NASA Technical Memorandum 85870

NASA-TM-85870 19840015587

A Study of Helicopter GustResponse Alleviation.byAutomatic ControlShigeru Saito

December 1983

liBRARY COpy

.LAI'lGLEY RESEARt;H CENTERLIBRARY, NASA

HAIVWTON, VIRGINIA

NI\SI\National Aeronautics andSpace Administration

https://ntrs.nasa.gov/search.jsp?R=19840015587 2020-05-11T08:01:51+00:00Z

NASA Technical Memorandum 85870

A Study of Helicopter GustResponse Alleviation byAutomatic ControlShigeru Saito, Ames Research Center, Moffett Field, California

NI\S/\National Aeronautics andSpace Administration

Ames Research CenterMoffett Field, California 94035

SYMBOLS

a lift slope, or nondimensional distance between midchord of the wing andelastic axis (positive aft)

b semichord, c/2

C,C~e'Co coefficients in equations (73) and (80)

C' lift deficiency function, equation (B6)

Cdo airfoil profile drag coefficient

CH,Cy,CT horizontal force, side force, and thrust coefficients

CMX,CMY , CMZ rolling, pitching, and yawing moment coefficients

C~ lag damper coefficient

c chord length

Di induced drag

E( ) expectation of ( )

e distance between elastic axis and feathering axis at blade root

eo distance between mass center and elastic axis of blade cross section

Fax,Fay,Faz aerodynamic loads on the wing section, equation (Bll)

FX,Fy,FZ net hub forces equal to (H,y,T)

hub forces vector of the

gust frequency

HhT

blade, (F ,F ,F )0x x Z N

g

H

h

gravitational acceleration or' the gust velocity vector

horizontal force in the rotor frame = p(Rn)2(~R2)CH

plunging motion of wing cross section

i

J

moment of inertia of a blade about flapping hinge

imaginary unit = I=I

shaft inclination angle (positive fore)

quadratic performance function, equation (72)

Bessel function of the nth order

iii

k

feedback gain vectors, equation (26)

polar radius of gyration of a blade per unit span

Kalman gain at n time-cycle

L blade lift per unit span

M blade pitching moment per unit span

Max,May,Maz aerodynamic moments on the wing section, equation (B12)

variance of the error before the measurement at n time-cycle

m

N

p

Q

R

R'

r

r1;

net rolling moment = p(R~)2(~R2)RCMX

net pitching moment = p(R~)2(~R2)R~

net yawing moment = P(R~)2(~R2)RCMZ

mass moment of blade = rR r(r - re)m drJre

hub moments vector of the Hh blade, (~,My,Mz)1

mass per unit span

generalized mass of the jth mode, equation (2a)

number of blades

angular velocity of a rotor along the XB-axis

variance of the error after the measurement at n time-cycle

angular velocity of a rotor along the YB-axis, Floquet transitionmatrix, variance of the process noise, or hub inertia moment vectorin equation (B3)

generalized force of the jth mode, equation (2c)

generalized coordinate of the jth mode

kth constant coefficient in the solution of L1q, equation (7)

perturbation generalized coordinate vector, equation (5)

rotor radius

angular velocity of rotor along the ZB-axis

radial coordinate of blade or variance of the measurement noise

position of flapping hinge

position of lag damper

iv

T

T

t

u

u

v

v

w

w

x

Y

z

8

e

thrust in the rotor frame = P(RrI)2.(7fR2.)CT

or period

transfer function matrix

matrix, equation (59)

time

velocity of rotor along the XB-axis

tangential, normal, and radial components of velocity at a blade element

radial deflection of a blade; process noise

horizontal component of gust velocity

velocity of a rotor along the YB-axis

lead-lag deflection of a blade; measurement noise

lateral component of gust velocity

velocity of a rotor along the ZB-axis; gross weight

weighting matrices, equation (72)

flap deflection of a blade

vertical component of gust velocity

gust amplitude

precone angle

nondimensional radial coordinate = r/R

side force in the rotor frame = p(R~)2(7fR2)Cy

measurements

angle of attack

Lock number = pacR~/I~

perturbation from a steady value, or small increment

Kronecker delta function

blade trim pitch input = e ol + et(x - 0.75) + ,e1C

cos ~ + e1S

sin ~

blade twist rate

collective pitch control

lateral pitch control

v

a

]1

p

(J

w·J

Subscripts:

B

b

c

G

H

I

j

n

R

S

o

longitudinal pitch control·

control vector, equation (59)

control inputs, equation (57)

inflow ratio = (vS + V sin is)/Rn

advance ratio = U cos is/Rn

mean induced velocity

air density

solidity = NC/TIR

torsional deflection of a blade, inflow angle, or phase angle

phase shift in figure 17

azimuthal angle

rotor rotational speed

gust angular velocity = 2TIfG

jth natural frequency of a blade

lag frequency ofa blade

body frame

blade frame

control or cosine element

gust

rotational hub frame

inertial frame

jth mode

Hh blade

n time-cycle

rotor frame

sine element

initial value, amplitude, uncontrolled value, or origin

vi

Superscripts:

C) trimmed value

() d( )/dt

() estimate of ( )

( ) , d( )/dX

( ) T transpose of ( )

vii

SUMMARY

Two control schemes designed to alleviate gust-induced vibration are analyticallyinvestigated for a helicopter with four articulated blades. One is an individualblade pitch control scheme. The other is an adaptive blade pitch control algorithmbased on linear optimal control theory. In both controllers, control inputs toalleviate gust response are superimposed on the conventional control inputs requiredto maintain the trim condition. A sinusoidal vertical gust model and a step gustmodel are used. The individual blade pitch control, in this research, is composedof sensors and a pitch control actuator for each blade. Each sensor can detect flap­wise (or lead-lag or torsionwise) deflection of the respective blade. The actuatorcontrols the blade pitch angle for gust alleviation. Theoretical calculations topredict the performance of this feedback system have been conducted by means of theharmonic method. The adaptive blade pitch control system is composed of a set ofmeasurements (oscillatory hub forces and moments), an identification system using aKalman filter, and a control system based on the minimization of the quadratic per­formance function. Calculations for the individual blade pitch control system showthat the thrust fluctuation can be reduced by more than 50% in the low gust frequencyrange. In addition, it was found that the adaptive blade pitch control system canalso be useful for high frequency gusts.

1. INTRODUCTION

The helicopter has the capability of flying close to the ground where the motionof the atmosphere may be thought of as turbulent flow. To avoid structural vibra­tions and unfavorable dynamic characteristics in flying and riding qualities causedby gusty winds, it is necessary to analyze the gust response of a rotary wing air­craft and to develop a scheme to alleviate such responses.

Studies on the gust response of a helicopter rotor are reported in refer-ences 1-9. Many different assumptions were considered in those analyses: for exam­ple, quasi-steady or unsteady aerodynamics; constant or variable induced velocity dis­tribution; rigid or flexible blades, with or without hub motion; and sudden or gradualpenetration into the gust. In references 5 and 9 the wind tunnel response of a rotorencountering a vertical gust was experimentally investigated. Experimental data forvertical gusts show that the rotor thrust response is most sensitive to the gustamplitude and frequency. For the sinusoidal gust with frequency wG' the thrustresponse characteristics have not only a frequency wG component, but also(wG + nQ) (n = Il, I2, .• \.) components in the fluctuation, where Q is the rotorrotational speed. In this thrust response, the wG component of the oscillatorycharacteristics is dominant in the comparatively low gust frequency range (wG « Q).Other researchers have investigated the influence of atmospheric turbulence on rotoraeromechanical stability. In references 10-18, rotor stability analyses are reportedfor the case of a rotor encountering a gust. Typically the turbulence is a randomgust with stochastic properties.

Attempts to alleviate gust-induced vibration have been made in the workdescribed in references 19-37. Among them, Briczinski and Cooper '(ref. 19) andBriczinski (ref. 20) investigated the effect of a rotor/vehicle state feedback sys­tem on the handling qualities of a helicopter, specifically the characteristics con­cerned with gust response. It was found that the feedback scheme of the rotor tip­path-plane or body-state was very useful as a means of gust suppression. Frick andJohnson (ref. 21) and Johnson (ref. 22) studied the performance of an optimal controlsystem applied to proprotor/wing response to vertical gusts. In that investigation,the von Karman model or the Markov-process gust model were assumed as gust models,and linearized state equations were assumed to govern the motion of the rotor andwing for the rotor model. Optimal feedback inputs were determined by using linearoptimal control theory, and state variables were estimated by a Kalman-Bucy filter.Significant and simultaneous reduction in the rotor and wing response was achieved.Ham et al. (ref. 23) and other investigators (refs. 24-26) have dealt with a classicparameter optimization method to alleviate the gust response of the tilt rotor air­craft. A von Karman gust model was also used in these studies. Zwicke et al.(ref. 27) and Taylor et ale (ref. 28) investigated the performance of the optimalsampled-data feedback system on the gust response. Zwicke also studied a suboptimalfeedback system derived from the above control system; a significant reduction in thegust response was achieved by using this suboptimal feedback system. Ham andMcKillip (ref. 29) and Ham (ref. 30) developed an individual blade control (IBC)design for gust response alleviation, which has also been applied to suppressingblade stall-induced vibration, lag damping augmentation, and other dynamic problems(refs. 31-36). Saito et al. (ref. 37) have also studied a simple feedback system toalleviate rotor gust response. In their control scheme, individual blade pitch anglecontrol, using scheduled feedback gains derived from analytical calculation, was used.

There has been significant progress in vibration reduction techniques for heli­copters, theoretically and experimentally, in the past decade (refs. 37-83). Inthese vibration reduction systems, control schemes, known as multicyclic (ref. 45),or higher harmonic control (ref. 61), were applied to reducing the inherent vibratoryresponse of a helicopter. Pitch inputs at harmonics of the rotor rotational speedare used. Typically, the helicopter is represented by a linear, quasi-static fre­quency domain model. The relationship between control inputs, e, and outputs, z(which can include loads, vibrations, and rotor performance parameters) are modeledby a transfer matrix TC' Theoretical and experimental results show that the levelof vibration in a helicopter can be significantly reduced by using controllers inwhich the transfer matrix is updated by a Kalman filtering scheme. These vibrationreduction systems are reviewed in detail tn reference 70. Investigations haverecently been extended to the analysis of nonlinear effects on controller perfor­mance (refs. 71 and 72). Other researchers investigated structural modification ofthe rotor blade or the fuselage body for vibration reduction (refs. 79-83).

In this report, a gust alleviation system is ,studied using two control schemes:first, an individual blade pitch control scheme, and second, an adaptive blade pitchcontrol scheme based on linear optimal control theory. The emphasis is on the appli­cation of these controllers to alleviate gust response as measured by blade flappingand hub oscillatory forces and moments. In this study, the local momentum theory(LMT) (ref. 84) is used to calculate timewise vibratory airloadings and moments atthe rotor hub. The LMT is based on the instantaneous balance between the fluidmomentum and the blade elemental lift at a local station in the rotor rotationalplane. Therefore, this theory can be used to evaluate timewise variation of airload­ing, aerodynamic moments, and the induced velocity distribution along a blade span.The LMT has been applied to study many rotary wing phenomena (refs. 9, 37, 85-87).An H-34 rotor model with four articulated blades is used. The blades have full

2

flexibility in the flap, lead-lag, and torsion directions. The perturbationalmotion of the fuselage is not considered, and the shaft inclination angle is isassumed to be zero (perpendicular to the flight direction).

The author expresses his great gratitude to Dr. Wayne Johnson and Dr. WilliamWarmbrodt for their review and valuable suggestions. Further thanks are extended tothe late Mr. John L. McCloud III and to Mr. Stephen A. Jacklin for their kind andencouraging discussions. The author also acknowledges computer operation support bythe people of NASA Ames Research Center. This study was made possible by the NationalResearch Council Associateship.

2. INDIVIDUAL BLADE PITCH CONTROL

In this section, an individual blade pitch control scheme is studied as a gustalleviation system. First, the harmonic method is applied to the theoretical calcu­lations of the rotor gust response with a simple feedback control system for bladepitch. A similar feedback control system for an individual blade was investigatedfor a rigid blade in reference 37. The analysis is extended here to include elasticblades; therefore, sensors to detect blade deflection (flap, lead-lag, torsion) areconsidered to study the sensitivity of the control system. A blade stability analy·­sis with the feedback control system was made using Floquet theory. Finally, for aspecified feedback gain, this control system is validated for a rotor penetratinginto a vertical gust.

Theoretical Calculation by the Harmonic Method

Theoretical calculations of rotor response tb vertical gusts in forward flighthave been made by using the harmonic method. Derivation of the isolated blade equa-·tions of motion and method of solution are based on references 88-95 with the excep-­tions that the hub motion (U, V, W, P, Q, R') is replaced by a vertical gust velocitywG. Coordinate systems used in this study are briefly explained in appendix A. Thedisplacements of an elastic blade are expressed as follows:

w v (1)

~here qj is the generalized coordinate of the jth coupled mode and Wj' Vj' and~j are the corresponding jth coupled mode shapes. When the associated jth modalfrequency is denoted by Wj, the following equation can be derived according to theRayleigh-Ritz approach (ref. 96):

where

(j = I, 2, 3, . . '.) (2a)

3

(2b)

Qj jiR ({Max _ m[eg + k 2 (e + Q2 6 + 2Q6v') + e6CeoQ2 - 2Qu)]}~jo

+ [Faz - meg + e6 - eQ2e) - 2mQW~V]Wj + {Fay + m[-2Qu + 2eQ(v' + 6w')

+ e66 + Q2(eo + 2e)] - 2mQU}Vj)dr (2c)

In this study, the radial extension of the blade is neglected in the blade equations.However, to take account of the Coriolis force caused by the flapping, u is approxi­mated as

u = {r - ~r [1 + (v~ + V,)2 + (w~ + W')2]dr} _ [r _ ~r (1 + V~2 + W~2)dr]

= -1/2 rr (v'2 + w'2)dr _ rr (v'v' + w'w')drJ_ J_ 00o 0

co

(3a)

where

(3b)

and w~ is the time-averaged flap displacement. The lead-lag displacement isomitted in equation (3). More detailed expressions of Max' Faz ' Fay' etc., shownin equation (2), are given in appendix B. For this study, consider the vertical gustcomponent wG as an external exciting source. By introducing wG into the aero-

. dynamic loads and eliminating W in reference 89, the blade equation of motion,including gust excitation, can be obtained. Equation (2) yields a set of simulta­neous differential equations for qj' A trimmed value of qj can be obtained bysetting all perturbing values from trim condition to be zero. When the gust velocitydefined in the rotor frame

(4)

is superimposed on the trim condition, additional blade motion will occur, where wGois a constant real number. If the gust-induced blade response amplitude is assumedto be small, the incremental blade motion can be described by a small perturbation6qj from the trim value qj' Furthermore the perturbation generalized coordinatevector, 6q, is defined as

(5)

It is then easy to show that 6q can be described by a set of second-order lineardifferential equations. Hereinafter, all quantities (i.e., elastic deflections,

4

aerodynamic forces, and moments, etc.) should be considered as perturbed values froma trim position.

In forward flight, coefficients of the differential equations become time­varying and can be written as follows:

3

Em=-3

00

{[Am exp(im~)]~q + [Bm exp(im~)]~q + [em exp(im~)]~q} = E Fn exp[i(wGt + n~)]n=-oo

(6)

where Am' Bro, em are time-varying, periodic, complex coefficient matrices, and Fnis a complex vector. These coefficients are shown in appendix C. Let us assume thesolution of equation (6) as

00

1:. qk exp[i(wGt + k~)]k=-oo

(7)

where qk is a complex, constant vector. If equation (7) is substituted into equa­tion (6) and the same harmonic components on either side are equated, we have aninfinite number of simultaneous equations involving qk (k = -00, ••• ,00).

00 00

~ ~ [-(wG + k~)2Am + i(wG + k~)Bm + Cm]qk exp[i(m + k)~]exp(iwGt)k=-oo m=-oo

00

E Fn exp(in~)exp(iwGt) (8)n=-oo

By eliminating exp(iwGt) from this equation, equation (8) becomes

00 00

I: I: l\nkClk exp[i(m + k)~]k=-oo m=-oo

00

I: Fn exp(in~)n=-oo

(9)

Equating (m + k) to be n on the left-hand side of equat,ion (9), and after mathemati­cal manipulation, the following expression results:

where

n+3I: Rn-kqk

k=n-3Fn (-00 < n < 00) (10)

For the practical calculations, we assume that

(ll)

for (12)

Then the following equation will be obtained:

5

R-k.-k

R_k+s .-k

R-k.-k+l

o

KL'lq = E qk exp[i(wGt + k1/J)]

k=-K

R_k.-k+s

o

=

~"-N'

.(13)

(14)

where n is truncated at n = N'. By solving equation (14). constant amplitudes of(2k + 1) generalized coordinates. qk' can be determined. and the blade perturbationmotion can be approximated by equation (13).

Hub Forces and Moments

In the previous section. the blade equation of motion and its solution using theharmonic method are discussed for a sinusoidal vertical gust with frequency wG.' Inthis section. hub forces and moments of a rotor responding to a gust are derived.Therefore. we can refer to these responses as "rotor impedance" in the general sense.In this study.airloads are obtained by integrating section loading from two­dimensional quasi-steady thin airfoil theory. The lift deficiency function C' isassumed to be unity.

The solution of the generalized coordinate ~q is expressed by equations (13)and (14). and perturbations of blade deflections from a trim position are given bysubtracting an equilibrium value from each expression in equation (1). Let vectors~F~ and ~M~ denote the variations in hub forces and moments. respectively. owing tothe ~th blade. The augmented vector [L'lFI.~Mi]T can be expressed in the followingform in the rotating hub frame:

(15)

where {Zl} is a complex vector which gives the loads of a completely rigid blade. and{Z2}' {Zs}' and {Z4} denote complex matrices that account for the effects of theelastic blade displacement. The terms {Zl} through {Z4} are all functions of theazimuthal angle of the ~th blade. 1/J~. and each of them has harmonics in 1/J~ up tothe third harmonic as follows:

6

3

{Zr} = E Zlm exp(imljiR,)m='-a

a{Z2} = E Z2m exp (imlji~)

m=-a(16)

a{Za} = I: Zam exp(imlji~)

m=-3

3

{Z4} = I: Z4m exp(imlji~)m=-a

Substituting equations (7) and (16) into equation (15) results in the following form:

(17)

where

(18)

Here Zlm and Z5m denote a constant vector and a constant matrix, respectively.

The hub loads in the rotating hub frame can be transformed to the rotor frame asfollows:

(19)

where THR is denoted by equation (A6) in appendix A. Introducing

(20)

7

gives the following form:

C:l- P~' m~3 {J-+ kt- KPZ,mqk exp{i[wG t + (k + m+ P)';tll}

1 (k+p)+3 00

+ 1: 1: 1: ~Z5,V-(k+P)qk exp[i(wGt + V1/!Q,)]p=-l v=(k+p)-3 k=-oo

v=n+m+p

v = k + m + p

(21)

where Kp is a complete matrix.

For a practical calculation, the limits of nand k are taken as N' and K,respectively. From equation (21), the following equation can be obtained:

M+lt

= 1: Zv exp[i(wGt + v1/!Q,)]v=-M-lt

where

1.

Tvn = 1: ~Zl,v-(n+p)p=-l

1

Svk = 1: ~Z 5, v- (k+p)p==-'-l

r- (N' > K)M =

K (K > N')

(22)

(23)

where Zv is a vector and denotes the vth harmonic component of the hub loads inthe rotor frame. Since the Fourier coefficients have been truncated at the N'th orKth harmonic in the blade motion, the hub loads should also be truncated at theN'th or Kth harmonic. The total hub loads can be obtained by summing up equation (22)for all blades. Using the multiblade summation formula

8

N {N exp(iv$*)1: exp (iv$R) = 0JI,=l

the following final expression can be obtained:

v/N = integer

otherwise(24)

(25)

where N denotes the number of blades, q is an integer, and $* is the azimuthalangle of the reference blade.

Individual Blade Pitch Control and Stability Analysis

The feedback control system which can control blade pitch individually is dis­cussed in this section. Let us consider the following feedback system:

1'18 = (K I'1w + K. I'1w + K.. I'1w) + (K I'1v + K· I'1v + K.. I'1v) + (K", M + Kl 1'1~ + K;k 1'1~)w w w v v v ~ ~ ~

T T+K

T l'1ep(r)= K I'1w(r) + K I'1v(r)w v <j>

where

K (Kw' K. , T= K.. )w w w

K (K , K. , TK.. )v v v v

K<j> (K<j> , K~,T

K")<j>

I'1w(r) [l'1w(r) , I'1w(r),.. T

= I'1w(r)]

I'1v(r) [l'1v(r), I'1v(r), I'1v(r)]T

L1</> (r) [M (r) , L1~(r), 1'1~(r)]T

(26)

Kw, Kv, and lCij> are feedback gains and I'1w{r), L1v(r), and L1<f>(r) are flap, lead-lag,and torsion blade deflections from trim position at r, respectively. By usingequations (1) and (5), I'1w, I'1v, and 1'1</> are given as follows:

00

L1w = 1: (wj I'1qj' I'1qj' .. )Tw. w. I'1q ..j=l J J J

00

I'1v = L (vj I'1qj' vj I'1Qj' v. .. ) T (27)j=l J

L1qj

00

L1<f> - L (tPj I'1q. , <P j I'1qj' <P j.. )T

j=l JI'1qj

9

For simplicity, let us consider only the flap defl~ction of a blade; that is,

K = K = 0v ep

Then ~8 becomes

M = K ~w + K ~w + K.. ~ww w 'w

and ~q is

From these equations, the following expressions are obtained:

K

~w = 1: (w1qlk + w2q2k + ... + wLqLk)exp[i(wG + k~)t]k=-K

K~w = i I: ,(wG + k~) (W1qlk + ... + wLqLk)exp[i(wG + k~)t]

k=-K

K~w = - I: (wG + k~)2(wlqlk + ... + wlqLk)exp[i(wG + k~)t]

k=-K

Substituting equation (29) into equation (20) results in the following form:

M = K ~w + K. ~w + K.. ~ww w w

K

= 1: ~8k exp[i(wG + k~)t]k=-K

(28)

(29)

~80 exp(iwGt ) + ~8_1 exp[i(wG - ~)t] + ~81 exp[i(wG + ~)t] + . . (30)

In equation (30), ~8k (k = -K, •. • , K) are considered as control inputs. There­fore, appropriate combinations of ~ek may enable us to alleviate the oscillatorygust response of a helicopter rotor. The extension of this idea to an optimal bladecontrol algorithm will be discussed in the next section.

By substituting equations (26) and (27) into equation (C5) (appendix C), thefollowing equation can be obtained:

10

(31)

where

A~ .1.J

c ~ .1.J

A •• - D. (K.. t"w + K•• t"v + K;h t,,~)1.J J w V 't'

= BiJ

. - D.(K. ~w + K. ~v + Kl ~~)J w V 't'

(32)

= JRD.J 0

In this feedback system, each sensor (that detects blade deflection) islocated at radial position r along the blade span. Signals from eachtrim values subtracted in order to get pertu~bation quantities such asBy multiplying these perturbation values by feedback gains, this systeminputs to be fed back to the blade pitch control. In this process, anythe sensor/actuator are not considered.

assumed to besensor havet"w, ~v, t,,~.

generatestime lags of

Now let us consider the stability of the blade equation of mot ion expressed byequation (31). In this equation, each coefficient, Alj, Blj' and G:lj' is time-variantwith the fundamental period T = 27f. In this study, F10quet theory (refs. :97-101} isused in order to analyze- the stability of a blade. Define a new variable set as

By this transformation, equation (31) becomes a first-order differential equationwith zero external force, as shown by the following:

[){] = -[E .. (t)][X]1.J

where

(33)

(34)

[[A~.][B~.]

[E .. (t)] = 1J 1J1.J -I

(35)

and I is a unit matrix. According to the Floquet-Liapunov theory, we can assumethe following ;solution:

(36)

The iG(t) is a periodic function with period T, and the nk are complex character­lstic numbers. The ~k are constants derived from initial conditions, that is:

11

The Floquet transition matrix [Q] of this system is defined by

[X(T)] = [Q][X{O)]

(37)

(38)

The stability. of the system, therefore, can be completely determined from the valuesof nk' Inserting the relation for {ak} of equation (37) into equation (36), oneobtains:

[X(t)] = [G(t)] [-exp(nkt)] [G(O) r 1 [X(O)]

Introducing the state transition matrix [~(t)] defined by

(39)

[XC t) ]

[~(O) ]

= [~(t) ](X(O)]}

= [I](40)

one obtains by comparison of equations (39) and (40)

[~(t)] = [G(t)] [-exp(nkt)] [G(O) ]-1

or

[G(t)] = [~(t)][G(O)][-exp(-nkt)]

Setting t to be T and considering [G(T)] = [G(O)] gives

Thus

(41)

(42)

(43)

[cj>(T)] = [Q] (t = T) (44)

Substituting equation (40) into equation (34) gives the following equation:

d[Ht)]/dt + [Eij(t)](~(t)] = [0] (45)

Equation (45) is solved by numerically integrating from t = 0 to t = T, with ini­tial conditions given by equation (40). Eigenvalues Ak are given, by using equa­tions (43) and (44), as

= [-exp(-nkT)] = [G(O)]-l[Q] [G(O)]}

= [~(T)]

Numerical Calculations

(46) ,

In this section, numerical calculations are conducted to determine the capabilityof the individual blade pitch control system. The properties of the blade used inthis study (an H-34 rotor blade) are given in table 1. The blade structural charac­teristics and rotor operating conditions are given in tables 2 and 3, respectively.

12

Gust shape and sensor locations on a blade are schematically depicted in figure 1.The hub forces (Fx ' Fy , Fz) and the hub moments (Mx , My, Mz) in the rotor frame areshown in figure 2. Blade natural frequencies and mode shapes were calculated usingthe Ho1zer-Myk1estad method (ref. 9). Figure 3 shows the natural frequencies andmode shapes for the rotor blade.

All quantities used in this calculation are expressed as complex numbers, forexample, wG = wGo exp(iwGt). The undeformed elastic axis is assumed to lie in theXHZH plane and to coincide with the aerodynamic center axis (eo = 0.0, a = -1/2).Eight coupled modes (~ = 8) were used (see fig. 2). In figure 3, the f1apwise bend­ing modes (Nos. 2, 3, 5, and 7) and the lead-lagwise bending modes (Nos. 1,~, and 8)correspond to those of a blade with collective pitch angle of 80

; the sixth mode isa torsional mode. The highest harmonic numbers are assumed to be truncated at 5 forthe gust expression (n = 5) and for the solution of the generalized coordinate(k = 5). Correspondingly, Iql ~ M/N reduces to q = 0, ±1 in equation (25). Underthe operating conditions, the equilibrium positions (in this case, the time­independent rigid body portion) of the blade deflection are as follows: the flapdeflection (w~) is 0.11 rad, the lead-lag deflection(v~) is -0.017 rad, and the tor­sional deflection (~o) is -0.0004 rad. These values were used in the calculation inorder to evaluate the effects of the in-plane Coriolis force. The flowchart of thecomputer code for the harmonic method is given in appendix D.

Figures 4(a) to 4(g) show the rotor response toa sinusoidal gust. The rotorimpedance is defined as the amplitude ratio between the additional responses of therotor from the steady responses and those owing to sinusoidal gusts with unity ampli­tude (wGo = 1.0 m/sec). Phase shift to a vertical sinusoidal gust is also shown infigure 4. The horizontal axis denotes nondimensional gust frequencies divided by therotor rotational speed, which varies from 0.0 to 1.0. Each figure (figs. 4(a)-4(g))includes the responses at frequency (wG) and (wG ±4Q). The response at frequency(wG) is hereinafter called the major response (shown by solid lines in the figures).The responses with frequency (wG - 4Q) and (wG + 4Q) are called the subharmonicresponse and the superharmonic response, respectively. In the case of the horizontalforce Fx (fig. 4(a)), each response increases gradually as nondimensional frequency(wG/Q) increases. The major response has a peak at about 0.7 and has a minimumaround 0.9. This phenomenon comes from the effect of the lag dampers. Without anylag dampers for an articulated blade, the responses will diverge at lag frequencYL(wL - Q). Tpe lag dampers suppress this extremely large response. The symbols (.)on the horizontal frequency axis denote the location of (wL/Q) and (wL - Q)/Q. Thesubharmonic response is slightly greater than the superharmonic response in the non­dimensional frequency range of 0.2 to 0.8. In the case of the side force Fy(fig. 4(b)), the tendency of each response is similar to that in the case of thehorizontal force. There is a sharper reduction in response at (wG/Q) = 0.9 for themajor response. For both figures 4(a) and 4(b), the major response is dominant upto 0.8. The subharmonic response is again greater than the superharmonic responsein the nondimensional frequency range of 0.2 to 0.8. For the vertical force Fz(fig. 4(c)), a significant difference can be seen relative to the other impedances(Fx ' Fy ). The major response decreases gradually as the gust frequency increases.The subharmonic response is almost the same as the superharmonic response. Theamplitude of the maj9r response of Fz is greater than that of Fx or Fy • Quali­tatively, the major response of the vertical force is from 2 to 100 times larger thanthe other responses below the nondimensiona1 frequency 0.5. As the frequencyapproaches 1.0, the three response amplitudes become similar.

13

Concerning the phase angle characteristics of the hub forces, the phase shiftof the subharmonic response is relatively insensitive to the gust excitation com­pared with the other responses. From these calculations, the angles are about +90°for the horizontal force, about -150° for the side force, and about -90° for thevertical force. On the other hand, the phase shift characteristics of the majorresponse depends on the gust exciting frequency. Those angles vary from -180° to+180° over the frequency range for the horizontal force. However, in the case ofthe vertical force, it seems to be insensitive to the vertical gust. It is almostconstant at about -10° to 20° below the nondimensional frequency 0.5. These resultsagree with experimental results (ref. 9). For the case of the side force, thoseshift angles vary from +90° to -120°.

For the hub moment responses (figs. 4(d) to 4(f», the trend of the impedanceis similar to that of the vertical force. ' Below the nondimensional frequency 0.5,the major response dominates the other responses. As with the case of the horizontalforce and side force, a minimum around the nondimensional frequency 0.9 can be seenin the major response of the rolling moment ~. In the phase angle characteristicsof the hub moments, the angles depend on the gust exciting frequency.

In figures 5(a) to 5(c), root loci of the blade characteristic equation withfeedback (the individual feedback gain is fed back independently) are shown. Floquet'theory is used in this stability analysis to solve the characteristic equation withtime-variant, periodic coefficients. The feedback gain is assumed to be a realnumber. Figure 5(a) shows the case in which only the flap deflection (~w) is fedback to the individual blade pitch angle. The feedback gain (Kw) varies from 0.0 to0.15 rad/m. The limit of the feedback gain is about 0.111, above which the feedbacksystem becomes unstable. Figure 5(b) shows the case in which only the velocity ofthe flap deflection (~w) is fed back. The feedback gain (Kw) varies from 0.0 to0.005 rad/m/sec. The point at which the system becomes unstable is about .0.00515 rad/m/sec. In figure S(c), root loci for the case of the acceleration feed­back (~w) is depicted. The feedback gain (Kw) varies from 0.0 to 5xlO- 3 rad/m/sec 2 •

In this region, the system is stable; for (Kw) > 5xlO- 3 the system is unstable.

Figure 6 shows the sensitivity of the phase angle of the feedback gain to thevertical hub force response at gust frequency 0.5 Hz (wG/0. = 0.265) and amplitude1.8 m/sec. The vertical axis denotes the amplitude of the oscillatory vertical forceor the nondimensional vertical force (thrust coefficient). The horizontal axis isthe phase angle of the feedback gain, ~, which varies from 0° to 360°, In this cal­culation, the amplitude of the feedback gain is assumed to be constant. The dashedline in figure 6 shows the vertical force response without any feedback system (baseline). The feedback system used here generates the control input to be fed back tothe blade pitch according to equation (26). Each sensor is assumed to be located i

at the blade tip and measures blade deflection independently. Each measured deflec­tion (~w, ~v, ~~) is assumed to be fed back to the blade pitch. The amplitude ofthe vertical force depends on the phase angle, and it has a minimum point at theappropriate combination of the amplitude and the phase angle of the feedback gain,that is, (Kw, ~), (Kv. ~), etc. From these calculations, the minimum point may berealized at ~ = 30° for ~w feedback, ~ = 30° for ~v feedback, and ~ = 210°for ~~ feedback. For ~~ feedback, IK~I = 5.0 was used because the tor~ionaldeflection of the blade is very small compared with the other deflections.

Figures 7(a) to 7(c)to the vibratory verticalure 7(a), the phase angle

show the sensitivity of the amplitude of the feedback gainforce response at the same operating condition. In fig-~ is assumed to be zero; ~ varies from 0.0 to

14

-1.6 rad/m. The amplitude of the vertical force at first rapidly decreases, thenslowly decreases as Kw further decreases. In this figure, the symbol 0 denotesthe calculation results by the local momentum theory (1MT). In the case of ~v

feedback (fig. 7(b)), the phase angle ~ is assumed to be 30°, and Kv varies from0.0 to -0.1 rad/m. The amplitude of the oscillatory vertical force linearlydecreases and reaches a minimum at about Kv = -0.04; then, it increases linearly asKv further decreases. For the ~~ feedback (fig. 7(c)), the general trend is simi­lar to that in the case of 6w feedback. However, K$ is several orders of magni­tude greater from that shown in figure 7(a). This difference is due to the magnitudeof the blade deflection; that is, usually the torsional deflection is very small com­pared with the flap deflection. From these three figures, it is found that the ver'­tical force response is not as sensitive to the variation of Kw or K~ for ~w or ~~

feedback as a variation in Kv for ~v feedback. It follows that it is desirableto select the 6w or 6~ deflection as the measuring signal in order to alleviate thevertical hub force response to vertical gusts.

Figures 8(a) to 8(c) show the effect of the sensor position on the hub responseto the specified combination of (Kw, ~), (Kv ' ~), and (K~, ~). Four sensor positionsare investigated: r = (1/4)R, r = (1/2)R, r = (3/4)R, and r = R. In these figures,the base line is depicted by a dashed line. An effect of sensor position is shown inthe vertical force and the yawing ~oment responses. Positions of the sensor fartheroutboard are more effective in the gust alleviation system. It should be noted thatthere is an increase in some responses because of the individual blade pitch control.However, this increase is small compared with the vertical force and the yawingmoment reduction.

Figures 9 to 14 show the hub gust response calculated by the LMT. Each figureincludes the three hub forces (Fx ' Fy ' Fz ), the three moments (Mx ' My, Mz), and alsothe pitch angle input of a reference blade. The same operating conditions used inthe harmonic method are used in this calculation; Gust shapes assumed here are sinu­soidal (figs. 9-12) and step (figs. 13 and 14). These gusts are appiied to the cal­culation after the thrust level has reached a steady-state trim value (usuallyrequiring six rotor revolutions for the LMT). The gust amplitude is 1.8 m/sec, whichis almost half the mean induced velocity of the rotor. The time history of the gustshown in each figure is measured at the rotor head. The hub responses are measuredfrom when the reference blade is located at zero azimuthal angle. The control pitchangle is fed back to the individual blade pitch actuator. Any time lag of the actua­tor or other mechanism is not considered. The physical limitation of the pitchactuator should be considered for the purpose of implementation. However, in thiscalculation, physical limits were set at the very large values because the emphasisof this investigation is on the feasibility of this kind of controller.

Figure 9 shows the hub gust responses for a vertical sinusoidal' gust without anycontroller; gust frequency is 0.5 Hz. Among these responses, the vertical force(thrust), the rolling moment, and the yawing moment (torque) respond noticeably tothe vertical gust compared with the others. In the case of the thrust response(fig. 9(a)), the amplitude of the fluctuation is about 10% of the steady stateresponse. It should be noted that the phase difference between the vertical forceand the rolling and yawing moments is about 180°. The vertical force respondsquickly to the vertical gust.

In figure 10, the hub gust response with the individual blade pitch control isshown for the same vertical sinusoidal gust. Feedback gain Kw is -0.5 rad/m. Flapdeflection (~w) at the blade tip is selected as a measuring signal. The reduction ofthe oscillatory thrust response is about 60% of that calculated for the uncontrolled

15

case (fig. 9(a)). From figure 7(a), it is expected that the reduction of the thrustresponse should be about 65%. This result implies that the theoretical calculationusing the harmonic method can reasonably predict the reduction of the thrust response.In the case of the rolling moment (fig. 10(d)), a significant reduction of the oscil­latory response can be obtained. However, from figure 10(f), it should be noted thatthe yawing moment response increases about 20% because of the individual blade pitchcontrol. Figure 10(g) shows that the pitch angle of the reference blade varies sinu­soidally with a phase difference of 180 0 relative to the gust. From this result, thechange of the pitch angle may be expressed as follows:

(47)

where Mo(t), Lie1C(t), and M1S(t) are control inputs generated by the controller,and all of them are time-variant. The Lie oc and Lieos terms are constant values,which are automatically decided once the feedback gain is selected.

Figures l1(a) to 11 (g) show the hub gust response for the vertical sinusoidalgust without control. The gust frequency of 2.0 Hz is used. The operating conditionis the same as in the case shown in figure 9. Compared with figure 9, the amplitudeof the uncontrolled oscillatory thrust response has decreased noticeably (about 60%).This decrease is in agreement with the result calculated by the harmonic method(about a 63% decrease; fig. 4(c)). Yet the hub forces and moments still respondnoticeably to the sinusoidal gust. '

Figures l2(a) to 12(g) show the hub gust responses for this vertical sinusoidalgust with the individual blade pitch control. The feedback gain is -0.5 rad/m. Inthis case, there is a noticeable reduction in the thrust, side force, and yawingmoment responses. However, in the rolling and pitching moment response, increasescan be seen. Contrary to the case of a 0.5 Hz gust frequency, a slight reduction ofthe yawing moment response can be seen in the case of the 2.0 Hz gust frequency.

Shown in figures l3(a) to l3(g) are the hub gust responses for a step gust. Thegust amplitude is 1.8 m/sec. The change in the thrust and in the yawing momentresponses is significant. The thrust response increases gradually to the new steadystate without any overshoot. The yawing moment response, however, shows an overshootand then reaches a steady state value. Other responses do not show any significantchanges. Figures l4(a) to l4(g) show the hub gust responses with the individualblade pitch control for this step gust. The feedback gain Kw used in this calcula­tion is -0.5 rad/m and the phase angle is assumed to be zero. The general tendenciesof the response are similar to those of the case shown in figure 13. For the thrustresponse, a reduction of about 70% is achieved. The difference between the steadystate value and the trim value for yawing moment (fig. l4(f)) is roughly twice thatshown in figure l3(f). A collective pitch angle change of about 0.6 0 is input tomaintain the trimmed condition of the rotor.

16

3. ADAPTIVE BLADE PITCH CONTROL

In this section, an adaptive blade pitch control designed to alleviate gust­induced vibration is analytically investigated for a helicopter with four articulatedblades. Multicyclic control (MCC) and higher harmonic control (RHC) have previouslybeen applied to the reduction of inherent responses of a helicopter, such as N-per­revolution vibratory responses. In these controllers, pitch inputs at harmonics ofthe rotor rotational speed are used as control inputs. For the gust response of ahelicopter, additional vibratory responses, such as gust harmonic (wG)' subharmonic(wG - It), and superharmonic (wG + It), appear in the response. Gust harmonic responseis the most dominant in the comparatively low gust frequency range (refs. 5 and 9).In this study, this dominant response is selected as the response to be reduced.

An adaptive blade pitch control system is based on digital optimal controltheory. This system is composed of a set of measurements (oscillatory hub forces),a control system based on the optimization of the quadratic performance function,and a simulation system of the helicopter rotor. In the following subsection, thedetails will be explained.

Helicopter Model

In this study, the helicopter is represented by a linear, quasi-static frequencydomain model relating the output z to the input e. Here, z is a vector of theharmonic of the gust-induced vibration in the rotor frame. The input e is at thefrequency corresponding to the gust frequency. It is assumed that the gust is sinu­soidal, and that the gust frequency is known a priori. When a rotor penetrates intothe gust, the response at the hub position can b~ considered to be composed of gustand control inputs, such as

(48)

Equation (48) can be rewritten by using the expression of the rotor impedance [TG]and rotor transfer function [TC] as follows:

{zG} [TGHg}

{zC} [TcHe}

That is, equation (48) becomes

(49)

where

{z} (m x 1)

[TG] = (m x 3)

{g} (3 x 1)

[TC] (m x n)

{ e} = (n x 1)

17

(50)

Once the characteristics of a gust are determined, [TG] becomes a constant matrix.Therefore, we may denote the first term of equation (49) as {zo}' Then equation (49)is

(51)

This form resembles the global model formulation of helicopter vibration (ref. 70).The following "local model" can be also taken into account:

(52)

where ~z means zn ~ zn-1' and ~e means en - en- 1•

From theoretical calculations using the harmonic method, the hub response to avertical gust in the rotor frame (XR' YR, ZR) can be expressed as follows:

+ Zs cos(wG + 4Q)t + Zs sin(wG + 4Q)t + H.O.T.

where Zo(t) are the inherent responses of a helicopter rotor represented by

Zo(t) = 20 + Zc cos(NQt) + Zs sin(NQt) + H.O.T.

(53)

(54)

In these responses, gust harmonic responses (Zl' Z2)T are the most dominant in thegust response. Oscillatory hub forces and moments at the gust frequency in the rotorframe are chosen here as measurements z:

(55)

where

(56)

Control inputs are selected at the gust frequency wG' subharmonic frequency (wG - Q),and superharmonic frequency (wG + Q) in the rotor frame as follows:

~8 = 81,cos(wGt) + e2 sin(wGt) + 83 cos(WG - Q)t + e4 sin(wG - Q)t

+ 85 cos(WG + Q)t + 86 sin(wG + Q)t

18

(57)

Hence the control vector e has six components:

(58)

These control inputs are superimposed on the blade trim pitch input.

Identification

Three cases are distinguished for the helicopter model, depending on the identi.­fication approach:

1. Identify {zo} only

2. Identify [Tel only

3. Identify {zo} and [Tel

For the gust response, the uncontrolled response {zo} is generally time-variant;the matrix [Tel depends on the operating flight conditions. Hence, it is necessarythat the transfer function [Tel be identified simultaneously with an uncontrolledresponse {zo}' Case 3 is taken into consideration in this investigation. For thelocal model, it is necessary that the [Tel matrix be identified for each time-cycle.The Kalman filtering technique (refs. 102 and 103) is applied to identify them. Forthe global model, equation (51) can be rearranged as follows:

where

T = m x (n + 1)

a = (n + 1) x 1n

matrix

vector

(59)

and the subscript n denotes the time-step at t = n fit. For this study, it isassumed that there is no noise in the measurement of en' The identification algo­rithm can be derived by considering the jth measurement as

(60)

where tjn is the jth row ofT, and Vjn is measurement noise, which has zeromean,

(61)

and variance

(62)

19

and a Gaussian probability distribution. Here Qnm is the Kronecker delta function,and the subscript j will be omitted to simplify the notation. The variation of theparameters will be modeled as a random process:

(63)

where un is a random variable, which has zero mean

(64)

and variance

(65)

and a Gaussian probability distribution. The minimum error-variance estimate ofis then obtained from the Kalman filter (ref. 103),

(63)

where

~ = Pn- 1 + Qn-l (67a)

P = M - M e e™ / (r + e™ e ) (67b)n n n n n n n n n n

kn Mnan/(rn + e~Mnen) (67c)

Here Mn is the variance of the error in the estimate of t n before the measurementand Pn is the variance after the measurement; ku is the Kalman gain vector. Tosimplify the calculation, it will be assumed that Qn and rn have the same timevariation for all measurements, and that Qn' r n , and Po are proportional to thesame function fj:

rjn = fjrn

Qjn =0 f.QJ n

P. fjPoJO

(68a)

(68b)

(68c)

Then it follows that Pjn = fjPn and Mjn = fjMn ; and that the gain ku is alsothen the same for all measurements. With the same gains, the rows can be combinedto form

For the local model, equation (69) becomes the following:

T = T + [(z - z ) - T (8 - 8 )]kT

n n-l n n-l n-l n n-l n

20

(69)

(70)

where

Pn = Mn - M 6e 6e™ f(r + 6e™ 6e )nnnn n nnn

kn = M 6e f(r + 6e™ 6e )n n n nn n

Controller

(7la)

(71b)

(7lc)

(7ld)

The Gontrol algorithm is based on the minimization of a performance index Jthat is a quadratic function of the input and output variables. The quadratic per­formance function to be used here is

(72)

where Wz , We' and W6e are weighting matrices, which are assumed to be diagonal andthe same value for all harmonics of a particular quantity. Then J is a weightedsum of the mean squares of the gust response and control. The matrix We constrainsthe amplitude of the control, and W6e constrains the rate of change of the control.

For the deterministic controller, the control required to alleviate the helicop­ter vibration is given by substituting for zn in the performance function J, usingthe helicopter model, and then solving for en that minimizes J. For the globalmodel (eq. (51)), the solution can be obtained as follows:

where

CM = DWM

D = (T~WzTC + We +W6e)-1

for the local model (eq. (52)), the solution can be obtained as follows:

(73)

(74b)

(74c)

(75)

In this derivation, the response z is assumed to be deterministic; therefore, it isreferred to as the,deterministic controller. When the parameter uncertainties aretaken into account, the cautious controller can be obtained by using the expectedvalue of the performance function:

21

where it is assumed that Wz is diagonal, and en remains deterministic. For thecase of the open-loop control (zo feedback), there follows

where

= I:j

(" + eTt )2 +wzj ZjO n jn t

j(71)

Mt

".].M .zz

(78)

= (n x m)

T= Mzt = (n x 1)

Mzz = scalar

So the performance function becomes

matrix

vector

J (79)

The solution for the control that minimizes J is then

(80)

where the gain matrices C and C~e are the same as for the deterministic controller,using the identified values of the parameters and with We replaced by

The neW constant term is

Co = -D[E w • (Mt ). 1j ZJ z J~

22

(81)

(82)

Similarly, for the case of the closed-loop control (zn-l feedback), the performancefunction is

(83)

.'

The solution is identical to that for the deterministic controller, using the identi.­fied values of the parameters, and with W68 replaced by

WM + (84)

Regulators

A controller combining recursive parameter estimation and linear feedback iscalled a self-tuning regulator. There are two fundamental options for the identifi-­cation: the use of either an invariable algorithm or an adaptive algorithm. Thereare also two fundamental options for the controller: open-loop or closed-loop.Hence, there are four possible regulator configurations (fig. 15). For the adaptivealgorithm, the parameters are recursively identified on-line, using a Kalman filter.For the open-loop algorithm, the control is based on the uncontrolled vibration levelZo (identified either on-line or off-line). For the closed-loop algorithm, the con­trol is based on the feedback of the measured vibration, zn-l. These four regulatoroptions are reviewed in reference 70.

In this study, two regulators are investigated for a gust alleviation system.Figure 16(a) shows an adaptive open-loop regulator algorithm, in which both {zo} and[Te] are identified by a Kalman filter. Figure 16(b) shows an adaptive closed-loopregulator algorithm. In this regulator, only [Te] is updated by a Kalman filter andthe output vector zn is contaminated by measurement noise, vn • In both regulators,the uncertainty of parameters can be taken into consideration by using the varianceof the error in the estimate of t n before the measurement, Mn+1 . (see eq. (81) or(84». These regulators consist of on-line identification of the parameters and cal­culation of the gain matrix, with feedback of the control based on the measuredresponses. It should be noted that the Kalman filter gains can be calculated in thetime interval between application of en ,to the helicopter and the measurement ofthe resulting zn.

Gust Model

In the past, studies that have dealt with gust suppression systems have usedgust models, such as the von K~rman model, Dryden model, step model, sinusoidal model,and sine-squares model. These models have some advantages as well as some disadvan­tages. The von Karman and the Dryden models are expressions derived from statisticaltechniques in the frequency domain. These expressions are close to the natural tur­bulent flows in the sense of the statistics. However, they are not able to show theindividual flow pattern of turbulence in the time domain. Therefore, these modelsare not suitable for timewise numerical calculation. On the other hand, the step andsinusoidal gust models, etc., are very simple yet different from actual gust shapes.These expressions are easy to handle in numerical calculations. By using these

23

gust models, it is possible to obtain the basic characteristics of the plcmt modelfor the gust response. In this study, the following gust representation is assumed:

(85)

where uGo' vGo' and wGo are horizontal, lateral, and vertical gust amplitudes,..respectively; and wG is the gust angular velocity. For the step gust, wG = O. Thepositive direction for the gust velocity is defined in the rotor frame. Most resultsare for a vertical gust only. For this study, the rotor is assumed to have theblades at 0°, 90°, 180°, and 270° when the sinusoidal gust is initially encounteredby the rotor (the gust velocity field is convected past the rotor by the helicopterforward speed). As in reference 20, the ratio of the magnitudes of the gust compo­nents is assumed to be

uGo : vGo : wGo = 3.0 : 2.0 : 1.8 (86)

The helicopter is represented mathematically by a simple linear quasi-static fre­quency domain model, relating zn and en. Therefore, a simple sinusoidal gust shapeis selected in order to investigate the feasibility of the gust alleviationalgorithm.

Numerical Calculations

Numerical calculations have been performed to determine the feasibility of theadaptive blade control algorithm. This adaptive blade pitch control algorithm con­sists of a set of measurements (oscillatory hub forces and moments), an identifica­tion system using a Kalman filter, a control system based on the minimization of thequadratic performance function, and a simulation system of the helicopter rotor. Theoperating conditions of the rotor are the same as the case of the individual bladepitch control (table 3). The initial estimate of the transfer function [Tel isobtained by using the harmonic method program. The detailed derivation of [TC] isgiven in appendix E. Tables 4 and 5 show the transfer function of the rotor forf G = 2.0 Hz and fG = 0.0 Hz (step gust)~ respectively. In the case of the stepgust, gust frequency is assumed to be zero and the control inputs ~e and the out­puts z are assumed as follows:

{z}(87)

"

where the subscript 0 denotes the mean value. The [TC] obtained by the harmonicmethod is regarded as the initial estimate in the adaptive blade pitch control. Inthe controller, the [TC] is updated by the Kalman filter at every time-cycle. Inthis calculation, it is assumed that there is no noise in the measurement of ~e.

However, measurements z are usually contaminated by noise. Measurement noise isgenerated by a random noise generator. Several sets of noise-to-signal ratios werestudied before these calculations were made. A noise-to-signal ratio of about 0.05was used in this study. After trial and error, the initial values of Po, Qo in

24

equation (67) were estimated as follows: Po = [9.8x10 1t ]N (or N-m), Qo = [9 .8x10 2 ]N(or N-m). The variance of the measurement noise, r, is' assumed to be 9.8N (or N-m).To keep the noise-to-signal ratio approximately constant, the following relationshipis taken into account (see ref. 28):

(88)

where I n is the quadratic performance function at the nth time-cycle. The weight­ing matrices of the quadratic performance function are taken as

We = [0.0] (89)

Since the dimensional values of the measurements z are used, the elements of theweighting matrix Wz become small. The pitch control en are unconstrained; thepitch control rates ben are constrained. The values given by equation ,(89) wereselected for good algorithm convergence characteristics.

In this gust alleviation system, the gust frequency f G is specified. There­fore, each time-cycle that updates the parameters must vary depending on the gustfrequency. In most calculations, a 2.0 Hz gust frequency is used. This frequency ismore than half of the rotor rotational speed, an~ somewhat larger than the typicalmean atmospheric turbulence (usually below 1.0 Hz). An updated parameter estimate isperformed every four rotor revolutions. During that time, the measurements z arediscretely sampled by the measuring system. Since the measurements z are data inthe time domain, it is required that they be converted from the time domain to thefrequency domain using the fast Fourier transform (FFT) (ref. 104). In the FFT, thesolution of the data in the frequency domain depends on the sampling time bt(Shannon's theorem); in this calculation, Qbt = 10°. Therefore, when the gust fre­quency fG falls between two points (that is, nbf < fG < (n + l)bf, n = integer,bf = frequency step), it is impossible to calculate the correct values by the FFT.In this analysis, correct values are approximated from known gust frequency fG andby using a cubic spline interpolation method. Furthermore, it is necessary to con­sider the calculation time of the optimal control, which determines the optimal con­trol for the next time-cycle using the output data from the FFT. One time-cycle is8TI rad (fig. 17). The sampling interval is {l28/144)(8TI) rad. Data are sampled atevery 10° (144 samples in four revolutions); 128 samples are used in FFTsince powersof 2 are typically computationally more efficient (2 7 = 128). Consequently therequired calculation time is assumed to be (16/l44)(8TI) rad. As a result, there isa phase shift (~I + ~II) in the measured response at the beginning of the next time­cycle relative to the phase at the end of the previous sampling interval. In theprogram code, this phase shift is accounted for in implementing the control asfollows:

25

(90)

where Z1' Z2 are coefficients of the cosine and sine term of the response, respec­tively. The subscripts n denote the nth time-cycle. If this phase shift is notconsidered, (Z1)n and (Z2)n coincide with (Z) and (Z ) By using the above. 1 n-l 2 n-l·equat10ns, the optimal control inputs for the next time-cycle can be determined. Aglobal helicopter model is used. For the open-loop controller, both Zo and [Tel areupdated by the Kalman filter. Only [Tel is updated for the closed-loop controller.

In figures l8(a) to l8(g) and figures 19(a) to 19(9), the hub gust responseswithout control are shown separately for the horizontal and the lateral componentsof the gust. The amplitude of the horizontal gust is 3.0 m/sec, and that of the lat­eral gust is 2.0 m/sec. The gust frequency of 2.0 Hz is used. Figures 18 and 19show little influence on the thrust response. In the horizontal and side forces andyawing moment responses, the effect of these types of gusts on the hub response canbe seen. Figures 20(a) to 20(g) show the hub responses without control for the threedimensional gust. The amplitude of the three gust components are uGo = 3.0 m/sec;vGo = 2.0 m/sec; and wGo = 1.8 m/sec. The three components of the gust have thesame frequency (fG = 2.0 Hz). The increase of the horizontal force, the side force,and the yawing moment responses can be seen by comparing figure 20 with figure 11.From these results, a vertical component of the gust is the most influential on therotor gust response (specifically in the thrust response).

Figures 2l(a) to 2l(g) show the hub gust responses with the adaptive blade pitchcontrol for a vertical gust component only (fG = 2.0 Hz). The controller is deter­ministic, and the global helicopter model was used •. Two inputs (8 1 and 82) and twooutputs (cosine and sine elements of the thrust response) are considered. Therefore,the dimension of [Tel becomes (2 x 2). The thrust response (fig. 2l(a» graduallydecreases, but the yawing moment response (fig. 2l(f» responds dramatically (seefig. ll(f». The other responses (Fx ' Fy \ Mx, Mv) are similar to that shown infigure 11. In this case, the aim of the controlier is to reduce the thrust fluctua­tion owing to the gust, and no attempt is made to reduce the other responses.

In figures 22(a) to 22(g), the hub gust responses are shown with adaptive bladepitch control, again for the vertical gust component only. In this case, the dimen­sion of [Tel is (6 x 12). As control inputs, six elements (8 1 to 86 ) are used toalleviate the responses. Measurements z involve the hub forces (Fx ' Fy , Fz) andthe hub moments (Mx ' My, Mz). Compared with figure 21, the reduction of the thrustresponse is significant (about 50% to 80%). From this figure, it is observed thatthe fluctuation of the thrust does not reduce uniformly; in other words, it sometimesdiverges and converges. This phenomenon is explained as follows. The control inputsgenerated by the controller depend on the measurements. When the hub responses aredecreased by means of the controller, the optimal control inputs necessarily becomesmall. However, the gust itself is unchangeable. Therefore, the effect of the con­troller on the reduction of the responses becomes weak. At the same time, theuncer­tainty of the identified parameters may increase, even if the noise to signal ratio

26

is kept constant by equation (88). In the yawing moment response (fig. 22(f)), thecharacteristics of the response are improved (see fig. 2l(f)), and the amplitude ofthe fluctuation is a little less than that of the case shown in figure 11. Similarto the thrust response, the response sometimes diverges and converges. This is dueto the change of the induced drag which is directly related to the thrust. Contraryto expectations, the responses (Fx ' Fy ' Mx, ~) show a slight increase in magnitude.In the pitch angle change of the reference blade (fig. 22(g)), the sinusoidal changewith the gust frequency can be observed, similar to that shown in figure 2l(g).

Figure 23 shows the time histories of the mean square thrust response. In thisfigure, the solid line denotes uncontrolled response. The dashed line correspondsto figure 21 (two inputs and two outputs) and the broken line corresponds to fig-ure 22 (six inputs and 12 outputs). As explained before, the parameters are updatedat every four revolutions. During one time-cycle, the control inputs are kept con­stant. The response for the case of thrust control only is at first constant, andthen gradually decreases. The response is reduced by almost 50% after 20 revolutionswith the controller engaged (five time-cycles). The response for the case of completehub response control decreases to a much lower value, although not continuously. A50% to 80% reduction of the thrust response is attained for the case of complete hubresponse control. From these results, it is found that the measurements z shouldinclude not only the thrust response but also other hub responses.

Shown in figures 24(a) to 24(g) are the hub responses for the vertical step gust.Operating conditions are the same as in the case shown in figure 13. The adaptiveclosed-loop controller is used. The measurements z involve only the longitudinalresponse of the rotor; that is, the horizontal force Fx ' the thrust Fz ' and thepitching moment My. The control inputs haye three elements (6 1 , 82 , 83 ) accordingto equation (87). All parameters (zo and [TC]) are updated at every rotor revolutionby the Kalman filter. Significant reduction in the thrust response (fig. 24(a)) isachieved by the controller (almost 100%). However, compared with figure 14, thehorizontal and side force respOnses transfer to a new steady state condition. Therolling and pitching moment responses (figs. 24(d) and 24(e)) transfer to new steadystate conditions. In the yawing moment response (fig. 24(f)), there is a significantchange in the steady state condition. This phenomenon is a result of the strongeffect of the controller on the gust alleviation system. The controller generatesthe optimal control inputs in order to reduce the change from the steady value. Ifthese control inputs are large values, a new trim condition could result. Referringto figure 24(g), the maximum change of the pitch angle is about 2°. The cyclic pitchangles for the trim condition itself are of the order of several degrees. Therefore,the pitch angle change by the controller has an effect on the rotor trim condition.In this case, the ratio between the control angles A6 and the trim pitch angle 8becomes more than 10%. The flight trim condition consequently changes and transfersto the new steady state condition.

In figures 25(a) to 25(g), the hub responses for the vertical step gust areshown. The adaptive closed-loop controller is used, and the operating conditionsare the same as those shown in figure 24. To investigate the sensitivity of thetransfer function to the controller performance, arbitrary small initial values ofthe transfer function are used in this calculation. Compared with figure 24, thethrust fluctuation (fig. 25(a)) decreases very slowly. For the other responsesexcept yawing moments the trim condition remains the same. In the case of the yaw-'ing moment (fig. 25(f)), the transfer to the steady state can be seen after showingan overshoot. This phenomenon is due to the induced drag, as mentioned before.From these results, it can be concluded that the performance of the controller

27

depends on the initial estimate of the rotor transfer function. If a more correcttransfer function is used, convergence of the responses can be obtained more quickly.As shown previously, the transfer function derived by the harmonic method gives areasonable initial estimate.

Figures 26(a) to 26(g) show the hub responses, again for the vertical step gust.In this case, the adaptive open-loop controller is used. The measurements z involvesix responses of the rotor; that is, three hub forces (Fx , Fy , Fz ) and three hubmoments (Mx ' My, Mz). Similar to the case shown in figure 24, all elements of [TC]are updated at every rotor revolution. The magnitude of the controller inputs isconstrained by a prescribed maximum value (~8max = 2.0°). Three components (8 1 , 82 ,

8 3 ) of the control inputs (eq. (87» are used in this calculation. In this thrustresponse (fig. 26(a», the deviation from the steady value of the thrust graduallydecreases after responding to the step gust. Compared with the case of figure 24,the pitch angle change is very slow and small. The other hub forces, together withthe rolling and pitching moments, remain steady. The yawing response deviates slowlyfrom the steady value because of the thrust response. In the case of the open-loopcontroller, the optimal control inputs depend on the uncontrolled response zOo Ifthe zo are small, then the optimal control inputs may be small. For this case, theoptimal control inputs at the beginning stage of controller operation are less than0.5° .

In figure 27, the time histories of the quadratic performance function J areshown. The solid line denotes the function J without control. The J' S with con­trol are shown for both the adaptive closed-loop control (corresponding to fig. 24)and the adaptive open-loop control (corresponding to fig. 26). For the adaptiveclosed-loop controller, the J decreases rapidly after showing a sharp increase.The J for the adaptive open-loop controller decreases moderately after showing thesame increase. Both control schemes show the effect the controller has on the reduc­tion of the hub gust responses.

Since deterministic controllers have been considered in this analysis, theuncertainty of the parameter identification has not been considered in the calcula­tion of the optimal control inputs. These cautious properties can play an importantrole for inherent helicopter vibration reduction schemes (ref. 68). In figures 28(a)to 28(g), the hub gust responses with the cautious controller are shown for the ver­tical gust (fG = 2.0 Hz). Referring to figure 22, there is no significant differencebetween these figures. The cautious controller involves the variance of the errorin the estimate of the transfer function rTC] before the measurement. The moreuncertain the parameter estimates, the larger the variance of the error. In thiscalculation, the term concerned with the cautious properties

in equation (81) was about l% of the term TeWzTc in equation (74c). Therefore,these calculated results show that the cautious properties have little effect on thecontroller performance in this study.

Results from applying adaptive blade pitch control for the three dimensionalgust are shown in figures 29(a) to 29(g). Operating conditions are the same as thosein the case of figure 20. The large reduction in the thrust response can be observedby comparing figure 28 with figure 20. The yawing moment response shows the same

28

characteristics as for a vertical gust only (fig. 22). In general, all the hubresponses are similar to those of the case shown in figure 22. In this calculation,the cautious properties have not been considered because of their minimal effect oncontroller performance.

4. CONCLUSIONS

Control systems to reduce the gust-induced vibration of helicopter rotors havebeen studied by two means. One is an individual blade pitch control scheme, and theother is an adaptive blade pitch control scheme. Among three gust components, thehorizontal and lateral components of gust have little influence on the hub gustresponse. However, the vertical component of the gust has a great influence on therotor hub response. In studying the gust response of the rotor, the thrust responseshows the most significant change compared with the other responses. Hence, reducingthe thrust response should be the aim of the gust alleviation system. Both controlschemes show significant effects of the controllers on the reduction of the gust­induced vibration of the helicopter rotor. From these theoretical analyses, thefollowing conclusions are drawn.

For the individual blade pitch control scheme:

1. A computer code that calculates the gust response of a rotor has beendeveloped by using the harmonic method. This code is applicable to the rotor impe­dance calculation for gust response or the transfer function calculation of the rotor.

2. Outboard measurement sensors are more effective in reducing the gust-inducedvibration than are inboard radial 'station locations.

3. The gust-induced thrust response determined by the harmonic method is about10% to 20% larger than that given by the LMT. However, the results by the harmonicmethod can qualitatively predict the characteristics of the gust response.

4. Individual blade pitch control can significantly reduce gust-induced vibra­tion. For example, reductions of the thrust response for the sinusoidal gust byalmost 60% and for the step gust by almost 70% can be obtained for a specified feed-­back gain Kw.

5. For the gust models used, an increase in the yawing moment fluctuation isobserved. A control scheme to include the reduction of the transient yawing responseshould be incorporated.

6. The magnitude of the control inputs is less than 1.0°. For these pitchangles, there is little effect on the trim condition of the rotor.

7. Because the f1apwise blade deflection is most sensitive to the gust (com­pared with the lead-lag and torsion deflections), it is a good measurement of thegust-induced rotor response for an articulated rotor.

For the adaptive blade pitch control scheme:

1. The major, the subharmonic, and the superharmonic inputs are considered inthe controller. As the gust frequency increases, higher frequency terms of the gust

29

response become large. To alleviate these high frequency terms in the vibratoryresponse, the number of terms included in the control inputs increases, making thecontroller more complex.

2. The performance of the controller depends on the initial estimate of therotor transfer function. When the exact transfer function is used, the convergenceof the hub responses can be achieved quickly. The transfer function derived by theharmonic method gives a good estimate.

3. In using the FFT to convert measurements from the time domain to the fre­quency domain, some approximations must be made because of the arbitrary gust fre­quency. This may increase the uncertainty of the measurement. A more accuratefrequency domain determination method is required.

4. For the case of a sinusoidal gust, the adaptive open-loop regulator is bestsuited for the gust alleviation system. Results show that a 50% to 80% reduction inthe thrust response can be obtained. The regulator studied in this report is shownto be applicable to a three dimensional gust.

5. For the step gust, the adaptive closed-loop regulator performs better thanthe adaptive open-loop regulator. The closed-loop regulator yields a rapid reductionof the gust-induced thrust response (almost 100%), even though it violates the trimcondition. The open-loop regulator shows that convergence of the thrust response isslow.

6. The uncertainty of the parameter identification has little influence on theimprovement of the regulator in this investigation.

7. In this adaptive control scheme, the. gu~t frequency is prescribed at theinitial stage of the calculations. Therefore, this type of regulator does not applyto random gust responses.

30

APPENDIX A

COORDINATE SYSTEMS

Five Cartesian frames are used in this study to describe the blade and hubmotions. The definition of these frames is briefly reviewed in this appendix.

Inertial frame (XIt YI, ZI)- This frame is inertially fixed and coincides withthe body frame under the steady flight condition.

Body frame (XBt YB, ZB)- This frame is fixed to the fuselage with its origin atthe center of gravity; XB is oriented to the 'direction of the steady flight t YB istaken to be positive in the right-hand direction, ZB is positive downward. Therelationship between the inertial frame and the body frame is as shown:

[I} fB} [BO}·Y1 = [TB1] YB +. YBO,ZI ZB ZBo

cos IJ!t -sin IJ!, ° cos 8 t 0, sin 8 It 0 °[TBI ] _. sin IJ!t cos IJ!t a a It a 0, cos q>, -sin q>

0 a , 1 -sin 8 , 0, cos 8 0, sin q>, cos q>

(AI)

(A2)

where (q>, 8, IJ!) are Euler angles of the body, and (XBo' YBo' ZBo) is the position ofthe origin of the body frame in the inertial frame.

Rotor frame (XR, YR, ZR)- This frame is fixed to the fuselage with its origin athub center; ZR coincides with the rotor shaft and XR is positive rearward. Thisframe is related to the body frame by the following transformation:

-cos; is' 0, -sin is

(A3)

o 1, a (A4)

sin is' 0, -cos is

where is is the shaft inclination angle and(XRo ' YRo ' ZRo) is the position of theorigin of the rotor frame in the body frame.

31

Rotating hub frame (XH, YH, ZH)- This frame is rotated about the ZR axis by~ = Qt. The coordInate transformation between the rotor frame and the hub frame isexpressed as

{:} = [T~{::}cos ~, -sin ~, 0

['fAA] = sin ~, cos ~, 0

0 0 , 1

(AS)

(A6)

Blade frame (Xb, Yb, Zb)- This is a local frame fixed to each blade section con­sidered. The blade section itself is assumed to be rigid; Zb is directed outwardalong the local elastic axis, and Yb is directed toward the zero-lift angle ofthe blade section. The displacements of the elastic axis are denoted by (u, v, w)in the hub frame. Angular changes related to the hub frame are given by the Eulerangles (6 + ~, _WI, VI) where ( )1 denotes d( )/dXH' The transformation between thehub frame and the blade frame is given as

t:} fb} {+U}= [TbHl :: + eo w+ v

1 , -VI, a 1 , 0, -WI 1, a 0

[TbH ] = VI, 1 0 0 1, 0 0, cos(6 + ~) , -sin(6 + ~)

o , 0 , 1 WI , 0, 1 0, sin(6 + ~) , cos(6 + ~)

1 , -VI _ 6w l -WI + 6v l

- VI, 1 - 82 /2 - 8~, -(8 + ~)

(A7)

(A8)

WI , (8 + </J) , 1 - 82 /2 - 8</J

In figures 30(a) and 30(b), four coordinate systems (except the inertial frame) areshown schematically.

Since motion of a fuselage is not considered (cfJ = 8 = IjI = 0), and shaft inclina­tion angle is zero (is = 0), three coordinate systems, (XR, YR, ZR)' (XH, YH, ZH)'and (Xb, Yb' Zb) are used in this study.

32

APPENDIX B

BLADE SECTION LOADS

In this appendix, total loads on the blade sectiori are defined (fig. 31).

Inertial loads- The inertial forces {p} and moments {Q} acting on a unit span ofthe blade are given as follows:

{p} (Bl)

{

PX} {ll -e(v' + 8w ' ) + 2Q[-v + e8(6 + ¢)] - Q2[r + u - e(v' + 8w')]

Py -m v - e8.~e +,,~) + 2Q[u - e(v' + 8w')] - Q2[eo + e.j. v - e8~] (B2)

. pz w+ e(8 + ~)

{Q} =n:} (B3)

{

e+ ¢ + Q2(8 + ~) + 2Q8V'}

= -mk2 -8v' + Q2 8v '

v' + 8w' - 2Q8(6 + ¢)

where

{

w -

me 8 [u

["- u

e[v + 2Qu - Q2(eo + v)] + e Q2~ }

_ 2Qv _ Q2(r + u)] _ rQ2~ 0

- 2Qv - Q2(r + u)] - rQ2e~ + e Q2(v'+ew')'o

(B4)

m = [ . dm ,chord e = ~ lhord n dm ,

. 1/2

k = (~ J:hord n2

dm) (BS)

are mass, distance between mass center and elastic axis, and polar radius of gyrationof the blade per unit span, respectively. Equations (B2) through (B4) are the iner­tial loads owing ,to the blade motion when the hub is not in motion •

.;Aerodynamic' loads- Aerodynamic forces and moments are derived from two­

dimensional quasi-steady thin airfoil theory (ref. 96) and assuming airfoil plunging

33

to be zero (h = 0). Blade lift, pitching moment, and profi~e drag are given asfollows:

L = 2TIpbUaC' [Ha + aUa + (1/2 - a)b(a + Qo)] + TIpb 2 (d/dt) [Ha + aUa - ab(a + Qo)]

(B6)

(Bn

(B8)

(B9)

where C' is the lift deficiency function and is assumed to be unity in this study.The quantities Di'~' and Cdo· are induced drag. inflow angle, and drag coefficient,respectively. It is noted that in equations (B6) through (B8) differentiation (d/dt)must be operated on Ua as well as Ha , Qo' and a. Quantities of Ua , Ha , Qo' and aare expressed as follows:

Ua = r~ + v + Uo sin ~ + v'Uo cos ~ + Uc sin ~ + Vc cos ~

(B10)

By using equations (B6) through (B9), aerodynamic forces and moments acting on theblade are

F::}1, 0 0 0 tW'L}

-: : Di[l:IA] 0, cos ~, sin </> -D - (Bll)

0, -sin </>, cos </> LFaz

fax} [~A]8 {V~M}May = - (BI2)

Maz 0 w'M

1 , -v' , 0 1 , 0, -w' 1 , -v' , -w'

[l:IA] = v' , 1 0 0 1, 0 - v' , 1 0 (B13)

o , 0 , 1 w' , 0, 1 w' , o , 1

34

The wing motions in the two-dimensional thin airfoil theory and the lift, drag, andpitching moment acting on the wing section are shown in figures 32 and 33,respectively.

Total loads- Total loads acting on the blade section can be obtained as the sumof inertial, aerodynamic, and gravitational loads as

(B14)

(B15)

where g is the gravitational acceleration.

35

APPENDIX C

COEFFICIENTS OF EQUATION (6): Am , Bm, Cm, AND Fn

The equation given from equation (2) is

(Aij Aqi + Bij Aq i + Cij Aqi) = Fj (CI)i

where

IoR= m.6 + _o Sj (_ C2$i)dr - (wj + abSj)C2(0_i- wi ab_i)dr (C2)Aij 3 ij

R

B13""= _o 2n_[$j(k2051 - eOui) + w'W.5o3 i + qj{ui + e(_l + 0E'.)}]drl

+ Sj -_ C2 _ _ dr- _j + (a + i) bSj Cl (20UT + Up)_ i - UT_ io o

foR -+ (i_ a)bUT$i + (i_ a)b(0 + _Wo)Vi}dr - (wj + abSj)C2{UR(6__ - w_)

f f: /. UT$i + 0Vi - abaCi}dr+ vjC_UTVi dr + _jCI (OUT + 2Up)W i - OUpVio

- (i - a)bUp$ i + (i - a)b(0 + _Wo)Wi}dr + C_'i(r_)vj-'(r_) (C3)

z b i -UTU R y a b_U _.• = _.m.6 + C2DUT_ dr - _j _ + ICi3 3 3 ij _ + + CI -o o

foR-+ (20UT + Up)U:_. + (i _ a)b(8 + _W'o)U:i + UT$ dr - (wj + ab_j)C 2

x {0U:I - 0:_ + 8U_i + UT$i}dr + fR _jCI{_(0UT + 2Up)U:_ + (i _ a)bUp_i "o

fR+ UTUp$ i + 0UpU_- (i_ 0b(_ + _W,o)U:_}d r + _j(C3UTU_.)dr (C4)o

36

Fj = J[R (wj + ab~j)C2[-wGJdr + .:R[{Wj + {~ - a)b~j}c,UT + VjC, (eUT + 2Up)

+ VjC, e-a)b(8 + Qw~)] ["'GJdr + [t [(wj + ab~j)cA + {Wj + G+ a)b~jlC,u~

t.'and where

C1 = 21TpbC'

C2 1Tpb 2

C3

= 2pbCdo

b = C/2

UR = U cos 1jJ0

UT rli + U sin 1jJ0

U =W - vp 0 0

6 = 6 0 + 61C cos 1jJ + 61S sin 1jJ

Here wG is a vertical gust and ~e is the additional control input to be fed backto the actuator. These equations can be expanded, using Euler's formula, to yield

where

cos 1jJ = exp(i1jJ) + exp(-i1jJ)2

sin 1jJ = exp(i1jJ) - exp(-i1jJ)2

When the two-dimensional quasi-steady thin airfoil theory is used, ,there appear har­monics of 1jJ up to the third in each coefficient; that is,

A.. = A exp(-i31jJ) + A exp(-i21jJ) + A exp(-i1jJ) + A + A exp(i1jJ)1.J - S -2 -1 0 1

3

= Em=-3

A exp(im1jJ)m

37

(C6)

Bij

3

= ,;.

m=-S(Cn

3

Cij = 1: em exp (iml/J)1ll=-8

(C8)

Each term of these coefficients has the ith and jth mode of the blade deflection;for example:

A-1 f R _ I R

= -2 mea5v.~. dr • 0ijo J J 0

(C9)

(CIO)

fR _·fR-2 mea7v.~. dr • 0iJ'

o J J 0

A = A = A = A = 0-8 -2 2 8

where

(ClI)

(Cl2)

a 6 = 80

+ 8t (x - 0.75)

1a 7 = I (81~- i8 1S )

rx=-R

and 0,8 t are the Kronecker delta function and pre-twist angle of a blade, respec­tively. For the practical calculation, both i and j are truncated by L. Then thefollowing equation can be obtained:

3 00

l: {[Am exp(iml/J)]~q + [Bm exp(iml/J)]~q + [em exp(iml/J)]} = l: F~ exp[i(wct + nl/J)]m=-8 n=-oo

(C13)

where Am' Bm, em are (L x L) matrices in which all elements are complex.

38

Now, consider the expression of a sinusoidal gust in the rotating hub frame atthe position of hub center. In the rotor frame, a sinusoidal gust is assumed to beexpressed as follows:

Then, it can be transformed at position (r,~) in the rotating hub frame as

where

00

= 1: (-i)nJn(wG/~· x/~)exp(in~)n=-oo

In this derivation, the following formula is used:

00

(CI4)

(CI5)

(CI6)

exp(iz sin 8) = 1:n=-oo

J (z)exp(in8)n

where I n is nth order Bessel function. By substituting equation (CI6) into (CI5),the expression of the coefficient Fn is given by

39

(CI7)

APPENDIX D

FLOWCHART OF THE HARMONIC METHOD

The flowchart of the computer code by the harmonic method is shown in figure 34.In this program, the feedback gains are specified in advance and read as data. It ispossible to analyze the stability of a blade and also to calculate the rotor trans­fer function using this code. Approximately 1.0 sec of CPU time was required forone execution of this code on the CDC 7600 computer.

40

APPENDIX E

TRANSFER FUNCTION OF A ROTOR

In this appendix, the derivation of the transfer function of a rotor isdescribed by using the harmonic method. First, gust amplitude is assumed to be zeroin the computer code. Second, the transfer function is assumed to be expressed inthe frequency domain as follows:

{z} = [Tcl{6} (E1)

where {z} and {6} are the response vector and the input vector, respectively; {z} and{8} are given by equations (55) and (58), respectively. Therefore, [Tel becomes a(12 x 6) matrix.

In order to calculate the transfer function of a rotor, .it is necessary to modifyequation (31). Let us consider the following control input ~8:

(E2)

The real part of the input ~e corresponds with equation (57). On the other hand,the response vector {z} can be obtained by equation (25). From assumptions used here(i.e., K = N' = 5), q becomes zero in equation (25). For simplicity, let us con­sider only the vertical response:

where ZT is the complex response vector. The real part of the ZT is ZTC andthe imaginary part is ZTS. The comparison of the real part of both response andinput gives the following relationship:

(E4)

where tij (i = 1, ••• , 12, j = 1, ••. , 6) are elements of the [Tel matrix.Simi­1ar relationships can be given for the other responses.

41

To determine all the elements of [Tcl matrix, the least-squares method is used.Let us consider the sum of the squares of errors:

NS = E

n=l(E5)

where N denotes the number of parameters (the dimension of Zj).parameters to be identified (the dimension of tj) is assumed to begreater than or equal to L. Here, the vector Zj and matrix e

The number ofL. N must be

are defined as

e = (E6)

The solution that minimizes S is the least-squares estimate:

"t.J

(E7a)

or

"T T ( T )-1t. = Z. e 8 8J J

Putting the rows together again gives

where

(E7b)

(E8)

z Tz.

Jz •• .]

n(E9)

In this study, both Nand L have values of 12. Therefore, equation (E8) canbe rewritten as follows:

(ElO)

Elements for the transfer function [TC]' calculated using equation (E10), are shownin tables 4 and 5.

42

1.

REFERENCES

Harvey, K. W.; Blankenship, B. L.; and Drees, J.copter Gust Response at High Forward Speeds.1969.

M.: Analytical Study of Heli­USAAVLABS, TR 69-1, Sept.

2. Drees, J. M.; and Harvey, K. W.: Helicopter Gust Response at High ForwardSpeed. J.Aircraft, vol. 7, no. 3, May-June 1970, pp. 225-230.

3. Bergquist, R. R.: Helicopter Gust Response Including Unsteady AerodynamicStall Effects. U.S. Army Air Mobility Research and Development Laboratory,TR 72-68, May 1973.

4. Arcidiacono, D. J.; Bergquist, R. R.; and Alexander, W. I., Jr.: HelicopterGust Response Characteristics Including Unsteady Aerodynamic Stall Effects.NASA SP-352, Feb. 1974.

5. Yasue, M.: A Study of Gust Response for a Rotor-Propeller in Cruising Flight.NASA CR-137537, Aug. 1974.

6. Judd, M.; and Newman, S.Gusts and Turbulence.

J.: An Analysis of Helicopter Rotor Response due toVertica, vol. 1, no. 3, 1977, pp. 179-188.

7. Mirandy, L.: A Dynamic Loads Scaling Methodology for Helicopter Rotors.J. Aircraft, vol. 15, no. 2, Feb. 1978, pp. 106-113.

8. Dahl, H. J.; and Faulkner, A. J.: Helicopter Simulation in AtmosphericTurbulence. 4th European Rotorcraft and Powered Lift Aircraft Forum,Sept. 13-15, 1978.

9. Azuma, A.; and Saito, S.: Study of Gust Response by Means of the Local Momen­tum Theory. 5th European Rotorcraft and-Powered Lift Aircraft Forum,Sept. 4-7,1979. Also J. Am. Helicopter Soc., vol. 27, no. 1, Jan. 1982,pp. 58-72.

10. Gaonkar, G. H.; and Hohenemser, K. H.: Flapping Response of Lifting RotorBlades to Atmospheric Turbulence. J. Aircraft, vol. 6, no. 6, Nov.-Dec.1969, pp. 496-503.

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43

14. Gaonkar, G. H.: A Study of Lifting Rotor Flapping Response Peak Distributionin Atmospheric Turbulence. J. Aircraft, vol. 11, no. 2, Feb. 1974,pp. 104-111.

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Shear and Moment Response of the Airplane Wing to NonstationaryNational Aeronautics Laboratory, Technical Report 404T, Japan,

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17.

18.

Lin, Y. K. ; Fujimori, Y. ; and Ariaratnam, S. T. : Rotor Blade Stability inTurbulence Flows. Part I. AIAA J., vol. 17, no. 6, June 1979,pp. 545-552.

Fuj imori, Y.; Lin, Y. K. ; and Ariaratnam, . S. T. : Rotor Blade Stability inTurbulent Flows. Part II. AIAA J. , vol. 17, no. 7, July 1979,pp. 673-678.

19. Briczinski, S. J.; and Cooper, D. E.: Flight Investigation of Rotor/VehicleState Feedback. NASA CR-132546, Jan. 1975.

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21. Frick, J. K.; and Johnson, W.: Optimal Control Theory Investigation ofProprotor/Wing Response to Vertical Gust. NASA TM X-62384, Sept. 1974.

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23. Ham, N. D.; Bauer, P. H.; Lawrence, T. H.; and Yasue, M.: A Study of Gust andControl Response of Model Rotor-Propellers in a Wind Tunnel Airstream.NASA CR-137756, Aug. 1975.

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26. Yasue, M.; Vohlow, C. A.; and Ham, N. D.: Gust Response and its Alleviationfor a Hingeless Helicopter Rotor in Cruising Flight. 4th European Rotor­craft and Powered Lift Aircraft Forum, Sept. 13-15, 1978.

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44

29. Ham, N. D.; and McKillip, R. M., Jr.: A Simple System for Helicopter Individua1­Blade-Control and its Application to Gust Alleviation. Proceedings of the36th Annual National Forum of the American Helicopter Society, May 1980.Also 6th European Rotorcraft and Powered Lift Aircraft Forum, Sept. 16~19,

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30. Ham, N. D.: A Simple System for Helicopter Individua1-B1ade-Contro1 usingModal Decomposition. Vertica, vol. 4, no. 1, 1980, pp. 23-28 •

•! 31. Rahnema, R. M.: Alleviation of Helicopter Fuselage-Induced Rotor UnsteadyLoads through Deterministic Variation of the Individual Blade Pitch. NASACR-166234, May 1981.

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34. Ham, N. D.; Beha1, B. L.; and McKillip, R. M., Jr.: A Simple System for Heli­copter Individua1-B1ade-Contro1 and its Application to Lag Damping Augmenta­tion. 8th European Rotorcraft Forum, Aug. 31-Sept. 3, 1982.

35. Ham, N. D.: A Preliminary Note on the Application of Individua1-B1ade-Contro1to Vertical Vibration Alleviation. VTOL ~echno1ogy Laboratory, MassachusettsInstitute of Technology, Cambridge, Mass., TR 196-5, Oct. 1962.

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38. Kidd, D. L.; Balke, R. W.; Wilson, W. F.; and Wernicke, R. K.: Recent Advancesin Helicopter Vibration Control. Proceedings of the 26th Annual NationalForum of the American Helicopter Society, June 1970.

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40. McCloud, J. L., III; and Kretz, M.: Mu1ticyc1ic Jet-Flap Control for Allevia­tion of Helicopter Blade Stresses and Fuselage Vibration. NASA SP-352,Feb. 1974.

41. McCloud, J.L., III: An Analytical Study of a Mu1ticyc1ic Controllable TwistRotor. Proceedings of the 31st Annual National Forum of the AmericanHelicopter Society, May 1975.

45

42. Biggers, J. 0.; and McCloud, J. L., HI: A Note on Multicyclic Control bySwashplate Oscillation. NASA TM-78475 , Apr. 1978.

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44. Brown, T. J.; and McCloud, J. L., III: Multicyclic Control of a HelicopterRotor Considering the Influence of Vibration, Loads, Control Motion. Pro­ceedings of the AlAA/ASME/ASCE/AHS 21st Structures, Structural Dynamics andMaterials Conference. May 1980.

45. McCloud, J. L., III: The Promise of Multicyclic Control for Helicopter Vibra­tion Reduction. Vertica, voL 4, no. 1, 1980, pp. 29-4L

46. Chopra, I.; and McCloud, J. L., III: Considerations of Open-Loop, Closed-Loopand Adaptive Multicyclic Control Systems. Proceedings of the AHS NortheastRegion National Specialists' Meeting on Helicopter Vibration Technology forthe Jet Smooth Ride, Nov. 1981. Also, J. Am. Helicopter Soc., vol. 28,no. 1, Jan. 1983, pp. 63-77.

47. Kretz, M.: Research in Multicyclic and Active Control of Rotary Wings.Vertica, vol. 1, no. 1/2,1976, pp. 95-105.

48. Kretz, M.; and Larche, M.: Future of Helicopter Rotor Control. Vertica,vol. 4, no. 1, 1980, pp. 13-22.

49. Kretz, M.: Active Elimination of Stall Conditions. Proceedings of the 37thAnnual National Forum of the American Helicopter Society, May 1981.

50. Sissingh, G. H.; and Donham, R. E.: Hingeless Rotor Theory and Experiment onVibration Reduction by Periodic Vibration of Conventional Controls. NASASP-352, Feb. 1974.

51. Young, M. I.: Optimizing the Cyclic Control Response of Helicopter Rotors.Vertica, vol. 1, no. 1/2, 1976, pp. 107-112.

52. Taylor, R. B.; Farrar, F. A.; and Mia!J' W.: An Active Control System for Heli­copter Vibration Reduction by Higher Harmonic Pitch. Proceedings of the35th Annual National Forum of the American Helicopter Society, May 1979.

53. Beiner, L.: Optimal Second Harmonic Pitch for Minimum Oscillatory Blade LiftLoads. 5th European Rotorcraft and Powered Lift Aircraft Forum, Sept. 4-7,1979.

54. Beiner, L.: Optimal Higher Harmonic Blade Pitch Control for Minimum Vibrationof a Hinged Rotor. 6th European Rotorcraft and Powered Lift Aircraft Forum,Sept. 16-19, 1980.

Jacob, H. G.; and Lehmann,Harmonic Rotor Control.Forum, Sept. 8-11, 1981.

55. G.: Optimization of Blade Pitch Angle for Higher7th European Rotorcraft and Powered Lift Aircraft

46

56. Lehmann, G.: A Digital System for Higher Harmonic Control of a Model Rotor.8th European Rotorcraft Forum, Aug. 3l-Sept~ 3, 1982.

57. Reichert, G.: Helicopter Vibration Control - A Survey. 6th European Rotor­craft and Powered Lift Aircraft Forum, Sept. 16-19, 1980.

58. Yen, J. G.: Higher Harmonic Control for Helicopters with Two-Bladed and Four­Bladed Rotors. J. Aircraft, vol. 18, no. 12, Dec. 1981, pp. 1064-1069.

59. Abramson, J.; and Rogers, E. 0.: Optimization Theory Applied to Higher HarmonicControl of Circulation Controlled Rotors. Proceedings of the 37th AnnualNational Forum of the American Helicopter Society, May 1981.

60. McHugh, F. J.; and Shaw, J.: Benefits of Higher-Harmonic Blade Pitch: Vibra­tion Reduction, Blade-Load Reduction, and Performance Improvement. Pro­ceedings of the American Helicopter Society Mideast Region Symposium on RotorTechnology, Aug. 1976.

61. Shaw, J.: Higher Harmonic Blade Pitch Control: A System for Helicopter Vibra­tion Reduction. Ph.D. Thesis, Massachusetts Inst. of Tech., Cambridge,Mass., May 1980.

62. Shaw, J.; and Albion, N.: Active Control of the Helicopter Rotor for VibrationReduction. Proceedings of the 36th Annual National Forum of the AmericanHelicopter Society, May 1980.

63. Shaw, J.; and Albion, N.: Active Control of Rotor Blade Pitch for VibrationReduction: A Wind Tunnel Demonstration. Vertica, vol. 4, no. 1, 1980,pp. 3-11.

64. Wood, E. R.; Powers, R. W.; and Hammond, C. E.: On Method for Application ofHarmonic Control. Vertica, vol. 4, no. 1, 1980, pp. 43-60.

65. Wood. E. R.; and Powers, R. W.: Practical Design Consideration for a Flight­worthy Higher Harmonic Control System. Proceedings of the 36th AnnualNational Forum of the American Helicopter Society, May 1980.

66. Powers, R. W.: Preliminary Design Study of a Higher Harmonic Blade FeatheringControl System. NASA CR-159327, June 1980.

67. Hammond, C. E.: Wind Tunnel Results Showing Rotor Vibratory Loads ReductionUsing Higher Harmonic Blade Pitch. Proceedings of the 36th Annual NationalForum of the American Helicopter Society, May 1980. Also, J. Am. HelicopterSoc., vol. 28, no. 1, Jan. 1983, pp. 10-15.

68. Molusis, J. A.; Hammond, C. E.; and Cline, J. H.: A Unified Approach to theOptimal Design of Adaptive and Gain Scheduled Controllers to Achieve MinimumHelicopter Rotor Vibration. Proceedings of the 37th Annual National Forumof the American Helicopter Society, May 1981.

69. Molusis, J. A.; and Bar-Shalom, Y.: Identification and Stochastic Control ofHelicopter Dynamic Modes. NASA CR-166425, Jan. 1983.

47

70. Johnson, W.:Vibration.

Self-Tuning Regulators for Mu1ticyc1ic Control of HelicopterNASA TP-1996, Mar. 1982.

71. Molusis, J. A.; Mookerjee, P.; and Bar-Shalom, Y.: Evaluation of the Effect ofVibration Nonlinearity on Convergence Behavior of Adaptive Higher HarmonicControllers. NASA CR-166424, Jan. 1983.

72. Mo1usis, J. A.: The Importance of Nonlinearity on the Higher Harmonic Controlof Helicopter Vibration. Proceedings of the 39th Annual National HelicopterSociety, May 1983.

73. White, F.; and Blake, B. B.: Improved Method of Predicting Helicopter ControlResponse and Gust Sensitivity. Proceedings of the 35th Annual NationalForum of the American Helicopter Society, May 1979.

74. Hall, W. E., Jr.; Gupta, N. K.; and Hansen, R. S.: Rotorcraft System Identifi­cation Techniques for Handling Qualities and Stability and Control Evalua­tion. Proceedings of the 34th Annual National Forum of the American Heli­copter Society, May 1978.

75. Fuller, J. W.: Rotor State Estimation for Rotorcraft. Proceedings of North­east Region National Specialists' Meeting on Helicopter Vibration, Nov. 1981.

76. DuVal, R. W.:Estimation.

Use of Mu1ticyc1ic Sensors for On-Line Rotor Tip-Path PlaneJ. Am. Helicopter Soc., vol. 25, no. 4, Oct. 1980, pp. 13-21.

77. DuVal, R. W.; and Mackie, D. B.: Identification of a Linear Model of Rotor­Fuselage Dynamics from Nonlinear Simulation Data. Vertica, vol. 5, no. 4,1981, pp. 317-335.

78. DuVal, R. W.; Gregory, C. Z.,Jr.; and Gupta, N. K.: Design and Evaluation ofa State-Feedback Vibration Controller. Proceedings of the American Helicop­ter Society Northeast Region National Specialists' Meeting on HelicopterVibra~ion Technology for the Jet Smooth Ride, Nov. 1981.

79. Done, G. T. S.; and Hughes, A. D.: Reducing Vibration by Structural Modifica­tion. Vertica, vol. 1, no. 1/2, 1976, pp. 31-38.

80. Walker, W. R.: Experimental App1icatlon of a Vibration Reduction Technique.Vertica, vol. 4, no. 2/4, 1980, pp. 135-146.

81. Taylor, R. B.: Helicopter Vibration Reduction by Blade Model Shaping. Pro­ceedings of the 38th Annual National Forum of the American HelicopterSociety, May 1982.

82. Friedmann, P. P.; and Shanthakumaran, P.: Aeroe1astic Tailoring of RotorBlades for Vibration Reduction in Forward. Flight. Proceedings of the AIAA/ASME/ASCE/AHS 24th Structures, Structural Dynamics and Materials Conference,May 1983.

83. Friedmann, P. P.; and Shanthakumaran, P.: Optimum Design of Rotor Blades forVibration Reduction in Forward Flight. Proceedings of the 39th AnnualNational Forum of the American Helicopter Society, May 1983.

48

84. Azuma, A.; and KawachitK.: Local Momentum Theory and Its Application to theRotary Wing. J. Aircraft, vol. 16, no. 1, Jan. 1979 t pp. 6-14.

85. Azuma, A., Saito, S., Kawachi, K.; and Karasudani, T.: Application of theLocal Momentum Theory to the Aerodynamic Characteristics of Multi-RotorSystems. 4th European Rotorcraft and Powered Lift Aircraft Forum,Sept. 13-15, 1978. Also, Vertica, vol. 3, no. 2, 1979, pp. 131-144.

86. Kawachi, K.: An Extension of the Local Momentum Theory to a Distorted WakeModel of a Hovering Rotor. NASA TM-81258, Feb. 1981.

87. Saito, S.; and Azuma, A.: A Numerical Approach to Co-Axial Rotor Aerodynamics.7th European Rotorcraft and Powered Lift Aircraft Forum, Sept. 8-11, 1981.Also, Vertica, vol. 6, no. 4, 1983, pp. 253-266.

88. Kato, K.; and Yamane, T.: A Calculation of Rotor ImpedanJe for HoveringArticulated-Rotor Helicopters. J. Aircraft, vol. 16, no. 1, Jan. 1979,pp. 15-22.

89. Kato, K.; and Yamane, T.: Calculation of Rotor Impedance for Articulated-RotorHelicopters in Forward Flight. J. Aircraft, vol. 16, no. 7, July 1979,pp. 470-476.

90. Kato, K.; and Yamane, T.: Numerical Prediction of Typical Articulated RotorImpedance. J. Aircraft, vol. 16, no. 7, July 1979, pp. 417-418.

91. Kato, K.; Yamane, T.; et a1.: Experimental Substantiation for Hovering RotorVertical Impedance Calculations. J. Aircraft, vol. 18, no. 6, June 1981,pp. 445-450.

92. Gessow, A.; and Myers, C. G., Jr.: Aerodynamics of the Helicopter. FrederickUngar Publishing Co., 1952.

93. Bramwell, A. R. S.: Helicopter Dynamics. Edward Arnold, London, 1976.

94. Johnson, W.: Helicopter Theory. Princeton University Press, Princeton, N.J.,1980.

95. Etkin, B.: Dynamics of Atmospheric Flight. John Wiley & Sons, Inc., NewYork, 1972.

96. Bisplinghoff, L. R.; Ashley, H.; and Halfman, L. R.: Aeroe1asticity.Addison Wesley Publishing Co., San Francisco, 1957.

97. Peters, D. A.; and Hohenemser, K. H.: Application of the Floquet TransitionMatrix to Problems of Lifting Rotor Stability. J. Am. Helicopter Soc.,vol. 16, no. 2, Apr. 1971, pp. 25-33.

98. Bielawa, R. L.: Dynamic Analysis of Mu1ti-Degree-of-Freedom Systems UsingPhasing Matrices. NASA SP-352, Feb. 1974.

99. Biggers, J. C.: Some Applications to the Flapping Stability of HelicopterRotors. NASA SP-352, Feb. 1974. Also, J. Am. Helicopter Soc., vol. 19,no. 4, Oct. 1974.

49

100. Hammond, C. E.: An Application of F10quet Theory to Prediction of MechanicalInstability. NASA SP-352, Feb. 1974. Also, J. Am. Helicopter Soc.,vol. 19, no. 4, Oct. 1974, pp. 14-23.

101. Gaonkar, G. H.; Simha Prasad, D. S.; and Sastry, D.: On Computing F10quetTransition Matrices of Rotorcraft. J. Am. Helicopter Soc., vol. 26,no. 3, July 1981, pp. 56-61.

102. Franklin, G. F.; and Powell, J. D.: Digital Control of Dynamic Systems.Addison-Wesley Publishing Company, San Francisco, 1981.

103. Bryson, A. E., Jr.; and Ho, Yu.-Chi.: Applied Optimal Control: Optimization,Estimation and Control. Hemisphere Publishing Corporation, Washington,1975.

104. Otnes, R. K.; and Enochson, 1.: Applied Time Series Analysis. Vol. 1. BasicTechniques. John Wiley and Sons, New York, 1978.

50

TABLE 1.- BLADE CHARACTERISTICS

f.

Rotor radius, RBlade lift slope, aBlade semi-chord, bNumber of blades, NRotor rotational speed, QBlade twist rate, 8tPosition of flapping hinge,Blade cutoff, rCC.G. position of blade, rCGPosition of lag damper, r sLag damper coefficient, CsBlade mass, illMoment of inertia of blade,Mass moment of blade, M~

Lock number, yWing sectionGross weight, W

8.53 m5.730.2185 m423.67 rad/sec_8°

r~ 0.3 m0.594 m2.74 m0.3 m1000.0 N-m-sec/rad106.4 kg

IS 1593.9 kg-m2

1659.1 kg-m2

8.84NACA 001262259.4 N

TABLE 2.- BLADE PROPERTIES

Length, GJ, Ely, Elz , LiI, Lim, Lie, e, I

I N-m2 N-m2 N_m2 kg-m2 kg deg mm x10 4 x104 x10 5 x10- 3 x10- 1 x10- 1 xlO- 2

1 0.003 34.13 22.37 26.68 94.71 3.063 0.0 0.02 .327 34.13 143.4 13.19 87 •.80 320.7 .0 .03 .254 12.97 20.28 6.801 59.35 144.9 .0 .04 .508 6.017 4.900 4.929 29.81 24.06 -.28 .41675 .610 5.018 4.214 4.567 31.57 38.50 -6.816 .280 40.37 24.21 -3.127 .787 47.02 62.13 -8.808 .457 47.82 39.67 -5.109 .737 43.22 58.05 -8.23

10 .406 44.71 35.29 -4.5411 .711 55.68 53.82 -7.9412 .762 38.35 57.75 -8.5213 .254 34.55 22.02 -2.8314 .610 47.02 48.13 -6.8115 .635 39.70 60.23 -7.0916 .356 23.03 29.17 -3.9717 .254 33.22 30.04 -2.8.418 .280 , . 7.448 22.02 -3.12

51

TABLE 3.- ROTOR OPERATING CONDITIONS

Advance ratio, l..lCollective pitch angle, 80Longitudinal pitch angle, 81SLateral pitch angle, 81CInflow ratio, A

0.188.0°

-1.48°.695°.0179

TABLE 4.- TRANSFER FUNCTION OF ROTOR FOR GUST FREQUENCY OF 2.0 Hz

-25.52 -576.5 -192.6 22.40 81.50 -11. 92

576.5 -25.52 22.40 -192.6 11.92 81.50

9.831 -59.62 -211.6 -44.92 50.86 53.35

66.51 15.42 3.635 -179.3 -88.02 59.16

34.46 -112.7 2397. -85.89 2561. 91.49

112.7 34.46 85.89 2397. -91.49 2561.[TC] = xl0 2

-63.17 2.580 -48.63 216.3 -56.87 37.14

.,.2.580 ..,.63.17 -216.3 -48.63 -37.15 -56.87

-1185. 4.166 96.97 2508. 87.10 -2618.

-4.166 -1185. -2508. 96.97 2618. 87.10

969.5 77 .35 -10.06 -293.7 4.302 202.5

-77 .35 969.5 293.7 -10.06 -202.6 4.305

Advance ratio = 0.18

Rotor rotational speed = 23.67 rad/sec

Gust frequency = 2.0 Hz

Gust amplitude: uGo = vGo = 0.0, wGo = 1.0 m/sec

{z} = [TcH 8}

{z} (TC' HC' HS' MyC ' MyS ' yC' yS' MXC ' MXS 'T= TS' MZC ' MZS )

{8} = (81

, 82 ' 8 s , 84

, 85

, 8 )T6 .

52

= [TC]{6}

TABLE 5.- TRANSFER FUNCTION OF ROTOR FOR STEP GUST

0.996 8.545 2.984

-0.496 8.004 2.077

[TC] 1.849 1.872 0.635 x10 3=.68.42 252.3 95.31

-16.62 289.0 81.92

8.599 17.79 5.652

Advance ratio = 0.18

Rotor rotational speed = 23.67 rad/sec

Gust frequency = 0.0 Hz (step gust)

Gust amplitude: uGo = vGo = 0.0, wGo = 1.0 m/sec{z}

{z} ={6} =

53

r--------------------------.-------------

X R, Y R, ZR

X H, Y H, ZH

SINUSOIDAL GUST

COORDINATE SYSTEMS

ROTOR:

HUB:

BLADE: x, y, z

(a) Gust shape and coordinate systems.

P, ~ P4: POSITION OF SENSOR

I I- ,'(l""":~f7JlC::'-":::=-_, "

P3II _-

I' LI_-I L _ --,1-,-r I I

I / , ,

y

x

(b) Blade deflection geometry.

Figure 1.- Coordinate systems and blade deflection.

54

Figure 2.- Mode shapes and mode frequencies of blade: ~ = 226 rpm, 60 = 8°, 6t = _8°.

6.)1 =0.233 n (L1) v 6.)5 = 4.831 n (F3) w

I I~~if>W, if>v

6.)2 = 1.027 n (F 1) 6.)6 = 6.911 n (T1)if>

IW

Iv,if> W,V

6.)3 = 2.695n (F2)W

6.)7=7.183n (F4) W

I::>z:: if>

I~C7I v

-=::::::::: v if>

6.)4 = 3.250n (L2)v 6.)8 = 8.119n (L3)

v

I~ W

I~ / w

~ :.?' if> "=7 if>

Figure 3.- Hub forces and moments.

55

WG RESPONSE WITHOUT

LAG DAMPER ~/'

/ \" \wG- 4D // \

\\

10

L L1 l..--__---IL...1.__--1 ---I.__---"L..-L -..J

o

g> -90"C

w..J{!J2 -180«w

~~ -270

-360 L-.__----JL-.%..__---l ---l__---LL-L~____'

o .2 .4 .6 .8 1.0

wGIn

NONDIMENSIONAL GUST FREQUENCY

(a) Horizontal force, Fx .

Figure 4.- Rotor impedance to gust excitation without feedback: wGo = 1.0 m/sec,11 = 0.18.

56

104 wG RESPONSE WITHOUT

LAG DAMPER ~ II

"/ \

/ \u

103wG -4D .... / \

QJ \./

VI

--- \E---2:w

//~0:J 102 /' ,/'I-

-- ---~~/..Ja..

~...-:2:<t

~ wG+4n

10

L L1'---__---JL-X._____l --1..__--:lL.l.. -l

o

g> -90"C

w..JC!)2: -180<twC/)

<tI -270

-----

-360 '--__----J~_____l __l.~_ ___.:L..L __J

o .2 .4 .6 .8 1.0

wdD

NONDIMENSIONAL GUST FREQUENCY

(b) Side force, Fy •

Figure 4.- Continued.

57

wG RESPONSE WITHOUTLAG DAMPER

10

L1L-----l-L__--L ...I.-__-L..L-__--l

o

---------------~------""""-L'"",,--360 '-----L...L---==~ _LL______:L...J._ _=_....J

o .2 .4 .6 .8

wdD

NONDIMENSIONAL GUST FREQUENCY

~ -90"C

w.J

~ -180«w~:::c -270Q.

(c) Vertical force, Fz •

Figure 4.- Continued.

58

WG RESPONSE WITHOUT

LAG DAMPER "'"

~/\I \

/ \..........._-r--.... \"/ \~_U---#-

WG-4n~~/~~// wG+ 4 .Q

...--"""'"--10

L L1L....-__----L..L-__--l.... ...l...-__..L-l.-__---.J

o ----------

-270

-180

""-"""------------g -90"0

w...JC)Z«w

~::ca.

_3600L..-------L....L.----e:---...l---..L-L-----l

.2 .4 .6 .8 1.0

Wd.Q

NONDIMENSIONAL GUST FREQUENCY

(d) Rolling moment, MX.

Figure 4.- Continued.

59

10

WG RESPONSE WITHOUT

)"AGDAMPER ~

// \ /\/ \ / \

\ / \\ / \, // \

~_~6 \\

L L1l-------L~----L..-___I_____L..I..______l

1.0

--~..........-- ~---- ~ ------------

.2 .4 ;6 .8

wdQ

NONDIMENSIONAL GUST FREQUENCY

-360 L.......-__----L.........__---'- ..........__.....L.-'--_..L..----J

o

o

g -90"C

w'...J

~ -180«w~J: -270c..

(e) Pitching moment, My.,

Figure 4.- Continued.

"

60

10

/\ wG RESPONSE WITHOUT/ \~ LAG DAMPER

/ \wG

L

L

-------

-------------------~

o

-360 L..-__--I.-1,;;_~_.1_ ...L___~..L.______I

o .2 .4 .6 .8 1.0wGIn

NONDIMENSIONAL GUST FREQUENCY

g> -90"C.W...I~Z -180«w~if: -270

(f) Yawing moment, Mz •

Figure 4.- Concluded.

61

5.0

IMAG

°

-5.0

GAIN SWEEP, rad/m

o I"Kw=O .1

0T1

0F3

F200- 0.047

0.023 (0.063) /

~ ~F1 L1 VO. 1/0.073

~/ ..-~!t::>F 1L1

.,/ 0.023 (0.063) Kw = 0.111

Kw =0.047 00-F2

F30

T10

-1.0 -0.5REAL

(a) ~8 = Kw . ~w.

°

Figure 5.- Stability analysis (root loci): ].l = 0.18.

62

5.0

IMAG.

o

-5.0

GAIN SWEEP, rad/m/seco I ~

Kw= 0 .005

~

T1

0F3

LKW = 000051\Kw=O

(.),. 'I

F1 L ~.

- 10 ~- L 01vF1~

F2

F30

T,~

-1.0 -.5REAL

.(b) M = Kw • !;,w.

Figure 5.- Continued.

63

o

5.0

GAIN SWEEP, rad/m/sec2o I.

Kw= 0 5 X 10-5

0.000092 ~F2

II II =0 F1IMAG.

Kw=OL1000

..(:) F1

L1

~F2

-5.0

I

-1.0I

-.5REAL

o

(c) ~8 = Kw • ~w.

Figure 5.- Concluded.

64

15

5

~I

I­(.J

w·~ 10ou..-I<C(.J

l-ccw>>-cco~-I-I(.Jenou..owo:::>I--I0-:E<C

zN

u..

10

5\

V

r',/ ,

I '/..(r-~ '"

/ ",P 'u.. \; ,

I ,

I ~_$ BAS~IN~ ~

/ WITHOUT FEEDBACK

flO = IK . fl Z = IK Ieirp • flZ

o oLO------9..L.O-------::1-:-80:-------;2:;2::0-----~360PHASE ANGLE, rp, deg

Figure 6.- Sensitivity of the phase angle to the vertical hub force, sensor at bladetip position: wGo = 1.8 m/sec, f G = 0.5 Hz, ~ = 0.18.

65

X 10-4 X103

I- 10u

10w'ua:oLL

..Je:!uI­a:w>>a:o 5l-e:!..J..J

UCI)

oLLoWCI:::>I-..Jc..2«

zN

LL

5

BASE LINE (WITHOUT FEEDBACK)

o HARMONIC METHOD

® LOCAL MOMENTUM THEORY

-1.6-1.4-1.2-.4 -.6 -.8 -1.0FEEDBACK GAIN, Kw' rad/m

-.2O'------I.----'----L.------'-----'-------L..-----'--_----Io

o

(a) ~W feedback, ~ = 0°.

Figure 7.- Sensitivity of the feedback gain to the vertical hub force, sensor atblade tip position: wGo = 1.8 m/sec, f G ~ 0.5 Hz, ~ = 0.18.

66

w(.)a:oLL

..J«(.)

I­a:w>>­a:~ 5«..J..J(.)CI)

oLLoWCl::>I-..JQ.

~«

o

zN

LL

10

5

-2

BASE LINE (WITHOUT FEEDBACK)

-4 -6FEEDBACK GAIN, Kv X 102, rad/m

-8 -10

(b) bv feedback, ~ = 30°.

Figure 7.- Continued.

67

-160-140

BASE LINE (WITHOUT FEEDBACK)

Ol...--_-L-__.......L ..I-__-.L. L...-__....L-__--I. _

o -20 -40 -60 -80 -100 -,120FEEDBACK GAIN, Kif>' rad/rad

10

2

NLL

X 10-4

I- 10uwU0:0LL

...J«UI-0:w>>-0:0 5I-«...J...J

U(J)

0LL0WC::::>I-...JQ.~.

«0

(c) ~~ feedback, ~ = 270°.

Figure 7.- Concluded.

68

X 10-4 X 104 X 104 X 10-4

WITH FEEDBACK, liw en20

~ 1/4R I-en r = Zw p w()

II rp =2/4R ~a: 2 2 00 2 ~u.~ rp =3/4Rco '> co

::>::> [2] R :I::I: r =>

p >a:: BASE LINE a:0 - - - -- 0

(WITHOUT FEEDBACK) l-I- N N-m ««..J ..J..J 10

..J

() ()

1 enen 1 1 00 - ----u. u.0 ----- 0w wC c::> ::>I-

~~v' :'~':l !::

..JA<

1~..J,. c..c..

:~~.

:E• < :E• ««

~~.>

~~:~:.0

v

0 00liFX liFy liFZ liMX liMy liMZ

(a) ~w feedback: ~ = 0°, Kw = -0.5 rad/m.

Figure 8.- Effect of the sensing points on the feedback system: wGo = 1.8 m/sec,f G = 0.5 Hz, ~ = 0.18.

69

~20ua::ou.a::l::>J:

>a::oI-«..J..J

U 10CI)

ou.owc::>I-..JQ..

~«

o

N

2

1

WITH FEEDBACK, ~V~ rp = 1/4R

lli»~'£f1 rp =2/4R

[:~':l rp =3/4R

lW;~fi;J rp = R

- - - - - BASE LI NE(WITHOUT FEEDBACK)

~Fy

(b) ~v feedback: ~ = 30°, Kv = -0.04 rad/m.

Figure 8.- Continued.

70

2

1

N-m

2

1

o

CI)I­Zw~o~a::l::>J:

>a::oI­«..J..J

UCI)

ou.owc::>I-..JQ.

:!:«

WITH FEEDBACK, Do¢>~ rp =1/4R

ff)~;"l rp = 1/4R

I',·,') rp = 3/4R

k\iJ rp = R

- - - - - BASE LINE(WITHOUT FEEDBACK)

en 20wua::oLL

ttl::>J:

>-a::oI-«...J...J

U 10~LLoWCl::>I-...JQ.:E«

o

N

2

1

DoFy DoFZ

2

1

N-m

2

1

o

~zw:Eo:Ettl::>J:

>­a::oI­«...J...J

UenoLLoWCl::>I-...JQ.:E«

(c) 6¢ feedback: ¢ = 210°, K¢ = -100 rad/rad.

Figure 8.- Concluded.

71

~ 2 .---=----,-----::;=--r--;;;=--..--::::=::::r--,E Ol-"":.-..p,.,....---h,&--+~-b"'--p..o:----h,&--+"""ell -2 1-_..l--=:::::::L.._--l----:='--_-I--=:::::::L.._--l---1

;= a) VERTICAL S'INUSOIDAL GUST

7,.---,----r--,---.,----r--,----,--,

'1xl­

e.>5'--_-1-_-'-_--'-__'--_-'-_-'-_--'-----'

b) THRUST

b 0 ,.---,----r--,---.,----r--.---,--,~

x

cJ -5 L--...L..---'-c-)-H-O-R"'-IZ-O-N-T...JAL-L-FO-R~C-E---'---l--l

5,.--,--....,-----,--,--,--....,-----,--,

;x>-e.>

-5 '--_..1-_-'-_--'-__"-_..1-_-'-_--'----'d) SIDE FORCE

x

~ _11-~+----t--'-+--+---'-t--+--_+_je.>

e) ROLLING MOMENT

f) PITCHING MOMENT

h) PITCH ANGLECHANGE

763 4 5TIME, sec

g) YAWING MOMENT

2

3

1o

,-0~

x 0>-::E

e.>-1

30

III25,

~xN::E 20

e.>

15

13

11

g> 9"C

:r:" 7e.>l-ii: 5

Figure 9.- Time history of hub gust responses without control for a verticalsinusoidal gust: WGo = 1.8 m/sec, f G = 0.5 Hz; ~ = 0.18, 0 = 226 rpm.

72

a) VERTICAL SINUSOIDAL GUST

! 2.----=-<----r----:=r---,--.-==---r---=,---,E 01-.L::....-.p....,..,---b<~...p..c----h-e.--P>...----b,L--P~

g; -2

7 .--...---.-----,--,--'---T---,r--;---l

5 L _...l-_--l.._--1__L-_...J-_--I-_-l---l

b) THRUST

X

tJ~ -5 L-...L--..l.C-)-H-O-RLIZ-O-N-T..JA'-l-FO-R...lC-E---l..--L-l

5.----,---.---.--.---,---.----,------,

\x>

tJ-5 L-_...l-_--l.._---.1__L-_...J-_--I-_-l--l

d) SIDE FORCE

\ 0x

~ -1 I--+--+---:.-t--+---+--l--+---ltJ

-2 '--_..l-_-L_--L__L-_..l-_-L_--L----I

e) ROLLING MOMENT

1,...x 0>:E

tJ-1 L_...l--.-..L--L----L-.-..L--.l...-----L---I

f) PITCHING MOMENT

76345TIME, sec

g) YAWING MOMENT

2

~ ~

~ ..... ~.-' ..... -

.-'~ ..... ~ ...../'~ .....

la)'ITCH ANGlE CHANGE1o

30 r--..,--A:--___r--f-~--r--I_1't-___r--,

It),250...

xN:E 20

tJ

15

13

11

:g> 9'0

:x:" 7tJl-ii: 5

3

Figure 10.- Time history of hub gust responses with individual blade pitch control fora vertical sinusoidal gust: wCo = 1.8 m/sec, f C = 0.5 Hz, ~ = 0.18, Q = 226 rpm,Kw = -0.5 rad/m.

73

a) VERTICAL SINUSOIDAL GUST

xI­

()

~ 2r-;:;--;;r-,,--,;r--;;:---,;r-~Ar,,~r-;;;--:;;r- -=~E Ot-L+-f-hI-+f-\-+Jt+lrf-\+H+1'-\-cf-+f-t-f-Jl,.-f-'k-H~ -2

7r---r----r--.,-----,---,--,---.---,

5'--_-l-_-'-_.........__"--_......._--L-_---J'---l

b) THRUST

1 0 ...--,----,----,-.-..,---,---,--,---,~xcS -5 L..----l----'-c-)-H-O-R.LIZ-O-N-T-ALL-F-O-R...JCL...E---'---.1--1

5r--,---.,....-----r-----r-----,--r---,--.

'kx>

()

d) SIDE FORCE

x

~ -1 t-+-/I'-~'-It_~-tJL._I'_---flJL......JL.-jfl'__'+_-jL--'L-jlL-f()

-2 I---......._--L-_---J__J..-_......._--l-__~

e) ROLLING MOMENT

f) PITCHING MOMENT

h) PITCH ANGLECHANGE

763 4 5TIME, sec

g) YAWING MOMENT

21o

1 00~

x>::E -1

()

-2

30It)

I0~

x 25N::E

()

20

13

11

5l' 9"C

:£ 7()l-ii: 5

3

Figure 11.-Time history of hub gust responses without control for a verticalsinusoidal gust: wGo = 1.8 m/sec, f G = 2.0 Hz, ~ = 0.18, Q = 226 rpm.

74

! 2E 06 -2~

1M

I0...

6xl-

e..>

5

'f 0~x

~:I: -5e..>

5

'f~

0x>e..>

-5

a) VERTICAL SINUSOIDAL GUST

b) THRUST

c) HORIZONTAL FORCE

." ~ ".,

'rr "" '11 II" 'II 'I' ..,." II' rr ''I' 'If or

d) SIDE FORCE

x

~ -1 1--JWLJI--4H--e..>

e) ROLLING MOMENT

ox>:iE -1 ~-+-J~'-Il-

e..>

-2 '-_...I-_-L._--l._--l__.l-._....l-_-l..--I

f) PITCHING MOMENT

30 ...--...,---,.--,...--...,---,.--,....--...,---,

'?o...

20 I-.....L-l....._--l.__i.-_-l....._--l.__i.-_-l.....-l

g) YAWING MOMENT

'\

h) PITCH ANGLECHANGE

13

11

g> 9''0

£ 7!:?;;: 5

3

1o 2 345

TIME, sec6 7

Figure 12.- Time history of hub gust responses with individual blade pitch control fora vertical sinusoidal gust: WCO = 1.8 m/sec, fC = 0.5 Hz, ~ = 0.18, Q = 226 rpm,~ = -0.5 rad/m.

75

u

~ 2E(; 0!: a) VERTICAL STEP GUST

7

M

fI

~x 6I-

()

5b) THRUST., 0

0~

XJ: -5(.)

c) HORIZONTAL FORCE

5,.0~

0x .....>

(.)

-5d) SIDE FORCE

,.00

~

xX::i: -1

(.)

-2e) ROLLING MOMENT

,.0

~x>::i: -1

()

-2f) PITCHING MOMENT

25It)

I

~0~

x 20

VN::i:

(.)

15g) YAWING MOMENT

hI PITCH ANGLECHANGE

13

11

~ 9

::t:' 7~ii: 5

3

1o 2 345

TIME, sec6 7

Figure 13.- Time history of hub gust responses without control for a vertical stepgust: wGo = 1.8 m/sec, ~ = 0.18, Q = 226 rpm.

76

a) VERTICAL STEP GUST

~ 2rf===f==p==f====f===r==F==F=16 ol-J-_--1-_-.l.__L-_--l-_-.l.__J..-_-L-J;;;

7,..-,.--.,--.,---,----,----.--r--->

XI-

U5L.-_J.-_...I.,.._..l-_...I.-_...I-_...I-_...L....J

b) THRUST

~ °EErr:El·'j,~u:r -5 J-lJ

c) HORIZONTAL FORCE5.---..,_-___r--~-..,_-___r--.,.......-~~

x>u

d) SIDE FORCE

ox

~ -1 t--'t'--l----l---f---l----l---f---+--Ju

-2 1..-_-'-_.....L..._--'-_.....l.._--l._.....JI.--_.1-..J

e) ROLLING MOMENT

ox

~ -1 {--I-+--+---l---+--+---l---+--Iu

-2 L-_J.-_..l-_...I.-_....L..._...I-_...l-_...L....Jf) PITCHING MOMENT

30 r----r--,---,..--....,--___r--.,---..,--.Ll'l

b~

g) YAWING MOMENT

h) PITCH ANGLECHANGE

13

11

g> 9."

:r 7~0:: 5

3

1o 2 3 4 5

TIME, sec6 7

Figure 14.- Time history of hub gust responses with individual blade pitch control fora vertical step gust: WCO = 1. 8 m/sec, )l = 0.18, r2 = 226 rpm, l<w = -0.5 rad/m.

77

INVARIABLE

OFF-LINE

ADAPTIVE

ON-LINE

OPEN-LOOP

CLOSED-LOOP I

Figure 15.- Four regulator options.

OISE

-1IIIIIIIII

I IL __~ ----lSELF-TUNING REGULATOR

~wG HELICOPTER FFT

DYNAMICS HARMONIC() Tn,Zon ANALYSIS

MEASUREMENT N,,/

Zn~()n()n

vn

, ,--------.~-----4--

_ ACTUATOR J- CONTROLLER .... KALMAN FILTER .....- KALMAN GAINS

AO I O~+1,J rn, an

IMn, Pn

/\

t ~nknI Zon

III I Mn+1

FEEDBACK GAINS ...-- __ JI Wz, W(), WA()

I C, C(), CA() CAUTION

(a) Open-loop regulator.

Figure 16.- Adaptive regulators for gust alleviation.

78

OISE

-1IIIIIIII

I IL --.JSELF·TUNING REGULATOR

~wG HELICOPTER FFT

DYNAMICS HARMONIC() Tn,Zon ANALYSIS

MEASUREMENT NZn <)()n

()nvn

~ t...

~ I---·~----~-----~--I

- ACTUATOR t CONTROLLERKALMAN FILTER +-

KALMAN GAINS~() ()~+1' J rn, an

I knMn, Pn

I ~ l~nI I

I FEEDBACK GAINS:Mn+1

IC,C(),C,:i() Wz, Wo, W,:iO

.. ___ ...J

I CAUTION

(b) Closed-loop regulator.

Figure 16.- Concluded.

CALCULATION TIME OFOPTIMAL CONTROL

RESPONSEPHASE 1>,

CYCLE (n) CYCLE (n+1)l-__l-_-J-__--L__.....J...-__..I..-_.---J

2 4 6 8 10 121r,rad

o

Figure 17.- Sequence of controller implementation. I

79

~ 2E 0(; -2it

7

t?0~

6xI-

(.)

5

'i 0~XJ: -5(.)

5v

J

~0X

>-(.)

-5

a) HORIZONTAL SINUSOIDAL GUST

b) THRUST

cl HORIZONTAL FORCE

..... oAo .J.. ..... III .... .L .. .., ..,... ''''I'' ..,. ..... 'V ..... ..,. v .... 'Y'""" ....

d) SIDE FORCE

'i 00~

xX~ -1

(.)

-2e) ROLLING MOMENT

'i 00~

x>-~ -1

(.)

-2f) PITCHING MOMENT

30In

I0~

x 25N

V~(.)

20g) YAWING MOMENT

h!PITCH AliiGLECHANGE

13

11

~ 9"0

:t:' 7~;;: 5

3

1o 2 345

TIME, sec6 7

Figure 18.- Time history of hub gust responses without control for a horizontalsinusoidal gust: UGo = 3.0 m/sec, f G = 2.0 Hz, ~ = 0.18, n = 226 rpm.

80

~ 2E 0c!J -2• 81 LATERAL SINUSOIDAL GUST

7

~6x...

(J

5b) THRUST

t 0...x:c -5(J

c) HORIZONTAL FORCE

5

"t:2 0x>

()

-5dl SIDE FORCE

"t 0:2xX:::E -1

()

-2e) ROLLING MOMENT

v - ~ .......... ~ ... ...~. _. - ..... _..~..... ... T "'" ..... v "".

h) PITCH ANGLECHANGE

\x> -1:::E

()

-2

30In

b...x 25N:::E

()

20

13

11

g' 9"C

:r:' 7()l-ii: 5

3

1o 2

f) PITCHING MOMENT

g) YAWING MOMENT

345TIME, sec

6 7

Figure 19.- Time history of hub gust responses without control for a lateralsinusoidal gust: VGo = 2.0 m/sec, f G = 2.0 Hz, ~ = 0.18, Q = 226 rpm.

81

~XI­

()

5'----'----'----L--'----'--_-'-_--L---lb) THRUST

xcJ -5 '----l----I.C-)-H-O-RJ...IZ-O-N-T...JA'-L-FO-R-LC-E----J----.Jl--J

5r--,-----,--,---,-----,--,-----r....

\x>­

()

-5 '--_-l-_--'-__~_ _'__ __'___-'----1-........

d) SIDE FORCE

ox

~ -1 1-..lf--II'-4JI-flf--1ll-fIl-+-fIL-I-flHlI-.ff-+-lII--1()

-2 '--_-.1...-_--1.__'--_-.1...-_......__'--_ _1_---'

e) ROLLING MOMENT

hI PITCH ANGLECHANGE

"'f 0~x>-:E -1

()

-2

30LO

I0~

x 25N:E

()

20

13

11

g> 9"0

:£ 7()l-i>: 5

3

1o 2

f) PITCHING MOMENT

gl YAWING MOMENT

345TIME, sec

6 7

Figure 20.- Time history of hub gust responses without control for three dimensionalgust: UGo = 3.0 m/sec, vGo = 2.0 m/sec, wGo = 1.8 m/sec, fG = 2.0 Hz, ].1 = 0.18,Q = 226 rpm.

82

! 2 ..---=---=r---;:-~~AI-:.....--;;r-;;:~-;;--..r~-;;r-';;:lE 0 1-L'W-H-Jr+-\-f+-f+f+f+f-Tt+t'+1L.rJI-hf-hHf? -2 L..---'!--l>!--><-""'--''-L''---=-=--''--=---''--''''-->'-A'''--'.. a) VERTICAL SINUSOIDAL GUST

7..-----r-----,---,.--.---.,---r---,.--,

b) THRUST

Xl­

e.>

x

6" -5 L..---'---'-cl-H-O-R...LI-ZO-N-T-'A'-L-F-O-R...LC-E--"----1---'

b O..----,----::r--;r--:::---:r-:--,-----,---,.-,~

. ,l".

5r---..,.....----,---,.--.-----..,.....--,---,..---,

d) SIDE FORCE

ox

~ -1 1-+-W--...m.-Il--!..:.--ji~~fl!_+-+....lIf-_+--'.l_jl'-ie.>

-2 1..-_-1-_-'-_--'__"--_......._-'-_--'--'

e) ROLLING MOMENT

- -

./

h) PITCH ANGLE CHANGE

f) PITCHING MOMENT

76345TIME, sec

g) YAWING MOMENT

21o

"t 0~x>-::E -1

e.>

-2

30III

I0~

x 25N::E

e.>20

13

11

2' 9

":£ 7(.lI-0:: 5

3

Figure 21.- Time historya vertical sinusoidalinputs, two outputs):

of hubgust.

WGo =

gust responses with adaptive blade pitch control forAdaptive open-loop regulator without caution (two1.8 m/sec, f G = 2.0 Hz, ~ = 0.18, Q = 226 rpm.

83

~ 2E 0(; -2~

7

l"lI:=

6xI-

U

5

,. 0:=x

::I: -5u

5,.:=

0x>-u

-5

a) VERTICAL SINUSOIDAL GUST

1\ A A .... ... 6L l.. .... ~

'\j V'lI V'" ... " .... " """T .. vr

b) THRUST

c) HORIZONTAL FORCE

d) SIDE FORCE

"':= 0x

~ _1 t---t--fI'-..U!..-fl--"~I'-I~f"--"--fl--4--fl--'L!.-fI'-IU

-2 '--_-'-_-J.._--I.__'--_-'-_........_--I.---'

0) ROLLING MOMENT

- -- - - -- -

--h) PITCH ANGLECHANGE

,.0:=

x>-::!: -1

u

-2

30It')

I0~

x 25N::!:

u20

13

11

g> 9"C

::1:' 7ul-ii: 5

3

1o 2

f) PITCHING MOMENT

g) YAWING MOMENT

3 4 5TIME, sec

6 7

Figure 22.- Time history of hub gust responses with adaptive blade pitch control fora vertical sinusoidal gust. Adaptive open-loop regulator without caution (sixinputs, 12 outputs): WGo = 1.8m/sec, f G = 2.0 Hz, ~ = 0.18, Q = 226 rpm.

84

15

WITHOUT CONTROL

ADAPTIVE OPEN·LOOP CONTROL(THRUST RESPONSE CONTROL ONLY)

ADAPTIVE OPEN·LOOP CONTROL(COMPLETE HUB RESPONSE CONTROL)

30

"-"" ...............

'- .... -...

---,,,,,,

10 15 20 25ROTOR REVOLUTIONS

c> CONTROLLER ON

c> GUST PENETRATION (SINUSOIDAL GUST)

0 .........._-~--_l--__--'- l--__-3

5

N1-'"<I+

NC,)I­<I

~ 102o55wa:I­(I)

::Ja:J:I-w 5a:«::Jo(I)

2«w~

Figure 23.- Time history of the mean square thrust perturbation: f G = 2.0 Hz,].l = 0.18.

t,

85

! 2E6 0~ a) VERTICAL STEP GUST

7l')

~~I:=x 6I-

(,J

5b) THRUST

'1' 0:=xJ: -5(,J

c) HORIZONTAL FORCE

5

'1':= ~

x 0 ...>

(,J

-5d) SIDE FORCE

'1' 0:=xX:E -1

(,)

e) ROLLING MOMENT

~

:\f\~

b) PITCH ANGLECHANGC

'kx 0>:E

(,)

-1

25

IIII 20:=xN:E 15

(,J

10

13

11

Ii!' 9"tl

J:' 7(,JI-0:: 5

3

1o 2

f) PITCHING MOMENT

g) YAWING MOMENT

3 4 5TIME, sec

6 7

Figure 24.- Time history of hub gust responses with adaptive blade pitch control fora vertical step gust. Adaptive closed-loop regulator without caution (threeinputs, three outputs): WGo = 1.8 m/sec, ~ = 0.18, ~ = 226 rpm.

86

a) VERTICAL STEP GUST

l 2r----.=1===f====r==F==I===F=====I=6 oL---l..i-_----L_--l__L-_...L-_-L_----l.--.J~

XI­

U

7

6If

""'"5

b) THRUST

"f 00...XJ: -5u

c) HORIZONTAL FORCE

5

"f~

0x>-u

-5d) SIDE FORCE

"f 00...xX:;;: -1

U

-2e) ROLLING MOMENT

"f 00...x>-:;;: -1

U

-2f) PITCHING MOMENT

25Ln

~,0... -x 20

IVN:;;:

(J

15g) YAWING MOMENT

h) PITCH ANGLECKANGE

13

11

g> 9

"£ 7uI-0: 5

3

1o 2 3 4 5

TIME, sec6 7

[Figure 25.- Time history of hub gust responses with adaptive blade pitch control for.a vertical step gust. Adaptive closed-loop regulator without caution (threeinputs, three outputs): WGo = 1.8 m/sec, ~ = 0.18, n = 226 rpm.

87

~ 2E6 0~ a) VERTICAL STEP GUST

7

M JIt,I 10~

6x...(,)

5b) THRUST

"f 0~x

''-\J: -5(,)c) HORIZONTAL FORCE

5

or~

0.w

X....

>(,)

-5d) SIDE FORCE

or~xX::!:

(,)

e) ROLLING MOMENT

or0~

x>::!: -1

(,)

-2

25III

I0~

x 20N::!:

(,)

15

f) PITCHING MOMENT

~

V .g) YAWING MOMENT

h) PITCH ANGLECHANGe"

13

11

i' 9"0

J:' 7

~a:: 5

3

1o 2 345

TIME, sec6 7

Figure 26.- Time history of hub gust responses with adaptive blade pitch control fora vertical step gust. Adaptive open-loop regulator without caution (three inputs,six outputs): WGo = 1.8 m/sec, ]..l = 0.18, rl· = 226 rpm.

88

WITHOUT CONTROL

- - -- ADAPTIVE OPEN-LOOPCONTROL

- - - ADAPTIVE CLOSED-LOOPCONTROLPGUST PENETRATION (STEP GUSTI

15

..,2'0i=C,,)

2:::>LL 10wC,,)2«2:c:::0LLc:::W0-

C,,)5i=«c:::

Cl«:::>0

o

PCONTROLLER ON

10 15ROTOR REVOLUTIONS

20

Figure 27.- Time history of quadratic performance function J: wGo = 1.8 m/sec.

89

------~-----------_._--------------------~

1\ IA I\. I... ~ k ... .\ V .... 'V"" ... ~ V--..; .... V

5

6XI­

<..>

! 2.---.c---=......,.-=-....-..,-~;;r--;......--;;;,...,.-;<r----,;--=--=E ol-L++-\-,I~l\-,A-,L\--A--A--I-4+1~'H--\-++-I--H~o -2 L~.JY-:::!.~~~~~~....::!....--C!.~~!...-,~~

;: a) VERTICAL SINUSOIDAL GUST

7

b...

b) THRUST"

"f 00...XJ: -5<..> cl HORIZONTAL FORCE

5....

I

~0X

>-<..>

-5d) SIDE FORCE

....I 00...xX2 c 1

<..>

-2e) ROLLING MOMENT

I

h) PITCH ANGLECHANGE

....I 00...x>-2 -1

<..>

-2

30l!l

I0...x 25N2

<..>20

13

11

ll1' 9"0

J:' 7<..>l-ii: 5

3

1o 2

f) PITCHING MOMENT

g) YAWING MOMENT

345TIME, sec

6 7

Figure 28.- Time history of hub gust responses with adaptive blade pitch control fora vertical sinusoidal gust. Adaptive open-loop regulator with caution (six inputs,12 outputs): wGo = 1.8 m/sec, f G = 2.0 Hz, ~ = 0.18, Q = 226 rpm.

90

XI­

(.)

5L.-_J.--_...I-_-'-_....L-_..l..._....I-_--l-.--J

b) THRUST

b 0 r---.r-~-..r......._:u--:l,,~;r_..-_r_-.r_k"~-,\,~

x

cS -5 '----L---'--c-lH-O-R-"I-Z-O-NT-A-'--L-FO-R..J.C-E--'------'--'

5r--....----r---r--,--..,---r---r---,

x>­

(.)

-5 '-_-L-_-'-_--'-__.L-_--'-_-'--_--'--'

d) SIDE FORCE

ox

~ -1 t--'f---ff--tI'--tt-'l'L......fI'-'f'-ft'--fl--fl--ilI--Jf--'f1f-1J1-I(.)

e) ROLLING MOMENT

763 4 5TIME, sec

g) YAWING MOMENT

f) PITCHING MOMENT

2

VI 00~

x>-::iE -1

(.)

-2

30It)

I0~

x 25N::iE

(.)

20

13

11

g> 9"C

£ 7(.)I-0:: 5

3

10

Figure 29.- Time history of hub gust responses with adaptive blade pitch control forthree dimensional gust. Adaptive open-loop regulator without caution (six inputs,12 outputs): UGo = 3.0 m/sec, vGo = 2.0 m/sec, wGo = 1.8 m/sec, f G = 2.0 Hz,~ = 0.18, ~ = 226 rpm.

91

(a)

(b)

.;

../ ~v---- - - - ..... u .. X

H~-.,- ::. - - - - ~P:"".3..."~.-__

. te systems.- 30 - CoordJ.uaFigure •

92

"

o

Figure 31.- Blade section loads.

z

-b r'----+---+--+---+-.,.---t--.. x L,/Uo

hIt)WING SECTION

Figure 32.- Two dimensional thin airfoil theory.

93

L

D

Ha

Figure 33.- Schematic view of lift and drag.

94

L

:E (A{J·~qi + B!J·~qi + C·/·~q·) = F! ... Eq. (31)i=1 1 IJ I J

START

if

READINITIAL VALUE

TRIM CONDITION

if

CALCULATION OF *AERODYNAMIC COEFFS.

Ai/j• Bjj. Clj• Fj

It

CALCULATION OFTHE GENERALIZED

COORDINATES

~q. (j = 1 - L)J

FORCES ANDMOMENTS FOR

ONE BLADE

,

HUB FORCESAND MOMENTS

SUMMATION

,

STOP

I{

BLADE PROPERTYOPERATING CONDITIONTRIM POSITIONFEEDBACK GAINS

AERODYNAMIC COEFFICIENTS

(FXR' FyR• FZR' MXR' MyR• MZR)

FOR ROTATING SYSTEM

(FXN' FyN• FZN' MXN ' MyN• MZN)

FOR NON·ROTATING SYSTEM

NFX'" =:E FXN

NFYH = L: FYN

••oN

MZH L: MZN

*OPTION

A) BLADE STABILITY ANALYSIS

B) CALCULATION OF THE TRANSFERFUNCTION OF THE ROTOR

... Eq. (26)

... Eq. (14)

... Eq. (17)

... Eq. (25)

Figure 34.- Flowchart of the programming code.

95

1. Report No. ,I 2. Government ACClISSion No. 3. Recipient's Catalog No.

NASA T11 858704. Title and Subtitle 6. Report Date

A Study of Helicopter Gust Response Alleviation ~ , lqR,:\

by Automatic Control 6. Performing Organization CodeATP

7. Author(s) 8. Performing Organization Report No.

Shigeru Sidto A-9578

10. WOrk Unit No.

9. Performing Organization Name and AddressT-5484AAmes Research Center 11. Contract or Grant No.

Moffett Field, California 94035

13. Type of Raport and Period Covered

12, Sponsoring Agency Name and Address Technical MemorandumNational Aeronautics and Space Administration

14. Sponsoring Agency CodeWashington, DC, 20546 505-42-11

15, Supplementary Notes

Point of contact; William G. Warmbrodf, iulle~ i{esearcn Genter, MS 247-1,Moffett Field, CA (415) 965-5642 or FTS 448-5642

16, Abstract

Two control schemes designed to alleviate gust-induced vibration areanalytically investigated for a helicopter with four articulated blades.One is an individual blade pitch control scheme. The other is an adap-tive blade pitch control algorithm based on linear optimal controltheory. In both controllers, control inputs to alleviate gust responseare superimposed on the conventional control inputs required to maintainthe trim condition. A sinusodia1 vertical gust model and a step gustmodel are used. The invidual blade pitch control, in this research,is composed of sensors and a pitch control actuator for each blade. Eachsensor can detect f1apwise (or lead-lag or torsionwise) deflection ofthe respective blade. The actuator controls the blade pitch angle forgust alleviation. Theoretical calculations to predict the performanceof this feedback system have been conducted by means of the harmonicmethod. The adaptive blade pitch cpntro1 system is composed of a setof measurements (oscillatory hub forces and moments), an identificationsystem using a Kalman filter, and a control system based on the minimi-zation of the quadratic performance function. Calculations for theindividual blade pitch control system show that the thrust fluctuationcan be reduced by more than 50% in the low gust frequency range. Inaddition, it was found that the adaptive b1adepitch control ,system canalso be useful for high frequency gusts.

17, Key Words (Suggested by Author(s) I 18. Distribution Statement

Individual blade pitch controland stability analysis Unlimited

Adaptive blade pitch control Subject Category: 08Gust model

19.. Security Oassif. (of this report) 20. Security Classif. (of this page) 21. No. of Pages 22. Price"

Un~l Unc1 58 A04

"For sale by the National Technical Information Service, Springfield, Virginia 22161

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