LECTURE NOTES

ON

HELICOPTER AERODYNAMICS

Academic Program On

Aircraft Engineering, Avionics and Manufacturing Technology

for HAL Design Trainees

G. Bandyopadhyay

Professor

Department of Aerospace Engineering

IIT Kharagpur

2

INTRODUCTION

Landmarks in the historical development of helicopter

15

th Century: Leonardo da Vinci sketched a machine for vertical flight using a screw type

propeller.

18th

Century: Sir George Cayley constructed models powered by elastic elements and made

some sketches.

19th

Century: The problem of cheap, reliable and light engine is still not resolved. W. H.

Philips (England, 1842) constructed a 10kg steam powered model.

Enrico Forlanini (Itali, 1878) built a steam driven model. It climbed to a height

of 12m and stayed aloft for about 20 minutes.

20th

Century: (i) Renard (France, 1904) built a helicopter using a two cylinder engine.

(ii) Paul Cornu (France, 1907) constructed the first man-carrying helicopter

with two contra-rotating rotors of 6m diameter. The weight was 260kg.

It used a 24Hp engine. It achieved a height of 0.3m for about 20seconds.

First successful manned helicopter flight.

(iii) Juan de la Cierva (Spain, 1920-30) developed autogyro. A propeller is

used for propulsive force. A rotor is developed for generating lift.

Cierva incorporated flapping hinges in his design. He was the first to

use flapping hinges successfully.

(iv) Igor Sikorsky (USA, 1939-41) built the modern helicopter (VS-300) in

1941. It was a helicopter with a 3-bladed main rotor and a tail rotor.

Several hundreds were produced.

(v) Piercy (1977) developed the shock-free transonic aerofoil and it became

possible to increase the forward speed of helicopter greatly.

3

Leonardo da Vinci’s vertical-lift machine, 15th century.

Courtesy NACA

Sir George Cayley’s helicopter and airplane, 1976.

Courtesy NACA.

4

Genealogical Tree

Flying machines can be broadly classified, as given below,

Flying Machines

Lighter than air Heavier than air

Air ships Fee Balloons Kit Balloons

Fixed Wing Aircraft Rotary wing Aircraft

Glider Sea-plane Land plane Amphibian

Autogyro Helicopter

Differences between Autogyro and Helicopter

Autogyro

1. Propeller connected to engine provides

thrust. Rotor driven by airflow

provides lift

2. Can not take off vertically

3. Can not hover

4. In case of engine failure parachute

effect is achieved without blade pitch

control and it can glide safely

Helicopter

1. Engine driven motor provides both

thrust and lift

2. Can take-off vertically

3. Can hover

4. Can only be done by suitable blade

pitch change.

5

Thrust Lift Thrust Lift

Autogyro Helicopter

Use of helicopters

Helicopters are used for both military and civil purposes :

Military : 1. Defense helicopter for direct support of infantry. These

Helicopters are provided with armaments such as machine

guns cannons, missies etc.

2. Air observation post

3. List Reconnaissance and communications

4. Search and Rescue (where hover is essential)

5. Anti-submarine activity

Civil : 1. Used as crane for constructional work of structural assemblies

2. Patrolling of highways, oil pipelines, electrical transmission lines

3. Forest patrolling, forest fire extinguishing

4. Agricultural operations in planting

5. Emergency rescue and medical aid.

Comparison between Helicopter and Aircraft

Helicopter Aircraft

1. Helicopter is less efficient in power and fuel

requirements.

1. It is more efficient in power and fuel

requirements.

2. Maximum speed achieved is 400 km/hour. 2. Speed achieved can be very high.

3. It can hover.

3. It can not take-off vertically

4. It can take off vertically. 4. It can not hover. In fact, a minimum speed

(stalling speed) is required.

6

Helicopter configurations

Helicopters can be classified based on rotors as shown below:

(1) Single rotor helicopter (with tail rotor) (2) Side-by-side helicopter

(3) Coaxial contra-rotating (4) Tandem Overlapping

(5) Tandem

7

Chapter - 1

Characteristics of Main Rotor

Characteristics of the main rotor in „Single–Rotor‟ helicopter configuration are described

in details in the following sections.

a) Pitch (θ)

The blade pitch angle (θ) is the angle between the plane perpendicular to the rotor shaft

and the chord line of a reference station on the blade (Fig. 1).

For a hovering helicopter, angle of incidence (i) is different from θ. As the rotor blade

rotates, a downward velocity (vi) is induced. The resultant velocity VR is a combination

of this induced velocity (vi) and the linear velocity (Ωr) in the plane of rotation at a

distance r from the hub, as shown in Fig. 1. The angle between induced velocity (vi) and

the linear velocity (Ωr) is defined as inflow angle φ and the angle of incidence (i) is

reduced from θ by the inflow angle φ.

It is common knowledge that the lift over an aerofoil is proportional to lift curve slope „a‟

and the angle of incidence i.e., a function of the aerofoil shape and angle of attack,

CL = a.i, CL = lift coeffi

where a is given by linearised theory as

a = 2π if i is in radian

a = 0.11 if i is in degrees

b) Azimuth ()

The helicopter rotor blade moves through 3600 azimuth. The azimuth position (Fig. 2)

is measured positively in the direction of rotation from its downstream position.

8

c) Change of Pitch

i. Collective change of pitch: Within limits of stall, CL increases with increase in. If

the pitch of all the blades are increased (decreased) simultaneously, the overall lift

and hence thrust increases (decreases). Therefore, changing the thrust to values more

than or less than weight will cause the helicopter to climb or descend. The means of

achieving this change of pitch of all blades simultaneously is called „Collective‟ pitch

change. The pilot uses collective pitch lever for this change.

ii. Cyclic change of pitch: With cyclic pitch lever, the pilot can increase the blade pitch

at one azimuth position (A) and decrease it at a diagonally opposite position (B), as

shown in Fig. 3. As a result all the blades coming to position A steadily have

increasing pitch values those receding from A and going to B have steadily

decreasing values. This causes increased angle of attack at position A and decreased

angle of attack at position B. This cyclic variation of pitch along azimuth position is

called „Cyclic‟ pitch change. This cyclic pitch change by using cyclic pitch lever

helps in many ways. It is one of the most effective way of changing the direction of

rotor thrust since this cyclic variation of pitch effectively amounts to tilting of rotor

cone.

Both collective and cyclic pitch change are accomplished by pilot by a swash-plate

system, described later.

d) Rotor Hinges

The development of the autogyro and, later, the helicopter owes much to the introduction

of hinges about which the blades are free to move. The use of hinges was first suggested

by Renard in 1904 but the first successful practical application of hinges was due to Juan

de la Cierva in the early 1920s. There are three hinges in the so-called fully articulated

rotor:

i. Flapping hinge

ii. Drag or lag hinge

iii. Feathering hinge

9

i) Flapping hinge: The flapping hinge solves the problem of rolling moment when the

helicopter is in forward flight. In hover, pitch is maintained the same throughout the

azimuth position. However, when the rotor moves forward horizontally at a velocity V,

the advancing blade (at = 900) is at a velocity V + r and the retreating blade (at =

2700) is at V-r. Thus, if the pitch is same, the advancing blade gives higher lift than the

retreating blade. This production of unequal lifts on either side of the helicopter would

result in undesirable rolling moment and excessive alternating air blades on the blade.

One way of correcting this is by setting the pitch on the advancing side lower and the

retreating side higher by use of some sort of lateral control. The alternative way is to

introduce flapping blades by use of flapping hinge (Fig. 4), as proposed by Cierva.

With the blades free to flap the moment problem is solved. The blades will move in such

a manner as to seek equilibrium; that is, in such a way as to make the summation of the

moments about the flapping hinge zero.

As the blade advances and develops more lift, it begins to flap upward. This then

introduces a downward vertical component of velocity in relation to the blade which

reduces its angle of incidence and hence the lift of the advancing blade. As it retreats, the

opposite is true, for a downward flapping of the blade produces an increased lift (Fig. 5).

The changes in speed in advancing and retreating blades are compensated by opposing

changes in angle of incidence (and lift) and net rolling moment about flapping hinge

becomes zero.

It is to be noted that this flapping motion is caused automatically by unequal velocities

only (i.e. without any control force by pilot) and it is referred to as aerodynamic flapping.

ii) Drag (Lag) Hinge:

The next important hinge is the drag hinge (Fig. 6). In addition to the flapping hinge, a

hinge is essential to cater for the lead-lag motion of the blade; this is the drag hinge.

10

The blade is hinged about a vertical axis near the center of rotation so that is free to

oscillate or “lead and lag” in the plane of rotation (Fig. 6). This flexibility makes the net

moment about drag hinge zero.

Both the flapping and drag hinge (in a so-called fully articulated rotor) is shown in Fig.

7c.

iii) Feathering Hinge:

Pitch of the blades can be increased or decreased by the pilot simultaneously or

differentially (collective and cyclic pitch change) by the use of feathering hinge.

e) Types of Rotors

Three fundamental types of rotors have been developed so far (Fig. 7) :

a. Rigid rotor: In these rotors, the blades are connected rigidly to the shaft. Such rotors

do not have either flapping or drag hinge. Usually, such rotors are two-bladed.

b. See-saw (or teetering) rotor: Rotors in which blades are rigidly interconnected to a

hub but the hub is free to tilt with respect to shaft. These rotors are two bladed. The

blades are mounted as a single unit on a “see-saw” or “teetering” hinge. No drag

hinges are fitted and therefore lead-lag motion is not permitted. However, bending

moments my still be reduced by under-stinging the rotor.

The principle of see-saw rotor is similar to that fully articulated rotor (having both

flapping and drag hinges) except that blades are rigidly connected to each other. The

“see-saw” hinge is like the flapping hinge located on the axis of rotation and because

of rigid interconnection between two blades, when the advancing blade, flaps up, the

opposite (retreating) blade flaps down.

c. Fully articulated rotor: Rotors in which blades are attached to the hub by hinges,

free to flap up and down also swing back and forth (lead and lag) in the plane of

rotation. Such rotors may have two, four or more blades, such rotors usually have

drag dampers which present excessive motion about the lag hinge.

11

f) Mechanics of Rotor Control

In the case of the conventional single rotor helicopter the control is achieved mainly by

the tilt of the main rotor thrust. Now the problem is to how to tilt the thrust of the main

rotor. One way to do this is to tilt the hub of either a rigid or flapping rotor with respect to

the fuselage. In the normal engine driven helicopter it is mechanically awkward to tilt the

hub since the hub is a rotating structure to which large torque loads are applied.

The most common way of achieving this is by means of a swash-plate system (Fig. 8).

This system consists of two parts: one rotating and the other fixed. This system provides

change in both collective and cyclic pitch.

i.) Collective pitch change: A collective pitch change is applied by raising the fixed

swash plate vertically, which raises the moving swash plate through the same distance

thus ensuring that pitch of all blades changes by the same amount. This change in pitch is

independent of azimuthal position.

ii.) Cyclic pitch change: If the fixed plate is tilted angularly, the moving swash plate is

also titled by the same value. This increases the pitch of one blade in one azimuthal

position and the pitch of the blade at diametrically opposite position will decrease by

same amount.

So this swash plate arrangement when displaced vertically up and down provides

collective pitch change and when titled angularly it provides for cyclic pitch change.

g) Limits of Helicopter Operation : Stall and Compressibility:

Of all, the aerodynamic characteristic peculiar to the helicopter, retreating blade stall is

perhaps the most interesting. Whereas the stall of the wing limits low speed

characteristics of the airplane, stall of the retreating rotor blade imposes a limitation on

high speed capabilities of the helicopter.

12

During forward flight the rotor blade encounters a velocity differential between the

advancing and retreating blades necessitating a change in angle of attack with azimuth

and correction for the resulting dissymmetry of lift. At some particular tip speed ratio, the

retreating blade will reach an angle of attack at which aerofoil will stall. The stall may

begin on any portion of the retreating region depending on the aerodynamic

characteristics of the rotor system. Experience has indicated, however, that for most

conventional rotors, the stall generally begins at the tip at an azimuth of 2700 to 300

0 and

propagates deeper into disc with increasing tip speed ratio ( = V cos / ΩR), as shown

by the shaded are in Fig. 9.

Another limit somewhat similar to stall but on the advancing side may be set by

compressibility effects. The tip of the blade on the advancing side will at high forward

speeds reach the critical Mach number of the blade, causing just like stall, a loss of lift

and an increase in blade drag. Usually stall and compressibility will cause severe

vibrations and excessive oscillatory blade loads. Stall also results in sluggish control and

sometimes a nose up and a slight rolling tendency.

When stall is encountered it may be quickly eliminated by one or a combination of the

following:

1. Decrease airspeed

2. Reduce main rotor pitch

3. Decrease severity of maneuver

4. Increase rpm, unless compressibility effects makes it impractical

Blade stalling can not be avoided entirely, but there are several ways in which it can be

delayed. Some of these methods are:

1. Twisting the blades towards tip

2. Reducing parasite drag, thereby reducing mean lift coefficient and tip angle attack

3. Increase the solidity

13

h) Rotor Blades:

The rotor blades are about 15 to 20 times as long as they are wide. Blades vary in

both planform (Fig. 10) and twist. Best blades from an aerodynamic standpoint

incorporate both twist and taper. Blades re of the following types:

1. All wood blades: are used frequently. They are usually built up from laminations of

several woods, heavier woods being used in the forward section and lightwood such

as Balsa being used in the rear ward portion. Such blades are relatively simple to

fabricate, especially if built with rectangular plan form and constant thickness.

Surfaces are aerodynamically clean and true to contour. However, such blades are

heavy, subject to moisture and deterioration.

2. Metal blades: are being developed at the present time by most manufacturers.

Blades can be built from pieces of sheet metal. It is probably safe to say that all

metal blades will eventually become standard for helicopter rotors.

3. Fabric covered blades: Most early rotor blades employed this type of construction.

The primary structural member of such fabric-covered blades consists of a steel

spar, which is usually step-tapered. Spars are drawn as one continuous tube. The

ribs are usually cut from plywood and are fastened to the spar by metal collars. The

leading edge is built up with solid wood – often with a metal strip to keep the blade

center of gravity forward. The entire blade is covered with fabric. The disadvantage

is that it is difficult to avoid surface irregularities and fabric distortions in flight.

4. Plywood covered blade: Most of the objectionable features of the fabric covered

blades can be overcome by using the same basic structure and covering the entire

blade with thin plywood. However, such blades require careful hardwork, do not

lend themselves to quantity production and are nor weatherproof.

i) Rotor Aerofoil Section:

NANA 0012 aerofoil was selected for early helicopter rotor application and was used

almost exclusively from 1937 to 1977. Alternate aerofoils were hardly considered during

this period because aerodynamic problems were secondary to many structural and

mechanical problems related to flight controls, power systems and structural life.

14

Later, efforts have been made to define more effective aerofoil sections for helicopter

rotors. This has resulted in designing aerofoils tailored to optimize hover, maneuver and

high speed performance simultaneously.

The aerofoil in the helicopter blade is subjected to the most varied and adverse type of

free conditions as it swings around the azimuth. The profiles of an aircraft wing are never

placed in such hostile atmosphere and relatively speaking have an easy life. The

optimization of a helicopter rotor sections is a more difficult task.

First of all, the rotor in hover has uniform flow conditions at all azimuth positions. It

mainly operates up to Mach numbers of 0.5 to 0.6 at the tip gradually reducing towards

the root. The hovering demands a high lift/drag ratio and low pitching moment

coefficient.

In forward flight, the conditions are entirely different. At low altitudes in the higher

ranges of velocity the Mach numbers are quite high at the tip of the advancing blade with

blade sections at low values of CL of 0 to 0.3. On the other hand, the aerofoil on the

retreating blade operates at high values of CL nearing or at stall at Mach numbers of 0.3

to 0.35. At the same time, at the inboard there is a region of reversed flow as well.

In summing up, the aerofoil of the helicopter rotors must show both favorable low speed

and high speed characteristics.

Favorable low speed characteristics include:

high CL at M = 0.5, low CD , low CMa/c ,high CL/CD

Favorable high speed characteristics include:

shock free flow, high MD, low CD, low CMa/c

15

Design requirement for new advanced aerofoils in tabular form :

Flight condition Operating condition Specification

Hover M = 0.6 and CL = 0.65 CL/CD = 72

1 4

0.02cMC

High Speed Advancing blade:

CL = 0 to 0.03

Retreating blade:

CLmax > 1.2

No separation

M > 0.85

Shock free flow

CD < 0.013

M = 0.3 to 0.35

Maneuver M = 0.5 CLmax > 1.35

Shock free flow

General 2D Test Condition Re = 5 106

The aerofoil designed must conform to the two transonic design requirements (high speed

flight and maneuver) which simultaneously satisfying stringent subsonic (hover)

requirements as mentioned in the above table.

There are two possibilities in the design process. One is to calculate a suitable shock free

shape for a high CL at Mach number 0.5 and then modify parts of such a basic aerofoil to

optimize towards hover. The other possibility is to start from low speed and optimize

towards high speed and maneuver.

The procedure usually adopted is to start from the high speed side. This is because the

high speed condition will determine a much larger part of the aerofoil counter. Also,

shaping for transonic shock free flow is a very delicate matter and one would certainly

like to leave this designing by some method as much as possible.

Two factors have helped in the designing of “advanced” aerofoils for helicopter rotor.

Firstly, the concept of super-critical aerofoil helped considerably in obtaining suitable

high speed characteristics. The initial stimulus for developing aerofoils with favorable

16

transonic characteristics was given by Piercy. That shock free transonic flow is a real

possibility was proved experimentally by Piercy. He found that shockwaves can be

reduced in strength and even eliminated by designing for a “peaky” type of pressure

distribution.

The first “advanced” aerofoil for helicopter rotor is due to Wortmann who applied the

“peaky” principle to improve the transonic characteristics. Since the Kemp & Piercy et al

have developed other aerofoils designed exclusively for helicopter rotor.

Secondly, though the flow through the rotor is of three-dimensional and unsteady nature,

it has been verified that the performance of a rotor depends strongly on the two-

dimensional steady characteristics of the rotor profiles. In other words, the performance

of a helicopter rotor in a given flight condition can be improved by improving the

characteristics of the rotor aerofoil selected in the two-dimensional steady flow condition.

Overall Design Features of Conventional Helicopter Rotor

1. Profile used : NACA0012, NACA23012, New advanced aerofoils

2. Thickness ratio : 9% - 18%

3. Disc loading : 8 to 48 kg/m2 depending on the type of helicopters

Majority are loaded between 10-20 kg/m2

4. No. of blades : 2 to 6

5. Plan form : Usually rectangular, sometimes trapezoidal blades are

used with taper ratio between 0.5 and 0.7

6. Twist : From root to tip usually between -50 to –12

0,

-80 to –10

0 mostly used.

7. Collective pitch at 0.75 R : 60 to 12

0 in powered flight, 0

0 to 3

0 in autorotation

8. Tip speeds : Between 150 to 220 m/sec

9. Mach number at blade tip : 0.92 – 0.97 achieved presently

10. Main operating CL : 0.4 to 0.6

11. Power loading : 2 to 7 kg/hp

17

Tail Rotor

The tail rotor is primarily to counteract the anti-torque due to engine torque. In the

absence of the tail rotor, the helicopter would tend to spin against the main rotor. Thus, it

is always necessary to have an anti-torque device in the form of a tail rotor, situated at a

distance from the center of gravity providing a convenient moment arm for a single rotor

helicopter.

In the conventional tail rotor the working is similar to the main rotor except that it is

much smaller in size. The tail rotor has no drag hinges but only flap hinges.

In the case of twin rotors contra-rotating coaxial main rotors cancel out the torque of each

other. Such helicopter does not need tail rotors.

18

Ωr

Fig. 1 Blade pitch (θ)

dr

Ω

ω

R

θ φ vi

VR i

19

Fig. 2 Azimuth position (ψ)

Fig. 3 Advancing (ψ =900) and retreating (ψ =270

0) blades

=900

=2700

= 00

=1800

Front of helicopter

Helicopter motion

A

Advancing

blade Retreating

blade

B

θ

θ

20

(a) Flapping hinge (without offset)

Ω

(b) Flapping hinge (with offset a)

Fig. 4 Flapping hinge (with and without offset)

a

21

(a) Advancing blade (flapping up reduces angle of incidence)

(b) Retreating blade (flapping down increases angle of incidence)

Fig. 5 Effect of flapping up and down on advancing and retreating blades

r

V+r

VR

r

VR

V-r

22

(a) Drag hinge with offset e

(b) Lead-lag motion due to drag hinge

Fig. 6 Drag hinge and its effect

e

23

(a) Two-bladed rigid rotor (no hinge)

(b) Teetering or see-saw rotor (flapping hinge only)

(c) Fully articulated rotor (flapping and drag hinge)

Fig. 7 Various types of rotors

24

Fig. 8 Swash plate system used in conventional single-rotor helicopter

25

=1800

=270

0

=0

0

Angle of attack contour plot Propagation of stall

Fig. 9 Propagation of stall with increase in forward speed of helicopter

V=70 mph

=0.23

=900

V=70 mph

=0.27

=900

Reversed

flow

V=70 mph

=0.27

=900

Reversed

flow

26

Fig. 10 Various rotor blade planform shapes

27

Chapter – 2

Performance Analysis of Helicopter in Hover & Vertical Climb

Using Momentum Theory

2.1 Introduction

Various motions of a helicopter can be, broadly, classified as :

a) Hover

b) Vertical Climb

c) Forward motion

d) Vertical Descend

e) Maneuver

The early development of the theory for helicopter motion in hover followed two

independent lines of thought:

i) Momentum theory

ii) Blade element theory

A combination of momentum theory and blade element theory has been developed later.

Identical equations may be derived by means of vortex theory, but it is believed that the

combination of momentum and blade element theory has greater physical significance

and can be easily grasped. The combined theory can be applied for performance analysis

of helicopter in hover, vertical climb and forward flight.

Notations

(i) Rotor

R = radius of rotor (m)

c = chord of the blade (m)

r = span wise distance of a section from center of rotation (m)

r = r/R

b = number of blades

a = flapping hinge offset

= pitch of the blade section

0.75R = collective pitch at 0.75 R

28

T = twist (linear) of the blade in degrees

= inflow angle at the blade element

Ω = angular speed of the blade (rad/sec)

i = incidence of the blade section

M = Mach number at the blade section

CL = coefficient of lift of the blade profile [f (i, M)]

CD = coefficient of drag of the blade profile [f (i, M)]

U = tip speed of rotor (m/sec)

Vi = vertically downward air velocity induced at the rotor disc (m/sec)

Vi∞ = vertically downward air velocity induced at infinity downstream (m/sec)

VR = resultant air velocity at the blade profile (m/sec)

dL = elemental lift

dD = elemental drag

dT = elemental thrust

dFx = elemental inplane force

dQ = elemental torque

w = swirl or rotational speed at the blade element (m/sec)

= solidity ratio (bc/R)

T = thrust of the rotor (N)

P = power absorbed by the main rotor

= traction coefficient (=2T/4U

2R

2)

S = rotor disc area (m2)

S = area at infinity downstream

k = slipstream contraction ratio (= /S S )

(ii) Flight condition

Z = attitude (m)

= density of air (kg/m3)

m = mass of helicopter (kg)

VZ = rate of climb (m/sec)

29

2.2 Momentum Theory for Vertical Climb

The momentum theory, started by Rankine and further developed by Froude and Betz,

stems from Newton‟s second law of motion F = ma. Although this theory does not

consider the geometry of blades, it results in the general, higher than the speed with

which airscrew advances in air. The increase in velocity of the air from its initial velocity

VZ to its value at the airscrew disk is called the induced or downwash velocity and is

denoted by Vi (Fig. 1). The thrust developed is then equal to the mass of air passing

through the disk in unit time, multiplied by the total increase in velocity caused by the

action of the airscrew.

If is the air density, S the disc area, the mass flow rate per unit time through the disc

equals ( )Z is V V . Equating thrust to the change in momentum gives

( )Z i iT S V V V (1)

Condition of continuity of an incompressible flow gives the relationship

( ) ( ) Z i Z iS V V S V V (2)

Where S and S are the areas of airscrew and wake respectively. Therefore, Eq. (1) may

also be written as

( ) Z i iT S V V V (3)

Now this work done by the thrust of the airscrew on the air per unit time is .( )Z iT V V .

This work must be equal to the increase of Kinetic energy of the slipstream per unit time.

This gives

2 21( ) ( ) ( )

2

Z i Z i Z i ZT V V S V V V V V

Substituting for T from equation (1)

21( ) ( ) ( ) 2

2

Z i i Z i Z i i i ZS V V V V V S V V V V V

or 2 21

2

i i Z

Z i

i

V V VV V

V

12

2i ZV V (4)

Eq. (4) reveals some interesting feature. The axial velocity (VZ + Vi) may be shown as

1

2Z i Z z iV V V V V (5)

30

This shows that the axial velocity through the disc is the average velocity of upstream

fluid VZ and downstream velocity in the wake (VZ + Vi).

Secondly, Eq. (4) also gives directly

1

2i iV V or 2i iV V (6)

This states that induced velocity at the disc is one-half of total increase in velocity

imparted to the air column.

2.2.1 Momentum Theory for Hover

Putting VZ = 0 for hover in Eq. (1) gives

( ) Z i iT S V V V

or, i iT S V V (7)

Using 2i iV V (Fig. 2) and 2S R

2 2 2.2 2i i iT R V V R V (8)

or 2/ 2iV T R (9)

In hover total thrust supports the weight so that T = W and

22 2

i

W DLV

R (10)

and 1 2

2 i i

DLV V

(11)

where DL is the “disk loading” equal to the helicopter weight divided by the disc area

(analogus to “wing loading” for a fixed wing aircraft). Variation of Vi∞ with DL is plotted

in Fig. 3.

This simple relationship illustrates that rotor thrust in hover may be increased by

a. higher density (low attitude)

b. larger disc area (greater rotor diameter)

c. higher downwash velocities (produced by higher collective pitch setting and/or

higher rpm.

31

2.2.2 Limitations of the Momentum Theory :

The analysis made by the simple momentum theory is idealised because it neglects

profile drag losses, non-uniformity of induced flow (including the energy losses due to

spilling of the air about blade tips, commonly known as tip losses) and slipstream rotation

losses. Thus an actual rotor would require more power to hover with a given load than an

“ideal” rotor (i.e., a rotor having zero profile drag and uniform inflow) and therefore

would be less efficient.

The order of magnitude of the rotor losses not considered by simple momentum theory,

expressed as a percentage of the total power required is as follows:

Profile drag losses : 30%

Non uniform inflow : 6%

Slipstream rotation : 0.2%

Tip losses : 3%

Lastly, it does not provide any information as to how the rotor blades should be designed

for a given thrust.

2.2.3 Rotor Efficiency :

Propeller Criterion of efficiency is given by

ZT Vuseful power

total power P (12)

In hover, it is not possible to apply this condition. Although power P is expanded in

producing thrust T, the transnational velocity VZ is zero, thus is always zero.

Obviously, lifting rotor needs some other standard of efficiency whereby its lifting ability

may be judged.

A very reasonable way to estimate the efficiency of a lifting rotor is to compare the actual

power required to provide a given thrust with the minimum possible power required to

produce that thrust (i.e., using an “ideal rotor”).

32

Consequently, the criterion of rotor hovering efficiency may be defined as

iT Videal power required to hoverM

actual power required to hover P (13)

where M is called the rotor figure of merit.

Using Eq. (4), Eq. (7) gives

22

T T

MP R

22

W W

P R

2

/

/2

PL power loading W PDLPL

DL discloading W R

For an ideal rotor, M is 1. A value of 0.75 is considered to be good for a rotor while M =

0.5 indicates poor performance (Fig. 4).

2.2.4 Further Development of Momentum Theory :

It is necessary to modify some of the simplifying assumptions made in the original

theory. It need not be assumed that air flow is uniform all over the disc. Considering an

elemental annular area, elemental thrust may be written as

2 Z i id T r d r V V V (14)

Integration gives

0

( ) 2 R

Z i iT V V V r dr (15)

Validity of Eq. (15) has not been established.

33

Chapter – 3

Performance Analysis of Helicopter in Hover & Vertical Climb

Using Blade Element Theory

3.1 Introduction

Two primary limitations of the momentum theory are that it provides no information as to

how the rotor blades should be designed, so as to produce a given thrust. Also, profile

drag losses are ignored. The blade element theory provides means for removing these

limitations.

The blade element theory, which was put in practical form by Drzewiecki, is based on the

assumption that element of a propeller or rotor can be considered as an aerofoil segment

that follows a helical path. Lift and drag are then calculated from the resultant velocity

acting on aerofoil, each element being considered independent of the adjoining element.

The thrust and torque of the propeller or rotor are then obtained by integrating the

individual contribution of each element along radius.

3.1 Blade Element Theory for Vertical Climb

Inflow angle (Fig. 5) is given by

tan

Z iV V

r w (1)

and the resultant velocity is given by

2 2 2( ) ( ) R z iV r w V V (2)

Elemental lift and drag can be written as

21

2 R LdL c V C dr (3)

21

2 R DdD c V C dr (4)

where c is the chord.

34

Resolving in the plane of the disc (Fig. 5)

21cos sin

2 R L DdT c V C C dr (5)

21sin cos

2 x R L DdF c V C C dr (6)

The total thrust can be calculated by

0

RdT

T drdr

2

0

1cos sin

2

R

R L DcV C C dr

for b number of blades

2

0

cos sin2

R

R L D

bT cV C C dr

3.2 Blade Element Theory for Hover

Elemental lift, obtained in equation (3) is

21

2 R LdL c V C dr (1)

For purposes of simplification, it can be assumed

sin =

cos = 1 (2)

VR = Ω r

The blade element lift coefficient may be expressed as

CL = a i = a ( -) (3)

where a = slope of the lift curve slope

With the aid of equations (2) and (3), eq. (1) becomes

21( ) ( )

2 dT dL r a cd r (4)

For (b) number of blades

21( ) ( )

2 dT b r a cd r (5)

35

For simplicity in integration it may be assumed that the pitch angle of a blade element

will vary with its radial position r as

t

R

r (6)

where t is the pitch angle at tip (Fig. 6). This pitch distribution results in a uniform

inflow distribution along the blade span. Such a distribution is therefore labeled ideal

twist, because it yields the minimum induced loss for a given thrust.

The ideal twist results in the following variation on inflow angle

t

R

r (7)

where t is the inflow angle at tip.

After substituting eq. (6) and eq. (7) in equation (5)

21( ) ( )

2 t t

RdT b r a cd r

r (8)

Integrating eq. (8) over the blade radius, assuming blade c to be constant, the thrust of the

rotor is

3

2 ( )2 2

t t

b RT a c (9)

Thrust coefficient may be defined as

2 2( ) TC T R R (10)

This gives ( )4

T t t

a bcC

R

(11)

The term solidity may now be introduced. The solidity, of a rotor having rectangular

blades may be defined as the total blade area to the rotor disc area. Thus

2

bc R bc

R R

(12)

Expression for CT becomes

( )4

T t tC a

(13)

36

In order to use Eq. (13) easily, it is necessary to replace t by parameters that are known

or easily determined. This is done as follows:

From momentum theory

2 22 2iV T R W R (14)

By definition / t iV R (15)

or, 2 42

t

W

R

(16)

Thrust, for a helicopter rotor in hover, can be determined using equation (13) together

with equation (16).

37

Chapter – 4

Performance Analysis of Helicopter in Hover

Using Combined Theory

4.1 Introduction

Performance analysis of a helicopter in hover and vertical climbing may be done very

conveniently using a combination of momentum and blade element theory. However,

hover and vertical climb treated separately. Analysis for vertical climb is given in the

next chapter.

4.2 Combined Theory for Hover

For hover (VZ = 0), momentum theory gives an expression for element thrust as a annulus

of radius r (Fig. 7) as

2 i idT r dr V V (1)

Continuity equation for hover can be written as

i iSV S V giving

1

2

i

i

V S

V S

1

2

Rgiving

R

(2)

This above expression may not be valid for a helicopter in hover, particularly for higher

thrust loading. It can be written as

2

i

i

V Sk

V S (3)

[Slip stream contraction factor k is to be evaluated using an iterative technique, starting

with an initial value 0.707].

Blade momentum theory gives an expression for thrust as

21( cos sin )

2 R L DdT cV C C dr (4)

where 2 2 2( ) 0 R i ZV r w V V for hover (5)

38

and tan

iV

r w (6)

The term solidity () of a rotor having rectangular blades may be defined as the ratio of

the total blade area to rotor disc area

2/ /bcR R bc R

Equating eq. (1) and (4) for (b) number of blades

212 . ( cos sin )

2 i i R L DdT r dr V V b CV C C dr

Using equations (3), (5), (6) this leads to

2 tan sin 2

2 cos sin

L D

Rk

r C C

(7)

Proceeding in the same manner and equating the elemental torque from the momentum

and blade element theory

22 2 ( sin cos )2

i L D R

bdQ r dr V wr c C C r V dr (8)

This gives after simplification

28 ( sin cos )

i

L D R

VR

r C C V

(9)

After further simplification with Vi from eq. (6) and VR from eq. (5)

sin 2

4 ( sin cos ) ( )

L D

R w

r C C r w

(10)

or, 24 ( sin cos )2

i L D R

bcwrV C C V

or, 2( sin cos ) 8

i

L D R

w V bc R

C C V r R

8

R

r

39

Solving for w gives

4 sin 2

1( sin cos )

L D

rw

r

R C C

(11)

The thrust is given by

2

0

( cos sin )2

R

R L D

bT cV C C dr (12)

Power is given by

2

0

( sin cos )2

R

R L D

bP Q cV C C r dr (13)

4.3 Method of Computation :

Pitch 0 (or ө0.75R ) is given :

1. Divide the blade into number of equal parts

2. Assume a contraction ratio factor k to start with (initial value can be taken as

1/ 2 0.707k )

3. Take one station and calculate local pitch using the formula (.75 ) ( .75) R Tr

4. Assume a value of to start with (initial value can be taken as = 0)

5. Calculate i, i = -

6. Local Mach number 0( )20.1 273

rM T in C

T

7. Find CL, CD for i and Mach number M from 2D polar of given profile

8. For the assumed value of k (contraction ratio) determine from the equation (7)

2tan sin 2

2 cos sin

L D

Rk

r C C

(7)

40

9. To get a converged value of , iterate steps between steps (5) and (8) where new

values of are assumed each time.

10. At each station calculate the rotational component of induced velocity from

equation

4 sin 21

( sin cos )

L D

rw

r

R C C

11. At each station calculate the axial induced velocity

tan ( ) iV r w

12. Calculate the resultant velocity

2 2 2( ) (1 tan ) RV r w

13. At each station calculate the elemental thrust 21( cos sin )

2 R L DdT cV C C dr

14. Repeat steps (3) to (13)

15. Calculate the total thrust by integrating

0

R

R

dTT b dr

dr

0

21( cos sin )

2

R

R L D

R

b cV C C dr

16. Obtain new value of k from the following figure

17. Compare the new value of k with assumed (old) value of k and repeat steps (2) to

(16) till desired accuracy in achieved in k.

18. Calculate the total power

0

21( sin cos )

2

R

L D R

R

P c C C rV dr

41

[Proof: for Eq.(7)

From Eq. (7), replacing the value of r, and K gives

2

2. .

( cos sin )2 2

4

i

R L D i

i i

VR R bcL H S K

bcV C Cr R V

V V

2

4

( cos sin ) 2

i i i

R L D i

R V V Vbc

bcV C C R V

2

2

2

( cos sin )

i

R L D

V

V C C

sin

cos2sin costan sin 2

. .cos sin cos sin

L D L D

R H SC C C C

22sin 1 cos 2

cos sin cos sin

L D L DC C C C

2

2

1 tan2

1 tan

2

1 2 tan

cos sin (1 tan ) ( cos sin )

L D L DC C C C

22

2 2

1 2

( cos sin )1 cos sin

i

i

V

r w i

VL DL Dr w

V

r w C CC C

2

2

2

cos sin

i

R L D

V

V C C ]

42

Chapter – 5

Performance Analysis of Helicopter in Vertical Climb

Using Combined Theory

5.1 Combined Theory for Vertical Climb

In vertical climb, the contraction ratio is not only a function of thrust loading but of the

rate of climb as well. As the rotor climbs from hover, the induced velocity Vi decreases

as VZ increases. However, for all practical purposes it is assumed that the induced

velocity Vi at downstream infinity is twice that at rotor disc.

Elemental thrust given by momentum theory for an annulus of width dr and at a radial

distance r

2 2Z i idT r V V V dr (1)

Flow conditions on a blade element will be similar as in the case of hovering except that

the axial inflow velocity will be VZ + Vi instead of Vi.

The inflow angle is given by

tan z iV V

r w

(2)

Resultant velocity VR is given by

2 2 2( ) ( )R Z iV r w V V (3)

or, 2 2 2( ) (1 tan )RV r w (4)

Elemental forces in the plane of the disc are, as for hover,

43

21cos sin

2 R L DdT cV C C dr

21sin cos

2 R L DdT cV C C dr

Equating the thrust from momentum and blade element theory (for b number of blades)

22 2 cos sin2

Z i i R L D

br dr V V V cV C C dr

or, sin 2 [( ) tan ]

4 ( )[ cos sin ]

Z

L D

r w VR

r r w C C

(5)

Similarly, equating elemental torques from blade element and momentum theories:

22 2 sin cos2

Z i L D R

bdQ r dr V V wr c C C rV dr

Rearranging,

sin 2

4 ( sin cos ) ( )

L D

wR

r C C r w

(6)

where, 4 sin 2

1( sin cos )

L D

rw

r

R C C

(7)

The thrust and power are given by same as in hover

0

2 ( cos sin )2

R

R L D

R

bT c V C C dr (8)

0

2 ( sin cos )2

R

R L D

R

bP c r V C C dr (9)

44

5.2 Method of Computation

1. Divide the blade into number of equal parts

2. 0.750 ( 0.75)

R Tr

3. Assume

4. i

5. Mach no. 0( )20.1 27.3

rM T in c

T

6. For i, M, find CL, CD

7. w4 sin 2

1( sin cos )

L D

r

r

R C C

8. check if equation (5)

sin 2 tan

4 ( sin cos )

Z

L D

r w VR

r R C C

is satisfied

9. If it is not satisfied, repeat (2) to (10) with new value of .

10. For next station repeat steps (2) to (10).

11. At each station, calculate tan ( )i ZV r w V

12. At each station, calculate 2 2 2( ) (1 tan )RV r w

13.

0

2 ( cos sin )2

R

R L D

R

bT c V C C dr

14.

0

2 ( sin cos )2

R

R L D

R

bP c V C C r dr

45

Chapter – 6

Performance Analysis of Helicopter in Descending Flight

6.1 Introduction

Four different flow states have been identified with descending flight.

(b) normal thrusting state

(c) vortex ring state

(d) autorotative state

(e) windmill brake state

Figs 8-9 give idealised representation of these states for a purely vertical descent

6.2 Normal thrusting state:

For hovering and fairly low rates of descent the velocity of air induced through the

disc (Vi) exceeds the rate of descent (VZ) itself. In this state all flow through the rotor

is downward but not necessarily of equal magnitude because of the non-uniform

condition of speed and angle of attack from root to tip. Thrust is quite steady.

6.3 Vortex ring state:

For descent rates upto 150% of hover, a condition of large variations in thrust is

experienced with accompany increased vibration and tendencies to produce even

higher rates of descent. This condition, known to the pilots, as “Settling with power”

is more formally called vortex ring state and is somewhat like flying in one‟s own

wake. As can be seen from the figure the high rate of descent has overcome the

normal downward induced flow on inner blade sections. The flow is thus upward,

relative to the rotor disc in these areas and downward at outboard.

This produces a secondary vortex ring in addition to normal tip vortex system. The

result of this set of vortices is unsteady turbulent flow over a large area of the disc,

with an accompanying loss of thrust and excessive thrust fluctuations even through

46

power is still being supplied from the engine. Pilots are warned to avoid situations

that create this condition (i.e., steep descent at high rates of descent).

6.4 Autorotative State :

Beyond the vortex ring state, things settle down again in terms of the intensity of the

turbulent, unsteady flow. There is some rate of descent in vertical flight between 150

to 180% of (Vi) hover where no power is required to maintain the rpm. This state of

autorotation is, of course, extremely important in cases of engine failure when one

wishes to produce thrust, equal to weight, in order to effect controlled equilibrium

flight to the ground at reasonable rates of descent. In essence, potential energy is used

at a rate just sufficient to provide the power requirement in vertical flight.

Helicopters are equipped with overriding clutches so that, in the event of power

failure, the rotor will not be restrained by the engine but will be free to rotate.

Immediately, after a power failure, the pilot must “dump” his collective pitch within 2

to 3 seconds. With decreased collective pitch, rotor will auto rotate as the helicopter

begins to descend; that is the aerodynamic forces on the rotor will cause it to rotate

even though no mechanical torque is present.

6.5 Windmill brake state:

If the rotor descends at rates in excess of 180% of (Vi)hover, it is necessary to brake the

system in order to maintain RPM. In this state, all the flow is “up” relative to the rotor

and energy may be extracted from the system. This is the state in which windmills

operate, extracting energy from the flow of air past them. This is not a normal

operating state of any helicopter.

47

Chapter – 7

Helicopter Control

7.1 Introduction

Control methods are discussed, first from the over-all point of view of the forces and

moments applied to the helicopter and second, from the point of view of the levers which

the pilot operates.

7.2 Control Requirements:

To control completely the position and attitude of a body in space requires control of the

forces and moments about the three axes (Fig. 10).

This involves six independent controls. For example, if the body drifts to a side, a force

may be exerted to return it to its original position. If it rolls, a moment may be exerted to

right it again. However, it will be exceedingly difficult for a human pilot to coordinate six

independent controls. Fortunately, it is possible to reduce this number by coupling

together independent controls, although such couplings involve some sacrifice of

complete freedom of control of position and attitude (in space).

The pilot of a helicopter does demand the ability to produce moments about all axes in

order to right himself, when disturbed by a gust. He does not, however, demand that he

be able to produce moments without producing an accompanying force. For example, if a

pitching moment is produced along with an accompanying force in longitudinal direction.

By this coupling of pitching moment with longitudinal force, the necessity for one of the

independent controls is implemented.

Actually, only four independent controls are adequate for the helicopter:

(a) Vertical control

(b) Directional control

(c) Longitudinal control

(d) Lateral control

48

a) Vertical control: Vertical control is necessary to fix the position of the helicopter

in the vertical direction, i.e., providing means for climbing and descending flight. It is

achieved by increasing or decreasing the collective pitch of the main rotor. By increasing

collective pitch we mean that the pitch of all the blades has been increased by the same

amount and that pitch is independent of azimuthal position of the blade.

b) Directional control: Directional control fixes the attitude of the helicopter in

rotation about the vertical axis, permitting the pilot to point the ship in any horizontal

direction. This is achieved by either by changing the pitch (thrust) of the tail rotor in

conventional single rotor helicopter or by obtaining differential torque in case of twin

main rotor helicopter.

c) Longitudinal control: Longitudinal control involves the application of both

moments and forces. Pitching moments are coupled with longitudinal force. When the

pilot operates the longitudinal control, a pitching moment is produced about the

helicopter C.G. which tilts the helicopter in forward direction. As a consequence of the

tilt, a component of the rotor thrust vector acts in the direction of the tilt. The application

of longitudinal control has therefore resulted in a forward tilt and forward motion of the

helicopter. In longitudinal control, thus, moment is coupled with force.

d) Lateral control: Lateral control is identical in nature to longitudinal control.

Lateral control results in rolling moment as well as sideward motion of the helicopter.

7.3 Pilot’s Control

There are generally, four control levels to be operated by the pilot. They are:

(a) The control stick (cyclic pitch lever)

(b) Pitch lever (collective pitch lever)

(c) Pedals

(d) Throttle

a) The Control Stick: Control stick is located in first of the pilot and is operated by his

right hand. In fact it is comparable to the stick used in fixed wing aircraft and is used for

49

longitudinal and lateral control. It controls the cyclic pitch of the main rotor. It can be

displaced fore and aft and sideways as well as combination of these two motions.

In the helicopter, the pilot pushes the stick in the direction he wishes to go – forward,

sideward or even backward. For example, if the pilot wants to go forward he moves the

stick in forward direction. Similarly, if he wants to go towards his right, he moves his

stick towards his right.

Forward motion: To go forward the rotor cone must be tilted forward. To tilt to rotor

cone forward, the rotor blade must flap up at = 00 and flap down at = 180

0. Thus,

with the blade flapping high over the tail and low over the nose, the rotor disc is

effectively tilted forward.

Sideward motion: Similarly, if the pilot wants to go to right, blade should flap up at

= 2700 and flap down at = 90

0 effectively tilting the rotor thrust towards right.

Phase-Lag: It is worth noting that when the pitch of a blade is increased the blade

does not flap up instantaneously. There is a phase lag between the application of the force

on the rotor and the ensuing displacement due to the inertia of the blade. The blade can

not deviate immediately from its path of rotation but does so 900 later. Therefore, to

create forward motion, blade pitch would be increased at = 2700 and decreased at =

900 and for sideward motion at = 180

0 and = 0

0 so that the desired flapping up and

down occurs 900 later.

b) Collective Pitch Lever: Collective pitch lever is situated on the left hand side of

the pilot and is operated by his left hand. It is used for up and down motion of the and for

adjustments as required forward flight. The pitch lever operates the collective pitch of the

main rotor. When pilot wants to go up he moves the pitch lever upwards, which increases

the collective pitch of the main rotor and the helicopter starts climbing because of

increased thrust.

50

c) Pedals: The pedals are situated at the floor in front of the pilot. They are two

numbers, left and right, and they are operated by the pilot‟s feet. They move

differentially, i.e., when left pedal is pushed, the right pedal comes out towards pilot.

Pedals are used for directional control of the helicopter. To point the helicopter towards

right, the pilot pushes the right pedal, to the left, the left pedal.

The pedals are connected to the pitch of the tail rotor. Under equilibrium, the antitorque

of the main rotor is balanced by the thrust of the tail rotor. When the pilot pushes the

right pedal, the pitch of the tail rotor is increased which gives a left force at the tail rotor

which in turn produces a nose right moment.

d) Throttle: The throttle controls the power output from the engine. It is usually

located near or on the pitch lever. Throttle adjustments are made by twisting a grip

located at the top of the pitch lever.

51

Vz

S

Vz+Vi

Vz+ Vi

S

Fig. 1 Flow through rotor disc in vertical climb

Vi

Flow field Velocity variation

Fig. 2 Flow through the rotor disc in hover

Vi

Vi =2Vi R

Vz=0

Vi R

52

Vi =

DL2 m/sec

Downwash

Velocity

Vi

DL

Fig. 3 Disc loading characteristics

PL M =1 (ideal rotor)

M = . 75 (good rotor) At sea level

M =. 5

(poor rotor)

DL

Fig. 4 Performance characteristics of a rotor

1000ft Sea level

53

dL

dT

dFx VZ + Vi

r W

dD

Fig. 5 Flow characteristics on a rotor blade element in vertical climb

Vi Vi

R

Tip section Inboard section

Fig. 6 Variation of pitch and inflow angle at different blade sections in hover

dT = 2 π dr ρ Vi Vi∞

Fig. 7 Flow through the rotor annulus

r dr

i

t

t

54

55

hoveri

i

V

V

B

A C VZ =0

D Vi

1 Hover VZ

Climb Vi

Vi decreases as

Vz increases

-3 -2 -1 0 1 2 3

hoveri

Z

V

V

Descend Climb

A = Windmill brake state

B = Autorotative states

C = Vortex ring state

D = Normal thrusting state

Fig. 9 Graph of Flow state

56