Category HELICOPTER AERODYNAMICS

Losses of the Real Rotor

We have been examining the operation of an ideal main rotor, i. e., a rotor in which all the power obtained from the engine was converted into work

in accelerating the air downward or in creating thrust.

Losses of the Real RotorWe have assumed that the entire area swept by the rotor participates in creating thrust. This means that the increased air pressure below the rotor and the reduced air pressure above the rotor (Figure 25a) acts on the entire main rotor area. In reality, as will be shown later, the entire swept area does not participate in creating thrust. The ideal rotor accelerates a uniform air jet downward with the same induced velocity for all the blade elements. The real rotor provides a swirling jet, and the induced velocities will vary markedly along the radius for the different blade elements (Figure 25b).

The ideal rotor does not expend energy in overcoming friction forces, while the real rotor experiences profile drag forces resisting rotation, and considerable power is expended in overcoming these forces. Moreover, the real rotor has the so-called tip and root losses. The essence of these losses lies in cross-flow of the air from the high-pressure region below the rotor into the low-pressure region above the rotor. This cross-flow takes place through the ends of the blades (tip losses) and through the root sections of the blades near the main rotor hub (root losses), where the structural part of the blade (spar) does not have a lifting surface. The concept of the end loss coefficient x has been introduced to account for the tip and root losses. With account for this coefficient, the actual area participating in creation of thrust is defined by the formula

For most main rotors x = 0.90 – 0.92.

Подпись:Since for the real rotor varies along the radius, we take as the induced velocity its value at the radius r = 0.7

To account for the influence of the profile drag forces, we assume that the real rotor power required for creating thrust is greater on the average than the ideal rotor power required by 25%.

With account for these losses, the thrust of the real rotor can be found from the formula

T~2yFoVl

Hence, it is easy to find the induced velocity in the hovering regime

7

2/Fp *

Knowing that

T = CTF-^-u

Losses of the Real Rotor
Losses of the Real Rotor Losses of the Real Rotor

we obtain

For most main rotors, the induced velocity in the hovering regime is V«8—10 m/sec, and Cx« 0.003.

An important characteristic of the main rotor is the relative efficiency Ni

The main rotor relative efficiency is the ratio of the power required to create the thrust of the ideal rotor to the total power supplied to the rotor. For modem rotors, the efficiency is 0.6-0.75.

HELICOPTER HORIZONTAL FLIGHT

§ 44. General Characteristics of Horizontal Flight /88

Rectilinear flight of a helicopter with constant velocity in the horizontal plane is termed horizontal flight. This is the primary flight regime for the helicopter. Since the Earth is a sphere, flight at constant altitude takes place along a curvilinear trajectory. But the radius of curvature of the Earth’s surface is so large that the curvature of the Earth’s surface can be neglected in flight.

Only in flights on supersonic airplanes at a speed 2-3 times that of sound is it necessary to consider the Earth’s curvature. We shall use an example to demonstrate this. An airplane is flying horizontally at a speed of 1000 m/sec or 3600 km/hr. The airplane weighs 10,000 kgf. Let us find the centrifugal force which arises as a result of curvature of the Earth’s surface

„3

г – mV Fc " R

where m is the airplane mass, kg;

V is the airplane speed, m/sec;

R is the radius of the Earth, equal to 6 370 000 m.

Then

F

с

 

= 1570 N,

 

or Fc = 161 kgf.

 

We see from the example that the Earth’s curvature should be considered when flying at 1000 m/sec, since the airplane lift becomes 161 kgf less than its weight. But for the same flight weight and a velocity of 180 km/hr the centrifugal force is 0.4 kgf.

Therefore, in the following we shall consider the Earth’s surface to be

flat.

HELICOPTER HORIZONTAL FLIGHT HELICOPTER HORIZONTAL FLIGHT

The following forces act on the helicopter in horizontal flight: weight /89

The horizontal flight conditions are expressed by the equalities

Y = G or G – Y = 0;

P =~X or P-X =0; par par

T = S or T – S = 0; t. r s t. r s

ГМ = 0. /L eg

The first condition ensures constant flight altitude, the second provides constant velocity, and the*third specifies linearity of flight in the horizontal plane.

The forces Y, P, Sg are the components of the main rotor thrust. Accordingly, the main rotor thrust in horizontal helicopter flight performs the functions of propulsive, side, and lifting forces.

What determines the minimal vertical rate of descent during /140 helicopter gliding?

Answer 1. The vertical rate of descent descent during helicopter gliding is found from the formula V^eg = V ^ sin 0, i. e., it depends on the velocity along the trajectory and the gliding angle. The velocity along the trajectory depends on the forward or aft tilt of the coning axis. The gliding angle is determined by the lift force and depends on the main rotor pitch: the larger

the pitch, the larger the lift force, and the smaller the gliding angle and the vertical rate of descent.

G2Vgl

Подпись: descent is the propulsive force per unit

What determines the minimal vertical rate of descent during /140 helicopter gliding?

Answer 2. During helicopter gliding the vertical rate of found from the formula V^eg = V ^ sin 0. Since during a glide force is the weight force = G sin 0, the work of the weight time will be the equivalent power supplied to the rotor

What determines the minimal vertical rate of descent during /140 helicopter gliding? Подпись: N, What determines the minimal vertical rate of descent during /140 helicopter gliding? Подпись: N, Подпись: mm

Hence

Thus, the minimal vertical rate of descent in a glide depends on the flight weight and altitude. The gliding velocity along the trajectory should be equal to the economical speed for horizontal flight.

Main Rotor Cone of Revolution

When horizontal hinges are used, the rotating rotor forms a "cone of revolution". If the flapping angle is positive, as the main rotor turns the blades travel along the generator of a cone whose apex is located at the center of the hub. The plane passing through the rotating rotor blade tips is called the main rotor plane of rotation [tip path plane] (Figure 36).

The line passing through the center of the hub perpendicular to the tip path plane is called the coning axis. The main rotor thrust vector passes lies along the coning axis.

Main Rotor Cone of RevolutionIf the blade flapping angle does not change azimuthally, the main rotor tip path plane is parallel to the hub rotation plane and the coning axis coincides with the main rotor hub axis. In this case the flapping angle 3 equals the coning angle a^. The coning angle is the angle a^ between the hub rotation plane and the generator of the cone of revolution. It varies from 0° to 10 – 12°.

The coning angle is larger, the larger the main rotor pitch. In – /53

crease of the pitch leads to increase of the blade thrust and moment about the horizontal hinge.

The main rotor cone of revolution is clearly visible if all the blades have the same flapping angle, i. e., there exists so-called co-conicity of the blades, absence of which leads to severe vibration of the main rotor. To obtain co-conicity it is necessary that the incidence angles of all the blades be the same. The technique for adjusting the rotor for blade tracking is given in the helicopter maintenance instructions.

Vertical Rate of Climb

Vertical Rate of Climb

During climb the helicopter center of gravity moves along the flight trajectory with the velocity V The climbing velocity vector can be repre­sented as the vector sum

where is the horizontal component of the velocity;

V is the vertical rate of climb.

У

The horizontal component of the velocity equals V ^ cos 0, but since the climb angle is usually small and does not exceed 10°-12°, then ^ V^.

This means that the same power is expended on horizontal displacement of the helicopter during climb as is expended in horizontal flight at the same speed.

Analyzing (32) from this viewpoint, we can say that the excess power AN = GVy is expended on vertical displacement of the helicopter. Hence, knowing the helicopter weight and the excess power, we find the vertical rate of climb

Vertical Rate of Climb

The excess power can be found from the power required and available curves for helicopter horizontal flight (see Figure 63a). The maximal excess power corresponds to the economical speed (for the Mi-1, = 80 km/hr or

about 23 m/sec). Therefore, climbing should be performed at the economical speed. Moreover, from the power required and available curves for the Mi-1, we can conclude that vertical climb is impossible when using rated engine power. This means that the static ceiling when using rated power is equal to zero, i. e., helicopter hovering is possible only in the air cushion influence zone.

When using takeoff power vertical climb is possible, but the rate of climb will be lower than when climbing along an inclined trajectory. Conse­quently, this once again confirms that climbing should be performed along an inclined trajectory. Vertical climbing is performed only when it is necessary to clear surrounding obstacles. It must be kept in mind that takeoff power can be used only for a brief period, not to exceed five minutes.

Increase of the flight altitude involves change of the power required and the power available, and, therefore, change of the vertical rate of climb.

Dual-Rotor Helicopter Control Principles

Control principle of dual-rotor helicopter with tandem arrangement of the lifting rotors. Longitudinal control of the helicopter is achieved by
deflecting the stick fore and aft. This leads to cyclic change of the pitch of the lifting rotors, as a result of which the axes of the cones-of-rotation are tilted forward or backward, i. e., in the direction of the stick (Figure 108). In addition to the cyclic variation of the pitch there is a differential change of the collective pitch, in which the thrust of one rotor is increased while that of the other is decreased.

Подпись: v>Подпись: Figure 108. Control of tandem twin- rotor helicopter. If the stick is deflected forward, the axes of the cones-of- rotation of the lifting rotors are tilted forward. The collective pitch of the front rotor is reduced and that of the aft rotor is increased. As the thrust force vectors tilt, there is a change of the thrust force arms relative to the helicopter transverse axis. The result is the creation of a diving moment equal to the difference of the thrust moments of the front and rear rotors.

Under the influence of this moment the helicopter nose will drop, increasing the flight speed. If the stick is moved aft, a climbing moment is created and the helicopter nose will rise, reducing the flight speed.

Lateral control of the helicopter is achieved by deflecting the stick to the right and left. This leads to simultaneous identical change of the cyclic pitch of the front and rear lifting rotors. A lateral control moment appears which then causes rotation of the helicopter around the longitudinal axis.

Directional control, or control of helicopter rotation around the vertical axis, is accomplished with the aid of the directional control pedals. Deflection of the pedals leads to differential change of the cyclic pitch of the lifting rotors. The axes of the cones-of-rotation deflect in opposite directions, forming the directional control moment as a result of side forces. If the right pedal is pushed, the coning axis of the front rotor is deflected to the left. The side components of the lifting rotor thrust forces create a pair, whose moment turns the helicopter to the right.

Control principle of the dual-rotor helicopter with side-by-side lifting rotors. The control stick is moved fore and aft for longitudinal control, and this causes the same change of the cyclic pitch of the lifting rotors and deflection of the axes of the cones-of-revolution in the direction of stick displacement. This creates a longitudinal control moment (just as in the case of the single-rotor helicopter).

Lateral control is accomplished by deflecting the stick to the right or left. This deflection leads to differential change of the collective pitch of the lifting rotors. If the stick is moved to the right, the collective pitch of the right rotor is reduced and that of the left rotor is increased. Change of the collective pitch causes change of the thrust forces. The difference of the thrust forces of the right and left lifting rotors leads to the creation of a lateral moment which then causes a bank to the right.

The helicopter with side-by-side arrangement of the lifting rotors has an auxiliary wing which gives the helicopter lateral stability.

Directional control of the helicopter is accomplished by the control moment which is created by differential change of the cyclic pitch and tilting of the axes of the cones-of-rotation in opposite directions: forward and

backward. If the right pedal is pushed, the axis of the cone-of-rotation of the right rotor is deflected aft while that of the left is deflected forward. The horizontal components of the thrust forces create the directional control moment which causes rotation of the helicopter to the right.

Control principle of dual-rotor helicopter with coaxial rotors. Longi­tudinal and lateral control is accomplished similarly to the control of the single-rotor helicopter, i. e., by cyclic change of the pitch of the upper and lower lifting rotors. When the stick is deflected, the axes of the cones-of – rotation deflect in the same direction as the stick, creating the longitudinal or lateral control moment.

Directional control is accomplished by deflection of the pedals, which leads to differential change of the collective pitch of the lifting rotors.

This does not cause any change of the overall thrust, but leads to a change of the reactive moments of the lifting rotors and the helicopter turns in the direction of the action of the larger reactive moment.

Control of the vertical displacement of all helicopters is the same.

When the collective-throttle lever is moved up, the collective pitch of all the lifting rotors is increased, which leads to increase of the thrust and upward displacement of the helicopter. If the collective-throttle lever is fixed, the collective pitch is reduced, the thrust force decreases, and the helicopter transitions into descent.

Basic Regimes of Operation

The operating conditions of the main rotor or its working regime are defined as the position of the main rotor relative to the air stream. Depending

on the position, two basic working regimes are considered, those of axial and translational flow.

The axial flow regime is the term used for the operating condition of the main rotor where the axis of the hub is parallel to the oncoming free stream flow. In the axial flow regime the free stream passes perpendicular to the plane of rotation of the main rotor hub (Figure 12a). This regime covers the hovering, vertical climb and vertical descent conditions of the helicopter main rotor. An important feature of the axial flow regime is that the loca­tion of the blade of the rotor, relative to the oncoming free stream, is not changed. Consequently, the aerodynamic forces on the blade as it moves around the circle are not changed.

The oblique flow regime is the term used for the operating conditions of the main rotor, where the airstream approaches the rotor in a direction not parallel to the axis of the hub. An important difference of this regime is that, as the blade moves around in a circle, it continuously changes its location relative to the flow approaching the rotor. As a consequence, the velocity of the flow at each element is changed and also the aerodynamic forces on the blade. The translational flow regime occurs in the horizontal flight of a helicopter and in flight inclined upwards and downwards.

From consideration of the operating conditions, one can see that the position of the main rotor in the airflow is important. This position is determined by the angle of attack of the main rotor.

The angle of attack of the main rotor is termed angle A, and it is formed by the plane of rotation of the hub and the flight velocity vector, or by the undisturbed flow approaching the rotor. The angle of attack is positive if the flow approaches the rotor from below (Figure 12b). If the flow approaches the rotor from above, the angle of attack is negative (Figure 12c). If the airflow approaches the rotor parallel to the plane of rotation of the hub, the angle of attack is zero (Figure 12d).

а). •
A-ffO

 

. d) Г

A*ffe

 

CD

пз

 

b) A-Яв’

 

=£3

 

Basic Regimes of OperationBasic Regimes of Operation

Basic Regimes of Operation

Figure 12. Operating regimes and angle of attack of the main rotor.

It is not difficult to observe the connection between the operating regime of the main rotor and the angle of attack:

In the axial flow regime, the angle of attack of the main rotor A = + 90°. In the oblique flow regime, А Ф – 90°.

If the angle of attack A = 0°, the operating regime of the main rotor is termed the planar flow regime.

Blade Coriolis force and its azimuthal variation

Answer 1. The blade Coriolis force is the force which develops as a result of the combination of two velocities: the circumferential velocity of

the blade center of gravity and the radial velocity which develops as a result of variation of the flapping angle.

The Coriolis force of the advancing blade is directed in the direction of rotation of the motor and reaches its maximal value at the 90° azimuth.

The Coriolis force of the retreating blade is directed opposite the rotor rotation and reaches its maximal value at the 270° azimuth.

Answer 2. The Coriolis force is an inertial force which arises in the forward flight regime as a result of the combination of the circumferential velocity of the blade center of gravity and the helicopter translational flight velocity. For the advancing blade, this force is directed forward and reaches its maximal magnitude at the 90° azimuth; for the retreating blade it is directed aft and reaches its maximal value at the 270° azimuth.

Answer 3. The Coriolis force is an inertial force which arises from

Подпись: /73flapping motions resulting from the combination of the angular velocity of motion of the blade center of gravity and the vertical flapping velocity.

For the advancing blade, this force is directed upward and reaches its maximal value at the 90° azimuth; for the retreating blade, it is directed downward and will be maximal at the 270° azimuth.

Main Rotor Autorotation Conditions and Regimes

The main rotor blade element may be compared with an airplane wing element. Let us examine the aerodynamic forces acting on an airplane wing and then trans­fer these forces to the blade. The flight force Y and the drag force X develop on the airplane wing at the angle of attack a. The geometric sum of these forces will be the resultant aerodynamic force R = Y + X. The angle between the lift force Y and the resultant aerodynamic force R vectors is called the aerodynamic efficiency angle (0 ). The larger the aerodynamic efficiency angle, the lower the aerodynamic efficiency of the wing, since the minimal aerodynamic efficiency angle corresponds to maximal wing aerodynamic efficiency ctg 0 =

= Y/X = K. Reduction of the aerodynamic efficiency angle means a sort of "attraction" of the resultant aerodynamic force vector AR to the lift force vector Y, i. e., it means reduction of the backward tilt of AR relative to the normal to the undisturbed flow.

Main Rotor Autorotation Conditions and Regimes

Figure 75. Blade element autorotati-ve conditions.

Now let us turn to examination of the forces acting on the blade. We draw three straight lines through the center of pressure of the blade element: a-а is perpendicular to the rotor rotation plane; b-Ъ is perpendicular to the blade element chord; y-y is perpendicular to the resultant velocity vector (Figure 75a). The angle between the lines y-y and b-b and the angle a are equal, since they are formed by mutually perpendicular sides. On this same basis, the angle between the lines a-a and b-Ъ is equal to the incidence angle ф.

The resultant aerodynamic force vector is applied at the blade element

center of pressure. We resolve this force into the lift force AY and the drag

force ДХ. The angle between the vectors AY and AR is the aerodynamic efficiency

angle The angle between the vector AR and the line b-b will be equal to

the difference of the angles (a – 0 ) . If this difference is less than the

blade incidence angle, then ф – (ot – 0 ) = у, i. e. , the angle у is positive. This

K.

means that the projection AQ of the force AR on the main rotor plane of rota­tion will be directed aft and creates a retarding moment which reduces the rotor rpm.

The main rotor will operate in the decelerated autorotative regime, which leads to stopping of the rotor. The larger the blade element incidence angle or pitch, the larger the angle y, and the larger the force AQ and its decelerating moment.

If the difference between the angle of attack and the aerodynamic effi­ciency angle is greater than the incidence angle, i. e., (a – 0 ) > ф, then

К

Ф – (a – 0„) = – y. The angle у is negative, which means that the vector AR /121

K.

is directed forward relative to the hub axis (Figure 75b). The projection AQ of the force AR on the hub rotation plane is directed forward and creates a turning moment which accelerates the rotor rotation. The main rotor will operate in the accelerated autorotative regime. The smaller the incidence angle ф, the larger the forward tilt of the vector AR, and the higher the speed at which the rotor turns.

If the difference (a – 0 ), then ф – (a – 0 ) = 0, i. e., the force AR is

К к

parallel to the hub axis and its projection on the hub rotation plane AQ = 0 (Figure 75c). In this case, the retarding or turning moment equals zero, and the rotor revolves at constant rpm, i. e., the rotor autorotative regime will be established.

From these examples we conclude:

— the tilt of the elemental resultant aerodynamic force vector depends on the blade element pitch;

— with reduction of the pitch, the force vector AR is deflected forward, and the main rotor autorotation becomes accelerated;

— with increase of the blade element pitch, the force vector AR is deflected aft, and the main rotor autorotation becomes decelerated.

The dependence of the autorotative regime on the blade element angle of

attack and pitch can be expressed graphically (Figure 76). This graph is

called the autorotation margin graph. The abscissa is the blade element

angle of attack, the ordinates are the incidence angles ф and the angles equal

to the difference a – 6 .

К

Let us examine the characteristic points in this figure. The ascending portion of this curve corresponds to blade element angles of attack below stall. The point В is the stall angle, and the descending portion corresponds to angles above stall.

If we draw a straight line paral­lel to the abscissa axis, it crosses the curve at the two points A and C.

In the figure, this straight line passes through the point on the ordinate corresponding to the incidence angle ф = 4° (such a straight line can be drawn through any point of the ordinate). What do the points A and C characterize? The point A corresponds to the blade element angle of attack (in our example a = 7°30′) which corresponds to steady state autorotation. Let us show that this actually is the case.

It was established above that steady state autorotation will occur when

the difference a – 0 = ф. In this case, the force AR of the element will be

К

parallel to the main rotor hub axis. In our example ф = 4° and а – 0^ = 4°. 180

This means that y= 0 (see Figure 75c). Therefore, in order for the Mi-1 helicopter blade to have steady state autorotation with ф = 4° the blade angle of attack must equal 7° 30′. Points of the curve for which Ct-0 <4 correspond to smaller angles of attack, i. e., the autorotation will be decelerated (see Figure 75a). At angles of attack between the points A and C all the points of the curve correspond to the inequality а-0^>ф, i. e., accelerated auto­rotation (see Figure 75b). But at angles of attack up to the stall angle, the accelerated autorotation will be stable, while at angles above stall flow separation takes place, and the autorotation is unstable.

The range of angles of attack between the points A and C is called the blade element autorotation margin. Since flight with blade element angles of attack above the stall angle is not feasible in practice, the autorotation margin will correspond to the angles of attack between the points A and in the figure.

With increase of the blade element pitch, the straight line AC shifts upward (A’C’). This means that the angle of attack range corresponding to decelerated autorotation increases, and the autorotation margin decreases.

With reduction of the pitch, the straight line AC shifts downward, and the autorotation margin increases.

Since the angles of attack are different for different blade elements, the autorotation conditions for these elements will also be different, and therefore the autorotation margin graph has a somewhat arbitrary nature, i. e., it serves for a qualitative evaluation of this process.

Let us return to the point A on the autorotation margin graph. In our example, it corresponds to ф = 4° and a= 7°30′. At these angles, the auto­rotation will be steady-state. But how can we obtain an angle of attack a = 7°30′ with an incidence angle ф = 4°?

Since а=ф+Да (see Figure 75b), then Да=а-4. This means that for our example the angle of attack increment caused by the vertical rate of

descent is &x=7°30,-4°=3°30′. Let us find the vertical rate of descent corresponding to this Да if r=0.7; ф=4°; ш=26 rad/sec. It is known that

If the vertical velocity < 7.9 m/sec, then for the given blade

element the angle of attack becomes less than 7°30′, and the autorotation will be decelerated;conversely, if > 7.9 m/sec, the autorotation will be

Подпись:accelerated. We must emphasize once again that the words "accelerated auto­rotation of the element" are arbitrary. They mean that under the given conditions the elemental resultant aerodynamic force vector AR is inclined forward relative to the hub axis and creates a turning moment. The larger the helicopter vertical rate of descent, the larger the angle of attack increment Да, the larger the forward tilt of the force vector AR, and the higher the main rotor rpm will be.

Characteristics of Operation of Coaxial System of Two Main Rotors

In the coaxial twin-rotor helicopter, the main rotors are positioned on a single axis — one above and the other below. Such a helicopter has certain operational characteristics. The area swept by the two main rotors is equal to the area swept by a single rotor

where Fc is the area swept by the system of coaxial rotors;

F^ is the area swept by a single rotor.

In this case, we have assumed that the diameters of the upper and lower /34

rotors are the same.

Let us examine the system of air jets passing through the areas swept by the upper and lower rotors (Figure 26). Increase of the distance between the hubs of the upper and lower rotors degrades the operating conditions of the lower rotor and complicates the construction of the entire system, while re­duction of this distance leads to the danger of collision of the rotor blades and increases helicopter vibration. This distance is h = 0.08D = 0.8m in the Ka-15 and Ka-18 helicopters. At this distance, the lower rotor has no effect on the operation of the upper rotor. The jet from the upper rotor con­tracts, and in the plane of rotation of the lower rotor its radius is 0.7R,

where R is the rotor radius. In this case, the lower rotor blade tips

operate under the same conditions as those of the upper rotor and draw additional air in from the side.

On this basis, we shall estimate the effective area of the entire system through which the air flows, just as for an isolated rotor in the hovering regime.

From the area swept by the upper rotor, we must subtract the root loss area (of radius 0.25R). Under conditions similar to those in the hovering regime, only the tips of the lower rotor blades operate. The area swept by these tips is

F ігтеЯ* _ тгО.72/?3.

1 V

Consequently, the effective area of both rotors through which the stream flows, as in the case of hovering of an isolated rotor, is found from the formula

Fc = *R2 _ *0.2S2/?2 + кГГ – – *0.7=i42 == 7гД* (1 _10.06 +

. +1 -0.49) = 1.45г,. ‘ ,

That portion of the lower rotor which operates in the jet of the upper rotor has lower efficiency. The angles of attack of the lower rotor blade elements are reduced as a result of the induced velocity of the upper rotor (see Figure 23b), which leads to reduction of the thrust. To reduce this effect, the incidence angles of the lower rotor blades are made 2-3° larger than for the upper rotor, but this does not eliminate entirely the harmful influence of the upper rotor on the lower. In the presence of this influence, the efficiency of the central portion of the lower rotor, which is in the jet from the upper rotor is reduced by a factor of two, in comparison with the efficiency of the tip area outside the jet from the upper rotor.

The swept area of the lower rotor, operating in the jet from the upper /35 rotor, is found from the formula

F± = kPFOJ2 – */?20.252 = = ^20.43-0.43Л.

Подпись:Подпись: iJ p p e r • main rotor Figure 26. Operation of coaxial rotor system. Since its efficiency is less than that of the upper rotor by a factor of two, the additional effective area of the lower rotor is

F = 0.43^0.5 = 0.22Fi.

e. 1 • –

The effective area of the entire

system isFe ^=lA5Fi–Q.22Fi = l.67Fi •

This formula shows that the thrust of

two coaxial rotors under the same conditions is greater than the thrust of an isolated main rotor of the same diameter by a factor of 1.67.

If the thrusts of the coaxial system and the isolated rotor are the same, then less power is required to create the thrust of the coaxial rotor system, which follows from ideal rotor momentum theory.

The power required to turn the ideal rotor is entirely converted into kinetic energy of the jet, i. e., N = TV^.

If we use Tc, Vc> F^, respectively, to denote the thrust, induced velocity, and effective area of the coaxial system of two rotors, and T^, V^, F^ to denote the thrust, induced velocity, and swept area of the isolated

rotor, then we have T = Tn.

5 cl

Characteristics of Operation of Coaxial System of Two Main Rotors

Consequently,

We know that

Fc = 1.67/7!.

Then

Hence, we find

T„ 2pFtVf V

Vc~ 2p 1.677?! = 1.67 ’

or

Characteristics of Operation of Coaxial System of Two Main Rotors0.78V!.

In order to obtain thrust on a system of coaxial rotors equal to the thrust of an isolated rotor of the same diameter, the induced velocity of the coaxial system must be less than the induced velocity of the isolated rotor.

Since the ideal rotor power required is proportional to 1Л, less power is required to obtain the same thrust for the coaxial system than for the /36

isolated rotor. This is the advantage of the coaxial system. The number

0. 78 ‘v is called the aerodynamic advantage coefficient, and is denoted by Using this coefficient, we express the power required for the coaxial system in terms of the power required of an isolated ideal rotor

This implies that for the same power the coaxial rotor system provides 13-15% more thrust than the isolated main rotor. Therefore, the helicopter with coaxial rotors has smaller dimensions than the single-rotor helicopter.

However, to date only light helicopters have been built using this scheme because of structural complexity and other problems.

Twin-rotor helicopters of other arrangements, for example, with the rotors placed longitudinally and with intermeshing rotors, also have an aerodynamic advantage in the axial flow regime. The aerodynamic advantage coefficient of these systems approaches closer to 0.8, the less the distance between the main rotor hub axes.