Study of rotor gust response by means of the local momentum theory

STUDY OF ROTOR GUST RESPONSE BY MEANS OF THE LOCAL MOMENTUM THEORY

by

Akira Azuma, Professor and

Shigeru Saito, Graduate Student University of Tokyo

Tokyo, Japan

STUDY OF ROTOR GUST RESPONSE BY MEANS OF THE LOCAL MOMENTUM THEORY

By

.Aki;ra Azuma and Shigeru Saito University of Tokyo

SUMMARY

The vertical gust response for helicopter rotor in cruis-ing flight is studied analytically by means of the local momentum theory (LMT) and experimentally in. wind tunnel installing gust generator. By introducing the unsteady aerodynamic effects into the LMT and by taking the elastic deformation into the rotor blade, the.vibratory characteristics of the flapping behaviour and the rotor forces can be obtained.

Since the LMT is possible to calculate the instantaneous load distribution along the rotor blade in desirable azimuthal or timewise interval, the effects of gradual penetration of the rotor into the gust can be studied and the Fourier analysis or power spectrum analysis of the rotor response can be applied to any kind of gust input.

Some results obtained analytically are verified by the experimental tests performed in the wind tunnel which generates step, sinusoidal and random vertical gusts by the motion of a series of cascaded vanes in the upstream of the flow.

1. Introduction

The helicopter rotor response due to vertical gusts has been treated in many weys theoretically and experimentally .1-5) The variety of the analyses can be seen in the setting of the following assumptions: quasi-steady or unsteady aerodynamics, constant or variable induced velocity distribution, rigid or flexible blade , with or without hub motion, sudden or gradual penetration into the gust, and so on.

Since the rotor blade is operating in the unsteady flow field including the periodic variation of the horizontal or tangential velocity component as well as the vertical velocity component, the unsteady aerodynami<:: effect must be introduced in the calculation of the airloading of a blade element working with high reduced frequency.

The LMT has been developed6-8) to obtain the instantaneous airloading acting on the rotor blade and the related downwash distribution on the rotor operating plane in a reasonable range of accuracy. The theory can evaluate the effect of the tip vortices trailed from the preceding blades by introducing the attenuation coefficient which expresses the degeneration of the induced velocity at a local station on the rotor operating plane after the blade element having passed on that station.

The idea can easily be applied to the gust response of the rotor blade by taking the vertical velocity component of the gust into the normal velocity component of the airflow at the blade element. This enable us to calculate the aero-elastic characteristics and the flapping behaviour of the

rotor blades penetrating gradually into any kind of gust shapes. Then the obtained thrust, for example, shows very vibratory characteristics which have not be seen in any other paper, and thus the Fourier analysis will be introduced for the random

gust.

The hub motion can also be introduced to the LMT by only adjusting the velocity components at the blade element by those given as the vector summation of the hub motion and the flapping motion.

The wind tunnel test can validate the theory even in a restricted condition that the rotor is bounded in a narrow test area and its hub is fixed to a point in that area without having any degree of freedom in motion except the rotor

revolution.

2. Rotor response for step and sinusoidal gusts

Let us consider the following two types of spatially frozen gust :

Step: w G(X,t)

=f~Go ~~~~

(1)

Sin.usoidal: wu(X,t)=wu

0sinK(X+Xo-VGt), (2)

where the coordinate system can be seen in Fig. 1. The trans-formation of the velocity components into the blade coordinates axes will be given in APPENDIX A. Then, following to the LMl' yields where

J

x. ~+1 1 .

Z

= {~V~c.a(6.-¢. )-prr(c.I2)2VN}dxl(x .. -x.) i X. 2 ~ ~ ~ ~ ~ ~T1 ~ ~

J

x. ~+1 . =

t

P1L lrrR(l-x ) }{RQ(lJsin,P+x)IV }/1~{(2x-l-x\l) X. V=1 ·v \1 V,c L\1 = V. = ~

v

v,c sv =

a.

= ~ 1(1-x )}2dxl(x.+ -x.) \) ~ 1 ~ 2

_{PS\IVV,c(v\ljk~v\l-1jk)+M\I(y\ljk-vV-1Jk)}

Vsin(,P0+k2rrlb+ji: 1vli,P)+RQ(x.+x.+ )/2

_v=o

_{~ ~}

1 =

RQ(o/o+k2rrlb+j~

1

vli,P)+RQ(l+x

)12

\1-o

v

rr{R(l-x\1)12}2, c. = c[x=(x.+x.+ )12] ]. l ]. 1 6[x=(x.+x.+ )12] ( 3)

a = (R/2} (l-x\1), b = c/2

$i = (VN+vim+vijk)/Vi

VN = Vsini+Vcosi•ScosljJ+R(x-xS)~ - wG

i _ j-1 ( j-1 n b • )

v7 - c7 v7 +.l:7 kr.7v. . k o7 •

t-ill <-ill <-ill

J.=" =" J.

J-1 <-ill

The equations of motion for vertical acceleration and the flapping motion ~or articulated blades can be given respectively by where z/g = T/W-l IS+MsQ2S = JR(f -f }(r-rs)dr rs a g b-1 n T = k~ {.~_-o

1

Z.-ms(z+r GEl} J.- 1 c I =

r

(r-r )2dm rs

s

R MS = Jrsr(r-rs)dm f = g cos8dm g (5)

(6)

fa=

~(RQ)

2

ac(K

6

e-~Gup/UT)

(7)

r G = !R rdm/!r dm c rs rs up = -RQ{ll tani+!:1.J cosljJ+vim/RQ-w 0/RQ+(

~/Q}(x-xS)}

UT = RQ(x+\lsinljJ) ll = Vcosi/RQ,

and where the lift defficiency function, Ke and Kw , showing the unstea<ly effects of the pitching motion of the

bla~e

and of the vertical gust, can be seen in APPENDIX B.

Fig. 2 and Fig. 3 show the time responses of load factor T/W, flapping angle S and tip path plane (S₁ , S

1 )

of: a rotor penetrating gradually into a step gust aiid a ~inusoidal gust respectively. In these examples of calculation it has been assumed that the blade is affected by the quasi-stea<ly aero-dynamic force and the rotor hub is either fixed (dotted line) or free for only heaving motion (solid line).

The vibratory characteristics of the load factor is obvious. For the step gust two peaks and one valley are observed in the flapping angle as well as the load factor before the response attains to the stea<ly state.

The first peak appears with time· delay of 0.23 sec. that is equivalent to the duration within which the rotor has fully

bathed in the gust or R/Vcosi

=

1/~Q. The valley is resulted from the downwash flow caused by the upward flapping. Then the response tends to recover the steady state with the inherent or natural frequency of the blade flapping motion, which is nearly equal to the undamped natural frequency,

(M~/I )Q ~ Q.

The inertial effect of the flapping motion brings the similar fluctuation in the load factor with the phase lag of TI. Due to the gradual penetration of the rotor into the upward gust the rotor tends to tilt backward initially, then right-ward, and finally falls into a terminal orbit.

Similar results can be seen in the response for the sinusoidal gust too. However, the initial rise in the response of the load factor and the flapping angle are more gradual than those obtained for the step gust, because of the moderate slope of the sine function. This is also verified by seeing Fig.

4

which shows the difference in thrust responses between the sudden and the gradual penetrations of the rotor into a step gust.

Allowing the heaving motion of the hub does not change the peak values appreciably but alleviates the load factor as well as the flapping angle in the terminal state for the step

gust.

It is, from the theoretical consideration described

in APPENDIX A, obvious that for the sinusoidal gust the response is, in either load factor or the flapping angle, comprised of three fundamental harmonic components, the angular velocities of which are wG and wG±Q respectively, as shown in Fig.

5.

3. Wind tunnel test of a model rotor

Wind tunnel experiments were conducted at National Aerospace Laboratory (NAL), using a model rotor subjected to vertical gust.

An overall view of the model rotor and mount set up in the wind tunnel is shown in Fig.

6.

A series of vanes which generates the vertical gust in any shape within the frequency of 20 Hz can be seen at the backside of the model rotor. Mechanical properties of the model rotor are also given in Fig.

6.

Shown in Fig.

T

is a systematic arrangement of the instruments for measuring forces, moments, angular velocities, and angles;

the rotor driving system; and the filtering, monitoring, and recording systems.

Two examples of test results for the step and the sinusoidal gusts are shown in Fig. 8 and 9 respectively. The vibratory characteristics were amplified,by uneven finish of the blades and the lack of rigidity of the model support system.

Fig. 10 and 11 show comparisons of thrust between the theoretical and experimental results at the initial stage for the step and sinusoidal gusts respectively. It can be seen that the satisfactory results were obtained. In order to stress the above fact there are prepared two drawings, Fig. 12 and 13. The former shows the time lags of the first peak of thrust responses for the step gust of (a) upwind and (b) down-wind respectively. Except a misterious case at jl=O.l5 for the upwind, the theoretical results show good coincidence with the experimental data. The latter shows the power spectrum of the thrust coefficient for a sinusoidal gust with angular frequency of which is WG = 25.1 rad/sec. The biggest peak at Q was resulted from the mechanical vibration of the overall rotor systems. Shown in Fig. 14 are comparison between the computed and experimental results on the deviation of the (a) thrust coefficient and (b) flapping angle from their mean values.

4. Refinement of the analyses

Three important factors which are supposed to give some effects on the gust response, (a) Lock number, (b) blade flexibility, and (c) unsteady aerodynamics, are treated here for the refinement of the analyses.

(a) Lock Number

Fig. 15 shows the theoretical comparison of time responses of the load factor for the step gust in three different Lock numbers, y=4.42, 8.84 and 17.68. As expected intuitively,

any noticeable change is not observed except that the larger Lock number has better damping.

(b) Blade Elasticity

The blade bending flexibility can easily be introduced to the dynamic analyses based on the LMI'. The flapwise and chord-wise bending and torsional deflections were expressed in mQdal expansion series and solved by Holzer-MYklestad method9,1°J.

The followings are the correction terms in blade pitch and velocities due to the blade deflections:

l\8 =

q,

LIUP = w

(8)

The equations of blade deflections can be seen in APPENDIX C. Shown in Fig. 16 a and b are the time responses of load factor of the rot.or penetrating gradually into (a) step gust and (b) sinusoidal gust respectively. The rotor blades are considered to be either flexible, shown by solid lines, or rigid, shown by dotted lines. It can be seen that the bending flexibility of the blade del~s the response and attenuates the load factor. The difference is appreciable for the

sinusoidal gust, specifically, in steady response. (c) Unsteady Aerodynamics

As written in APPENDIX B, the unsteady lift of a two-dimensional wing which has a simusoidal pitching and surging motions and is operationg in a sinusoidal gust· can be obtained analytically. The principal parts of K

0 and

KWG

in APPENDIX B can be approximated by the following complex expressions, lift defficiency functions for complex sinusoidal inputs,in the small reduced frequency k :

KG~ Jo(ka2)C(k)e-j(ka2cosW)

Kg

~ _{Jo(ka2 )S(k)e-j(ka2COSW-k)}

G

(9)

where C(k) and S(k) are the theodorsen and Sears functions respect-ively and Jo(ka2) is the 0-th order Bessel function.

The above lift defficiency functions, K₈ and KgG' are respectively shown in Fig.l7 a and b, in comparison with the Theodorsen and Sears functions. As seen from equations (9) and these figures the lift difficiency functions are very close to the original functions, C(k) and S(k).

Shown in Fig.l8 is time responses of the load factor for a sinusoidal gust in comparison of the unsteady aerodynamics with the quasi-steady aerodynamics. A little phase difference

can be visible but the dynamic behaviour is almost same for the both results because of small reduced frequency. However, higher reduced frequency does not bring any appreciable thrust change because the wave lengh of the gust approaches to the rotor diameter.

5. Conclusion

The local momentum theory(LMr) has been applied to analyze, the gust response o:f helicopter rotor ·and verified by the wi·nd tunnel tests.

Since the LMr is possible to calculate the instantaneous load distribution along the rotor blade in desirable timewise in-terval, the e:f:fects o:f gradual penetration o:f the rotor into the gust can be studied and the Fo\].rier analysis can be well applied to the random gusts.

The blade :flexibility was an important :factor :for determining the :flapping behaviour o:f the rotor blade in the vertical gust, whereas the unsteady aerotynamics o:f the rotor blade did not bring any appreciabe change :from the quasi-steady aerodynamics because the reduced :frequency in the sintisoidal gust was considered to be small.

Nomenclature

•

• litt slope _~ • airloa.ding

b • ving semi-chord and blade number

"

• blade section mass

j

c. elm • attenuation coe:tticients n .;. load factor

C(k) • Theodorsen tunction r • radius

c.,

• airfoil profile drag coetticient S(k) • Sears fUnction

CT • thrust coertieient • T/pnR1(R0)2 T • thl"U3t end tension

•

• ving chord t • ti:me

E • Young's modulus

_llz·'l>

• tangential and normal componenta ot

•

• distance between mass and elastic o.xia velocity at a.blade.ele~nt

••

• distance at root betveen elastic v • :fozve.rd.velocity

axis and :t'ea.therlng a.xia _VG • gust :torvard.velocity

G • shear modulus _VG • vertical gust velocitr

•

• gravitY. acceleration _vc, ;. gust amplitude

~~· Iy • blade croas-section moment ot inertia

a

• flapping angle • 130 T a1ccoslj.l + B18sinljl

•

• inclina£ion angle ot tip-path-plene y • Lock number·• pacR1/I

J • torsional rigidity constant 6 • ti. function

.J.(z) • Bessel function ot n-tb order e • pitch angle ot a "bod1"

j • itne.gine..ry ,.. r-r K • vave number • Zrr/).

K • unstea~ attenuate-coefficient tor A • va.ve length VG

vertica.l gust p • air density

Ke • unstea~ attenuate coetticient tor ¢(a) • Wagner function

pitching motion ot a blade Y(ol • Kussner tunction

k • reduced trequency >/1 • e.dmuth angle

kA • blade cross-sectiOn pol&r radius >/lo • ini tia.l azimuth engle

APPENDIX A Expression o~ sinusoidal gust in the rotor cordinate system

Let us assume that the rotor advances diagonally with the yawing angle, '¥, to a two~dimens·ional~sinusoidaJ. gust. The gust' given by equation (2) can be expressed in the. rotor coordinate system by applying the·successive coordinate trans~ormations such as

(i) trans~ormation ~rom the hub coordinate axes to the stationary

coordinate axes

T1 =

(~~~!~ ~!!~ ~

)

{A~1)

0 0 1 '

(ii) trans~ormation ~rom the rotor coordinate axes and the hub coordinate axes, ( cos (Qt+lj!o) T2 = sin(Rt+lj!o) 0 ~sin(Rt+lj!o) cos (Rt+lj!0 ) 0

~

)

.

(A~2)

Then, a point along XR axis, r, is expressed in the (X,Y,Z) axes as ~ollows:

(X,Y,Z)T =

T1•T2•(r,O,O)T+T1·(~Vt,O,O)T+(Xo,Yo,Zo)T, (A~3)

where the detailed expression o~ the X is given by

X= rcos(nt+lj!o~'¥)~Vtcos'¥+ Xo. (A~4)

Substituting this into equation (2) yields wG(r,t)/wG

0 = sin[K{rcos(nt+lj!o~'¥)~(Vcos'¥+VG)t+x0}] = cos(wGt+~1)[2Jo(a _s lJ1(a )cosnt _c

+2

E

{2J1(a )J₂ (a )cos(nt)cos(2nnt)+(-l)n n=1 c n s

J0(a )J + (a )cos(2n+l)Qt}+4

E

{(-l)nJ + (a )

s 2n 1 c n=1 2n 1 c

cos(2n+l)Qt•

E

J (a )cos(2mnt)}-2J₀(a )J1(a )

m=1 2m s · c s

sin(Qt)-2 _n-1

!

{J0(a )J _{c 2n+1} (a )sin(2n+l)Ut+(-l)n _s

2Jl(a )J (a )sin(nt)cos(2nlli)}-4

E

{(-l)nJ (a)

s 2n c _{n=1 2n c}

cos(2nnt)•

E

J +(a )sin(2m+l)nt}]+sin(wGt+~1)

m==t 2m 1 s

[Jo(a )Jo(a )+2

E

{Jo(a )J (a )+(-l)nJ0(a )J

c s n=1 c 2n s s zn

(a )}cos(2nnt)+4

!'

{(-l)nJ (a )cos(2nQt)•

00 ~ J _{la )cos2(mnt) lt2J1 (a )Jl la )sin2lli} m~l ·2)11 s c s +4!1: 1 {Jl Cac)J 2n+l (fts lcos(Utl.sin(2n+l)Slt+(-l)nJ1 (a )J + (a )sin(nt)cos (2n+l)nt+(-l)nJ . (a ) S 2n 1 C 2n+1 C 00 cos(2n+l)Qt• L' J (a )sin(2m+l)S&t}] m=1 ·2m+1 s

(A-5)

where a = a(x) coslj!1 c a = a(x)sinlj!l s a(x) = KRx = Kr

(A-6)

lJ!r

= lJ!o-'¥ <f>r = XoK

APPENDIX B Unsteady aerodynamics for rotor blade

When a rotor blade is operating in a vertical gust, the quasi-steady airloading can be given by

(B--l)

where

UT = Rrl(x+\lsinrlt). (B--2)

However, if the variations of the blade pitch and the gust

are not slow, the unsteady aerodynamic loading must be considered. Then, by applying Duhamel's integration after having multiplied the time derivatives of Wagner function ~ for the pitch change and the Kiissner function '¥ for the gust variation, the following

unsteady airloading can be obtained.

where q>( ) = ~/" C(k) jks dk s 21T _oo jk e '¥(s) = ~foo S~k}ejk(s-l)dk 21T -00 _Jk (B--3) l t R s = s(T,t) = b/TUT(T)dT = (b){rlx(t-T)-~{cosQt-cosrlT)} and where C(kf)and S(k) are Theodorsen and Sears functions respectivelyl •

By considering the first harmonic feathering oscillation, a = 8o+,8 cosw+a sinw,

1 C l.S (B-5)

and a random gust written in Fourier expansion series, wG(t) = wG +

E

(wG . cos·w t+wG sinw t) (B-6)

, o m=1 ,me m ,ms m

equation (B-3) yields

t /-

₂1p(RQ)2ac

=

Kn·0+K

·(w

/U)

=

K

·0+~~

·W (B-7)

us o wG G T 0 ~-wG G

The detailed expression of the above equation can be seen in Table B-1.

APPENDIX C Modal equations of motion

Equations of blade chordwise and flapwise bending deflections, v and w, and torsional deflection

2

)$' can be

given respectively by the following equationsl :

where

-(Tv' )'+{{EI -(EI -EI )sin2_8}v"

z z y

-(EI -EI

){~w"sin28-$v"sin28+<jlw"cos28}

)" = z y 2

-(Tw' )'+[{EI +(EI -EI )sin28}w"

y z y +(EI -EI ) {! 2v"sin28+$v"cos28+<jlw"sin28}]" = L z y w -[(GJ+Tk 2<jl1 _{)'+(EI -EI ) [}1_{(w")2_-(v")2 }sin28 A z y 2 +v"w"cos28)

=

M<P • • • L = F +m[-20U+2e0(v'+8w')+e8B v ey +02_{(eo+2e))-m[v-e8~-n}2_(v-e8$)) L

=

F -m(q+eB-e028)-m(w+e~) w az

M$

=

Max-m[eg+kA2(B+028+208~')+e8(e

0

02-2nU))

-me[w-8v+802v+e002<jl+r02(w'-8v')) and where <'>=_£_() dt

()'=~()

dr

(C-1)

(C-2)

(C-3)

(C-4)

(C-5)

Fay=

~UT

2

ccd

0

+~ipi

F _{az .} • Z.(see eq_.(3) l · M

_p

=

~u

2c2c 2 T

m

REFERENCES (C-6)

1) Harvey, K. W., Blankenship, B. L. and Drees, J. M. : Analytical Study of Helicopter Gust Response at High Forward Speeds. USAAVLABS Technical Report 69-1, AD862594, September 1969. 2) Bergq_uist, Russell R. : Helicopter Gust Response Including

Unsteady Aerodynamic Stall Effects. USAAMRDL Technical Report 12-68, ADI63951, May 1913.

3) Judd, M. and Newman, S. J. : An Analysis of Helicopter Rotor Response due to Gusts and Turbulence. Vertica, Vol. 1, 1911, pp.ll9-188.

4) Yasue, Masahiro, Vehlow, Charles A., and Ham, Norman D. : Gust Response and Its Alleviation for a Hingeless Helicopter Rotor in Cruising Flight. Paper No. 28, Fourth European Rotorcraft and Powered Lift Aircraft Forum, September 13-15,

1918.

5) Dahl, H. J. and Faulkner, A. J. : Helicopter Simulation in

Atmospheric Turbulence. Paper No. 29, Fourth European

Rotorcraft and Powered Lift Aircraft Forum, September 13-15,

1918.

6) Azuma, Akira and Kawachi, Keiji : Local Momentum Theory and Its Application to the Rotary Wing. Journal of Aircraft, Vol. 16, No. 1, January 1919, Page 6-14.

I) Nakamura, Yoshiya and Azuma, Akira : Rotational Noise of Helicopter Rotor. To be published in VERTICA.

8) Azuma, Akira, Saito, Shigeru, Kawachi, Keiji, and Karasudani, Takahisa : Application of the Local Momentum Theory to the Aerodynamic Characteristics of Tandem Rotor in Yawed Flight. Paper No. 4, Fourth European Rotorcraft and Powered Lift Aircraft Forum, September 13-15, 1918. 9) Myklestad, N. 0. : Fundamentals of Vibration Analysis.

McGraw-Hill Book Company, Inc., New York, 1944.

10) Isakson, G. and Eisley, J. G. : Natural Frequencies in Coupled Bending and Torsion of Twisted Rotating and Nonrotating Blades. NASA CR-65, July 1964.

11) Bisplinghoff, R. L., Ashley, H. and Halfman, R. L. :

Aeroelasticity. Addison-Wesley Publishing Company, Inc. Reading, Mass. 1955.'

12) Hodges, D. H., Ormiston, R. A. and USAARDL: Stability of Elastic Bending and Torsion of Uniform Cantilever Rotor Blades, in Hover With Variable Structural Coupling. NASA TN D-8192, April 1976.

Item

Rotor radius,

No. ot' blades,

Blade chord,.

Rotor rotational speed, Blade twist angle, Collective pitch angle, Cyclic pitch angle,

Position of flapping hinge, Blade cut off,

C~ G. position of blade,

Blade mass,

Moment of inertia of blade, Mass moment of blade, Lock number,

Wing section Advance ratio, G:ross weight,

TABLE 1 Rotor dimensions

RotOr used in Rater used in

theoretical calculation experimental test

R b c r c I y

w

8.53 m 4 0.417 m 23.67 rad./sec. -8 deg. 8 deg, 0 deg. 0 deg. 0.3 m 0.594 m 2.74 m 10~86 kg sec.' m-1 162.6 kg m sec! 169.3 kg m sec! 8.84 NACA 0012 0.18 6,353 kg 0.75 m 2 0.085 m 104.7 rad./sec. o deg. 2 - 6 deg. -6 - 6 deg. -6 - 6 deg. 0.03 m 0.086 m 0.2475 m 0.058 kg sec.2 m-1 0.0125 kg m sec! 0.01638 kg m sec.' 1.45 NACA 0012 0.15 - 0.3

Tabl~ ~·

l\UI/~p(Rn)'ac • K0.a + Kv •vn/U'l'

n

....

";; • l

a,,,

• (ka,• KO,c' t;on,l•(Ae, n,,., &u)T

+ (Kv • Kv .' Kv , ... )•(vr.~· "a,,c• a, a,1 l r.,111 v ,•••IT a,,ll K •0 _o + K-_·1i' •\i 0 G

(Ke,• Ka,c' Kou• 1\01<'' _{Kou• K0 1}<'' Ko.l·

(Oe, Oleo 0,11 , 0:<', Ou, O,c, Oss)T

"v ·"v ·"ii ·"ii ,"")•

G,JSol G,lllo2 OolBoJ G,2t'ol

(iio.,,• W'Ghz' WO•Ic•l WG•Ic•2 iiG,Ic•I'""}T

vhere W₀₀₁ • x(v 0/Rn) ii 002 • ll (v0/Rfl) iiGem, • x(vGmc/Rfl) ii 08,1111 • x(vca/Rfl)

wGe,lll2 • iiGe,llll. ~<"am/Rfl)

\iGr!,m2 •-WGII,IIIJ • ~(vQDa/RI1)

w •w

.,

.

w

.,

• ""'

.

w _•• • """' • X • r/R u • Veomi/Rn • • !1t fc(u,w,t) • Jo(azk)F(k)cou(lllt-a2keos¢'} ge(l!,w,t) • Jo(a2k)G(k)sin(lllt-azkeos!l>) n- J {a2k(n~+l)) F {ll,w,t) • !' (-1).1.!?-[ n "' e n•odd n•1n.,. n(~)+l

x{F{k(n~+l) )sin(k(n~+l) _{w w} (a, t-a2cou'41))

+G(k (n~+1) _w )cos (k (ng+l )( 111 t-azcos'41))} _w

n

J {a k(n--1)) n n

n"' {F(k(n-1)sin(k(n-1)

n(wl-1 "' w

>:(a, t-azco:~l/1) )+G(k(n9-l) )eos (k( n9-l)

w w

{a2k(n§+l)) n{9)+1

w

-O{k ( ng+l)) sin(k{ ng+1 )(a, t-azcos'41))} _{w w}

J (a2k(n9-l))

+ n n w F(kfn£-1l)t'os(k(n§-1l

n(-_w l-1

I<{"' t-n1cosll>l l-a(k (ng-1) lsin(k( ng-1) _{w w} ><(a,t-ll2<'0sl/I))}J

r.<u,~oa,t.). J,(aak)f(kl•tn(wt.atkcoat) 1₁ (u ,~oa,t I • J,(ark )O(k l<!ol(ldt·&akcoatl

.!!y1. J (a,k{n!!+ll)

r!u,~oa,tlao4d•I' (.1) t" ~.~

a n n 111••• n!£l+1

II{F{k(n£+1) )co•(k(n£+1 Ha1 tga,coat)) •O(k ( n£-11)11 n{k { n£+1) la1 t•&aeoat) )

J (aalr.(a!l.1)) n

+ D h ld fF{k(n-1))coa{k(n!!..1) nlwl-1 Iii Iii

•I •t t-a•cost) ).O{k( n~1 ) }It n(k ln£-1)

•latt-a.~tllll

!L J {&Jk(.J!+lll

f' {U,III,tl • I' (-1)~1 n 111

• n-.'"n n l\1••• n(S)+1

-f1"(k(~1) ).tn(lr.( n£•1 l ( •1 t-aacoaljl l l

-tO(k(~ 1 )co. (k ( ~+1 )( •1 t-aacolt) ) } J lark{o£.1))

+ D d 111 _{[F{t(?1)dn(k(n£-1l} nlwl-1

o(q t•atco.t) )+O{k{ng-1) )eoa{k 1?1) lO(a,t-a,c:o.tllll

rc,C(u,w,tl • J,(a,k)F₀(k)coa(wt•aakcoa~k)

1

0Cu,w,tl • Jel•aklC0(k)dn(wt-aakcoll ... k) <!, - J <••k(ng.1)) F (U,III,t) '

•J'

{-l}.nr-[.JD'--:11":-"--e,G n-ode! 1 11••• n{B)+1 •[F 0(k(ng.1) )dn(t(n£+1 )(&Jt-a,co•~1)) +G (t(A1l)coa(k(A1l(aJt•&tcOa .... 1))}

'

.

.

J (a1t(n!!..1ll n n n n ~oa rr 01klnw1llstn(k(nwll nlwl-1 •( a1 t-atcos~1) )+G0 lk (ng..1) l coa (k (n2-1) >(al t•&aeoa'(.-1) l} 1 n J (a:klng+1)} 1" (u w tl • f' I-lly[ n e,G ' ' n..,ven n •• ,... nl£1+1 •{F 0Ck Cn£+1 ) lee• ( k { n£+1) ( 111 t-•• collt-1) l

-o₀iklnE+l l lllin(k(nE+l l C "I t-llzeoa!j;-1) l I

J (a2k(n!L1Jl n n

+ n w [F

0(kln--lllcoa(k(n-1) n(!!)-1 w w

• n n

~<(a₁ t-a1coa"'"l) l-G0(kfnw-1) lllinlklnw-11 ><( ljt•lltC011"'"1) J} I

r₁₁•₀(u,~oa,t) • J,{azk)F₀1k)atn(wt-atke011 .... k}

11

1,0(u,w,t) • J,(azk)00Cklcoa{wt.aakeo•t-k)

.!!.;1. J Catlr.(n!!.u)) F lu,w,t) • !• 1-1) [ n 111

a,G n-odd 1'1"1 n... n.{~)+1

>c( F

01k (ng+ 1) )eo• (k (n.g+1 )( 111 t-az eoa~l} )

-o₀ ( k{ n£•1) }ai n{k (ng.1) (a I t_,coat-1 ) ) } Jn(atlr.{n~l)) n n il r₀(k(nw1))coa(k(nwll nlwl-1 x( 111 t-azcoat-1) }.o 0 (k (ng..1) )ein(k{ n2--l) x(al t•lzCOII~1)))] n J (aalr.{n~+1)) F (u,w,t) • f• {-l)Y[ n w

e,G n-.ven n IU••• n(~)+1

1nd wheTe

>c(F 0 (lr. (n~+l) lain(k (n~ 1 ){ BJ t-&a cot1 ... l} ) +0

0 (k( n3+1) )cos (k(ng.-1 ) { &J t-aa coe¥-1) ) }

J (a,t(J!.-1)} n n + n tl 111 _(F

0{k{n-1))dn{k(n.-l) n(wl-1 Ill w

1<( &J t-aa coat-1 ) )+0

0 {k(ng..1) } eoa (lr.( n3-1) x(al t-&IC!OIIt-1))})

a, • 2.1.R/c

Jn(z} : nth Benl!l t\lnctton of" the tint kind C(k) • ,f(k)+iG(k) : Tbeodoraan tlmetton S(k) • F₀{k)+to₀(k )• C{k) (J,(k)•iJJ (k) ]+iJI (k)

: Seara FUnction

Figure l. (j(jst sbapes and the reht..d ~oordll!llte systl!af,.

o.

i (\

f"'c, /"· WITHOOT 6 ...

~

:rv

d \ : : \ ·'\HEAVING KOTIOH ~

·t

I\\

,.J\';

t

WITH HEAVING IIJTIOH -.,.} .. f

·-_{o 4"r ar 12.,.} -::: _.=

AZI!lJTH AHGL£ ,t

0 0.5 1.0 1.5 2.0

TifiE, t. sec.

I fJ K:• (leg, RIGHT TILT

-o>-•--:,c-===--,, ...

,5 -/31$, cJeo,

~I

i

-4,5

!(-TAATI~

•.•. 'I -5

Ft.,. l. 11ot1w rnp~~~t~e M to 1 llnusol411 JiUit.

~'r'~'~o-_• ______________________________ ___

""

9 c; ~ !; n-w, ~ fl+w10 t; ~

2.1l

~ ~ 4.11 ~ 1

2.1lj

_lvt'W&

3.11 4.11-;:- llfl~w 0 3 FREQUENCY I 1l

Figure 5, Pow.~ spectn. of tlU thNSt.

Wc:JITITlllii!!!TI:IT t• we. 1 8 m/sec.

GUST SHAPE '

GUST PENETRATION

TIME. t. sec.

(1 ) Time ruponus of load h~to~ uo flapping •"gle

RIGHT TILT GUST PENETRATION 8,000

I

SUDDEH

PEI<ETAATI~

'··-1.8 ""'"·

8

'J

bGUST PfNETRATIOH

l

.I \ . , , , \/'-.!•-1\J\ .... 7, I " 1 ~, "( I" I v '('--... ''"• ~ ('J \jl..-1

~

'·

~

rJ

GRADUAL PENETRATION ,M;• 1,8 11/sec,

0 0

'"

AZJI'I.ITH AMGl£ , t 0.2 ARTIOJLATED BLADE TIP£ • t • sec.

Figure 4. Dlffere~~U In tlln.st lftiiOMft llebiiMII tilt ...,. 1111111 ln. ~~ Ptftltl'tltiDIIS Of ~otor hltO I Step 1111t.

...

- GUS! !iHIERA10R -bLACl HUB BALANCE SYSTEM WWER IS RE:-ltlV(!ll

(1, 1~11tl\!lti"U[R FOR ~Ml !'lllH ~ WSI GlHlAAIOR. (1,

HIHIIIRI: FnR LOIIGI!UDIIIAL MU l'fRIIC~l \~UI(JI !ES,U .~

~

~1-r---1-..

1 sec. (t) Sttp gust 1 W& {b) Thrust 1 T

~~

0 {c:) Torq11• , Q § "'!111"'

:;I .... .;; II Ill R:Zts&

_•

0

(d) Fhpphg Ingle, /3

F1gure 11. Gust rHpCMH 111 wtlld ~1 for • stap gust.

.,

_~1

~

""'""' lv----

_I

_{__j"}

(a) IItva for• of gus\ Ill wtnof tunntl

(b) hparl•tnUI rnult

0

_{;.!!t':ftlflhwv}

: I

/W'f'{V V' I • .

(c) Tlltarttlc ruult far 1

MHhtMJttCJlly lttP gult

.,

I!"CO!Ul£R MGIIIT IAI't 1l4<:hl I ""• (b) Til rust 1 T (c) Torq11e~a

-

-(d) FhJptng tllgle • IJ

~I

0

o ••

U!JD

0 0 0 0.3 p. • 0.15 ~ ~ A

"

A II! p. • 0.2 ;:: i!i 0.2 D Q Q ~ p. • 0.25 ~ II! ~ : TliEORY 0 0.1

_"

: EXPERII£NT D 0 2

•

6 COLLECTIVE PITCH ANGLE. 8,'

....

(•) Upwind o .•

rnm

0.3

r-u ~ ,.,. • 0.15 ~ II! _{p.• 0.2} ;:: i!i 0.2 -~ D D

o---iii

~ _p. • 0.25 0.1 I 0 2

•

6

COLLECTIVE PITCH, ANGLE, 80 • deg,

(b) Downwind

Ulrll!llt ~lponU fort 5ti'P gust.

X 10 ~G 36

N.

_v ~ 16 !> ~ ~

"''

~ fi+WG ~

1

~ fl~wG

I

2 !liWG 2fl +WG 2!1 0 2

'

FR£0lEHCY I fl

Figure 13. Pow.r sptet,_ of the thrust respon!le for • sfiiUsoldal Ollft,

•r'~'o~--·---,

0 _{- - - · 9o.} z•

••

6"

0 0.05 0.1 0.15 NOODU£HSIOKAL 6UST FREQI.ENCY, WI ,Q

(l) Thrust X 10' 4 6~---, 5 p. • 0.15 ---THEORY, 90•6 (QUA$ I STEADY ) o 0.05 0.1 0,15

NONDI/£NSIONAL GUST FPfQUEHCY , w I fi {b) fhpplng angh

" ll!

~

~

f ')SUDDEN GUSt I'EHETRATtOii fliDUJI~~;· _{1.$ ""'"·}

' GUST SHAPE t

-l

1.1

!

·-·-·

!

l.O Y₁•lf,l{2•0.5Y~ '>2 •3.811 '>;-i1.6f .. •2.0)f 0.9 0 0 ;v

...

I2V

,.,.

AZIHIJTH AH6l£

•

ftiJII,. \$, Efl'«:t of tilt' I.OC:k l'll,albtr (11'1 Hit hwl hctor hr

stiddtn !**tntlon Into t $Up !JQ"·

•••

o.25

,

•.

0 ~0.25 ke ,u ..-).0 JL•O,l SEARS FUNCTION

!It) f~r nrtlul gu1t

...

k•O.Ot

...

c "'

"'

11!

"

3 " "' 0

"

if 0 .§ l.l

p

GUST PEiiETM.TlCff

JflDlii

~G•l,Sm/sec.

I

GlJST SI!Al'E ·

-M

i

l

o.9

I

IUGlD Bl.Are 0 ₀ l.l l.C o.s 0 !d A stnn~;~td•l oun GUST PEIETMTlON n

'"'f

1,0 o.9

(b) LOU ftUOr bUd on llllllltb lforodyu,ah:t GUST PENETRATJOH

n

... !

r

I. 0 [ Vl.o.~r.J""