Statespace Models of Tilt Rotor

7/28/2019 Statespace Models of Tilt Rotor

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APPLICATION OF A STATE-SPACE WAKE MODELTO TILTROTOR WING UNSTEADY AERODYNAMICS

Martin Stettner Daniel P. SchrageGraduate Research Assistant Professor

Georgia Institute of TechnologyAtlanta, Georgia 30332

David A. PetersProfessor

Washington UniversitySt. Louis, Missouri 63130

Abstract

The Peters/He Finite State Wake Model isdescribed in its application to fixed wing aerolasticity.Expressions for coupling this model with a wing,

aerodynamically represented by a flat plate with atrailing edge flap, are developed, and fidelity issues arediscussed. An application is presented where thewing/wake system is coupled to a proprotor model.The effects of unsteady wing aerodynamics ondamping of this system are investigated. It is foundthat wake effects are small as a result of generally lowdamping levels in the system due to wingaerodynamic damping.

Notation

AR aspect ratioC linear differential operator

Cnm, Dnm coefficients, eq. (6) and (7)D drag/unit length

Di, Di* coefficients, eq. (34)

L lift/unit length; linear diff. operatorM moment/unit length; max. order of

radial polynomials[M] wake mass matrix[N] wake damping matrixP pressure discontinuity

Pnm, Qn

m associated Legendre Polynomials of

first and second kindR disk radiusS wing semi span

U freestream velocityV disturbance velocity

anm, bn

m wake states

b wing semi chordi coordinate index; index

Presented at the American Helicopter Society Aeromechanics SpecialistsConference, San Francisco, California, January 19-21, 1994. Copyright 1994 by the American Helicopter Society Inc. All rights reserved.

n, j polynomial numberm, r harmonics numberf frequency [Hz]k reduced frequencyq perturbation velocityr radial coordinatet timeu, w chord/beamwise wing deflectionx, y, z cartesian coordinates relative deviation, (value-ref.)/ref. wing sweep angle pert. pressure; acceleration potential airfoil pitch angle0 airfoil pitch angle, steady state flap deflection angle wake induced velocity

i,jr

integrals, eq. (21)

, , ellipsoidal coordinates azimuthal coordinate density

chordwise coordinate pressure discontinuity at the disk

nm wake generalized forcing function normalwash on airfoil/flap coordinate in freestream direction wake skew angle

Subscripts:,i derivative wrt. coordinate id flap hinge locationh horizontalv verticaln,j polynomial number, derivative in freestream direction circulatory

Superscripts:A acceleration partV momentum flux partc cosine partitionle leading edgem, r harmonics numberte trailing edge


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Symbols:

( )* derivative wrt. nondimensional time

normalized value; for ... by ...:

length Rspeed Utime R/Ufrequency U/R

lift/unit length U2R

pressure U2R2

moment/unit l. U2R2

[ ] matrix{ } column vector

Introduction

The tiltrotor aircraft currently receives a lot ofattention as a possible solution to airport congestionproblems. Recent preliminary design trend andoptimization studies like reference [1] show that thisconfiguration has the potential of being

economically competitive with turboprop aircraft.These studies usually include only an approximaterepresentation of the effect of proprotor whirl flutter,a coupled rotor-wing instability and major designdriver for this configuration. The current state-of-the-art computer program for V/STOL aircraft,VASCOMP [2], for example, accounts for thisphenomenon by placing the first three wing naturalfrequencies at particular fractions of rotor rpm. Thisapproach does not account for any coupling betweenthese modes, rotor dynamic characteristics, or flutteralleviation by active controls. Previous research inthe latter areas, on the other hand, did not includeintegration with aircraft sizing/mission performanceanalysis and optimization ([3], [4]). A current researcheffort at Georgia Tech's School of AerospaceEngineering therefore focuses on integrating aircraftsizing, wing structural design, wing aerodynamics,rotor aeroelasticity and control system design into aMultidisciplinary Optimization (MDO) framework[5].

Two areas are particularly under investigation.The first is an improved structural model of the wingwhich accounts for dynamic tailoring through the useof composite materials, without requiring excessivecomputational effort. The Equivalent Laminated Plate

Solution ELAPS [6] proved to be sufficientlyaccurate for this task, while being computationallymore efficient than Finite-Element analyses. Thesecond focus is on inclusion of unsteady wing/flapaerodynamics. Previous studies included only a quasi-steady representation, although the natural frequencyof the flutter mode found by van Aken [7] for an XV-15-type wing-rotor configuration translate intoreduced frequencies around 0.16. A closerinvestigation of the effect of unsteady aerodynamics

on the proprotor whirl phenomenon seems thereforenecessary.

Most fixed-wing flutter analyses employ k-typeaerodynamics like the Doublet-Lattice or Vortex-Lattice Method, Kernel Function approaches, or StripTheory with Theodorsen Function correction. Since

k-type aerodynamics are formulated for simpleharmonic motion, they are not directly applicable tomodal analysis of an aerodynamically damped systemand subsequent flutter suppression controller design.In the case of the first task, iteration on the imaginarypart of the eigenvalues is required (p-k-Method). Forthe second task, the aerodynamic influencecoefficients for simple harmonic motion (purelyimaginary eigenvalues/k-type) need to be expandedinto the complex plane using Pad-Approximation [8]or Minimum-State rational function approximationtechniques. Both approaches require calculation of theunsteady aerodynamic influence coefficient matrix forseveral reduced frequencies. In summary: Utilizationof k-type aerodynamics turns conversion of theaeroelastic system to state-space form into aninconvenient process. For aeroservoelasticapplications, an unsteady aerodynamics modelformulated in state-space form is clearly preferable.

Peters/He Wake Model

Such a model was developed by Peters and He forrotary wing applications [9]. It is currently beingimplemented in the aeroelastic stability analysis ofrotary wing computer codes like CAMRAD and2GCHAS for its simplicity and accuracy. Nibbelink

and Peters [10] showed also its applicability to fixedwing aeroelasticity. For a lifting-line lift model and apressure distribution assumed constant along thechord, rectangular, large aspect ratio wing planform,and simple harmonic motion the results correlatedacceptably with Theodorsen Theory. The followingparagraphs are meant to provide an overview of thegeneral philosophy of the approach and a review ofreferences [9] to [11], rather than a detaileddescription.

For incompressible flow with smallperturbations, the continuity and the momentumequation can be written in index notation:

qi,i = 0 (1)

qi* - V qi, = - ,i (2)

where V is the nondimensional freestream velocity(divided by rotor tip speed in rotary wingapplications; for fixed wing, this term is unity), qi are

the perturbation velocity components and qi*, qi, ,


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and qi , i their derivatives with respect to

nondimensional time, along freestream direction, andalong coordinate direction, respectively. The form ofequation (1) suggests separation of the perturbation

pressure into a part resulting from acceleration A

and a part stemming from the momentum flux V,

so that

= A + V (3)

where

V,i = V qi, (4)

and

A,i = -qi* (5)

By differentiating (2) with respect to i and

applying the continuity equation (1) it can be shownthat both parts of the perturbation pressure mustsatisfy Laplace's equation and therefore resembleacceleration potentials. One solution for Laplace'sequation is known as Prandtl's acceleration potentialfunction for circular wings in the ellipsoidalcoordinates, , and

(, ,, t) = m,n

Pnm() Qn

m(i) (6)

[ Cnm(t) cos (m) + Dn

m(t) sin (m) ]

using the Legendre Polynomials of the first and

second kind, P nm and Qnm,and coefficients Cnm and

Dnm

. On the rotor disk ( = 0, = 1 - r2 ) this

function models a pressure discontinuity with apressure difference, , between upper and lower sideof the circular lifting surface*

(r,, t) = - 2 m,n

Pnm() Qn

m(i0) (7)

[ Cnm(t) cos (m) + Dn

m(t) sin (m) ]

If the perturbation velocity normal to the rotordisk, qz, is denoted by , Equations (4) and (5) can be

rewritten in the form

* in the following called 'rotor disk'; it is, in fact, a circular surface inwhich pressure discontinuities may occur. For fixed wing applications, itis a disk circumscribing and encompassing the wing, for rotory wingcases it is simply the rotor disk.

(r,,t) = 1V

V

z

0

d L[V] (8)

* = -

A

= 0C [A] (9)

L and C are linear operators on A and V.Provided that these two operators are invertible, a firstorder differential equation in can be written:

C-1

[*] + L-1

[] = A

+ V

= (10)

The inversion is possible if the induced velocityis expanded in terms of harmonics azimuthally andarbitrary functions radially, e.g.

(r,, t) = m,n

Pnm()/ (11)

[anm(t) cos (m) +bnm(t) sin (m) ]

introducing the inflow states anm and bnm

ascoefficients of the azimuthal harmonic, m, and theradial expansion function, n. The bar over theLegendre Polynomial symbolizes normalization to aunit integral over the interval = [0,1]. Substituting as in Equation (11) into the differential equation

(10), premultiplying by P nm

and cos(m) andintegrating over the rotor disk yields a set of first

order ordinary differential equations in air :

Mc

{ajr}

*+ N

c {aj

r} = 12

{nmc} (12)

and an equivalent equation for the bjr , multiplied by

sin(m) (replace superscript c by s). Since the inflowdynamics (i.e. the rotor wake, or the perturbationvelocities induced by a wing) are expressed in state-space form, it is immediately obvious that the modelis equally applicable to harmonic and non-harmonicexcitation.

The wake 'mass' matrix [M] and 'damping' matrix[N] are known functions of the wake skew angle, .[N] is in fact the inverse of another matrix originatingfrom the operator L. Wang [11] developed closed formsolutions for this inverse in edgewise flow ( = 90)for an infinite number of wake states. For a yet notfully understood reason these solutions are alsoaccurate for two special kinds of truncated systems:(a) cos-partition, maximum order of azimuthalharmonics odd; and (b) sin-partition; maximum order


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of azimuthal harmonics even [11]. Since in fixedwing aeroelastic analyses symmetric andantisymmetric modes are usually treated separately(i.e. either the cos- or the sin-partition is used), theseconditions do not create an empty set of options forthe analyst striving to reduce computational effort bybypassing a numerical inversion. The following

discussion concentrates on the cos-partition, i.e.modeling of symmetric problems.

The right hand side of equation (12) representsthe wake system excitation through an imposed

pressure discontinuity, P (here nondimensional),across the rotor disk.

nmc = 1

0

2

P(r,,t) Pnm

()/

0

1

dr cos(m) d (13)

or, expressed in cartesian coordinates (refer to Fig. 1):

nmc = 1

-S

S

P(x,y,t) Pnm

()/

xle

xte

cos(m) dx dy (14)

In the special case m = 0 (uniform inflow), theexpression is divided by 2.

Wake System Forcing Functions

Obviously, the actual form of these forcingfunctions depends on the way the pressurediscontinuity (or lift) is distributed over the disk, or

on the distribution of pressure over the liftingsurfaces. One of the key features of the Peters/Hewake model is the separation of wake dynamics andlifting surface aerodynamics, as displayed in equation(13). The inflow model is therefore concerned withthe effects of shed vorticity only. As a result, (a) only

circulatory lift enters the system through P, and (b)the effects of bound circulation are filtered outthrough choice of a chordwise vorticity distributionwhich does not induce any velocity on the airfoil(where the wake induced velocity is sought - [9], [10],[12]). Reference [9] lists a number of these candidatedistributions. One of them is the solution for the flatplate airfoil,

P(x,y,t) =L(y,t)

b(y)tan

2(15)

where L(y,t) is the nondimensional circulatory lift asa function of the spanwise coordinate, and thenondimensional chordwise coordinate, , for a wingplanform as depicted in Fig. 1 is

b(y)

x le x te

x

z

x

y

x te

x le R = 1

S

b

Fig. 1: Coordinate Systems

cos = (x - y tan ) / b(y) (16)

Another possible solution, however, has beenpreviously chosen [12]. This particular distributionwas selected here since it allows a simplificationwhen expanding the integral (14):

P(x,y,t) =L(y,t)

b(y) sin (17)

First, L(y,t) is expanded. The normalwash at alifting surface chord due to airfoil shape and motion,

denoted by (x), and the induced velocity (x) can beexpanded in a Fourier Series:

(x) = n cos (n)n = 0

; (x) = n cos (n)n = 0

(18)

The nondimensional circulatory lift can now beexpressed in the following form:


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L(y,t) = 2b(y) (0(y,t) + 12

1(y,t)

-0(y,t) - 12

1(y,t)) (19)

which allows separation into a wing forcing part(bound circulation) and a wake feedback part.Recalling the definitions for the Fourier Coefficients,

0(y,t) := 1

(x,y,t)

0

d

1(y,t) := 2

(x,y,t) cos

0

d (20)

and defining of the integrals

j,0r,c

(y) := 1

Pjr()

0

cos(r ) d

j,1r,c

(y) := 2

Pjr()

0

cos(r ) cos d (21)

the circulatory lift, equation (19) can be rewritten

L(y,t) = 2b(y) (0(y,t) + 12

1(y,t)

- j, 0r,c

(y)+j, 1r,c

(y) ajr(t)r,j

) (22)

Including the pressure distribution (17) in the

wake forcing functions (14) and transforming thechordwise coordinate form x to (using equation(16)) yields

nmc = 1

L(y,t)

-S

S

1

Pnm

()

0

cos(m) d dy (23)

Notice that the inner (chordwise) integral has

exactly the same form as j, 0r,c

(y) in equation (21), due

to the choice of (17) for the chordwise pressuredistribution. We finally obtain

nm(t) = 2 b(y) n, 0m,c

(y) 0(y,t) + 12

1(y,t) dy

-S

S

- 2 b(y) n, 0m,c

(y) j, 0r,c

(y) + j,1r,c

(y) ajr(t)r,j

dy

-S

S

(24)

The question is how the integrals (21) are to becomputed. Previous applications in rotary wingaeroelasticity reduced to high aspect ratio rotor bladeswhere a lifting line approximation was justified.Nibbelink [11] also shows for moderate aspect ratiofixed wing cases (AR = 5) acceptable performance.

The fact that, in case of a lifting line model, theintegration simplifies to evaluation of the expression

j, 0r,c

(y) :=Pj

r()

cos (r ( - /2)) (25)

while j,1r,c

(y) = 0, certainly makes this option very

attractive.

For the next level of fidelity, a large but finiteaspect ratio is assumed, and the approximation ofsmall azimuthal deviations from the wing halfchord is

made for points on the airfoil:

= ( - /2) +b(r)

rcos := ( - /2) + (r) (26)

If the chordwise integration is finally approximatedby an azimuthal one, the integrals yield BesselFunctions of the first kind as aspect ratio correctionfactors:

j, 0r,c

(y) :=Pj

r()

J0(r ) cos (r ( - /2))

j, 1r,c

(y) :=Pj

r()

J1(r ) cos (r ( - /2))

(27)

It must be noted that (21) and (25) only convergeif the sweep angle approaches zero and the aspectratio is very large; the same can be shown for thehigher order approximation (27). Fig. 2 shows the

deviation of2, 01,c

(y) as calculated with equation (27),

from the benchmark result from a numericalintegration using (21), normalized by this referencevalue (). Besides the obvious increase in deviationtowards the wing root, due to the growing error in thesmall angle assumption (26), there is also a constantoffset - for the first harmonic, m = 1, the ratio

between the results from (21) and (27) is about thecosine of the sweep angle. This appears to relate thedeviations directly to the fact that the actual chordwiseintegration was approximated by an integration alonga line perpendicular to the wing's halfchord line.Unfortunately, cos can only serve as a correctionfactor for the first harmonic; for higher m, theinfluence of the polynomial index n becomes morepronounced, and the search for a generally validcorrection factor for sweep was not successful. In con-


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y

Fig. 2: Deviation of High Aspect RatioApproximation from Numerical Integral

(aspect ratio of 6, sweep angle 20)

w

u

d

U + Vh

Vv

b

x

z

Fig. 3: Flat Plate with Trailing Edge Flap

clusion, both the lifting line and high aspect ratioapproximation are, strictly speaking, only valid forunswept wings. Since the values of the integralsgenerally decrease with increasing harmonic andpolynomial number, a correction by cos for m = 1

and restriction to < 10 appear to be reasonable.

Coupling with Structural System

In order to couple the wake system, equation(11), with a structural system, the generalized forceson structural and wake system must be formulated interms of the generalized coordinates (or states) ofwake and structure. The aerodynamic forcingfunctions on the structure in z - and x - direction, andthe pitch up moment, respectively, can for examplebe found in reference [12]. The drag expression,equation (29), has been linearized.

L(y,t) = 2b(y) ( 0(y,t)+12

1(y,t) - 0(y,t) ) (28)

+ b2(y) ( 0

*(y,t)-1

22

*(y,t) )

D(y,t) = -2b(y) 0 ( 20(y,t)+12

1(y,t) (29)

- 20(y,t) - 12

1(y,t) )

M(y,t) = b2(y) ( 0(y,t)-1

22(y,t) - 0(y,t) ) (30)

+8

b3(y) ( 1

*(y,t) - 3*(y,t) )

Note the difference in the lift (28) and the

circulatory lift L(y,t)(19) used in the wake forcingfunctions: Equation (28) includes non-circulatoryterms, which cancel part of the circulatory terms.Johnson [13] developed expressions for sums ofinduced flow Fourier coefficients for a flat wake;based on these expressions, one can show that ingeneral

1 + b (0*

- 1

22

*) = 0

2(n+1)n+1 + b (n*

- 1

2n+ 2

*) = 0; n > 0 (31)

which eliminates 1 from the lift expression,equation (28).

Finally, the normalwash Fourier Coefficients, i, are expressed in terms of the structural geometry.For the flat plate airfoil with a trailing edge flap asdepicted in Fig. 3, the normalwash, , is

d < < :

(,t) = ( 1 + Vh(t) - u*(t) ) (t) - w*

+ b cos *(t) + Vv(t)

0 < < d : (32)

(,t) = ( 1 + Vh(t) - u*(t) ) ((t) + (t)) - w*

+ b cos (*(t) + *(t)) - b cos d

*(t)+ Vv(t)

Application of the Fourier transform to (32) andlinearization yields the following coefficients:

0 = -w* - 0u* + + D0 + D0*

b *+ V v + 0Vh

1 = b * + D1 + D1*

b *

(33)

i = Di + Di*

b *; i > 1

D0 = 1

d; D0* = - 1

d cosd - sind

D1 = 2

sind; D1*

= 1

d - 12

sin(2d)

D2 = 1

sin(2d); D2*

= 1

2 sin d - 1

3sin 3d (34)

Di = 1

i-1

isin(id); i > 2

Di*

= 1

i

sin (i-1)d

i - 1-

sin (i+1)d

i + 1


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Application to a Tiltrotor Wing:Effect of Unsteady Aerodynamics on

Proprotor/Wing Eigenvalues

The wake model as described above has beenimplemented in a FORTRAN program, PWAKE, andincorporated into the Tiltrotor Integrated Design andAnalysis Tool, tridat!. The package includesfurthermore the wing structural analysis ELAPS [6]and the Proprotor Aeroelastic Stability AnalysisPASTA [13]. Fig. 4 provides in overview of the dataflow. The package uses UNIX shell scripts forprocess control and data filtering.

It is instructive to first investigate the behaviorof the uncoupled (open-loop) wake system. Table 1shows the ten eigenvalues of a system with amaximum order of radial polynomials M = 7 (refer toreference [11] regarding the harmonics-polynomial

ordering scheme). The eigenvalues are ordered byundamped natural frequency. The first column might

require some explanation: The eigenvalues, , arenormalized by the ratio of forward velocity, U, to diskradius, R. In order to obtain the frequency in Hz, thevalues in Table 1 must therefore be multiplied byU/R. This means that the eigenvalues areproportional to U; as a result, all wake eigenvaluespass through the range of structural naturalfrequencies with increasing speed. The fundamentalmode with a damped nondimensional frequency of0.42092 is the last one to leave the vicinity ofstructural modes, and therefore has the mostsignificant regarding coupling of wake and structure.

Observe that for an unswept, untapered wing thisnondimensional natural frequency, f, is directly relatedto the traditional reduced frequency, k, like

k = 2 fAR

(35)

This means that for a wing with aspect ratio AR= 2 at a reduced frequency of k = 0.49235 (equivalentto the undamped natural frequency of the first wakemode) the basic wake mode passes through a circle inthe complex plane with the radius of the oscillationfrequency. In other words, above this reducedfrequency, coupling with wake modes is unavoidable.With increasing reduced frequency, the number ofwake modes entering this circle increases, and thusmore and more wake-structures couplingopportunities occur.

It is reassuring to note from Fig. 5 that the basic(lowest frequency) wake mode in fact resembles theinduced flow field around a lifting surface in steadyflow. The higher order modes are not that easy to

elaps.in pwake.dat pasta.dat

wing.dat elaps.out

pwake.in

pwake.out

pasta.in

pasta.out

pasta_prep

pwake_prep

pasta_post

pwake_post

elaps

pwake

pasta

Fig. 4: Data Flow, coupled Proprotor/Wing Aeroelastic Analysis

Table 1: Wake Eigenvalues, M = 7

Damped Frequency *R/U Damping [% critical]

0.420920.295350.324310.702710.919610.658781.036220.533061.403692.34010

51.8782.4091.9958.0829.2174.3440.5690.6428.4463.87


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Fig. 5: Amplitude, Wake Mode f = 0.42092

Fig. 6: Amplitude, Wake Mode f = 0.53306

interpret, but it seems that the highly dampedeigenforms show a large increase of inflow amplitudein the vicinity of the wing tip, like in Fig. 6. One istempted to see here the effect of a tip vortex, and theassociated dissipation of energy; however, the wakemodel is purely inviscid and incompressible. Arigorous interpretation of this effect can not be givenat this time.

As an application study, the effect of wingunsteady aerodynamics on proprotor/wing eigenvalueshas been investigated. The key data of theconfiguration are summarized in the Appendix. Testruns have been conducted with quasi-steady

aerodynamics (in all plots the solid lines), lifting linemodel/numerical integration, and maximum order ofradial polynomials of 7 and 15. Wing mode 1 isdominated by beamwise bending, with a positivepitch contribution (forward sweep effect); mode 2 isprimarily chordwise motion, with a small negativebeamwise and negative pitch contribution (nacelle

pitch mode); mode 3 is the second chordwise, ornacelle yaw mode; and mode 4 is the secondbeamwise bending, or nacelle roll mode. Analysis ofthe mode shapes emphasizes the dominance of thenacelle inertias in the dynamic characteristics. Thewing is only slightly modified from a design thatshowed stability up to a forward flight speed of about780 ft/sec without aerodynamic damping of the wing[15]; thus, no instability was expected in theinvestigated velocity range.

Fig. 7 shows the frequencies as a function ofvelocity. Wing modes 3 and 4 are located around 23and 48 Hz, respectively, and are therefore off thescale. Wing aerodynamics do not appear to affect thesystem behavior at all, since results for both wingunsteady aerodynamics and no wing aerodynamicsdiffer only insignificantly from the quasi-steadyresults.

The effect of unsteady aerodynamics becomesmore apparent in the damping plots, Fig. 8 - 10.Note that numerical integration and lifting line resultsmatch closely for this configuration with moderateaspect ratio and small wing sweep angle. Dampinglevels of the wing modes are generally low, with amaximum value around 8% for the first bending

mode. This explains the very small effect on thefrequencies. Fig. 8 and 9 indicate that wake feedbackreduces the aerodynamic damping, particularly of themodes that are dominated by beamwise bending. Thereduction in damping is more pronounced in the caseof the higher resolution wake model (Fig. 9 and 10).Experience with the wake model has shown, that thismust not necessarily be a sign for convergence;correlation of the results obtained using PWAKE istherefore mandatory.

Fig. 8 to 10 show negligible affect of fixed wingaerodynamics on the rotor-dominated modes. Thereason for this somewhat disappointing result is seen

in the fact that the chosen velocity range ends at least100 ft/sec below the anticipated flutter speed. Seriouscoupling between wing and rotor modes - in otherwords: initiation of proprotor whirl flutter - will onlybe detectable in the vicinity of the flutter boundary.Hence, wing damping hardly affects the rotor modes.


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700600500400300

0

2

4

6

8

10

12

Velocity [ft/sec]

Frequency

[Hz]

x No Wing Aerodynamics

o M = 15, numerical

fwd. inplane

shaft torsion

1st wing chord

gimbal fwd.

gimbal bwd.

1st wing beam

bwd. inplane

coning

Fig. 7: Frequencies - No /Quasi-Steady / Unsteady Wing Aerodynamics

700600500400300

0

2

4

6

8

10

Velocity [ft/sec]

Damping

[% critical]o Lifting Line

+ Numerical

1st wing beam

shaft torsion

fwd. inplane

1st wing chord

wing 3&4

Fig. 8: Damping - Quasi-Steady and Unsteady, M = 7


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700600500400300

0

2

4

6

8

10

Velocity [ft/sec]

Damping

[% critical]

o Lifting Line

+ Numerical

1st wing beam

shaft torsion

fwd. inplane

1st wing chord

wing 3&4

Fig. 9: Damping - Quasi-Steady and Unsteady, M = 15

700600500400300

0.0

0.4

0.8

1.2

1.6

2.0

Velocity [ft/sec]

Damping

[% critical]

o Lifting Line

+ Numerical

1st wing chord

wing 3 wing 4

Fig. 10: Damping - Quasi-Steady and Unsteady, M = 15 (Zoom)


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Conclusions

The Peters/He wake model appears to performwell in modeling fixed-wing unsteady aerodynamics.The results show anticipated tendencies, however,correlation with established methodologies ismandatory. This is especially the case for low aspectratio wings, where concerns regarding convergenceremain.

For the investigated wing-rotor system, unsteadyaerodynamic effects are detectable. Yet, due to anoverall low damping level and choice of a small rangeof velocities below the anticipated flutter speed, theeffects are not significant. A closer look at the modalbehavior around the stability boundary must be taken.

Appendix:Key Data of Proprotor/Wing Model

Wing

Span 50.1 ftAspect Ratio 6Taper Ratio 1.0Sweep -5Thickness/Chord Ratio 0.23Ply Thicknesses:0, root 0.54 in0, tip (linearly tapered) 0.14 in+45 (constant) 0.036 in- 45 (constant) 0.015 inSpar Cap Areas (two spars, upper spar only):

root 0.9 in2

tip (linearly tapered) 0.57 in2

Rotor/Nacelle

Nacelle Weight 3720 lbRotor Radius 18.4 ftNumber of Blades 4Rotor rpm 300Natural FrequenciesGimbal Tilt 1.02/rev

Blade, 1st inplane (uncoupled) 1.6/rev

Blade, 1st out of plane (uncoupled) 1.3/rev

Acknowledgments

Implementation of the unsteady aerodynamicsprogram PWAKE and development of the TiltrotorIntegrated Design and Analysis Tool, tridat!, wasfunded by the Sikorsky Aircraft Division of theUnited Technologies Corporation.

References

[1] Schleicher, D.R., Phillips, J.D., and Carbajal,K.B., "Design Opimization of High-SpeedProprotor Aircraft," NASA-TM-103988, April1993

[2] Schoen, A.H., Rosenstein, H., Stanzione, K.,and Wisniewski, J.S., "User's Manual forVASCOMP II, The V/STOL Aircraft Sizingand Performance Computer Program," BoeingVertol Company Report D8-0375, Vol. VI,Third Revision, May 1980

[3] Frick, J., and Johnson, W., "Optimal ControlTheory Investigation of Proprotor/WingResponse to Vertical Gust," NASA-TM-X-62384, September 1974

[4] Nixon, M.W., "Parametric Studies for TiltrotorAeroelastic Stability in High-Speed Flight,"33rd AIAA/ASME/ASCE/AHS Structures,Structural Dynamics, and Materials Conference,Dallas, Texas, April 13-15, 1992

[5] Stettner, M., and Schrage, D.P., "An Approachto Tiltrotor Wing Aeroservoelastic Optimizationthrough Increased Productivity," 4th AIAA/U SA F/ N A SA / O A I Sympos i um onMultidisciplinary Analysis and Optimization,Cleveland, Ohio, September 21-23, 1992

[6] Giles, G.L., "Equivalent Plate Analysis of

Airdraft Wing Box Structures with GeneralPlanform Geometry," Journal of Aircraft, Vol.23, No. 11, November 1986, p. 859-864

[7] van Aken, J.M., "Alleviation of Whirl-Flutteron Tiltrotor Aircraft using Active Controls,"47th Annual Forum of the American HelicopterSociety, Phoenix, Arizona, May 6-8, 1991

[8] Liebst, B.S., Garrard, W.L., and Farm, J.A.,"Design of a Multivariable Flutter Suppression

/ Gust Laod Alleviation System," Journal ofGuidance, Vol. 11, No. 3, May-June 1988, p.220-229

[9] Peters, D.A., Boyd, D.D., and He, C.J.,"Finite-State Induced-Flow Model for Rotors inHover and in Forward Flight," Journal of theAmerican Helicopter Society, Vol. 34, No. 4,October 1989


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[10] Nibbelink, B.D., and Peters, D.A., "FlutterCalculations for Fixed and Rotating Wings withState-Space Inflow Dynamics," 34th AIAA /ASME / ASCE / AHS / ASC Structures,Structural Dynamics, and Materials Conference,La Jolla, California, April 19-22, 1992

[11] Wang, Y.-R., "The Effect of Wake Dynamicson Rotor Eigenvalues in Forward Flight,"Ph.D. Thesis, Georgia Institute of Technology,May 1992

[12] Peters, D.A., and Su, A., "An IntegratedAirloads-Inflow Model for Use in RotorAeroelasticity and Control Analysis," 47thAnnual Forum of the American HelicopterSociety, Phoenix, Arizona, May 6-8, 1991

[13] Johnson, W. Hel icopter Theory, Princeton

University Press, Princeton, New Jersey, 1980,p. 477 - 478

[14] Kvaternik, R.G., "Studies in Tiltrotor VTOLAircraft Aeroelasticity," PhD Dissertation,Department of Solid Mechanics, Case WesternReserve, June 1973

[15] Friehmelt, H., "Design of a Tiltrotor Wing for aEUROFAR-type Configuration," DiplomaThesis, Technische Universitt Braunschweig/Georgia Institute of Technology, March 1993

Documents

Statespace Models of Tilt Rotor