Analytical study of dynamic response of helicopter in autorotative

FOURTEENTH EUROPEAN ROTORCRAFT FORUM

Paper No. 71

ANALYTICAL STUDY OF DYNAMIC RESPONSE OF HELICOPTER

IN AUTOROTATIVE FLIGHT

Yoshinori Okuno and Keiji Kawachi

Research Center for Advanced Science and Technology

University of Tokyo

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

Akira Azuma

Metropolitan Institute of Science and Technology

and

Shigeru Saito

National Aerospace Laboratory

20-23 September, 1988

MILANO, ITALY

ASSOCIAZIONE INDUSTRIE AEROSPAZIALI

Analytical Study of Dynamic Response of Helicopter

in

Autorotative Flight

Yoshinori Okuno a.nd Keiji Ka.wa.chi

University of Tokyo Akira Azuma

Metropolitan Institute of Science a.nd Technology and

Shigeru Saito

Na.tiona.l Aerospa.ce Laboratory

A nonlinear optimal control problem minimizing the touchdown speed of a. helicopter in the a.utorota.tive flight following power failure is studied. The motion of the helicopter is expressed by the longitudinal three degree-of~freedom equations) a.nd the rotor a.erodyna.m.ic peiforma.nce is ca.lcula.ted by using bla.de element theory combined with empirically modified momentum theory. This modified model has the ability of estimating the rotor thrust, H-force &nd torque properly even in vortex-ring sta.te, including the effects of bla.de stall. This optimal control problem is, then, solved by using a. numerical solution technique called SCGRA{1_{l. Results}

indicate that pilots should postpone the collective pitch flare. The effects of the time· delay in the collective pitch reduction and/or of the climbing rate at the moment of power failure on the minimum touchdown speed are also presented. In a.ddition, height-velocity dia.gra.ms for various gross weights and density altitudes are calculated by using the other numerical solution technique, dynamic programming. Results show good agreement with flight test data..

Notation

_C>R

""

angle of attack of rotor disk,

lift-curve--slope of blade element

defined by Eq. ( 3)

a

_""

"•

""

a.ngle of a.tta.ck of blade element a.t 0.75R,

B

_""

tip loss factor _{defined by Eq.(l7.4)}

c,,

_""

profile drag coefficient of blade element

"s

_""

stalling angle of blade element

c,,

_""

dra.g coefficient of bla.de element _{3,

""

longitudinal fla.pping a.ngle,

in stall region _{positive nose down}

D

₌

drAg

'Y

=

flight path angle, positive climbing, or

h

₌

height from ground, or _{Lock numbeZ.} distance above CG, see Fig.7

_e

""

pitch a.ttitude, positive nose up

h,

₌

density altitude

_e.

₌

_{longitudinal cyclic pitch, positive nose up}

J

₌

nondimensiona.l ground contact velocity, _8,

₌

_{blade twist angle, positive twist up} defined by Eq.(l5)

_e,

₌

_{collective pitch a.t 0.75R}

L

=

lift ,\

""

inflow ratio, defined by Eq.(ll.3)

l

_""

diata.nce behind CG, see Fig.7

I'

_""

a.dva.nce ratio, defined by Eq.{ll.l) m

₌

mass of helicopter _v

₌

_{rate of de5cent, defined by Eq.(11.2)}

n

=

load factor _u

₌

_{rotor solidity}

P.

₌

a.va.ila.ble engine power ₀

₌

_{rotor roh.tiona.l speed}

q

=

pitch ra.te, positive nose up

R

=

rotor radius

Subscripts

s

=

rotor disk area., or

reference a.r:ea. F

=

fuselage

=

time from power failure

I

=

time of touchdown

u

=

horizontal velocity H

=

horizontal stabilizer

"

=

control vector

R

=

rotor

"s

=

limitation of forward speed a.t touchdown 0

=

time of power fa.il ure

v

=

induced flow (nondimensiona.lized by RO) ii

=

reference velocity defined by Eq.(2)

w

=

weight of helicopter

w

=

vertica.l velocity, positive descending

Introduction

ws

=

limita.tion of sinking rate at touchdown _{In case of power failure, a helicopter can safely land}

"'

=

sta.te vector using a.utorota.tion. The control sequence during this a.u~

•s

=

r:a.dius of bla.de sta.ll region, torotative flight is genera.lly composed of the following five defined by Eq.(S.l) or Eq.(8.2) _{stages as shown in Fig.l:}

f-::c

'-"

w

::c

POWER LOSS, COLLECTIVE DOWN

~

\ CYCLIC MANEUVER

~

... STEADY AUTOROTATION

~

CYCLIC FlARE

--~

'-...

COLlECTIVE FLARE, TOUCHDOWN

DISTANCE

Fig. 1 Autoroto.tive la.nding following power failure.

1) Red~dion of the collective pitch just after power fail-ure.

2) Control of the forward speed to achieve the minimum sinking rate.

3) Steady autorota.tive flight towlt.rds a. la.nding point. 4) Nose up of the body to reduce the sinking ra.te and the

forwa.rd speed.

5) Colledive flare, tha.t is, the rlLpid increa.se of the col-lective pitch in order to convert the rota.ting energy of the rotor blade into the thrust.

The delica.ie timing and the appropriate amplitude of the control a.re required throughout the entire flight stages. Especially, the stages 4) a.nd 5) are importa.nt to land in safety, and the miscontrol in these stages ca.uses the hard landing. The flight test is usually used in order to verify the safety of landing from the specified initial conditions. It, however, requires the expensive cost and risk.

In this pa.per, the analytical method using an optimal control theory b.a,sed on variational method is developed in order to make clear the most sa.fe sequence of. control a.nd/ or flight pa.th during the a.utorota.tive flight following power failure. These optimal informations are useful to reduce the cost and risk of the flight test. The applica--tion of the optimal control theory to the a.utorota.tive flight was first proposed by KomodaJ2

1,

a.nd the recent numeri-cal method wa.s tried by Lee£3

_1.

_{These analyses assume the}

simple point-mass model of the helicopter a.s well a.s the simple a.erodynamic model, which reduces the accuracy of the solution and it's effectiveness. In this pa.per, a. more realistic model is used to improve the accuracy of the so~ lution, owing to the advancement of the computer a.nd the numerical solution technique.

In addition to the study of the most safe control and

flight path, the height-velocity diagrams (H-V diag-rams)

a.re a.lso a.nalyzed. The single engine helicopter ha.s the avoidance flight region a.s schernatica.lly shown in Fig.2, which is usually obta.ined by the flight test. The empirical equa.tions concerning the sa.fety bounda.ries were proposed by Peg-g[8], but. it neglected the fundamenta.l parameters, such M, the speed limit of the la.nding gears and/or the

own characteristics of the helicopter motion. In this study, dynamic programming is applied to predict the H- V dia.-grams1 because this method makes it possible to investi-gate the minimum touchdown speed for the various initial conditions without ca.lcula.ting ea.ch optima.! pa.th.

HIGH HOVER POINT

!50

VELOCITY

Fig. 2 Typical height-velocity dia.gra.m.

Rotor Performance in Autorotation

In the intermediate descending range between normal working state and windmill sta.te, the induced flow is di-rected a.ga.inst the uniform flow a.nd does not extend in-finitely. Therefore, momentum theory cannot be applied theoretically to analyze the rotor performance in the a.u-torota.tive flight following power failure. However, a.s far a.s

the rotor angle of attack is not so large, momentum theory works well even in so-called vortex-ring sta.te, empirica.Uy[4_l.

In this paper, a modified model of the induced flow is de-veloped a.s follows:

Simple momentum theory is expressed a.s

where

•=

JcT/2

aR=tan-1 (vfp.)

(2) (3) The solution of Eq.(l) is indic~ted in Fig.3 by

dot-ted lines. It is observed that this solution is discontinuous a.nd smaller tha.n the experimental results(4

] in vortex-ring

sta.te. In order to overcome these defects, Eq.(l) is modified introducing two coefficients 01,02 as follows:

where

(5.1) (5.2)

The solution of Eq.(4) is also indica-ted in Fig.3 by a solid line. It is observed that this modified model can give more realistic induced ftow over the entire ra.nge of the ilight conditions. The effect of the modific~t.tion becomes negligible •mall and Eq.(4) nearly equah to Eq.(1) when the rotor ~t.ngle of ~t.tta.ck is not so large.

Referring to Wolkovitchl6_{l, the vortex-ring state occurs}

when the relative velocity o{ the vortex cores normal to the disk fa.lls to zero, tha.t is,

-v+v/2=0 (6)

Fig.4 shows the result ofEq.(6) combined with Eq.(4). The broken lines indicate the boundaries of the region where the fiuctua.tioru of the thru:!lt were observed in the experi-ments(11. It is observed tha.t Eq.(6) gives the condition for the most severe vortex-ring sta.te, but the ftuctua.tions of the thrust decrea.se a.:3 the rotor angle of a.tta.ck decrea.ses.

'l'////, EXPERlMENTS(1₁ MOMENTUM THEORY MODIFIED MODEl

...

~···

4

3

2

1 L _ _ _ _ J _ _ _ _ -L---'0 3 2 1

u

RATE OF DESCENT,

vfv

Fig. 3 Induced flow in vertical flight.

0

5

0 --' u.. 0 w u :::> 0

z

•

0

"'

u 30 "'

~

u. 0 UJ 6 0 <:3 z <t

"'

0 1-L---~--~----L-~go

o

0 "' 2 1 RATE OF DESCENT,

vfv

Fig. 4 Boundary of vortex-ring state.

Thrust, H-force, and torque coefficients are described by using blade element theory as

1 { (B3 - " 3) (B2 - " Z) Cx = -a.u 5 80 - 5 .:\ 2 3 2 (B2 - x 2) }

+

₂ s _~ "8 _s (7.1) Gq

=

~u{

(1-x))G,,

+

x~G,s}

1 [ (B3_-xs3 ) (B2-xs2)

+

-au { 8o - ,\ 2 3 2

(B2-x,') t8 )-'

{- (B<'-x,')8

+

₄ I s + 8 s

_ (B

2

~

x,')

p.>. _

(B'

~

x:)

p,)p,] (

_{7 _3)}

where :r: s is the radius of blade stall, which is given by

for the simple model,

for the modified model,

x -

1 [

{as-

(eo-

:!.

4

e,)}

2

+

48,,\ s - -28, (8.1)

The solutions of the present modified model t~.nd the simple momentum model are calcula.ted by using Eq.(l) &

Eq.(8.1). &nd Eq.(4) & Eq.(8.2), respectively. Fig.5 &nd Fig.6 show the collective pitch angles and the torque coef~ ficients required to generde the constant thrust,

Gxfu

=

0.08 calculated by using these two models, which are com-pared with experimental da.ta[s]. The re~mlt by using the present modified model shows the better agreement with the experimental da.ta. over the entire ra.nge of the flight condition, from hovering to windmill sta.te.

~ 0 EXPERIMENT['] (.!J _{SIMPLE MODEL} UJ 0

₁₂

MODIFIED MODEL 0

"'

w

₈

--'

_{··· ···}

(.!J

z

<>:

4

Cr(u

=

0.08 :J: u

!::

0

a.. UJ 0.04 0.08 0 RATE OF DESCENT, v <>:

_-4

--' cc

Fig. 5 Blade pitch a.ngles for consta.nt thrust in vertica.l descent.

Dynamic Model of a Helicopter

The motion of the helicopter is considered to be limited in the longitudinal pla.ne a.s shown in Fig.7. The equations of motion a.re described a.a follows:

dh

-=w

dt du 1 (T . ) -d = - - sm6+H cos6 +DF cos a, t m dw 1 ( . )

- = - -

Tcos6-Hsm6+Dpsino

+g

dt m '

dO=-~·(Q-P.)

dt [R 0

de

dt=q where T=CxpSR2₀2 H=CHpSR202 Q=CqpSR302 (9.1) (9.2) (9.3) (9.4) (9.5) (9.6) (10.1) (10.2) (10.3)

The coefficients OT1 GH1 a.nd Cq a.te given by

Eqs.(7.1-3). The a.dn.nce ratio p.1 the descending ra.te v, and the

inflow ra.tio ..\ a.re given by

/' =

{cu -hR •

q) cos(6-,8,) - (w +lR · q)sin(6-

p,)

}fRO v=

{cu-hR

·q)sin(6-p,)

+

(w +lR · q) cos(El-

p,)

}fRI'l A= -v +v (11.1) (11.2) (11.3) 0 EXPERIMENT['!

xlo-

1 _{•••···•••• SIMPLE MODEL} ct

4

- - MODIFIED MODEL

u

~---z

UJ u "-UJ 0 u UJ :::>

a

a::

0

1--2

0

0.04 0.08 \ RATE OF DESCENT, v

·.

-2

_{··· ...}

-4

Cr/u

=

0.08

Fig. 6 Necessary torque for constant thrust in vertica.l descent.

T

w

Fig. 7 Helicopter model.

The longitudinal flapping angle {3, is given by

The lift of the horizontal stabilizer LH is given by 1 2 ' ) LH = 2p(u

+

w SHCLH C L~e

=

aJCo • 21nn 1 . 2 an

"'" =

6+tan-1{(w +lH · q)fu} (13.1) (13.2) (13.3)

where a no is the lift-curve-slope of the horizontal stabilizer at a" =0.

Finally, the horizontal a.nd vertical components of the dra.g of the body are given by

a, =tan-1

_(w/u)

1 2

Dp

cos a,~

2pu

SpGD,

Dp sin

a,~ 0

(14.1)

(14.2) (14.3)

These &ta.te equations cannot be explicitly solved be-cause GT, t11 p., v, A,

f3,

are dependent each other. Therefore,

Formulation of Optimal Landing Problem

The prima.ty objective of the a.utor:ota.tive ilight ioliow~ ing power failure is to la.nd in safety with a.llowa.ble touch~ down speed. Therefore, the maneuver in a.utorob.tive flight is a.na.lyzed by using optimal control theory to solve the problem of minimizing the touchdown speed. The maxi~ mum value of the allowable touchdown speed depends on the landing gea.r design. In the case of wheel-type landing gea.x, hothon:ta.l component oi: the ground contact velocity is allowed to be f.a...i.cly large.

In this 6tudy, the perforrna.nce function is defined as

and the boundary conditions a.re given by

h(O), u(O), w(O), fl(O), e(o), q(O) ;oiven

h(t, ), e(t,)

=

o,

q(t,)

=

0 ; specified (15)

(16.1) (16.2) The ra.nge of the collective pitch angle and the cyclic pitch angle ate mecha.nica.lly limited, and the range of the

pitch attitude is limited by the handling qua.lity. In

addi-tion, two more limits a.re imposed, the one is the limit on the &ng!e o! &tta.ck o! the bl•de element &t 0.7SR, •nd the

other is the limit on the loa.d fn.ctor. These iive limits a.re

for:mula.ted "a.s inequality constraints a.s follows:

w

8smi• :5 8s ~ 8sm.u:

e'";"

:5

e.::;;

emu

4

a,.

=

8o -

3"-

5

C1m4Z

dw

dt

2:9

(1-nm .. )

h

HIGH HOVER POINT

a (17.1) (17.2) (17.3) (17.4) (17.5)

CRITICAL SPEED POINT

LOW HOVER POINT

···~

... .

For the usa.ge of va.Iia.tiona.l method, these inequality constraints are transformed into equality constraints by in-troducing five additional control va.ria.bles ll.nd two addi-tional state va.ria.bles ca.lled «slack va.ria.bles"[9_{] a.s follows:}

dq .

--

=

e,(-e~

sine,+ e, cos e,)

dt

(18.1) (18.2) (18.3) (18.4) (18.5)

In all, the dimension ofthe state vector becomes- 8, a.nd i.he dimension of the control vector becomes 7. They ll.Ie

given by

:c== (h, u, w,

n, e,

q, eo:~:, E>d)x

u=:: (Do, 8 Sl Bod,

e

Sdl ed, a,z, n,z)T

(19.1) (19.2) a.nd, the following two diifetentid equa.tions for the new sta.te va.ria.bles a.re added to the state equa.tions Eqs.(9.1-6): a

w

de,

.

--=e;

dt

ae, ..

"dl

=

e,

h

d

h

w

(20.1) (20.2) a b d

1L

d 'U b a

Theoretical Results

in Comparison with Flight Test Data

The flight paths to minimize the touchdown speed a.re a.nc.lyzed for a.n exemplified helicopter, the specifications of

which a.re presented in Table 1, with the four initial condi-tions shown in Fig.8. This nonlinea.x optimal control prob-lem with equality constraints is solved by using a. numerical solution technique called SCGRA[1_{1. The optimal control}

sequences for these four initial conditions are discussed a.s follows:

Landing from High Hover Point

Fig.9 shows the optimal solution and the flight test da.ta.[7] for landing from high hover point. Theoretical re-sult& a.re in good agreement with the :flight test data. in the pitching motion but not in the collective pitch input. Fig.lO compares the optimal path with the predicted flight test pa.th which is calculated by using the time history of the collective pitch input of the flight test da.ta.. The pre-mature collective flare results

in

the hard ground contact beca.use of the loss of the rotor rotational energy.

Fig.ll shows the time history of the load factor ;,.nd the blade angle of a.tta.ck during the optimal landing ftom a. high hover point. It is observed that the blade pitch during the collective flare is limited by the loa.d factor for the ea.rlier period and limited by the blade a.ngle of a.ttack for the later period.

Helicopters cannot avoid to encounter the vortex-ring sta.te if the power fails in hover. Fig.12 shows the locus of optimal la.nding path in a a -v/V plane. Numera.b in

the iigure denote the time elapsed from the power fa.ilure in second. In this case, the helicopter is in the vortex-ring state for the first two seconds. Duvortex-ring this period, the collective pitch is reduced to the minimum vll.!ue a.nd the cyclic pitch holds the consta.nt va.lue a.s shown in Fig.9. Therefore, it is a.ssumed tha.t the vortex-cing does not cause the loss of control.

Table 1. Specifications of the exemplified helicopter. ma.ximum gross weight,

w

₌

5900 (kg)

rotor radius,

R

=

8.534 (m) blade chord1 c

=

0.381 (m)

number of bla.des, b

=

4 rotor rota.tiona.l speed,

n

=

25.03

(rad/s)

rotor moment of inertia,

_h

₌

7107 (kgm2)

location of CG, [R

=

0.0254 (m)

""

40

u.i,...

,_u

<(~

0

"'-:r"'

vw f--2

a:

_-40

u.i

₂₀

0 :J ;-~

0

-"'

f--w f--Cl ..:~ J: <D

-20

v

!::

_-40

a.

:r"

v

₁₅

!::

_a.~

"'

10

ww >Cl -~ f-- 0

5

v., w -' -' 0

₀

u 0::

₁₀

:r

u~

!::"'

..

a.w

₀

!::!e.,

-' u

>-

_-10

v

c

30

o"~ wu ww

₂₀

Q.V1

"'-c:o

oc2

f--~

₁₀

0 c:

00

·.

·.

--THEORY

··· FLIGHT TEST DATA (FAA-ADS-84, FLT. No. 46 W= 4120 kg, hd= 1950 m ho= 102m, Yo= 0 m/s) •

...

...

···•··

··•···

...

···.

_··:

4

8

12

TIME, t (SEC)

Fig. 9 Comparison of optimal solution with flight test la.nding from high hover point.

0

;3 Ws

>-"

t:

u

3

10

UJ

>

--'

..:

u

i=

a::

20

UJ

>

HORIZONTAL VELOCITY,

u (mfs)

0

10

20

30

..

····--] " ' ; -·\NON-OPTIMAL

/

··("

n

~·~-~

l

OPTIMAL \ _~'C,..--· ~---:::::::.----

7

0 :;::: Oo

.,~

~-·.'/-'" . STEADY AUTOROTATION

Fig. 10 Comparison between optimal a.nd non-optima.l la.nding trajectories !rom a. high hover point.

"

ri: 0 f-u

:.::

0

<

0 -' lOAD FACTOR

BLADE ANGLE OF ATTACK

2

--t···

I

10

I I

--..._I

/ _ _ _ _ ... J 1

5

_ / /

00

L

_ L _ _ L _ L _ _ _ L _ _ _ j _ . . . J

0

4

8

12

TIME, t (SEC)

Fig. 11 Time history of optimal landing from

high

hover point.

0

11 10 ~ 30 ;: I 7

,..,

'

I

..

...--\

I ,J 1/ I I I I I I I 60 I 1 I 2 _I I I 1 1 I I I I 0 I 90

2

1 0 RATE OF DESCENT,

vfv

u..~

o:B

UJO --' ~

"'

z

<

UJ

"'

<

--' co

G'

_w 0

•

0

,c

u

~

_<

u.. 0 w -'

"'

z

<

0:: 0 f-0 0::

Fig. 12 Locus of optimal landing tra.jedory from

high

hover point.

.,.

w·u

f-w

"""'

"'---

-'-' _~w uo f-~

a:

:i

u f-a:~

'-'

ww >Cl -~ f- 0 u., w -' -' 0 u

.;

:i

u~

t:<.:J

a.w

~e

-' u

>-u

c

a"~ wu ww a_Vl

"'---o:Cl

oc:'i

f-~ 0

"'

40

0

-40

15

10

5

0

10

0

-10

30

20

10

0

0

THEORY

FLIGHT TEST DATA

(FAA-ADS-84, FLT. No. 43

4

W= 4130 kg, h•= 2070 m ho= 34 rn, V0= 13.7 m/s)

8

12

TIME,

t

(SEC)

Fig. 13 Compa.ri:son of optimlL! :solution with flight test landing from critical speed point.

5

10

FLIGHT VELOCITY, V (m/s)

Fig. 14 Minimum la.nding velocity from aitica.l height veraua flight speed a.t time of power fa.i.lure.

Landing from Critical Speed Point

Fig.l3 show~J the result for critical speed point. In this ca.se the flight time from the power failure to the touch-down is much shorter tha.n that of the landing from the high hover point. Therefore, the influence of neglecting the time dela.y in the control inputs a.re not small especially in the pitching motion. It follows tha.t the theoretical results estimate lower critical speed than the flight test da.ta..

Estimation of critical speed point is important for de-cision of the takeoff tra.jectory. However, the flight tests for H-V diagrams a.re conducted generally with the level flights a.t the time of power fa.ilure. Fig.l4 shows the effects of the fl.ight pa.th a.ngle a.nd the time dela.y in initia.l collective pitch reduction on the minimum touch down speed. The ground conta.cl velocity increases with the climbing ra.te at the time of power failure a.nd also the influence of one second dela.y in the collective pitch reduction is more rema.rkable as the climbing ra.te increases.

Landing from Low Hover Point

Fig.15 shows the results for low hover point. The test pilot didn't reduce the collective pitch following power fa.il-ure in-contra.st to the optimal solution. This difference is ca. used by neglecting the time delay in the collective pitch opera.tion in the present a.na.lysis.

Landing from High Speed Region

Fig.l6 shows only theoretical result for high speed re-gion. Experimental data are not ava.ila.ble for this rere-gion. In this case, the cyclic pitch is controlled to nose-up through-out the time from the power loss to the touchdown because not only the sinking rate but a.lso the forward speed is crit-ical for the sa.fe landing.

In high speed region, the initia.l kinetic energy of heli-copter is enough for the sa.£e la.nding, therefore estimation of the longitudinal ma.neuvera.bility including pilot's reaction time is significa.nt for estima.tion of the a.voida.nce region.

THEORY

FLIGHT TEST DATA (FAA-ADS-84, FLT. No. 46 W= 4110 kg, hd= 2010 m ho= 4.8 m, V0= 0 m/s)

"'

40

u.i...-.

f-u

OO::W

"'~

0

..

::r:'-"

u"'

f-2-0:

-40

w·

20

Q

:J f-~ ... t •• !

-<.:J

₀

f-w

f-Q

..:~

::r:<D

-20

u

t:

"'

-40

::r:·

u

₁₅

f-a:~ w'-"

₁₀

>"'

_Q

f-~

u

0 w"'

5

..J ..J 0

₀

u

.;;

₁₀

::r:·

u~

_...

_···:

t:<.:J

'

Q.W

₀

~e. ..J

u

>--10

u

30

c

6~

··

...

wu

ww

₂₀

··.

_··

..

~ Q,Vl

Vl--"'Q

0~

f-~

₁₀

0

"'

0

0

4

8

12

TIME,

t

(SEC)

Fig. 15 Compt~.rison of optimal solution with flight test landing from low hover point.

,.

w·u

f-w

""'"'

"'--:r:"

<.,)Ul

t::2.

a. u.i 0 ::> ~--~ -\.:J f-UJ f-0 ...:~

40

0

-40

20

0

THEORY (W= 41~0 kg, h•""o m ho= 5 m, Vo= 25 m/s)

:r:

<D

-20

<.)

f-0::

:i

<.) f-o::~

w'-'

>w

_{_o}

~--~ <.) 0

w"'

..J ..J 0

u

.,·

:r:

<.)~

t:<:J

a.w

~2.

..J

u

>-<.)

c

a"~

wu

ww

O.Vl

"'--

0

"'<

0"' ~--~ 0

"'

-40

15

10

5

0

10

0

-10

30

20

10

0

0

4

8

12

TIME,

t

(SEC)

Fig. 16 Time history of opti.ma.lla.nding irom high speed region.

Analytical Prediction of H-V Diagram

Using Dynamic Programming

Height-velocity diagram ca.n be obtained as a. contour line in H-V pla.ne on which the minimum ground conta.ct velocity J equals 1. The optimal control theory based on va..ria.tiona.l method gives the good results a.s mentioned in the previous section when calculating a pa.rticula.r optimal pa.th. However, this method requires many ca.ses of cd-cula.tion with the various initial heights and velocities to obtain the entire H-V diagrams. Here, the other optimal conttol theory called dyn.a.mic programming is lUed to esti-mate the H-V dia.gra.ms. Although this method costs more computa.tion time (several times in this siudy) thut vaci~

a:tional method when calculating pa.rticula.r optimal paths, dyna.mic programming makes it possible to ca.lcuh.te the minimum touchdown speed for the various entry conditions without clt.lculating eZJ.ch optimal pa.th. Therefore, dynamic programming is selected in order to estimate the H~ V dia-grams in this study.

Although recent development in the super computers e~tends the applicability of dynamic programrning1 it is still

required in this study to simplify the equa.tions of motion o! the helicopter. The inertia. of the pitching motion is ne-glected and the helicopter is a..ssumed to be a. point-ma.ss. Dyna.mic progra.mming ha.s the merit to formulate inequa.l-ity constra.ints on the control variables e.nd/or on the state

v~ria.bles without transformation into equality constraints . This makes it possible to a. void the increase of t.he dimen-sion of the control vector. Aa a.. Iesult1 the sta.te va.t:ia.bles

a.nd the control variables are

x=

(u,w,O)T

u=

(&o,e)T

(21.1) (21.2)

Height h

ia

selected to the independent varia.ble be-ca.uae it is necessary for dyna.mic programming that the teuninal time is £.xed a.nd the sta.te va.ria.bles a.re not spec-ified a.t the terminal time.

Fig.l7 shows the H-V diagrams of the exemplified he~ licopter with three combina.tions of gross weights a.ud den-sity altitudes. The solid lines indica.te the results with the present modified model and the broken lines indicate the re~ sults with simple momentum model. It is observed that the present modified model shows the better agreement with the flight te!St data.. Theoretical results estimate the avoid-ance region smaller tha.n the flight test data because the time dela.y in the pitching motion is neglected in this study. In a.ddition1 the possibility of the non-optimal control is

150r---. 150,---~ 150 ,---, 100

1-:c

t.:J

w

:c

50 Coo

0 FLIGHT TEST DATAl') SIMPLE MODEL MODIFIED MODEL 0 ~

..§.

1-100 :r: t.:J

w

:r: 50 0 0

FLIGHT TEST DATAI7₁

SIMPLE MODEL MODIFIED MODEL 100 1-:r: t.:J

w

:r: 50 0

o FliGHT TEST DATAI7_l

- - - SIMPLE MODEL - - - MODIFIED MODEL 0 0'---"1'::-0 ---2::':0:---'30 VELOCITY (mfs) 0 0 _{10 20} VELOCITY 30 (m/s) 10 20 VELOCITY 30 (m/s) (I) W

=

4130 kg, hu

=

1620 m (2) W = 4580 kg, hd = 1510 m (3)

w

= 4530 kg, hd = -210 Ill

Fig. 17 Compa.rison of height-velocity die.gra.ms.

Conclusion

Nonlinea.t optim~Ll control theory based on variational method is well applied to a.na.lyze the minimum touchdown speed in autoroh.tive flight of a. helicopter following power fa.ilure. In a.ddition, dynamic progra.mm.ing is applied to predict height-velocity dia.gra.ms. It is indicated tha.t these a.pplica.tion:s of the optimal control theory ha.ve the pos-sibility to improve the maneuver procedure shown by the flight tests a.nd to extend the safety bounda.ry empirically determined.

The following results a.re a.lso dra.wn:

1) Pilots tend to conduct collective fia.re earlier tha.n the theoretical optimal timing.

2) Critical speed increases with climbing ra.te a.t the mo· ment of power failure.

3) The modified model of the rotor performance can give the more rea.listic prediction of the height-velocity di-a.gra.ms.

References

[1) Wu, A. K., and :Miele, A.: uSequentia.l Conjugate Gra-dient Restora.tion Algorithm for Optima.l Control Prob-lems with Non-Differential Constraints a.nd General Boundary Conditions, Part

1/'

Optimal Control Ap-plications and Methods, Vol.!, 1980.

[2] Komoda, M.: "An Analytical Method to Predict Height-Velocity Dia.gra.m and Critical Decision Point ofRotor-cra.R," NAL TR-245 (in Japanese), 1971.

[3] Lee, A. Y., Bryson, A. E., Jr., a.nd Hindson W. 5.:

11

0ptimal Landing of a. Helicopter in Autorotation, n Journal of Guidance and Contra~ Vol.ll, No.1, 1988. [4] Wa.shizu, K., Azuma., A., Koo, J., a.nd Oka., T.: "Ex-periments on a. Model Helicopter Rotor Operating in the Vortex-Ring Sta.te," Journal of Aircraft, Vol.3, No.3,

1966.

[S} Castles, W., Jr., a.nd Gra.y, R. B.: "Empirical Rela..-tioMhip Between Induced Velocity, Thrust and Rate oi Descent of a. Helicopter Rotor a.s Determined by Wind Tunnel Tesb on Four Model Rotors," 'NACA TN 2474, 1951

[6] Wolkovitch, J.: "Analytical Prediction of Vortex-Ring Boundaries for Helicopters

in

Steep Descents," Journal of the. American Helicopter Society, Vol.l7, No.3, 1972.

[7] Hanley, W. J.1 DeVore, G., and Ma.rtin, S.: "An

Eva.lu-a.tion of the Height Velocity Diagram of a. Heavy Weight, High Rotor Inertia., Single Engine Helicopter," FAA-ADS-84, 1966.

{8] Pegg, R. J.: "An Investiga.tion ofthe Helicopter Height-Velocity Di&gra.m Showing Effecl& of Densily Allilude •nd Gross Weight," NASA TN D-4536, 1968

{9] Jacobson, D. H., and Lele, M. M.: 11

A Tta.nsiorma.-tion Technique for Optimal Control Problems with a.

State Va.ria.ble Inequality Constra.int,'1 _IEEE

Transac-tions on Automatic Control, Vol.AC-14, No.5, 1969.