# Category HELICOPTER AERODYNAMICS

## Programmed Testing Questions and Answers

In Chapters 1 and 2 we have examined concepts which are of considerable importance in themselves and ensure further successful study of helicopter aerodynamics. We shall present some questions and answers to test the readers’ knowledge of this information.

The objective is to select the most complete and correct answer from three or four possibilities. Some of the answers given are completely incorrect, most of the answers are simply incomplete.

Question 1. Definition and purpose of blade geometric twist.

Answer 1. Geometric twist involves variation of the incidence angles of the blade elements. Twist is provided to distribute the loads uniformly over the blade and increase main rotor thrust.

Answer 2. Geometric twist involves variation of the blade element inci­dence angles along the main rotor radius. The root elements have larger incidence angles,.and the tip elements have smaller angles. Twist gives the blade elements angles of attack close to the optimal values and increases the main rotor thrust by 5-7%. Twist results in more uniform loading on the individual blade elements and delays flow separation from the tip portion of the blade.

Answer 3. Geometric twist is the difference between the incidence angles at the root and tip sections of the blade. Twist provides minimal incidence angles at the root elements and maximal angles at the tip elements. This is necessary to obtain higher rotor efficiency, increase thrust, and achieve more uniform loading on the different parts of the blade.

Question 2. Main rotor operating regime coefficient. /37

Answer 1. The main rotor operating regime coefficient is the dimension­less number p, equal to the ratio of the helicopter flight speed to the blade tip induced velocity

V

Answer 2. The main rotor operating regime coefficient is the number p, equal to the ratio of the projection of the flight velocity on the main rotor hub axis to the blade tip circumferential velocity

V sin A

JJ* rj ‘ •

‘ ID/?

Answer 3. The main rotor operating regime coefficient is the number p, equal to the ratio of the helicopter flight speed to the blade tip angular velocity

■л = —

* o> ■

Answer 4. The operating regime coefficient is the number p, equal to the ratio of the projection of the helicopter flight speed on the main rotor hub plane of rotation to the blade tip circumferential velocity

V cos A ^ ” <»R

Question 3. What is the connection between the operating regime coefficient and the main rotor operating regime?

Answer 1. The larger y, the larger the main rotor induced velocity and the closer its operating regime approaches the axial flow regime.

Answer 2. If у = 0, this indicates the axial flow regime. The larger y, the more effectively the properties of the axial flow regime manifest themselves.

Answer 3. Increase of the coefficient у indicates increase of the main rotor angle of attack and approach of its operating regime to the axial flow regime.

Question A. What is the connection between the main rotor angle of attack and its operating regime?

Answer 1. The main rotor angle of attack is the angle between the flight velocity vector and the hub rotation plane. In the axial flow regime, the main rotor angle of attack A = + 90°, in the inclined flow regime А Ф + 90°.

Answer 2. The main rotor angle of attack is the angle between the flight velocity vector and the hub axis. If the main rotor angle of attack A = 90°, the rotor is operating in the axial flow regime. However, if A ^ 90°, it is operating in the inclined flow regime.

Answer 3. The main rotor angle of attack is the angle between the plane of rotation of the main rotor and the vector of the undisturbed flow approach­ing the rotor. For A = 0° the inclined flow regime is present; for А ф 90° the flow regime is axial.

Question 5. What is main rotor thrust, and on what does it depend?

Answer 1. Main rotor thrust is the aerodynamic force which arises during rotor rotation as a result of the difference of the air pressure on the rotor blades

7*=– CT Z7 -— /Л

The thrust depends on the rotor area thrust coefficient, flight speed, and air density. The thrust coefficient depends on rotor rpm and blade element pitch.

Answer 2. Main rotor thrust is the aerodynamic force directed along the main rotor axis and formed as a result of the difference of the air pressures below and above the rotor

T = CTF-^-W<»-

The thrust depends on the thrust coefficient, main rotor area or radius, air density, and main rotor rpm. The thrust coefficient depends on the pitch.

Answer 3. Main rotor thrust is the aerodynamic force which arises as a result of the difference of the air pressure below and above the rotor

T = Cr5-|-u>/?/? or 7 = 2?FV].

The main rotor thrust depends on the thrust coefficient, area swept by the rotor, rotor pitch, and rotor rpm. The thrust increases with increase of the pitch and rpm.

Question 6. What is the main rotor reactive torque, what does it depend on, and how does it act?

Answer 1. The reactive torque is the torque opposing rotor rotation

M = Q, Rk
г Чэ

It retards rotor rotation and yaws the helicopter opposite the direction of rotation. The reactive torque depends on the rotor rpm, air density, rotor pitch, and flight speed.

Answer 2. The reactive torque is the moment of the forces of resistance to rotation about the hub axis. It is defined by the formula

Mr=VQk*

It depends on the rpm, pitch, air density, surface condition and flight speed. It retards rotor rotation and yaws the helicopter opposite the direction of rotor rotation.

Answer 3. Reactive torque is the moment of the forces of resistance to rotation, directed opposite the rotor direction of rotation, retarding rotor rotation and yawing the helicopter opposite the direction of rotation. It depends on the flight speed, rpm, and air density

Mr = 2Qbrk.

Question 7. Power required to rotate the main rotor and the constant rpm conditions.

Answer 1. The power required to turn the main rotor depends on the rpm, pitch, flight speed, and air density

If N = N the rpm is constant; if N > N the rpm increases, sup req sup req

Answer 2. The power used to overcome the reactive torque depends on the rotor pitch, rpm, and flight speed

= M ш.
r

 the rpm

 increases

Answer 3. The power required to turn the main rotor and overcome the retarding action of the reactive torque depends on the main rotor thrust, rpm air density, and flight speed

When N = N the rpm is constant; when N > N the rpm increases sup req sup req

Question 8. What is the rotor blade element angle of attack and how is it changed?

Answer 1. The blade element angle of attack is the angle between the blade chord and the resultant velocity vector. It depends on the blade ele­ment pitch, induced velocity, and helicopter flight speed. The larger the induced velocity, the lower the angle of attack. The larger the vertical climbing velocity, the lower the angle of attack.

Answer 2. The blade element angle of attack is the angle between the blade element chord and the resultant velocity vector. It depends on the flight speed and induced flow downwash angle. With increase of the induced velocity, the angle of attack increases, with increase of the flight velocity it decreases.

Answer 3. The blade element angle of attack is the angle between the chord and the circumferential velocity vector. It depends on the pitch and helicopter flight speed. With increase of the vertical descent velocity, the angle of attack increases. With increase of the vertical climbing velocity, the angle of attack decreases.

## Characteristic Helicopter Horizontal Flight Spe

The characteristic horizontal flight speeds define to a considerable degree the helicopter flight qualities. Calculation of these speeds, and then their verification in flight, is one of the important problems of helicopter aerodynamic design. The calculation of the characteristic speeds is most often made using the power method, first suggested by Zhukovskiy.

To construct the power required and available curves we use the curve of Figure 62. We find the power available at rated and takeoff engine rpm from the formulas

N.. = N I,
avail e

We use these values to plot a curve (Figure 63a), which permits deter­mining the following characteristic horizontal helicopter flight speeds:

Figure 63. Power required and available as a function

 of helicopter we ight and flight 1) N a for H = 0, n = 2200 rpm; 2) N a for H = 0, n – 2050 rpm; 3) Nh for H = 0, G = 2200 kgf; 4) Nh for H = 0, G = 2300 kgf.

1. The maximal speed, which corresponds to the point of intersection of the power required and available curves. When using rated engine power, the maximal horizontal flight speed at sea level will be about 165 km/hr; when using takeoff power, this speed will be 208 km/hr.

2. The optimal velocity at which the longest flight range is obtained.

This speed corresponds to the point of contact of a tangent drawn from the coordinate origin to the power required curve (for the Mi-1, V t = 90-95 km/hr),

3. The economical speed, i. e., the speed corresponding to minimal power required (for the Mi-1, Vgc = 80 km/hr).

4. The minimal speed. When using rated power at a flight weight of /94

2200 kgf, the helicopter cannot hover at sea level. It can only perform hori­zontal flight with a speed of about 20 km/hr. However, heavy vibration develops at low speed; therefore, the constructor has established speed limi­tations: minimal permissible is 40 km/hr, maximal permissible is 170 km/hr.

5. The horizontal flight speed range, i. e., the speeds at which horizon­tal flight is possible. When using rated power the speed range is

ду = V – V. = 165 – 20 = 145 km/hr.

rat max mxn

When using more than rated power, the total speed range is V t = 170 – 0 = = 170 km/hr.

6. The excess power, i. e., the difference between the power available and required for horizontal flight at a given speed. Each speed is associated with a given excess power AN = ^ – N^. The maximal excess power will

occur for flight at the economical speed.

## Landing

Landing is transitional flight from a height of 25 – 50 m with reduction of the velocity and subsequent touchdown. Helicopter landing may be performed in helicopter style, in airplane style, in the autorotative regime, from a glide along an inclined trajectory, and with flare-out.

The helicopter-style landing is the primary technique for landing with the engine operating. It includes the following stages (Figure 91):

(1) glide with reduction of the speed along the trajectory and vertical rate of descent;

(2) hover at a height 2 – 3 m above the landing area;

(3) vertical descent;

(4) touchdown.

During the landing approach the helicopter performs steady-state descent along an inclined trajectory with the engine operating. At a height of 40 – 50 m reduction of the speed along the trajectory is initiated while maintain­ing a constant descent angle. In this stage the motion of the helicopter is governed by the following conditions:

О > O’

77/’//У/У//77Ї7УУ’ "/7/ / 77^/~ґ7?~77^7/УУ777У/у///’//////////У/У///,

і О*

 Figure 90. Takeoff on air cushion.

Figure 91. Helicopter-type landing.

Y = G^ = G cos 0 (constant descent angle);

P + X > G_ (reduction of the speed); x par 2

T = S (absence of lateral displacement); t. r s

M = M and EM = 0 (constant direction of flight, i. e., absence of r t. r eg

rotation about the helicopter’s primary axes).

Deceleration of the helicopter is achieved by tilting the main rotor
thrust force vector aft and increasing the thrust component P. Upon reaching

a speed of 50 – 60 km/hr, the vertical rate of descent is reduced by increasing main rotor pitch and its thrust force. The helicopter deviates from the descent path angle and travels parallel to the surface of the ground at a height of 2 – 3 m. During this inertial motion the speed decreases to zero and the helicopter hovers above the landing area, orienting itself relative to the center of the area. If the landing approach was not made directly into the wind, the helicopter is turned about the vertical axis to take up a heading into the wind. Then a vertical descent is made at a low rate in order to avoid rough contact of the wheels with the ground.

The airplane-type landing is made under the same conditions as the take­off of the same type. It includes the following stages (Figure 92): glide

from a height of 15 – 30 m, flare, holdoff, touchdown, and rollout. During the descent altitude is lost, but a constant speed and descent angle are maintained. The descent is made with the engine operating. The flare is initiated at a height of 7 – 10 m by increasing the thrust and lift forces.

 touchdown and rollout. transition1 flare’ glide

Figure 92. Airplane-type landing.

The flare is terminated at a height of 1 – 0.5 m and the horizontal component of the velocity diminishes at this point, since the weight force component decreases to zero. After the flare the helicopter still has a relatively high speed, which is then reduced during the holdoff period. The

touchdown is made on the main gear wheels at a speed of 30 – 40 km/hr. In this procedure care must be taken that the tail does not get too low, since damage to the tail skid and the tail rotor can occur. The touchdown is followed by rollout, during which the main rotor thrust is decreased. An airfield or a smooth area with firm soil is necessary for the airplane-type landing.

Landing under special conditions. If the landing area is surrounded by obstacles, a helicopter-type landing is made without utilization of the "air cushion." Hovering is performed into the wind at a height of 5 – 10 m above the obstacles to align the helicopter with the center of the area. Then a vertical descent is made at a rate of descent of no more than 2 m/sec in order to avoid the vortex ring state. During this descent the vertical velocity of the helicopter is reduced to 0.2 – 0.3 m/sec at the moment of touchdown. Therefore, this type of landing can only he made if there is sufficient power margin available for hovering outside the "air cushion" influence zone. This type of landing is used only in case of extreme urgency, since a safe landing is not guaranteed in case of engine failure at a height of more than 10 m (in the danger zone).

Landing in the main rotor autorotation regime with glide along an inclined trajectory. We have established above that in case of engine failure flight in the main rotor autorotation regime should be made along an inclined trajectory rather than vertically.

Landing from a glide along an inclined trajectory is similar to the landing of an airplane and requires a smooth area with solid ground. It consists of the following stages (Figure 93):

(1) glide at constant angle and constant speed;

(2) deceleration (reduction of glide angle and vertical rate of descent through use of the kinetic energy of the helicopter and the main rotor);

 Figure 93. Autorotative landing.

(3) touchdown;

(4) rollout and reduction of main rotor pitch.

The advantage of this landing is that the helicopter has considerably less /147 vertical velocity and better controllability. In addition, during flight along the inclined trajectory the helicopter has considerable kinetic energy, which is used to reduce the vertical velocity prior to touchdown. Therefore, the landing is safer and simpler when the helicopter approaches in a glide along an inclined path. This type of landing utilizes the kinetic energy of the entire helicopter, as well as the kinetic energy of the main rotor.

During the glide a constant descent angle is assured by the condition

У — Gt = G CO? 0,

and constant velocity is assured by the condition (see Figure 92)

X + P. par x‘

The gliding speed is close to the economical speed for level flight, but it changes as a function of the wind velocity and direction. When gliding і nto the wind the helicopter speed should be higher, the higher the wind velocity.

The height for initiation of the deceleration or flare is different on different helicopters: the higher the disk loading, the higher this height.

For example, this height for the Mi-1 is 15 – 20 m; for the Mi-4 it is 25 – 20 m. Deceleration is obtained by tilting the main rotor coning axis aft. This leads to increase of the main rotor angle of attack and increase of the angle of attack of each blade element, which then leads to increase of the main rotor thrust force and its rpm. Therefore, both the glide angle and the velocity along the trajectory decrease. After the coning axis is deflected aft, the main rotor collective pitch must be increased to the maximal value. This leads to further increase of the thrust force and reduc­tion of the vertical rate of descent. The helicopter will travel for some time parallel to the ground surface, similar to the motion of an airplane during transition. The height at the end of this motion decreases to 0.5 – 0.3 m and the helicopter touches down with vertical velocity close to zero.

During the flare and transition the helicopter nose is high. Touchdown cannot be made in this attitude because of danger of damage to the tail rotor. Therefore, prior to touchdown the helicopter’s nose is lowered by deflecting the cyclic control stick forward. The increase of main rotor pitch leads not only to increase of the thrust force, but also to increase of the resis­tance to rotation; the rotor rpm decreases and the coning angle increases (blades flap up). If the rotor pitch is not reduced after touchdown, the rotor blade may descend abruptly and strike the tail boom.

Landing in the autorotative regime with vertical descent. If the landing /148 is made on a small area which is surrounded by obstacles, the landing must be made from a vertical descent. It was established earlier that the vertical rate of descent during flight in the autorotative regime along a vertical

trajectory is found from the formula = 3.6 tF and amounts to 14 – 20 m/sec.

Touchdown at such a velocity leads to damage to the helicopter and does not guarantee safety of the crew. Therefore, the velocity is reduced prior to touchdown by utilizing the kinetic energy of the main rotor. Such a landing is called a landing with flare-out. The essence of the operation is as follows.

At a height of 20 – 25 m the main rotor pitch is increased to the maximal

value, and the main rotor rpm should be as large as possible in order to

impart maximal kinetic energy to the main rotor (the kinetic energy is propor – 2

tional to to ). As the pitch is increased there is a marked increase of main rotor thrust flare-out, which leads to reduction of the vertical rate of descent to 3 – 5 m/sec. Such a velocity can be abosrbed completely by the gear shock absorbers, and a safe landing can be made.

The landing is easier when there is a wind. In this case the helicopter is turned into the wind and transitioned into the inclined glide regime (slope 45°). The effect of the wind is to carry the helicopter aft, and its trajectory relative to the ground will be nearly vertical. The helicopter controllability is better in this type of glide, and the thrust force is increased somewhat as a result of forward flight.

A vertical landing in the autorotation regime requires considerable skill and coolness on the part of the pilot. The following errors are possible in this type of landing:

(1) early flare-out (reduction of vertical velocity at a high altitude);

(2) late flare-out, as a result of which the vertical velocity is not reduced and hard contact with the ground may result.

It is clear from this discussion that the velocity is not entirely arrested in landing from a vertical descent. Even when the flare-out is

performed correctly, the final velocity may be quite large — 3-5 m/sec or more. This is explained by the fact that the helicopter, which has a rate of descent prior to flare-out of 15 – 20 m/sec, has considerable kinetic / C7V?

energy! s= —2^eSj • e4uiva3-ent kinetic energy of the main rotor must be

expended to arrest the vertical velocity completely. But the main rotor

kinetic energy is not entirely utilized for braking the helicopter. A large

part of the rotor energy is expended in overcoming the profile and induced

drags, on friction in the transmission, and on the blade end losses. Only

one fifth or sixth of the total kinetic energy of the main rotor is used for

deceleration. This means that the main rotor must have 5-7 times the kinetic /149

energy of the helicopter in order to decelerate the helicopter completely.

In reality the main rotor kinetic energy is about three times the helicopter kinetic energy.

Therefore, the vertical velocity cannot be arrested completely, but it must be reduced as much as possible, and to this end the main rotor rotational energy is increased; this energy is proportional to the blade mass and the square of the angular velocity of revolution

where is the rotational kinetic energy;

j is the main rotor moment of inertia;

9.b’L p~

JV •

Substituting the value of the rotational moment of inertia into the formula, we obtain

We see from the formula that heavier blades are required for safe landing in the autorotation regime.

CHAPTER XI

## Main Rotor Coning Axis Tilt

With variation of the flapping angles the plane of rotation and the coning axis deflect backward and to the side in the direction of the advancing blade through the angle т (Figure 38a). As a result of the tilt of the coning axis backward by the angle a^, there is an increase of the blade flap­ping angle to 8 = + a^ at the 180° azimuth and a reduction to 8 = a^ – a^

at the 0° azimuth (Figure 38b). Tilting of the cone axis to the side by the angle b^ leads to change of the flapping angles: at the azimuth 90° 8 = ад – b^; at the azimuth 270° 8 = a. Q + b^ (Figure 38c).

 Figure 38. Blade flapping motions and tilt of main rotor cone axis. 1 – cone for p = 0; 2 – cone for у > 0.

Tilting of the cone axis backward by the angle a^ leads to deflection /55

through the same angle of the thrust vector and the formation of the longitud­inal thrust component H (Figure 39a). This force is the projection of the main rotor thrust on the hub rotation plane. Since it is directed aft, it is a drag force and is analogous to the induced drag of an airplane wing. The

 Figure 39. Main rotor thrust force components.

larger the flapping motions, the larger the backward tilt of the cone axis and the larger the longitudinal force H resisting helicopter forward motion. Consequently, the flapping motions in the forward flight regime must be restricted.

If the deflected thrust T is projected on the hub axis, we obtain the force required for helicopter flight

Ty =.T cos au

In view of the smallness of the angle a^(2 – 3°) we can take a^ « 1. /56

Then T и T.

У

The sideward tilt of the cone axis (Figure 39b) leads to the appearance of the side force Sg, which is the projection of the main rotor thrust on the hub rotation plane

S = T sin b.• s 1

Since this force is directed to the left, this direction is unfavorable for single-rotor helicopters. Therefore, the blade flapping motions must be restricted in order to alter the sideward tilt of the cone angle from the

left to the right. Moreover, restriction of the flapping motions is also necessary to reduce main rotor vibrations.

## HELICOPTER DESCENT ALONG INCLINED TRAJECTORY

§ 54. General Characteristics of the Descent Regime /10{

Rectilinear flight at constant velocity along an inclined trajectory is termed the helicopter descent regime with operating engine. A characteristic of this regime is the possibility of controlling the vertical rate of descent and the speed along the trajectory by varying the power supplied to the main rotor.

In this regime the following forces act on the helicopter: weight,

main rotor thrust, parasite drag, and tail rotor thrust (Figure 70).

The helicopter motion takes place along a trajectory which is inclined to the horizon at the angle 0, termed the descent angle.

We resolve the weight force G and the main rotor thrust force T into components perpendicular and parallel to the flight trajectory. We obtain the weight force components G^ = G cos 0 and G^ = G sin 0. The main rotor thrust components will be the lift force Y perpendicular to the flight trajectory, and the force P parallel to this trajectory. The force P may be directed

X X

either opposite the helicopter motion direction or in the direction of this motion.

The direction of the force P^ depends on the position of the cone axis and the main rotor plane of rotation. If the cone axis is perpendicular to

 Figure 70. Forces acting on helicopter in descent.

the trajectory, then P = 0. If the cone axis is inclined aft relative to the perpendicular, then P^ will be directed opposite the helicopter motion and will retard this motion. If the cone axis is tilted forward, the force P^ will be directed along the motion and together with the component G2 will be a propulsive force. The cone axis direction is connected with the position of the rotor plane of rotation and, consequently, with the main rotor angle of attack. Most frequently, the main rotor angle of attack is close to zero or has a small negative value. During flight with a large descent angle, the angle of attack is positive and the force Px is directed opposite the motion.

Steady state descending flight is possible under the following conditions

The first condition assures rectilinear flight and constant descent angle. Consequently, by varying the lift force Y we can alter the helicopter descent angle. When the lift force is increased, the descent angle decreases, and vice versa. The second condition assures constant helicopter speed. Let us compare these conditions with those for climb along an inclined trajectory.

The first condition is the same for descent and climb. The second condi­tions differ fundamentally from one another: in climb, the propulsive force

is the main rotor thrust component P, while in descent this force will be the weight force component G^. The thrust force component P^ may be either a part of the propulsive force or a part of the retarding force, depending on the position of the main rotor cone axis. The third and fourth descent conditions are analogous to the same conditions for the other flight regimes.

## HELICOPTER VIBRATIONS

§ 79. General Analysis of Vibrations

Periodic reciprocating motions of the elements of an elastic system can be termed vibrations or oscillations. The problem of helicopter vibrations remained unresolved for a long time ; therefore, large-scale helicopter flying was not possible. Experimental flights performed prior to the middle 1940’s frequently terminated in accidents as a result of severe vibrations.

Several hundred different vibrations of individual parts and of the entire helicopter as a unit can he counted on a helicopter.

Parameters of oscillatory motions. We consider an elastic plate with one end clamped and a small weight on the other end (Figure 109a). If the end with the weight is deflected and then released, oscillations of the plate develop. This will be the simplest example of vibrations (Figure 109b). The oscillatory motions are characterized by three basic parameters: period,

frequency, and amplitude. The period is the time for a complete oscillation (T).

Frequency is the number of periods per unit time

/ if

Amplitude is the largest deviation of an oscillating point from the neutral position

(y).

Oscillatory motion modes. With regard to nature of onset, oscillatory motions can be excited.

Forced vibrations are those which are caused by periodic external forces. Such forces are exciting. Forced vibrations take place with a frequency equal to that of the exciting forces. Damping forces or forces which attenuate the vibrations arise during all vibrations. The damping forces may be either internal or external. The internal damping forces arise as a result of elasticity of the material itself from which the structure is fabricated. External damping forces arise as a result of resistance of the medium in which the vibrations take place. The larger the damping forces, the faster the vibrations decay.

Natural vibrations are those which continue after termination of the action of the disturbing forces. The basic characteristic of natural vibrations is that each structure has a very definite vibration frequency, which is independent of the exciting force and is determined by the mass and stiffness of the structure.

The larger the mass of the structure, the lower the natural vibration frequency. The greater the structure stiffness, the higher the natural vibration frequency.

With regard to nature of the amplitude variation, vibrations can be divided into damped and increasing. If the amplitude decreases, in the course of time, the vibrations will be damped. Natural vibrations are always

damped. If the amplitude increases with time, the vibrations will be increasing. Increasing vibrations develop at resonance.

Resonance is coincidence of the frequency of the exciting forces with /179

the frequency of the natural vibrations of the structure. Vibrations of helicopter parts are most often forced vibrations.

## OPERATION OF THE MAIN ROTOR IN THE AXIAL FLOW REGIME

During the rotation of the main rotor, a thrust force arises that creates lift and motion forces for the helicopter. The question of the origin of this thrust force is a basic question in the study of the main rotor operation. Several theories exist that explain the origin of the thrust force. We consider the physical aspects of two such theories.

§ 8. Impulsive Theory of an Ideal Rotor

In this theory an ideal rotor is considered — that is, a rotor that operates without losses. Such a rotor receives its energy from the engine, and all of it is transformed into work by displacement of the air mass along the axis of rotation.

If the rotation of the rotor in the hovering regime is considered — that is, when there is no translational motion of the helicopter and its speed is zero — the air is attracted by the rotor from above and from the sides (rotor induced flow) and it is deflected downwards (Figure 14). A flow of air is established through the area swept out by the rotor. The parameters of this flow are characterized by the inflow velocity (the speed of the flow in the plane of rotation and the main rotor), by the downwash velocity of the flow of VD (the speed of the flow at a certain distance from the plane of rotation of the main rotor), by the increase of pressure in the flow ДР, and by the change of speed along the axis of rotation.

 Figure 14. Operation of an ideal rotor according to impulsive theory.

By deflecting the air downwards with a force T, the rotor receives a force from the air in the upward direction (action equals reaction). This force will he the thrust force of the rotor. But from mechanics, it is known that a force equals the product of the mass of a body times the acceleration that the body receives under the action of the force. Therefore,

T = msa%

where m = mass of air per second, flowing through the area swept out by the rotor;

a = acceleration in the flow.

The mass of air per second is determined from the formula:

ms = pFVt,

where p = air density;

F = area swept out by the rotor;

= induced flow velocity (inflow velocity)

As is known, the acceleration is equal to the change of velocity in unit time, /17

dV dt »

where dV = increase of flow velocity dt = time

If we take dt = 1 second, the acceleration a is numerically equal to the velocity increase, that is, a = dV.

Let us clarify the value of dV. Considering the operation of the rotor in the hovering regime, it is not difficult to see (Figure 14) that the air at a certain distance from the rotor is stationary — that is, its velocity relative to the rotor is zero.

Beneath the rotor, the air moves at the inflow velocity, which means that the velocity increase dV = V^. Then utilizing Formulas (1) and (2) we obtain

1 ■ "s> ■ Vd ■ рРЇЛ • (3>

In order to arrive at a final conclusion, it is necessary to ascertain the relationship between the inflow velocity V. and the downwash velocity V^. We use the law of conservation of momentum: ‘The impulse of a force equals

the increase of momentum’.

It will be recalled that the impulse of a force is the product of force and time. If the time dt = 1 second, then the impulse of the force numeri­cally equals the force.

 The product of the mass of a body and the velocity increase is termed the increase of momentum: m dV = m V^. This means that, based on the fundamental /18 s s D ‘ — law of conservation of momentum, we obtain the thrust force;

 (4)

 T = mV. s D

 The work per second by the main rotor with respect to the downwash will have the value;

 N = TV.

 (5)

 But because we are considering an ideal rotor — that is, a rotor without losses — then, as a consequence, all of the work is changed into kinetic energy of the flow leaving the rotor. The kinetic energy is determined by the formula .2

 Ek =

 Utilizing Formula (4), we find;

 (6)

 -¥v

 Equating Formulas (5) and (6) on the basis of the theory of an ideal rotor, we obtain

 or 2V. = V_ • і D

 Thus Formula (4) finally takes the form

 2 FV^. P і

 T = FV. V = FV.2V. or T p і D p і і

The conclusion is that the thrust force, developed by the main rotor, is proportional to the air density, the area swept out by th – rotor and induced velocity squared. –

In order to determine on what the induced velocity dc ^.nds, it is necessary to consider another theory that explains the ori* 1 of the thrust force of the main rotor.

## Hovering Regime. General Characteristics

The helicopter hovering regime is that flight regime in which the velo­city equals zero. Hovering can be performed relative to the air and relative to the ground. If the air is stationary relative to the earth, i. e., the wind velocity equals zero (u = 0), the helicopter hovering relative to the air will be at the same time hovering relative to the Earth.

If the wind velocity is greater than zero, when hovering relative to the Earth (when the nose of the helicopter is pointed into the wind), it will perform flight relative to the air with the velocity of the wind. In this case the main rotor will operate in the forward flight regime. When the helicopter hovers relative to the air, the main rotor operates in the axial flow regime.

If during hovering relative to the air, there is a wind and the heli­copter’s nose is pointed into the wind, the helicopter will move backward with the velocity of the wind.

If in the presence of a wind the helicopter plane of symmetry is at an angle of 90° to the wind direction, the helicopter will displace to the side relative to the Earth (when hovering relative to the air) or relative to the air (when hovering relative to the Earth).

Hovering is performed in every flight during takeoff and landing. In addition, hovering is performed during unloading and loading when it is not possible to land (for example, over water, brush, rough ground, and in other such situations). Therefore, hovering must be performed relative to the Earth. In this case the pilot maintains the helicopter stationary relative to some point on the ground at a height of no more than 10 meters. Hovering at a height of more than 10 m and less than 200 m is hazardous, since in case of engine failure a safe emergency landing is not assured. Hovering at higher altitudes is performed only relative to the air, since the pilot cannot main­tain the helicopter stationary relative to the ground from a high altitude.

The helicopter speed relative to the air must not be less than that which can be indicated stably by the airspeed indicator meter (40 km/hr).

## Gliding

Rectilinear flight of the helicopter along an inclined trajectory with

the main rotor operating in the autorotative regime is termed gliding

(Figure 78). In this flight regime the helicopter is subject to the forces:

helicopter weight G, main rotor thrust T, parasite drag X, and tail rotor

p er

thrust T.

t. r

We resolve the helicopter weight force into two components: directed

along the flight trajectory, and G^ perpendicular to the trajectory.

We resolve the main rotor thrust force into the lift force Y and the drag force P.

The steady-state gliding condi­tions will be expressed by the equalities:

Y = G^ = G cos 0;

= X + P = G sin 0; par x

T = S ;
t. r s

V M = 0,

c. g

The first condition ensures constant gliding angle and rectilinear flight; the second assures constant speed along the trajectory. The tail rotor thrust is directed in the opposite direction in comparison with the thrust in the descent regime with the engine operating. The velocity along the trajectory and the gliding angle can be altered by tilting the thrust

force vector forward or aft, and also by varying the main rotor pitch. But we recall that flight takes place in the autorotative regime, and therefore the /127 pitch can be altered only within the permissible rpm limits. The pitch cannot be increased markedly, since the rotor may transition into the decelerated auto­rotation regime, and the rpm may become less than the minimal permissible value.

The main rotor autorotation conditions in a glide are much more complex than in a vertical descent. This is basically the result of two factors: azimuthal variation of the flow velocity over each blade element and the presence of blade falpping motions caused by transverse flow over the main rotor.

In a vertical descent each blade element has a constant velocity

In a glide this velocity depends on the blade azimuth and

changes continuously. In a vertical descent we can assume the absence of flapping motions, which simplifies the analysis of the blade element auto­rotative conditions. The flapping motions must be considered in a glide. But the derivations of the autorotation conditions which were carried out for the vertical descent remain valid for the gliding conditions as well. Let us recall these conclusions.

The autorotation conditions depend on the blade element pitch and the pitch of the entire main rotor: the lower the pitch, the greater the forward

tilt of the force vector AR and the higher the main rotor rpm.

The larger the angle of attack increment caused by the vertical descent velocity, the larger the forward tilt of the force vector AR and the higher the main rotor rpm.

The latter conclusion is particularly important in clarifying the autoro­tation conditions in a glide, therefore, we shall examine the diagram in Figure 79. We see from the figure that the angle Да is equal to the angle between the lines y-y and a-a (y-y is perpendicular to the resultant velocity vector, a-a is perpendicular to the plane of rotation, or parallel to the hub rotation axis). Consequently, if Да>0 , the force vector AR will be tilted forward relative to the hub axis by the angle – y, and the rotor autorotation will be accelerated. The larger Да, the higher the main rotor rpm.

Keeping this analysis in mind, we turn to examination of the auto­rotation characteristics in a glide.

We first examine the influence on the autorotation conditions of the azimu­thal variation of the resultant blade element velocity. Since the resultant velocity varies continuously in azimuth we cannot analyze this variation directly. Therefore, we take the two most characteristic azimuths, 90 and 270°, and we compare the autorotation conditions at these azimuths (Figure 80). The flight direction along the trajectory is shown in the figure by the arrow DF. The main rotor hub rotation plane is horizontal, the angle of attack of the main rotor lies

between this plane and the gliding velocity vector and equals the gliding angle A = 0. At the 90° azimuth in the hub rotation plane, the flow approaches the blade element with the velocity г.’1–!- У ros 0 (Figure 80a). The blade ele­ment angle of attack increment caused by the vertical descent velocity can be found from the formula

V. sin 0

gl _______

шг + V, cos 0
gl

At the 270° azimuth the blade travels aft relative to the direction of flight, therefore, the flow approaches the blade element in the hub rotation plane with the velocity шг – V ^ cos 0 (Figure 80b). In this case, the angle of attack increment is

V sin 0

_Jgl__________

шг – V, cos 0
gl

 Figure 80. Dependence of autorotative conditions on azimuth and flapping motions:

a, c – ф = 90°; b, d – ip = 270°

Comparing diagrams a and b, and also Formulas (43) and (44), we conclude that the angle of attack increment at the 270° azimuth is larger than at the 90° azimuth. Consequently, as a result of the variation of the resultant velocity in the gliding regime, there is a change of the angle of attack increment and tilt of the resultant aerodynamic force: at the 90° azimuth

the tilt is aft; at the 270° azimuth the tilt is forward.

Now let us examine the influence of flapping motions on the autorotation conditions.

As the advancing blade flaps up, the angles a and Да decrease. At the 90° azimuth, where the upward vertical flapping velocity reaches its maximal value (Figure 80c), the angle of attack increment becomes minimal

V sin 0 – T gl fl

шг + V.. cos 0
gl

Therefore, the maximal aft tilt of the force vector AR and the maximal blade retarding moment occur at this azimuth. The maximal down vertical flapping velocity will occur at the 270° azimuth. Therefore, in accordance with the formula

V і sin 0 + Jf. gl fl

юг – V, cos 0
gl

the blade element has the largest angle of attack increment (Figure 80d). The maximal forward tilt of the elemental force and the maximal turning moment will occur at this azimuth.

Thus, we draw the following conclusion. During gliding, the autorotation conditions of each blade element and the entire blade as a whole vary during a single revolution of the rotor. The advancing blade creates a retarding moment, which reduces the rotor rpm. The maximal retarding moment is created at the ф = 90° azimuth. The retreating blade creates a turning moment whose maximal value occurs at the 270° azimuth, where the angle of attack increment Да becomes maximal. This means that during a glide, the blades alternately accelerate and retard the rotation, and on the whole, the main rotor operates under steady-state autorotation conditions. The rotor rpm is regulated by the pitch: the lower the pitch, the higher the rotor rpm.

## MAIN ROTOR OPERATION IN FORWARD FLIGHT

§ 19. Characteristics of Main Rotor Operation in Forward Flight

We recall that the term forward flight refers to operation of the main rotor in an undisturbed stream which approaches the rotor nonparallel to the hub axis (see Figure 12c). While in the axial flow case, the rotor imparts to the air mass traveling along the axis additional momentum in the same direction, in the case of forward flight the rotor also imparts to a definite air mass some additional momentum, only this time not in the direction of the undisturbed approaching stream, rather in the direction along the rotor axis, downward. This leads to the appearance of the so-called downwash (Figure 27a).

The downwash magnitude is connected directly with the magnitude of the thrust which the main rotor develops in the forward flight regime.

In accordance with wing and propeller vortex theory, developed by Zhukovskiy in the 1905-1921 period, the wing lift and the main rotor thrust in the forward flight regime can be determined using the same formulas.

We imagine a stream of circular cross section, flowing past a wing (Figure 27b). The stream approaches the wing with the velocity V. As a result of the formation of the induced vortices, the wing imparts to the air /40 mass per second mg the vertical velocity u, termed the induced velocity.

Vortex theory shows and experimental aerodynamics confirms that there is a gradual increase of the induced velocity behind the wing.

 Figure 27. Operation of main rotor in forward flight regime.

At a distance equal to about 0.5Z (wing half-span) the induced velocity

reaches the value 2u (Figure 27c). Thus, the air acquires from the wing

additional momentum equal to m 2u.

s

The energy conservation law states that the momentum increase equals the impulse of the force. The impulse of the force per second will be simply the wing lift. Consequently,

Y — tns2a. (із)

Let us find the magnitude of the air mass flowrate mg. The stream section area F^, normal to the vector V^, equals the area of a circle of diameter equal to the wingspan l.

The velocity vector Yx=s]/ ц (V is the undisturbed flow velocity, and u is the induced velocity). Then

Substituting this value of the mass flowrate into (13), we obtain

Y=2?FnV1u.

Thus, the wing lift depends on the air density, wingspan, flight speed, and the induced velocity with which the wing deflects the stream downward.

From (15) we find the magnitude of the induced velocity /41

Г

U~ 2PFnVx ‘

Since the stream induced downwash angle is small, we can assume that

The downwash formed by the main rotor (see Figure 27a) is similar to the downwash due to a wing with span Ъ = D.

The air approaches the rotor with the velocity V and is deflected downward as a result of the induced inflow velocity V^. The resultant rotor velocity will be equal to the vector sum of the velocities of the undisturbed stream and the induced velocity

Vi^V + Vf.

The angle є between the vectors V and is the induced downwash.

Continuing the comparison with the airplane wing, we can say that the air mass flowrate ftls — pFiiV і passes through the area F normal to the resultant velocity vector Vl:. Since the rotor is taken to be a wing with span Ї = D, then

л-

i. e., the area perpendicular to the vector Vi,. will be equal to the area swept by the main rotor = F.

In the forward flight regime the downwash velocity is also equal to twice the inflow velocity. On this basis and using ideal rotor momentum theory, we find the thrust in the forward flight regime using (4)

T = m V, = m 2V.. s dw s x

Using (14), we can write

If F„ = F, then N

T = tyFViV,.

If we account for tip and root losses, this formula can be written in the form

T= 2xpFVxVt.

Consequently, main rotor thrust in the forward flight regime depends on air density, rotor pitch, and flight velocity.