elastic mode for each degree of freedom, and a nonlin-ear analysis conducted in the same time period,39;121 showed thatsmall amounts of structural damping (1% or less) are also suf� -cient to eliminate this instability, even when elastic coupling is setequal to zero. Figure 17 depicts a typical stability. Combinationsof rotating � ap and lag frequencies N!F1 and N!L1 inside the ellipse-like region represent unstable blade con� gurations for values of µc

given on the curves. Here, µc is the critical collective pitch settingat which the linearized system becomes unstable.For the fully non-linear system121 regions of stable and unstable limit cycles are alsoshown in Fig. 17. Structural coupling Rc D 1:0, or small amounts ofdamping completely eliminate the unstable, ellipse-like regions forpractical values of blade frequencies.

In addition to the theoretical studies on the � ap-lag type of insta-bility, Ormiston and Bousman105 have performed an experimentalstudy that validated the theoretical results. Their � ndings indicatedthat under static-stallconditionsan unexpectedtype of blade motioninstability for torsionally rigid hingeless rotors was encountered.Elastic coupling was not successful in eliminating this type of in-stability, as indicated in Fig. 18. Similar conclusionswere obtainedby Huber.122

The � ap-lag stability problem, without elastic coupling, exhibitsexaggerated sensitivity to small effects that in� uence the lead-lagdamping. For example, although dynamic in� ow can affect � ap-lag stability when elastic coupling is set equal to zero, with elastic

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Fig. 18 Stall-induced � ap-lag instability; dimensionless lead-lagdamping vs collective pitch (300 rpm, Rc = 0.96, ¹!F1 = 1.62, and ¹!L1 =1.28); O, experimental data.

Fig. 19 Convergence of � ap-lag stability boundaries in hover, whenthe number of modes and elements is varied; region above curves isunstable: ¾ = 0.10, ° = 5.0, and ¹!F1= 1.15.

coupling present the effect of dynamic in� ow on this instability inhover is small.61;117 An interesting result85 based on a � nite ele-ment model for a hingeless blade, shown in Fig. 14, is presented inFig. 19. The blade undergoes only � ap and lag motion, and eachtype of motion is represented by a varying number of elastic globalmodal degreesof freedom(between 1 and 3, per type of motion): theunstable region of the stability boundaries is denoted by the letterU on the boundary.The � gure illustrates the combined effect of thenumber of modes used in the analysis and the elastic coupling pa-rameter Rc : when Rc D 1:0, elasticcouplingis present;when Rc D 0,elastic coupling is neglected;and values of 0:0 < Rc < 1:0 representvaryingamounts of elastic coupling.From Fig. 19 for Rc D 0:0, it isalways the fundamentallead-lagmode thatyieldsthe lowest stabilityboundary, and for Rc D 1:0 the � ap-lag instability is virtually elim-inated. The interesting results shown in the � gure are the unstableregionsassociatedwith the secondlag mode, which for intermediatevalues of the elastic coupling parameter Rc D 0:60 becomes unsta-ble at lower values of the critical collectivepitch angle than the � rstlag mode. The implicationof this result is that the second elastic lagmode should be retained in the stability analysis of hingeless andbearingless rotor blades.

After understanding the hover problem, several studies on the� ap-lagstabilityproblemin forward � ightwere completed.106;108;123

These studies were also performed on both the offset-hingedspringrestrained model of a hingeless blade123 as well as the fully elas-tic blade.106;108 Among these, Ref. 106 was the most realistic be-cause it contained the effects of trim and reversed � ow, which affectblade behavior in forward � ight. Later studies have illustrated thesensitivity of � ap-lag stability in forward � ight61;62 to dynamic in-� ow. This sensitivity,which is associated primarily with regressingmode damping, was exaggerated because of the absence of elasticcoupling. Subsequently, a theoretical and experimental study64 hasshown that with elastic coupling the effect of dynamic in� ow isrelatively small.

This descriptionindicates that it took more than a decade to com-pletely understand � ap-lag stability in hover and forward � ight. Inhover it is a fairly weak instability, which can be eliminated byelastic coupling and small amounts of damping. A stronger insta-bility can be triggered on highly loaded rotors operating in stall. Inforward � ight the problem is sensitive to small terms; and, as willbe shown next, in forward � ight it is safer to consider the coupled� ap-lag-torsionalproblem.

F. Flap-Lag-Torsional Problem in Hover and Forward FlightThe coupled � ap-lag-torsional aeroelastic problem provides a

more realistic representation of hingeless blade behavior than the� ap-lag problem. However, understanding the � ap-lag instabilityhas an important implication for the complete coupled � ap-lag-torsional behavior. The � rst basic studies on coupled � ap-lag-torsionalaeroelasticbehaviorof hingelessrotorbladesin hoverwerecarried out between 1973 and 1980 (Refs. 54, 104, 122, and 124).This research has shown that soft-in-plane hingeless rotor bladesare usually stable. A typical � ap-lag-torsional stability boundarytaken from Ref. 104 is shown in Fig. 20. The main item of interestin this � gure is the bubble-like region of instability present at lowvalues of collective pitch. This instability occurs only in the pres-ence of precone and is a � ap-lag type of instability.Sometimes it iscalled the precone-induced� ap-lag instability. It was also obtainedin Ref. 54. The unstable region decreases as the fundamental tor-sionalfrequency N!Á1 is increasedfrom4.5 to 6.0per revolution.Verysmall amounts of structural damping in lag (´SL1 D 0:0025, 0.25%of critical in lag) reduce this unstable region for the torsionally softblade and completely eliminate it for the stiffer blade ( N!Á1 D 6:0).Other results not shown here indicate that droop and sweep (seeFig. 3) can have a strong bene� cial as well as detrimental effecton the hingeless blade stability. In addition, offsets between cross-sectional elastic axis, aerodynamic center, and center of mass canalso in� uence blade stability. A similar study clarifying the effects

Fig. 20 Coupled � ap-lag-torsional stability boundary illustratingcombined effect of precone and structural damping for Rc = 1.0, ¯p =3 deg; XA = 0, ° = 9.0; and ¹!F1 = 1.14.

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Fig. 21 Blade stability, lag degree of freedom, soft-in-plane con� gura-tion, effect of nonlinear terms, and comparison of � ap-lag and � ap-lag-torsional analysis (CW = 0.005, ¹!L1 = 0.732, ¹!T1 = 3.17, ° = 5.5, and ¾ =0.07).

of modeling assumptions on the coupled � ap-lag-torsional stabil-ity of a stiff-in-plane hingeless blade, including comparisons withexperimental data, was conducted in Ref. 125.

Other results, not shown here, also indicate that unsteadyaerody-namic effects,wake, and compressibility53;126 can have a signi� canteffecton thecoupled� ap-lag-torsionalaeroelasticstabilityof hinge-less rotor blades in hover. Another study127 has shown that by com-bining three-dimensional tip loss and unsteady in� ow effects witha conventional moderate-de� ection theory remarkable agreementbetween theoretical and experimental results was obtained. Mostanalyses described in this section combine geometricallynonlinearstructural models of the blade with linear aerodynamic theories.Therefore, an important key to substantial improvements in aeroe-lastic modeling capability of rotor blades is linked to improving theunsteady aerodynamic models.

Studies aimed at modeling coupled � ap-lag-torsionalbehavior inforward � ight started to appear in the early 1980s and continuedthroughout most of the decade. The � rst comprehensive study ofthe coupled � ap-lag-torsional dynamics of hingeless rotor bladesin forward � ight was presented in Ref. 97. This study, based onthe equations derived in Ref. 81, clearly demonstrated the role ofgeometric nonlinearitiesand trim for this important problem. It wasalso concluded that usually forward � ight is stabilizing for soft-in-plane blade con� gurations. However, forward � ight caused severedegradation in stability of stiff-in-planecon� gurations.

Figure 21, from Ref. 97, illustratesa number of important effects.The label CFLT on the curves denotes the results from coupled � ap-lag-torsional analysis. The label � ap-lag denotes results obtainedfrom a � ap-lag analysis. The full lines are results from a convergednonlinear analysis, the dashed lines are results for the case whengeometricalnonlinearitiesare neglected.The results shown, depict-ing the real part of the characteristic exponent for lag as a functionof ¹, are for propulsive trim. From the � gure it is evident that bladestability increases with forward � ight for soft-in-plane con� gura-tions. The importance of the geometrically nonlinear terms is alsoevident from the � gure. Comparing the stability margin (as repre-sented by ³L AG ) from a � ap-lag analysis with that obtained from a� ap-lag-torsional analysis it is clear that damping from a � ap-laganalysis is 250–300% lower than that obtained from the accurate� ap-lag-torsional analysis. Therefore, � ap-lag analyses in forward� ight can be misleading and should be avoided in trend studies.

Subsequent research on hingeless rotor stability in forward� ight,128 as well as more recent research,86;129 has con� rmed theconclusions presented in Ref. 97. In Refs. 128 and 129 the effectof dynamic in� ow was also included and was found to be relativelysmall. All of the studies mentioned indicated that stiff-in-planecon-� gurations are destabilized by forward � ight, whereas the stabilityof soft-in-planecon� gurations increases with forward � ight.

Fig. 22 Effect of number of degrees of freedom used in trim analysison lead-lag damping vs ¹ (¹!L1 = 1.40, ¹!T1 = 3.0, ¹!F1 = 1.15, ¾ = 0.10,and Rc = 1.0) in propulsive trim.

The results presented in Ref. 97 were based on � ap trim only. InRef. 86, the in� uence of this approximate trim procedure on bladestability was determined. It is evident from Fig. 22, taken fromRef. 86, that the effect of coupled trim on blade stability is small.The destabilizing effect caused by forward � ight on stiff-in-planerotors is also evident from Fig. 22.

The important trend that emerged from the studies on � ap-lag-torsional stability of hingeless blades in forward � ight was thatforward � ight destabilizes stiff-in-planecon� gurations.

G. Air and Ground ResonanceThe ground-resonance problem of articulated rotors has been

quite well understoodas indicatedearlier in this paper. When asym-metry in the rotor support system or in the blades themselves exists,the classical treatment36 is inadequate.Hammond130 has consideredthe ground resonance of an articulated rotor with one lag damperinoperative, using Floquet theory. This was also the � rst paper todemonstratetheconvenientapplicationofFloquet theory to the classof coupled rotor-fuselage problems involving asymmetry. Dissim-ilarities introduce periodic coef� cients in the equations of motion.Blade-to-bladedissimilaritieshave been consideredby McNulty.131

The effect of nonlinearities on ground resonance has also beenconsidered. Bellavitta et al.132 considered the ground resonance ofan articulatedrotor helicopterwhere the landing gear had nonlinearcharacteristics; the solutions were obtained using direct numeri-cal integration. The in� uence of nonlinear damping on helicopterground resonancewas studied by Tang and Dowell.133 The analyti-cal model includeda three-bladed,articulatedrotor,with each bladehaving only lead-lag motion, combined with a fuselage that couldpitch and roll. The formulation contains both a nonlinear damperand a nonlinear landing gear damping. The analytical results werecomparedwith experimentsconductedon a model, and good agree-ment was obtained.

There is also evidencethat the aerodynamicloadingon the bladescan have a signi� cant role on the ground-resonanceproblem of hin-geless and bearinglessrotors.For such con� gurationsaeromechani-cal studies frequently involve both air and ground resonance,whichare considered next.

The advent of hingeless and bearingless rotors has generatedstrong interest in analyses capable of modeling coupled rotor-fuselage problems. Several studies conducted in the late 1970s and1980s have made important contributionstoward the understandingof air-resonanceproblems.

One of the � rst comprehensive theoretical studies of the aerome-chanical stability of bearingless rotors was conductedby Hodges134

using the computer program FLAIR, based on the mathematicalmodel describedin Refs. 91 and 92. This study134 and its companionone135 deal mostly with soft-in-plane con� gurations using quasi-steady aerodynamics. The analytical results were also compared

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with experimental data on bearingless main rotors, and good corre-lation was obtained. FLAIR was also used to study hingeless rotoraeromechanicalstabilityby comparingthe theorywith experimentaldata obtained by Bousman.136 Overall agreement between theoret-ical predictions and experimental data was good. When FLAIR isused for hingeless rotors, it uses an offset-hinged,spring-restrainedmodel.

Bousman conducted a careful experimental investigation of theeffectof elasticcouplingson the aeromechanicalstabilityof a hinge-less rotor helicopter.136 Five different con� gurations were tested todetermine to what extent pitch-lag couplingand structural couplingcan successfully stabilize the air-resonancemode. This experimen-tal data set has been widely used during the last decade as a basisagainst which many analytical models have been validated. Theseexperimental results were compared � rst against theoretical resultsof Hodges.91;134 The measured lead-lag regressing mode dampingagreed well with theory. Comparison of the theory and experimentfor the damping of the body modes showed signi� cant differencesthat were attributed by Bousman to dynamic in� ow.

Johnson137 compared Bousman’s results136 with analytical pre-dictions of ground resonance,using the model described in Ref. 93.The calculationswere performed with and without dynamic in� ow.Use of dynamic in� ow improved the correlationwith experimentaldata. He obtained the remarkable result that in� ow dynamics in-troduces an additional “in� ow mode,” which explained previouslyunresolvedquestions about the correlationbetween test and theory.

Venkatesan and Friedmann83;94 developeda mathematicalmodelfor aeromechanical problems associated with multirotor vehicles.A subset of this model consisting of a three-bladed, offset-hinged,spring-restrainedmodel of a hingeless blade with � ap-and-� ag de-grees of freedom for each blade (see Fig. 16) mounted on a gim-bal, which could pitch and roll, was used to simulate the experi-mental data obtained by Bousman.136 The results obtained usingquasi-steadyaerodynamics138 were in good agreement with the ex-perimental data, except that the quasi-steady aerodynamic modelwas incapable of predicting the dynamic in� ow mode found byJohnson.137 Subsequentlyboth perturbation in� ow and dynamic in-� ow aerodynamicswere incorporatedin the coupled rotor-fuselagemodel,139 and the results obtained with dynamic in� ow producedgoodagreementwith the experimentaldata.Furthermore, the in� owmode obtained by Johnson was also reproduced. Results illustrat-ing this unsteady aerodynamic effect are shown in Figs. 23 and 24.Figure 23 shows the variation of modal frequenciesas a function ofrotor speed at zero collective pitch setting, using quasi-steadyaero-dynamics. All frequencies except the one corresponding to 0.7 Hzare predicted well. When perturbation in� ow and dynamic in� oware included,the results shown in Fig. 24 indicate that with dynamicin� ow all frequenciesare predictedwell.140 Furthermore, the in� owmode associated with the augmented states introduced by the dy-namic in� ow model is also predicted.It is shown in Ref. 139 that the

Fig. 23 Variation of modal frequencies with . µ0 = 0, con� guration4, where ³R , regressing lag mode; µ, pitch mode; Á, roll mode; and ¯R,regressing � ap mode.

Fig. 24 Variation of modal frequencies with , µ0 = 0, con� guration4, where ¸, in� ow mode and all other designations of the curves areidentical to Fig. 23.

identi� cation of this mode is relatively complicated. In addition tothese results, very good agreementwith the regressingmode damp-ing was also obtained.

Another interesting aspect of the coupled rotor-fuselageaerome-chanicalproblemin hover was studiedby Loewy and Zotto.140 Theystudied the effect of rotor shaft � exibility and associated rotor con-trol couplingon the ground/air resonanceof helicopters,which is ofinterest for certain advanced helicopters that have a relatively � exi-ble shaft. Numerical results were obtained for a four-bladed articu-latedrotor resemblingtheOH-58D helicopter.It was foundthat shaft� exibility/control couplingadds new modes of instability to groundresonance.Thesemodelscouldbeeasilystabilizedby small amountsof structural damping. Air-resonance type of instabilities,however,were found to be more susceptible to shaft � exibility/control cou-pling, and the instability in this case was stronger.

A comprehensive analytical study of the air-and ground-resonance characteristics of simpli� ed hingeless rotor helicopterswas undertaken by Ormiston.141 The study examined the effect ofnonoscillatorybody modes on air resonance; the effect of high rotorspeeds and high Lock numbers was also considered.The study wasrestricted to hover.

The studies mentioned were primarily for the case of hover, andthey did not clarify the role of the torsional degree of freedom onthe air-resonance problem. These items were carefully studied inRefs. 116 and 117. The mathematical model derived for this cou-pled rotor-fuselage system also has a provision for including anactive controller capable of suppressing air resonance.The blade isa simple, offset-hinged,spring-restrainedmodel, with coupled � ap-lag-torsional dynamics for each blade attached to a rigid fuselagewith � ve rigid-bodydegreesof freedom.Unsteadyaerodynamicsarerepresentedusing dynamic in� ow of forward � ight.63 In this modelthere is complete coupling between trim and the aeroelastic analy-sis. This mathematical model was used to analyze the behavior of afour-bladedhingelessrotor helicoptersomewhatsimilar to theMBB105 helicopter, with an arti� cially induced unstable air-resonancemode. The system is described by 37 states. In air-resonance prob-lems the lead-lag regressingmode is the critical degree of freedom.Therefore, the essential features of this instability are described bydampingplots for thisparticulardegreeof freedom.Figure 25 showsthat neglecting the torsional degree of freedom on the nominal con-� guration increases the instability of the lead-lag regressing mode.The trend of the two curves also tends to diverge at high advanceratios. The addition of torsion ampli� es the effect of the periodicterms. At high values of advance ratio, the � ap-lag-torsion modelshows a much greater differencebetween the constant and periodicstability analysis than does the � ap-lag analysis. The air resonanceof hingeless rotors in forward � ight was also studied in Ref. 142.Clearly, the neglectof the torsionaldegree of freedom is not prudentin air-resonance simulations.

It is remarkable that in one decade (1978–1988) consider-able progress was made in modeling and understanding airand ground resonance. Reliable models need to include coupled

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Fig. 25 Effect of torsional � exibility on lead-lag regressing modedamping, air resonance, and soft-in-plane rotor.

� ap-lag-torsionalblademodelswith geometricnonlinearities.Fuse-lage pitch and roll combined with at least two translationaldegreesof freedom are required. The inclusion of simple unsteady aero-dynamics, as represented by dynamic in� ow, is also essential. Forcertain con� gurations pitch link � exibility and shaft bending mightalso be important.

H. Composite Blade ModelingDevelopment of structural dynamic and aeroelastic models for

composite blades undergoing moderate or large de� ections, alongwith their applicationto aeroelasticityof hingeless,bearingless,andtilt-rotor blades, as well as of coupled rotor-fuselageproblems, hasbeen a particularly active area of research. Because of its impor-tance, this research topic has also been addressed in several surveypapers.9;143;144 Most modern rotor blades are built from compositesbecause this type of constructionguarantees essentially in� nite lifecompared to metal blades used previously that had to be replacedafter a few thousand hours of operation.

The � rst studieson compositeblade modeling started to appear inthe late 1970s. Mans� eld and Sobey145 initiated the � rst pioneeringstudyof thisdif� cult subject.Theydevelopedthe stiffnesspropertiesof a � ber-reinforcedcomposite tube subjected to coupled bending,torsion, and extension.Because transverse shear and warping of thecross section were not included in the model, it lacks some of theingredientsnecessaryfor compositerotor-bladeaeroelasticanalysis.

In the seminal work of Giavotto et al.,146 prismatic compositebeams were modeled making use of the St.-Venant principle,whichallowed the “interior” or “central” solutionsto be expressedin termsof polynomials in the beam axial coordinate.Using the virtual workprinciple, a two-dimensional, � nite element based cross-sectionalanalysis was then developed that supplies a fully populated 6 £ 6matrix [see Eq. (21)] of cross-sectional elastic constants (whichdetermines the shear center location) and stress recovery relationsfor the cross section in terms of stress resultants.The blade analysisof Ref. 147 uses the stiffnesses from this analysis supplementedby those from Ref. 148 to account for the trapeze effect. Theseworks were part of the development of a comprehensivehelicopteranalysis in Italy. Despite the generality of this approach, it waslargely unknown in the United States until the late 1980s.

The � rst structuralmodel that was actually incorporatedand usedin an aeroelastic analysis of a composite rotor blade in hover wasdeveloped by Hong and Chopra.149 In this model the blade wastreated as a single-cell, laminated box beam composed of an arbi-trary layup of composite plies, and the cross-sectional propertieswere found analytically.The strain-displacementrelations for mod-erate de� ections were taken from Hodges and Dowell,41 which doesnot include the effect of transversesheardeformations.Each laminaof the laminatewas assumed to haveorthotropicmaterial properties.The equations of motion were obtained using Hamilton’s principle.

A � nite element model was used to discretize the equations of mo-tion. Subsequently this analysis was extended to the modeling ofcomposite bearingless rotor blades in hover,150 and a systematicstudy was carried out to identify the importanceof the stiffness cou-pling terms on blade stability with � ber orientationand for differentcon� gurations.In this model the composite� exbeamof the bearing-less rotor blade was represented by an I section consisting of threelaminates. In addition to aeroelastic stability studies of compositerotor blades in hover, Panda and Chopra129 also studied the aeroe-lastic stability and response of hingeless composite rotor blades inforward � ight using the structural model presented in Ref. 149. Itwas found that ply orientation is effective in reducing both bladeresponse and hub shears.

The accuracy of analytical structural modeling was improved byReh� eld and coworkers,151¡154 culminating in the study of the freevibration of composite beams.155 This model provided insight intothe role of couplingsand improvedupon the model used in Ref. 149.It was used as the blade stiffnessmodel in Refs. 156 and 157 to treataeroelasticstability for isolatedhingeless,composite rotor blades inthe hovering � ight condition, using a mixed � nite element methodbased on Ref. 158. Parametric studies are presented to investigatethe effects of composite elastic couplingand the thrust conditiononthe aeroelastic stability, especially that of the lightly damped lead-lag mode. The stability of some of the elastically coupled casesstudied was sensitive to the nonclassicalcouplings.When bending-shear coupling was neglected, for example, this led to signi� canterrors, especially at high thrust levels. Another signi� cant effectstems from changes in the equilibrium solution for elastic twistcaused by extension-twistcoupling.The necessityof includingsucheffects in the blade model for general-purposeanalysis was noted.

A more comprehensive analysis was developed by Kosmatka159

for the structuraldynamicmodelingof compositeadvancedprop-fanblades, which, with some modi� cations, were also suitable for thegeneral modeling of composite rotor blades. The associated cross-sectional stiffness properties and shear center location were ob-tained froman accompanyinglinear two-dimensional� nite-elementmodel, which takes into account arbitrary cross-sectionalgeometryand generallyorthotropicmaterials.The initiallytwistedbladecouldundergomoderatede� ections.Numerical results for frequenciesandmode shapes obtained from this structural dynamic model were ingood agreement with modal tests on conventional and advancedpropellers.160

Bauchau and Hong47;161 developed a series of large-de� ectioncompositebeammodels thatwere intendedfor rotor-bladestructuraldynamic and aeroelastic analysis. The � nal version of this theory iscapableof modelingnaturallycurvedand twistedbeams undergoinglargedisplacementsand rotationsand small strain and is a precursorto the beam model in DYMORE (see what follows).

Atilgan and Hodges162 presented a theory for nonhomoge-nous, anisotropic beams undergoing large global rotation, smalllocal rotation and small strain, using nonlinear-beam kinematicsbased on Ref. 48. They used a perturbation analysis to obtaina two-dimensional linear cross-sectional analysis governed by aset of equations identical to those of Ref. 146, uncoupled fromthe nonlinear, one-dimensional global analysis.163 This work wasused164 to study the aeroelastic stability of rotor blades using acomputer program based on Ref. 146 for determining the bladestiffnesses.

Almost all of the work doneby 1990was restrictedto closedcrosssections. The early 1990s seems to have marked a turning point inthe approach to composite blade modeling used by researchers inRWA. Whereas most of the community continued along the linesof modeling blades in terms of simpli� ed geometries (e.g., box-beams, thin walls, etc.), a subset of the community began insteadto focus on modeling of realistic blades (principally Hodges andcoworkers) and incorporation of those models into comprehensiveanalyses (Bauchau and coworkers).

A detailed description of all of the currently available compositeblade theories is beyond the scope of this paper. However, somebackground information is necessary to put into perspective howcomposite blade theories developed from the early 1990s. In the

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usualapproachto beamtheorythe in- and out-of-planedeformationsof the cross-sectionalplane (denotedas warping) are either assumedto be small or neglected.Actually, one can only neglect the in-planewarping (Poisson contraction and antielastic deformation) if thestress � eld is uniaxial. Although isotropic, prismatic beams haveuniaxial stress � elds, that is not the case for composite beams ingeneral.

In view of the relative smallness of the warping, the earli-est structural models for composite rotor blades determined thecross-sectional warping and elastic constants based on linearanalyses.146;160;165;166 The linear, two-dimensional, cross-sectionalanalysis is developed based on the assumption that it can be un-coupled from the nonlinear,one-dimensionalglobal analysis for thebeam. The sectional analysis is then done once at each of severalcrosssectionsof a nonuniformbeam; the more rapidly the beamsec-tion changes in the spanwise direction, the more sectional analysesare necessary.

Unlike the mere assumption that the cross-sectional analysis isuncoupled or linear, with the use of asymptotic methods one canformulate conditionsunder which these assumptions actually hold,and a basis for extending the analysis beyond classical cases can befound.167 Indeed, asymptotic analyses have indeed shown that thisuncoupling holds for most cases affecting analysis of rotor blades.Necessary conditionsincludesmall strain, linearly elasticmaterials,and h ¿ `, where h is a typical cross-sectional dimension and `is the wavelength of deformation along the beam axis. Even so,suf� ciency is more dif� cult to establish. For example, the trapezeeffect,which satis� es thenecessaryconditions,can onlybe obtainedfrom a nonlinearcross-sectionalanalysis. In any case it is clear thatthe discussion of composite rotor-blade structural modeling can bedivided into two categories: cross-sectional modeling approachesand beam structuralmodels that use one-dimensionalbeam kineticsand kinematics.

Rotorbladesare typicallymodeledas beamsfor aeroelasticityanddynamicsanalysesbecauseof the simplicityof beamtheoryvs otherapproaches.Whereasthree-dimensional� niteelementmodelinghastremendouscapabilities,to model rotor blades in that manner wouldbe extremely expensive, requiring millions of degrees of freedomand an immense amount of setup labor. Not just any beam theoryis suitable for composite rotor-blade analysis. A typical structuralmodel in this category should at least include geometric nonlin-earities and initial twist. Theories in which only strain is assumedto be small147;158;168 (sometimes referred to as “geometrically ex-act” theories) are now the ones most promising for general-purposeanalysis.

These methods require cross-sectional elastic constants as in-put, however, and the determination of these constants is pre-cisely where one encounters the most dif� culties in composite-blade modeling. For accurate determination of the cross-sectionalelastic constants of composite blades, two distinct characteristicsmust be present: 1) the resulting theory must be elastically cou-pled and 2) the cross-sectional deformation must be suf� cientlygeneral.

The � rst requirement, to handle elastic coupling, is exhibitedin the strain energy per unit length, a quadratic form involv-ing certain beam generalized strain measures (e.g., the exten-sion of the reference line °11, the elastic twist ·1 , and the elas-tic bending curvatures ·2 and ·3). For prismatic beams made ofisotropic materials, this quadratic form can take a simple form,namely,

2U D

°11

·1

·2

·3

T E A 0 0 0

0 G J 0 0

0 0 E I2 0

0 0 0 E I3

°11

·1

·2

·3

(19)

When the beam is initially twisted and curved and is made of gener-ally anisotropic materials, the strain energy per unit length instead

becomes of the form

2U D

°11

·1

·2

·3

T S11 S12 S13 S14

S12 S22 S23 S24

S13 S23 S33 S34

S14 S24 S34 S44

°11

·1

·2

·3

(20)

where the Si j constantsdependon initial twist and curvatureas wellas on the geometry and materials of the cross section. There are twoalternative models that are commonly used. When the generalizedstrains accounting for transverse shear (2°12 and 2°13) are includedin the beam model,

2U D

°11

2°12

2°13

·1

·2

·3

T S11 S12 S13 S14 S15 S16

S12 S22 S23 S24 S25 S26

S13 S23 S33 S34 S35 S36

S14 S24 S34 S44 S45 S46

S15 S25 S35 S45 S55 S56

S16 S26 S36 S46 S56 S66

°11

2°12

2°13

·1

·2

·3

(21)When a generalized strain accounting for the Vlasov or restrainedwarping effect (· 0

1) is included in the beam model,

2U D

°11

·1

·2

·3

· 01

T S11 S12 S13 S14 S15

S12 S22 S23 S24 S25

S13 S23 S33 S34 S35

S14 S24 S34 S44 S45

S15 S25 S35 S45 S55

°11

·1

·2

·3

· 01

(22)

The beam generalized strain measures are, in general, nonlinearfunctions of the beam displacement and rotational variables.169

The second requirement for accurate modeling of compositeblades is that the calculationof elasticconstantstake into accountallpossible cross-sectional deformations, including transverse shear.Here an important distinction must be made. Because transverseshearing is taken into account in the cross-sectional analysis, thisdoes not imply that one needs explicit transverse shearing general-ized strains in the beam strain energy density. Moreover, whetheror not a separate warping degree of freedom needs to appear ex-plicitly in the resulting beam theory also depends on the appli-cation. The most basic model for closed cross sections, namely,Eq. (20), when its cross-sectional constants are calculated prop-erly, takes transverse shearing into account. Although it has neithertransverseshearingnor a separatewarping variable, it is suf� cientlyaccuratefor analysisof static or low-frequencydynamicbehavior.170

The additionof generalizedstrain measures for transverseshear de-formation [see Eq. (21)] will increase the accuracy of second andhigher modal frequenciesassociated with bending, and retention ofa cross-sectionalwarping variable [see Eq. (22)] will predict moreaccurately the behavior of thin-walled, open-section beams. It isimportant to distinguish between interior warping (also called St.Venant warping), which affects the values of the elastic constantsofall three of the preceding models, and restrained warping, which isonly present in Eq. (22) as an additional generalized strain measureand which changes the boundary conditionsof the one-dimensionalproblem. Indeed, among the most commonly held misconceptionsis that one always improves a beam theory by adding more defor-mation variables in the beam equations, some theories having asmany as nine.171;172 Results obtained from the simplest theory arefrequently as good or better, provided the simplest theory has thecorrect elastic constants.173 Although there are theories based on adhoc assumptionsthat do a good job for certain classes of blade crosssections,174 the only way to guarantee that a cross-sectionalanalysiswill always predict correct elastic constants is to make certain thatit is asymptotically correct in terms of a small parameter, that is,in fact, small. Asymptotically correct means that the approximatesolution agrees with an expansionof the exact solution (in this case

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three-dimensionalelasticity) in terms of the small parameter up toa speci� ed power of the small parameter.

Cross-sectional analyses can be classi� ed as either analyticalor � nite element based. The analytical ones can be further clas-si� ed as ad hoc or asymptotic. The ad hoc analyses have becomequite sophisticated,174¡180 all of which are restricted to the thin-walled case, except Ref. 174. However, asymptotic analyses canyield closed-form results for section constants and stress/strain re-covery for beams with thin-walled geometries. As expected, theyare the most accurate of the thin-walled analyses, as shown inRefs. 181–184. The ad hoc analyses generally invoke assump-tions that do not hold in the general case, such as ignoring thehoop stress or hoop moment or ignoring shell bending measures.The most accurate and powerful of the ad hoc methods appearsto be Ref. 174; although it is only applicable to speci� c cross-sectional geometries, it yields results that compare favorably withthose from � nite element based analyses. The � nite element basedanalyses can be derived either from the point of view of St.Venant’s principle146;160;165;166;185;186 or from that of asymptoticmethods.167;187 In addition to the variational asymptotic method,which provides a variationally consistent result, there are analyesbased on standard asymptotic methods.188 Cross-sectional anal-yses are usually linear, but an exception is the trapeze effect,which requires either an initial stress approach148 or a nonlinearanalysis.189;190

One computercodefora � niteelementbasedcross-sectionalanal-ysis thathas shownconsistentlyto bequiteaccurateis theVariationalAsymptotic Beam Section Analysis (VABS), originally developedby Cesnik, Yu, Hodges, and their coworkers.167;170;187;191¡196 VABShas promise for meeting industry’s requirements for an ef� cient,reliable analysis tool for composite blades. Validation studies showthat it has accuracy and analysis � exibility comparable to morecostly, general-purpose three-dimensional � nite element analysessuch as ABAQUS and can reduce computational effort by two tothree orders of magnitude relative to such tools. VABS can performa classical analysis [i.e., producinga model of the form of Eq. (20)]or a Timoshenko-like analysis [i.e., producing a model of the formof Eq. (21)] for beams with initial twist and curvature.VABS is alsocapable of capturing the trapeze and Vlasov effects, which are use-ful for speci� c beam applications. Finally, VABS can recover thethree-dimensionalstress and strain � elds for � nding stress concen-trations, interlaminarstresses,etc. VABS is a two-dimensional� niteelement analysis with a typical element library (triangular elementswith 3–6 nodes and quadrilateral elements with 4–9 nodes). It ismodular and can be easily integrated into any CAD/CAM software.VABS input is highly compatible with formats used in commer-cial � nite element packages, and so any two-dimensional meshedmodel of a cross section constructed in PATRAN or ANSYS can beconverted into an input for VABS with very little effort.

The last 15 years have exhibited a lot of progress in compositerotor-blade modeling. In summary, currently available composite-blade theories that were developed and have been used in rotary-wing aeroelastic applications can be separated into three groups:

1) Theories in the � rst group are those in which some ad hoccross-sectionaldeformation is assumed, which leads to a set of one-dimensional equations governing behavior of the blade. Althoughthis is themost common approachfor bladesmade of isotropicmate-rials, it can lead to grossly inaccurate results for composite blades.Such assumptions as “plane sections remain plane” or “the crosssection is rigid in its own plane” or the uniaxial stress hypothesiscan all lead to serious errors. Examples of such errors are presentedin Refs. 173 and 197.

2)The secondgroupof theories is basedonequationsfor the bladeas a one-dimensional continuum (frequently written in a canonicalform), the cross-sectional properties of which are obtained from aseparate source. The canonical form of the one-dimensional equa-tions has been known at least since the mid-1980s (Refs. 147, 156,and 168) and typically takes as input a fully populated 6 £ 6 ma-trix of cross-sectional elastic constants. Methods for � nding theseconstants vary. Unfortunately, this approach lacks a rigorous basisfor extension to include effects other than extension, shear, torsion,

and bending (such as the Vlasov effect) and nonlinear effects suchas the trapeze effect.

3)Theoriesin the thirdgroupare thosein which theequationsgov-erning cross-sectional deformation and the one-dimensional equa-tions governingbehaviorof the blade as an equivalent beam are rig-orously reduced from the common framework of three-dimensionalelasticity theory: This is the newest and most general approach.Ex-amples include Refs. 163, 167, 187, 198–200. It provides the bestpossible cross-sectional properties, quite accurate strain and stressrecovery,197 and yields the geometricallyexact canonical equationsof motion for beams.147;158;168 It has been extended to include suchthings as the trapeze190 and Vlasov184 effects. The trapeze effectaccounts for the increase in effective torsional rigidity from axialforce, important in rotating beams. The Vlasov effect is importantfor thin-walledopen cross sections,examples of which are typicallyused in bearingless rotor � exbeams.

These theories have been applied to a large number of rotary-wing aeroelastic problems, as indicated next: 1) aeroelastic behav-ior of composite hingeless and bearingless rotor blades in hoverand forward � ight.156;157;176;180;185;186;201;202 2) air and ground reso-nance of helicopters with elastically tailored composite blades178;and 3) tilt-rotoraeroelasticperformance,stability,and responsewithelastically coupled composite rotor blades.203¡227

I. Swept-Tip, Hingeless, and Bearingless RotorsThe preceding sections have provided considerable information

on hingeless rotors. Therefore, it is interesting to discuss the aeroe-lastic behavior of swept tip, or advanced geometry rotors schemat-ically depicted in Fig. 26. “Swept tip” implies both sweep andanhedral. Furthermore, the tip can have a tapered geometry. Thestructural modeling of swept-tip rotor blades represents an impor-tant practical and complicated theoretical problem. An approxi-mate aeroelastic model for swept-tip rotor blades was developedby Tarzanin and Vlaminck208; however the approximate represen-tation of the structural, inertia, and aerodynamic coupling effectscause the model to be unreliable. The � rst consistent model for ahingeless rotor with a swept tips was presented in Refs. 100 and101. The hingeless blade was modeled using a Galerkin � nite el-ement technique, and a special element for structural, inertia, andaerodynamicproperties of the swept tip was developed.Both hoverand forward � ight were considered. It was found that sweep andpreconecan be used to modify the aeroelasticbehaviorof the blade.The aeroelastic analysis of swept-tip rotors was also consideredin Ref. 209 with transonic aerodynamics and a free wake model.Structural dynamic tests and correlation with a moderate de� ec-tion theory were undertaken by Ref. 210. Subsequent correlationby Ref. 211 showed conclusively that geometrically exact analy-ses correlate much better. Parametric studies for such rotors werecarried out.201 Recently, Maier et al.212 conducted correlations be-tween experimental data and simulations using the CAMRAD II213

computer code good agreement was obtained between theory andtest for the case of hover. For forward � ight the agreement betweensimulation and test data was considerably worse than for the caseof hover.

One of the most important modern rotor systems is the bear-ingless rotor, schematically depicted in Fig. 4. The analysis of abearingless main rotor (BMR) is complicated due to the redundantstructuralcon� guration in the root region. Mathematicalmodels forsuch rotors made their � rst appearance in the late 1970s. One ofthe earliest analyses of such a rotor con� guration was incorporated

Fig. 26 Typical hingeless blade with advanced geometry tip.

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into the FLAIR code.91;92;134 Subsequently, Sivaneri and Chopra103

developed a useful � nite element model for bearingless rotors. A� exbeam-type bearingless rotor is modeled using regular beam � -niteelementsfor the outer portion,a rigid clevis,andmultiplebeamsto represent the � exbeam and the torque tube, as shown in Fig. 27.Special displacement compatibility conditions are enforced at theclevis. This model represents essentially a special redundant rootelement for the � exbeam.

A sectiondealingwith the propertiesof hingelessand bearinglessrotors would be incomplete without mentioning an insightful studyby Weller,214 which provides a comparison of the aeromechanicalstability characteristics, in hover, for two models of convention-ally designed soft-in-planemain rotors. One model is a bearinglesscon� guration, simulating the Bell helicopterM680 main rotor. Thesecond model is a hingeless rotor similar to the MBB BO-105 mainrotor.The purposeof the studywas to compare the test data obtainedfrom the two models, identify their respective aeromechanical sta-bility characteristics,and determine the design features that have aprimary effect on the air- and ground-resonancebehavior in hover.

In Ref. 214 two Froude-scaled models, one hingeless and onebearingless,were tested. One was an MBB-105 1 : 4 scale rotor, andthe other one was a 1 : 4 scale bearingless rotor resembling the Bellbearinglessrotor.The rotorswere testedon the AdvancedRotorcraftExperimental Dynamics system, which can provide body pitch androll degrees of freedom, at both low and high thrust conditions.Theresults obtained indicate that the hingeless rotor concept offers bet-ter stability margins at moderate-to-high-thrust conditions because

Fig. 27 Idealized � nite element model for the root region of a bearing-less rotor.

Fig. 28 Description of the Comanche bearingless main rotor, including both elastomeric and Fluidlastic® damper con� gurations.

of its aeroelastic characteristics; thus, the hingeless rotor is morestable at 1g thrust and above. For low thrust conditions, however,the bearingless rotor is better because of its larger structural damp-ing caused by the elastomeric lag damper. In these comparisons itis also important to keep in mind that the hingeless rotor had nolag damper, and its damping was caused by its inherent structuraldamping.

An outstanding study is Ref. 215, which describes in detail theaeroelastic stability wind-tunnel testing of the Comanche BMR andpresents correlations with an analytical model. This BMR con� g-uration is depicted in Fig. 28. A series of wind-tunnel tests wereperformed on a 1

6 Froude-scaled model of the RAH-66 ComancheBMR at the Boeing vertical/short takeoff and landing wind tunnel.The tests had two objectives:1) establish the aeromechanicalstabil-ity characteristics of the coupled rotor-fuselage system and 2) cor-relate the experimental data with analytical stability predictions sothat the methodology can be used with con� dence for the full-scale aircraft. An initial test of the rotor with elastomeric dampers,shown in Fig. 29, uncovereda limit-cycleinstability.This instabilitymanifested itself for the minimum � ight weight con� guration.Figure 29, taken from Ref. 215, depicts the frequency anddamping of the coupled rotor-body system with elastomericsnubber/dampers. The presence of the body degrees of freedomand their coupling with the blade degrees of freedom modi� es sig-ni� cantly the dynamic characteristics compared to the isolated ro-tor case. A frequency coalescence between the lag regressing andthe � ap-regressing/body-roll mode now exists. Near this coales-cence, the damping is low, and a limit-cycle oscillation occurs atthe regressing lag frequency. Closer examination of this nonlinearphenomenon215 revealed that this problem might also be presentwhen � ying with the prototype � ight weight. A decision was madeto replace the elastomeric snubber/damper by a Fluidlastic® snub-ber/damper, which is also shown in Fig. 28. The Fluidlastic snub-ber/damper is similar to the elastomeric dampers, except that it in-cludes a chamber within the � at elements, which is � lled with sili-cone � uid to providethe blade lead-lagdamping.As the elastomericelements that constitute the wall of the chamber � ex in shear, the� uid is forced to � ow around a rigid diverter protruding into the� uid, thereby generating a damping force.

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Fig. 29 Hover air resonance of the minimum � ight weight con� gura-tion with elastomeric dampers at 8-deg collective pitch.

Fig. 30 Hover air resonance at 9-deg collective with Fluidlastic®

damper.

Further study revealed that nonlinearitiesin the stiffness and lossfactor of the elastomeric snubber/dampers were the cause of thislimit-cycle behavior. As shown in Ref. 215, the stiffness of the Flu-idlastic damper is nearly linear, and using it eliminates the limit-cycle instability. Figure 30 shows the hover air response character-istics of the prototype � ight weight con� guration with the Fluid-lastic dampers at 9 deg collective. The test data for both frequencyand damping are also compared with analytical results obtainedfrom the UMARC/B code, which is a Boeing modi� ed version of

UMARC.216 The correlations between the results for the code inboth hover and forward � ight are quite good.

During the last threedecades,thehelicopterindustryin the UnitedStates and abroad has invested a very substantial amount of re-sources in the developmentof productionhingelessand bearinglessrotor systems. Hingeless rotored helicopters, such as the MBB BO-105 and the Westland Lynx, have been in production for almost25 years. However, successful bearingless rotored helicopters havegone into production only during the last decade. Typical exam-ples are the MD-900 Explorer,217 the Comanche bearingless mainrotor (BMR),215 and the Eurocopter EC135.218 The MD900 andthe Comanche have � ve-bladed rotors, whereas the EC135 is four-bladed. This is an indication that BMR technology has matured inthe last decade, and substantial gains in the understandingof aeroe-lastic and aeromechanical aspects of these rotors have been made.Therefore, one can view the BMR systems that are currently in pro-duction as the crowning achievement of RWA during the last twodecades.

J. Comprehensive Analysis CodesThe complexity of the RWA problem has motivated the devel-

opment of computer codes that have the capability of solving bothisolated-blade,as well as coupledrotor-fuselageproblems.Once thelarge effort required was invested, other calculations in the area ofperformance and � ight mechanics were also included in the code.Such codes became known as comprehensive analysis codes. Per-haps the � rst of these codes, known as REXOR, was developedby Lockheed in the early to mid-1970s (Ref. 219). One of the � rstsuccessfulcodeswas CAMRAD developedby Johnson,82;220 whicheventually became CAMRAD/JA.221

Another important comprehensive analysis code initiated in theearly 1980s and completed in the 1990s was the Second Gen-eration Comprehensive Helicopter Analysis System, also knownas 2GCHAS,222¡228 which was developed with funding by theU.S. Army Aero� ightdynamics Directorate (formerly known asthe Aeromechanics Laboratory), as a second generation replace-ment for REXOR. Similar codes were developed by various heli-copter companies, two of the better known ones are RDYNE229 andCOPTER.230

Another very useful code is UMARC,216 developed at theUniversity of Maryland. The UMAR code developedat the Univer-sity of Maryland has also enjoyed considerablesuccess, as studentswho graduatedhave taken the code with them and started using it inan industrial setting. Subsequently, a more advanced and improvedversionof CAMRAD/JA was developed:CAMRAD II.214 The threemost advanced,CAMRAD II, 2GCHAS, andDYMORE, have takenadvantage of multibody dynamics that facilitate the effective treat-ment of complex con� gurations.231¡233

Among the various comprehensive helicopter analysis codes,CAMRAD II is perhaps the most widely used, both in the UnitedStates as well as Europe and Japan. The code has been slightlymoresuccessfulthan its competitorsin correlatingwith experimentaldata.The wide acceptanceof this code is evident from a recent paper thatdescribes the design aspects of a new production bearingless mainrotor used on the European EC135. This rotor has excellent damp-ing margins throughout its operation envelope. Modal damping forthis rotor in level � ight is shown in Fig. 31. The dots are from the� ight test, and the solid line is the result of a calculation performedby CAMRAD II. The agreement between theory and test is good.The damping amounts to approximately 2:5% in the rotating sys-tem. This rotor is equipped with an elastomeric lag damper andapparently the code reproduces its behavior well.

The 2GCHAS code has also undergone considerable validationduring the last � ve years, and overall the correlationsindicategener-ally satisfactorypredictivecapabilityfor a fairlywide rangeof rotor-craft problems. A modi� ed and improved version of the 2GCHAScodehas recentlybecomeavailable;it is denotedby the name Rotor-craft ComprehensiveAnalysis Code (RCAS). In addition to consid-erable improvements that enhance its computationalef� ciency andreduce the run times required, the code has the added advantage ofbeingable to run on PC platforms using the Linux operatingsystem.

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Fig. 31 Regressing lag mode damping in forward � ight and compari-son with CAMRAD II.

The nonlinear beam element has been signi� cantly improved overthe older 2GCHAS codeby makinguse of embeddedframes similarto the method used in GRASP.

The multi-� exible-body code DYMORE by Bauchau andcoworker has extensive capability in modeling of systemhardware.168;231

IV. 21st Century—Period of Re� nement(2001–present)

It is evident from the various papers published since the turnof the new century that the interest in studies dealing with aeroe-lastic stability has diminished during the last few years. Thereis some interest in tilt-rotor aeroelastic stability,234 aeroelasticscaling,235 active control for stability augmentation,234 and aeroe-lastic analysis of rotors with trailing-edge � aps used for vibrationreduction.236 At the last European Rotorcraft Forum (28th Euro-peanRotorcraftForum, Bristol,United Kingdom,September2002),there was a total of seven dynamics sessions, where 21 differ-ent papers were published and presented. Not a single paper dealtwith any signi� cant aspect of the rotary-wing aeroelastic stabilityproblem.

A recent paper8 clearly indicates that the primary interest hasshifted towards active vibration reduction, load correlation, and re-� nement of existingcodes and analyses to providebetter agreementwith existing experimental databases or new experimental test datagenerated.

V. ConclusionsThis historicalperspectiveillustratesthat the progressmade in the

area of rotary-wing aeroelasticityduring the last 60 years has beenspectacular.Given the complexityand diversityof the problems thathave been considered, as well as those that still persist, it is fair tosay that the accomplishmentsof this relativelyshort period comparevery favorably with what has been accomplishedby the � xed-wingcommunity during an entire century. Furthermore, considering thedisparity in funding between these two areas of endeavor over suchan extended time period further ampli� es the accomplishments inRWA.

AcknowledgmentsThis paper is dedicated to the memory R. H. Miller, who passed

away on 28 January 2003 at the age of 87. The authors gratefullyacknowledgethe fundingof the U.S. Army ResearchOf� ce, NASA,and the National Rotorcraft Technology Center over a long timeperiod. The � rst author is grateful to his student Dan Patt for his

invaluable help in preparing this paper. Without his help this paperwould have never made the deadline.

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73Peters, D. A., “Toward a Uni� ed Lift Model for use in Rotor BladeStability Analysis,” Journal of the American Helicopter Society, Vol. 30,No. 3, 1985, pp. 32–42.

74Petot, D., “Differential Equation Modeling of Dynamic Stall,” La Re-serche Aerospatiale, Vol. 5, 1989, pp. 59–71.

75Beddoes, T. S., “A Synthesis of Unsteady Aerodynamic Effects Includ-ing Stall Hysteresis,” Vertica, Vol. 1, No. 2, 1976, pp. 113–123.

76Beddoes, T. S., “Representation of Airfoil Behavior,” Vertica, Vol. 7,No. 2, 1983, pp. 183–197.

77Leishman, J. G., “Two-Dimensional Model for Airfoil Unsteady DragBelow Stall,” Journal of Aircraft, Vol. 25, No. 7, 1988, pp. 665, 666.

78Leishman, J. G., “ModellingSweep Effects on Dynamic Stall,” Journalof the American Helicopter Society, Vol. 34, No. 3, 1989, pp. 18–29.

79Leishman, J. G., and Beddoes, T. S., “A Generalized Model for Un-steady Aerodynamic Behavior and Dynamic Stall Using Indicial Method,”Journal of the American Helicopter Society, Vol. 34, No. 3, 1989,pp. 3–17.

80Kaza, K. R., and Kvaternik, R. G., “NonlinearAeroelastic Equations forCombinedFlapwise Bending,ChordwiseBending,Torsionand ExtensionofTwisted Non-Uniform Rotor Blades in Forward Flight,” NASA TM 74059,June 1977.

81Shamie, J., and Friedmann, P. P., “Effect of Moderate De� ections onthe Aeroelastic Stability of a Rotor Blade in Forward Flight,” Proceedingsof the Third European Rotorcraft Forum, Aix-en-Provence, France 1977,pp. 24.1–24.37.

82Johnson, W., “Development of a Comprehensive Analysis for Rotor-craft, Part I: Rotor Model and Wake Analysis,” Vertica, Vol. 5, No. 2, 1981,pp. 99–129.

83Venkatesan, C., and Friedmann, P. P., “Aeroelastic Effects in MultirotorVehicles with Application to Hybrid Heavy Lift System, Part I: Formulationof Equations of Motion,” NASA CR 3822, Aug. 1984.

84Warmbrodt, W., and Friedmann, P. P., “Formulation of CoupledRotor/ Fuselage Equations of Motion,” Vertica, Vol. 3, No. 3, 1979,pp. 254–271.

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86Reddy, T. S. R., and Warmbrodt, W., “Forward Flight Aeroelastic Sta-bility from Symbolically Generated Equations,” Journal of the AmericanHelicopter Society, Vol. 31, No. 3, 1986, pp. 35–44.

87Crespo da Silva, M. R. M., and Hodges, D. H., “Nonlinear Flexure andTorsion of Rotating Beams with Application to Helicopter Rotor Blades—I.Formulation,” Vertica, Vol. 10, No. 2, 1986, pp. 151–169.

88Crespo da Silva, M. R. M., and Hodges, D. H., “Nonlinear Flex-ure and Torsion of Rotating Beams with Application to Helicopter RotorBlades—II. Response and Stability Results,” Vertica, Vol. 10, No. 2, 1986,pp. 171–186.

89Millott, T. A., and Friedmann, P. P., “Vibration and Reductionin Hingeless Rotor Using an Actively Controlled Trailing Edge Flap:Implementation and Time Domain Simulation,” Proceedings of 35thAIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Mate-rials Conference, AIAA, Washington, DC, 1994, pp. 9–22.

90Millott, T. A., and Friedmann, P. P., “Vibration Reduction in HelicopterRotorsUsing an ActivelyControlledPartial Span TrailingEdgeFlap Locatedon the Blade,” NASA CR 4611, June 1994.

91Hodges, D. H., “Aeromechanical Stability of Helicopters with Bearing-less Main Rotor, Part II: Computer Program,” NASA TM 78460, 1978.

92Hodges, D. H., “Aeromechanical Stability of Helicopters with Bearing-less Main Rotor, Part I: Equations of Motion,” NASA TM 78459, 1979.

93Johnson, W., “A Comprehensive Analytical Study of Rotorcraft Aero-dynamics and Dynamics, Part I: Analysis Development,” NASA TM 81182,July 1980.

94Venkatesan, C., and Friedmann, P. P., “Aeroelastic Effects in MultirotorVehicles with Application to Hybrid Heavy Lift System, Part II: Method ofSolution and Results,” NASA CR 4009, Feb. 1986.

95Nagabhushanam, J., Gaonkar, G. H., and Reddy, T. S. R., “AutomaticGeneration ofEquationsforRotor-BodySystems with Dynamic In� ow for a-Priori Ordering Schemes,” Proceedings of the Seventh European RotorcraftForum, 1981.

96Hodges, D. H., Hopkins, A. S., Kunz, D. L., and Hinnant,H. E., “Introduction to GRASP—General Rotorcraft Aeromechani-cal Stability Program—A Modern Approach to Rotorcraft Modeling,”Journal of the American Helicopter Society, Vol. 32, No. 2, 1987,pp. 78–90.

97Friedmann, P. P., and Kottapalli, S. B. R., “Coupled Flap-Lag-TorsionalDynamics of Hingeless Rotor Blades in Forward Flight,” Journal of theAmerican Helicopter Society, Vol. 27, No. 4, 1982, pp. 28–36.

98Straub, F. K., and Friedmann, P. P., “A Galerkin Type Finite ElementMethod for Rotary-Wing Aeroelasticity in Hover and Forward Flight,” Ver-tica, Vol. 5, No. 1, 1981, pp. 75–98.

99Straub, F. K., and Friedmann, P. P., “Application of the Finite ElementMethod to Rotary-Wing Aeroelasticity,” NASA CR 165854, Feb. 1982.

100Celi, R., and Friedmann, P. P., “Aeroelastic Modeling of Swept TipRotor Blades Using Finite Elements,” Journal of the American HelicopterSociety, Vol. 33, No. 2, 1988, pp. 43–52.

101Celi, R., and Friedmann, P. P., “Rotor Blade Aeroelasticity in ForwardFlight with an Implicit Aerodynamic Formulation,” AIAA Journal, Vol. 26,No. 12, 1988, pp. 1425–1433.

102Sivaneri, N. T., and Chopra, I., “Dynamic Stability of a Rotor BladeUsing Finite Element Analysis,” AIAA Journal, Vol. 20, No. 5, 1982,pp. 716–723.

103Sivaneri, N. T., and Chopra, I., “FiniteElement Analysis of BearinglessRotor Blade Aeroelasticity,” Journal of the American Helicopter Society,Vol. 29, No. 2, 1984, pp. 42–51.

104Friedmann, P. P., “In� uence of Modeling and Blade Parameters on theAeroelastic Stability of a Cantilevered Rotor,” AIAA Journal, Vol. 15, No. 2,1977, pp. 149–158.

105Ormiston, R. A., and Bousman, W. G., “A Study of Stall Induced Flap-Lag Instability of Hingeless Rotors,” Journal of the American HelicopterSociety, Vol. 20, No. 1, 1975, pp. 20–30.

106Friedmann, P. P., and Shamie, J., “Aeroelastic Stability of TrimmedHelicopter Blades in Forward Flight,” Vertica, Vol. 1, No. 3, 1977,pp. 189–211.

107Peters, D. A., and Hohenemser, K. H., “Application of the FloquetTransition Matrix to Problems of Lifting Rotor Stability,” Journal of theAmerican Helicopter Society, Vol. 16, No. 2, 1971, pp. 25–33.

108Friedmann, P. P., and Silverthorn, L. J., “Aeroelastic Stability of Pe-riodic Systems with Application to Rotor Blade Flutter,” AIAA Journal,Vol. 12, No. 11, 1974, pp. 1559–1565.

109Friedmann, P. P., Hammond, C. E., and Woo, T., “Ef� cient NumericalTreatment of Periodic Systems with Application to Stability Problems,” TheInternationalJournal for Numerical Methods in Engineering, Vol. 11, 1977,pp. 1117–1136.

110Dugundji,J., and Wendell, J. H., “Some AnalysisMethodsforRotatingSystems, with Periodic Coef� cients,” AIAA Journal, Vol. 21, No. 6, 1983,pp. 890–897.

111Friedmann, P. P., “Numerical Methods for Determining the Stabilityand Response of Periodic Systems with Applications to Helicopter RotorDynamics and Aeroelasticity,” Computers and Mathematics with Applica-tions, Vol. 12A, No. 1, 1986, pp. 131–148.

112Peters, D. A., and Izadpanah, A. P., “Helicopter Trim by PeriodicShooting with Newton-Raphson Iteration,” Proceedings of the 37th AnnualForum of the American Helicopter Society, 1981, pp. 217–226.

113Borri, M., “Helicopter Rotor Dynamics by Finite Element Time Ap-proximation,” Computers and Mathematics with Applications, Vol. 12A,No. 1, 1986, pp. 143–154.

114Biggers, J. C., “Some Approximations to the Flapping Stability ofHelicopter Rotors,” Journal of the American Helicopter Society, Vol. 14,No. 4, 1974, pp. 24–33.

115Hohenemser, K. H., and Yin, S. K., “Some Applicationsof the Methodof Multiblade Coordinates,” Journal of the American Helicopter Society,Vol. 17, No. 3, 1972, pp. 1–12.

116Takahashi, M. D., “Active Control of Helicopter Aeromechanical andAeroelastic Instabilities,” Ph.D. Dissertation, Mechanical, Aerospace andNuclear Engineering Dept., Univ. of California, Los Angeles, June 1988.

117Takahashi, M. D., and Friedmann, P. P., “Active Control of Heli-copter Air Resonance in Hover and Forward Flight,” Proceedings of the29th AIAA/ASME/ASCE/AHS/ Structures, Structural Dynamics and Materi-als Conference, AIAA, Washington, DC, 1988, pp. 1521–1532.

118Torok, M. S., and Chopra, I., “A Coupled Rotor Aeroelastic AnalysisUtilizing Nonlinear Aerodynamics and Re� ned Wake Modeling,” Vertica,Vol. 13, No. 2, 1989, pp. 87–106.

119Tran, C. T., and Falchero, D., “Application of the ONERA DynamicStall Model to a Helicopter Blade in Forward Flight,” Vertica, Vol. 6, No. 3,1982, pp. 219–239.

120Ormiston, R. A., and Hodges, D. H., “Linear Flap-Lag Dynamics ofHingeless Helicopter Rotor Blades in Hover,” Journal of the American He-licopter Society, Vol. 17, No. 2, 1972, pp. 2–14.

121Friedmann, P. P., and Tong, P., “Non-Linear Flap-Lag Dynamics ofHingeless Helicopter Blades in Hover and Forward Flight,” Journalof Soundand Vibration, Vol. 30, No. 1, 1973, pp. 9–31.

122Huber, H. B., “Effect of Torsion-Flap-Lag Coupling on Hingeless Ro-tor Stability,” Proceedings of the 29th Annual National Forum of the Amer-ican Helicopter Society, 1973.

123Peters, D. A., “Flap-Lag Stability of Helicopter Rotor Blades in For-ward Flight,” Journal of the American Helicopter Society, Vol. 20, No. 4,1975, pp. 2–13.

124Hansford, R. E., and Simons, I. A., “Torsion-Flap-LagCouplingon He-licopter Rotor Blades,” Journal of the American Helicopter Society, Vol. 18,No. 4, 1973, pp. 2–12.

125Sopher,R., and Cassarino, S., “Effect ofAnalytical Modeling Assump-tions on the Predicted Stability of a Model Hingeless Rotor,” Journal of theAmerican Helicopter Society, Vol. 33, No. 4, 1988, pp. 15–27.

126Anderson, W. D., and Watts, G. A., “Rotor Blade Wake Flutter: AComparison of Theory and Experiment,” Journalof the American HelicopterSociety, Vol. 21, No. 2, No. 2, 1976, pp. 31–43.

127Hodges, D. H., Kwon, O. J., and Sankar, L. N., “Stability of Hinge-less Rotors in Hover Using Three-Dimensional Unsteady Aerodynamics,”Journal of the American Helicopter Society, Vol. 36, No. 2, 1991,pp. 21–31.

128Panda, B., and Chopra, I., “Flap-Lag-Torsion Stability in ForwardFlight,” Journal of the American Helicopter Society, Vol. 29, No. 4, 1985,pp. 30–39.

129Panda, B., and Chopra, I., “Dynamics of Composite Rotor Blades inForward Flight,” Vertica, Vol. 11, No. 1/2, 1987, pp. 187–209.

130Hammond, C. E., “An Application of Floquet Theory to the Predic-tion of Mechanical Instability,” Journal of the American Helicopter Society,Vol. 19, No. 4, 1974, pp. 14–23.

131McNulty, M., “Effects of Blade-to-Blade Dissimilarities on RotorBody Lead-Lag Dynamics,” Journal of the American Helicopter Society,Vol. 33, No. 1, 1988, pp. 17–28.

132Bellavitta, P.,Giorgi,C., and Galeazzi, M., “Some Aspects ofMechani-cal InstabilityProblems forFullyArticulated RotorHelicopter,” Proceedingsof the Second European Rotorcraft and Powered Lift Aircraft Forum, 1976,pp. 14.1–14.40.

133Tang, D. M., and Dowell, E. H., “In� uence of Nonlinear Blade Damp-ing on Helicopter Ground Resonance,” Journal of Aircraft, Vol. 23, No. 2,1986, pp. 104–110.

134Hodges, D. H., “An Aeromechanical StabilityAnalysis forBearinglessRotor Helicopters,” Journal of the American Helicopter Society, Vol. 24,No. 1, 1979, pp. 2–9.

135Hodges, D. H., “A Theoretical Technique for Analyzing Aeroelas-tic Stability of Bearingless Rotors,” AIAA Journal, Vol. 17, No. 4, 1979,pp. 400–407.

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136Bousman, W. G., “An Experimental Investigation of the Effects ofAeroelastic Couplings on Aeromechanical Stability of a Hingeless RotorHelicopter,” Journal of the American Helicopter Society, Vol. 26, No. 1,1981, pp. 46–54.

137Johnson, W., “In� uence of Unsteady Aerodynamics on HingelessRotor Ground Resonance,” Journal of Aircraft, Vol. 19, No. 8, 1982,pp. 668–673.

138Friedmann, P. P., and Venkatesan, C., “CoupledHelicopter Rotor/BodyAeromechanical Stability Comparison of Theoretical and Experimental Re-sults,” Journal of Aircraft, Vol. 22, No. 2, 1985, pp. 148–155.

139Friedmann, P. P., and Venkatesan, C., “In� uence of Unsteady Aero-dynamic Models of Aeromechanical Stability in Ground Resonance,”Journal of the American Helicopter Society, Vol. 31, No. 1, 1986,pp. 65–74.

140Loewy, R. G., and Zotto, M., “Helicopter Ground/Air Resonance In-cluding Rotor Shaft Flexibility and Control Coupling,” Proceedings of the45th Annual Forum of the American Helicopter Society, 1989.

141Ormiston, R. A., “Rotor-Fuselage Dynamics of Helicopter Air andGround Resonance,” Journal of the American Helicopter Society, Vol. 38,No. 2, 1991, pp. 3–20.

142Wang, J. M., Jang, J., and Chopra, I., “Air Resonance Stability ofHingeless Rotors in Forward Flight,” Proceedings of the 14th EuropeanRotorcraft Forum, 1988, pp. 18.1–18.18.

143Hodges, D. H., “A Review of Composite Rotor Blade Modeling,” AIAAJournal, Vol. 28, No. 3, 1990, pp. 561–565.

144Jung, S. N., Nagaraj, V. T., and Chopra, I., “Assessment of CompositeRotor Blade Modeling Techniques,” Journal of the American HelicopterSociety, Vol. 44, No. 3, 1999, pp. 188–205.

145Mans� eld, E. H., and Sobey, A. J., “The Fiber Composite HelicopterBlade—Part I: Stiffness Properties—Part II: Prospects for Aeroelastic Tai-loring,” Aeronautical Quarterly, Vol. 30, 1979, pp. 413–449.

146Giavotto, V., Borri, M., Mantegazza, P., Ghiringhelli, G., Car-maschi, V., Maf� oli, G. C., and Mussi, F., “Anisotropic Beam Theory andApplications,” Computers and Structures, Vol. 16, No. 1–4, 1983, pp. 403–413.

147Borri, M., and Mantegazza, P., “Some Contributionson Structural andDynamic Modeling of Helicopter Rotor Blades,” l’Aerotecnica Missili eSpazio, Vol. 64, No. 9, 1985, pp. 143–154.

148Borri, M., and Merlini, T., “A Large Displacement Formulation forAnisotropic Beam Analysis,” Meccanica, Vol. 21, 1986, pp. 30–37.

149Hong, C. H., and Chopra, I., “Aeroelastic Stability Analysis of a Com-posite Rotor Blade,” Journal of the American Helicopter Society, Vol. 30,No. 2, 1985, pp. 57–67.

150Hong, C. H., and Chopra, I., “Aeroelastic Stability Analysis of a Com-positeBearinglessRotorBlade,” Journalof theAmerican HelicopterSociety,Vol. 31, No. 4, 1986, pp. 29–35.

151Hodges, R. V., Nixon, M. W., and Reh� eld, L. W., “Compar-ison of Composite Rotor Blade Models: A Coupled-Beam Analysisand an MSC/NASTRAN Finite-Element Model,” NASA TM 89024,March 1987.

152Reh� eld, L. W., “Design Analysis Methodology for Composite Ro-tor Blades,” Proceedings of the Seventh DoD/NASA Conference on FibrousComposites in Structural Design, 1985.

153Reh� eld, L. W., and Atilgan, A. R., “Shear Center and Elastic Axis andTheir Usefulness for Composite Thin-Walled Beams,” Proceedings of theFourth Technical Conference on CompositeMaterials, American Society forComposites, 1989, pp. 179–188.

154Reh� eld, L. W., Atilgan, A. R., and Hodges, D. H., “Nonclassi-cal Behavior of Thin-Walled Composite Beams with Closed Cross Sec-tions,” Journal of the American Helicopter Society, Vol. 35, No. 2, 1990,pp. 42–50.

155Hodges, D. H., Atilgan, A. R., Fulton, M. V., and Reh� eld, L. W.,“Free-Vibration Analysis of Composite Beams,” Journal of the AmericanHelicopter Society, Vol. 36, No. 3, 1991, pp. 36–47.

156Fulton, M. V., and Hodges, D. H., “Aeroelastic Stability of Compos-ite Hingeless Rotor Blades in Hover: Part I: Theory,” Mathematical andComputer Modeling, Vol. 18, No. 3/4, 1993, pp. 1–18.

157Fulton, M. V., and Hodges, D. H., “Aeroelastic Stability of Compos-ite Hingeless Rotor Blades in Hover: Part II: Results,” Mathematical andComputer Modeling, Vol. 18, No. 3/4, 1993, pp. 19–36.

158Hodges, D. H., “A Mixed Variational Formulation Based on ExactIntrinsic Equations for Dynamics of Moving Beams,” InternationalJournalof Solids and Structures, Vol. 26, No. 11, 1990, pp. 1253–1273.

159Kosmatka, J. B., “Structural Dynamic Modeling of Advanced Com-posite Propellers by the Finite Element Method,” Ph.D. Dissertation, Me-chanical, Aerospace and Nuclear Engineering Dept., Univ. of California,Los Angeles, June 1986.

160Kosmatka, J. B., and Friedmann, P. P., “Vibration Analysis of Com-posite Turbopropellers Using a Nonlinear Beam-Type Finite Element Ap-proach,” AIAA Journal, Vol. 27, No. 11, 1989, pp. 1606–1614.

161Hong, C. H., “Finite Element Approach to the Dynamic Analysis ofComposite Helicopter Blades,” Ph.D. Dissertation, Rensselaer PolytechnicInst., New York, Oct. 1987.

162Atilgan, A. R., and Hodges, D. H., “A Geometrically Nonlinear Anal-ysis for Nonhomogeneous Anisotropic Beams,” Proceedings of the 30thAIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materi-als Conference, AIAA, Washington, DC, 1989.

163Atilgan, A. R., and Hodges, D. H., “Uni� ed Nonlinear Analysis forNonhomogeneous Anisotropic Beams with Closed Cross-Section,” AIAAJournal, Vol. 29, No. 11, 1991, pp. 1990–1999.

164Fulton, M. V., “Stability of Elastically Tailored Rotor Blades,” Ph.D.Dissertation,Schoolof Aerospace Engineering,Georgia Inst. ofTechnology,Atlanta, GA, June 1992.

165Borri, M., Ghiringhelli,G. L., and Merlini, T., “Linear Analysis ofNat-urally Curved and Twisted Anisotropic Beams,” Composites Engineering,Vol. 2, No. 5-7, 1992, pp. 433–456.

166Ie, C. A., and Kosmatka, J. B., “Formulation of a Nonlinear Theoryfor Spinning Anisotropic Beams,” Recent Advances in Structural DynamicModeling of Composite Rotor Blades and Thick Composites, Vol. AD-Vol. 30, edited by, P. P. Friedmann and J. B. Kosmatka, ASME, 1992,pp. 41–57.

167Cesnik, C. E. S., and Hodges, D. H., “VABS: A New Concept forCompositeRotorBlade Cross-SectionalModeling,” Journalof the AmericanHelicopter Society, Vol. 42, No. 1, 1997, pp. 27–38.

168Bauchau, O. A., and Kang, N. K., “A Multibody Formulation for He-licopter Structural Dynamic Analysis,” Journal of the American HelicopterSociety, Vol. 38, No. 2, 1993, pp. 3–14.

169Hodges, D. H., “Finite Rotation and Nonlinear Beam Kinematics,”Vertica, Vol. 11, No. 1/2, 1987, pp. 297–307.

170Hodges, D. H., Atilgan,A. R., Cesnik,C. E. S., andFulton,M. V., “On aSimpli� ed Strain Energy Function for Geometrically Nonlinear Behaviourof Anisotropic Beams,” Composites Engineering, Vol. 2, No. 5–7, 1992,pp. 513–526.

171Centolanza, L. R., and Smith, E. C., “Re� ned Structural Modelling ofThick-Walled Closed Section Composite Beams,” Proceedings of the 39thStructures, Structural Dynamics and Materials Conference, AIAA, Reston,Virginia, 1998, pp. 552–568.

172Chandra, R., and Chopra, I., “Structural Response of CompositeBeams and Blades with Elastic Couplings,” Composites Engineering, Vol. 2,No. 5–7, 1992, pp. 347–374.

173Volovoi, V. V., Hodges, D. H., Cesnik, C. E. S., and Popescu,B., “Assessment of Beam Modeling Methods for Rotor Blade Applica-tions,” Mathematical and Computer Modelling, Vol. 33, No. 10–11, 2001,pp. 1099–1112.

174Jung, S. N., Nagaraj, V. T., and Chopra, I., “Re� ned Structural ModelforThin-and Thick-walledCompositeRotorBlades,”AIAAJournal, Vol. 40,No. 1, 2002, pp. 105–116.

175Rand, O., “Periodic Response of Thin-Walled Composite HelicopterRotor Blades,” Journal of the American Helicopter Society, Vol. 36, No. 4,1991, pp. 3–11.

176Smith, E. C., “Aeroelastic Response and Aeromechanical Stability ofHelicopters with Elasticity Coupled Composite Rotor Blades,” Ph.D. Dis-sertation, Dept. of Aerospace Engineering,Univ. of Maryland,College Park,July 1992.

177Smith, E.C., and Chopra, I., “Formulationand Evaluationof an Analyt-ical Model for Composite Box-Beams,” Journal of the American HelicopterSociety, Vol. 36, No. 3, 1991, pp. 23–35.

178Smith, E. C., and Chopra, I., “Air and Ground Resonance With Elasti-cally Tailored Composite Rotor Blades,” Journalof the American HelicopterSociety, Vol. 38, No. 4, 1993, pp. 50–61.

179Song, O., and Librescu, L., “Structural Modeling and Free VibrationAnalysis of Rotating Composite Thin- Walled Beams,” Journal of the Amer-ican Helicopter Society, Vol. 42, No. 4, 1997, pp. 358–369.

180Tracy, A. L., and Chopra, I., “Aeroelastic Analysis of a Compos-ite Bearingless Rotor in Forward Flight Using an Improved WarpingModel,” Journal of the American Helicopter Society, Vol. 40, No. 3, 1995,pp. 80–91.

181Berdichevsky, V. L., Armanios, E. A., and Badir, A. M., “Theory ofAnisotropicThin-WalledClosed-SectionBeams,” Composites Engineering,Vol. 2, No. 5–7, 1992, pp. 411–432.

182Volovoi,V. V., and Hodges, D. H., “Theory ofAnisotropicThin-WalledBeams,” Journal of Applied Mechanics, Vol. 67, No. 3, 2000, pp. 453–459.

183Volovoi, V. V., and Hodges, D. H., “Single-and Multi-Celled Compos-ite Thin-Walled Beams,” AIAA Journal, Vol. 40, No. 5, 2002, pp. 960–965.

184Volovoi, V. V., Hodges, D. H., Berdichevsky, V. L., and Sutyrin, V.,“Asymptotic Theory for Static Behavior of Elastic Anisotropic I-Beams,”International Journal of Solids and Structures, Vol. 36, No. 7, 1999,pp. 1017–1043.

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186Yuan, K. A., and Friedmann, P. P., “Aeroelasticity and Structural Op-timization of Composite Helicopter Rotor Blades with Swept Tips,” NASACR 4665, May 1995.

187Yu, W., Hodges, D. H., Volovoi, V. V., and Cesnik, C. E. S., “OnTimoshenko-Like Modeling of Initially Curved and Twisted CompositeBeams,” International Journal of Solids and Structures, Vol. 39, No. 19,2002, pp. 5101–5121.

188Buannic, N., and Cartraud, P., “Higher-Order Asymptotic Model for aHeterogeneous Beam, Including Corrections due to End Effects,” Proceed-ings of the 41st Structures, Structural Dynamics and Materials Conference,AIAA, Reston, Virginia, 2000.

189Hodges, D. H., Harursampath, D., Volovoi, V. V., and Cesnik, C.E. S., “Non-Classical Effects in Non-Linear Analysis of AnisotropicStrips,”International Journal of Non-Linear Mechanics, Vol. 34, No. 2, 1999,pp. 259–277.

190Popescu, B., and Hodges, D. H., “AsymptoticTreatment of the TrapezeEffect in Finite Element Cross-Sectional Analysis of Composite Beams,”International Journal of Non-Linear Mechanics, Vol. 34, No. 4, 1999,pp. 709–721.

191Cesnik, C. E. S., and Hodges, D. H., “Stiffness Constants for Ini-tially Twisted and Curved Composite Beams,” Applied Mechanics Reviews,Vol. 46, No. 11, Pt. 2, 1993, pp. S211–S220.

192Cesnik, C. E. S., and Hodges, D. H., “Variational-AsymptoticalAnalysis of Initially Twisted and Curved Composite Beams,” Interna-tional Journal for Engineering Analysis and Design, Vol. 1, No. 2, 1994,pp. 177–187.

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