Effects of difference in induced velocity distribution on the

THIRTEENTH EUROPEAN ROTORCRAFT FORUM

Paper No. 6

EFFECTS OF DIFFERENCE IN INDUCED VELOCITY DISTRIBUTION ON THE HELICOPTER MOTION

Yoshinori OKUNO, Akira AZUMA and Keiji KAWACHI

Institute of Interdisciplinary Research Faculty of Engineering, The University of Tokyo

4-6-1, Komaba, Meguro-ku, Tokyo, 146 Japan

September 8-11,1987

ARLES,FRANCE

EFFECTS OF DIFFERENCE IN INDUCED VELOCITY DISTRIBUTION ON THE HELICOPTER MOTION

Yoshinori OKUNO, Akira AZUMA apd Keiji KAWACHl The University of Tokyo

Tokyo, Japan

ABSTRACT

Limits or allowable range of application of

momentum theory is made clear in comparison with

momentum theory,[lJ,[2J.

the the

simple local

As far as concerning the helicopter motion in high ~

flight, there is no difference in the results calculated by the above two theories. However, for the helicopter motion in low~ flight and for the blade deflections in either low or high ~ flight, some discrepancies can be observed in the results based on the above two theories, because of a difference in the induced velocity distribution.

1. Introductio~

Every blade of helicopter rotor is operating under influences

of induced velocity distribution generated by preceding blades

and of the blade motion coupled with the helicopter motion. As

far as concerning an isolated rotor motion, the vortex theory

gives a precise information on the induced velocity distribution on the rotor disc. However, if the rotor motion is coupled with

the helicopter motion, then the calculation of the induced

velocity on the rotor blades is not easy because of the complexity, divergent tendency and long computation time in the calculation.

Thus, in the actual calculations by combining with the

blade element theory, the simple momentum theory (SMT) which

assumes a constant induced velocity distribution on the rotor

disc is very effective to give simply analytic solutions on the blade motion and the helicopter motion. However, this method of

calculation occasionally brings erroneous conclusions on the

stress analysis of the rotor blade and the resulted helicopter motion, specifically in low flight speed.

A purpose of this paper is, therefore, to calculate more precisely the spanwise and timewise change of the induced velocity distribution of a helicopter rotor operating in unsteady conditions

than the momentum theory by using the local momentum theory

CLMT). And another purpose is to compare the results obtained by the above two methods, the SMT and the LMT, and to make clear the limit or allowable range of application of the SMT to the actual problems.

2. Equations of Motion

Let us consider a helicopter motion with six-degrees of

freedom disturbed by external forces and moments which are

calculated by either the SMT or the LMT.

<1) Body Motion u=Fx/m-qw+rv -(laJ ~=Fy/m-ru+pw -(lb) w=F 2/m-pv+qu -(lc) p=[<Mx-<Iz-Iylqr+qpJzxliz•<Mz-<Iy-IxJqp-qrJzxlJzxl 2 I<Ixlz-Jzx J -<2aJ ·(2) Blade Motion flapping ·• • 2 - R I f3 fJ+KiJ fJ+MfJ Q {J+KfJ <f3-f3₀

J-S

rl <r-r f3 Jdr c lead-lag .. · 2 - R It;.';+K(.';+Mt;Q t;+Kt;<t;-t; 0

J-S

rd(r-rt;Jdr R . c -2

S

rdr-r t; J (r-r f3) Q f3 {Jdm=O (3) Blade deflection flatwise

[(EJ +(El -EI Jsin28lw"

y z y

+(El -El )(v"sin28/2+¢v"cos28+¢w"sin28)l"

z y - <Tw') '+m(i,;+e¢ J .. 2 =F -m(g+e8 -eQ 8 J az -(3a) -(3b) -(4a)

chord wise

[{EI -CEI -EI )sin28}v"

z z y

+CEI -EI )(w"sin28/2-¢v"sin28+¢w"cos28)J"

z y - C Tv' ) '+m [

v-

e 8

'¢-

Q 2 C v- e 8 ¢ ) l • . .. 2 =F _ay +m[2Qu+2eQCv'+8w'l+e88+Q ce₀+2ell torsion -(4b)

-[(GJ+Tk 2J¢'l'+CEI -EI )[Cw"2-v"2Jsin28/2+v"w"cos28J

A z y

+mk2 (

¢

+ Q 2¢) +me

[w-e

v+

e

Q 2v+e Q 2¢ +r Q 2 ( w'-

e

v') l = M - m [ eg+ k 2 C

B

+ Q 2 8 + 2 Q 8 ,;. ' ) + e 8 CeO Q 2-2 Q

u)]

ax

-(4C)

(4) Inflow in the Simple Momentum Theory steady A = tLtan i + CT/2/A2+tL2 unsteady

.

T A +A = L CT - (5a) -(5b) 3. Trim Analysis

By selecting adequate control inputs such as collective

pitch control of main and tail rotors, and longitudinal and

lateral cyclic pitch controls, a trimmed state of rotor and

helicopter motion can be obtained for a given set of advance

ratio and flight path angle of a helicopter, detailed dimensions of which are given in Table 1.

For the calculation of the induced velocity, three methods of computation coupled with the blade element theory are considered here;

(i) simple momentum theory CSMT), Cii) local momentum theory CLMT).

For the calculation of airloading along the blade span and the flapping motion of the blade, two methods are considered;

Cal Under assumptions that the blade is infinitely rigid

except a part of equivalent flapping hinge and the flapping motion is comprised of the first harmonic motion, in addition

to the constant induced velocity given by the SMT, the

(b) By disregarding the above assumptions the airloading and the blade motions are calculated by the Runge-Kutta method.

Shown in Fig.l is a set of trimmed quantities calculated by three methods; combination of (il&Ca), (i)&(b), and (ii)&(b),

for a hingeless-rotor helicopter flying with various flight

speeds in comparison with experimental data. It can be seen that

there is not any appreciable discrepancy in these quantities

among the above three methods and the experimental data.

However, as shown in Fig.2 and Fig.3, there are some

differences in the angle of attack distribution of the blade over the rotor disc between two methods, (il&(b) and (ii)&{b) or between the SMT and the LMT, specifically for low ~ flight.

4. Control Response

Shown in Fig.4 and 5 are control responses by input of -0.75 degrees for one second (-0.75 deg x longitudinal cyclic pitch for the helicopter flying advance ratio of ~=0.05 and ~=0.20 respectively.

an impulsive 1 sec) in with trimmed As seen from Fig.2 appreciable differences are observed in

almost all quantities for the low ~ flight. Specifically, the

rolling motion, which is typically observed in the

hingeless-rotor helicopter for the longitudinal cyclic pitch input, is

stressed more in the method of Cil&Cbl than in the method of Ciil&(b). However, as seen from Fig.3 in the high ~ flight there are no differences for all quantities between the two methods of

calculation because the effect of the induced velocity on the

angle of attack distribution is small.

5. Gust Response

Shown in Fig.6 through 8 are gust responses for sinusoidal vertical gusts in frozen state with the frequency of 2.0 and 0.5 Hz for the helicopter flying with advance ratio of ~=0.05 and 0.20.

Like the control response, the difference of the method of calculation between (i)&(b) and (ii)&(b) or the difference of the

induced velocity distribution, is appreciable in the low ~

flight as seen from Fig.6 and 7 and in the high frequency gust as seen from Fig.6 and 8. It can further be observed that the method of (i)&(b) tends to overestimate the hub moment in the low ~ flight.

6. Consideration on the Dynamic Inflow

In the calculation of so called ''dynamic inflow'', [3],[4], the induced velocity given by equation (5b) is considered to be related to the rotor disc but not related to the blade, and the development of the induced velocity has a time lag due to the inertia effect of the added mass related to the rotor motion.

Therefore in the gust of small wavelength over the rotor radius such as i\/R;i!l.O the above concept can not be applied.

For large wavelength over the rotor radius i\/R>>l.O the dynamic inflow is effective only for high frequency or high

u

flight because the frequency is proportional to the flight speed over the wavelength. However, in the high

u

flight the effect of the induced velocity on the airloading is not predominant.

In the control response, however, the control input with high frequency for the helicopter flying with low

u

is one of subjects of the dynamic inflow.

7. Blade Deflection

The equations of blade deflections, equation (4al-<4cl, are solved by the Holzer-Myklestad method, (5],[6].

Shown in Fig.9a-d are induced velocity, airloading,

flatwise bending moment, flatwise deflection, as a function of

the spanwise distance, for the hingeless-rotor helicopter flying with u=0.20. Some discrepancies in the above quantities can be observed between the method of (il&(b) and (iil&(b) even in this advance ratio of u=0.20, at which there is little difference in the helicopter motion between the two methods as stated in the previous sections.

8. Dynamic Analysis by Means of Linearized Equations

By introducing a perturbed motion, the following

linearized equations of motion can be derived;

mux

x"xxllz xq-mu₂₀ xe

x

/1. xP

y X +muy r o Xq, "x

muy _z

_"x

_z

_l1z

_{zq -m/l.Xo ze Z Uy Zp -mlly0}

z

_r Zq,

_llz

1 yq M M M 0 M

_Uy

MP M 0 q

u

x "z q r

e 0 0 1 0 0 0 0 0 e

muy y y y

"x

l1z

q Ye Y

Uy

YP

+muxo

Y r

-mu

Xo y<l> Uy

IxP L L L 0 L

Uy

L L 0 p "xll.zq p r I 2r

N

_u

_x

N

_{"z q}

N

0

N

Uy

NP N r 0 r <!> 0 0 0 0 0 1 0 0 <!> -(6)

T

The external forces and moments F=<X,Y,Z,L,M,Nl generated by the main rotor, the tail rotor, and the fuselage are expanded by the Taylor series as follows;

dF =~ + dCT + dCB _-(7a) d8 d8 d8 d8 dF =~ ₊ dCT + dCB _-(7b) df! d/1 d/1 d/1 dF = 5 + dCT + dCB _-(7c) d£ d£ d£ d£ dF dC₀ = -(7d) dE dE dCR ₌

_{(_!___}

_{+ dl3,_!!_ +} ~~ ₎

_c

dO ao dO a 13 dO a A R -(8a) a a dA a a dA a = _{tao +}

< ao+ do·aA >13 a13 +do·aA} CR

dCR _{= (} d.u.~+ dA.~)

_c

d,u d.u a .u d.u a A R -(8b)

= d.u (

_!_

+ d13.~ )+ dA(~ + d~~)}

c

d.u a.u d.u a 13 d.u BA dA a 13 R

dCR ₌

_{(_!___}

₊ dl3 a _{+ d.C.:~t dA.~)}

_c

·

-de Be de a 13 de a .u de a A R

-(8c)

a a d.u a dA a a

ct.u

a dAB

= { - + ( - +

- · - +

- · - )13'-t _{de· a .u+ de·a A} CR}

Be Be dea.u de a A a 13

where the advanced ratio /1, the inflow ratio A, and the flapping angles ~ are considered to be intermediate variables for other

independent variables and for themselves. The derivatives

specified in equation (7a)~(7d) are, therefore, given by the

analytic expressions of the total derivatives which are also

consisted of the partial derivatives as given by equations

<Bal~<Bc).

Shown in Fig.10 and 11 are control responses for lateral

cyclic and longitudinal cyclic pitch inputs respectively in the

helicopter flying with /1=0.20. By applying two methods of

calculation, (il&(a) and (ii)&(b), it can be revealed that as seen from Fig.10 there is no distinctive difference between the two methods within a short period, but as seen from Fig.1l the damping of oscillation in the method of (i)&(a) is much smaller

Fig.l2 shows roots of the characteristic equation of the helicopter motion in a complex plane. The method of (i)&(a) gives roots of less damping than those given by the method of (ii)&(b). Fig.l3a-e show mode ratios based on the pitching angle

e

for the longitudinal motion and the rolling angle

t

for the

lateral motion. The difference between the two methods of

calculation appears in the coupling motion, specifically in the phase angle rather than the amplitude.

9. Conclusion

Three methods of calculation, essentially based on either the simple momentum theory or the local momentum theory for the

induced velocity distribution, and combined with either the

analytic method or computational method for the helicopter and

blade motions, are applied to get control responses and gust

responses of a hingeless-rotor helicopter.

There are some discrepancies in the helicopter motions

given by two methods based on the SMT and the LMT in the low ~ flight but not in the high ~ flight. However some differences

can be observed in the bending moment and deflections of the

blade even in the high ~ flight.

Analytic method based on the linearized equations of motion

gives appropriate solutions for the helicopter motion but the

dampings of the motion are underestimated in comparison with the method of the LMT and the numerical integration. For the coupling

motion, the above analytic method gives some phase difference

from the method of LMT.

References

[lJ A. Azuma and K. Kawach i , Local Moment urn Theory and Its

Application to the Rotary Wing. J. Aircraft 16, 6-14, 1979; also AIAA Paper 75-865, 1976.

[2] S. Saito, A. Azuma, K. Kawachi, andY. Okuno, Study of the Dynamic Responses of Helicopters to a Large Airplane Wake, 12th European Rotorcraft Forum, Paper No.42, Sep. 1986.

[3] P.J. Carpenter and B.Fridovitch, Effect of Rapid Blade Pitch Increase on the Thrust and Induced Velocity Response of a Full Scale Helicopter Rotor, NACA TN-3044, Nov. 1953.

[4] D.M. Pitt and D.A. Peters, Theoretical Prediction of Dynamic-Inflow Derivatives, Vertica Vol.5., 1981.

[5] N.O. Myklestad, A New Method of Calculation Natural Modes of Uncoupled Bending Vibration of an Airplane Wings and Other Types of Beams, J. Aeronautical Science, Apr. 1944.

[6] N.O. Myklestad, Fundamentals of Vibration Analysis,

NOMENCLATURE

C = (C , C , C , C , C , C ) R T H Y Q L M E=(<l>,6,'1') El e F= (F , F , F , M , M , M ) X Y Z X Y Z GJ IX, IV, lz I ltl

i'

J Kzx Kll Ktl

K<

k'

M Mil

'

Ill tl Ill R< r c r tl r T< u.v,w

IJ= (,'lo, /is, ,'lc)

tl ,'lo e=(p,q,r)

'

_<o B= (Bo, Bs, Be) ,\ .p

"'

Q

: :

~

:

;

~

::s

_()T ( ) '

nondimensional forces and moments of rotor

euler angle of the helicopter bending stiffness of blade

distance between mass and elastic axis external forces and moments

torsional stiffness of blade

moments of inertia of the helicopter moment of inertia about flapping hinge moment of inertia about lead-lag hinge

inflow angle

product of inertia of the he! icopter flapping spring stiffness

flapping damping constant lead-lag spring stiffness lead-lag damping constant

polar radius of gyration per unit span mass moment about flapping hinge

mass moment about lead-lag hinge

mass of the blade about flapping

mass of the blade about lead-lag

blade radius

blade cut-off radius flapping hinge offset

lead-lag hinge offset

- Az s R

- u rmrdr (in equation(4}}

hinge hinge

flight speed in body coordinate system displacement of elastic axis of the blade blade flap angle

flapping angle. angle of side slip pre-coning angle

angular velocity lead-lag angle pre-lag angle

blade pitch angle inflow ratio advance ratio

nondimensional flight speed time constant

elastic torsional deformation of the blade azimuth angle

rotor rotational speed

quantity concerning airloads quantity concerning fuselage (including tail planes} quantity concerning gravity quantity concerning main rotor quantity concerning rotor shaft quantity concerning tail rotor time derivation

Table l Dimensions of a hingeless-rotor helicopter

Gross mass, m(kg)

Moment of inertia of body, ! {kgn()

X I ( kgn{) y I CkgnO z J ( kp;n() for Main Rotor

Rotor radius, R<m> Number of blades, b Blade chord, c(m)

Blade twist, 8 <deg)

t

zx

Rotor rotational spee, O.<rad/s)

Blade mass, m (kg}

p

Moment of Inertia of blade, Jnclinallon of rotor shaft, Equivalent hinge

Equivalent hinge

offset, x{J

XI; stiffness,

PreconJng angle, {J(deg) Longitudinal position, IR<m>

Hight of main rotor, h (m)

For Tail Rotor

Rotor radius, R Cm) T Number of blades, b T Blade chord, c (m) T

Blade twist, Bt Cdeg)

T R I fJ (kgm') i Cdeg>

'

k.IJ <Nm/rad) k <Nm/rad> I;

Rotor rotational speed, Qcrad/s) Blade mass, m

T

Moment of inertia of blade I (kgm")

' PT

0 hinge, 0 (deg} 3 3

Longitudinal position, IT<m> Hight of tail rotor. h (ml For Fuselage

Area, S <m'> F

Drag coefficient, For Horizontal Tail

Wing area, S <m'> H Span, b <m> H Chord, c (m) H Aspect ratio, AR c D F T H Long\tudinai position, l (m) H

Hight of horizontal tail, h <m> H Angle of Incidence, I (deg} For Vertical Tail

WI nr. area, s <TTl'> v Span, b (m) v Chord, c <m> v

Aspect ratio, ARV

H Longitudinal position, 1 <m> v Hight of Angle of vertical tall, hv<m> Incidence, I (deg} v 2,850 2,380 7,314 5,560 1. 057 5.5 4 0.32 -8.0 40.15. 31.95 212.66 5.0 0. 129 o. 145 1 49.0 816.0 2.5 -0.135 1. 50 0.95 2 0 .}8 0.0 227.2 0.94 0. 28 45.0 6.48 1.64 3.68 0.315 l.O 2.5 0.4 2 • 1 7 5. 06 0.62 -1.5 2. 2 4 1. 28 1. 76 0.73 5.27 0.22 -7.0

12

8

4 -8

-4

0

4

0 -4

8

4

0

0

COLL. PITCH - DEG

LONG. CYCLIC - DEG

_.·

LAT. CYCLIC - DEG

0 0

_: .. ::....:.·..:...·.:.:.:.:..._

0

0 0

PITCH ATTITUDE - DEG

.

---0. 1

0.2

--

..

0.3

Advance ratio

J1

Method of ( i i J&(bl Method of ( i ) &(b) Method of ( i ) &(a) 0 Experimental values

__, I

"'

I ... ... 41 5

FWD

_FWD

5 4 \ 3

(a) Method of < i) &<bl (b) Method of (iil&(b)

-.) I

"'

_{,_.}

I

"'

FWD

7 7 ~·

""'

'

_'

·'-,'\ 4 I 3 I 2 I 1

FWD

(a) Method of ( i ) IHbJ _{(b) Method of ( i i J&(b)}

Figure 3 Angle of attack distributions of the helicopter

0.3 0.0 -0.3 10.0 0.0 -10.0 10.0 0.0 -10.0 10.0 0.0 -10.0

...,

I w 3.0 I ~ 0.0 w -3.0 3.0 0.0 -3. 0 3.0 0.0 -3.0 NORMAL A((. (G)

ROLL ATilT. COEGl

- --1

PITCH ATTIT. COEGJ

'tAW ATTIT. COEGJ

LONG. VEL C"/Sl

-LATR. VEL. Cld/S)

VERT. VEL. (IUS)

0.0 1.0 2.0 3.0 4.0 5.0 T !ME CSECJ 6.0 5.0 '. 0 2.0 0.0 -2.0 2.0 0.0 -2.0 '. 0 2.0 0.0 2.0 0.0 -2.0 2.0 0.0 -2.0 6.0 3.0 0.0 CT X!(h"3 CH X!~u4

t-c----'

CY X!0u4 CO X10••4 CL X\0•"4 C~i X\0u4

~

FLAP t..NGLE COEG)

~IMfNV• 0.0 1.0 2.0 3.0 4.0 5.0 TIME CSECl 0.3 0.0 -0.3 10.0 0.0 -10.0 10.0 0.0 -10.0 10.0 0.0 -10.0 3.0 0.0 -3.0 3.0 0.0 -3.0 3.0 0.0 -3.0 NORMAL ACC. tG) ~"'l'tMr--·VM.-.!~,_,1

ROLL ATTIT. COEGJ

PITCH ATilT. COEGl

YAW ATT!T. COEGJ

LONG. VEL OUS)

LATR. VEL. (IV$)

VERT, VEL, (M/5) 0.0 !.0 2.0 3.0 4.0 5.0 T !ME CSECl 6.13 5.0

'·"

2.0 0.0 -2.0 2.0 0.0 -2.0 ' . 0 2.0 0.0 2.0 0.0 -2.0 2.0 0.0 -2.0 6.0 3.0 0.0 (T Xl(hd

~\M-W,..r·-...."'"~'tt'livA~

CH X\0u4

- .

Cf X {0u4 CO X !0u4 CL X!0 .... 4 • . .-...V<y,._N'y~~.~J (~1 X l0u4 '""""'-1'~~

FLAP ANGLE COEG)

1-MJ-/o.1fi,/\J,~~·~,~.N\ ... ~

0.0 !.0 2.0 3.0 4.0 5.0

TIME CSECl

(a) Method of < i) &<bl _{(b) Method of < i i J&(b)}

e. ~ 0.0 -0.3 10.0 0.0 -HL QJ 10.0 0.0 -10.0 10.0 0.0 -10.0 3.0 0.0

_,

I -3.0

"'

I ...

....

3.0 0.0 -3.0 3.0 0.0 -3.0 NORMAL ACC. CG)

~-

---ROLL ATilT. COEGJ

P1TCH ATTIT. COEGJ

YAW ATTIT. COEGJ

LONG. VEL C M/Sl

-LATR. VEL. CM/Sl VERT. VEL. (M/5) 0.0 1.0 2.0 3.0 4.0 5.0 TIME CSECJ 6.0 5.0 ' . 0 2.0 0.0 -2.0 2.0 0.0 -2.0 4. 0 2.0 0.0 2.0 0.0 -2.0 2.0 0.0 -2.0 6.0 3.0 0.0 CT XL0u3

G

v

---CH XL0u4 ~ CY Xi0u4 CO Xl0u4 CL Xl0u4 L (!.1 Xl0u4 ~ FLAP ANGLE COEG)

W-N'NI'/'fW~MW#/-11 0.0 1.0 2.0 3.0 4.0 5.0 TIME CSEC) 0.3 0.0 -{L3 10.0 0.0 -10.0 10.0 0.0 -10.0 10.0 0.0 -10.0 3.0 0.0 -3.0 3.0 0.0 -3.0 3.0 0.0 -3.0 NORMAL ACC. (G) I - - '

---ROLL ATilT. COEGJ

PITCH ATilT. CDEGJ

YAW ATTIT. COEGl

LONG. VEL CM/Sl

-LATR. VEL. CM/SJ VERl. VEL. CM/5) 0.0 1.0 2.0 3.0 4.0 5.0 TIME CSECJ 6.0 5.0

'·"

2.0 0.0 -2.0 2.0 0.0 -2.0 4.0 2.0 0.0 2.0 0.0 -2.0 2.0 0.0 -2.0 6.0 3.0 0.0 CT X\0u3 CY Xl0H4 CO X[0u4 CL X10••4 C\tl X!0u4

FLAP ANGLE C OEG)

0. 0 I. 0 2. 0 3. 0 4. 0 5. 0

T !ME CSECJ

(a) Method of (i) &(b)

(b) Method of (ii)&(b)

Figure 5 _{Control responses of the helicopter}

...,

I w I ~

"'

CT X\0u3

"'l~

ROLL RATE CDEG/5)

_~

: :

~.~~lill~l~l+-~if~A

10.0 _0.0

t

10.0

CH X\0u4 0.0

-10.0 _1.0 -10.0

PITCH RATE COEG/S) 0.0

10.0 -1.0 10.0

0.0 _{CY X10*-4} 0.0

-1\3.0 _{1. 0} -10.0

YAW RATE C OEG/S) 0.0

f-·~·/vl·'I-JI.y,~.

, ,

10.0 _-1.0 10.0 0.0 CO X\0-u4 0.0 -10.0 _{'. 0} -10.0 LONG. ACC. CGJ 2.0 0.3 _0.0 0.3 0.0 _{CL X\0u4 0.0} -0.3 2.0 -0.3

LATR. ACC. CGJ _0.0

¥VvYrAN·

0.3 -2.0 0.3

0.0 (1-i X\0u4 0.0

-0.3 2.0 -0.3

VERT. ACC. CGJ 0.0

::~~~~~

-2.0

L

0.3

FLAP ANGLE CDEG) 0.0

-0.3 5.0 -0.3

3. 0

<v~..v-Jvi"Jv.~-ROLL RATE COEG!S)

~~-·

PITCH RATE COEG/S)

YAW RATE COEG/S)

LONG. ACC. CG) LATR. ACC. CGl i VERT. ACC. CGJ

~....,.,~~-'1-'Vi''lt"'

5.0 5.0 4 . El 1.0 0.0 -l.IZl 1.0 0.0 -1.0 4.0 2.0 0.0 2.0 0.0 -2.0 2.0 0.0 -2.0 5.0 3.0 (T X![hd

~tl~~~v~'f!,AIJ,f~1~\~.·

CH Xt0u-4

....

~. CY Xt0u4 "" , .. ,"w/v'..,VV'II/<"' CO X\0u4 CL X\0u4 1-v--"""""<-"'y"v'"\\''V\"\1, Ol Xl0u4 i(l',>/'lo!<V'V'o.""-.ii'W•Vi"'

FLAP ANGLE C OEG)

~~,_. 0.0 1.0 2.0 3.0 4.0 5.0 0.0 _{0.0 1.0 2.0 3.0 4.0 5.0 0.0} TIME CSECl (a) 0.0 1.0 2.0 3.0 4.0 5.0 T !ME CSECl Method of ( i ) &(b)

Figure 6 Gust responses of the helicopter

TIME CSECl

' '

0.0 1.0 2.0 3.0 4.0 5.0

TIME CSECl

.._, I

"'

I ~

"'

CT X!l}..r·3 6;0 5.0

I

ROLL RATE COEG/S)

!

ROLl RA 1£ C Q£01'5) 4.0 10.0 _{CH X!0H4} 10.0 0.0 2.0

_

1

~::

l\IVVVVVVVV1

-10.0 _0.0 -2.13 P1TCH RATE COEG/SJ 10.0 10.0 0.0 2.0 0.0

vvvvvvvvv

-10.0 -te.e 0.0

YAW RATE COEG/SJ

YAW RATE CDEG/SJ -2.0 _10.0

10.0 CO X 10 ... 4

0.0 ~'-/'

0.0 _{'. 0}

-10.0

-10.0 2.0

LONG. ACC. CGJ 0.0 LONG. ACC. CGJ

0.3 CL X10..,4 0.3

0.0 2.0

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Figure 7 Gust responses of the helicopter flying with advance ratio of ~=0.20

T1ME CSECJ (b) Method CT Xl(hd S.il 5.13 '. 0 1.0 0.0 -!. 13 1.0 0.0 -1.0 I CO Xl0·•+4 4.0 2.0 0.0 I CL X10••4 2.0 0.0 -2.0 2.0 0.0 -2.0 6.0 3.0 0.0 0.0 1.0 2.0 3.0 4.0 5.0 TIME CSECJ of (ii)&(b)

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~igure 8 Gust responses of the helicopter

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Figure 10 Control responses of the helicopter flying with advance ratio of ~=0.20

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Figure 13 Mode ratios of the helicopter motion flying with advance ratio of g=0.20

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