NASATechnicalMemorandum102267 USAAVSCOMTechnicalMemorandum90-A-004

A Flight-Dynamic Helicopter

Mathematical Model with a Single

Flap-Lag-Torsion Main Rotor

M. D. Takahashi

February 1990

National Aeronautics and

Space Administration

UvSIAATRMIo _

SYSTEMS COMMAND

AVIATION RESEARCH AND

TECHNOLOGY ACTIVITY

https://ntrs.nasa.gov/search.jsp?R=19910009728 2019-02-15T11:07:39+00:00Z

NASA Technical Memorandum 102267 USAAVSCOM Technical Memorandum 90-A-004

A Flight-Dynamic Helicopter

Mathematical Model with a Single

Flap-Lag-Torsion Main Rotor

M. D. Takahashi, Aeroflightdynamics Directorate, U.S. Army Aviation Research and

Technology Activity, Ames Research Center, Moffett Field, California

February 1990

National Aeronautics and

Space Administration

Ames Research Center

Moffett Field, California 94035-1000

AVIATION RESEARCH AND

TE C HNO LOGY ACTIV ITY

MOFFETT FIELD. CA 94035-1099

CONTENTS

NOMENCLATURE ...................................................................

SUMMARY ..........................................................................

INTRODUCTION ....................................................................

HELICOPTER EQUATIONS OF MOTION ............................................. 2

Fuselage ......................................................................... 2

Main Rotor ....................................................................... 6

Inflow ........................................................................... 11

Tail Surfaces ..................................................................... 13

Tail Rotor ........................................................................ 14

Main-Rotor Downwash ............................................................ 15

Mass Matrix ...................................................................... 17

Multiblade Coordinate Transformation ............................................. 21

OPERATIONS ...................................................................... 22

Basic Setup ...................................................................... 22

Initialization ...................................................................... 24

Trim ............................................................................. 24

Integration ....................................................................... 26

Linearization ..................................................................... 28

Page

V

1

1

RESULTS ........................................................................... 29

CONCLUDING REMARKS .......................................................... 34

REFERENCES ...................................................................... 35

TABLES ............................................................................ 37

FIGURES ........................................................................... 54

PRi_CEDIi_G PAGE I_LANK NOT FILMED

,o,

111

NOMENCLATURE

A vector is denoted with an over-arrow (e.g., the position vector gab), and the column matrix representation of a vector, or any other type of column matrix, is represented with an underbar (e.g., _).

Points in the system are designated by lowercase letters, so the position vector Tab in the above example

is read "the position from point a to point b." Coordinate frames are designated by capital letters (e.g.,

d:A is the z unit vector in the A system). Coordinate-frame rotation rates are represented as _F/I,

which is read as "the rotation rate of frame F relative to frame I." Rotation vectors are always resolved

in the frame of the first subscript. Nondimensional quantities are represented with an overbar and are

nondimensionalized by the blade mass, rotor radius, and nominal rotor rotational rate.The sine and

cosine functions are abbreviated to S O and C O, where the subscript is the argument. In the following

list, the abbreviation DOF is used for degrees of freedom.

a

asnd

aTR

aOTs, boTs, aTS, bTS

b

bTR

Coo,CD , CD , CO3,Co,, Cos

CFz,CFu,CFz

CL s , CL I , CL 2

CT, CM, CL

CTS

Cu,Cz

CorR, ClrR

d 0, d l, d2

e,f

eTR

f,f,]

Lq

Fx,Fy,Fz

9

h

iAero

if(ni:)

ifeq(nfeq)

ii(ni ])

i](ni])

iu(niu)

iy(ni u)

io(nif_)

ix(nix)

blade-section lift-curve slope

speed of sound

tail-rotor lift-curve slope

tail-surface dynamic pressure loss angles

blade-section semichord

tail-rotor semichord

tail-surface-drag coefficient values at break angles

fuselage-mount damping constants

tail-surface-lift coefficient values at break angles

main-rotor thrust, pitch, and roll coefficients

tail-surface dynamic pressure loss factor

hinge damping constants

tail-rotor drag function constants

blade-section drag function constants

first and second offsets

tail-rotor flapping-hinge offset

fixed-DOF array and derivatives

fixed equations

applied-forces components at mount in F system

gravity constant

shaft length

apparent-mass-term flag

fixed DOF indices (number selected)

fixed equation indices (number selected)

first-derivative fixed DOF indices (number selected)

second-derivative fixed DOF indices (number selected)

fixed control indices (number selected)

feedback fixed-output indices (number selected)

auxiliary fixed-output indices (number selected)

feedback X-output indices (number selected)

{MIII.,._ IN11_N'IrlON,l_l.l,,¥IlI:AII_

pR_.'C_.DI:.':G PAGE _LA,"IK NOT FILIVIED
iN(niR)

il(nil)

IbT R

IB

IF

IR

jr(njr)

j÷(nj÷)

jw(nj_)

j&(nj_)

Jo(Wo)

K(nK)

KbTR

KF_, KFu, KF:

K ,Ky, Kz

KITR

MB

_IBTR

2l,IF

AIR

toO, ml

nAbld

'nA yu s

n b

'nbT R

nf

T_hLI N

_hTRM

TLr

TLs

72 u

7%u A _ 72U C _1%Up _ 7_U S _ 1%U G

nw

_z A _ TlX C _ 'nXp _ l_X S _ ?2X G

ny

Try A _lty C _nyp _rty S _ny G

TlO_rnid _ TtO_en d

72_mid, n_en d

no

nl

auxiliary X-output indices (number selected)

first-order variable/equation indices (number selected)

tail-rotor flap inertia about flapping hinge

blade inertia matrix

fuselage inertia matrix

rotor-hub inertia matrix

rotor DOF/equation indices (number selected)

first-derivative rotor DOF indices (number selected)

feedback-rotor output indices (number selected)

auxiliary-rotor output indices (number selected)

rotor control indices (number selected)

rotor control blade indices (number selected)

tail-rotor flap-spring constant

mount spring constants

hinge spring constants

tail-rotor tangent of _3

inflow matrices

blade mass

tail-rotor blade mass

fuselage mass

rotor-hub mass

blade-section moment function constants

number of blade accelerometer measurements

number of fuselage accelerometer measurements

number of blades

number of blades in tail rotor

number of fixed DOF

number of harmonics in periodic system linear coefficient matrices

number of harmonics in trim blade equilibrium solution

number of rotor blade DOF

number of empennage surfaces

number of fixed inputs

number of controls in blocks A, C, R S, and G

number of rotor outputs

number of states in blocks A, C, R S, and G

number of fixed outputs

number of outputs in blocks A, C, R S, and G

number of divisions in middle/end region of a for surface downwash

table

number of divisions in middle/end region of/3 for surface downwash

table

number of rotor blade inputs

number of augmented states

vi
?"co

rl,r2

_r,r_',£

T_eq

R

RTR

R___ T = [Rmz Rm v Rmz] T

Sf

STa, BLa, WLa

STc, B Lc, W Lc

STm, B Lm , W Lm

STr, BLr, WLr

STs, BLs, WLs

STt , B Lt , W Lt

STS

tfINT

TRBRK, TRLoss

r_, T_

U_

XA

Xb, Yb, Zb

XbT R

Xl,X2

X--eq

X__l

O_D1, O_D2, O_D3, O_D4

O_Ls_ O_L1, O_L2

O_LOBND, O_LO, Olup, O_UPBN D

/3LO B N D ' /_LO, 3UP,/3UPBND

9p,

ADW

A!,A] -

At_, At_"

A_U

Axl

A0_

_e

Os, c)s

OtTR

Ozo, Oyo, Ozo

trim Fourier coefficients matrix [(2nhrnM + 1) by nr] of the blade

DOF

integration limits in downwash table calculation

blade DOF arrays and derivatives

rotor equations

rotor radius

tail-rotor radius

translation DOF of fuselage

reference area of fuselage aerodynamic forces

aerodynamic force position a in station-buttline-waterline coordinate

frame

fuselage center-of-mass position c

translational DOF position m

tail-rotor position

tail-surface position

shaft tilt position

tail-surface reference area

final time of response

tail-rotor loss function constants

blade linear twist constants

fixed input array

blade-section torsion axis distance behind 1/4 chord

blade center-of-gravity position components in B system

tail-rotor center of mass from hinge offset

aerodynamic forces integrated from x l to x2 along _B

first-order equations

first-order augmented state array

tail-surface-drag break angles

tail-surface-lift break angles

downwash table a ranges

downwash table/3 ranges

flap, lag, and torsion blade DOF

hinge offset angles

nearest a to wake layer for surface downwash table calculations

finite-difference perturbation for fixed DOF and derivative

finite difference perturbation for rotor DOF and derivative

finite-difference perturbation for fixed inputs

finite-difference perturbation for augmented states

finite-difference perturbation for rotor-blade inputs

rotor inputs

shaft tilt in pitch (positive back) and roll (positive right)

tail-rotor twist per unit length

zero moment angles at mount

vii
O_T = [Ox,Oy,Oz] T

A

#

PA

Px , Py

O"x ; O_y

_f

_TR

For position vector _:

T

!

_(F)

÷

fuselage Euler angles

inflow at main rotor

advance ratio

air density

tail-rotor orientation angles

tail-surface orientation angles

rotor DOE _k = _ + _27r

rotor fundamental frequency

nominal rotor rotation rate

tail-rotor rotation rate

time derivative

time derivative of a vector relative to vector frame

nondimensional time derivative

a vector resolved in a particular frame

special matrix form of a vector, i.e.,

0 rz -ry

= --rz 0 rx

ry -rx 0

°°°

VII1
SUMMARY

A mathematical model of a helicopter system with a single main rotor that includes rigid, hingerestrained rotor blades with flap, lag, and torsion degrees of freedom is described. The model allows

several hinge sequences and two offsets in the hinges. Quasi-steady Greenberg theory is used to

calculate the blade-section aerodynamic forces, and inflow effects are accounted for by using a threestate nonlinear dynamic inflow model. The motion of the rigid fuselage is defined by six degrees of

freedom, and an optional rotor rpm degree of freedom is available. Empennage surfaces and the tail

rotor are modeled, and the effect of main-rotor downwash on these elements is included. Model trim,

linearization, and time-integration operations are described and can be applied to a subset of the model

in the rotating or nonrotating coordinate frame. A preliminary validation of the model is made by

comparing its results with those of other analytical and experimental studies. This publication presents

the results of research completed in November 1989.

INTRODUCTION

Future requirements for helicopter flight performance place greater emphasis on high-bandwidth

flight-control systems than current requirements. Low-order rigid-body models of helicopter systems can

be used successfully to design feedback control systems, but these models impose bandwidth limitations

on the controller, making them unsuitable for the design and evaluation of high-bandwidth systems

(ref. 1). To meet the new requirements, higher order helicopter models must be used in the design of

the flight-control systems. The higher order effects of primary interest are those of the rotor dynamics

and the rotor inflow. In addition to capturing the blade-flap and lag dynamics, the effect of torsional

dynamics on the control design problem needs to be understood. This last item is relevent to rotor

designs using hingeless and bearingless rotors and to advanced concepts, such as servo-flap control,

which require torsionally soft rotor blades (ref. 2).

This report documents a higher order helicopter mathematical model created to fill these needs. The

helicopter model is described in detail in the next section. The rotor consists of rigid blades with flap,

lag, torsion, and pitch motions; and with linear hinge springs and dampers. This gives the model the

rotor dynamics of interest and allows for approximate modeling of hingeless rotor systems. The flap (f),

lag (I), torsion (t), and pitch (p) hinge motions can have the order flpt, fplt, pfit, lfpt, lpft, or plft (only

flpt and lfpt are currently available), and two hinge offsets are available to model various articulated

configurations. The blade-section aerodynamics are linear, and the quasi-steady Greenberg model is used

(ref. 3). Unsteady inflow effects are included using the three-state nonlinear Pitt/Peters dynamic inflow

model (ref. 4). An rpm degree of freedom (DOF) of the main rotor is also available, making analysis

of engine/rotor interaction possible. The rigid fuselage model has six degrees of freedom and includes

an equivalent-drag-area aerodynamic model, a tail-rotor model, and an empennage-surface model. The

empennage-surface models have main-rotor downwash effects that are calculated from the flat-wake

model described in chapter 2 of reference 5. The tail rotor is the quasi-static flapping model described

in reference 6. For greater flexibility, the spring/damper model, the blade-section aerodynamic model,

and the fuselage aerodynamic model can be replaced with user-defined models.

Following the description of the model is the Operations section, which details the operations that

are performed on the model. There are three operations: trim, linearization, and time-integration. Trim
placesthe helicoptersystemin someuser-definedflight conditionby satisfyingthe systemequationsof

motion. After trimming, the modelmay be linearizedaboutthe trim condition,or the model may be

initializedinthetrim conditionandthentime-integrated.Linearizationusesnonrotatingrotorcoordinates

througha multibladecoordinatetransformation(MBCT) (ref. 7). In forwardflight, the periodicsystem

matricescan be extractedfrom this linearizationprocess. The time-integrationusesrotating blade

coordinatesandallows for arbitraryinputsto the modelcontrols.Additionally,the integrationprocess

hasbeencast in a generalblock form, allowing the introductionof user-specifiedactuator,sensor,

feedback,and precompensatordynamicsto the system.

In the Resultssection,a preliminary validationof the helicoptermodel is madeby comparing

resultswith thosefrom otheranalyticalandexperimentalstudies.Therotor modelresultsarecompared

with trimmedrotor-responseresults,from reference8, of an isolatedrotor. Rotor/fuselageresultsare

comparedwith theexperimentalfrequencyanddampingdatapresentedin reference9. Thefull helicopter

model is validatedby comparingtrim and time-historydataextractedfrom flight dataon a UH-60A

helicopter(ref. 10). Additionally,a comparisonwith thequasi-staticeight-statelinearizedmodelfrom

the GEN HEL program(refs. 11,12)is made.

HELICOPTER EQUATIONS OF MOTION

Fuselage

The fuselage is modeled as a rigid body whose motion is defined by the six degrees of freedom

shown in figure 1. The fuselage coordinate frame F is body fixed, with its origin at point m. The

location of point m is chosen by the user. The body translational motion is defined by the components

of the inertial position vector g'im, and body rotation is defined by the Euler angles. Other relevant points

on the body are shown in figure 2. The positions of these points are chosen by the user in the body-fixed

coordinate frame defined by the station, the buttline, and the waterline. The forces and moments at

m result from mounting restraints, which can be set to simulate any mounting condition, such as in a

wind-tunnel test. Point c is the center of mass of the fuselage, where inertial and gravitational forces

are resolved, and point a is where fuselage aerodynamic forces are applied. The ns empennage surfaces

produce resultant forces at the sj points, and the resultant tail-rotor force is at point r. There are nAil, s

points designated as Yn, where accelerations are output measurements. Point h is the attachment point

of the main-rotor hub, which can be set at tilt angles about point t. These tilt angles are defined by

the transformation, shown in figure 3, which tips the hub coordinate frame H by constant pitch and roll

angles, Os and Cs.

The equations of motion for the fuselage are formed by summing the forces created by the main

rotor, fuselage inertia and gravity, fuselage aerodynamics, the fuselage mount, the tail rotor, and the tail

surfaces. This gives the vector equation

ns

OH+_)IG+Q,A+Q,M+OR+_OSj =0 (1)

J

Summing the respective moments about point m gives

f_H q- f_IG + J_A + ff"%'I + LR + iS = 0 (2)

2
where

= JH+ (' mt+ 'th) × 0n

f"IG = :I + (r'mc x 0IG)

LA=fA+(FmaXOA)

n8

J

(3)

(4)

(5)

(6)

(7)

The final fuselage equations are resolved such that they correspond to the fuselage generalized coordinates. That is, if a Lagrangian approach had been used, the same fuselage equations would have resulted.

This resolution of the equations facilitates the extraction of the system mass matrix, which is discussed

in a subsequent section. The definition of the equations and their location in the fuselage-equation

vector f--eq are

feq(1) = _ moments about XF (8)

feq(2) = Z moments about Y0y (9)

feq(3) = _ moments about z0z (10)

feq(4) = _ forces in xl direction (11)

feq(5) = _ forces in _)i direction (12)

feq(6) = _ forces in zI direction (13)

Using the position vectors and coordinate frames as defined, the velocities and accelerations can

be determined. At the main rotor hub, the angular velocity and acceleration are given by

_H/I = _H/F + _F/I (14)

?ZF/X = _-"F/X (15)

_H/I = _tH/I = _tH/F + _H/I x WH/F + _ZF/I (16)

Taking advantage of the constant components in the position vectors, the velocities at selected points

are

flit = rim q- _F/I × _mt (17)

rzh = r_t +_H/I × r'th (18)

risj = rim + _F/I x r'msj (19)
r3_= r3,n+ _F/I × rm_

_a = _3_+ _F/I ×rma

;3c= #_ + ZF/I × ¢_c

#_. = #i., + Zr/r ×¢,ny

and the acclerations at selected points become

Fit =t-'i,,, +w'r/I x Fret +gr/I x (gF/I x Fret)

_ih=_'it+__/H/i XFth -FdH/l x (OH I Xrth )

_'_c= _-_m+ J'y/z × ¢_c + Zr/z × (_F/Z × _mc)

_y_ = r3m + J'r/I × rm_ + _Y/I × (_F/I × _'my_)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

(27)

The forces and moments of the fuselage are discussed separately, beginning with the inertial and

gravity effects. At the fuselage center of gravity, the inertial force and moment vectors are

,_ = --_F" ;°_F/I- WF/I x _F" WF/I) (29)

where the inertia tensor is defined in the F system.

The fuselage aerodynamic effects are based in the wind-axis coordinate frame W defined in figure 4.

The velocity of the air relative to point a is

Va = VG - Fia (30)

which also defines the direction of the J:W component of the wind-axis frame. The angles af and ,3f

are defined from the relative velocity as

o_), tan-! (-V_a'Z_F) = -r < o_F <_ 7r (3l)

\--Va

tan-l ( -17a" YF (32)

_y k

The fuselage aerodynamic force and moment are

1

QA = D:?W + Y_]W + Lzw-" "_PlIValI2Sf (CDf2gIV -F Cyf_lVff-+-CLfZIV) (33)

JA = -R:rW + A-f_)w- N_?W = pllVal 2 of _,3/2 (-CRi:hv + C,;fs_w-C,xy_w) (34)

4
which canbeconvertedto the F framevia thetransformationTFW, which is defined in figure 4. The

force and moment coefficients are functions of af and _f, but the default fuselage-aerodynamics sets

all force coefficients to zero except CDI, which is set to unity, making the fuselage reference area Sf

the equivalent drag area.

The fuselage mount moments are modeled as linear nonorthogonal springs, and the mount forces

are treated as given quantities. These forces can be used during trim to satisfy a constraint condition,

such as setting a certain amount of thrust. The parameters defining the mount effects can be set to zero

for free-flight analysis. The force and moment expressions for the mount are given by

where

Mx = -KF_ (Oz - Ozo) - CF_Oz

.Mz = -KF z (Oz - Ozo) - CFzbz

QM = + Fyg +

The measurement outputs from the fuselage are acceleration, velocity, and angular rates.

accelerometer outputs are expressed as

Y(j)= (r-iuj - g_I) " I1 11 j= l'2"'"nAl_=

fj = fzjYCF + fyj_lF + fzjz'F

(35)

(36)

(37)

(38)

(39)

The nAlus

(40)

(41)

where the direction of the measurement is selected through the components of ._. The velocity of the

fuselage center of mass is resolved in the fuselage fixed frame and is available as a measurement, as

are the fuselage angular rates. The components of the velocity and angular rate vectors are stored in y

according to

(42)

(43)

(44)

(45)

(46)

(47)

y(nAl_ + l) = =" ric. 3cF

y (nAlus + 2 ) = ric- • _IF

y (nAy_s + 3) = :"ric" ZF

y (hA.f,., s + 4) = CSF/I " 3CF

y (nAyus + 5) -- _F/I" _IF

y (nAyus + 6) = gF/I" ZF

5
Main Rotor

Fixed to the main rotor are n b coordinate frames, one for each blade. Figure 5 shows the rotor

coordinate frame R for the kth blade, in which several additional blade coordinate frames are defined.

The additional frames are shown in figure 6. The first hinge is offset a distance e from the hub center,

and point e corresponds to the origin of the link coordinate frame L. The second hinge is offset a

distance f from point e. The origin of blade coordinate frame B is at point f and is fixed to the blade.

The significant points on the blade are b, at the blade center of gravity, and ,r, at a distance ,r along

the ?/3 axis. Also on the blade are 7_.4btd points designated Wn, which locate the blade accelerometer

measurements.

The hinge sequences that define the orientation of the blade coordinate frames can be divided into

two groups. The first group is flap-lag-pitch-torsion (flpt), flap-pitch-lag-torsion (fplt), and pitch-flaplag-torsion (pflt), in all of which the flap motion precedes the lag. The second group places lag before

flap in lag-flap-pitch-torsion (lfpt), lag-pitch-flap-torsion (lpft), and pitch-lag-flap-torsion (plft) hinge

sequences. Only the flpt and lfpt sequences are available; they are defined in figures 7 and 8. An

intermediate coordinate frame P, which is not shown in figure 6, is included in these figures. Frame

P allows the introduction of hinge effects such as precone and _53. The angle tip in figures 7 and 8 is

the precone angle, and (p is a cant in the flap hinge that geometrically introduces the _53 effect. By

definition, the variable _3 is not an Euler angle (ref. 13), but it is related to the cant Euler angle by

tan 63 = - tan (p cos tip for the flpt system and tan 63 = - tan ((p + (k) cos tip for the lfpt system. The

remaining hinge sequences are defined such that for the flap-before-lag sequences, flk rotates about the

coordinates axis -0.p; (k, about £L; and ¢_k, about .?/3. For the lag-before-flap sequences, (k rotates

about the axis _.p;/3 k, about --OL; and Ok, about _'B. The sequence of rotations of the blade angles is

summarized in table 1. The items in the flpt row and R to P column are read as follows: starting in

frame R, rotate through an angle of Ap about the negative OR axis; then rotate through an angle of (p

about the £3p axis, to the P system. Each element of the table is read similarly.

The forces on the main rotor are shown in figure 9 with only the kth blade depicted. A rigid body

representing the rotor hub mass contributes only inertia to the system. The link between points e and f

is massless, contributing no inertial forces to the system. Each blade contributes inertial, gravitational,

and aerodynamic forces and moments, which are related to the loads at hinges f and e by

ff f : -(ff-'lG + ffA) : --fiT (48)

3If : --(A_IIG + 3_4) : --3_T (49)

;, =;i (50)

The vectors fit and _QT are the sums of the blade inertial, gravitational, and aerodynamic forces and

moments at point f. The hub forces and moments of equations (1) and (2) for the main rotor are then

nb

k=l

(52)
rib

k=l

This gives the rotor equation

feq(7) = Tq + JH" "_R = moment about 5,R (54)

where Tq is the main-rotor torque. For the flpt, fplt, and pflt systems, the flap, lag, and torsion equations

are

_(k, _)_--[_, +_ +% ×_]. _ (55)

req(k,2) = [My + MT] " zL (56)

.,_(k,3)= [_?I+_] _ (57)

For the Ifpt, lpft, and plft systems, these equations are

..,(k,,) =- [,_ +._.]. _ (58)

req(k,2) = [!Qe + fliT + ?'e I x fiT]" zP (59)

req(k, 3) = [fill + fliT]. XB (60)

These rotor blade equations have been written such that they are equal to the ilk, (k, and _k equations

that would result from using a Lagrangian approach. The total aerodynamic forces and moments at the

hub center are

nb

ffFA = E (ffA) k (61)

k=l

nb

SZle[A = E [ I_A "q- (_he + ?'el) x FA] k (62)

k=l

The angular velocities and accelerations are

_R/I = _:R/H + J:n/I (63)

J:P/I "" _P/R + _R/I (64)

gL/I = Y:L/P + _P/I (65)

_:B/I = J:B/L + _TL/I (66)

_ZR/I = w"tR/I = w'tR/H + WR/I × WR/H + _H/I (67)

_Zp/I = °-'-'tP/I = w-'tP/R + _P/I x wP/R + '_'R/I (68)

_ZL/I = °-'-'L/I = W-'L/P + Y_L/I x _L/P + _ZP/I (69)

YZB/I = WDB/I = w'tB/L + J_B/I × J_B/L + _ZL/I (70)

7
For the bladeaerodynamiccalculations,the velocity of point x results from

r-_e = r'_h q- C_R/I x r'he

r'if = r'ie + GL/I x r'ef

• :_

r-ix = riy -t- WB/I x _fx

For the mass matrix calculation, the velocity of the blade center of mass is

rib -- rif q-WB/I x r'fb

(71)

(72)

(73)

The accelerations of the blade center of mass, the points Wn, and the point x along _B result from

• - o_

_'_s= ,'i_+ _o's/_r× e'#+ _s/z × (,zr_/z× e#)

riwn = rif q-JB/I

o_ ••

ri: =eiy + S'B/Z

(74)

(75)

(76)

(77)

(78)

(79)

The rotor hub center of mass is at point h, and the rotor hub orientation is referenced to the k = 1

rotor coordinate system. The inertial and gravitational effect of the rotor hub is

and the inertial moment is

(80)

RZ = -[_R" _aIR/I q-_R/I x ( IR " _R/!)] (81)

Likewise, for each blade, the inertial and gravitational force and moment are

;IG = MB (-rib + gz'z) (82)

= -[h, +<,/, ×(h, <,/,)] + ×

where the moment is about point f.

Blade hinge-spring restraining moments are of two types. The first type is nonorthogonal, and for

the flpt, fplt, and pflt sequences the moments are

-]Qe " YP = -Ky_k - Cy_k (84)

-_s _s= -g_¢k - c,4k (85)

-@' J:B= -K:,:¢k- cxck (86)
For the Ifpt, lpft, and plft sequences, the nonorthogonal moments are

(87)

(88)

(89)

The second type of spring restraint is orthogonal and is used to approximate hingeless rotor-blade

systems. These orthogonal moments are valid only for (1) no second offset (f = 0), (2) no cant angle

((p = 0), and (3) with the pitch inboard or outboard of the flap and lag degrees of freedom. For the

flpt and pflt sequences, the moments are

+[-_,_(-9_++_s_)-c_(-_ +_s.)] _

+[-_<:(_c_ + _ _)- c_(_c_ + '

(90)

For the lfpt and plft sequences, the moments are

(91)

The total blade aerodynamic effects are found by integrating the sectional aerodynamic forces and

moments along the span according to

where

PA= ff_ fA dx

1

(92)

x fA + rfiA] dx (93)

fA = fgcflC + f;.c_C (94)

r_ A -- rnkc_c C (95)

The integrations are carried out using Gaussian quadrature numerical integration. Blade-section forces

are resolved in a chord coordinate frame C shown in figure 10, which is rotated from the blade coordinate

frame B by a geometric twist. The linear twist function is defined by

OT(X ) -- TwxX +Twc (96)

9
The componentsof thesectionalforcesandmomentsareexpressedin termsof the angularvelocity and

accelerationof thechord sectionandof thecomponentsof the relativevelocity and accelerationof the

air massat the origin of the C frame. The lattertwo are

- - riz (97)

•. ..

Ac = VG - riz (98)

where the vector,_i is the induced air velocity whose derivative has been ignored in the air-mass acceleration. The induced air velocity results from the dynamic inflow model, which is described in a

subsequent section.

The sectional aerodynamic model is based on quasi-steady Greenberg theory, which is a Theodorsen

theory modified to account for lead-lag motions (refs. 3,14). The lift and moment about the torsion axis

are given by

L = pAab [VQ + P_ZAero (99)

where

(

= (J_ + Va)" - xab8 (102) P

The variable iAero is a switch to remove the apparent mass effects, which can be ignored when the

order of magnitude of the mass of the airfoil section is much greater than the order of magnitude of the

mass of the air it occupies (ref. 14, chap. 5). Drag and additional moments due to camber are added by

D = d o + d 1 aef f + d 2 a2ff (103)

:_fcam = rno + ml aeff (104)

Figure 11 shows the application of the sectional aerodynamic model in forward and reverse flow conditions. The variables in the Greenberg theory are interpreted as in reference 15, where the quantity

J_ + Va is interpreted as the normal air velocity at the torsion axis. The derivative of this quantity is then the derivative of the -_c component of the air velocity t7'c, which can be extracted from

-_ = (;'_ + _c/_ × f_c.

Blade measurements are stored in w__in the following manner.

acclerometer outputs are given by

•- bn

w [k+ ,_b(,_- 1)]= (_'iw.- g_z)" II_ II

-bn = bx.xB + by._lB + bz.ZB

For the kth blade, the nAbld

n = I, 2,..., nAbld (105)

(106)

10
wherethedirectionof the measurement is selected through the vector bn" Blade moments at the hinges

are also in w__.They are given by the following expressions when flap is before lag:

w [k -t-nbnAbtd ] = (]_I T -t- r'ef X fiT)" _)P (107)

w [k +nb (nmb,a + 2)] = MT" zB (109)

and by the following expressions when lag is before flap:

w [k + nbn4btd] = ]_T" !tL (110)

w [k + n b (nAbtd + 1)1 = (-MT + _ef x FT)" z. (lll)

Inflow

The rotor inflow is modeled using the three-state nonlinear Pitt/Peters model of reference 4, which

is applied in the actuator-disk coordinate frame A shown in figure 12. This frame is tipped to the left

by the sine of the flapping angle and tipped forward by the cosine of the flapping angle. The velocity

and acceleration of the air relative to the hub center are

I_h = 17_7- r"/h (113)

._ ..

Ah = VG - 7ih (114)

The advance ratio, the free-stream inflow, and the total average inflow can be found by using the velocity

vectors; they are

_/(I_h" ZA) 2q- (Vh" t)A) 2

/.z = R_ (115)

r_h" £'A (116)

Af s - Rf_

A = Afs + A0 (117)

where the total average inflow A is the sum of the free-stream inflow plus the constant induced-inflow

component, which is one state of the dynamic inflow model. The inflow vector for the blade aerodynamic

calculation, in terms of the inflow states, is

,_i = Jiz.4 : A0 -J- Jls_S_bk "1" "_A

(118)

11
whereAo, Als, andAlc arethe threeinflow statesthat definetheinflow in the A system. These inflow

states are governed by the first-order inflow equations

Xeq(3) Vlc

+ Vls - LI {.}CL

Vlc CM

(119)

The v's in equation (119) are coefficients that define the inflow in the same way that the A's define the

inflow in equation (118). However, the v's define the inflow in the D system, which is aligned with the

velocity in the disk plane. The v coefficients and the A coefficients are related by

v 0 ---

b'lS =

Vlc --

W0 --

*

/./is ---

,

Vlc =

),o

AlsC_h + AlcSflh

AlcC3h + AlsS3h

AO

f2

- ),xcWC_h + --_'Sflh + _ (--/_l'sSflh + .'_leO_h)

(120)

(121)

(122)

(123)

(124)

(125)

where the side-slip angle and rate are

/3h=tan-1 _ "_A

/3h "- (_h" "ZA)(Ah" t)A)- (.Ah" "_A)(Vh" _).4.)

(126)

(127)

The thrust, roll, and pitch coefficients in equation (119) are given, in terms of the main-rotor aerodynamic

forces and moments, by

SF a " _.A (128)

C T - 7rpAR4_.12

CL _.. (S,_'._A " OA) Sflh + (g'VIA " if:A) Cflh "_/foff (129)

7rpAR5f22

(ffMA " _la) 63 h -- (ifMa " X,A) SJh ,>..[off (130)

C M = 7rpA.R5O2

12
The matricesfor the Pitt/Petersmodelare

L I =

l 0

0 -4

ts_

U-:VT,-_,

0 (131)

mo(ma.ss ) 0 0

-16 0

0

-16

0 0

(132)

where

#2 + :_(_+ ,x0)

u = (133)

_/#2 4- A2

uT = _//.t 2 -k- A2 (134)

as = tan-I (_) (135)

The element m 0 takes on two possible values. One value corresponds to a zero-inflow boundary 128

condition at the hub center, mo(O ) = 75'7; the other value, too(l) = 37, does not enforce this condition.

Tail Surfaces

Up to five tail surfaces are available (the subscript representing each surface will be omitted in

the following discussion); they are oriented as in figure 13. Each surface can be tipped away from the

horizontal by an angle ax, and the pitch incidence can be adjusted through ay to define the zero point

of the input iTS. The velocity of the air relative to the surface is

= -- ris (136)

where dTS is the downwash from the main rotor. Each tail surface has its own downwash table that is

a function of the angles at which the air strikes the hub center, the magnitude of the air velocity, and

the thrust coefficient (i.e., dTS = CTfDWTs (av, fly, V_) _-s). The moment Js at the surface is zero,

but the force at the tail surface is

(_s 1

=_PA IIv, II2 STS{[CL(_=)S_,--CD(c_-)C_,] _

+ [-cL (_z)c.. - co (_)s..] S,}q_o,,SR (137)

13
where,for forward flow, az and S R are defined by

SR= az - as1 ftfor_,_<as<,- - _

and, for reverse flow, they are defined by

a z _ a --Tr /

SR = 0.8 )f°r'_<as<¢r

SR = 0.8 ff°r-rr<as <-

(138)

where

as = tan -1 " (139)

The lift and drag coefficients are functions of az and are defined by specifying the break points shown in

figures 14 and 15. The dynamic pressure loss due to fuselage blockage is approximated by the functions

qtoss- 1- (_,_ 2

\ "V_ /

Vs - CTSexp -2 aTS /

(140)

(141)

When the angles a/and ff are at aOT s and bOTs, respectively, the loss is a maximum.

Tail Rotor

The tail-rotor orientation is shown in figure 16; the Euler angles Px and py have the same sequence

as the tail-surface orientation. The relative velocity of the air is given by

= - fir (142)

and the downwash dTR, like the tail surface downwash, is interpolated from a table. From the figure,

the advance ratio and total inflow are

#TR = RTRf2T R

ATR -" f_TR tan ar nt- CTTR

2@-'2 R + k2 R

(143)

_ AIST R + CTTR (144)

2_/#2 R + A2 R

The wind orientation angles with respect to the hub are defined as

/3r = tan-I (145)

14
The force andmomentfor the hub from the tail rotor are

(146)

fR= (LTRC:3,- - MTRS3,-) x'T+ (LTRSI3,. +MTRC3,-) (IT+QTR: T (148)

All components of the force and moment expressions result from the quasi-static flapping model of

reference 6, and are functions of the total inflow and the blade harmonic components. The inflow

comes from equation (144) and the blade harmonics result from

[/_']a_- ] = 0 (149)

where _a is a vector containing the collective, the first sine, and the first cosine components of the blade

flapping motion. The precise definitions of the matrices and the rotor forces and moments are given in

appendix C of reference 6. In the present report the total inflow is defined such that positive inflow

is down through the rotor, whereas in reference 6 total inflow is defined such that positive inflow is

up through the rotor. Both equations (144) and (149) are algebraic nonlinear equations and are solved

using the Newton method. Once acceptable convergence is reached, the values of a__and ATR can be

used in the force and moment equations. The loss factor fblk in equation (147) is included in order to

model the interference effect between the rotor and the surface to which it is mounted. Loss factor fbtk

is defined as follows:

fblk=(TRloss-1)x[1 [ - ( /ZTR \12+ 1

\ YRbrk ]

fblk = 1

I_tTR < T.Rbr k (150)

#TR > TRbrk (151)

The variable TRtoss is the percentage of thrust available at zero advance ratio, and TRbr k is the advance

ratio at which fbllc becomes unity and remains unity at advance ratios above T.Rbr k.

Main-Rotor Downwash

The downwash of the main rotor on the tail surfaces and tail rotor is calculated using the flat-wake

model described in detail in chapter 2 of reference 5. The wake model assumes a rotor in edgewise

flow, as shown in figure 17; the coordinate frame is that of reference 5. The downwash velocity is

determined using the Biot-Savart law, assuming cycloidal vortex filaments shedding from the disk plane.

This flat-wake assumption is considered valid in forward flight when V/Rf2 >_ 1.63x/-_. The x, y,

and z components of the velocity are calculated from

vj('xI'YI'Zl) -- 47rV I

j = x, y, z (152)

15
where A£,j results from

A0m : Ax_x (_1,Yl, _1) (153)

A_y : A_y I (_1,Yl, _'1)

-- !}2 {_--'_ [i_/" (_:1' _1' _1) -b "/_/"(OO' _1' 31)] q- [l_ (_1' _1' _1) "k- fie" (_'/)1' _'1)] (154)

_0: = -_ 1 {_V I_'/x I, yl,Zl)+_'(oO,_l,._l) 1 + [.LIYrl _l,Y.l)+L(c_,Z)l,_.l)l} (155)

The functions in equations (153) through (155) are pretabulated and depend on the nondimensional

distance from the hub center _ and the components of the point of interest, through

Xl-"-- ; Yl =- ; Zl'-z'_ (156)

P P P

A typical example of each of these functions is shown in figure 18 for Yl = 0.6. The functions need to

be tabulated only at the positive values of _:1, Yl, and £'1 because of the symmetric and antisymmetric

properties of the functions. A summary of the odd and even behavior of the functions is shown in

table 2 along with the equation numbers and figure numbers that define the functions in reference 5.

The downwash components are expressed in terms of the function 1', which is the average circulation

over a rotor revolution at a radial distance from the center of the rotor. As in appendix C of reference 16,

this function is approximated as

F(P) = r0(1 - _)_2 (157)

FO = 20rrCT (158)

5(r4-r 4) -4(r25-r 5)

where the constant Fo is a function of the thrust coefficient. This relationship between the circulation

and the thrust coefficient comes from evaluating

(159)

where

E

1 f2,'r -_pd_, = _AF(_)ff

which results from the Joukowsky lifting law.

(160)

The wake model can be used to generate a downwash table for each tail surface and tail rotor. The

wake is oriented along the relative air velocity vector in the :_V - OV plane, shown in figure 19. The

S-frame is the tail-surface frame, but the tail-rotor frame T can be substituted for it. The velocity at the

hub is

I?v = VG + A0Y-A - r"/h (161)

16
and theorientationanglesare

av = tan-1 1,7"v• _:r-/

/3v = tan -! \_i? v

(162)

(163)

For the given tail surface (tail rotor) fixed in the S (T) frame, downwash values fDWrs (fDWTR) at

different combinations of av,/3v, and Itl?vllare calculated and stored in a table. These values are the

downwash velocities in the _.S (_.r) direction divided by CT. This division of the downwash velocities

by C T is based on equations (152) through (158), from which the proportionality of C T to the downwash

velocity can be deduced. For the tail rotor, the downwash vector is

and for each tail surface the downwash vector is

(164)

dTS -- CTfDWTs 119 ,11) (165)

The fDW functions are linearly interpolated from values tabulated from a given set of av,/3v, and I?v.

The points that make up the interpolation tables are selected in the following manner. The l[IT"vll's

are chosen directly by selecting the number n_,ow and the particular values. The c_v's are chosen by

selecting a set of points apT and defining av = apT + avO; similarly, the/3v's are chosen by selecting

a set of points/3pT and defining/3v =/3pT +/3vO. The quantities avo and/3v0 are the orientation angles

that center the wake on the given surface (i.e., when Vv is parallel to Fhs ). The points apT are chosen

by setting the boundary angles OtLOBN o < ctLO < -ADW < 0 < ADW < aup < aUPBN D, and the

points/3pT are chosen by setting the boundary angles/3LOBNo < 13LO < 0 < /3Up < /3UPBxo" These

boundary angles are defined for the condition in which the flat wake is centered on the given surface. For

the a boundaries, the number of divisions in the end regions, [aLOBND, aLO ] and [aup, aUPBNO],

is selected through na_,_a, and the number of divisions in the middle regions, [aLO,--ADw ] and

[ADw , aup ], is selected through nam_a. The value ADW is a small value and precludes evaluation

of the downwash on the wake layer, where the vortex singularities are present. For the/3 boundaries,

the number of divisions in the end regions, [/3LOBxo,/3LO] and [/3UP,/3UPBN#]' is selected through

n3_a, and the number of divisions in the middle regions, [/3LO, 0] and [0, 3UP], is selected through

72_mid"

Mass Matrix

The equations of motion are formed numerically, making the model described in this report implicit.

Because of this, the system mass matrix is not readily available as it would be if the equations were

derived explicitly. Allowing for different hinge sequences precludes the explicit calculation of the mass

17
matrix, sinceeachof themanyhingesequenceswouldrequirea separatecalculation.To avoid this, the

massmatrix is implicitly calculatedwith the condition that no acceleration-dependent appliedforces,

suchasthe apparentmasstermsin blade-sectionaerodynamics,arepresentin thesystem.Suchforces

contributeto the massmatrix, making its extractionmoredifficult. It is well known from analytical

dynamicsthat the form of thekinetic energyis

1.T

Tki n = T 2 + T 1 + TO -= 5q [t_,[qq(q,t)] q_+ C(q,t)T(t + r(q,t) (166)

where L.a,lqqj is the mass matrix of the system and q_represents the system generalized coordinates. In

the model presented,

= 1

1

/32,.'',t3nb, _l, _2,..., _nb, ¢1, _2,..., _nb:Ox, Og, Oz, Rmz, Rmy, Rrnz, _1

J

(167)

and the quadratic contribution T2 to the kinetic energy is given by

T 2 _ r2fuselag e

nb

k=l

(168)

For the fuselage, the velocity and angular rate are given by

_(F)

--F/r

WE

= TF+i + F/Z ,Lmc F/±:rFZ,

RF

(169)

(170)

which is a re-expression of equation (22) using matrix notation instead of vector notation. The elements

of the column matrix/_ are the time-derivatives of the three translational degrees of freedom describing

the motion of point m on the fuselage; the elements of _ are the time-derivatives of the three ruler

angles of the fuselage. For the coordinate system F defining the fuselage motion,

&F/I =

1 0 -Sou

0 Cox SoxCoy

0-Sox CoxCou

(171)

The velocity and angular rate give the T2 contribution to the kinetic energy from the fuselage

(172)

18
Likewise, for the rotor hub mass, the matrix representation of the velocity and angular rate is

= _F/I -F ,ZH/F 'F _R/H = TRFdZF/IO--- -F &R/FY) =, TRFS,'F/I : O:S,'R/F , ..._,

r--ih •IF/ = TF/IR---• + +lr/_ mh F/I--o = r.IF/_, [rmh_F/s:TF/I:O] {_}

F;R

(173)

(174)

where

_R/F = 0 'g' = WR/FW

1

Equations (173) and (174) arise from equations (14), (18), and (63).

contribution to the kinetic energy from the rotor hub is

(175)

As with the fuselage, the 7'2

T

T2,o,or= _

6

0

(176)

The same approach is used to find the kinetic energy for the kth blade. For the kth blade the angular

rate is

,(B)

_B/I

,(B) ,(B)

= _R/I q- _L/R "F W__B/L

= [TBF_F/I i 0 i TBR_'R/F i TBL&L/R i_B/L]

t

WB k

= TBF_F/I _ -q- TBR_:R/F_, q- TBLWL/Ra_L/R -'F _;B/LaB/L

V;

hL/R

hB/L

(177)

and from equation (74) the velocity of the center of mass becomes

.(R) = ÷(R) .(R)

r-ie '-ih + r-he W---R/I

÷(L)_,,

I c9

S_

=-iI T_yb --B/S= M3_F/I iTBt iM2_R/F iMt_L/R rib "B/LJ _

_ ALIn

RBk h__B/L

(178)

19
where

.(B),,,

_1' = TBLr_Lf ) + "fb " BL

T ÷(R) Af2 = M1TLR + BR he

M3= :V2TRF+TBr_mF)

Since no particular hinge sequence has been stipulated, the hL/R and hB/L are generic, representing

the generalized coordinates in the transformation indicated in the subscripts. For the flpt, fplt, and pflt

hinge sequences, the terms are defined as

_L/R =

0

/3k = _;L/R_L/R (179)

_B/L =

0 1

Sok+¢k 0

Cok+o k 0

_k

{ Ck } ='_B/L_B/R (180)

and for the lfpt, lpft, and plfl hinge sequences, the terms are defined as

_L/R = _k = &L/RhL/R (181)

_B/L

Sip 1

Cok+okCCp 0

SOkWCkC(p 0

,3k $k } ="'L/R_B/R _ •

(182)

Using the velocity and angular rate, the T2 contribution to the kinetic energy from the kth blade becomes

(T_'"')k = 2 "a_/_

-_B/L

T

[Msn;knB_+ wB_z_%]

0

a-L/R

_B/L

(183)

For the inflow model, the mass matrix associated with the first-order states ,__vT = [A0, Als , Alc ] can be

easily extracted using equation (119) and equations (123) through (125). The mass matrix for the inflow

model is

3_IDI=-_LI._VII 1 I,°0 C3 h S3h

°1 (184)

0-Szh C_ h

20
If there are no acceleration-dependent applied forces, the total mass matrix of the system can then be

related to the equations of motion by

The matrix Mqq is formed using the T 2 terms in equation (168), by adding the contributions of the

corresponding fuselage, rotor hub, and blade elements. The equations included in r_.eqare the rotor-blade

equations, equations (55) through (57) or equations (58) through (60); the equations included in f--eq are

the fuselage equations, equations (8) through (13) and equation (54). Finally, the equations included in

x__eqare the inflow equations, equations (119). The reader will note that these equations have been cast

such that they are identical to equations that would result from a Lagrangian formulation. Thus, the

mass matrix calculation is compatible with the equations of motion and allows the relationships

-Mqq = -._ f--eq (186)

0

--3IDI = _ [£eq] (187)

(188)

Multiblade Coordinate Transformation

For rotor systems with three or more blades, a multiblade coordinate transformation (MBCT) is

available to transform the rotor rotating coordinates to nonrotating coordinates (ref. 7). For a set of

rotating variables, each of which represents the same physical quantity for each blade,

x_ = [xl,x2, x3,...] (189)

The nonrotating representation for an even number of blades is

x_TN = [xO, Xd, Xls,Xlc, X2s,...] (190)

The generic variable x 0 is the collective, while x d is the differential, which is not present if the number

of blades is odd. The xns terms are the nth sine variables, while the Xnc terms are the nth cosine

variables. An example is the flap degree of freedom, for which x R is a vector whose components are

flapping angles. The transformation between variables is linear and time-varying, so the transformation

from rotating to nonrotating systems is given by

x N = TNRXR

iN = TNR&R + 7'N RXR

_N = TN R_R + 27"N R&R + _'N RXR

(191)

(192)

(193)

21
andthe transformationfrom nonrotatingto rotatingsystemsis

x_.R = TRNX. N

:R = TRN&N+ TRNXN

X-R = TRN_N + 2J'NRJCN + TNR_ZN

(194)

(195)

(196)

The form of the transformation matrix for models with an even number of blades is

TRN =

l-1 s_, c,, s2_,

l 1 s,_ c,2 s:_2

l--I _3 C_'3 "'"

1 1 s_4

C2_I

(197)

and its inverse is given by

1

1 I 1 1

-1 1 -1 1

2SVa, 2S_, 2 2SW3

2Cw, 2Cw2 2C_ 3

2S2_1 2S2_2 "'.

2C2_,1

(198)

The Ck terms are the azimuth positions of the blades:

_bk = _Pl + (k- 1) 2n k = 1,2,...,n b (199)

nb

These positions all have the same time derivatives. For models with odd numbers of blades, the second

column of equation (197) and the second row of equation (198) are not present.

OPERATIONS

Basic Setup

A driving routine performs four basic operations on the helicopter mathematical model: initialization, trim, linearization, and integration (fig. 20). Initialization sets the basic data that define the

helicopter configuration and conveys to the driving routine all the data necessary to operate on the

model. The trim operation sets the helicopter configuration into some prescribed steady condition. A

harmonic balance technique is used, which simultaneously balances the fuselage forces and moments

and finds the rotor-blade equilibrium solution. Once the model is trimmed, two possibilities are available. One is linearization, in which a two-point difference formula is used to linearize the model about

22
the trim condition. The other possibility is time integration,which gives the systemtime response,

initialized in trim condition,for givencontrol inputs.

Eachoperationusesa subsetof the model,whosegeneralstructureis shownin figure 21 and

representedin equationform asfollows:

X_.eq

I T = [_T, S T, _T, r_..r, fT, _T r_T fT

f

k

=o

(2Ol)

,0_T ' u_T] (202)

Variables and controls are represented in a single vector X, which acts as the input to the model, whereas

the equations and measured quantities are represented in a single vector __FeTq= [FTyT], which acts

as the output. Both the inputs and the outputs can be categorized into three types: rotor-blade, fixed,

and augmented. The rotor-blade inputs/outputs are items associated with the rotating coordinate system,

such as the flap degrees of freedom or the blade pitch; the fixed inputs/outputs are items associated with

the nonrotating system, such as fuselage degrees of freedom or fuselage acceleration measurements. The

augmented inputs/outputs are associated with any additional first-order equations necessary to complete

the model. The structure in figure 21 allows the use of rotating or nonrotating inputs and outputs.

The variables are transformed by the multiblade coordinate transformation; the choice of the coordinate

system depends on the particular operation. This basic setup is general and the operations can work on

any model set up in this manner.

For the model described in the Helicopter Equations of Motion section, the rotor-blade degrees of

freedom in the rotating system are

r-T-- [31,/32,"',(1,(2,'" ,01,02,'"] (203)

When using nonrotating coordinates, the vector r__is defined according to equation (190) for each degree

of freedom. For a model with an even number of blades the rotor degrees of freedom in the nonrotating

system are

(204)

For a model with an odd number of blades, the differential coordinates are not present. Using nonrotating

coordinates for the input variables requires that the flag ISNROT = 1, which transforms the nonrotating

coordinates to the rotating system via equations (194) through (196).

The rotor-blade controls in the rotating system are

0 T = [01,02,'",01,(02,"',01,02,'"] (205)

23
andaswith the rotor-bladevariables,the functionof the flagIINROT is to transform_0to the rotating

systemif theseinputsaregivenin the nonrotatingsystem.For a modelwith anevennumberof blades,

therotor-bladecontrolsin the nonrotatingsystemare

0--T = [00, Od, Ols, 01c,""", 00, 0d, Ols, 01c,""", 00, Od, Ols, 01c,-..]

For a model with an odd number of blades, the differential coordinates are not present.

variables are the fuselage degrees of freedom and the rotor azimuth angle

(206)

The fixed

fT = [oz, oy, Oz, Rmx,Rmu,Rmz, W]

and the augmented variables are the three dynamic inflow coefficients

(207)

X T = [A0, AIs, Alc] (208)

The fixed controls u_u_are

u_.T = [OOT R , Yq, Vgx , Vgy , ?)gz , iTS1, iTS2,..., iTSn s]

(209)

The center block of figure 21 is the model, described in the previous section, that calculates the helicopter

equations of motion. The equations in r eq are the rotor-blade equations, equations (55) through (57)

or equations (58) through (60), and the equations in f-eq are the fuselage equations, equations (8)

through (13) and equation (54). The equations in Z__eqare the inflow equations, equations (119). The

vector __is defined by equations (40) through (47), and w is defined by equation (105) and equations (107)

through (112). The vector w__is in the rotating frame but can be transformed to the nonrotating frame

by using the flag IONROT.

Initialization

Initialization performs two basic functions, one of which is to set up any data not given in X

that define the model. These data are the physical parameters defining the model that was previously

described; they are summarized in table 3. The main-rotor downwash tables are either generated and

stored in files or read in from previously generated files. The second function of initialization is to

transfer basic system data to the driver so that subsequent operations can be carried out on the model.

These data, summarized in table 4, are primarily size data of the full model. Included in the table are

the current values of these data for the model described in the Helicopter Equations of Motion section

of this report.

Trim

The trim operation places the helicopter in the desired flight configuration by using a harmonic

balance technique that is described in reference 8, chapter 3. The method casts the helicopter equations in

an algebraic form by defining the rotor-blade variables as a Fourier series. The coefficients of this series

expansion, along with the fixed and augmented state variables and inputs, are adjusted to force the rotorblade, fuselage, and augmented state equations to zero. The rotor-blade equations are also expressed as

a Fourier series, so that forcing the rotor blade equations to zero means forcing the coefficients of this

24
seriesexpansionto zero.It is assumedthat thehelicoptermain rotorhasaconstantrotationrate_ = f2

andthat the rotor-bladecontrolsareexpressedin the nonrotatingframe(i.e., IINROT = l). All blades

are assumed identical, and information from only one blade, expressed in terms of rotating coordinates

(i.e., ISNROT = 0), is needed in the trim. An IMSL nonlinear algebraic equation solver, ZSPOW, is

used to find the solution to the following equation (ref. 17):

__F(V) = 0 (210)

The vector of inputs, _V, is defined as

E

n=1,2,

f [if(n)] n = 1,2,...,nil

X[il(n)] n= 1,2,. .. ,ni,

O__{K(n) + nb[Jo(1) - l]} n= l,Z,...,nK

0__{K(n)+n b [jO(njo) - 1]} n= 1,2,...,n K

n=

reo[n, jr(1)] n = 1,2, ..., (2nhTRM

rco [n, jr(nj_)] n = 1,2,..., (2nhrnM

+1)

+1)

(211)

which is composed of fixed, augmented, and rotor-blade inputs. (Matrices are stacked by columns

when represented in column vectors.) The blade variables are represented as coefficients of the Fourier

expansion,

r__[k+nb( j - 1)] =rco(1,j) +

rthTR M

rco(2n, j)cos(nCk) + rco(2n + 1,j)sin(n_k) (212)

n=l

The variable k is the blade number, and j is the blade degree of freedom (e.g., j = 2 is the lead-lag;

see eq. (203)). For the model of this report, the coefficients are represented in matrix form as

rco

Zo

fllc

ills

fl2c

flrl hT RM c

flnhTRM s

(ts

: :

(213)

25
The elements of __Vare adjusted by the equation solver after the user chooses determining indices. For

example, the choice of nil = 2 with if(l) = 2 and if(2) = 3 directs the solver to adjust angles 0 v and

Oz to find the trim. Given the information in __V,the input vector X can be formed and the model can

be evaluated. The resulting equations can then be used to form __F,which is defined as

E(E) =

ff_ t2,'r/_ '0 G [ dt

f_ r2rr/fi

r_ _0 _q[il(n)lwl(n)dt

n = 1,2,..., (2nhTnM + 1)

n = 1,2,..., (2nhTnM + 1)

n= 1,2,...,nfe q

n= 1,2,...,nil

(214)

where

rfo(1,j ) =

rfo(2, j ) =

rfo(3, j ) =

f_ f2_/_

JO r_eq [1 +nb( j - 1)] dt

f_ f2_/fl

JO r__eq[1 +nb( j -- 1)] cos(flt)dt

fl f2 /fl

7 Jo r_.q I1 + rib( d - 1)] sin(flt)dt (215)

rfo(2nhTRM ' J) =

7rJO

rfo(2nhrnM + 1,j) = --f_ f2_/fi

7rJO

r__eq [l + rib( j -- l)] COS(TZhTRMf_t)dt

r_.eq[1 + nb( j - 1)] sin(nhTnMf_t)dt

These equation are the Fourier coefficients of the first blade equation, where

req [k + nb( j - 1)1 = rfo(1,j ) +

rlhTR M

rfo(2n, j)cos(nzPk) + rfo(2n + 1,j)sin(nWk) (216)

n=l

The choice of nhTRM determines the number of rotor coefficients that are adjusted to force the same

number of coefficients of the blade equations to zero. The fixed and augmented equations are averaged

over one rotor revolution in equation (214) to zero only the constant harmonics of these equations. A

summary of the inputs that control the trim procedure is shown in table 5. Indices and their number (in

parentheses) are selected to choose the adjustable variables and the equations that are to be satisfied. If

an adjustable variable is set to a value, the value acts as an initial condition to the solver. If the variable

is not adjusted by the solver, it remains at the set value throughout the trim. An example of the use of

this feature would be to trim the model at a constant climb rate.

Integration

The time response of the system shown in figure 22 to given inputs _Upcan be obtained by numerical

integration. Since it is difficult to generalize helicopter control/actuator models, these systems have

26
been broadly defined as first-order equations that can be defined by the user. The P block in figure 22

represents precompensation, and the A block is any actuator/mixer dynamics; feedback dynamics are

represented in the C block and sensor dynamics are represented in the S block, which has the restriction

that the output of S cannot explicitly depend on Us. The helicopter model is represented in block G with

nonrotating rotor inputs (IINROT = 1) and rotating-blade degrees of freedom and outputs (ISNROT = 0,

IONROT = 0). By default, P and A are identity feedthrough and C and S are zero, giving the response

of the open-loop helicopter model G. Any subset of the model represented by equation (200) can be

integrated, where the state of G is defined by

__{n+nb[j÷(m )- 1]} n = 1,2,...,nb;

] [i/(m)] m= 1,2,...,ni]

z__a = x [il(m)] m= 1,2,...,ni_

r_{n+nb[Jr(m )- 11} n = 1,2,...,rib;

Z [if(m)] m = 1,2,...,nil

m= 1,2,...,nj÷

m = 1,2,...,nj,.

(217)

As in the the trim procedure, the selection of the variables and equations is done by choosing indices.

If the rotor degree of freedom is not selected, a constant rotor rate is used, by default, in the integration

of the model (i.e., _ = f_). The inputs are defined as

u G = O_{K(n)+nb[JO(rn)- 11} n = 1,2,...,nb; m= 1,2,...,njo l

_u[iu(m)] m= 1,2,...,niu

(218)

The feedback outputs are

w__{n + nb_w(m ) -- 11}

YG = 9_[iy(ra)]

X [ix(m)]

n = 1,2,...,nb; m= 1,2,...,ni w I

m= 1,2,...,niu

m= 1,2,...,nx

(219)

and the auxilary outputs are

_w + nbF (r-) - 11}

za = 9_ [i9 (m)]

n = 1,2,...,nb; m = 1,2,...,ni_, ]

m= 1,2,...,ni9

m = 1,2,... ,nsc

(220)

The equations integrated are

r_eq {n + nb_Ji,(m ) -- 1]}

if-G = f__eq[i](m)]

z__eq[il (m)]

n = 1,2,...,nb; m= 1,...,nj÷ I

m= l,2,...,ni]

m = 1,2,... ,nil

From equation (200) the form of the subsystem that is integrated is

(221)

- M G (Z_G,U_G)_4.(7 + G__G(Z__G,U__G)= E.G (dG,z--G,U--G) (222)

27
where

f_{n + nb[ji.(rn ) -- 1]}

n = 1,2,...,rib; m= 1,2,...,nj÷

m = 1,2,...,nil

m = 1,2,...,nil

(223)

Two methods are available for extracting the mass matrix M G. One method uses equation (222)

directly, in which it is clear that the jth column of M G is __FG, with _dG equal to zero except for the

jth element set to unity, minus FG(O, XG, U_G). If AIG is of dimension n:'tlc = nbnj_ + nil +rtil,

one evaluation of the model is necessary to get FG(O, z_G, uG), and n3i c additional evaluations of the

model are necessary to get M G. This approach always works; however, it requires many evaluations

of the model and thus is inefficient. The other method is to use the corresponding elements from the

mass matrix, which is calculated in and passed from the model, to form the submatrix M G. Once M G

is available, the equation

A/GI (Z_.G,UG) G__G(Z_G, UG)

x-G = __In + nbji. (m) -- 1] n (224)

can be found; this is the expected form of block G in

= 1,_,..., 9 rib; m= 1,9_,..., nj÷

= 1,2,...,nil

figure 22.

If the helicopter model is trimmed, the inputs and outputs of G are available at the initial time. The

initial state XGo, initial input UGo, and initial output YGo can be used to extract the initial conditions

of the blocks A, C, P, and S (fig. 22) as follows. It is assumed that 5:A = :/:C = :/:P = :/:S = 0 and

it is required that nuc - nup, which gives an equal number of equations and unknowns. With these

conditions, a nonlinear equation f(v) -- 0 can be set up, where f and v_ are defined as in figure 23.

The variables on the left-hand side of the figure are adjusted until the variables on the right-hand side

are forced to zero. The values at which this occurs are the initial conditions of the system of figure 22.

The solution is extracted with the modified Newton method used in the trim procedure.

With the initial conditions available, the entire system of figure 22 is integrated according to the flow

it* diagram in figure 24. The S block is evaluated to get its output YS using -5', which is not necessarily

the the correct value at the given time. However, the output of S is specified to be independent of

_uS, so the output value is correct. With this input to block C, the values on the fight-hand side of the

figure can be evaluated from the given information on the left-hand side of the figure. With this setup,

and the input up defined by the user, the integration is carded out using the predictor-corrector method

in the subroutine DVERK from the IMSL package (ref. 17). A summary of the integration inputs is

shown in table 6. As in the trim, the variables to be integrated are selected through indices. The input

is _UP = _U_P0+ A_.U_p, where _U_.p0is determined in the trim. The term A_U_ is the input perturbation

about this trim.

Linearization

After trim, it is also possible to linearize a subset of the helicopter model (e.g., block G of fig. 22)

in the nonrotating system (ISNROT = 1, IINROT = 1, and IONROT = I). Linearization is done about

28
the trim solutionusingthe two-pointfinite-differenceformula,

oZ t + - Z(xj- xj) (::5)

Oxj 2Axj

The partial-derivative matrix of the helicopter model is defined in figure 25; the rows and columns are

selected in the same way as in the integration process. For any given time, the partial-derivative matrix

can be calculated and the linear system can be evaluated according to the following equation:

s(t) =

......

I[OJ = [A(t) B(t)

[C(t) D(t).

(226)

The harmonics of the linear model matrix can be found using

_f f 2_/w S(t)dt So=_j o(227)

S.c = w--_ff2_/_f S(t) cos(nwft)dt (228)

7r JO

S._ = __Af 2_/w S(t)sin(nwft)dt

7r Jo

(229)

These integrals are evaluated numerically using a Gaussian quadrature procedure. The periodic model

matrix is then expressible as

AO B 0 iv Ant Bnc cos(nwft) + sin(nwft)

S(t) = Co Do + Cnc Dnc Cns Dns

n=l

(230)

Since nonrotating coordinates are used, the helicopter model matrix will be constant when the model is

in hover. In forward flight, the model matrix has periodic coefficients, which become more important as

the forward flight speed is increased. The fundamental frequency _f depends on the number of blades

in the helicopter system. If n b is odd then wf = nbf_, whereas if it is even, a;f = _f_. A summary of

the linearization procedure inputs is given in table 7.

RESULTS

The model results are compared with other experimental and analytical results as a preliminary

validation. The first comparison is with blade-equilibrium responses in trim, of a flap-lag-torsion rotor

model used in reference 8. Unlike the model described in the present report, the equations in reference 8

were symbolically generated out to their final form. An omission in the reverse-flow region was

29
discoveredin the model of reference8: the modeldid not accountfor the sign changein Xab in its

aerodynamic formulation (see fig. 11). This error is introduced into the model described herein so the

comparison would be valid. The parameters of the model, presented in table 8, are of a four-bladed,

soft in-plane rotor with a lead-lag frequency of 0.7/rev. The rotor trim is calculated with six harmonics

(nhTRM = 6) in each blade degree of freedom, and the fuselage mass is chosen to give a weight

coefficient of 0.005. The flap, lag, and torsion equations are satisfied for the blade equilibrium while

simultaneously satisfying the roll-moment, pitch-moment, x-force, and z-force equations. The trimmed

blade-equilibrium response in flap, lag, and torsion is shown in figure 26. Three response curves are

presented for each blade degree of freedom: (1) a curve from the model from reference 8, which has

an erroneous reverse-flow formulation; (2) a curve from the model of the present report, which uses the

same reverse-flow formulation as that used in reference 8; and (3) a curve from the model of the present

report, which uses what is believed to be the correct interpretation of the aerodynamic formulation

in the reverse-flow region. The two models with the incorrect reverse-flow interpretation show close

agreement, which serves to verify the analysis. The alternative reverse-flow interpretation greatly affects

the torsion response of the system, although its effect on the flap and lag responses is still very small.

The results from the linearized helicopter model are compared with experimental hover results from

reference 9. The experimental setup consisted of a three-bladed rotor mounted on a mast that allowed

pitch and roll motions. Each blade had metal plates with notches, which acted as concentrated hinge

springs to closely simulate the flap-lag rigid-blade approximation. Torsion springs on the body mount of

the experimental rotor were adjusted to simulate an air-resonance condition. The parameters necessary

to model this system are given in table 9. Measurements were made of the damping and frequencies

of the system modes, and it is to these data that a correlation is made. Figure 27 shows the modal

frequencies and the lead-lag regressing damping plotted against the rotor rotation rate, with zero blade

pitch. The simulated air-resonance condition occurs when the body roll-mode frequency nears that of

the lead-lag regressing mode, which is near a rotor rotational rate of 700 rpm. Excellent correlation with

the experimental data is seen in this condition. The body pitch and roll damping at zero blade pitch is

shown in figure 28. Good correlation between experiment and theory is also seen here, although the

model tends to overpredict the damping at the higher rotor rates. The lead-lag regressing damping at

9 ° blade pitch is shown in figure 29; again, the model does a good job of predicting the damping in the

resonance condition. Detailed cross-plots of the lead-lag regressing damping with respect to the blade

pitch are shown in figure 30. Good agreement is seen between experiment and theory, although the

damping tends to fall short at high and negative blade-pitch angles at the higher rotor rates. Figure 31

shows cross-plots of the body damping at 650 rpm; good agreement in pitch is seen at positive bladepitch angles. The body roll damping tends to be overpredicted, getting worse at higher blade-pitch

angles.

The main-rotor wake model used to calculate the downwash on the empennage surfaces is tested

by correlating it with experimental results from reference 5, page 77. A correlation is done in the

reference and is also done in the present report by using the flat-wake model described in the Main

Rotor Downwash subsection. Unlike the model used in reference 5, the model described herein assumes

a circulation distribution. In reference 5, a blade-element approach is used to find the circulation

distribution. The integral in equation (152) is evaluated by using a Gaussian quadrature integration

method along with linearly interpolated values from the pretabulated functions. Figure 32 shows the

vertical nondimensional downwash at various azimuths at 0.1R below the rotor. The experimental rotor,

30
with 10° bladetwist, is immersedin edgewiseflow at CT = 0.006. Using the flat-wake model gives

good correlation results; the asymmetry of the downwash at the 90 ° and 270 ° azimuth positions is

captured.

With the rotor and downwash models validated, it is now possible to validate the full helicopter

model. The model is configured to simulate a UH-60A helicopter in order to compare model responses

with the flight-test data that are available in references 10 and 18. Reference I0 presents a validation

of the real-time model presented in reference 11, which is also known as the GEN HEL model. Reference 18 presents a validation of an updated version of the GEN HEL model, which is referred to as

the Ames GEN HEL model. In the model of the UH-60A used in this report, all automatic controls

are off (SAS, PBA, etc.). The fuselage aerodynamic representation in the model was generated from

curve-fitted equations of the static aerodynamic characteristics. The exact equations are given in reference 19, pages 2-3, and are used in place of the equivalent-drag-area fuselage aerodynamic model.

The lag damper is modeled according to reference 11, page 5.1.10, and this representation is substituted

for the spring/damper model. The control mixer is modeled according to reference 11, pages 5.5.14 to

5.5.17, without the servo models. The tail-rotor parameters are extracted from reference 20, where a

detailed description of the UH-60A tail rotor is given. References 11 and 12 are used to extract other

physical parameters of the helicopter. Table 10 summarizes of all of the inputs to the model. The data

followed by a question mark are specified for each of the following correlations. From table 10 it can

be seen that the model has a flap-and-lag main rotor and a six-degree-of-freedom fuselage at constant

rpm. The dynamic inflow model is included, and the collective pitch, sine pitch, cosine pitch, and

tail-rotor collective pitch act as the inputs to the helicopter model. No feedback outputs are selected,

and accelerometer outputs are taken from the fuselage at the positions corresponding to those on the

flight-test vehicle.

Trim results calculated with and without the effect of the main-rotor wake in straight and level

flight are compared with experimental results in figure 33. The trim data were calculated at 25-knot

increments with the horizontal tail surface at the following settings: 40 ° for 0, 25, and 50 knots; 11.3 °

for 75 knots; and 4 ° for I00, 125, and 150 knots. The vehicle was trimmed by satisfying all six fuselage

force and moment equations, the flap and lag rotor equations, and the inflow equations (see table 10).

Between 0 and 50 knots, the angle Oz was fixed, while the angle Ox was adjusted (i.e., if = 1,2 in

table 10). Between 50 and 150 knots, the angle Ox was fixed, while the angle Oz was adjusted (i.e.,

if = 2, 3 in table 10). The collective, longitudinal, lateral, and pedal inputs are presented in percent

of total motion; the total motion is 10 in. for the stick inputs and 5.38 in. for the pedals. The model

tends to deviate significantly from the experimental data above 120 knots, where aerodynamic effects

such as dynamic stall and radial flow, which are not accounted for in this model, can become important.

Below 120 knots, good correlation is seen and the results are similar to those presented in reference 18,

p. 28. The lateral and longitudinal stick positions are predicted more accurately by the Ames GEN HEL

model, but the pedal position is closer for the model described herein. Overall, the results from the Ames

GEN HEL model are slightly better, probably because that model used tabular data for blade-section

aerodynamics, as well as other refinements particular to the UH-60A configuration. The flat-wake model

enhances correlation in all but the collective and lateral stick positions. The collective stick position is

hardly changed, whereas the lateral stick position is degraded by 1/4 in. at 150 knots.

Flight test time-histories from reference 10 are compared with the responses of the UH-60A model,

at 1 knot (figs. 34-37) and at 100 knots (figs. 38-41). Four sets of time-responses are presented at each

31
flight speed;they correspondto I in. adjustmentsto the collective, longitudinal, lateral, and pedal

inputs. The inputsto the modelarethosefrom theflight data,referencedaboutthe trim of the model.

The trim valuesgeneratedby the model areindicatedby points c, e, a, and p, corresponding to the

collective, longitudinal, lateral, and pedal inputs, respectively. The angular rates of the fuselage (roll,

pitch, and yaw) and the accelerometer outputs from the fuselage (longitudinal, lateral, and vertical)

are also presented for each case. The accelerometer position corresponds to the approximate center of

gravity of the helicopter; the precise center-of-gravity location is given as POSFUS in table 10. Each

plot has three curves: the test data, the model response, and the linearized model response. The test

data are from reference 10 and the test numbers are identical to the numbers used in the reference. The

model responses come from integration of the full nonlinear system, and the linearized model responses

come from integration of the linearized model. The responses of the linearized model are added to the

trim values to give the total responses presented. Since the helicopter is in straight and level flight,

the angular rates are zero at trim, and the linearized responses are identical to the total responses. The

accelerometer responses contain gravity effects, and the nonzero trim values of the accelerometers must

be added to the linearized accelerometer outputs.

At 1 knot (figs. 34-37), the on-axis angular rates agree well with the test results. The off-axis

responses of the model do not correlate as well, with the pitch axis being the most troublesome by

diverging from the test data in tests 212, 203, and 209. However, this is no worse than the results

from the GEN HEL model, which is refined specifically to the UH-60A configuration and includes an

engine model with a rotor rpm degree of freedom and actuator dynamics. At 100 knots (figs. 38-41) the

on-axis responses have fair agreement with experimental results, and again, these results are consistent

with results from the GEN HEL model. Other tests were run at 60 knots and 140 knots, and the results

show a strong similarity to the results from the GEN HEL model. Another source for comparison is

reference 16, in which the results of a linearized model are compared with the same flight data. The

results from reference 16 are also similar to results obtained with the model of this report. Taking all

these results together, it is concluded that the present model gives results that are consistent with those

obtained from the GEN HEL program and the linearized model of reference 16.

A final comparison is made of the linearized model with the eight-state quasi-static model generated

from the Ames GEN HEL program. The linearized GEN HEL model is generated with f't = 27 rad/sec,

STm = STc = STa = 29.583 ft, WLm = WLc = WLa = 20.683 ft, .M"F = 492.13 slugs, iaero = 0,

and Cry = [0, 0, 0] rad. The linearized model of this report is generated and arranged according to

= rA rAr,l{ r} + u__

_s LAs AHJ x-$ BS

(231)

_y= [q. c:] (x_r) "--: +

(232)

The vector z r contains all rotor states and inflow states, and the vector :rf contains all fuselage states.

The inputs are the longitudinal, collective, lateral, and pedal positions (u_T = [fie, 6c, 6a, 6p]T). The outputs _yare chosen to correspond to the states of the GEN HEL model (y_T = [u, w, q, A0, v, p, A¢, r, A_]).

For comparison with the GEN HEL model, the derivatives of the rotor and inflow states :/:r are set to

32
zero,yielding

_f = AredX__f + BredU__

y = CredX__f + Dred u

Are d = A f f - A frArr 1Arf

Bre d = Bf - AfrArr l Br

Cred = C f - Cr Ar) Ar f

Dre d = D - CrAft I Br

(233)

(234)

(235)

(236)

(237)

(238)

The state variables used to describe the fuselage motions are not precisely those used in the GEN HEL

model; however, the information needed to make the transformation to the GEN HEL states is found in

the equation

v t)y

y= _ v ' = [Cred] ' Rrny (239)

O I Rmz

L_¢ I Oz r Oy

, L_h Oz

The quantities u, v, and w are the velocities along the fuselage, and p, q, and r are the body angular

rates. The quantities A0, A¢, and A¢ are the pitch, roll, and yaw angular perturbations. With this

choice of output quantities, it is clear that Dre d = 0, and thus equation (239) holds. Equation (239) is

then used to transform the model to the final form

d

dt

U

W

q

AO

l)

P

A¢

r

=Fi

U

W

q

A0

P

A¢

r

..} +G I, 6c

['°#

(240)

where Aga has been removed from the system.

This operation was carried out on a linearized model at 1 knot; the resulting F and G matrices

are presented in table l l along with the system matrices from the Ames GEN HEL program. The

33
eigenvaluesandnormalizedeigenvectorsareshownin figure 42 for both models. The calculations are

based on the F matrix represented in dimensions of feet, seconds and degrees so as to scale the angular

rates to an order of magnitude comparable to the velocities. Real eigenvalues have real eigenvectors

(figs. 42(d) and 42(e)); complex eigenvalues have complex eigenvectors composed of a real part and

an imaginary part. The eigenvector corresponding to the complex conjugate of the eigenvalue is the

complex conjugate of the eigenvector. The greatest difference between models arises in the higher

frequency modes, shown in figures 42(a) and 42(b). The model of this report tends to introduce more

coupling between roll and pitch than the Ames GEN HEL model. The mode in figure 42(c) is dominated

by w and r in both models, although the content differs. In figures 42(d) and 42(e) the two modes are

very close, although the damping is much higher in the model of this report. Differences in the damping

and frequencies of the eigenvalues are also noted; they are likely caused by the different inflow models

used in the two analyses. Selected Bode plots of the system are shown in figure 43. The 6c/W plot

shows good agreement and the 6p/r gain agrees well with the Ames GEN HEL model. The 360 ° phase

shift in the 6p/r plot is due to a zero in the right half of the s-plane that is not unstable in the model of

this report. The sharp peak in the 6a/P plot is due to the light damping that is predicted by the Ames

GEN HEL model.

The entire process is repeated at 100 knots, resulting in the F and G matrices given in table 12. The

eigenvalues and eigenvectors are presented in figure 44; the eigenvectors agree more closely between

models in this case than in the 1-knot case. The on-axis Bode plots of the 100-knot case are given in

figure 45. The two models do not agree closely, but the general trends in the models are similar.

CONCLUDING REMARKS

A single-rotor helicopter mathematical model for the design of high-bandwidth flight control systems has been described. The flap, lag, and torsion rotor model allows the approximate modeling of

hingeless rotor systems, and the various hinge sequences that are available allow the modeling of many

different articulated rotors. Quasi-steady Greenberg aerodynamics are used in conjunction with the

nonlinear Pitt/Peters dynamic inflow model to calculate the blade aerodynamic forces. An rpm degree

of freedom of the main rotor is available, along with a six-degree-of-freedom rigid fuselage. The tail

surfaces have main-rotor downwash effects that arise from the fiat-wake simulation. Operations on the

model include trim, linearization, and time-integration, which have been formulated so that they may

be applied to any consistently cast rotorcraft mathematical model. This features allows for modification

of the helicopter model without requiring significant changes in the operations. Preliminary validation

of the model showed reasonable correlation with experimental and analytical results.

Further validation of the model is necessary. The higher order linearized models need to be

compared to experimental results, and flap-lag-torsion experimental data are needed to validate the

torsional effects of the rotor model. Also, additional work is necessary to extend the range of valid flight

conditions, which are currently limited to low-speed, lightly loaded helicopter systems. The limiting

factor is the blade-section aerodynamic model, which should be improved to account for dynamic

stall, radial-flow effects, and compressibility effects. Finally, the linearization process is limited to the

helicopter model itself. A useful additional operation would be the linearization of the remaining blocks

of the system and the combination of these linearized blocks into a single set of system matrices.

34
REFERENCES

1. Chen, R. T. N.; and Hindson, W. S.: Influence of High Order Dynamics on Helicopter Flight

Control System Bandwidth. J. Guidance, Control, and Dynamics, vol. 9, no. 2, Mar./Apr. 1986.

2. Curtiss, Howard C.: Modeling Effects of Blade Torsional Flexibility in Helicopter Stability and

Control. Forum of the Amer. Hel. Soc., St. Louis, MO, May 1983.

3. Greenberg, J. M.: Airfoil in Sinusoidal Motion in a Pulsating Stream. NACA TN-1326, 1947.

4. Gaonkar, G. H.; and Peters, D. A.: Review of Dynamic Inflow Modeling for Rotorcraft Flight

Dynamics. AIAA Paper 86-0845, 1986.

5. Baskin, V. E.; Vil'dgrube, L. S.; Vozhdayev, Ye. S.; and Maykapar, G. I.: Theory of the Lifting

Airscrew. NASA TT-F 823, 1976.

6. Talbot, P. D.; Tinling, B. E.; Decker, W. A.; and Chen, R. T. N.: A Mathematical Model of a

Single Main Rotor Helicopter for Piloted Simulation. NASA TM-84281, 1982.

7. Hohenemeser, K. H.; and Yin, S. K.: Some Applications of the Method of Multi-Blade Coordinates.

J. Amer. Hel. Soc., vol. 17, no. 4, 1972, pp. 1-12.

. Takahashi, M. D.: Active Control of Helicopter Aeromechanical and Aeroelastic Instabilities.

Ph.D. Dissertation, Dept. of Mechanical, Aerospace, and Nuclear Engineering, UCLA, Los

Angeles, CA, 1988.

. Bousman, W. G.: An Experimental Investigation of the Effects of Aeroelastic Coupling on Aeromechanical Stability of a Hingeless Rotor Helicopter. J. Amer. Hel. Soc., vol., 26, no. 1, Jan.

1981, pp. 46-54.

10. Kaplita, T. T.: UH-60A Black Hawk Engineering Simulation Model Validation Transient Response Comparison Data. United Technologies/Sikorsky Aircraft, SER-70983, Contract No.

NAS2-11570, June 1984.

11. Howlett, J. J.: UH-60A Black Hawk Engineering Simulation Program. Vol. I. Mathematical

Model. NASA CR-166309, 1981.

12. Howlett, J. J.: UH-60A Black Hawk Engineering Simulation Program. Vol. II. Background Report.

NASA CR- 166310, 1981.

13. Bramwell, A. R. S.: Helicopter Dynamics. John Wiley and Sons, 1976.

14. Bisplinghoff, R. L.; Ashley, H.; and Halfman, R. L.: Aeroelasticity. Addison-Wesley Pub. Co.,

Inc., 1955, pp. 251-281.

35
15. Johnson,Wayne: Applicationof UnsteadyAirfoil Theory to RotaryWings. AIAA J. Aircraft,

vol. 17,no. 4, Apr. 1980.

16. Zhao, X.; and Curtiss,H. C., Jr.: A Studyof HelicopterStability and Control Including Blade

Dynamics.TR 1823T,Dept. of Mech. Engr.,PrincetonU., Princeton,NJ, Oct. 1988.

17. IMSL Library ReferenceManual,9thedition. IMSL Inc., Houston,TX,1982.

18. Ballin, M. G.: Validation of a Real-TimeEngineeringSimulationof the UH-60A Helicopter.

NASA TM-88360, 1987.

19. Hilbert, K. B.: A MathematicalModel of theUH-60 Helicopter.NASA TM-85890, 1984.

20. Gerdes,W. H.; Jackson,M. E.; andBeno, E. A.: Directional Control DevelopmentalExperiencesAssociatedwith the UH-60A Utility Helicopter. U.S. Army Researchand Technology

LaboratoriesReportNo. USAAVRADCOM-TR-82-D-26,May 1983.

36
Table 1. Rotation sequences of blade coordinate systems

R to P P to L L to B

flpt :3p : -OR 3k : -.(/P (,k. " f L

(p ' -=3p (p : --f3 OX.+ oZ. ",f¢

fplt 3p : -OR 3k " -OP

(p ' i 3p (p ' - f 3

01. " ._(p

pflt 3p • -OR 3k " --OP

81,. : 5c3z' Cp • - f 3

(_p : -_0

lftp 3p" --OR (,'k + q'p : fP

lpft 3p " -OR (k : fP

3k : --_)L

_p " -f3

Ok + ok " ,fC,p

,4:-Or.

_p : -:-3

^

Ok : xcp

plft 3p : -,(tp (k + qp : -fP 3k : --_)L

Ok : [c3p (_p" -:-3

0# : ,f_;l,

Table 2. Function odd/even properties

Function Even/odd (.f 1,0 l, -"1) Equation number Figure number

._Xu, (E,O,E) 2.10 2.15

._k_-_u (O,E,E) 2.9 2.14

3I (O,E,O) 2.37 2.25

.V (O,E,E) 2.40 2.26

.g (0,0,0) 2.38 2.28

K (O,O,E) 2.39 2.27

37
Z:

%,

m

d.

"T'.

>,.

g.

I

i

°--

i

°--

-- ,,.,

_'_. m

,.., ,- z_ _,_,__ _xx_-.,x{_.<,-, _,,.,..,< _

T'l

I

38
o_

• o

I

_=

"i

m

I

M

I

-._ '_ _

.__ _. '_ ,, _

- o = .= ._ _ ._ _ "_ ._-: =

i

=

39
°_

m

i

I

°_

m

m

o_

i

!!

i

r_

i

q

_-(- _)(_

--

4O
A

m

I

J

Z :_ -_" --: LI Z :_

: ._ .__ _ -_

-_ _ _ .-_ ._ .

. _- _ = ._ ._ ,_.--"

--. Z ,.._ _ _ ",-_ ,_" ,.-." _-_

_ _ ._. - _ _._ _ "_.

-- _ _ _ _ .__

i. __ _ _ _ _- _- w -_

zZ

41
°_

j

J

j

._

,-n

z

.-- ,., v

,.., _--

,., .-- _ _--

' _ _

_ Z_._

Z_

<,-,

Z_

H

2.--

l

m

m

H

,--2"

5¢;.

L

z

42
.=_

=

>

m

=_-

o_

m

J

N

i

+

• o

• o

++

Z ._. _- ._. _ _ Z Z _ _ _

¢.

_, -,_ "=.--_ 5 5

_ M

I I

ZZZ Z Z Z Z Z_ _-

43
2..

:<

i8

m

>

"rV2.

L"2

_vH©

44
a..

<

,,,,,n

z

m

°_

.,.-

>

F

N

.m

0

J

A

m

a..

=z

m

_X_

-

= , g.= g_- ~

X

Z

r

X

Z

,_ < a_

J

Z_-ZZZ 0

Z_-_ZZ -z _

-<

.4

n

..e

G

II

4S
_c

m

Z

m

J

t_

A

v

m

X_-_

? _ Zzzz-E--==

_ _ ×_z_zzzz

-- - -->-x=_

z=- _Zz Zz _ z_ E E < - - -_ _ = = __, ;_

_ _______= _ _ _ -. _L_m__m.=__m.m.Em.=

Z _'-

__ Z

m

-4

v

u

46
Table 8. Flap-lag-torsion rotor nondimensional data

Variable Value Variable Value

Rotor Trim

n b 4

mass 1

0.0001

IB 0

0

-:So:: :

i.4er o 1

.li B 1

R 1

f2 1

.r b 0.5

.r I 0

.r 2 1

a 27;"

PA 4.8248

do 0.01

I'..'x 0.0024

I_'y 0.088542

/(- O. 16333

3if 38.1045

h 0.2

,3"

s:

o o] 0.083333 0

0 0.083433

n hrT t_._t 6

MPTTRM 30

n jr

n iy

ni 1

n jo

nK

nfeq

jr

if

i l

JO

Ifeq

1

3

1

3

4

[1,2,31

[21

[1,2,31

[31

[l,3,4]

[l,2,4,6]

47
Table 9. Flap-lag rotor/fuselage data

Variable Value Variable Value

Rotor/fuselage Trim

n b 3

3I b 0.209 kg

R 0.811 m

f2 ? rad/sec

[o o

I B 0 0.010069

0 0 o] 0 kg m 20.010069

mass 0

- Soll 1

e 0.0851 m

9 9.81 m/sec 2

i.4e ro I

x b 0.186 m

x 2 0.726 m

b 0.02095 m

MPT 10

a 5.73 1/rad

P.4 1.2761 kg/m 3

Twc 0.02618 tad

IOPT1 1

d o 0.0079

IOPT2 3

ICy 6.691 kg mZ/rad sec 2

Ix'. 30.659 kg m2/rad sec 2

Cy 0.003538 kg m2/rad sec

C: 0.007574 kg m2/rad sec

I(fz 68.03 kg m2/rad sec 2

I(fy 104.3 kg m2/rad sec 2

Cfx 0.08117 kg m2/rad sec

Cfy 0.4200 kg m2/rad sec

0.183

I F 0

0

M F 20.83 kg

h 0.241 m

n HTR. w 2

MPTTRM 30

njr 2

f(4) 0.01 rn/sec

t_ i l 3

0(I,3)

?2jo

nK

nfeq

Jr

il

Jo

00 rad

1

2

2

[1,21

[1,2,31

[31

°°l

0.633 0

0 0

kg m 2

K

t feq

nHLIN

MPTLIN

n j/.

n jr

3,

hie

i/

nif

if

ni z

il

_r

Af

Ax

[2,31

[1,21

Linearization

0

10

2

[1 21

2

[1,2l

2

[1,21

2

[l,2]

3

[1,2,31

[10 -5, 10 -5] rad/sec

[10 .6 . 10 .6 ] rad

[10 -5. 10 -5] rad/sec

[10 -6. 10 -6] rad

[10 -6. 10 -6" 10 -6 ]

48
Table10. UH-60A configurationdata

Variable Value I Variable Value

UH-60A

n b 4

R 26.83 ft

x b 10.833 ft

x2 24.78 ft

a 5.73 1/rad

do 0.01

Ttr_ 0.043595 rad

5' 32.1 ft/sec 2

mass 0

MPT 10

IOPTI 4

It3 [oo 0

0

573.42

0 573.42 o0 1

STt 28.433 ft

WLt 26.250 ft

Os -0.05236 rad

STc ? ft

WLc ? ft

MAXTR 40

px 1.22173 rad

RTR 5.5 ft

EPSTR 0.1 D- 3

bTR 0.40625 ft

.VbT n 4

arR 5.73 1/rad

eTR 1 ft

I6rn 3.0 slug ft 2

WLr 27.058 ft

STr 61.000 ft

<rx [0,0,- 1.5708] rad

_rv [?,?,0] rad

CL s [ 1.025,1.025,0.820]

CL l [0.75,0.75,0.890]

C L 2 [0.85,0.85,0.8001

aD1 [0.262,0.262,0.175] rad

aDz [0.349,0.349,0.524] rad

& D3 [0.524,0.524,0.698] rad

&D4 [ 1.047,1.047,1.047] rad

slug ft 2

f2

X 1

e

b

d2

Tw_

/9.4

- ZoSS

iAero

IOPT2

3IF

SS

sT,-,,

W L m

IF

ST,,

II/'La

QTR

3,I13T n

Corn

I£brn

],'ITR

OtTR

XbTn

TR13rk

TRLosS

IOPT4

n8

STs

BLs

WLs

STS

IOPT5

aOT s

bOTs

aTS

8.003 slug

? rad/sec

4.12 ft

1.25 fl

0.865 ft

1.2

-0.011258 rad/ft

0.002030 slug/ft 3

1

,)

5

? slug

1 ft 2

oft

?ft

[?1 slug ft 2

?ft

?ft

124.62 rad/sec

0.467 slug

0.01

-9920.8 slug ft2/rad sec 2

0.7002

-0.05458 rad/ft

2ft

0.8

0.7

1

3

[58.367,58.367,57.9171 ft

[-3.5,3.5,01 ft

[20.367,20.367,22.750] ft

[22.5,22.5,32.3] ft 2

[2,7,21

[0,0,0.081 rad

[0.12,-0.12,01 rad

[0.12,0.12,0.12] rad

49
Table 10.Continued

Variable Value I Variable Value i

UH-60A

CD o

C'D1

CD 2

C'D3

CD4

CD5

CtLO

Q U PB :v D

3LO

3_" PB ,\" P

110_m2 d

l13ml d

rl

ADW

Dt'D_ V

POSFUS

[0.01,0.01,0.02]

[0.1875,0.1875,0.043]

[0.3625,0.3625,0.360]

[0.4250,0.4250,0.580]

[0.9000,0.9000,0.875 ]

[ 1.200,1.200,1.100]

-0.4 rad

1.4 rad

-0.4 rad

1.4 rad

8

8

1.25 ft

0.08 rad

6

32.417

-2.5800

17.308

1

0

0

32.417

-2.5800

17.308

0

1

0

32.417

-2.5800

17.308

0

0

1

bTS

CTS

CtL,

QL 1

QL2

CtLOB.VD

aUp

,3L 0 B .V D

,3Up

l)0%n d

r23e n d

MPTDW

r2

Y.

n 4 f u._

[0.12,0.12,0.12] rad

[0.5,0.5,0.51

[0.262,0.262,0.349] rad

[0.524,0.524,0.436] rad

[0.786,0.786,0.698] rad

- 1.4 rad

0.4 rad

- 1.4 rad

0.4 rad

3

3

20

26.83 ft

[20,60,100,140,180,220] ft/sec

3

Trim

I_HTR M

i,,

if

il

Jo

IC

iu

ifeq

3

[1,2]

[?,?l

[1,2,31

3

[1,3,4]

[1,2]

[1,2,3,4,5,6,7]

f(4)

n j,.

niy

rli 1

njo

nK

ni u

nfeq

Linearization

groundspeed = ? ft/sec

2

2

3

3

3

9u

7

__S

?'_Jb

Eij r

nil

n if

ni 1

nje

nK

r'ti u

n i,Tt

10-5

2

2

6

3

3

1

3

1

6

MPTLIN

J_

jr

i:

if

il

Je

If

it,

if,

2O

[1,21

[1,2]

[1,2,3,4,5,61

[1,2,3]

[1,2,31

3

[1,3,4]

[11

[1,2,3,7,8,9]

50
Variable

Table 10. Concluded

Value ] Variable

Integration

Value

flint

TOLINT

r_je

F/jr

nil

rTi f

Djo

n I£

rl i u

ni 0

8

0.001

2

2

6

3

3

1

3

1

6

TI tt.4

NPTINT

J_

jr

i¢

iI

il

Jo

K

i u

iO

4

401

[1,2]

[1,21

[1,2,3,4,5,6]

[1,2.31

[1,2.3]

3

[1,3,4]

[11

[l,2,3,7,8,9]

51
J

J

Z

N

M

J

+..

,--,,+

"+_2_ = +_

__-_

_ _-- __" _-- _ '

_1 I t I

mll t I

__L__

$3
0

0

.,a

o

z

,m

L.

_ _ _ _ _ _ _ _ t_

_ tt _

_ __-

I1_ _ _ _ _ _ _ _ _ _ _ _ _ _ --

_ _--. _-- _ _ _

II I I

_._-_._

I II

_ I _ _ '- ..... _ "

u. ..... _ . ._-.4,_.,,___=_

I I I I _ I I I I

:_ aa

53
7

x I

RIGHT tiy

. /

'! Yl . ^

t)x _Oz/,, YOz' YCy

. .,c, 7/Am

i

_,r,,,. %_,,+Rr.yY,+Rr..I,_ dO,.v_,r_,._o.

zF Oz

Ce_Ce: ce ,,Se: -so,,

[TFI]= Se_Se_Ce:-Ce.Se: Se_Se_Se=+Ce_Ce: Se_Ce_

Ce_Se,Ce: + Se.Se: Ce_Se,Se: - Se_Ce: Ce_Ce,,

ZF/_ = (0. - O_So_)_F+ (4Ce. + O:Se_Ce_)OF+ (#:Ce.Ce, -4Se.l@

5' F/ I = ( O. -- _8:C8, -- 8"zSOu )z F

+(O;Co. -- O.eOgSOz + O'zS_zC_y + gzOxCo_Co_ -- O:_g'-'COz SO_t)OF

+(#:co.co, - 4O:se. co,, - #:Oyce_so, - Ouse_- #yg_ce_)@

Figure 1. Fuselage-fixed coordinate frame (F).

54
i

o <

uj_

_ 1-_ 1_ 0 _ _

I.I. Z

I-- u. >-l--

m_

Z

<a=- _j w I-"

I__. _ 1,'-_

o

,,'d,o 1-_

" i/ t=-

_ 4 .4 4 I

"-" I I ---

_ _q +

+ + + + _..

_ E._~

_ "',

_q I

II il II II <_

_-_ _ +

+ + + + :_

+ + + + +

II II II II II

=

m

I

I II

I

_q

+

=

I

I

II

b,l

+

II II

0

t_

0
A A

YF, ¥_s

xo s, x H

__ _/H

_F "t

_H

A

ZOs _F

[THF] =

I COs 0 -S_s -_

S_s SO s C_ s S,_s COsl

I

C_ s SO s -S_ s C_ s COs_j

CCH/F = 0

..:,

_H/F = 0

Figure 3. Rotor-hub-axis coordinate frame (H) and its transformation from the F frame.

56
R D -V a xw

N

zw,

-v a.zF ;_f, _

/ _

.-. ,,, -Va'X F \ Yw "_",_, -Va'Y F

= VELOCITY OF THE AIR RELATIVE TO POINT a

a

-C_fC_3 f -Cef S_f S_f 1

T FW = -S(3f C/3f 0

-S_f C_f -Sef S_f -C_f

Figure 4. Fuselage wind-axis coordinate frame (W) and its transformation to the F frame.

57
A

XH -"'

A

z R

')R

A

L, YH

A

-C_k S_k 0 1

S_k C_k 0

0 0 -1

--_ ° A

_CR/H = _ z R

-L_ "'A

C_R/H = _) z R

Figure 5. Main-rotor coordinate frame (R) and its transformation from the H frame.

58
A

z R

A

zB

"-_ A

rhe = e x R

-=ref = f _<L

"-_ A A A

rfb = XbXB +ybyB +zbzB

A

rfx = x x B

A A A

= XB + YB + ZB r fw n Xwn Ywn Zwn n = 1, 2,.. nAbld

Figure 6. Main-rotor-blade coordinate frames.

59
<x .rx

QO ÷

<N

<_Q" ®

e_

I

i i

H

I I

il

_J

e"

ee-

¢-,

t_

¢..,

_z

rr_

6O
.=J

'2

÷+ _

.._ Q'_ _i_ _. "_-4-

_ _ ÷

."r_

_ _ I + +

-t- _ ._'_

_ ÷ ÷ ÷

I

I I

_ ___ ._ :_

I I

i i

It II

m

I

I I

r-,

i

o_

-s

61
BLADE AERO

HUB

INERTIA/GRAVITY

A

R'II _ -Me -__.e __f_ -" l-Ff_ _. 1

e BLADE

..,. f INERTIA/GRAVITY

-QH

-JH

HUB MASSLESS BLADE

LINK

Figure 9. Main-rotor forces.

62
.,_ zB A

! d S, ,jv/ ?_,

co°T SQ°T 1

-SeT COTJ

A

zc

_c11 = _°B/I

c_C/I

QUARTER

_(/CHORD

/).

/_.,.,wrrw77/ll?_/I// _////7"_,./_ Vc

/_'rfTv//////////[//l_/ll VII/I///..J

- ._ /_\

" A

x c

Figure 10. Blade-section chord coordinate frame (C).

63
o

A,

<_>o

U=l

U.l >

LU

n,"

<N

't

<

X

<>.. >

<+

Xu

,>

C_l ,_I

il II

e_

<_.

X ,..

Iii

>u

<X

"7

e' _..

II I N

.,oI X

II X

_-_>0

=

\/

<_>O -_

¢N

u._

NO lr

i1

'r

<N>+>

<X

II

<N

<>. >+

>

<N>

_J

!1

tN<N>+

O

c,q< >.

.3I

II II

>

<N>

II >+

,e-

<N >Q+

>

(.I

>1 I

N I_1<_ ¢'_ u > = <X _ <x + <N

_

-_>

o° ,, , , , >+

E

:E+

:E

'7

II

<xE

E

:E

.tIE

II

<XE

A>3I >

<X

5"I

II

<N

.>

II >+A

<X>iI >U

<N

II

.>

It >+

=O

t--

e-OU6

64
A

YA

A

XH

t31c

A

xA

A

zA

Vtlc

A

YH

TAH]

[ -I CJlc S_ls S/31c S_lcC_ls

0 C_1 s -S_1 s

-S_lc S_lsC_lc C_lsC_lc]

-/Jls C_lc XA - _1c _/A + _ls S_lc ZA

CT ^

_A _h _ )

Vh = VELOCITY OF AIR RELATIVE

TO THE HUB POINT

Figure 12. Actuator-disk coordinate frame (A).

65
YF

xs

/k

YS

[TsF]

Ii -Cox S(°y+iTs)l (°;+iTS) S<xx S(uy+iTS) Cox So x

S(oy+iTS) -So x C(oy+iTS) Co x C(oy+iTS) J

zs

(Oy+iTS)

Figure 13.

A

z F

0 x

&

zs

Tail-surface orientation and aerodynamic forces.

66
C D

0

CD 3

C_D1 C_D2

CD 5

_D 3 _D 4 _--

y c_z

C D =

. o!

-OD2 < a- < aD2

(°_- _:) (°=-°_')(°:- °_")c

< a: < --OD_

a D2 < a- < _ and 2 - - -

Figure 14. Tail-surface drag coefficient break points.

67
-O_L2 -O_L1 -C_Ls

-CL 2

-CL S

CL S

0 _

C_Ls C_L1 °_L2 7

\GL 1 O'L.,

\aL., ]

C'L = (CL' -CL") (_z--CtL,) +CL.

\ C_LI _L_

-<o.<

- C_Ll <_ a: <_ --C,L. _

OL . < c_z < all

_L1 <--o.z < _-

Figure 15. Tail-surface lift coefficient break points.

68
//_F

xT

Py [TTF]

Cpy Sp x Spy -CP x SPy1

0 Cp x SPx

Spy -SPx Cpy Cp x Cpy j

A

xT

Figure

//£F

Px

zT

16. Tail-rotor orientation and aerodynamic forces.

69
(C(v) X

/

Figure 17.

y(-Zv)

FLAT WAKE

/

/

;x _(_1 ,_1,E1) ;z

Vy

Flat-wake model coordinate system.

70
--dL --.'_ I " i_- :'_,L o i_ '_

_-- ' _ L,-- " 4

l I ; i ._1 ._. j L-_L_!=

(_ CO 0'1 _ LO O0

' ' ' '7 '7 '7

[...I.._!...1 ,! i ._ .,I

e-

:l  ̧ :/_

'l

_L . . '_r_" ',, o

0 rI

C_

I!

I I I i

x

I I

_4

e0

eeG

0

E

°_

71
A

x V

A

YH

x H

TVH

Vv

FLAT

rhs

Py

Figure 19.

DINT OF

DOWNWASH

dw VELOCITY

A

Zs

_k

#Z F

Wake orientation for generating a surface downwash table.

#_F

Px

A

YS

72
I'N'T'A"ZE _:_MOOE'I

r

I TR,MI oo LL

NXT_TRM = 1 _ NXT-TRM = 2

LINEARIZE

_<= Ax+BU

y = Cx+DU u.I

09

Z

©

o.

o')

u.I

rc

k

M_

Figure 20. Overview of operations.

73
X:

r

ROTOR

BLADE

rrr

ISNROT _,_

T

VARIABLES

FIXED AUGMENTED

f f

I

_r

CONTROLS

r

ROTOR FIXED

BLADE

XX 6 U

IINROT

1

MODEL DYNAMICS

Feq:

Figure 21.

IONROT (_

req feq Xeq w y [ M ]

EQUATIONS MEASUREMENTS

General input/output model structure used by trim,|ineadzation, and integration operations.

74
• Z G

Up..----..

P A G S

Xp = fp(xp, Up)

yp = hp(Xp, Up) YA = hA(XA'UA) YG = hG(XG'UG)

xs = fs(xs,Us )

YS = hs(xs)

YS

YC

C

!

XC = fc(xc'Uc ) !_.

YC = hc(xc'Uc) I UC

Figure 22. General system structure for integration.

75
ADJUST v YGo TO ZERO f

xS

Figure 23.

Up

YA = UG

UG o

Yp

(UG-UGo)

Flow of calculation to find the initial conditions of the system of figure 22.

76
xC

Xp

x A

x G

x s

Figure 24.

xS Us*

Ys = Uc

Yp

l Yc

)

_r

w%

U A

YA = UG

YG = Us

YS

Xp

ZG

= xs

Flow of calculation to integrate the system of figure 22.

77
?8

X

n!u" ''_'L : w

O!u .... Z 'L = w

)_u .... Z 't = u

[(L-(W) _)!)qu+(u)_i ] _"

J'!u .... Z 'L = w

_!u ...._ 'L = w

qu .... _ 'l = u

[(L-(UJ)'4")qu+u] J_'

L!u'" '_'L =w

[(w)L!] x_

•_!u'"'Z'L="'

[("').h] _

"J!u .... Z 'L = uJ

qu .... Z 'L = u

[(L-(,.u)fi)qu+u] .J_

L!u" 'Z'L =w

[(w)L!] x.

•_!u. "Z'L =uu

=,

"_!u' "'L'L =w

qu .... Z 'L=U

r(L-(W).J!)qu+u] ?.

,.. ,,_

c

H

II

CO C)

Z

e,

.-i-

_=.

r-.

,..o

c

+

._,--

+ ,

4- II

.:._

e- Q.

ell

X

Z

÷

.,__-

¢-

+

,,o

rii

Z

Z

I!

_, c " = ,'-

E

q =q

>.

II

er

II

(_

Z

II

÷

-4-

c ," '- _,- E _-

4- II II _ II e,,- II

= = E _',.- E ._ E

_r _r _r

(3

w,-

+:TT T,

" " E .-_E .__ E

>.

,N

o_

t_

r',

.m

o

E

.J

E

>

>

t_

-?

t'N

o_

rr.
5

4

. 3

LU

z

O

a.

2

UJ

rr

CLJ

.J

-1

0

| I I I I f I I I

_F- = 1.125 _ = 0.07 /J = 0.35 _.

-_ ] -"'70 -50 MODEL: f \

_ _L1 - u. ")'- " CORRECT\ / \

_T = 5.0 Cw = 0.005 REVERSE "/ \

1 MODEL: FLOW / \

- INCORRECT /' / \

J REVERSE / _ J...._....._._...= \

i / \

REF. 7

-.T, _" _,-.-- - _ 7.:._.-........

I 1 [ I I I I I , I

.1 .2 .3 .4 .5 .6 .7 .8 .9

REVOLUTIONS

Figure 26. Blade responses from the present model compared with those from a flap-lag-torsion rotor

in trim.

79
N

-r-

>.-

Z

taJ

0

I,.I.1

e,e"

ii

8

7

6

5

4

3

2

1

0

I_/O © EXPERIMENT, REF. 9

THEORY

- LEAD-LAG /

_ BODY ROLL _ ^

L J v i J J

'7

z

r_

-I.0

-.5

.5

LEAD-LAG REGRESSING

I _ A I J

200 400 600 800 1000

ROTOR RATE, rpm

Figure 27. Modal frequencies and lead-lag regressing damping data from the present model compared

with those from a flap-lag rotor/fuselage system at zero blade pitch.

80
-2.5

-2.0

r_

_' -1.5

d

z

E

_; -1.0

a

-.5

-3.0

-2.5

"T

" -2.0

z

E

=E -1.5

a

-1.0

-.5

BODY PITCH _O _O

I t I I I A I ] J I

BODY ROLL

/

S

O EXPERIMENT, REF. 9

THEORY

0 0 = 0°

| | I I L J _ t _ I

200 400 600 800 1000

ROTOR RATE, rpm

Figure 28. Body pitch and roll damping data from the present model compared with those from a

flap-lag rotor/fuselage system at zero blade pitch.

81
-1.0

-.5

'7

d

z 0

<

a

.5

1.0

LEAD-LAG REGRESSING

(_ O EXPERIMENT, REF. 9

_ THEORY

I ] I I I I

500 600 700 800 900 1000

ROTOR RATE, rpm

Figure 29. Lead-lag regressing damping data from the present model compared with those from a

flap-lag rotor/fuselage system at 9" blade pitch.

82
O

0

O

650 rpm

720 rpm

760 rpm

0

.6

-.6

-.4

O

l O EXPERIMENT, REF. 9

-"-'- THEORY

900 rpm 0

0

O

-' - 2 0 2 4 6 8 10

BLADE PITCH ANGLE, deg

Figure 30. Lead-lag regressing damping data from the present model compared with those from a

flap-lag rotor/fuselage system at various rotor speeds and blade pitches.

83
-3.0

-2.5

-2.0

'T

z -1.5

O.

(:3

-1.0

-.5

0 0

-3.5

BODY PITCH

650 rpm O Q

LJ ©

O EXPERIMENT, REF, 9

THEORY

0 2 4 6 8 10

BLADE PITCH ANGLE, deg

-3.0

-2.5

Z -2.0

E.

=E

C3

-1,5

-1.0

©

BODY ROLL

650 rpm

8

0 t _ -5 t = _ I

-4 -2 -4 -2 0 2 4 6 8 10

BLADE PITCH ANGLE, deg

Figure 31. Body pitch and roll damping data from the present model compared with those from a

flap-lag rotor/fuselage system at 650 rpm.

84
>.

Irj

O

_1

I.iJ

>

"t"

o'3

<(

Z

O

r_

.-I

<(

Z

O

Z

LU

=Z

O

Z

Figure 32.

.02

0

-.02

-.04

-.06

-.08

.02

0

-.02

-.04

.02

0

-.02

f_

© EXP, REF. 5

THEORY

= 0.15

h = 0.1R BELOW DISK PLANE

I I I

_) = 45 °

-.04

-.06

.5 -1.0 -.5 0 .5

NON-DIMENSIONAL RADIUS

C, = 135 °

.0 .5

Downwash from the wake model compared with experimental results.

85
86
9

z 3

0 ""

! 1 ;

' I !

, i COLLECTIVE

r ! LATERAL i '

r

I " LONGITUDINAL/ ; I

I i 4 _ '

J

i i r

, p I i PEDAL

T--- ...... _-- ........ _ ............... ,-- .............. [..............

_- i I' i i

L ! , _ !

0 L , , , . , , J

==

-o 3

i,-

n,.

•.-= 0

-I

o

, 1 i MODEL, !

i

t ' !

I , I

-3 ..... ] .... L .........

• _ - MODEL I

<---- - ! i ,

w"

"t_3

u

I--

eL

-6

30,

i

20

er

0 ,

, , , b i , i , ,

I

_"_ _"_._'__ AE FJA TEST 212

..... ,,, ,4

:-'" 1......Mo,_,_,.: ,:,N"........

, MODEL

2 3

AEFA TEST 212

I

i _ LINEARIZEDI

L i_ I MOOEL _2

' 1

4 5 6 8

TIME, sec

Figure 34. UH-60A model responses with 1-in.-up collective at 1 knot.

87
N

4 r

I.-.

z k

o

L

_2 t , . .

E

2i

4 i

-3O

i AEFA TEST 212 J

• I

__ n r--- , ......

LINEARIZED i

MODEL

L

' I

I .

N

_-35

,¢

I,-

i,g

:>

-4O

r

! j LINEARIZED

MODEL

MODEL

AEFATEST212

I

_ t

rJ i i i

o

f !

,f' EST

/ 1

2 3 4 5 6

TIME, sac

r

MODEL

\

LINEARIZED

MODEL

_2= 26.882 rad/sec MF= 467.36 slugs ST m = ST c = ST a = 29.95 ft

38512 -1882

-1882 38512

f(4) = 1.68 ft/sec

WLm=WLc=WLa=19.29ft if =1,2

_y = 0.0698, 0.0698, 0 rad

iAero = 0 IOPT5 = 1,1,1

Figure 34. Concluded.

88
O

fz

O

cJ

c i COLLECTIVE_ __ 1 i

-_a,e I JI I LATERAL L LONGITUDINAL - ' I

. I ( i : j

I PEDAL

30 r

20

"0

P" 10

..I

..J

0 0

iT"

-10

! I =

\

! AEFA TEST 206 [/ _ t\ ",L

_'_--_.-_"_ -_--_---:_-_--- _. LINEARIZED

! I MODEL

i , I , , , [ ....

I

MODEL. 1

i

l

20 r

10 l

"r

_._--lo I

-20

i'

__10 =

,¢

i=

>.

-20 I I

f

1 2

_" _ _" . i t LINEARIZEO

AEFA TEST 206

3 4 5 6

TIME, sec

Figure 35. UH-60A model responses with l-in.-aft stick at 1 knot.

89
P4

.=

d

{3

Z

0

t_

o 3

u

h-

-3O

-33

G

_ -36

i11

-39

1 '"-.

L _ A r _-_ TEST 206 /'_'" '"' ........ i

,T i _, . : IVIUUC L

" " MODEL _

t MODEL

;

i

0 2 3

]

LINEARIZED i

I MODEL /_ I _ -

._ _ . .'.. I _ f "\ I 1/

"_ ?' '_ "i

//,-

[_""_ • MODEL .......... ,.

i TEST 206 / 1"-f v

i I I

4 5 6 7

TIME, sec

_= 26.987 rad/sec MF= 457.71 slugs STm = STc = ST a - 29.93 ft

[o 1

IF= 4 9 0 0

38512 -1882

-1882 38512

iAero = 0

WL m=WL c =WL a =19.29 ft

Oy = 0.0698, 0.0698, 0 rad

IOPT5 = 1,1,1

f(4) = 1.68 ft/sec

if = 1,2

Figure 35. Concluded.

9O
.J

O

kz

O

(J

I

i I

(.,C LCOLLECTIVEI i LONGITUDINAL I i ' i

L t _ !.

_a,e i ,____ _ _........-......._....

I I LATERAL : I

b

, t I i

/

_p- -, .............

I

PEDAL ' I :

................. t ---- _ ....

20

"_ 10

"O

_ 0

z

_- -lo

Q.

-2O

t ,! LINEARIZED _-" i

MODEL _-" _ MODEL

i 1

AEFA TEST 203

; I

I LINEARIZED MODEL

[ [MODEL

0

=.,

"_ -10

w"

k.-

-20

>.

-3O

Figure 36.

_'_"__AEF A TEST 203

', LIN EAR IZE/D " _ .'_'"_

i i MODEL .. --',_._

MOoEL""' ,_ t I

2 3 4 5 6 7

TIME, sac

UH-60A model responses with l-in.-left lateral stick at 1 knot.

91
<

z __ 0

..i

o L

i

! MODEL 4 .. I

i d i i i , i , 1

9 I

6

d

(J

<

P- 3

<

.J

/

i T 203_ ! AEFA TES

' I MODEL

i ,.'T-,

_- i ,';:Z_.,/ _ ....._ - ..........

-30

d

o -32

<

hrr

MJ

>

-34

0

I i h,

- ! i '_

- .... [............ ,/ \i_.. ,:..::,

2

T 1 "

I _ f'_ Y",,

! AEFA i _I _ _'_'_,

i TEST 203 _,--/" _)_t

:" '_'_ L._'."._, i . MODEL "Y'.":'

,:, .._:,,,,,,..;,; I,.L_._.,.I':.', ._ , . :...'.. . _

,_"_:A-.--.:_ ._i,:':%..':,':: F, :'i,:'_ii, , ._!!:,::- ",I

:' ,[r-,ej,-.a__,.,:..,L.. _ .-_ '_

_/ ! ; LINEARIZED_

! MODEL _ \

3 4 5 6 7 8

TIME _,ec

_ = 26.996 rad/sec M F = 458.64 slugs STm = STc -- ST a = 29.94 ft

0 38512 -1882

0 -1882 38512

f(4) = 1.68 ft/sec

WLm=WLc=WLa=19.29ft if =1,2

Oy= 0.0698, 0.0698, 0 rad

iAero = 0 IOPT5 = 1,1,1

Figure 36. Concluded.

92
=-; 6

o

rr

Z

o 3

0 L , , , ,

I

;_'C i i // COLLECTIVE E

, _ _ -

, _ _ 1

.... -:_ _ _--_--,---- ........ - ..... i- ....... -i ........ + .... 7"---1- .....

"_'-: a,e .... 1 ...... 7.............. ,......... ;).... i ................................... s...... [

' LATERAL

i LONGITUDINAL

, , L ......... i i

LINEARIZED" _

MODEL

1 i

LINEARIZED

IMODEL--I.....

AEFA TEST 209

MODEL "

r i

oi !

r

-30 q

i'

!

-6o i

o 1

I I i i

AEFA TEST 209

I i LINEARIZED

I _'_'_,...! !MODEL i

.

4 5 6 7 8

, h , = , , , , =

2 3

TIME, sec

Figure 37. UH-60A model responses with 1-in.-left pedal at 1 knot.

93
_4

u"

CJ

,¢

C3

Z

0

, 1 J ,,,t E

iN

d

<C

I-,

6 ,

4

i MODEL ..

. _2_" ---_ ........... /

..... / _ _ LINEARIZED

AEFA TEST 209 _/_ MODEL

-3,t

! r I

4 f

i

J_ ':'.. ! MODEL I LINEARIZED /-_

1 2 3 4 5 6 7 8

TIME, sec

iN

-32

u"

¢J

-33

"_'_"'_...... _' '"_'_"_'_ "t

t

! t

-34 f ,

0

_I= 26.925 rad/sec

iAero = 0

MF= 458.64 slugs ST m = ST c = ST a = 29.942 ft

o oj 38512 -1882

- 1882 38512

WL m=WL c =WL a =19.29 ft

qy = 0.0698, 0.0698, 0 rad

IOPT5 = 1,1,1

f(4) = 1.68 ft/sec

if = 1,2

Figure 37. Concluded.

94
9 i

O

I,.-

z

O 3

o_

I i

J I

I a,e I COLLECTIVE : ;q I

,/ I / 1 _ l

L ..................... . ...... _ ,........ _j_ -_ i i

c " ' / LONGITUDINAL : LATERAL I j

L/_............. L...... L............... ,.__,.:.... .......... ,

"_-_ p _ f i='EDAL I '_ .... l-- i

] i , , * i , , J . , 1 ' J

lOT

i i L i I

/ ! I .AE.ATEST,0_ i

1 LINEARIZED! _'-.

MODE, _ii,."

! MODEL,

h , , I h , i i

'

i h h i i

t , J i , 1 1

or--------- _-

=t- I

¢ r-

..i -10 t

°f og

-20 .... i

5 F

i

_ o_ .

"0

i= -5

,,_

>..

-10

0

1 r

........ _ _..<_.._ ._;_

I

11

1 2 3 4

TIME, .._lc

r"

t

i

i FA TEST 106

I

_-_ LINEARIZED

i I " ":> "" MODEL

i I ,\ I ,,

/

't,

! MODEL

5 6 7 8

Figure 38. UH-60A model responses with l-in.-up collective at 100 knots.

95
¢N

U

Z

0

-3

,- ; i /" MODEL

I AEFA TEST 106 .... _

i , _ ._

! i LINEARIZEDi

_- MO DE L

k

i I

i

i

N

U

_J

I-

<_

-3

i

I

I MODEL

LINEARIZED i

MODEL !

\

-20 _

¢N

-30

d

w

> L

-50 ]

o

I

1 4

TIME, sec

i j/ I

i ,;'_ LINEARIZED

MODEL

5 6 7

TEST 106

_Z= 27.018 rad/sec MF= 456.15 slugs ST m = ST c = ST a = 28.942 ft f(4) = 153.0 ft/sec

38512

-1882o -188238512o1

iAero = 0

WL m=WLc =WL a =19.29 ft

_y= 0.0698, 0.0698, 0 rad

if = 2,3

IOPT5 = 5,5,5

Figure 38. Concluded.

96
9_

E 6

J

0

DE

z

0 3

U

oi

a I !

De....!............. i...........

.... L........ r ......

p i

LATERAL

i!

i h , , i , , , , h , , , , k

l I

, J , i J

COLLECTIVE

/ L

t i

i .... i .... i

AEFA TEST 706

LINEARIZED ,.--_.,_.

MODEL

MODEL

20

==

.._ 10 i

m

= ol

-10

i •t _:_:''•'

/ __ L._ _ Z__

__ . ] /-.

LINEARIZED "

MODEL

"t3

I--

e"

-5

,.={

>.

-10

I AEFA TEST 706 ! LINEARIZED|

t ...... MODEL

0 2 3 4 5 6 7

TIME, sec

Figure 39. UH-60A model responses with l-in.-forward longitudinal stick at 100 knots.

97
¢N

u" o

L)

Z

0

-I

r

i

• _-j •

1

4

i

t

-3 , , i h i k , , I

i I

t

i

i i

,_ I , IMODEL i

AEFA TEST 706

I

i i h , , , _! _ .

¢N

u" o

cJ

<_

I-

,¢

..J

-3

AEFA TEST 706

I

, , i , n

-20

i

> -50

-60 .... 1

0 1

AEFA TEST 706

_2= 26.887 rad/sec M F = 448.05 slugs ST m = ST c = ST a = 29.325 ft f(4) = 175.6 ft/sec

IF= I4659 0 0 1

0 38512 -1882

0 -1882 38512

iAero = 0

WLm=WLc=WLa=19.29ft if =2,3

_y= 0.0698, 0°0698, 0 rad

IOPT5 = 5,5,5

Figure 39. Concluded.

98
e;

.J

O

m

H

z

o

u

9 ,

_a

3>°

0 I

t i'

COLLECTIVE 1 LATERAL LONGITUDINAL/

..... - .... -- 7 "_-"- .......

\ /

!" .i

............... I _ _._ _j .... ,...... :_,

I ! PEDAL , f

20_

_= 10

111

I-- 0 _----'_'_i <{

m

-I

-10

_20 I I

2O

=.

"_ lO

"0

=-

I-'

<{

R-

= 0

U

I-,

E

-10

10

==

o "0

1-

,<

-10

<

>.

L

LINEARIZEO

MODEL

p

• ', MODEL

_ } i I,'.

"_\ AEFA TEST 107 i '

'1

I

T T 1 B /

AEFA TEST 107 i MODEL >"

"_ LINEARIZED

, , , I MODEL L ,

-2O

Figure 40.

LINEARIZED _.

MODEL

I

3 4

TIME, see

I

.I ___,'._" _- .!/MODEL

5 6 7

UH-60A model responses with 1-in.-left lateral stick at 100 knots.

99
04

U

,¢

z

O

, !

i i '

P4

u

u

)-

<Z

.J

-3

r

1 I LINEAmZED

I [ MODEL

.MODEL

AEFA TEST 107

f

-2O

I,-

rr -40

_J

_>

-50

0

"_ -30 "_ _ _ _.._.._

LINEARIZED \\'"

E _'-_ MODEL ",_ ..MODEL

=

i _ k h i i i

2 3 4 5 6 7 8

TIME, $ec

_ = 26.798 rad/sec M F= 465.50 slugs STm = ST c = ST a = 29.058 ft

FI, -1882

38512 0 -1882

38512O1

WL m=WLc =WL a =19.29 ft

_y = 0.0698, 0.0698, 0 rad

iAero = 0 IOPT5 = 5,5,5

f(4) = 159.8 ft/sec

if = 2,3

Figure 40. Concluded.

IO0
9, ! l

i : i I

a,e _ LATERAL I COLLECTIVE I I ..... t

6 i_ t i t _ I ' _,,, "

j '_- ...... "_........ "r........ "'-- _ ..... ........ _-- .....

0 ¢-...,.. C ;

P"°: _-........ " " i J.............

o 3 ; I , / "t ....

........... _ - _ ! PEDAL _......

u L- _ i

_- =: J I

0 L .... ; , ,

LONGITUDINAL ! I

J i

J

i I

.. ....

I

-10 J .... _ ....

AEFATEST 10:

i , i i

1 /

1 /r'

LINEARIZED //

MODEL /

/

l/

v

\ /

\ " -/ I

\\ ' / / 1 .."

,,,-, L ,i

20

_.- 0

>.

-2O

I _ i ! LINEARIZED , _---_,.

.,,, .r / ' _,. iMODEL .- \,. / "I_'_ , ,"

//AE ATEST "%. I ' " ! .--"-----'--

I I

I I I , , I , ,

3 5 6 7 8

TIME sac

Figure 41. UH-60A model responses with l-in.-right pedal at 100 knots.

lOl
5 i

¢N

=.

£.i

z

o

-1

-5

i

"- !

i

b- i

I

i

MODEL '" !

LINEARIZEI_--______., _.. AEFA TEST 102

MODEL .... i

G

.J

-5

_ -20

_J

_ -40

>

-6O

I

L- !MODEL

. I J: i 1 ,/

..... i ." --,/"

L i AEFA TEST 102\

i , !

i

i . I J

i • 7

i j'

LINEARIZEDA

MODEL ._

.' ]

y

.,_ t i

I

I

[

2 3

LINEARIZED:

4

TIME, sec

_2= 27.023 rad/sec M F = 465.81 slugs ST m - ST c = ST a = 29.058 ft

38512 -1882

-1882 38512

iAero = 0

WL m=WL c =WL a -19.29 ft

e_y= 0.0698, 0.0698, 0 rad

IOPT5 = 5,5,5

f(4) = 161.6 fttsec

if = 2,3

Figure 41. Concluded.

102
,.=1

<

u.I

a:

GEN HEL ,_ = -0.008 -

MODEL ;_=-0.1332 :b4982

.____ L --L__,_

AMES GEN HEL *FEET, SLUG, SECOND, DEG

AMESGEN HEL,_ =+0.1289:1.4617

MODEL X = +0.2140 ---j.4201

el"

(a)

j ..L__..L__I J

--.¢¢-.

(b)

__J__..L 1 .I_._._L.._J._ I l

I--

CI:

,_1

ud

iP/J_ ,= _J_

AMES GEN HEL X = -0.2022 -- j.0384

MODEL ,_ = -0.2178 ± j.0266

1 I I I I l I I

U=l

,T,

AMESGEN HELX =-3.459

MODEL ;_ = -5.747

m L

(d)

I L I I I l I

u w q _ v p _,_ r

Figure 42.

1 knot.

AMESGEN HELX =-0.9497

(

._L ._ _ L l _ J

u w q _ v p _ r

Eigenvalues and eigenvectors of this report's model and the Ames GEN HEL model at

103
INPUT: ,5c, ins

OUTPUT: w, ft/sec

,. 0

MO E,

0

,x,

I _ _llll I L I 1 11111 I _ I I LIIII -40

INPUT: 5e, ins

OUTPUT: q, rad, sec

IIIIIII IIIr _ I_X% _ ,% "% • :% ,_

I 1 I I IIII1 1 J ] I Illll l t I 1 I till

-180 _ 200

I

I

I | 1 |till -270' , t = =L==LJ I I I Illlll I I _200 I I | I IIIIH I * 1111111 I

20

-2O

-4O

"D

z"

(D

INPUT: 'Sa' ins

OUTPUT: p, rad/sec

I,

/ \

! \

/11_

I ! 1 I I llli I I I I I Ill I ] I I I LI ILl

z

<¢

{3

-20

INPUT: _Sp, ins

OUTPUT: r, rad/sec

l

II

'"l XX

I I I i IIII L l I I IHII I J | I I111[

-200

"0

-300

,¢

"1-

o.

-400

.01

s _

i _ -200

"0

,<

"r

-400

l I 1 llllll I I 1 iJlltl I i 1111111

.1 1 10

FREQUENCY, rad/sec

Figure 43.

!

t

!

!

|

I

I

%

-600 L 1 i iiiiii i i i i11111 J | l *lllJl

.01 .1 1 10

FREQUENCY, rad/sec

On-axis Bode plots of present model and Ames GEN HEL model at l knot.

104
t--

t_

<

u.J

I t

AMES GEN HEL _ = -0.4906 : jl.634

MODEL ,_ = -0.5887 : 11.952

t t L £ A l

*FEET, SLUG, SECOND, DEG

AMES GEN HEL ,_ = 0.1295 : j 0.2229

MODEL 3 = 0.0415 : j 0.1862

l J i I £ I i

I--

la)

--_ 1 t t j j

(b)

{

r [l

I I I I I 1 £

GEN HEL ,_ = 0.1216

, = ,

(c)

GEN HEL ,_ = -3.660

MODEL ,_ = -5.000

(d)j l £ t i + l

GEN HEL ,_= -0.2215

= _ .

(e)

u W

_,_ AMES GEN HEL ,_,= -1.872

MODEL ,_ = -2.108

(f)

q ,._ v p _, r u w q __0 v P .3_ r

Figure 44. Eigenvalues and eigenvectors of present model and Ames GEN HEL model at 100 knots.

105
.,=

z

2O

INPUT: '_c'in.

OUTPUT: w, ft/sec

..... \

1 I 1 1 IlJll 1 I t l Ilil_ I 1 I | ll[ll

z

-20

-40

-60

INPUT: _e, m.

OUTPUT: q. rad/sec

t

p

J

.s

l I t t LIII I 1 I 1 I iIIl 1 I I I IILJI

-100 400

200 -2oo-

_" 0

_300 I l I i i i llll L i i I I llll 1 I i I IIIII -200

INPUT: 6a, in.

0 OUTPUT: p, rad/$ec

1/ s'

i i i i l iii i i i I i illl i i i i i i ill

INPUT: 6p, in.

OUTPUT: r, rad/sec

e_

z"

(3

-10

-20

-30

I

/

,/

z

i J i i i I iii i l l i i l ii! I i i l IIIII

-20

\',,

v v

l n I t I III i _ i l I llll I I l l lllll

-50

"%% # %_

"_ j x \

! i I I lllJl t l i i IIIII I J I i lll_a

.01 .I 1 10

FREQUENCY, rad/sec

400

0=

200

3:

0

.01

I

_,1

, _ '="_I_ l l I I Ill I i l i I fill l I I l I I Ill

.I I 10

FREQUENCY, rad/sec

Figure 45. On-axis Bode plots of present model and Ames GEN HEL model at 100 knots.

106

I. Report No. 2. Government Accession No.

NASA TM- 10226?

USAAVSCOM TM-90-A-004

4 Title and Subl_tle

A Flight-Dynamic Helicopter Mathematical Model with a Single

Flap-Lag-Torsion Main Rotor

7.Author(s)

Marc D. Takahashi

9.PerformingOrganizationName and Address

Report Documentation Page

Ames Research Center, Moffett Field, CA 94035-10(30 and Aeroflightdynamics Directorate, U.S. Army Aviation Research and Technology Activity

Ames Research Center, Moffett Field, CA 94035-1099

12. Sponsonng Agency Name and Address ' ' "

3. Recipienrs Catalog No.

5. Report Date

February 1990

6. Performing Organization Code

8. Performing Organization Report No

A-90037

t10. Work Unit No.

National Aeronautics and Space Administration

Washington, DC 20546-0001 and U.S. Army Aviation Systems

Command, St. Louis, MO 63120-1798

505-61-51

11. Contract or Grant No.

13. Type of Report and Period Covered

Technical Memorandum

14. Sponsoring Agency Code

15.Supplementary Notes

Point of Contact: Marc D. Takahashi, Aeroflightdynamics Directorate, USAAVSCOM

Ames Research Center, MS 211-I, Moffett Field, CA 94035-1099

(415) 604-5271 or FTS 464-5271

16. Abstract

A mathematical model of a helicopter system with a single main rotor that includes rigid, hingerestrained rotor blades with flap, lag, and torsion degrees of freedom is described. The model allows several

hinge sequences and two offsets in the hinges. Quasi-steady Greenberg theory is used to calculate the bladesection aerodynamic forces, and inflow effects are accounted for by using a three-state nonlinear dynamic

inflow model. The motion of the rigid fuselage is defined by six degrees of freedom, and an optional rotor

rpm degree of freedom is available. Empennage surfaces and the tail rotor are modeled, and the effect of

main-rotor downwash on these elements is included. Model trim, lineafization, and time-integration

operations are described and can be applied to a subset of the model in the rotating or nonrotating coordinate

frame. A preliminary validation of the model is made by comparing its results with those of other analytical

and experimental studies. This publication presents the results of research completed in November 1989.

17. Key Words (Suggested by Author(s))

Rotorcraft mathematical model

Flap-lag-torsion rotor

Helicopter

19.Secunty Classif. (of thisreport)

Unclassified

; 18 Distribution Statement

20. Security Classif. (of thispage)

: Unclassified-Unlimited

i

l Subject Category - 01

21 No.oi Pages 122. price

116 I A06

Unclassified

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