NASA Technical Memorandum 78459

N78-18043(NASA-TM-78459) AEROMECHANICAL STABILITY OF HELICOPTERS WITH A BEARINGLESS MAIN ROTOR. PART 1: EQUATIONS OF MOTION (NASA) 102 p HC A06/MF A01 CSCL 01C Unclas

GS/05 05988

Aeromechanical Stability of Helicopters with a Bearingless Main Rotor -

Part I: Equations of Motion Dewey H. Hodges

February 1978

NAA

https://ntrs.nasa.gov/search.jsp?R=19780010100 2019-04-13T21:54:30+00:00Z

TABLE OF CONTENTS

Page

SYMBOLS. . . . . . . ...

SUMMARY . 1

1. INTRODUCTION . . .i..........

- 2. PHYSICAL MODEL 3

3. DESCRIPTION OF THE DERIVATION . . . ........ 4

4. COORDINATE SYSTEMS ..... 6

5. DERIVATION OF GENERALIZED ACTIVE FORCES. ...... ... 105.1 Aerodynamics .............. 105.2 Gravity . . . .............. 145.3 Body Springs .............. 165.4 Flexbeam Structure .... ..... .... 175.5 Structural Damping ............ 18

6. DERIVATION OF GENERALIZED INERTIA FORCES ....... 196.1 Motion of the kth Blade . .......... 196.2 Motion of the Body .... ..... .... 20

7. FORMATION OF THE EQUATIONS OF MOTION . . ... 217.1 Equilibrium Solution ........... 237.2 Flexbeam Stiffness Coefficients.. ...... 287.3 Transformation to Fixed System Coordinates 287.4 Linearized Perturbation Equations of Motion 30

APPENDIX A: FLEXIBLE TORQUE TUBE......... 67

APPENDIX B: CANTILEVERED PITCH ARM WITH FLEXIBLE PITCH LINK 68

APPENDIX C: SNUBBER AND PITCH ARM WITH FLEXIBLE PITCH LINK ANDSNUBBER CONNECTION ........... 74

APPENDIX D: STRUCTURAL DAMPING COEFFICIENTS ....... 76

REFERENCES . .................. 78

FIGURES.................. . 80

i3i

SYMBOLS

A

a

aB *

ak '

B

Elf dimensionless flexbeam f-lapping stiffness, -

IP02t

rotor blade airfoil lift curve slope

acceleration of the body mass center, m/sec2

acceleration of the kth blade mass center, m/sec2

EIc dimensionless flexbeam chordwise stiffness, io2£

[B] transformation matrix relating the blade-fixed axis system Nxk, Nyk, Nzk to the flexteam tip axis system nxk, nyk , nzk

b

C

[CA]

number of blades ( 3)

dimensionless flexbeam torsion stiffness, GJ

IQ 02£,

damping matrix contribution from aerodynamics

[CD] damping matrix contribution from structural damping

c rotor blade airfoil chord length, m

C L

cl­ steady component of clk, m

c k components of the direction, i =

kth rotor blade mass center in the 1,2,3, from point J, equation (16),

n.k

m

ci steady component of cik made dimensionless by L

cX damping coefficient for uncoupled X motion, N-sec/m

CYdamping coefficient for uncoupled Y motion, N-sec/m

c damping coefficient for uncoupled

N-m-sec

C motion (blade lead-lag),

CxXdamping coefficient for uncoupled 'x motion, N-m-sec

C'y damping coefficient for uncoupled Dy motion, N-m-sec

c( ) cos( )

Cd0 rotor blade airfoil profile drag coefficient

0v V

ppiYEDING PAGE BLANK NOT FIUASI

EAQ-- ; also, rotor bladeD dimensionless flexbeam axial stiffness,

drag per unit length, N/m IQ02

d cd a

rotor blade mass center in the nikdik components of the kth

direction, i = 1,2,3, from point 0, equation (58), m

EA flexbeam axial stiffness, N

EI e flexbeam chordwise bending stiffness, N-m2

flexbeam flap bending stiffness, N-m2

Elf

e hub radius, m

eek

el,e2,e, space-fixed axis system, figure 1

el,e2,e unit vectors parallel to el, e2, e33

[F] transformation matrix relating the flexbeam axis systemnk,n2k,n3k to the rotating coordinate system N1k, N2k, N3k

Fr generalized active force for the rth degree of freedom, r = Uk' Vk' Wk' k' 8k' 0k' X, Y,x' 'yY,

Fr* generalized inertia force for the rth degree of freedom,r = uk' Vk' Wk' k' 8k' 0k' X, Y, ,x y

function to be minimized to produce rotor equilibrium solution

FB force acting on the body, N

FB* inertia force acting on the body, N

Fk force acting on the kth blade, N

Fk* inertia force acting on the kth blade, NI e02

FR force at the flexbeam root, made dimensionless by 0in 02

Ft force at the flexbeam tip, made dimensionless by £

FuFvFw components of external force in the direction of u,v,w,respectively, acting on the flexbeam tip

vI

I

F4,FsF

[G]

GJ

[GI]

[Gk]

g

go

go

[H]

[Hk

h

Ii1I21I3

,12T

Ix'Iy

Ix'IY

J 3 3 ,j 1 3 ,j 2 3

[K]

[KA]

components of external moments in the direction of C,8,Orotations, respectively, acting on the flexbeam tip

[B] [T]

flexbeam torsional stiffness, N-m2

gyroscopic matrix contribution from inertia forces

[B] [TkI

rotor thrust per unit mass of total aircraft, m/sec2

g

g02L

acceleration of gravity, m/sec2

Q02L

[G] [F]

[Gk] [F]

height of rotor-hub center above aircraft reference center, m

hL

two-dimensional airfoil section plunge velocity, m/sec

rotor blade flapping inertia for point J; I = 12 + mL2xb2

rotor blade mass moments of inertia for the blade mass center for axes Nxk, Nyk, Nzk, respectively

I ,x = 1, 2, 3

body mass moment of inertia for the body mass center for axes NA and NB respectively, kg-m 2

Ix Iy

I

geometric variables, equation (66)

flexbeam stiffness matrix

stiffness matrix contribution from aerodynamics

Vi3

[KI stiffness matrix contribution from structural damping

[Kg] stiffness matrix contribution from gravity

[KI] stiffness matrix contribution from inertia

[KS] stiffness matrix contribution from structural loads

KxI Kz landing gear stiffness for each of four landing gear springs in the NA, NB, and NC directions, respectively, N/m

KxL 2 KYL2 KzL 2

xi0 2 1iQ02

L blade length, m

LC two-dimensional airfoil circulatory lift per unit length, N/m

LNC two-dimensional airfoil noncirculatory lift per unit length,N/m

£ flexbeam length, m

L

Y£x, y longitudinal and lateral distances, respectively, from front to rear and from left to right landing gear, figure 7, m

vertical distance of aircraft reference center above the land­ing gear, figure 7, m

Tx'Y'£Z xx LT'

PyTL

yzTL

M mass of aircraft; M = bm + mf, ,kg; also two-dimensional airfoil pitching moment per unit length, N

ML2

I

f[A] mass matrix contribution from aerodynamics

MB moments exerted on the body, N-m

MB* inertia moments exerted on the body, N-m

[MI] mass matrix contribution from inertia

Mk moments exerted on the kth blade, N-m

viLiL

Mk* inertia moments exerted on the kth blade, N-m

Mk aerodynamic pitching moment per unit length on kth blade, N

k1k rLfaMk d., N-m

MR moment at the flexbeam root, made dimensionless by IQ0 2

Mt moment at the flexbeam tip, made dimensionless by 10 2

m mass of one rotor blade, kg

tmL2

I

mf mass of the fuselage, kg

mfL2

mf I

NA,NBNc body-fixed coordinate system, figure 1

NANBN C unit vectors parallel to NA' NB, NC

N k,N2k,N3k rotating coordinate system for kth blade, figure 8

Nik,N 2k,N3k unit vectors parallel to NIkN2k,N3k

NxkNykNzk coordinate system fixed in the kth blade, figure 8

NxkNkNzk unit vectors parallel to NxkNYkNzk

nk ,nk ,n3k coordinate system fixed in the kth flexbeam root, figure 8

k'n2k,n3k unit vectors parallel to nlk,n2k, k

n k,n k,nzk coordinate system fixed in the kth flexbeam tip, figure 8

n.k,nyknzk unit vectors parallel to nxk nyknzk

qr dummy symbol used to refer to the blade degrees of freedom

R rotor radius; R = e + k + L, m

r position vector of point along flexbeam, made dimensionless

by P,

rt position vector of flexbeam tip, made dimensionless by k

S component of aerodynamic force per unit length along the chord­line in a two-dimensional airfoil section, N/m

ix

Sk component of aerodynamic force per unit length along the chord-lle(Nyk al)o h

line (kaxis) of the rotor blade, N/m

Sk rLSk dx, NSf J0

Smk JXSk dx, N-m

s arc length of deformed flexbeam elastic axis made dimensionless by £

S() sin( )

T component of aerodynamic force per unit length perpendicular tothe chordline in a two-dimensional airfoil section N/m; alsokinetic energy, kg-m2/sec2 , also tension in flexbeam, N

9" rotor thrust, N

k I[T] steady part of [T

Tk component of aerodynamic force per unit length perpendicular tothe chordline (along Nzk axis) of the rotor blade, N/m

[Tk ] transformation matrix relating the flexbeam tip axis system

nxk,nyknzk to the flexbeam root axis system nk,n2 ,n3k

Tfk LTk dx, N

TmkoxTk dx, N-m

U total velocity of two-dimensional airfoil section, m/sec

U1k components of rotor blade velocity with respect to air in

n kaxis system, m/sec

UPk component of rotor blade velocity with respect to air in Nzk

direction, m/sec

UTk component of rotor blade velocity with respect to air in Nyk direction, m/sec

u steady component of uk, m

S+ ­

component of flexbeam axial deflection due to bending alone,u5 made dimensionless by 'Z

x

I3PAGFORIGNA OF POOR QUAIX"

b

ic bE

k=1

U'k Cos 'k' m

uk axial deflection of the kth flexbeam tip, m

"k 1 +-lT + Fl

Uk unsteady component of Uk, m

us component of flexbeam axial deflection due to stretching, made dimensionless by k

b

us

k=1

U k siln *k' m

ut . assumed value for u dimensionless by

in equilibrium deflection scheme, made P,

UlU2,m3 coordinate system along the deformed flexbeam principal axes

V

V1k

VP

free-stream velocity of two-dimensional airfoil section, m/sec

rotor blade mass center velocity components in n1 k axis

system, equation 16, m/sec

velocity of any point P

v steady component of vk, m

~X +F21£

b

c E k cos ik, m

k=1

vi induced inflow velocity, m/sec

vk chordwise deflection of kth flexbeam tip, m

Vk vk-+ eiF21

vk unsteady component of vk, m

xl

s ~2

b

Ervb.

k=1

Vk sin m

Vt assumed value for v

dimensionless by

w steady component of

w+ eF31

c b

Wk cosk'

k=1

in equilibrium deflection scheme, madek

wk, m

wk

wk

flapwise deflection of

wk -T + "F31

kth flexbeam tip, m

Wk unsteady component of wk, m

b

Ws b , Wk sin Ik,

k=1

wt assumed value for w dimensionless by

in equilibrium deflection-scheme, made P,

X time integral of body velocity component in NA direction, m

x distance along N.k axis from point J, m

xa chordwise distance from Nxk axis to aerodynamic center, made dimensionless by c, positive when aerodynamic center is ahead of Nxk axis

xb axial distance from point J to blade mass center along Nxk axis, made dimensionless by L

xc chordwise distance from Nxk axis to blade mass center made dijensionless by c, positive when mass center is ahead of NxV axis

Y time integral of body velocity component in NB direction, m

z vertical distance from aircraft reference center to body mass center, positive when body mass center is below reference center, m

X11

-

Bsteady

ab

Oc

f

Bk

k

Ss

Ot

Y

epitch

b

zc

f

k

z

two-dimensional airfoil angle of attack, rad

component of k, rad

built-in coning angle of the blade with respect to the flex­beam, positive tip up, rad

b

bE 0 k cos 'k' rad

k=1

built-in coning angle of the flexbeam with respect to the hub,positive tip up, rad

elastic flap rotation of the kth flexbeam tip positive tipup, rad

unsteady component of 13k' rad

b

bE k s 'k' rad

k= 1

assumed value for B in equilibrium deflection scheme, rad

peeL4dimensionless airload parameter, I

angle of two-dimensional airfoil section, rad; also,perturbation of flexbeam tip loads; also, flexbeam middlesurface strain, equation (73)

steady component of k' rad

built-in sweep angle of the blade with respect to the flexbeam,

positive tip leading, rad

b ORIGINAL PAGES

b E Zk cos Pk' rad OF POOR QUAL]WII

k=1

built-in sweep angle of the flexbeam with respect to the hub,positive tip leading, rad

elastic lead angle of the kth flexbeam tip, positive tip

leading, rad

X111

k unsteady component of Ck' rad

b

zs b

k=1

k sin k' rad

Ct k

Til

assumed value for in equilibrium deflection scheme, rad

geometric parameter, equation (18)

TI

Xn,'IxN'P y

X

isolated blade lead-lag structural damping ratio

fuselage damping ratio for uncoupled motion in the

and $y direction, respectively

X, Y, Ox,

0b

steady component of 8k' rad built-in pitch angle of the blade with respect to the flexbeam,

positive leading edge up, rad

b

Oe _ E

k=1

ek cos k' tad

Of built-in pitch angle of the flexbeam with respect to the hub, positive leading edge up, rad

0k elastic twist of the up, rad

kth flexbeam tip, positive leading edge

0k unsteady component of 6k' rad

6s b

2 E isi k' rad

k=1

at assumed value for 0 in equilibrium deflection scheme, rad

e3/4 pitch angle at the blade 3/4-radius, rad

Kcurvature in the local flexbeam flap direction, made dimension­

less by 1

curvature in the local flexbeam chordwise direction, made 1dimensionless by ~

p air density, kg/m 3

xiv

a rotor solidity --

T cirvature in the local flexbeam torsional direction, made

dimensionless by

x time integral of angular velocity component in rad

NA direction,

4y time integral of angular velocity component in rad

NB direction,

inflow angle at the blade 3/4-radius from momentum theory, equation (22), rad

lPk azimuth angle of kth blade

Q rotor angular velocity, rad/sec

Q0 nominal rotor angular velocity, rad/sec

0o

MB

wk

wIk

angular velocity of the body, rad/sec

angular velocity of the kth blade, rad/sec

components of angular velocity in NxkNykNzk

equation (18), rad/sec

axis system,

xv

AEROMECHANICAL STABILITY OF HELICOPTERS WITH A BEARINGLESS

MAIN ROTOR - PART I. EQUATIONS OF MOTION

Dewey H. Hodges*

Ames Research Centerand

Aeromechanics LaboratoryU.S. Army Aviation R&D Command

SUMMARY

Equations of motion for a coupled rotor-body system are derived for thepurpose of studying air and ground resonance characteristics of helicoptersthat have bearingless main rotors. For the fuselage, only four rigid bodydegrees of freedom are considered; longitudinal and lateral translations,pitch, and roll. The rotor is assumed to consist of three or more rigidblades. Each blade is joined to the hub by means of a flexible beam segment(flexbeam or strap). Pitch change is accomplished by twisting the flexbeamwith the pitch-control system, the characteristics of which are variable.Thus, the analysis is capable of implicitly treating aeroelastmc couplingsgenerated by the flexbeam elastic deflections, the pitch-control system, andthe angular offsets of the blade and flexbeam. The linearized equations arewritten in the nonrotating system retaining only the cyclic rotor modes; thusthey comprise a system of homogeneous ordinary differential equations withconstant coefficients. All contributions to the linearized perturbation equa­tions from inertia, gravity, quasi-steady aerodynamics, and the flexbeamequilibrium deflections are retained exactly. Part II describes a computerprogram based on these equations of motion.

1. INTRODUCTION

The general problem of helicopter aeroelastmc stability involves couplingbetween the motion of the individual blades through control system dynamicsand the rotor wake, as well as coupling between the rotor and fuselage of the helicopter The complexity of the problem poses a challenge to the analyst,both in developing an analytical model and in understanding its physicalbehavior. An important part of analyzing the general rotor-body dynamic sys­tem involves the study of the dynamic behavior of rigid-body fuselage motionscoupled with the rotor motion. The well-known phenomena of air and groundresonance occur in this way. The low-frequency rotor modes interact with the

*Research Scientist, Army Aeronautical Research Group, Aeromechanics

Laboratory.

rigid-body fuselage motions to produce instabilities both in the air and on

the ground (refs. 1-4).

Helicopter rotors without hinges to allow rigid-body flap and lead-lag

motions of the blades are commonly called hingeless rotors. Hingeless rotors

have distinct advantages over the more common hinged (articul-ted) rotors;

these include fewer moving parts, light weight, and more control--power. In

recent years there have been efforts to develop hingeless rotors without pitch

bearings to further reduce mechanical complexity and weight. These rotors,called bearingless rotors, rely on a torsionally soft portion of the blade

called the flexbeam (or strap) that is twisted by the pitch-control system to

provide changes in pitch. There have been several totally different designs

proposed within the helicopter industry (refs. 5-9). Of these designs, one

has a snubber (ref. 6); some have cantilevered pitch-arms without snubbers

(refs. 5, 7, 8); and one has a torsionally stiff torque tube that is very

flexible in bending and acts much like a speedometer cable (ref. 9). Both

hingeless and bearingless rotors have had some aeroelastic stability problems.

In fact, most production hingeless rotors still must rely on an auxiliary

lead-lag damper to suppress ground and air resonance instabilities One

reason for this is believed to be the lack of suitable analytical capability.

The analytical treatment of hingeless rotor air and ground resonance has,

for the most part, been limited to an equivalent-hinge, spring-restrainedrigid blade model for the rotor blades (refs. 1-4). The need arises when

applying these analytical models to somehow arrive at the proper orientationsfor the spring-restrained hinges so as to give the proper values for the aero­

elastic couplings that arise due to blade elastic deflections, angular offsetssuch as precone or sweep, and the pitch-control geometry. Even with this

apparent drawback, these relatively simple analytical models have been success­ful, to some degree, in assessing the air and ground resonance characteristics

of some configurations. Bearingless rotors, however, have coupling parametersthat change significantly as a function of the operating condition. Forexample, when the rotor is at high thrust the flexbeam may be highly twisted;whereas, at low thrust it may be untwisted. An alternative to a rigid-bladeformulation is that of an elastic blade. Again, there is the need to do aseparate analysis, here to obtain the free vibration modeshapes. This can be

time consuming for a bearingless rotor because the coupled modeshapes willchange as a function of operating condition. Also, these analyses are muchmore complicated than the rigid-blade approximations Bielawa has developed

such an analysis for the bearingless rotor, but only for the hub-fixed case(ref. 10). Johnson has developed a rotorcraft aeroelastic stability program(ref. 11) that possibly could be adapted to treat bearingless rotors; however,this modification has not been made to date An analysis that possesses thesimplicity of the rigid blade model but integrates the treatment of aeroelasticcouplings would appear to be very useful. Such an analysis has been developedand is the subject of this report.

In this report, equations of motion are derived that are suitable for usein studying both air and ground resonance as well as the hub-fixed dynamicbehavior of helicopters with a bearingless main rotor. The blade is modeledas rigid but connected to the hub with a structurally flexible appendage to

simulate the flexbeam portion of a bearingless rotor blade. The analysis is

2

ORIGINAL PAGE IS OF POOR QUALITY

restricted to rigid-body fuselage motions and all forces that would act alongthe flexbeam are neglected. Only rotor cyclic modes and body pitch, roll, andhorizontal translations are considered. Each blade has six degrees of freedomin the rotating reference frame. This means a total of sixteen degrees offreedom in the analytical model. The analysis is tailored to treat the con­figurations mentioned in references 5-9 in an approximate way

In section 2, the physical model is described. In the text, the config­uration described in reference 9 is analyzed; modifications necessary to treatother configurations are given in the appendmxes. In section 3, Lagrange'sform of D'Alembert's principle is introduced as a means of deriving the equa­tions of motion. The coordinate systems used in the derivation are defined insection 4 and expressions for certain kinematical quantities are derived. Thegeneralized active and inertia forces are derived in sections 5 and 6. Theequations of motion are formed, linearized about equilibrium, and then writtenin the fixed system in section 7. Particular attention is given to the flex­beam structural representation in this analysis. In the appendixes are foundthe details of including blade and body structural damping and the modifica­tions to the analysis that are required to treat the different pitch-controlgeometries.

An independent derivation of the equations of motion was performed byDr. Donald L. Kunz of the Aeromechanics Laboratory in order to check the pres­ent derivation. His effort is gratefully acknowledged.

2. PHYSICAL MODEL

In this section the physical model used to represent the helicopter isdescribed. Only those elements believed necessary to model air and groundresonance phenomena are retained. The aircraft dynamical system is composedof two parts: the fuselage and the rotor. The fuselage is assumed to be arigid body. When in contact with the ground, the fuselage is suspended by aspring system to simulate the elastic restraint to fuselage motion in anactual helicopter imposed by the landing gear system. When the aircraft isairborne in hovering flight, it is unrestrained elastically. The rotor con­sists of three or more rigid blades attached to the hub by means of slenderelastic beam segments. Both the fuselage and rotor are described in moredetail below.

A schematic of the fuselage/hub is shown in figure 1. The hub, mast, andlanding gear are all included in the mass and inertias of the fuselage. Thetotal fuselage mass is mf and the moments of inertia for the mass center areIx and Iy, respectively, for the X and Y directions. The aircraft referencecenter, shown in figure 1, is a distance z above the body mass center and adistance h below the hub center. In the study of air resonance (in hover)and ground resonance, vertical translation and yaw rotation of the bodyuncouple from the other body motions and are, thus, not significant. Theother four body degrees of freedom X, Y, 4 are included.ix, and (Dy Thesequantities are time integrals of velocity and angular velocity components inthe body fixed axis system, described in more detail in section 4. The

3

landing gear provides stiffness restraining the body motion depending on thelanding gear geometry shown in figure 2.

The rotor blades are attached to -the hub with flexible beam segmentscalled flexbeams and rotate at constant angular velocity Q. A schematic ofone rotor blade is shown in figure 3; here, the details of the pitch controlsystem are not shown

The blade pitch angle may be changed by twisting the flexbeam with any ofthree types of pitch control mechanisms or by using built-in pitch angles efand 8b. In this report four cases are considered. Case I is the simplestcase, with no pitch control system at all. This configuration corresponds toan experimental rotor set up to operate only at discrete pitch angles or to anoperational rotor with a disabled control system The equations of motion aredeveloped for this case throughout the text. The modifications for Cases IIto IV are relatively simple and are given in appendixes A-C, respectively.Case II, shown in figure 4, has only a flexible cable-like torque tube tor­sionally stiff enough to twist the flexbeam. The cable is assumed to beflexible enough in bending to put only a pure twisting moment on the tip ofthe flexbeam. This configuration corresponds to that of reference 9 and also,approximately, to the pinned-pinned torque tube of reference 5. Case IIIshown in figure 5 corresponds to configurations with a pitch link and canti­levered pitch arm described in references 6, 8, and 9 Pitch change isaccompanied by bending deflections and vice versa, giving sometimes largeaeroelastic couplings Case IV, shown in figure 6, is identical to Case IIIexcept that a snubber, intended to reduce aeroelastic couplings, is addedThe snubber is modeled as an additional nonfunctioning pitch-link/pitch-armassembly. The pitch link is connected to the rotating swashplate and thesnubber link to a point stationary in the rotating reference frame. All thepitch mechanism and snubber mechanism flexibility should be lumped into thepitch link and snubber link, respectively. The mass and inertia of thesecomponents should be included with the blade. Parameters for each configura­tion are discussed in the appendixes.

3. DESCRIPTION OF THE DERIVATION

The equations of motion are derived through an application of Lagrange'sform of D'Alembert's principle (ref. 12). This principle reduces to Lagrange'sequations when all the degrees of freedom are generalized coordinates. Thisis the case only for the blade degrees of freedom in this analysis. The bodydegrees of freedom are based on components of velocity and angular velocity inthe body fixed axis system (fig. 1) and thus are not generalized coordinates.They are "quasi-coordinates" and require special considerations, The analystmay choose to use the special form of Lagrange's equations for quasi­coordinates (ref 13), Lagrange's form of D'Alembert's principle (ref. 12) orNewton's laws.

Kane (ref. 12) has summarized the laws of motion in the form of a singleequation

4

Fr + Fr* = 0 r = l, 2, ... , n (1)

where Fr are the generalized forces associated with gravity, springs, con­

tact forces, and aerodynamics. The Fr* are the generalized inertia forces

for the n degrees of freedom. The generalized forces Fr are defined for

the above physical model as

b b

F = + b D-- + + b • 6 r = , 2 ... , nF B BrB S'q Yk ' 5- + MB Mk *w&k=i k=1 (2)

The vectors FB and MB are the forces and moments acting on the body at a

certain point P. The vectors Fk and 1k are the forces and moments actingVP on the kth blade at a certain point Qk" The velocities and VQk are

defined at the pointk P and Qk in an inertial reference frame. The angular

velocities nB and w are written for the body and for the kth blade,

respectively, also with respect to an inertial reference frame. The degrees

of freedom qr are Uk, vk, Wk, 4k'k, k for k = 1, 2, . ., b, and X, Y, 4x' 0y"

Similarly the generalized inertia force Fr* is defined as

B*I * . B + (k +Fr* = FB* - r -+ MB • - (3)

B STq_ k B aj E k 3 Sk=r k=1 r

where the inertia forces are

FB* = -mfaB

-mak*Fk* =

* k * The accelerations are aB of the body mass center and a of the kth

blade mass center. The inertia moments are given according to Euler's dynami­

cal equations, written in detail below. The velocities VE and V are for

the body and blade mass centers, respectively.

In section 5, the generalized active forces due to aerodynamics, gravity,

body springs, the flexbeam structure, and structural damping are derived. In

section 6, the generalized inertia forces are derived. Before proceeding with

the actual derivation, however, it is necessary to describe the coordinate

systems used in the report and to develop expressmons for certain velocities

and angular velocities needed in the derivation. This is done in section 4

immediately following.

5 ORIGINAL PAGE IS

OF POOR QUALIT

4. COORDINATE SYSTEMS

As an inertial reference frame, the space-fh3xed axes e,, e2 , and e.shown in figure 1 are chosen, e3 being positive down. The velocity of theaircraft reference center is defined to be

VC = MA + NB(5)

where the unit vectors NA and NB are parallel to the body-fixed axes NAand NB of the NA, NB, NC axis system also shown in figure 1. The angularvelocity of the body is defined to be

0B = xNA + ;yB (6)

Note that X, Y, x, and ')y are not geometric distances and angles, they are,instead, quasi-coordinates (ref. 13). In this report, the quantities X, Y,4)x' and y are assumed to be infinitesimal Thus,

Nf E = K e 2 (7)

C y - (xI -I e 3l

where e,, e2, and e, are unit vectors parallel to el, e2, and e3,respectively.

The rotating axes N1k, N2k, and N 3k associated with the kth rotorblade are also shown in figure 1. The angle 'k = Qt + 2w(k - 1)/b. Theassociated unit vectors are related by the following transformation

{E5}[k: S (8)kstNlikc The kth rotor blade is attached to the hub (at hub radius = e) by means

of an elastic beam segment. This flexbeam is built in at point 0 (canti­levered) along the n1k, n2k, n3k axes, shown in figure 7. The associatedunit vectors are related by the following transformation constructed from asequence of angular rotations Cf, of, and Of

U1 k N(

=~ SofS f-c f c CC ~s Of cOsfI M = [F] 12k

L-ofC fs SC +sof f Sof f- s fcof cofcof N3k3 k 0 j I j{I5} [ : f + f : C f

(9)

The flexbeam of the kth rotor blade is capable of all six beam deforma­

tions at the tip. The vector of translations is 6 = uknlk + vkn2k + Wk 3k

There are also three angular rotations, Ck, Sk, and 0 so that the flexbeamk , ktip is along the nxk nvk, nz axes shown in figure 7. The associated unit

vectors are related by the following transformation constructed from asequence of angular rotations kl Bk, Ok:

(nxklC cCcC ckst s~ ik Xi

~nyk~ = osk - Ck k cekc~k - s~ks0ksok k Skek]{2} - [Tk{n2k1

Izk k + 0k k Okck - skskcek ckc8 k1U 3k

(10)

, NzkThe kth blade is built in at point J along the axes Nxk, NYk

shown in figure 7. The associated unit vectors are related by the followingtransformation constructed from a sequence of angular rotations b, 8b'and Ob

NXk C bC~b c~sSb S b luxyk [B yk

= -SebS~bc~b-bCb e beCc b Scb Sob Seb C b Se8 BI

zU _:cobsobscb +sebs b -sobb - s b s b e b C bleub Uzi

(11)

The Nxk, NYk, Nzk axes are the blade-fixed axes. Note that all the blades,flexbeams, and angular offsets are identical to each other. This is reflectedin the absence of the subscript (or superscript) k in the [F] and [B]matrices. In the undeformed state [Tk] = [I], the identity matrix. Thefollowing additional matrices are needed in the derivation below

[Gk] = [B][Tk]

[Hk] = [Gk][F]

[T] = [Tk]I k= ,Sk=,Sk= for k=l,2,...,b (12)

[G] = [B][T]

[H] = [G][F]

Extensive use will be made of these matrices in defining velocities and angu­lar velocities needed in the derivation.

The velocities of several points in the system and the angular velocitiesof the body and kth blade are now developed. The velocity of the aircraftreference center is, by definition

7

V' - kA + * (5)

The angular velocities of the body and kth blade are, respectively,

WD = iN'+ ;Y(6)

6W.= ±"A+IN + W3+ e3 k - Bl(cCl 2 k - S I1 + k (13)

The sequence of blade angular rotations, k, Bk, and 6 k, is shown in figure 8 from which k s determined by inspection. Thus, the velocity of the body center of mass is

9 C + z!y)NA + C .)-N (14)

The position vector of the kth blade center of mass from the aircraft refer­ence center is

r k l hN 3 + ezIlk + (k + Uk)cl k + VI12 k + wkn3 k + LXbNx k + CXcI(y k (15)

" The velocity of k may be established by well-known laws of kinematics

3 Vk* vxu k (16)

1=1i

where

V3k= kv h ± (Wk + c3k)k 2h - (Vk + c2k)kwjh + k + aek

V2k k v2 h + ( + clk" k h - (wk + ck)klh + i k + C2kk

V~k= kv h + (vk + c2k)kwlh - (uk + clk)kw2h + wkt + 3k

kv I -[(k - hy)c ok- (C + h$,)s~ klFis - [(i Ey)s(k i­

+ Ct -hx)ckFiZ i = 1,2,3 (17)

k~ih ---($Xc* - ISk)Fi1 + (4'XS~k + lr*)212 + a x- 1,2,3

= + GF i = 1,2,3 (D

2 c 1 * 1=1,3 alk

k 6clk = cikc x k+ - k k+ DO =1,2,3k 8k

k k k

ORIGINAL PAGE IS OF POOR QUAUIJM

It is necessary to express the angular velocity of the kth blade in theNxk , Nyk , Nzk axis system

Wk= WhkNxk + m 2 kNyk + w3 kNzk (18)

where

Nt-k = Gk k h + k k3 C=kGi3I

J=1 ij ja3 ki +eki(19)

kk ki k = kB12 - SO B13 = CG - SkC 6

2T1 kk k'i k'1 ) Also, the velocity of the flexbeam tip J is needed. This is available

by inspection from equations (17) with cik = 0i

VJ =1 JkvIk (20)

where

kvIJ = kvIh + + wkk2h - vkkw3h

+kv2J = kv2h + k kk3h - wkklh (21)

kV3J = kV 3h + + kk'hkk - ukk2h

Finally, in developing the aerodynamics, the velocities of all pointsalong the blade NXk axis are needed. The blade is moving in space and, inaddition, there is an induced flow field assumed to remain along the e3axis. The magnitude of the induced inflow velocity is taken from momentumtheory (as in ref. 14); for simplicity, it is assumed to vary linearly withthe radius.

vi = Q(e + £ + u + x)4 0 < x < L (22)

where

uk = u + uk(t)

1113141)ir, [l= 6 + - ] sgn(03 /4) (23)

03/4 = H23

9

When the induced inflow is superposed on the blade velocity the result is

3

V k UIk k (24) 1 = ~1

where

U1k I h Uk XGil + (w k +XGl3) W2 h - (£Vk +XGl) kW3h= •Uhk++x k k k -k

= kkkh kkh2 k + xG12 + k + xG11) W3 - (kk + xG13) (i

hk = kh + k + xG + (Z% + xGkk)k ( + xG k hk -3 + +r 13±t9k 12) W1 - (92ilk +s 1 1 ) en2

(25)

%Ih = -- h$y + viy)c~k (Y+ h$x -vx)s JFil

+ [X - hy + viy)sk + ( + h x - viDx) c~k]F2

+ viF3 i = 1,2,3

k= 9 + uk + eF1 l

Z7k = vk + eF 2 1 (26)

+ eF31k k

5 DERIVATION OF GENERALIZED ACTIVE FORCES

In this section, the generalized forces due to aerodynamics, gravity,

body springs, the flexbeam structure, and structural damping are derived

Some details are omitted for the sake of brevity.

5.1 Aerodynamic Loads

The aerodynamic loads are derived from a quasm-steady version of

Greenberg's equations (ref. 15) for lift and pitching moment. Also included

is a quasi-steady profile drag contribution The circulatory lift per unitlength is given by

LC= paVc [ + Ve + c (1 - 2Xa) (27)

i0

12

The noncirculatory lift per unit length is

LNC ac h + Vc + v + c (1- 4xa)C] (28)

The pitching moment per unit length is

- 2xa) + VT(l - ± hi - 4xa)pac' [2V (± a

32 a) + e~pa - xa ) (

(c 2x + 4xa2),]+ PaVc2xa[ ( - 2xa)] (29)

The air density is p and the free-stream velocity is V. The airfoil sec­tion is pitched at the angle e with respect to the free-stream as shown infigure 9. In accordance with small disturbance airfoil theory, we may set

Up - h - VP

The blade airfoil velocity components are UT and Up, expressed in the N k

and Nzk direction, respectively. Substitution of equations (30) into (2T)to (29) yields

ILC pacU [up +c ( -2xa)j

2 (31)-4Xa)R[-Up +* (1LNC pac + I- xE

pacxaJUP pac 3 [U (I ­32

8xa + 16xa2)2 a

- Up(l - 4Xa) + -Xa + XaJa (32)

Next, the total aerodynamic force per unit length on the blade airfoil

section is considered The noncirculatory lift is taken to act in the direc­tion normal to the chord line as shown in figure 10. The circulatory lift is

taken normal to the resultant velocity U. An aerodynamic profile drag forceper unit length D, acting parallel to the velocity resultant, is included,based on a constant profile drag coefficient cd0.

PCd CU2

2 (33)

The force components and directions as shown in figure 10 give the followingexpressions for S and T

11 ORIGINAL PAGE IS

OF POOR QUALTY

S =L C sin a - D cos a (34)

T = LC cos + INCI D sin a

where, from figure 9

Cos = U (35)

UP sin a = -T--

Substitution of equations (31), (33), and (35) into equations (34) yields

S = -- [U 2 _i (1 - 2a)Ui --- UUT 2 [(36)

2 • Cdo T -

_2

-U T + 2

(1 - 2x)UT aT

- iU + C

( - 4x)s - a)

­ -­a

UU

Since Cd /a is small with respect to unity and because the magnitude of the aerodynamic pitching moment is small, it is permissible to set U = UT inS, T, and M. The aerodynamic force per unit length acting along the blade is

k= Skyk + Tkz (37) aero

The aerodynamic pitching moment per unit length is

(k'aero = MkNxk (38)

Now, for the kth blade

U k2 k2] Sk pac Up -jc( i-2a)Uektl1=k ---- 1 2 Ukk -guTJ

= aU -T ( 2X)UTlk I\

Tk -2-pac kU c k + 1 k + (1 - 4xa)IkJ

Xk 2 {x a c U Pk U T k k4) [UT -~ 8xa +1x W _1- x pk (39)

8 -+3 a a

where d = edo/a has been neglected with respect to unity, and has kbeen replaced with the component of angular velocity of the blade in the Nx

k .direction w1 The velocity components Upk and UTk are given by

12

3Up= S ulkck1

i=1

k 31 I2Ulk(;k (40)

wh r th co pone ts t = 1=1veel cit

where the velocity components Uzk are defined in equations (25). The aero­dynamic force and moment resultants at the point J are given by

(Fk)aero :Sf 1fy:k+TfkSz (41)

(Mk)aero = Mmkxk _ TmNyk + SmkNz

where

sfk= Sk dx Smk = f XSk dx

o o

Tfk= f Tk dx Tmk f XTk dx (42) 0 0

IoLMk f MkC dx

0

The generalized forces due to aerodynamics now follow immediately fromequations (2) and (4)

(FUk)aero, (uk) 1eo-U

eere k= (Fk)a

(FVk)aero

aer

(F k)ae M)aer o (ck 2 sklk

ero

ORIGINAL PAGE I \(kIer \it/aero = nik OF POOR QUALMIT

13

b

( x)aero T (Fk)aero * A k= i

b

= aeroY aero

k=i

(F_ ( z: F1 1I \ktaero (s(F) 2 ) kk

aero k=1b(43

+ [z~k(cok F31 - S4k)32) - £Wk(C* kF2 1 - skF22)]!ul

+ [zTk( 4 ) 1! ­ S4k)12) - £iik(CV) F3 1 - s,k F32)] 2k(3

+ [Pk(cpkF2 1 SkF22) - £Vk(Ck F1 1 - Spkl2)] }3k concluded

+ (Mk)aero. NA)

b 3

> {h (C) Fil - S4k )k)(F) E (F r aero k==

+ [CwkQkF21 + C4) F22) - Qvk(s k F31 + cokF3?)lnlk

+ [zIk(s* kF3 1 + cpk 32) - £wqk(s$kF11 + c4k)12)1U2k

+ [z4 k(s ) kF11 + c.*1F2) - £iik(s4)kF2 1 + c4k22)13}

-NB)+(Mk)aero

Equations (43) define the contribution of aerodynamics to the generalizedforces.

5.2 Gravity

In this section the gravitational contribution to the generalized forcesis expressed. For a system of rigid bodies, gravity applies a force actingthrough the mass center of each component. For the body

14

(FB)g = Mfg e 3 (44)

and similarly, for the kth blade

(Fk)g = mg e3 (45)

The subscript g refers to gravity. The symbol g is the rotor thrust perunit mass of the aircraft (.-/M) if the aircraft is airborne. When the air­craft is in ground contact, the analysis must be slightly modified. This isdone at the end of section 5.3.

The generalized forces due to gravity follow immediately from equa­tion (2): 3

mgejmg (Fr e -nk(uk)g m(e Wk/g 3 a kI

FVk)g = mg e, • k k

a(FBk) mge 3 e kgkg

gFk~ i=31 0

(Fkg = Mg e3 * NA 3= @k 30 Ik

i1g k(Fy)g = Mg e3 N B

(4x) = (mgbh - mfgz)e 3 1 B - mg e 3 • [ e+ c3k)(c F21 - sF 2 )

k= i kg

- (Vk + C czk Fs31 -S 4 k 13)]l+ [(Zk+ clk)(c* F3 1 - s, kF32) (46)

- (zvk+ c3k)(c kFl - k12)l + [(sk+ c2k)( k - 4k 1)

- (Piik~cl k)(c F2 1 - S41 F.)]u

(F = -(mgbh - mfgz)e3 •A + mg a • { k+csk)(skF +C1 kF)22 1

g

- (vk+c2k)(shF31+ckF 1 + k+clk)(skF31+ckF532)

- (wk +ck)(s1kF,1+ c*k F 2)1n2 + E(vk+ c2k)(s*kF,I +ckF12)

- (uk+ clk) (s kj21 + 4*k1)] us)

15

Equations (46) define the generalized forces due to gravity when the air­

craft is airborne. If the aircraft is hovering at thrust = weight, g becomes

the acceleration of gravity g. When the aircraft is on the ground, the bodyforces Fx and Fy- need slight modification.- This modifidation is closely

connected with the body spring forces treated in the next section.

5.3 Body Springs

When the aircraft is in ground contact, a system of springs is introducedinto the mathematical model to account for restraint of fuselage motion due tothe landing gear flexibility. These springs, shown in figure 2, are symmetricabout the aircraft reference center both in the longitudinal and lateraldirections In deriving the generalized forces, the analysis will be carried out in only one plane, the el, e3 plane and the results extrapolated to thetotal system.

Consider the two-dimensional system of figure 11. The velocities ofcertain points are needed in order to derive the generalized forces. Thereference center velocity is

VC = kIA + NC (47)

and the body angular velocity is

M= (48)

The velocity of points A1 and A2 is, respectively,

V A 1 = (i + ky) + Z ; N

( - 2 --sy)N C= (k + +Zt')NVA2

The force applied by the springs at each of the attachment points is

2 -FA1 = -2Kx(ej + zy)e, - 2Kz(e 3 + i ty)e3

9IX (50)

FA2 - 2 Kx(e + £z4y)e, - 2Kz(e 3 - yD2- Y (0

where el and e3 are the space-fixed deflections of the points A, and A2along the e1 and e2 axes. The vertical degree of freedom is retained only toaid in modifying the gravity terms in Fx and Fy for the ground contact case.When the aircraft is on the ground, there is a steady vertical deflection ofthe ground springs that varies with rotor thrust and gross weight. Thus, forfirst order in X and 4y the thrust _V and weight Mg0 must be includedfor this case. Each of these forces may be assumed to act at the referencecenter for the purpose of calculating spring forces.

16 O IGMAL PAGE IS

OOR QUALITYOF

FB, = -NC + Mg0 e 3 (51)

From equation (2)

Fx = -4Kx(e + £zy) + 4K. e3#y - Mg0oy

Fz = -4K. e3 - -+ Mg0 (52)

FIy= -(4Kxkz 2 + Kzx 2) y - 4Kxkz e1

For the equilibrium of the coupled rotor-body system (all degrees of free­dom = constant), Fz = 0. Therefore,

4Kze 3 = Mg0 - - (53)

Also, the deflections along el and e3 are identical to X and Z for infini­tesimal 0 y* Thus, for X and 0y degrees of freedom

(Fx)spr = -4Kx(X + kziy) + (Mg0 - Y-)ty - Mg04'y

= -4Kx(X + tzoy) - 90y (54)

(y)ly = -(4KLz9 2 + KzZx2)Oy - 4KxZzX

spr

Similarly, for Y and Ox degrees of freedom

(Fy)spr = -4Ky(Y - kzox ) + 9-Ox

2 ±K 2)p + 4 KykzY = -(4y(F(D zp2+zJspr

Equations (54) and (55) express the generalized forces due to bodysprings. The gravitational terms in and Fy are replaced by rotor thrustFx terms when the aircraft is in ground contact. This is equivalent to replacingg in Fx and Fy with g* where

g* = (56)

when the aircraft is on the ground and replacing g in all other terms with

go­

5.4 Flexbeam Structure

The structural loads exerted by the flexbeam are represented symbolicallyby the flexbeam stiffness matrix [K]. The matrix [K] is determined numeri­cally by perturbing the equilibrium solution. This operation is described in

17

section 7.2 below. With the stiffness matrix formulation, the generalizedforces are

6qik -_ KI q 1 k k = Ukekk, kk' k = 1,2,...,b (57)

1=flex

The fact that all the b flexbeams are identical is reflected in the lack ofdependence of the [K] matrix on k.

When 2 = 0 and the flexbeam is undeformed, the matrix [K] is given by

EA 0 0 0 0 0

12EIf -6Elf

[K] = 23 0 6E 0 (58)

4 EIc0 0

4EIf 0

where EA, EIc, Elf, and GJ are the flexbeam axial stiffness, chordwise bend­ing stiffness, flapwise bending stiffness, and torsional stiffness, respec­

tively. When the rotor is at some general operating condition, the matrix [K]fully couples the blade equations structurally.

5.5 Structural Damping

In the coupled rotor-body system only the blade lead-lag motion and somebody modes have small enough damping from aerodynamics to be significantlyaffected by a small amount of structural damping. The non-zero generalized forces due to structural damping are

ORIGINAL PAGE IS OF POOR QUALITY

18

(Fdamp =-cJ

(F )damp = -Cxi

(Fy)damp = cyY (59)

(F4) omp = -4x$x

(F4Y)damp -c~y~y

The explicit form of c , cx, Cy, Cox, Cy is derived in appendix D in

terms of damping ratios for both blade and body motion.

This concludes the development of the generalized active forces. In thenext section the analysis continues with the generalized inertia forces.

6. DERIVATION OF GENERALIZED INERTIA FORCES

In this section, the generalized inertia forces due to body motion andblade motion are derived. As in the last section some details are omitted forthe sake of brevity.

6.1 Motion of the kth Blade

Generalized inertia forces that are associated with generalized coordi­nates, as the blade degrees of freedom are, may be derived directly from thekinetic energy. For the rotor system the kinetic energy may be expressed as

Vk * k *b * • * 1 3 b (0

T z V~ r Ii

m m

(60)k +1 Idk22 k=1 2= k=1

Once this quantity has been expressed in the system degrees of freedom, thegeneralized forces for the blade degrees of freedom are given by

kF*k= T d T qr = U , , (61)% q k --dt k)~k r k~kw'k k' rk , 2, ... ,b

This operation is straightforward and the details are not given here.

19

6.2 Motion of the Body

Unfortunately, the generalized inertia forces associated with body motioncannot be derived from equations (61). This rs beticuse the degrees of freedomdescribing body motion are not generalized coordinates. They are, instead,quasi-coordinates and require special considerations. The portions of bodygeneralized inertia forces that are linear in blade variables are alreadyknown because of symmetry considerations and equations (61) Hence, onlyterms linear in X, Y, tx, and $y and their time derivatives need to beretained in this section.

The accelerations of B* and k* used in equation (4) to define theinertia forces are needed. These may be derived from equations (5) to (17)by standard laws of kinematics

B *a = (X + Zty)NA + (Y - Zq~x)N B +...

a k * I (h + dikF13 jY; NA+ Y + h + F dkFixNB (62)

3

+ (syx + 2QNY) Fa djk(Filc + F3.+

i=1

where

dlk =ik + clk

2 £Vk + c (63)2

+d3k = £k c3k

Here, the dots refer to blade terms and nonlinear terms in X, Y, tx, and yand their time derivatives. Also, in equation (3) the inertial moments mustbe written. These may be expressed directly from Euler's dynamical relationsand the angular velocities in equations (6)

MB* = -IxtxNA - IytDNB + Nk* = [w2km3k(I2 - 13) - *kI ]Nxk

2323~i ix(64)

+ [m3kwl1k(13 - I) -2 k12 ]Nyk

+ [wlkf 2k(11 - 12) - I3k]Nzk 0pPIGINA EP&GE-IS

OF pOO QJAL'a20

Now the body terms of the generalized inertia forces for X, Y, 4)x, and

$3y may be written from equations (3), (4), (62), and (64)

Fx B Fk*) +A k=i

b

F x z-NB°FB* + NA "1B + E kJ k=1

(65)

+ b Rh + 3 d kF 1 )NB+ 3 dlk Sk + Fi2c*)N] FkF,~ NE" 3N + I (Fl kkk

Fy ZNA FB* +B N* + =Iy*ZA EB EBN + k)

d k+ ~ [(h+AdikFiNA + E 1 1cek -FFkk= [( =1 diF3 = 1 ic k.

Equations (65) are the body terms of the generalized inertia forces for thebody degrees of freedom. In the next section, the equations of motion arewritten from equation (1).

7. FORMATION OF THE EQUATIONS OF MOTION

The equations of motion are to be written from equation (1) using all thecomponents of the generalized forces in sections 5 and 6. The ultimate pur­pose of this analysis is to provide a means of assessing the linear stabilityof small motions about equilibrium. In this section the equilibrium solutionis discussed separately before the final linearized perturbation equations arewritten. Because from the outset the blade and flexbeam properties have beenassumed identical for all blades, the equilibrium values of uk, Vk, Wk, k,

8k, 6k are identical constants for all k. Thus, the blade degrees of free­dom have steady and oscillatory components

+uk = u + k(t) ; Ck = k(t)

= +vk = v + k(t) ; k k(t) (66)

0wk = w + k(t) ; 2 k = + k(t) I

21

Fecall that the equilibrium values of X, Y, x, and 4y are zero so that

X = X(t)

Y = Ytt)

(67)= x (t ) 4)x

qy y (t)

The tilde refers to an infinitesimal perturbation motion. Collection of termsfrom sections 5 and 6 yields a set of relations like the following

Fk + Fk= Fu - Fu (t) = 0

F*Fvk ++F F v -Fv(t) 0

wk wk wkw

F + Fk F -Pk(t) 0Fk k k

F k + t F - F k(t) 0

(68)(t)=0 Fe + F F -F k Ok 6 k

= - (t) =0+F*Fx

Fy + Fy* - b2Fy(t) = =00F~ *x = 2 9x(t)F

y y f y

F +4 =- 7 Fb (t) =0

Here all perturbation quantities have been linearized in Uk, , Wk, Zk,kk, R,X,, x, and y and their first two time derivatives Below, the

rotor equilibrium solution is obtained by performing an iterative staticstructural analysis for the flexbeam. The flexbeam structural stiffnessmatrix is next obtained from a numerical perturbation of the flexbeam equilib­rium position. The perturbation equations, although linearized in all theperturbation degrees of freedom, contain terms with periodic coefficients

(Shk c*k). In order to most efficiently solve the system of linear, ordinary

differential equations, the system is transformed to fixed (nonrotating)coordinates by the so-called multiblade coordinate transformation (ref. 16).

voO?

22

7.1 Equilibrium Solution for the Flexbeam

The equilibrium generalized forces Fu, Fv, Fw, F, FO, and Fe are,physically, the components from aerodynamics, gravity, blade inertial loads,and flexbeam structural loads, of force and moments acting at the tip of theflexbeam. Explicit expressions for these generalized forces are known exceptthat expressions for the flexbeam structural loads are unknown. This meansthat all loads external to the flexbeam are known explicitly in terms of thedeflections of the tip. When the details of the generalized forces arecarried out from sections 5 and 6, equations (43), (46), and (60), the flex­beam tip external force Ft and moment M are given byt

Ft = Funi + F.vn 2 + FwPa

3

Mt = M3 i=1

C S (9M = - F + s +SCC F (69)

M3 =F

The bars over Fu, Fv, Fw, F, FO, Fe indicate that these are-external forcesand moments, excluding the structural part. These quantities are listed belowin nondimensional form. Forces are nondimensionalized with respect to I00

2 /tand moments by I02

= + - - h)] - mgZFl3 + i(Sf 0G2 1 +ITf 31)

Fv = ;-21[ ; + C2 - E)] ­- F2 3 (J3 3 g!F2 3 + Z(Sf 0G22 + Tf0G32)

w=;2 + - F3 3 (J3 3 - h)] - i7F3 3 + I(Sf0G22 + Tf0G 32) (70)

3

+ -c -E ­P = 2 i H 3 1 + ;Q2 [u + c 1) - + (V 2 )

i=I1

- 3E3 DJ33 aJ33 + (Zw + c3) (J03 3 -h)J - g-- +Gi3Mm -G23Tm0+G33Sm0

23

3r

= 2 D~~~~i-3 + -2 IU -j) + (- +- EC)W T 2)U E 31

= 1­-

J333c3 J3 3 + (2w + c3) - - 8 (J3 3 - 0 + 2Tm 0- n S m0

3 (70) con-~D __ H3. mgE a 2

=6 2Z + ;gK+-c + (v+_c) 3 2 cluded

i=1

- 3 3'33 ] 3J33

+ (Zw+c 3) - 0 (J3ae -mg T+ BiiMm0- B21Tm +B31Sm0

where

-IL

Ci L

(1j 33 =h_ K+KF 3 ~+cF 3 + 3 F 3

=J33 E[+ (T-u + Cl)FI3 + (71)( v + c2)F23 + (w+ c3)F33

g02L

S - [Vpo-dV2+3(U V dUT1 VT++12 dtT2) 0oP T

- (1 ­ 2x)i O(VPO + 2UPj0)

24 ORIGINAL PAGE IS

OF POOR QUALITYK,

U 0UT-60 Y35 [PVT, +2 (up+ VT0 ± Vp 1 ) ± 3UpQUT1

- (1 - 2Xa)wjO1 0V + 2T )]

= 6_0 poVT0 +2 oMM - T0X6 [V VT + 4 1 UUV + Vp Uhr ) + 31P111J0

YE2 01 + 2UT0)( - 8Xa + 16Xa 2 )

-Tm0 pOVTO+ + VpoUTT) + 2UP

- 2 (1- 2xa) 10(VT + 3T0)]3­

Sm+ [V8D - dV2 0 + (uPVP - dUTfVT) + 2(%

2E 2XaVml0 + 3UpO-T (1- 2Xa)lp (71)

con­- pacL4 cluded

I

- CL

W1o = 13

UP= (- 2 3H3 1 + 3 + £ H33)

UT = (-j23H21 + j + -uH 23)

Ve0 =p(H 3 3 - H2 3) O g C 5 ix

VT0 = 23 H33 )

J13= (iF1I + iF2 1 + WF3 1)

= (5F12 + ;F2 2 + 7iF32)

Each of the force and moment components acting at the tip of the flexteam areknown functions of u, v, w, C, 0, and 0 which are unknown. In order tosolve for u, v, w, C, 8, and 8 we assume a set of values ut, vt, wt, Ct,

25

8t and Ot We can then calculate the forces and moments Ft and Mt based on, .

equations (69) and (70). The force FR and moment MR at the flexbeam root

are then determined from statics

= FtF R

MR = Mt +rt X Ft

(72)

rt = (I + ut)ni + vtn2 + wtn 3

MR = Mt + (vtFw - wtv)ni + [wtPu (l+ut)Fw]n 2 + [ (l+ut)Fv - VtFuns-

At any station s along the deformed flexbeam length, the force F and

moment M are

F = FR

M = MR- r X FR (73)

r = [s + ub(s)lnl + v(s)n2 + w(s)n 3

where ub(s) is the geometric component of u(s) due to bending only. The

bending moments in the principal axis system for the flexbeam cross section

ul, u 2 , and u 3 are

M • u2 = AK (74)

M • = BA Ju 3

where A and B are flap and chord flexbeam bending stiffnesses made dimen­sionless by I02. The curvatures are K and X, made dimensionless by 1/9.The torsion moment is, according to reference 17,

M • U + + (A + B)T]T (75)

where, from purely geometric considerations, it can be shown that curvaturesand slopes are related by the following­

ds= Kce + As6

d- Ks6 + Ac6 (76)ds eC

de= T d--Sfds

26

76

I

and where

E 4(A 2 + B2) (for a rectangular cross section)5DI (77)

T = F •

The torsion stiffness C is made dimensionless by IR02i and the axial

stiffness D is made dimensionless by I 02/Z. The longitudinal strain is

given by

2du (dv\ fdw\ 1tdv l dw2 1 duf2 (78)ds -- ds) ds) 2\ds) 22ds) 2 \ds/

which is equal to, according to reference 17,

T2s =T + B (79)

We define u(s) as being

u(s) = Ub(s) + us(s) (80)

where us is due to axial stretching and ub is a purely geometric deflec­

tion due to bending only. We then set the strain due to bending equal to

zero and obtain

dUs dUs (81)

In combination with equation (79), we have

d__ /I + B\ T2 (82)

Also, from purely geometric considerations (see the appendix of ref. 17)

dub

-ds-cc -cl

dv (3ds =c~s¢ (83)

dwds

Now, with initial conditions us = ub = v = w = = = 6 for s = 0

equations (76), (82), and (83) may be integrated numerically, using equa­

tions (73) to (75) and (77), from s = 0 to s = £ + us(s). This will yield

27

ORIGINAL PAGV If

a set of values for u, v, w, , $, and 8 at the tip of the flexbeam which

will be the same as the set of ut, vt, wt, Ct, 8t, and Ot assumed if andonly if the trail set is the correct set. Since the correct ut, vt, wt, t,Ot, and Ot are unknown, the following function bay be minimized

min

utvt,wt, = (u-ut)2+ (v-vt) 2 + (w-wt) 2 + ( - t)2+ ( - )2+ (8-t)2

Ct'st'ot (84)

using the modified Levenberg-Marquardt algorithm (ref. 18). The minimumS-= 0 is the exact solution for the rotor equilibrium deflections.

7.2 Flexbeam Stiffness Coefficients

The external equilibrium generalized forces at the flexbeam tip Fu, Fv, Fw, F , F5, F8 = Q1, i = 1, 2, .. ., 6 may be used to obtain the flexibility coefficients for the flexbeam structure. The Q3 may be perturbed in succes­sion, starting with Q1, by a small number s This will yield six different sets of the deflections u, v, W, , a, 0 = x1 , i = 1, 2, .. ., 6 only slightly different from the equilibrium values. Flexibility influence coefficients are simply

xF(Q3 + e) - x (Q ) F - (85)

The smaller value of e that is chosen, the closer the coefficients will beto a pure linear perturbation. If the deflectiqns are calculated to Nsignificant digits, s should be chosen at 10-N/ 2 to minimize both nonlineari­ties and truncation errors, simultaneously.

The stiffness matrix for the flexbeam is simply

[K] = [F- ] (86)

When = 0, the structural stiffness matrix is given by equation (58). The numerical scheme described above gives this stiffness matrix to five signifi­cant figures when 5 = 0 and exhibits a fully coupled system structurally, for general flight conditions.

7.3 Transformation to Fixed System Coordinates

-Although the perturbation equations are linearized in k, 6k, wk, Zk,Sk, 8k, X, Y, Ox , y and their first two time derivatives, the equations haveperiodic coefficients in the form of sin *k and cos 4k- These terms may beeliminated with the multiblade coordinate transformation (ref. 16) when b 2 3. It is necessary to retain only the rotor cyclic modes since the collectivemodes couple only with vertical translation and yaw rotation, which are alsodecoupled from the retained degrees of freedom in hover. The differentialcollective modes and the warping modes (for b 2 5) are uncoupled from all

28

other degrees of freedom in hover. This reduces the 6b rotor blade degrees

(two cyclic modes per blade degree of freedom); thus aof freedom to only 12 total of 16 degrees of freedom remain for the coupled rotor-body analysis.

The 12 rotor degrees of freedom, in the order that the matrices below are

written, are

bb_ UkcoslCkkb k= 1

b us b Uk sin lk

b= , k coslk

b k= 1

Wc =b = Wkcos lk

2b

Ws= bL wk sin k k= 1

2 ECC = k Cos

b

C= -i;F k si *k

0k2b b

k= 1

b

8s = b ' k sn *k2 b

PAGE IB = ek cos kOc L

k= OF POOR QUAIXf

aes b L2b O_k sin k

29

The use of the variables in equations (87) will eliminate all terms withperiodic coefficients in the body equations. The transformation is carriedout on each kth blade equation in the following manner,

b

u 2=E Fuko 0 k=1

b

Fu =si2 u sins b E- FUk0

k=1

(88)

b =1Fc 22 ZekF~k cos 7Pk = 0

k=1

b2= =sin hk 0

k=1

where FUc' Fus' ." F, Fpe , X' FY' Fpx and Fy become linear equations

with constant coefficients in Uc, us, Vc, V5 , WC, Ws, c' s, 0 c, 41s , Y, Ox, $y and their first two time derivatives upon substitution of equa­tions (87). The details of these operations are omitted and only the final results are given in the next section.

7.4 Linearized Perturbation Equations of Motion

The linearized perturbation equations of motion for infinitesimal motionsabout equilibrium are expressed in matrix form as

CA KS[MI + M]AXM + (GI + + cD]f} + [K I + + Kg + KA + KD]fx} = 0 (89)

where

[MI ]I mass matrix due to inertial forces, symmetric

[MA] mass matrix due to aerodynamic forces

[GI ]1 gyroscopic matrix due to inertial forces, antisymmetric

[CAI damping matrix due to aerodynamic forces

30

[C ] damping matrix due to structural damping, diagonal

[K ] stiffness matrix due to inertial forces, symmetric

[KS stiffness matrix due to springs and structural forces, symmetric

[Kg] H stiffness matrix due to gravitational forces, symmetric except as specified

[K ] stiffness matrix due to structural damping, antisymmetric

Thus, M I is the c inertial term in the 6c equation The elements ofthe above matrices are given below. Only nonzer6 elements are specified, formatrices with symmetry or antisymmetry, only the terms on or above the diag­onal are given.

Modifications to the above analysis for different pitch-control geome­tries are described in appendices A to C.

MI I = I,1 H2 , 1 2 NII

I -- E1 I =MM 1,7 = mZ - T M213 1,14

I -- 3EI I M1 ,9 = mz --- M2,14 = -MI,13

MIII -3- M2 ,15 1,16MI _I I1,13 = M2, 1 6 =M1,15

HI = = 2

1,14 MnF12 3 ,3

I = - I - E2

1,15 =m 1 2 M 3 ,7 = mZ _T-­

1 ,16 = mz-11 N3 , 9 = taP.

2,= M I, M 3 1 1 = m T.

MI I I = F 2 ,8 = 1,7 M3 , 1 3 =

MI I MI kF2 2,10 = H 1,9 3 ,1 4 22

31 ORIGINAL PAGE ISOP POOR QUALRY

MZ = M3,15

mUj2 2 H51 5,16

= mZJ31

if3,16 = M5,5

I M44,4

= I 3,3

I H6 ,8

= MI 5,7

I I 4,8= M 3 ,7

T I M4,1O = M 3,9

I

6,1o

I

M 6 ,12

=

I M5,9

MI

5,11

I, I M4,12= M3, 1 1

I = =6,

I 5,14

I 4 ,13

MI

4,14

=

=

IM3,14

I 3,13

I6,14

I 6 ,1 5

= I 5,13

I = M 5 ,1 6

MI 4,15 =

I 3,16

16 6,16 5,15

I

=,16 H3,1 5

@1=1E

14I

7,7 =9

_

+

3

15, = m t IEI

MI 5,1

=

=

t 7

m

mP "­32

I--­

31

3

m = -mE 3 - ' c, G

5,15 1=1 j=

I

32

3 -

7,11 DE1

1=1

3E 1 -30

3

1i=1

11113 -E

3

j

G­1 3j

3

M7 ,13 .r 1 Fl

3

M7,14 =imn 1=1

4 F12

3 3

M7315 = 1 2 ­ IG H

7,16 =

3

JLl +

3

Ii 3 H1 2

H 8, 8 = II 7 , 7

I m 8 1 0 =

I M 7 ,9

I 8,12 =

I 7,11

MI

8,13

I

7,14

I 8 ,14 =

I-M 7 ,13

IH 8 ,15 =

IN7 ,1 6

I M 8 ,16 =

I 7,15

= if~(5Z8 :

3

= L 3

Jjj

33

I

33 3 J :i3jII ...

M ,= m2. 1 ac1 I -EaaE

j=1==1

I3

M9,13 m -- F1

MI

9 , 1 4 = -

3,=

3 -

1=1

C F 2

9t15 =

3

mn . j i2 +

3

IlniH1 1

33

I =I 9,1H1 0,10

~lM 1 ,1 = -M9,13i

I I = 9,14H 1 0 ,1 6

= -9,1MI0,13 II

MI I =M91H1 0 , 1 5

I I = -91M10,15

=13.=1

3

F1 1 I1,13 -m

i= I

ORIGINAL PAGE34

OF POOR QUA'='

3

1 4 r - ­r iI1l1,14 m/ .i

3 3

Ut 2-ili

3 311i,16 =1 I+E IIH2

12,12 1i,

I MNM 12,13 11,14

1 41M12,14 -M11,13

II

12,16 11,15

2Rmi b13,13

13,16 2 - 2iJ33

I IM14,15 =M13,1s

15,15 b + ii[2J2 3 - (J33- 6)2 + (Th + -i)2 + (' + Z2 )

3

+ (TiW + 6) 2] +Z 13_J( - HZ3) il=1

35

MI 6 1

16,16

2=

=-b-+ M1, 1 5

2 X

b

I

l2,0=

I

1,9

II -11I,2 = 2QM1,1 G2,11 =

-I -1,12

IG1,3 = -25R2F3 3

GI 2,12 =

I 1,11

IGI=

1

01,7 =

2-2F3

( 3

23

DcG

--­2 F 3 3)

I2,15 =1

G2 1

I,=_G

I ,1

1,16

01, = 2 2Mf,7 I

G3,4I =-2I =

1 ,

=

@E 3

2,92

GG = [M, -

3 F )23

F33

G I

I

5 = -2Wpi2F13 1

@c 1 3

i~1i9

I

G = 2

'c3 (Th -F

ac2 F3 3

33,8 =2 ,

0112 1M11

03 I=2 i,i9 - F 3 3 - 1F)

0115 = GIGI,'

01,16 =

-2 fiF 13J 1 3

II

2Mik 1 3 J2 3

I3,10

3,11

=

=

=

2M,93,9

-­ i

2 F)33 -

I

F1

GI 2,4

I G1,3

GI

3,12 3,11

I I=G

GI2 3,15

F23

23 13

2,7 = 0G,3,16 22EFJ 3

I I G I

2,8 = 1,7 04,6= 3,5

2

2 ,9

= GI

G,10

36

GI

G4, 7 I

-G3,8

ORIGINAL PAGE 18 OF POOR QUALITY

I I 4,= G3,7

G06,8

I = 05,7

G,9 = -GI3,10 I

6,9 I

= -G5,10

I04,10 =

I G ,9 G6 l5G6,10 = 5,9

GI 4,11

I

G4,12

GI

G4,15

= -GI 3,12

I

= G3,11

=

G3,16

IG6,11

IG612

IG6,15

I = -G5,12

I = G5,11

I = G5,16

I4,16

GI 05,6

= I 1

2I 5,5

II 6,16

078 =

5,15

2-M,72nM 7 ,7

= 2F 1 - -­ 23) 1 7,9 =2

[i3IIH1 3 30

I = 05,8

I = G5,9

2M,5,7

2W -F-

F F1 3

- -F2aF 3 2)

3

1=1

1 .

I = 2 M, 5Go5,109

5 = F­G5'I FI225,7

5,11

GI = - ­5,15 2FF33JI 3

­ -- F2 3

2,9

()­

o 25, I5,122=

cF3 C +F 2 3 s -

5,16 2QiFZF 3 J2 3

GI06,7

= GI1 5,8

37

I 07,11

= =I111I3

1 1 13 -­ n LE 13 + (F1 3T 12 - F2 3T11 )

=i=1 i=T

GI = 25M 1

7,12 7i

3 3

15 13 32

GI 07,16

= -

2 i - JC - 1H12

aH D

13+

+ F 1 I I 08,9 = 7,10

I I G8,10 = 07,9

GI ,I I

-7 ,12

I GI G8,12 7,11

I I G8,15 G7,16

GI8,16 =

I -G7,15

0I , = 23129,1,9

G9,11 = 25- Sc E1F1 3 Zi H 13 s311)

+ (F + F23 c $ 3

3

] 4 ORIGINAL PAGE ISG0 = 2M,9,12 9i OF POOR QUALITY

38

F 3 I3a33 13 i

G = 2 I+Zi~l + H -- + (F1 2 S- F22c,)2

381 3$---J3 tl33GI9,16 2a,fF -- L3-- J23 Hi2 - + (FIIS c 21-]- F 2 1 c )

I I10,11 9,12

GIGI

10,12 = G9,11

I IG10,15 = 09,16

I I010,16 = -9,15

I - I 11,12 1I2,

3J3 3H

G = 2 -m 13 ++ Hl + (T1 IFI2 + TI2 F22 + TTF3) 3 i=l1 1=1 J

F t t­3 3 1I Di33 IH13 Ici%1

11,16 = -J 2 3 liH 2 31 + (TIIFII + TI2F21 + T1 3F31) S Ti i =1 i=1l~

I I

G12,15 = 011,16

I = IG12,16 G11,15

2I, = 24 i[(J33 - ( 2_( +E )] 1T1 3

K 1 = - 22(2 - F2)

K(I = p22 FI 3 = 2 F13 23

39

1,4= G 310= KI,9

K1'5 KI2,11 i-KI1K, 12 K = EZ2iZF1 2-F 3

KI I KI I1 62,12 = KI,1 I

--(22J33KI, ff pj - F137 -2 2 /K, , = -ffikz2(2- F23)

IS= GI7 =22F

1,8 5,7

I =K1T,9 2-F13

)M 33I ,l = i 129 3 , 7 = 09,F23 2 8- '

K ,1/ =

=~~ F132 =e

KK,9 2 -iiit- F2 3 2­1,12 1,11 -9

1,11 = -KI, 4 -2 2_m KaFI I a 3 1 3 )-03,7I =I F23K2, 2 =K K 3,

G (KIII I= KI 1K,42,4 1,31 K3,II=a12,i=3,12 G31 F2 3

K I I = -K I I

2,5 "1,6 4,4 KK3,3

K2 I I = -KI2, = K,5 K45 3,6

I I2,7 1,8 4,6 = K3,5

I I 'I IK2,8 = K = -K3,8

KI I=-KI

2,9 = , 3,7 ORIGINAL PAGE IS

OF POOR QUAUT1

I T 4,9 = -K3,10

IK4 ,16

I =K3,9

I I

4,12KI 3,11I

K5

K1 5,7

= -iC252(2 - F2)

(J 3 3

-­2 1\Q-F F3 3 -

2 2

a33

KI K5,8 =

-I 5, 7

5s,9 K Z8 2J333$\ M3-1--F

K5,10 ,9

*I5I

*I 5,12

I 6

=

=

=

h2 (

GI,

5 1

I KS,5

U333 F F 3 3

2 -j-)

IK6

I = -K, 8

IK6, 88 =

I5,7

K6, 5,10

I

6,1

KI

6,11

=

=

I

Ko5,9

I

5,12

41

I IK6,12= K5,11

=2 3 22 5 2 -Ca1g )KT,7 3 2 F13 + Fi + D2-

7- (i+E 3 ) 7_2, "

33_2 12E 321 K, 132 3 11

-i+Z2) 13(-+E2 j3C 2(3: 36 "3 3--2z

7,9 { 3 F + E#Fi

3

- (2+ ) 31+3$2 ( 2E3 - I)(-)­_2_O (0+E 2 '33

38 c 34 3 11j3 313$jj* / H3,3

7 lo 7,9

17,11 =3 FJ Fcae

I3, 2E -2 3U2 E3_ 2Ej 33 19,1]1

1 ) kD36+34 3 13

( DG 3 0DG__/ -(~)H3 H3]

O PO-Q 42

I -IK7,12 = G7,11

I Ia8,8 = K7, 7

8, - 7,10

I IK8,10 = 7,9

I I8,1 = - 7,12

I I8,12 7,11

9 9 = 2 -E) aJ33 '+ F3_K(99 =-- F3 + F - (I + I

i"Cl

- ( +c 2) a2~ (a 3 2 ZG 2 + _i,--_­as2 (f)+ F3)3 a - 2 F+

2=1 3=1

-1-(H 3)2 aH-3

H13a. a

9,1 1$ 33

a,2262 a EG3 I

-(Qu+Ej) - -a-g - (Qv+c2 3S3 -3 (ii+aF3 ) a - 2 asa K I -2I1 - l I

\/aH 1 3 \faH13 a~n

KI I~9,12 9,1

=9,90 ORIGINAL PAGE IS

OF POOR QUALIIX43

II =K 0 ,11 -K9 ,12

I I K10,-12 9 il -0

3/

I =- J33_ 'S) + -( ii+~ 3Q21

KI 12 ,1 2

- ( +E2 ) a2 2E 2 ae2

(ae ( H , 22

IK1 1,l1

ii + E3 )

i3-

H- ]

2­

3+

e

+ 2

2

i2EE,3-)

+0

3 ac 2

),

Kg 1 5 -ngTF1 2

2,15

2,16 =

1,16

Kg ,15

*5g, -a~ F22

Kg3, 1 6 =-MiZF 2 1

4,15

4,16

3,16

3,15

5,15

5,16

= -mgF32

33

44

- ---

3 K9 K9 __ a El

6,15 5,16 Kg, = 28-8 F.1 3

9,1 6, 16 5=1

6,16Kg,1 =-m,-15Fl3~ ;2 K

Kg7, = mg_.. --7 F 3 ~iii~ l

1=l3 3E12­

2E K9, = - F

Kg = F1 3 16i­138

7=1

K3 Kg = F

K g 32E1 K= Ka 137,1 1 g 10,12 9,1111

K mg F 10,15 9,16 K7,15 = 1L2

g = _K9l=IK 10,16 9,15

3

Z -1-- 92E _F1

=7= Kl,11 mg . 2 F 1 3 1=1

Kg = K1 18,8 7,7 K9 =i~ 1=1 E 12

81 7 11,165Fl

K9 = Kg __ az l =8,12 7,11 11,16 -mg __ o-'-F

:1=lK9 = Kg8,15 7,16

Kg = K9l i

Kg Kg 12,12 11 1= 8 ,16 7,15

K8 =3 16 - K2,15

K g =1 982 K(53 Kg

K12,16 ,

45ORIGINAL PAGE IS OF POOR QUALM!

Kg 2Mg Kg = 0 b ' 16,1313,16

KgKg . -K g = 0 ­14,15 13,16' 16,14

K g g=K

16,16 15,15

Recall from section 5 2 that when the aircraft is airborne the parameter

is given by Ng0 where

gogo = 02L

and N is the load factor = thrust/weight. When the aircraft is in "I-G" hovering flight or in simulated hovering flight on a gimballed model test stand, N = 1 and g = g0 " When the aircraft is in ground contact is given by

g0 in all elements of [Kg] except

g - 2 -­

K13,16 -K14,15 b NMgo

where N 1.

All elements of [CD ] and [KD ] are zero except

CD CD c7,= C8, 8 =

D 2 D 2C13,13 b -X ' 14,14 b -

D 2 . D 2

C1 5,1 5 b Cx 16,16 b Cy

The above structural damping terms are derived in detail in appendix D.

The structural terms are

KS SK -,3_K212 = K3 , i,j = i, 2, • ., 6 ,22 1-1,2j-1 K2i,2j Kij i

The following elements of [K ] are zero except when the aircraft is in contact with the ground:

S 8 S 8K 1 3 13 = K14,14 - '

46

KS 8-- S =8-­

1 3 ,1 = b K 1 4 ,1 5 - Ky- z

K 15,15 = 2 (4R 5 Z2 + £zy2)

KS =2 ( 4 .j2 + kz 2)16,16 b OZ + X

DPIT1,1 2,2 3

p MA MA -ap2

p1,3 2,4 32

,M'= M'- -G31 a1,7 2,8= =4 5 2 p -2 _ - 13

T2PDPDpiapI a =119 2,10 = n - T11

MA1 = MA, 1 = (B2 -pt - B11 P3

MA =1H p ­1,13 31 Dt-J

MA 14 =-H ­32 a £

1 115( 432au , 2 H1

Mtm a p

aA 22 ~­ 1

2kJH Hi631

M= MA213 1,14

214 113 ORIGINAL PAqGt IS o4 PooR QUALS*

M2,15 1,16

MA=_MAM2,116 115

Rows 3 and 4 of MA are identical to rows 1 and 2, respectively, except that PI is replaced by P2 '

Rows 5 and 6 of 11A are identical to rows 1 and 2, respectively, except that P, is replaced by P3"

Rows 7 and 8 of it are identical to rows I and 2, respectively, except that P, is replaced by QCf/L.

Rows 9 and 10 of MA are identical to rows 1 and 2, respectively, except that P1 is replaced by Qa/s.

Rows 11 and 12 of Mt are identical to rows 1 and 2, respectively, except that P1 is replaced by Q6/i.

13,1 031 D- £

MA ap13,2 G-G31

DP

13,3 =-32

ap

ap

M36 -G33 pMtAi = -G p

13,4 -G23 V I 13,5 033 3-

MA = - aPx x13,7

G23 MA13, -+G013P

13, 23 - G13

48

Da 1' xMA13,9 = 2 3p"l30

13,10 = -12 - fl

3Px DPx

Mt3 -B -. +B-wx13,11 = 21 + BVp1

MAPy py 13,12 21 hVp B 1 ai

-yMA 13,13 31 -p x H32 aUp

p ap

x- H- yM13,14 H 32 313 p

M13,15 J32 a-p +2 pH18 J31 UpY H22 VpY H1 YX x1 HMA X + 1h 1

3 2 1' 13'21' 1

M H 3Px1 Px DPx H4 aPy Py aPy

MtJ3311 3fi~H 22 -~­ 1 2 j-+ 132 y~+ H1 r -H 1 1 1316 2 p pI Ap AG

14,1 =9MA13,2

Mt4 ,2 1t3 11

1t4 ,3 Mt3 , 4

M4A _~MA 14,4 13,3

NAM 414,5 13,6

4,6 =- ,5 ORIGrINM PAGE IS

BOOR QUALIMMA AOF

Mt _MA IA1 =-_MA =14,8 13,7 14,13 13,14

MA >MA A4r 13A14)9 13,10 ,f A14 m1-1

M MA MA _~MA 14,10 13,9 14,15 13,16

=MA 4 1 = MA,1 MA 1 5 14,11 13,12146 135

S=MA 14,12 13,11

Rows 15 and 16 of MA are identical to rows 13 and 14, respectively, except that Px and Py are replaced by Qx and Qy, respectively.

A /a p1 ap1) apC1, 1 + Gj2 - 1 (FH - F1 1 H3 2 )

C1 ,2 iticA =2

C api+ Da 1) + (F1 P11,3 = (32 U + 22 (F2 2H 3 1 2132)

A014 = 2 MA 1,3

CA aZ20 al -F H'1,5 = + G223 32H31 31132SU 23 UTaT aUp

CA = 25S

1 ,6 1,5

CA 7 - G33 -G 31 a23 -au pV a aiaP "I a-I--0ap V p H3ai

CA, = 2 M, A aP( aP PI UP a a ap1 13 I ap_o

1 9 + 2 3UT +In3 DVp ,8"I aw P 3-B

I,10 1,9CA =2 i '

50

0 a "I -_a 13 _

A = c1,11

-i B - - Bn 1- e U -- ae /21 (2 B31 1aT l awl _ a 9

A

CI, 12 =2QM1, 11

CA D-+ H21 - 5H ­

-p+11,13 H31 al'Up 2 au T 32 aU

A 14 (/ aPI p a H2 2 --- 2H3 1

1,15 -J3 2 --- H21 ; 22a H31 T +V al

H - at' a'1 apl(h4H 32 _ + (H22 + 4H32)S' _- 5H12

alUp aVp

A H ;1_IH atP1 aI' atPI-I

1,16 31 - + 22 - 21 --- H3 2 HI22--SUp @Vp ;UT aVT 1

9p - at' p1H- at'1 + (H31 + -r-q+ (H21 + 1HI)2 RV- HI- l

A A2,1 = -C1,2

A AC2,2 = I,1

A = AC2,3a,1

A AC2, = Ci,3

A AC2,5 1,6

A A 2,6 C-,5

= ct 8 ORIGIN P" G I A _ A oFpOQUALM,

c2,7 01,8O

51

--

A A A AC2 8 = C1,7 C2,13 = C1,14

A A A A- -02O29 = C14 = -C ,1-

CA A A A2,10 = C1,9 C2,15 = C1,16

A A A A02,11 = -C1,12 C2,16 -Ci,151

A A C2,12 11,11

CARows 3 and 4 of are identical to rows 1 and 2, respectively, exceptthat P1 is replaced by P2 "

CARows 5 and 6 of are identical to rows 1 and 2, respectively, exceptthat P1 is replaced by P3*

CARows 7 and 8 of are identical to rows 1 and 2, respectively, exceptthat P, is replaced by Q/.

CARows 9 and 10 of are identical to rows 1 and 2, respectively, exceptthat PI is replaced by QO/I.

CARows 11 and 12 of are identical to rows 1 and 2, respectively,except that P1 is replaced by Q/Z.

A - [G aPx Px Px 1y 1

A11 = 1 + 1 -- _ (F12H31 _ FH11 3 2) + 2 G3 1 Y-]L, p M T aUP

-Py PA Py +113,2 =[2G31 -31 - G21 -- an (FI2H31 - F11H3 2a1P G Up aUT aUP

A ar@x ­ aPxC 3 = £ +22 ­3 Q -.-- (F2 2H 3 1 - F2 1H 3 2 ) + 2MG32 -pF13, a2 Dup3U 1

apx 9Py 2Py ­ yUp an-ap(F221H314A3,4 = G32 p . G32 .. G22 + - F21H32]

P+AF Px G ax DPx aP]C13, = + 3- -r (F 3 2H3 1 - F3 1113 2) + 2 G33 -fp

T P p.

52

A =- 33 - 23 +U y13,6 [233 -- G33 23 -+ --a Y (F32H 31 - F31H32

L P p ulT P

cA @Px aPx @PxW UP aPx aV P - all aPx0 +

17-G3-p + G3 T GI U +m p a 013,7= 023 - 03 13 - + 0 - 0 13

1 p a v

A -250-:- .. .

\237G

vpap @VT1 p

a1 hp 3H o

2HP0 Sy -081 ~av T~ awl

)_+13= _+ _ + x+ ar)-25 VpX G3 aVT C13'ii T +IGm ay - Upy c@ a p 0 ap2 + p __3_y+T1u p13,10 = - 25 T -.p +1 ny a - 1

3 0' y M1 y1P an an

aVp 0 aPy 11113 UyA ar 2 -p - +x aupo3p + p@ a l?

+P xC13,11 -B2 n U1 1 +2 +-31pp 313av-1p 9av T 1awvl Do D__3 pDo a

25- BJ 21) 3? hvF 1

cA 31'

X DX aK 0 X a xy -u 13

13,121 5B, - B21 l B31 a- anl 36 a .Tv 3I DV Tpw38 53

avpo0 apy - 3R13 31'),

p aq @1

53

ORIGINAL PAG OF PQQR QUALIMY

A1 31' /P 31' 31'C13,13 -131 H21 H

a1' D T 31'

-A. @P +p ap + DP3 -H 2 2 C13,14 = 32 Up + -31 312 32

31' 31' @up 31'

A H Px + a3Px + H P aPx H013,15 J32 Up 3V J22 -UT HVH21 H31 HI

'Px TP ' Py

aVp J1--p + - (H2 2 + H 3 2 )5--- + H2 1H1 a---Y

Hm 3pY' - Py "'y +H

3 VT MI2 £ 3 ) amUpH32 - H112 ±- ( 31 + 32p

Py- PH I1 -I

A H Px+XPX H aPx31 313,16Dup H22 aVp J21 aUT H32 VT + H12

Px - Px H aPYyPY

211', 1a ~ u

H 111 y- - (ili(TU11 - H­3 T 11M1 1 p

+ al) 12

= 13,2

A AC14,2 1-C3,1

cA A14,3 13,4

A A14,4 = C1 3,3

54

H32­31'

- -31' u22

1'lapa T

Ha-_

31

H1 ary

21

a1'

Px - (H1)5 I iOp 3

Py

H

31a'

3y

H2 Hp T

(Hy (121 + (H3 1 )

a

-aVp

311

H­

32)

@

-

a'

H y

DUaT

(( + 4"H 3 2 52 p

22 3

A AC14,5 = C13,6

A A C14,6 = -C13,5

A AC14,7 = 13,8

CA A14,8 = -C13,7

A A C14,9 = C13,10

A A014,10 =-C13,9

Rows 15 and 16 of except that Px and Py

Fy

PICA [aTHp (F<2 21±p 12E

1,2 @up

11,2 G 3 1 + K, - Uj-- (F22H3 1

'AIT p

LUp

A C14,11

A= C13,12

A A C14,12 = -C13,11

A A C14,13 -C13,14

A A C14,14 C13,13

A A14,15 = -C13,16

A A C14,16 = C13,15

CA are identical to rows 13 and 14, respectively,are replaced by Qx and QY.

- 2+pi-BP (F12H21 - F11H2 2) + G -p1 p I1 aP]I32

[ aua atap 1 3

UT 2 1 U (F22H31 - F211H32) - F2 1 H3 2) +- (F22H21 - F2 1H2 2) + 5G32

PI 1

e UT @u

KA =- I-203+ --0 - -(F3R3- 1H) =

P (F32 31 - F31H32) + - (F-21 - I + 033 PC'p-N PI DPI I BI

[@up BU U T V~

K [, 3 + -OO - '2 ­"16 aup an1T anp

-A- 0p 1I C al/ 0aPOIGNL13 AE1Pi ~2=av-Pa 0 a 1 T0 a 1 1 I anD @VP an-1, 4 2yP +v aT acT+

+23 - -0(13 -£

@VP3 a6!I ORIGINAL PAGE IS 55OF POOR QUALIMY

--

A GP 3P l G13 ap UP0 aPI aVP 0 aPI A 8 = G ap l 33 aav ,T 3 pG 1a a p {p-

aUP0aPi VP 0 1 + TlV3P, 'T DPI = -- + - NI ++-K- U8 aV 8 TTaI 0 TUT aV

api ap ,

apBPI @PI I au P0 api av P0 @ Pl

P-E T 1 p pV K 2,10 1 ,10 a+ av 3

+~21ois~~52( BP a,3

V p w a 3 UU 3/K 11 n

-

1,12 ~ 21 31-lQH 3 pBp1A rUp , BaU T vT p ap5

14,12 1 p P)pi=p

+ 1) 1 pP p21 - B 3 1 pV B 1 1 a- 36 T)3 - 3­

[/ ap1 ap1 / @p1 Dp1 p1 =11 iTZp-+ H112 ~-F +--I +2 a.1,1 [\L a av/ + uTa av 22 an

AKA

KA2,4 1,2

AKA

2,3 - 1,4

56

H 3 'PI,

c a i

31113 DP1 -

38

D HI 3 PI

ow13 1a

3111 1

p/M

3HVp

+r 11 P 1

a'a] HavO

KA AtA KAK6 = 1 , 5 2,11 1,12

KA= A 2,1 1,A2,7 1, K212 K11

8 K1,7 2,15 1,18

K A =K&A KA -KA 2,9 1,10 2,16 1,15

2,10 1,9K KA

Rows 3 and 4 of KA are identical to rows 1 and 2, respectively, exceptthat P, is replaced by P2 "

Rows 5 and 6 of KA are identical to rows I and 2, respectively, exceptthat P1 is replaced by P3.

Rows 7 and 8 of KA are identical to rows 1 and 2, respectively, exceptthat P, is replaced by QC/.

Rows 9 and 10 of 1+ are identical to rows I and 2, respectively, exceptthat P, is replaced by Q/.£.

Rows 11 and 12 of Kt are identical to rows 1 and 2, respectively, excepf that P1 is replaced by Q/.

,A[aPx DPx aPx

GI3, - (F2H31 - F1IH 3 2 ) + - (FI2H2 1 - F1 1H2 2 ) + -U

2IJT aP

aPy IPy _P -UG31 ­ uG21 + (F 1 2 1 3 1 - FIIH 3 2 )-U

,p, T P

APx - P rx @ yK1 3 2 = [I-T 1 U 21 - Q(FI2 31 - FIR32) a+ -- (231 - FIH32)

aUp apU 1 2 2 ) + kl TFp- (F 1 2 H2 1 - F 1

ORIGINAL PAGE IS OF POOR QUALIT

57

43, = -£ F22H3 - F21H32) +-( F22H31 F1 ­(F2 2H 2 1 F2 1H2 2 ) + G32 ­aup 3uT a

aa G 32 -Fp ) Y

­

aUpy

- @U-yG 22 + 2(F2 2 H31 - F21H32 ­

13,4 = p G 32 + 22 - Q(FH - F2 1H 32 ) + ap (F22H31-F21H32

3Px - SPSl

+ - (F22 H2 -1 F2 1H2 2 ) + G 32 "p

T u1+a rap - + 3*SP x -p px

H -(F H; F H + 0135 L up (F32H31 - F31 32) +0U4T 2H21 - 31H22) 33 ap

aPy _ py apY- 033G-- - G23UT + Q(F 3 2 H 3 1 - F3 1H 32 ) -e]

13'6 3 2 H3 1 F 3 1H 32 ) +­

= [uppPX G33 G+ an 23 - f(F (F32H 31 F31H32)au UT -

Pu P@SPySF

a (32H21 - F3 1H2 2 ) + 3 I

Up 0 V0 33 Tp1

KA @Px DPx U + x 3H13 SPx13 ,7 Px + + +aU Sp SUT SV@ T S 1+ I

Spx I_ . 013..

a+ 52 23 ap G13 a i/ .

223 @ya FP- G33 -VT pG13 51'p@Vp DVT aw1

ac0

aVP0 pY) + HI3 3Py

58

UP0 aPy VP_ 0 P_ T0 Py VT0 MY M13 MY

91 aUp Do Vp a n 1UT A VT 31 DWI

0 aPy Sy

23 aQp 1331)

nVnPU P 0 Px UT 0 px VT0 3Px H13 aPxp - -x+ -o + + + Q

K13,9 = aB au 3$ Vp B aUT a B 8VT ao T1

_ 2( ANxAp aPx @Py iPy @Py DUPA Pyasa

aAVPI/0 31' 36 ai a 1 pp

APy 1 i H 13 Py

-wF4-1 Aj n2 -1 asY 8P + 5P TVeT k 0 1 13?AV +P 0 V)

(VTUP0 Py VP0 Ry UTo Py ANy MH aPy0 1 3

Da I o AVF a T u O A pW a

+ (1_'py- Il I

+ p ail)'

DUP 0 AN AVP0 sPx auT0 x T0 Ax 3H13 AN+K13,11 = x + Do S ?+ 30 30 a8T + T + @1

IN XPjP@_ D 3 uP0 3Py X&- -( B21 - B 1 -- B 1

3 i 6+ 21 B1 1 AVpA p 3 VT BlAl t ,p

AVp03P3TY) i H13 35.30 A%/ 30 ai1p +

059 0

K1 ,1 1~A9B 2 1 p

-- 3 1

31 aBIIlB1 -@Up

9­30e­p @ Vp

-­0 a3 3 ap

p p

0pUPPy V 0p y aUT 0 Pya VT -Py

yv p ae 30 aUT @a @VT

M3 Py -2BB aPA

21 1

: , n/ + m-+ v/ + Sapx at'x( aP apx 3px aP15--Y -(£u + HPY32 P- PY 22 £ Y Y3Ip avp/ 32 a01 + Hg1L\ o7 ~u +v- 3' 2+t T + T) HP + UHP + iP

+,Hy -. + a a+')H y 3, 3aP 11p31~ Z 21an av j 1§4 I~uj (u+ 33d )31 6 3 1

apa x ap c/ a a p x a a p upa \ a a av 22 a vp

44,213 ,2 +Kt"H3 ly

-+,---H-+-ihr+-), 31 Kkj 4,1 13,2

K'4, = KAI',I,41-KtS2-- (,,- (aKA 6

K14,36 ,

14,4 13,3KA 1A

14,8 13,7

60

° 1 4 9 = 10 14,12 = -= 3,11

KA14,10 -13,9 A= =- K 1+= _ K,11410139 14,15 13,16

KA KA KA A+14,11 13,12 14,16 13,15

Rows 15 and 16 of K+ are identical to rows 13 and 14, respectively,except that Px and Py are replaced by Qx and Qy, respectively, and the fol­lowing terms are added to the elements indicated:

K+ =POPF + KA [_15,1 z 12 y 13 16,1 15,2

1~A =~~ A-PA KA~ 15,2 Z OF11 Px 13 + -16,2 15,1

KA =POF P0 OF + KA15,3 2 22 y 23 16,3 15,4

KA =PF PX0FgS + 1 K46,4 =-1 5 ,3

A =A~- 0 +1 115,5 z 32 - py0F33 16,5 15,6

=5 Pz 0 F3 1 - Px 33+ 16,6 -K15,5

The quantities appearing in the elements of the above matrices not defined elsewhere in the report are defined as follows:

J12 = hF1 2 + (iv + E2)F31 - (z* + 63)F21

ill = hFll - ( Z + E2)F2 + (i + 3)22

J22 = hEF2 2 + (Cz + Z3)FII - (zu + EI)F31

J21 = hF21 - (DW + c)F12 + (Lu + I)F32

J32 = hE32 + (Lu + EI)F21- (Qv+ 2)FI1

J3 1 =hF 31 - (ii + I)F22 + (ZT+ 2)F12

M =hi + F6

61

--

- mfLmf=I

Z

IxIx T

Iyy T

J13= (iu + E1)F + (Zv + Z2)F21 + (ZQ + Z3)F31

23 (i+ + + (i + 22 + 3)F32Pi G iSf + G1Tf

Qc= GSMm -G 23 Tm + G33Sm

Q8 = -IMm+ 2Tm - n3Sm

Q= BIIM - B21Tm m + B3 1 Sm

Px = H 2 1 Sf + H31T f

Py = H2 2 Sf + H32Tf

Pz = H 2 3Sf + H 3 3Tf

Qx =H1 M1m- H2 1Tm + H3 1Sm - J3Py + P23Pz

Qy = H2Mm - H2Tm + H32Sm + j33Px - j3Pz

J33 = E + i(EF1 3 + VF23 + iqF3 3)

Sf [VP0+ 2UP -p E ( - 2xa)olp + [P + Up

38 (1 - 2Xa)tl)]VP - Ld(V + 2UT0)UT

- (VTo + 2 UT)VT- I- (i - 2xa)(Vp + 2

62

T= (VT0 + 2UT)UP -- j- (VT0 + ' U [VP + 2UP0- (I - 2Xa)d1]UTUT)VP ­

- i + - (-UP0 ( wVT + 1 (1 - 2xa) + 2UT

i-6 VP + y (1 - 4 Xa)OI

TVT) Lc - 2aVT 0 UTP -A UP0

Sm 3 (- UP0 8(1 - + Up0+ + (( - 2Xa)ali1]V

+ 5ulPo+(YYx(VTo + 12UT)UP - 1 +TYE 3 )VT ( 2)(VTo + - ]i(v )) + 0 2 1[ TMWY(VO UOUp - 81 VT0 + - UT0)Vp P0 + Uo

T 2 T 6 No 2 PO m

+2)a)Ul-+ UT-x( T EaPUP( - -3-(8+ ( - 2x 1]VT

V 3UT l TE p ­ E + y42 (+Yc 1 2 + " --J 1'i2(i-2a)2 0) T6 T4 -64 (1 - 4Xa)( j

Mm = 4 TO + 2UT0 Up - T + YE

(i 8xo Ur6 ax P02 V6Xa 74 P 2U- (

+ Wl 162]+ u0 -3 (I 8xa + T+6a

4 a f p + Y6-4 l 4 a V 16x a2)10 + c2­

~~~+ ­( VT 0 2UT 0) (1 8x a + yE264

3yE3 ( 16Xa 32x2a2- VU

256 3 + 3 I

v 21 ORIGINAL FAGE 18

OF POOR QUALIP, 31

63

-HJ 3 2 = J33H32 23 H33

J31 33H31- - -3H3

H-H J22 = J3322- j23H23

H-Hj2l = J 3 3H2 1 - 113H23

=O21 3G3iI0 f + ­

1 a-F Tfo

8G21 3G31

1x S 0 +Tf 0 3-a+P. =Sf0 3G

=m -F TMf0 8 - 0 Q C 3 G23 @G33

C = Mm Tm B +0- TM0Sm0G13 _ G23 3G33

_- + Smo3G13 3G23G33

0 39 ­ ao0 + S 0

Q= 0

Q8 = m0 (S6 B1 2 + c6 B1 3 ) - Tm0(SB 22 + cB 2 3 ) + Sm0(sOB32 + CB33)

Qo = 0

QO = 0

64

M313H21 34S 3= Tf Px Sfo ac+toa

3H21 31

Px Sf0 -to H31Sf H210

0 3336 1Sf0

_ 34 +Tf o 34

3H323H22

32Tf3y H328H22 33Sf0 323 0

Tf H32H22 3 0 113p Sf0

0 c +Tfo ac= -to D Tfo 3a

3H33S M23 + f H33PZ= fH23

366+f0

Mm311M1210 l 311313 2333 c - + Sm -Tm4

0l233 m0 3 2 Q 6 =MTft5 o + Sa 0 DO + ­ t 3Pyj23

=c 0 ac m 0 9 13 -3

4 311 3H21 M31 zQx Mm -- To0 34 + 4+ j23 P a j33Py

0

3H122 311 _j c+d pcQY= M 31112 - M @32 P 8 - j P +Qx =Mmin Tm + 13 z 33 x

0x8 8M T 0 3a +

H323H22 aH12 --- + Salo-D8 - Jl3Pz + J33PxQY = M 0 8 T 0

65 ORIGINAL PAGE IS OF POOR QUALITY

Px0 = Sf0H21 + Tf 0H31

Py0 = Sf0H + Tf H 322 2 -

Pz0 = Sf 0H2 3 + Tf0H33

Ames Research CenterNational Aeronautics and Space Administration

andAeromechanics Laboratory

U.S. Army Aviation R & D CommandAmes Research Center, Moffett Field, Calif. 94035, Jan. 1978

66

APPENDIX A

FLEXIBLE TORQUE TUBE

The effect of a pitch-control system similar to that of the Boeing BMR(ref. 9) can be easily incorporated into the analysis. This device has theeffect of placing an additional twisting moment at the tip of the flexbeam.This moment is proportional to 8 - 80 where 6. is the control input (seefig. 4). We assume that the torque tube is sufficiently flexible in bendingso that no bending moments are applied by the control system at the flexbeamtip. With torsion stiffness K8 , the additional moment in the equilibriumgeneralized force expression is

Fe = K(e(0 - 0+ . (Al)

K8 2where = K6 /I 0 . By changing 80, any pitch angle that is desired can be

obtained from the iterative process described in the text. The stiffnessmatrix is also modified so that

KS =KS

ii,1i 12,12 = K66 + K0 (A2)

Equations (Al) and (A2) show the modifications necessary to account for aflexible torque tube.

67 QOOMJ

67 O

APPENDIX B

CANTILEVER PITCH ARM WITH FLEXIBLE PITCH LINK

A pitch-control system with geometry similar to those of references 7and 8 may be easily added to the analysis described in the text. In figure 5,a schematic of this configuration is shown, and in figure 12 a more detailedschematic of the pitch-link geometry is shown. Here the blade and flexbeam

= are shown with f = Of = Of = u = v = w = C = a =Cb = = b =Ob 0. Theparameters ul, v1 , w, define the position of the swashplate-end of the pitchlink S with respect to point 0 in the n , n2k , n3k snsyte (threark axis system (there areno k's, indicating that the geometry is the same for all k). The parametersx,, yl, and z, define the pitch-arm/pitch-link junc ion J with respect topoint 0 in the nxk, nyk, nzk axis system. Geometric constraints may be for­mulated by writing the position vector of P in two different ways One wayis by the flexbeam tip through the pitch arm:

P/=(+Uk)11lk kkk

P / 0 r = (I + + Vkf2 k + Wkl3k + Xflx k + y1ny + zink (Bl)

Another way is by the point S and through the pitch link

r/0 k k krP/0 = Ulmlk + vln2 + wln 3 + (P0 + Pk)(sin Pk sin 6k, k + cos k sin 6k2 k

6k3k ) + cos (B2)

Here, all lengths are made dimensionless by £ and the pitch-link length isp0 (undeformed) + Pk (deformation) The angles Ik and 6k define the orien­tation of the pitch link, shown in figure 13. These equations yield geometricconstraints as follows:

kkk k_ _=

fx = 1 + uk + xT1 +y1 T21 + zT31 u1 - (P0 +pk)sin Pk sin 6k = 0

k +xTk + k + Tk -v )cos uk sin 6 k = 0 (B3)y k 1 12 Y1T2 2 1 32 PO+ k

k + kk + X Tk +

fz x 1T3 Y 1T2 3 z 1T -W - (P k

where f k is obtained by taking the dot product of (Bl) and (B2) with n1 k;fkis obtained by taking the dot product of (El) and (E2) with mk ko ianed by taking the dot product of (El) and (B2) with n3k 2 fz is

These geometric constraints form relationships between the geometric vari­ables and enable us to eliminate superfluous variables from the analysis.Meirovitch (ref. 13) has shown that the generalized forces due to the con-Straints are

68

+=A k + l~--k Zk k cLr = UkVk ,Sk k ,kP (B4)aqrk k afx k 3 fYk kfzk krqr 'r k'r

The xk , xyk zk are Lagrange multipliers and the aq k are the generalizedforces. Thus r

k = iaUk

aVk =xyk

zkM.k =

kxVx 12+y 2-x 2)+ ( 3 x32)k Z +k j k

\ k\kk k\()k' kk -k- jT1 ~Tk ­ zTk)+

1 zekT2 1 y \'32 Z 2 2) I\33 1 23

= x k

a xA (yITl + A A yTk zTk)(5

a"k (P0 + Pk)sin 6k()Yk sin "k- Cos I = 0

sinkoS + X kcos k Azk s = 0

=pk-xk sin Pk sin 6k -A cos Pk sin k - Cos 6 k=Kpk

I0 2

Here, a6k = a a - Pk 0 because the mass and inertia of the link are

neglected. From ajk = ask = aPk - KpPk = 0, the Lagrange multipliers, which

are the unknown reactions at the point P, are

xk = -KpPk sin Uk sin 6k

AYk = -KpPk cos Pk siln 6k (B6)

Azk = -pPk cos 6k

69

The relations (B6) and (B3) are six relations between the twelve variablesk .

k' 0k' ek, 1k' 6k, Pk, %Xk Xyk, and z Thus, the blade systemUk, vk' w' retains its six degrees of freedom. The scheme for calculating the equilibriumdeflections needs to be modified slightly. Given qi-, vi-, w-, X-, y-'- zp, andP0" and an estimate of u- V, w, C, , and 6 we can calculate v, 6, p equi­librium components of Pk: 6k, Pk, respectively, from (B3) and then X k, Xyk

and Xz from (B6). Now the total generalized forces including au, cv' etc.,from (B5), are known. Integration along the flexbeam will yield slopes anddeflections to be included in the minimization scheme, equation (84). Theiteration may include a variation in w , to achieve the desired pitch angleor thrust as an extra independent variale in the minimization scheme.

The perturbation equations are obtained by expressing equations (B3)-(B6)in perturbation from and eliminating Xe, IXs' AYe' Xys' Azc AZs ae, Us,

vc, Vs, Wc, and is. This operation yields

T T T[RTR]{XI + [RTCR]{X} + [R KR + D]{X} = 0 (B7)

where

[M] =-[MI + M]

[C] = [GI + CA + CDI

[K] = [KI + KS + Kg + K + KD ,

X} = Lc's'c,&s, pc,Ps,cc,cs,oc,os,ac,es,XY, x,y]T

R2 R1 0 (B8)

[R]=0 16 0

0 0 1 4

[14] = 4x4 identity matrix

[161 = 6x6 identity matrix

ORIGINAL PAGE IS

OF POOR QUALITY

70

R11 0 R12 0 R13 0

1 l 1

0 0 R. 0 R3

1 11 13

R2 1 0 R22 0 R23 0

[R a 1 1 = 1, 2 0 1 R21 10 R22 10 R23 1

R31 0 R32 0 R 33 0

R R1 32 2 R0I31 0 R32 0

R1I = XlT + YT +ZT32

R12 = c(xlTl3 + y1T2 3 + ZIT 3 3)

R13 = ZlT21 - YlT31

R11 = -(Xl + Y21 + zlT 3 1)

R22 = s(x IT1 3 + YIT23 + z1T 3 3) (B8) con­

= tin-R23 = zIT 2 2 - YlTued

R31 = 0

RI2 = -xlcs + s0 (Yls6 + ZlC0 )

Rl 3 = ZlIT23 - ylT33

11 = (P0 + p)cos v sin 6

R12 = (p0 + p)sln p cos 6

R2 = sin V sin 6 13

R21 = -(P 0 + p)sin p sin 6

22 (P0 + p)cos p cos6

R23 cos V sin 6

71

R2= 031

R 2 =-(P32 (Po+pOsin6

R = cos 6331

[D]=l 22

DI 0 0 D3 0D 1 2

0 U 0 U D1 2.1

D1210 D13

U 0 U 0D2322[D

Symmetric D22 0 D23I

D33 0

I

D33 B8)

con-D = (p0 + P)(Kpp + Az cos 6) tin­

ued

D = 0

1DU = 0

22= (Po +P

U1 =0oD23 0

D 33 = KP

1D2 = -AxR1 +yR OX m11 x 21 y 11 uvAL PAG4i

12 A 2 + X RI ~i QAJX 2 y 12

U2 = -xRI +AXR 1 13 X 23 y 13

D22 = -(Xc + A s )R32 + X (c1R2+ s4 R 2z 1 22)

72

-3 -(Axc + Ays33s z (cr$j t S0213)

3= x(ylT 2 1

xxk = X x + xk

Xyk y=k y + yk

+ z1T3 1) + Xy(T 2 2 + z1T 32) + Az(y1T2 3 t ziT 3 3)

k x +xk

b %Xc~s x c

p8)

'd

As =

b xxk sln k

73

APPENDIX C

SNUBBER AND PITCH ARM WITH FLEXIBLE PITCH LINK AND SNUBBER CONNECTION

For the case of a pitch arm and snubber as in the Sikorsky UTTAS tailrotor (ref. 6), the geometry may be treated as two flexible pitch links withtwo pitch arms and two different sets of geometric variables* ul, v1 , wl, x1 ,Y1, Z1 , and p01; and u2, Y2, w2, x2, Y2, z2, and po2 . The two dimensionlessflexibilities are K, and K, and the deflections, analogous to appendix B,are Plk, 6 1k, p1k and P2k, 52k, p2k. A typical configuration of this type isshown in figure 6.

The mathematical development of the generalized forces and the iterativeequilibrium solution is identical with that of appendix B except that thereis a respective contribution from both pitch arms.

fk k k k k k kfX =+uk ul+xlTi+ylT2 1 +z1 T3 1 - (p +P )sin p sin 61 = 0- 1

k k k k k k kx = +Uk-+ - )sin V2 sin 62 =02 u2 +x 2T1 1 +y 2T2 1 +z 2T3 1 (p0 +P 2

etc

X k kauk xl X2

etc

z l iY T 3 1 - xa k =Xl(xyTII- XxjTI2)+YI(XI T2 I- AxiT 2 2 ) + lT 32 )=k k k z 1 X IT (C)

k k kk kk k k kk kk+ X2(y2T1 1 - X2T 2 ) + 2 (y2T 2 1 - x2T 2 2 ) + z2T31 -xT )32

etck =sn k kc k

XK -sin p sp = 1, 2

etc.

Therefore, the equilibrium scheme is altered in a way similar to that ofappendix B except that both pitch arms contribute. One pitch link will, ofcourse, change the pitch angle where the other one remains fixed as a snubber.

The perturbation equations are given by

[M]{x} + [C]lf} + [K + Q]{x} = 0 (C2)

74 ORIGINAL PAGE IS OF POOR QUALYL,

where

[Q] = Q1+Q2

000

F> S+ aF 0 1 5S( 0 0 0 0 0

ST

[Q 1] = [D2 ]i = 1, 2

where the subscript i on the matrix [D2 ] refers to subscripting all the

parameters in [D2] sub i (xl, yl, zi, Axi, ' z., etc.) and similarly for

the following

[Sl] = -[Ri] l = 1, 2

[El] = [A1 I

1A ] = 1, 2

(C4)

[Al] = [R2] = 1, 21

[C3] = [Di]ll,= 1, 2

See appendix B for the definitions of the matrices on the right-hand side of

equation (C4).

75

APPENDIX D

STRUCTURAL DAMPING COEFFICIENTS

In section 5.5, the blade and body structural damping coefficients areused in the analysis. We first consider the rotor blade for £ = 0 in vacuoIt is desirable to specify a structural damping ratio isolated blade chordwisemotion. The single blade motion is given by

Fl2 Ti1 [lI2B 01Dl22+0 ++F -6B1 Likxb W o cjjJL6B 4jj4=](l

where cc is unknown. We wish to choose c so that the motion is criticallydamped and then multiply by q., a dimensionless number given by the struc­tural damping ratio. For critically damped motion we require that the eigen­values of (Dl) be a pair of complex conjugates -a ± i and a pair of negativereal numbers -a, -a. The characteristic equation for such roots is

2 ) s4+ 2(a+a)s3+ [(a+G)2+2aa+O2]s2+ 2[aa(a+c)+a2]s+a2(G2+ = 0 (D2)

The characteristic equation for equation (Dl) is

3 2S4 + P3cCsS + p2s + Plccs + P= 0 (D3)

where

= PlBP0

12B i 21 = (l - rnxb 2 )

P 2 =) p1 (1 + FalXb + 2 ) D

1 = 1 ­

(nxb

When the coefficients of equations (D2) and (D) are equated, it can be shownthat

3a4 + P2a2 - P 0

cc = 2p 3 (D5)C ~ 2a p3

76 ORIGINAL PAGE 1$ OF POOR QUAITI

2where a is the single real, positive root of

/3PI~l \ /PlP lP2 a 6 + a4 3 p + a 2 -~p3p) p (6

(P a 3

-

It is difficult to achieve an accurate numerical solution, for small 2, toequations (D5) and (DO). An asymptotic expansion, however, presents no prob­lem numerically and yields a simple formula accurate to second order in £

cC= rB + mZxb + ! (l 1 ixlhI2) + 0(a23)] (7

which gives accuracy within 1% for £ < 0.27.

The structural damping value is thus

c= C + fZxb + -- (1 3

where typically 0.005 < :50 03

For the body motion, we choose structural damping coefficients as ifX, Y, ox, and y motions were uncoupled

cx = 2 13,131MX 1 3

=n~ S I cy = 2% K1 5 14 ,1

c4x = K5,15M15,15

KS= 2m 4616 Coy 2noy 16,16M16,16

Here nX' fy, no., and roy are the body structural damping coefficients,

typically 0.005 : nx, fy, flox, noy 0.05

77

REFERENCES

1 Donham, R. E ; Cardinale, S V., and Sachs, I. B.: Ground and Air Reson­

ance Characteristics of a Soft Inplane Rigid Rotor System. J. AmericanHelicopter Soc., vol. 14, no. 4, Oct. 1969.

2. Lytwyn, R. T., Miao, W.; and Woitsch, W : Airborne and Ground Resonance

of Hingeless Rotors. 26th Annual Forum of the American Helicopter

Society, Washington, D.C , preprint no. 414, June 1970.

3. Miao, W.; and Huber, H. B.. Rotor Aeroelasticity Coupled with Helicopter

Body Motion. NASA SP-352, Feb. 1974, pp. 137-146.

4 Young, Maurice I.; Bailey, David J ; and Hirschbein, Murray S • Open and

Closed Loop Stability of Hingeless Rotor Helicopter Air and GroundResonance. NASA SP-352, Feb. 1974, pp. 205-218.

5. Bielawa, Richard L ; Cheney, Marvin C., Jr.. and Novak, Richard C.Investigation of a Bearingless Helicopter Rotor Concept Having a Com­posite Primary Structure. NASA CR-2637, 1976.

6. Fenaughty, Ronald R.; and Noehren, William L. Composite BearinglessTail Rotor for UTTAS. 32nd Annual Form of the American HelicopterSociety, Washington, D.C , preprint no. 1084, May 1976.

7 Harvey, K. W.: Aeroelastic Analysis of a Bearingless Rotor AmericanHelicopter Society Symposium on Rotor Technology, Essington, Pa.,Aug. 1976

8. Shaw, John, Jr., and Edwards, W Thomas The YUH-61A (UTTAS) Tail Rotor

The Development of a Stiff Inplane Bearingless Flexstrap Design.33rd Annual Forum of the American Helicopter Society, Washington, D.Cpreprint no. 77 33-32, May 1977.

9. Harris, Franklin D., Cancro, Patrick A.; Dixon, Peter G C * The Bear­ingless Main Rotor. Third European Rotorcraft and Powered Lift Air­craft Forum, Aix-en-Provence, France, paper no. 4, Sept. 7-9, 1977.

10 Bielawa, Richard L. Aeroelastic Analysis for Helicopter Rotor Bladeswith Time Variable, Nonlinear Structural Twist and Multiple StructuralRedundancy - Mathematical Derivation and Program User's Manual.NASA CR-2638, 1976.

11. Johnson, W : Aeroelastic Analysis for Rotorcraft in Flight or in a Wind

Tunnel. NASA TN D-8515, 1977.

12. Kane, T. R.: Dynamics. Holt, Rinehart, and Winston, New York, New York,

1968.

13. Meirovitch, Leonard. Methods of Analytical Dynamics. McGraw Hill, New York, New York, 1970. ?\

78

oy voO

14. Ormaston, Robert A.; and Hodges, Dewey H. Linear Flap-Lag Dynamics ofHingeless Helicopter Rotor Blades in Hover. J American HelicopterSoc , vol. 17, no. 2, April 1972, pp. 2-14.

15. Greenberg, J. Mayo: Airfoil in Sinusoidal Motion in a Pulsating Stream.NACA TN 1326, 1947.

16. Hohenemser, K. H., and Yin, S. K.: Some Applications of the Method ofMultiblade Coordinates. J. American Helicopter Soc., vol. 17, no. 3,July 1972, pp. 3-12.

17. Hodges, D. H.; and Dowell, E H Nonlinear Equations of Motion for theElastic Bending and Torsion of Twisted Nonuniform Rotor Blades.NASA TN D-7818, 1974

18. Brown, K. M., and Dennis, J. E. Derivative Free Analogues of theLevenberg-Marquardt and Gauss Algorithms for Nonlinear Least SquaresApproximations. Numerische Mathematik, vol. 18, pp. 289-297, 1972.

79

N...­k N AI RCRAFT

CENTER

BODY­z" c- FIX ED

AXES

e BODY MASSeI CENTER 4y Y N

SPACE-FIXEDe2

e3 AXES Z, Nc

Figure 1 - Rotorcraft fuselage model.

ORIGINAL PAGE IS OF POOR QUALITY

80

Kx K¥ Ky K×

AIRCRAFT REFERENCE

(a) TOP VIEW ~CENTER.

K z ONE SIDE OF THE AIRCRAFT"7_ K 7_

IS SHOWN, SPRING sYSTEM

LATERALLY SYMMETRIC

(b) SIDE VIEW

Landing gear geometry.Figure 2 ­

81

BLADE CENTER OF MASS

Nk AERODYNAMIC CENTER AXIS

TWIST xJ XaC AXIS

a

FLEXBEAMconfiguration.

bladeRotor3.-

Figure

HUB

kNAL IG

BO AQ OF p0O

L

~FLEXBEAM

HUB

'MOTIONPITCH CONRO'I

PITCH ARM

Figure 5.- Cantilever pitch arm, Case III.

FLEXBEAM

SNUBBER LINKAGE

HUBSNUBBER ARMPITCH *INK-

PITCH CONTROL

PITCH ARM

Figure 6.- Cantilever pitch arm with snubber, Case IV.

83

Nk N 3

N 2­

3 n2 nz nk Nz <k -

Nk

Of -- -fN nx Nk

BLADE-FIXED AXES

Figure 7.- Rotating rotor blade coordinate systems.

k n3

k

nk

2 k k

nfl33

k nz

k

k

nk 1

kk nx

Ok

knk

x

Figure 8.- Sequence of angles describing blade orientation.

84

C9f

x

kN N k

z

z

-UpT

-u

LNC

U

loading.of aerodynamic

of componentsOrientation10.-Figure

5 ORIOINAi, PAGE B oF PooR QUAUTYJ

V

Figure 11.- Longitudinal plane of body-spring configuration.

86

Y, (t +)

(a) TOP VIEW

wT+2 z (t +)

PITCH LINK, AXIAL STIFFNESS Kp

S (b) REAR VIEW

Figure 12.- Detailed schematic pitch link configuration.

87tNAW PAGE M

A' DOOR QTJALTX

87

1k P

Bk

n2n

n1

Figure 13.- Angles Pk and 6k describing pitch link orientation.

88

1 Report No 2 Government Accession No 3 Recipient's Catalog No NASA TM-78459

4 Title and Subtitle 6 Report DateAEROMECHANICAL STABILITY OF HELICOPTERS WITH A BEARINGLESS MAIN ROTOR - PART I: EQUATIONS OF 6 Performing Organization Code MOTION

7 Author(s) 8 Performing Organization Report No Dewey H. Hodges A-7301

10 Work Unit No9 Performing Organization Name and AddressAmes Research Center, NASA, and Aeromechanics Q9-21-O1-9O-1Laboratory, U.S. Army Aviation R&D Command, Ames 11 Contract or Grant NoResearch Center, Moffett Field, Calif. 94035

13 Type of Report and Period Covered12 Sponsoring Agency Name and Address National Aeronautics andSpace Administration, Washington, D.C. 20546, and Technical Memorandum

U.S. Army Aviation R&D Command, Ames Research 14 Sponsoring Agency Code Center, Moffett Field, Calif. 94035

15 Supplementary Notes

16 Abstract

Equations of motion for a coupled rotor-body system are derived forthe purpose of studying air and ground resonance characteristics of helicop­ters that have bearingless main rotors. For the fuselage, only four rigidbody degrees of freedom are considered; longitudinal and lateral transla­tions, pitch, and roll. The rotor is assumed to consist of three or morerigid blades. Each blade is joined to the hub by means of a flexible beamsegment (flexbeam or strap). Pitch change is accomplished by twisting theflexbeam with the pitch-control system, the characteristics of which arevariable. Thus, the analysis is capable of implicitly treating aeroelastmccouplings generated by the flexbeam elastic deflections, the pitch-controlsystem, and the angular offsets of the blade and flexbeam. The linearizedequations are written in the nonrotating system retaining only the cyclicrotor modes; thus they comprise a system of homogeneous ordinary differentialequations with constant coefficients. All contributions to the linearizedperturbation equations from inertia, gravity, quasi-steady aerodynamics, andthe flexbeam equilibrium deflections are retained exactly. Part II describesa computer prograi based on these equations of motion.

17 Key Words (Suggested by Author(s)) 18 Distribution Statement Bearingless rotor, Helicopter rotor, UnlimitedStructural dynamics,Beam (nonlinear bending torsion),Aeroelasticity, STAR Category - 02Air/ground resonance

19 Security Classif (ofthis report) 20 Security Classif (of this page) 21 No of Pages 22 Price*

Unclassified Unclassified 104 $5.50

-For sale by the National Technical tnformation Service, Springfield Virginia 22161

ORIGINAL PAGE IS OF POOR QIJALTY