5. Helicopter Aeroelasticityocw.snu.ac.kr/sites/default/files/NOTE/Aeroelasticity... · 2018. 5. 24. · • Stall flutter in rotorcraft aeroelasticity (Dowell Sec. 7.2) - Single

Active Aeroelasticity and Rotorcraft Lab., Seoul National University

5. Helicopter Aeroelasticity


Helicopter Aeroelasticity

- Single D.O.F. flutter

blade motion (flap, lag, torsion) coupling

hub

divergence, flutter (freq. coalescence)

(e.a., c.g., whirl flutter)

- isolated blade instability (hover) collective pitch

- stall instability (forward flight) cyclic pitch

- aeromechanical instability (rotor, fuselage)

- active control

longitudinal lateral



Blade angle Flapping θ β

“90o phase lag”

Resonance system Retreating side Advancing side



“hysteresis” feed energy into system

- limit cyclic not destructive

- vibration load

Rotor Type

a) Semi-articulated or teetering(2-bladed,bell)

b) Fully-articulated

c) Hingeless feathering hinge only

d) Bearingless

flapping, pitch, lag elastomeric bearing

13o 14o

16 ~ 17o

Cl

2∏

NACA 0012


Flapping hinge

z

Ω zB y, yB

xB

r

βs ---- Flapping angle

• Isolated Blade Dynamics

- Articulated Rigid Blade Motion

First let’s consider a single blade with only flapping motion

(hinge located at the axis of rotation)

• Rotor blade structural blade dynamics modeling

- rigid blade + spring concentrated at hub

- elastic blade + geometric coupling

box-beam


xB yB zB ---- rotating axis (Ω) x, y, z ---- βs inclination

x



- complete cross-section shape

Assumption

From dynamics

A

R

PPP

P

I

FdrHH

Mdt

Hd

0

The blade is slender, assume

I : Inertia : Angular momentum PH

BB zyB III (rotational inertia)

0BxI

If

.

B B B B B B B

P B B B B B B B

p i q j r k

H I I q j I r k

flapping, pitch, lag

elastomeric bearing



From dynamics

To relate with Ω,

...232221

131211

kIjjIjiIj

kIijIiiIiI(dyadic)

ByI

A

R

BBBBBBBBBB FdrkqprIjrpqI

0)()(

s

0

0

0

0

cos0sin

010

sin0cos

s

ss

ss

B

B

B

r

q

p

opposite to yB

B

0 0 0

0 0


c

θ

Φ Ωr

s

zB

yB

dL

dD

v= inflow


For aerodynamics “Hover”

and the equation of motion

sBsBsB rqp cos;;sin

A

R

BssBBsssB FdrjIiI

0

2 )sin2()sincos(

Strip theory “blade element theory” Inflow Momentum theory

r



Introduce non-dimensional variables

≈ small

2

2

0

( )

1

2

1

2

A B B

L

d

d F dL k dD dL j

dL r c dr C

dD r c dr C

Profile drag co-efficient

r

r s

4

L

B

rx

R

v

R

C c R

I

: inflow parameter

: Lock no. …. blade motion sensitivity


V

V+v

V+

A1

AR

A2

v


From total thrust ‘T’

dxxC

C

R

rI

dxxxR

rIdL

L

dB

sB

20

2

22

23

2

0

L

T

l

C

C

dLbT b – No. of blades

R

bc

where, : Solidity ratio

22 )( RR

TCT

: Thrust co-efficient

Momentum theory, 2

TC



Flapping eqn. of motion

0

220

4

8 3

242

8 3

R

sA B

d s s sBB

L

r d F j

CIk

C

22 4cos sin

8 8 3

s s s s

< Hover >

and linearizing it 2

2 4

8 8 3

s s s

Damping ratio

16

C.F. : Nat. freq. ω2 = Ω2 Resonant System



• Coupled Flap-Lag Equation “Lead-lag instability”

PB

R

APBP aMEFdrHH

0

This accounts for the acceleration

at hinge point

where, MB : mass of the blade

: acceleration of hinge point =

cos sin 0 0 0

sin cos 0 0 0

0 0 1

B s s

B s s

Bs

p

q

r

As before

Pa2

2

Id E

dt

EEEEEE

)()(0 0

βs

e Flap

e Lead-lag

ξs

e: Hinge offset ~ 5% R


)( EEE

vvdt

vda pp

p

I

p

EEEdt

Edv

I

p


• Neglect the hinge offset for aerodynamic calculation

Then,

(1)

0

0

21[( ) ]

2 ( )

ss l

s

rdL r c dr c

r

, ?pE a



The eqn. of motion for 2 D.O.F system

2

2

31 2

2

4 42

8 3 3

s s s s

s s

eCoriolis’ coupling

2

220 0

32

2

8 4 4( ) 2 2

8 3 3 3

s s s s

s d s d

l l

e

C C

C C

where, R

ee



Linearize about equilibrium (small perturbation)

0

0

s

s

( = coning angle)

3

4

2

318

0

e

0

200

24 12

3 312

Qd

l l

CC

C Ce e

CQ : rotor torque co-efficient



The perturbed equations are

0

3 41 2 2 0

8 2 8 3

s e

00

4 3 82 2 ( ) 0

8 3 2 3

d

l

Ce

C

2 2

2 2

3 31 , ,

2 2

ee e e

R


Lead

iω

“ Geometric coupling” θs

(pitch –lag)

θξ >0 (positive coupling) ------ θξ <0


Fan Diagram

Ω (rpm)

(Hz)

Flap



• Stall flutter in rotorcraft aeroelasticity (Dowell Sec. 7.2)

- Single DOF instability in rotorcraft ….. “ Stall flutter”

- primarily associated with high speed flight, maneuvering

- does not constitute a destructive instability, but rather

produces a limit cycle behavior

- Rotor disc in forward flight ….. AOA on the advancing side

is considerably smaller than those on the retreating side

(Typical AOA distribution at μ = 0.33 …. Fig 7.13

- Retreating side ….. large AOA and changes rapidly with

azimuth angle airload prediction needs to include unsteady

effects




- Since the stalled region is only encountered over a portion

of the rotor disk, it will not be continuing unstable motion

- complexity of the flow field around a stalled airfoil

experimental data

A.O.A distribution of helicopter rotor at 140 knots (μ = 0.33 )




- Single DOF blade motion assumed

Energy eqn. multiply by dα and integrating over one cycle

- Fig. 7.14 ….. time history of the pitching moment coeff.

normal force coeff.

as a function of AOA for an airfoil oscillating at reduced

frequency typical of 1/rev at three mean AOA’s.

)(

2

22

2

MCI

cR

dCI

cRM )(

222

2222

2




- large hysteresis loop ….. in the dynamic case when the

mean AOA is small




- Pitching moment behaviour ….. in the vicinity of lift stall

average pitching moment increased markedly

(“moment stall”)

time history looks like figure ‘eight 8’

change in energy over one cycle ~

- Value of this integral = area enclosed by the loop

loop is traversed in a counter-clockwise direction …..

integral is negative, dissipation, positive damping.

dCM )(



large torsional motion , 2.3 cycles of the torsion


Moment stall occurs statically … no net dissipation of energy

or energy is being fed into the structure (integral is positive)

“stall flutter”

-loss of damping at stall

-marked change in average

pitching moment




Fig. 7.15 ….. Typical time history of blade torsional motion

when stall flutter is encountered.

marked increase in the vibratory loads in the blade pitch

control system



• Aeromechanical instability in rotorcraft (Dowell Sec. 7.3)

Fig. 7.28 ….. Typical Coleman plot of the rotor-fuselage

coupling. Will induce ground/air resonance.

Documents

5. Helicopter Aeroelasticityocw.snu.ac.kr/sites/default/files/NOTE/Aeroelasticity... · 2018. 5. 24. · • Stall flutter in rotorcraft aeroelasticity (Dowell Sec. 7.2) - Single