Helicopterdynamics Chapter3 111208131938 Phpapp02

Helicopter Flight Dynamics

Chapter 3: Rotor Flapping Motion

Introduction

1. The trim, stability and control of helicopter is different from the fixed-wing aircraft.

2. Flapping motion of rotor blade is the key in helicopter stability and control.

3. Factors of flapping motion: forward speed, controls and angular velocity.

4. Pilot controls the helicopter through flapping motion.

Topics

1. Equation of rotor flapping motion

2. 3 origins of rotor flapping motion

3. The effect of flapping hinge offset on the rotor flapping motion.

4. Hub moments

5. Flapping motion of hinge-less rotor.

Flapping Motion Equation

Assume: rigid blade; flap hinge is at the center of hub

Thus:In the flap plane, the forces applied on the blade segment include aerodynamic, centrifugal, inertia forces and gravity. The moments of these forces about the hinge are:

桨叶微段上在挥舞平面内的外力，包括升力、离心力和重力。如果计入挥舞惯性力，则认为桨叶对挥舞铰的力矩始终保持平衡。

取向上挥起的角度和力矩为正，

rmgG → GrM G → gMMM s

R

Gg 0

2 rrmC → CrrCM c sin → 2

0 b

R

CC IMM

rrmF → rFM F → ,0

b

R

FF IMM

rTM T → R

T rdTM0

上述所有力矩对挥舞铰的合成力矩为零，即

由此得到挥舞运动方程

(3-1)

对比典型的质量—弹簧系统的运动方程

可知桨叶的挥舞运动是在作用下的谐振。该系统的刚度，

显然来自桨叶旋转的离心力；自然频率，恰是旋翼的旋

转角速度。也就是说，旋翼的挥舞运动是在每转一次的不平衡空气动力激振下的

共振。

Hub plane

Rotor shaft

Flapping Motion Equation

The summation of these moments should be zero. That is,



→ →

→ →

→ →

→


0 TFCG MMMM


(3-1)





共振。



→ →

→ →

→ →

→



gMMII sTbb 2 (3-1)





共振。

We have the following equation of rotor flapping motion.

Recall the motion equation of mass – spring system

)(tfkxxm

By comparing, the natural frequency of flapping motion is:

b

bn I

I

m

k 2

The natural frequency of flapping motion is the same as rotor rotation speed. If the frequency of exciting force is also rotor rotation speed. The rotor flapping motion will be resonance.

Coefficients of Flap Motion

2sin2cossincos 22110 ssss babaa

sincos 110 ss baa

Assuming:

Neglecting the harmonic components larger than two:

Assuming the flapping motion of each blade is the same. The geometric meaning of above equation is an upending cone.

0a

1a

1b

: Cone angle

: Backward tilted angle

: Sideward titled angle

Aerodynamic Forces on Blade Element

rbauyT T 2

2

1

：空气密度

a ：翼型升力线斜率

b ：叶素弦长

：叶素迎角 T

p

u

utg 1

：桨距角 sincos 1110 BAR

r

0 ：桨根安装角

1 ：桨叶扭度

1A ：横向周期变距操纵量

1B ：纵向周期变距操纵量

Air density

Longitudinal cyclic pitch

Attack angle

Slope of airfoil lift curve

Blade twist

Chord of airfoil

Pitch angle

Pitch angle at blade root

Longitudinal cyclic pitch

Hub plane

Aerodynamic Forces of Blade Element

sincos sT Vru → )sin( R

rRuT

：前进比R

V s

cos

sin)coscos(sin 1 ssp VrVu → rRup 10 )cos(

0 ：流入比R

V s

sin0

1 ：旋翼诱导速度 )cos1(11 R

rd

d1 ：旋翼等效诱导速度 RCT

d

22

1

12

1 ：总入流 Rd /101

]sin)(cos)([sin

1111110

ss aBbA

R

r

R

r

Rr

]sin)()[()cos1( 1011

R

ra

R

r

R ssd

]sin)(cossin)(cos 211110 ss aBbAa

R

T

R

T bdraurTrM0

2

0 2

1

Advancing ratio

Inflow ratio

Induced velocity

Equivalent induced velocity

Inflow

Hub plane

Side View

Top View

Solution of Flap Coefficients

sincos0 TSTCTT MMMM

TS

TC

STb

M

M

gMMaI

0

0

002

21

12

12

00)(2

3])()

6

51(

5

3)1(

4

3[

6 R

gR

RBa d

Sb

12

2

110

21

2

11

2

31

]3

4[

2

11

2B

Ra d

ss

11

02

1 ]3

4[

2

11

1A

Rab d

s

Flap Motion Due to Forward Speed

Relative speed

Flap Motion Due to Forward Speed

Flap Motion Due to Pilot Control

Blade pitch angle:

Cyclic control results in the variation of blade aerodynamic force and flap motion For central articulated rotor, the flap coefficients are

Although the variations of pitch angle at advancing and retreating side are same the aerodynamic force is different, the variation of aerodynamic force in the advancing side is greater then retreating side due to the higher relative speed in advancing side. Thus, the flap is slight larger than cyclic pitching angle.

In hover, the flap is the same as feathering. We call it as ‘equivalence of flap and feathering”

Pilot controls the helicopter by flap motion due to control:

Equivalence of Flapping and Feathering

Interpretation of flapping and and feathering coefficients

Equivalence of Flapping and Feathering

1101101

2

2

1101101

2

11

2

31

Abbbb

Baaaa

cs

cs

longitudinal lateral

Flap Motion Due to Fuselage Angular Velocity

The angular velocity of fuselage will produce additional flap motion because the angular velocity causes the additional velocity and Coriolis Force at the blade segment.

For example,At first, producing additional velocity and attack angle

Resulting in the variation of lift and flap Secondly, producing Coriolis force as

The Coriolis force applies additional flap moment

Flap Motion Due to Fuselage Angular Velocity

Similarly, the roll rate causes ,

Lock number:

Summary:1. Damping effect: Pitching nose up produces rotor disk tilting forward, rolling right res

ults in rotor disk tilting left.2. Gyro effect: There exists a serious corss-coupling, the magnitude of the cross-couplin

g is about the half of primary motion, the tilting of rotor disk leads 90 degree with respect to angular velocity.

3. The flap motion due to angular velocity varies with forward speed.4. The flap motion due to angular velocity decreases with the increment of rotor RPM

Exercise

sincos 1110 BAr

sincos 210 rkrkvv

Assume the induced velocity distribution of a see-saw rotor helicopter in hover is:

The controls applied are:

To determine the rotor flapping coefficients: ss banda 11

Effect of Flapping Hinge Offset on Rotor Flapping Motion

Hinge offset = 0

Hinge offset ≠ 0


We have discussed the flapping motion of rotor without hinge

offset• Flapping frequency is exactly the rotor rotation speed

How about the flapping motion of rotor with hinge offset ? • Flapping frequency ?• Flapping damper ?• ……

Effect of Flapping Hinge Offset on Flapping Frequency

The balance of flapping moments, produced by aerodynamic, centrifugal and inertial forces is about flapping hinge other than rotor rotation center.

The equation of flapping motion becomes

Tsbb MeMII 2

Effect of Flapping Hinge Offset on Flapping Frequency

Natural frequency is

I

Me s

n 1

For blade with uniform mass

e

en

12

31

When %5e 04.1n

Conclusion:

• There exists the effect of hinge offset on the rotor flapping frequency• The effects is small for the practical helicopter

Physical meaning ?

Effect of Flapping Hinge Offset on Rotor Flapping Damping

Reviewing mass-damper-spring system

• Damping: c

• Critical damping:

• Damping Coefficient:

m

k

cf(t)

x

)(tfkxxcxm

ncr mc 2

crc

c

Effect of Flapping Hinge Offset on Rotor Flapping Damping

Flapping damping

Damping

Average damping ratio

If the damping ration is about 84% of central hinge rotor

Flapping motion produces velocity at blade element , which results in er

the variation and produces the damping moment

Tu

er

er

r

erabdrrM

R

e

a

sin1

sin12

1 2

aMc

eec

c

ncr 3

111

1

163

%5R

re

Physical meaning ?

Effect of Flapping Hinge Offset on Rotor Flapping Phase

Non-resonance, the phase of output to input

n

90

2222

2

)(4]1)[(

]1)[(cos

nn

n

232 )]3

11()/1(

16[4)

/1

/

2

3(

/1

/

2

3

cos

R

eRe

Re

ReRe

Re

b

%5/ Re

8b85

For linear system

For uniform blade

Special case


Producing hub moments

Hub moments comes from centrifugal force and is proportional to flapping hinge offset e

0a

0a

21

21

2

2

GX s s

GZ s s

kM M eb

kM M ea


Notes about hub moment:

• For flapping motion due to control, the large e produces the great control moment and improves the control capability and maneuverability.

• For flapping motion due to forward speed, the large hub moment provide the velocity stability and dihedral effect. However, increases the instability of attack angle.

• In order to obtain the necessary control power, we need e in a certain

extent. While we have to limit the value of e to avoid the instability of attack angle. Generally, the suitable value of e is about 5%R.

Flap Motion of Hinge-less Rotor

For hinge-less rotor, the flapping motion is implemented by the elastic deformation of blade root. We can treat the hinge-less rotor as an articulated rotor with equivalent flapping hinge offset. By setting the first order flapping frequency of hinge-less rotor equal to that of the articulated rotor, the equivalent flapping hinge offset is:

Exercise

The rotation direction of Black hawk helicopter rotor is anticlockwise from the top view. Please answer the following questions:

1. How the rotor flaps when the helicopter comes up gust in hove? How to adjust the controller to keep the hover condition?

2. How the blade changes its feathering angle following the pilot’s adjustment?

3. During the left and right turn flight with constant speed and altitude, is there any different for pilot control? Why?

Documents

Helicopterdynamics Chapter3 111208131938 Phpapp02