# Category HELICOPTER AERODYNAMICS

## 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.

## Effect of Helicopter Weight and Flight. Altitude on Performance

With increase of the helicopter weight there is an increase of the power

* ^ ^

required for horizontal flight, since TV/ — —. Figure 63b shows curves of the power required for the Mi-1 helicopter for flight weights of 2200 kgf and 2300 kgf. In comparing these curves we can say that with increase of the flight weight:

 the maximal horizontal flight speed decreases; the minimal speed when using rated power increases; the economical and optimal speeds increase, although only slightly; the horizontal flight speed range decreases; the excess power decreases; hovering of the helicopter outside the air cushion influence zone is impossible even when using takeoff power.

 These variations of the helicopter flight characteristics should always be taken into account, particularly in those cases when a large fuel supply is carried. If the flight performance is based on takeoff weight, the values obtained will be too low. Therefore, if a large fuel supply is carried, the flight performance is based on the average flight weight with consideration for the fuel consumption

 Gfuel 2

 G = G av to

 /95

 where

 G is the average flight weight; cLV G is the takeoff flight weight; (Jfuel is the fuel weight (tanks completely full).

 Effect of flight altitude. The helicopter flight characteristics depend on the flight altitude and also on the air temperature and humidity. The air density decreases with increase of the altitude; therefore, the parasite drag decreases, as does the power required for motion

 where

Since the power required for motion has a large value at speeds above the economical speed, a change of flight altitude will have an effect on this speed

In studying the hovering regime, it was established that the thrust developed by the main rotor depends on the flight altitude, i. e., this thrust decreases with increase of the altitude, and this means that the lift force will decrease. But since the horizontal flight conditions specify that Y = G, it is necessary to increase the induced velocity V_^. Consequently, the induced power N. = GV. will increase in proportion to 1/Д, i. e., yV. = //, . The

і і *il *0 La

profile power changes very little with increase of the altitude.

Thus, with increase of the altitude the power required for motion decreases, while that required for creating the lift force increases. These conclusions are illustrated by the plot of power required for different altitudes (Figure 63c). This figure shows also how the power available varies with altitude.

For the supercharged engine the effective power increases up to the critical altitude and then decreases. As a result of this variation of the power available and the variation discussed above of the motion power and the induced power, we can say that with increase of the altitude up to the critical altitude:

1. For speeds lower than optimal, the power required for horizontal flight increases owing to the increase of the induced component of this power.

2. For speeds above optimal, the power required for horizontal flight decreases as a result of decrease of the motion power.

3. The magnitude of the optimal speed changes very little with change of the flight altitude.

г

4. The maximal and minimal horizontal flight speeds increase.

5. The excess power increases up to the engine critical altitude and then decreases.

Consequently, if flight must be accomplished at high speed, this should /96 be done at high altitude.

Increase of the air temperature is equivalent to increase of the altitude, since the air density decreases as its temperature increases. Increase of the air humidity leads to reduction of engine power and of the maximal horizontal flight speed.

All these conclusions are valid if we ignore the factors which restrict the maximal horizontal flight speed.

## HELICOPTER BALANCE, STABILITY, AND CONTROL. Helicopter Center of Gravity and Balance

The helicopter center of gravity is the point of application of its weight force vector. The center of gravity is the nominal point about which the helicopter rotates. The three principal axes of rotation (body coordinate system) passing through the helicopter center of gravity are used to charac­terize the rotational motions (Figure 94a). The 0 – x^ longitudinal axis lies in the plane of symmetry and runs along the fuselage parallel to the main rotor hub rotation plane. The 0 – z^ transverse axis passes through the center of gravity perpendicular to the plane of symmetry and is directed to the right. The 0 – y^ vertical axis passes through the center of gravity, lies in the plane of symmetry perpendicular to the longitudinal axis, and is directed upward.

If the external force acting on the helicopter passes through the heli­copter center of gravity, its moment will be zero and the helicopter will not have any rotational motion. If the external force passes outside the center of gravity, it creates a moment relative to some axis, under the influence of which the helicopter will rotate.

Of the cargo is attached rigidly to the helicopter the center of gravity does not move regardless of the attitude the helicopter assumes in the air.

If the cargo moves, the center of gravity will also move. Therefore, we need

to know precisely where the helicopter center of gravity is located. The center of gravity location is determined by balancing the helicopter. The helicopter balance point is the distance x from the main rotor huh axis to the center of gravity, expressed in millimeters, and the distance у from the center of gravity to the hub rotation plane (Figure 94b).

The distance x is the horizontal eg location and the distance у is the vertical eg location.

If the center of gravity is located ahead of the hub axis the eg is termed forward and denoted by +x.

If the center of gravity is located behind the hub axis it is termed aft and denoted by – x. Every helicopter has strictly defined eg travel limits. The forward eg limit is considerably greater than the aft limit.

For example, for the Mi-1 the

forward eg limit is +x. . = 150 mm,

lim

the aft limit is – x,. = -53 mm.

lim

The helicopter eg location must be known prior to every flight. The eg location changes with variation of

helicopter loading. The locations where the heaviest cargo is to be located is indicated in the operating manual for every helicopter. This manual also defines the sequence for finding the eg location, which amounts to the follow­ing. The basic helicopter weight (weight at a definite loading) and the basic eg location must be known. These data are presented in the helicopter specifications. Moreover, the weights and locations of the cargo must be

known. The distance from the main rotor hub axis to the cargo is measured in meters. The total moment about the main rotor hub axis is calculated and the new helicopter weight is determined as the sum of the basic weight and the weights of all the cargo. The new eg location is found from the formula

XM

X – £0′ •

Example of eg calculation of Mi-1 using the data:

(1) basic helicopter weight 1930 kgf;

(2) basic eg location 123 mm;

(3) eg limits +150, -53;

= 85 kgf (at distance – 1.2 m ahead of hub axis);

= 38 kgf (at distance = 1.4 m behind hub axis);

G^ = 105 kgf (at distance = 0.5 m ahead of hub axis);

(5) cargo G^ = 72 kgf removed, was located at distance = 0.6 m aft

of hub axis.

Solution. We find the moments of the basic weight and the weight of each cargo

Consequently, flight cannot be made with this eg location; the helicopter will be uncontrollable. Some of the cargo must be shifted aft.

Let us find how far the cargo G = 105 kgf must be moved aft to locate

° car

the eg at +150 mm.

Solution.

1. We find the moment required to shift the eg 34 mm

AM = G Ax = 2086-0.034 =70.9 kgf. new

2.

We find the distance which the cargo must be shifted

That state of the helicopter for which it travels in a straight line with constant velocity and does not rotate about its principal axes (about the center of gravity) is called equilibrium.

The equilibrium conditions follow from the definition. According to Newton’s first law, a body moves uniformly and rectilinearly if no external forces act on it. Therefore, it is necessary that the sum of the forces acting on the helicopter be equal to zero

ZF =0. eg

The second equilibrium characteristic (absence of rotation) will hold if the sum of the moments of the forces acting on the helicopter equals zero

EM =0. eg

Moments relative to the о – z transverse axis are termed longitudinal

(M ). Under the action of this moment the helicopter pitches up (nose rises) z

or pitches down (nose descends). The moments about the 0 – x^ longitudinal axis are termed transverse or rolling moments (M^). The moments about the 0 – y^ vertical axis are termed directional (M^). A general remark on the sign of the moments: a positive moment causes clockwise helicopter rotation

if we look along the direction of the axis.

Equilibrium of the helicopter exists in all the steady-state flight regimes. The steady-state flight conditions, which we examined previously, are the equilibrium conditions written in expanded form. It is true that these conditions were written in application to the velocity coordinate system. The velocity or wind coordinate system is a system fixed with the flight velocity vector. In this system the longitudinal axis is denoted by 0 – x and coincides in direction with the velocity vector (see Figure 94a). The angle between the axes 0 – x^ and 0 – x of the body and wind coordinate systems is equal to the main rotor angle of attack A. The angle between the longitudinal axis of the velocity coordinate system and the helicopter plane of symmetry is called the sideslip angle. If the flight velocity vector is in the plane of symmetry, the sideslip angle equals zero. In the absence of sideslip, the transverse axes of the body and velocity coordinate systems

coincide. The angle between the vertical axes 0 – y^ and 0 – у of the body and velocity coordinate system equals the angle of attack of the main rotor.

We take for example the conditions for horizontal helicopter flight

We see from these equalities that the sum of the forces acting on the helicopter along the vertical, longitudinal, and transverse axes of the velocity coordinate system equals zero.

Consequently, these three equalities express the first equilibrium

condition EF =0. The fourth horizontal flight condition (EM = 0) eg eg

expresses the second equilibrium characteristic, i. e., the absence of rotation about the center of gravity.

## Blade Flapping Motion Restriction and. Flapping Compensator

The blade flapping motions are limited by the action of the centrifugal force moment. Moreover, the flapping motions themselves create aerodynamic limiting of these motions. The essence of the limiting amounts to the follow­ing. As the blade flaps upward (Figure 40a), the blade element angle of attack is reduced by – Да as a result of the vertical flapping velocity V^, which leads to reduction of the blade thrust and moment and, consequently, to more rapid restoration of equilibrium about the horizontal hinge. When the blade flaps downward (Figure 40b), the angle of attack increases, which leads to increase of the thrust and limitation of the downward flapping motion.

But the restriction of the flapping motions as a result of centrifugal forces and aerodynamic limiting is not sufficient. Therefore use is made of the so-called pitch control arm compensation or flapping compensator.

The essence of the flapping compensator lies in a special positioning of the blade pitch control elements. It was established earlier that the blade pitch (incidence angle ф) changes with rotation of the blade about its longitudinal axis. Blade rotation is accomplished with the aid of the axial hinge, on the body of which there is the "blade pitch horn" lever. The /57

vertical rod from the main rotor tilt control is connected to the blade horn arm. Connection of the cyclic control rod with the blade horn is accomplished by means of the pitch horn hinge.

If the tilt control rod moves upward, the blade incidence angle is increased (Figure 41a).

A<x

Blade element angle of attack change.

If the control rod moves downward, the blade incidence angle is reduced (Figure 41b). The location of the horn arm hinge relative to the main rotor hub horizontal hinge is of fundamental importance. It may be located on the axis of the horizontal hinge (Figure 41c) or it may be shifted relative to this axis by the distance a (Figure 41d).

In the first case, during flapping motion the horn rotates together with the blade about the horizontal hinge axis so that it does not hinder blade rotation. The blade chord displaces parallel to itself and the incidence angle remains constant.

In the second case, the shift of the horn hinge relative to the horizontal hinge axis leads to change of blade pitch during flapping motions. Thus, when the blade flaps upward the horn hinge, remaining stationary, holds back /58 the blade leading edge, i. e., it causes reduction of the pitch (Figure 4ld).

When the blade flaps downward its pitch increases. This sort of pitch change leads to limiting of the flapping motions. For example, when the blade moves upward the blade pitch is reduced and its thrust and moment are also reduced. Therefore, equilibrium is restored more rapidly and the flapping angle is reduced.

When the blade flaps downward, the increased pitch leads to increased thrust and the flapping angle is restricted. The effectiveness of the flapping compensator action depends on the flapping compensation coefficient K.

The compensation coefficient is the ratio of the distance a between the pitch horn hinge and the horizontal hinge axis to the distance b between the pitch horn hinge and the blade longitudinal axis (Figure 41d).

a

T*

The larger the compensation coefficient, the larger the blade pitch change with variation of the flapping angle and, consequently, the more the blade up and down flapping is restricted.

For most helicopters the compensation coefficient is about 0.5.

By increasing the compensation coefficient we can limit the increase of

the flapping angle for the advancing blade to a point where maximum flapping will not occur at the ip = 210° azimuth, as we have noted above, but rather at the ip % 160° azimuth. In this case, the minimal flapping angle of the retreat­ing blade will occur at the ip ^ 340° azimuth. With this change of the flap­ping angles, the main rotor cone axis will be deflected aft and in the direc­tion of the retreating blade, and the side force will be directed to the right.

## Thrust and Power Required for Helicopter Descent

The thrust developed by the main rotor during flight along an inclined

trajectory must provide the necessary magnitudes of the lift force Y and its

component P parallel to the motion trajectory. In accordance with the dia – X

gram of the forces acting on the helicopter, the thrust force f = ]/"У2 – j – p*..

From (33) we find Y = G = G cos 0, and from (34) +P = G~ – X = G sin 0 –

1 – x 2 par

– X, then par

Since in most cases the descent angle 0 is small (less than 10°) cos 0^1. This means that the first term of the radicand in (35) is close to one and the second term is close to zero. Hence, we can conclude that the thrust required for helicopter descent along an inclined trajectory will be practically equal to the helicopter weight. For large descent angles, when the angle 0 approaches

90°, the first term of the radicand of (35) approaches zero, and the difference

о»

G sin 0 – X r approaches in magnitude the weight, i. e., T ^ G.

We came to the same conclusion in studying helicopter vertical descent. Comparing the thrust force required in the different helicopter flight regimes, we can say that the thrust required for flight in any regime is practically equal to the helicopter weight.

The power required for descent along an inclined trajectory consists of three parts: the motion power N , the power required to create the lift

force or inductive power N^, and the power required to overcome the profile drag If during descent the velocity and rpm are the same as in horizontal

flight, the profile power is the same in both cases. The induced power is found from the formula = YV^ = G cos 0 V^, and for descent angles up to 10° is practically equal to the induced power for horizontal flight, since in this

<Vi

case cos 9 ^ 1.

The motion power for descent along an inclined trajectory is found from the formula

N „ = P V.
mot x

par

The parasite drag forces are practically the same for descent and horizontal flight (for the same speed). Therefore

X V = N. par mot.

We denote G^V = ДЫ; then

i. e., the power required for descent is less than the power required for horizontal flight. Comparing (28), (32), (37), we can say that the most power is required for climbing flight and the least is required for descending flight The speed dependence of the power required for different flight regimes can be shown graphically with the aid of the Zhukovskiy grid, or by curves of the power required for various flight regimes, which are plotted for a given altitude (Figure 71a).