# Category HELICOPTER AERODYNAMICS

## Self-excited Vibrations

Self-excited vibrations are those which arise under certain conditions

when constantly acting forces are transformed into periodic forces and a steady motion becomes oscillatory. To these conditions we must also add coin­cidence of the periodic force frequency with the natural vibration frequency.

There are three characteristic forms of self-excited vibrations in heli­copters: "ground resonance," "helicopter auto-oscillations in flight," and

vibrations of the flutter type.

The combination of main rotor blade oscillations relative to the vertical hinges with oscillations of the entire helicopter as it moves over the ground can be termed ground resonance. The amplitude of these oscillations increases very rapidly.

Vibrations of the ground resonance type are not observed on helicopters having main rotors without vertical hinges. In this case the blades are positioned symmetrically, and the center of gravity of the entire main rotor is located on the hub axis. As the main rotor turns the circumferential velocity of the center of gravity equals zero; consequently the main rotor centrifugal force also equals zero.

When vertical hinges are used, the blades perform oscillatory motions /182

as a result of change of the moments of the rotational drag force and the Coriolis force. Such oscillations lead to shift of the main rotor center of gravity away from the hub axis. The center of gravity begins to travel along a sinuous curvilinear trajectory (Figure 112a). The main rotor centrifugal force N appears. The appearance of this force can be explained in a different way. If the main rotor blades are positioned symmetrieally relative to the hub, the resultant of the centrifugal forces of the blades taken individually equals zero. If the blades are positioned asymmetrically, the resultant of the blade centrifugal forces will be the centrifugal force of the entire main rotor.

Under the action of this force a moment is created relative to the landing gear wheel support point, which causes alternate deflection of the gear shock struts and tires (Figure 112b). Rocking of the helicopter on the gear develops. If the frequency of these (natural) oscillations coincides with the main rotor rpm, resonance occurs. The amplitude will increase

 г

 7??/Z^7777777//?77/7.

 I

Figure 112. Occurrence of "ground resonance."

rapidly. As a result of tipping of the helicopter, moments of the blade weight forces about the vertical hinges develop. These moments amplify the oscillatory motions of the blades and lead to increase of the centrifugal force of the entire main rotor. The increase of the oscillation amplitude can lead to overturning and destruction of the helicopter. Moreover, the vibrations are amplified by the action of the gyroscopic moment of the tail rotor. The tail rotor turns at high speed and has a large gyroscopic moment.

It tends to retain its axis of rotation in a fixed position. During oscilla­tions of the helicopter the position of the tail rotor axis of rotation changes, and therefore large torsional moments develop in the fuselage tail boom and vertical fin. The actual picture of ground resonance is more complex and depends on many factors other than those examined here.

Ground resonance develops most frequently when taxiing the helicopter over rough ground, during takeoff roll and landing runout when making takeoffs /183 and landings of airplane type. But ground resonance can also occur when the helicopter is parked with the main rotor running. The basic operational causes are the following:

Low or different tension of the vertical hinge friction dampers;

Incorrect charging of the landing gear shock struts and tires.

The first factor leads to increase of the oscillatory motions of the blades and the second leads to change of the gear stiffness. As a result of the stiffness change, there is a change of the helicopter natural vibration frequency, and conditions for resonance are created.

When resonance occurs, the main rotor rpm must be decreased and the engine shut down.

Helicopter auto-oscillations in flight. Helicopter auto-oscillations are similar in nature to ground resonance. These vibrations combine oscilla­tions of the main rotor blades relative to the vertical hinges and oscillations of the elastic elements of the helicopter fuselage. During the blade oscilla­tions a centrifugal force of the main rotor arises, and this leads to whipping of the shaft and deformation of the members of the frame supporting the gear­box and the structural elements of the fuselage.

Auto-oscillations occur very rarely on single-rotor helicopters; only in case of failure of individual structural elements of the fuselage, which results in reduction of the fuselage stiffness, or when the vertical hinge dampers are out of adjustment. These vibrations are observed more frequently in the dual-rotor helicopters with tandem arrangement of the lifting rotors.

The bending stiffness of the fuselage of this helicopter in the horizontal plane is comparatively low. If there are large oscillations of the blades of both rotors, large bending moments are created, which as they change their direction cause marked bending vibrations of the fuselage.

Rotor blade flutter. Vibrations of the flutter type are the most hazardous. They are encountered on the main rotor blades and are theoretically possible on the tail rotor blades, but in view of the high stiffness of the latter they are not encountered in practice.

Main rotor blade flutter may be of two types: bending-torsion and flap­

ping. Bending-torsion flutter in the pure form is observed most frequently for blades with rigid restraint at the hub. Blades with hinged support usually show the combined type of flutter.

In accordance with this theory, each element of the blade is considered as a small wing, which moves in a circular trajectory with speed u = 0)r (Figure 15a). If the profile of the blade were symmetrical and the incidence angle ф = 0, there would be no deflection of air downwards, and and T would be equivalent to zero.

For an asymmetrical profile and ф > 0, the airflow approaching the blade element is deflected downwards. This deflection, and as a consequence, the induced velocity will be larger, the larger the incidence angle of the element, and the greater the angular velocity or the rotation of the main rotor (Figure 15b).

Adding the vectors of circular and induced velocity, we obtain the result – /19 ing vector w = u +

The angle a between the chord of the blade element and the resultant velocity vector is termed the angle of attack of the blade element. The aerodynamic forces, arising from the main rotor blade, depend on this angle.

Examining the spectrum of the streamlines around a blade element, it can be observed that the streamlines have the same form as the spectrum of a wing (Figure 15c). On this basis we can state that the air pressure on the blade upper surface will be less than on the lower surface. Owing to the

24

 Figure 15. Development of thrust force according to blade element theory

difference in pressure, there arises an element of thrust force AT (Figure 15c). If all the elements of force are summed, we obtain the thrust force of the entire rotor

T = Tg к

where к = number of blades

T – thrust of blade; T = EAT

в в

To determine the force of the main rotor, it is possible to utilize the formula for the lift force of a wing:

T=CTF-t-(o>R) (8)

where CT = thrust coefficient.

Because uiR = u (u is the circular velocity of the blade tip), it is pos­sible to write the formula for the thrust force in the following form:

T=CrFu?. (9)

The conclusion is that the thrust force of the main rotor is proportional to the thrust coefficient, the area swept out by the rotor, air density, and the square of the circular velocity of the blade tip.

For a given rotor at a constant air density, the thrust depends on the /20 number of revolutions and the thrust coefficient. The thrust coefficient depends on the pitch of the rotor (Figure 16).

The conclusions that have been outlined according to "impulsive theory" and "blade element theory" do not contradict each other, but are mutually supplementary. On the basis of these conclusions, it is possible to state that, in order to increase the thrust force of the main rotor, it is necessary to increase the pitch or the revolutions, or both of them at the same time.

Besides the thrust force, the rotation of the rotor gives rise to forces that resist rotation. We will consider these forces in the next section.

## Diagram of Forces Acting on Helicopter and Hovering Conditions

In the further study of the hovering regime we examine helicopter hovering relative to the air with the main rotor operating in the axial flow regime.

In order to avoid complicating our understanding of the question, we shall assume that the wind velocity is zero.

During hovering, it is necessary to observe the general conditions which /75 characterize any steady-state flight regime, i. e.,

£F =0 and ZM = 0. eg eg

The following basic forces act on the helicopter during hovering (Figure 53a):

 Figure 53. Forces acting on helicopter in hovering regime.

helicopter weight G;

main rotor thrust T;

tail rotor thrust T ;

t. r

parasite drag X

The parasite drag force arises as a result of air flow from the main

rotor over the fuselage and other parts of the helicopter. This force is

very small and amounts to about 1-2% of the helicopter weight. The main

rotor thrust increases somewhat as a result of the air flow over the fuselage,

which means that the effect of the force X decreases, and it can hereafter

par

be neglected.

The main rotor reactive moment during hovering is balanced by the tail rotor thrust moment = T L. This is necessary to prevent the helicopter from turning about the vertical axis. But in this case the unbalanced force T r acts on the helicopter and the helicopter displaces to the side. To prevent lateral displacement it is necessary to balance the tail rotor thrust by a force directed oppositely. To this end the main rotor thrust vector is deflected to the side opposite the direction of the tail rotor thrust. For helicopters with right hand rotation of the main rotor (as seen from above) the tail rotor thrust is directed to the left (Figure 53b). As a result of tilting of the cone axis to the right through the angle 6, the main rotor side thrust develops 110

Ss — Г sin 8,

which balances the tail rotor thrust. The vertical component Y=TCOS d of the main rotor thrust will be balanced by the helicopter weight.

The angle 6 does not exceed 3-5°. And since COs5°!^i, we can say with /75 adequate precision that У Я# T. Thus, the helicopter hovering conditions are expressed by the equalities

Y = G or Y – G = 0,

Since there are no forces acting along the helicopter longitudinal axis in the hovering regime, EM = 0 is assured.

In view of the equality Y~T we can write the first hovering regime condition in the form T = G. We shall use this equality hereafter. There­fore, for helicopter hovering it is necessary that:

T = G (constant hovering height);

T = S (absence of lateral displacement);

L » Г S

EM = 0 (absence of rotation about the center of gravity).

The hovering regime is a characteristic flight regime and defines to a considerable degree the helicopter’s flight characteristics.

Example. A helicopter is hovering. G = 2200 kgf, = 575 hp, t, = 0.78,

n = 249 rpm, L = 8.65 m. Find: N, M, T, and S.

t. r ’ r t. r

Solution. 1. The power expended in rotating the main rotor is defined with account for the power utilization coefficient

N = N£ = 575 XO 78 = 450 bp.

2. From (11) we find the main rotor reactive moment M = r.

r —

Ш

The main rotor angular velocity is found from the formula

-i2ff-150k8£-

t. r

4. The tilt of the thrust vector is found from the equality

150

2200

arcsin 0.068 = 4°,

## Vertical Rate of Descent in a Glide

The vertical rate of descent is the altitude through which the helicopter descends per second (Figure 81). This rate is found from the formula

V і sin 0. gl

The vertical velocity will be constant for constant glide angle and constant velocity along the trajectory. This means that the propulsive force in a glide is the weight force component = G sin 0. The work of this force per unit time will be power, equiva­lent to the power required for horizontal flight at a velocity equal to the gliding velocity. Consequently,

G„V = N, = G sin 0 V. 2 gl h gl

 or

 Nh =

 (45)

This means that the work per unit time of the helicopter weight force will be equivalent to the power supplied to the main rotor shaft in horizontal flight at the same velocity with which the helicopter glides in the autorota­tive regime. From (45) we can find the vertical rate of descent during gliding

 75N __ h G

 (46)

is the power required for horizontal flight, hp; is the helicopter weight, kgf.

We see from (46) that the minimal vertical rate of descent will be achieved

when gliding at the economical speed, since Vgc corresponds to the required

power N. , i. e.
min

75N.

 G

For rough calculations we can use the approximate formula = l/2Nrat,

then we must introduce the power utilization coefficient since = N £ Taking the value of this coefficient to be? = 0.8, we find

We note that the ratio G/N = q is called the power loading, and the equality takes the form

‘a

des . q.

mm rat

q I’P = E., is the helicopter energy efficiency; ratr M

We substitute this value into (48)

30

des . E

mm M

The average value of the energy efficiency is20 • Then the formula for determining the minimal helicopter gliding vertical rate of descent will be

V, = у? =

des . 20

mm

We recall that this formula is approximate, but it yields adequately precise results, although somewhat low.

Comparison of the vertical rates of descent in glide and vertical descent makes it possible to say that the vertical velocity in the glide will be
2-2.5 times less than in the vertical descent. Therefore, gliding is used in all cases if surrounding obstacles do not interfere.

However, gliding can he performed with other velocities along the trajec­tory rather than the economical velocity. A special graph — the helicopter glide trajectory polar curve — is constructed for determining the gliding velocities and angles. We use the power required and available curves for horizontal flight to construct this graph. If we draw on these curves a straight line parallel to the horizontal axis, it crosses the power required curve at the points A and В (Figure 82a). The point A corresponds to the horizontal flight velocity V^,and the point В corresponds to the velocity 4 .

We take these velocities as the horizontal components of the gliding velocity (Figure 82b): V1 = V ; V„ = V. Since these velocities correspond

to one and the same power required for horizontal flight, we use (46) to find the vertical descent velocity. It will be the same for gliding with the velocities V and V. After determining V, and V, we find the gliding

Xj X£ QcS X

velocity

Then we find the gliding angle

Conclusion: the same vertical rate of descent corresponds to two gliding

regimes with large and small gliding angle, with high and low velocity along the trajectory. We usually select gliding with the lower angle but the higher velocity along the trajectory. After making these calculations for several vertical velocities, we plot the helicopter glide polar (Figure 83). From this plot we can find:

— the minimal vertical rate of descent;

the minimal gliding angle which corresponds to the velocity V , numerically equal to the optimal horizontal flight speed;

— the vertical velocity in a vertical descent (point of intersection of the polar curve with the vertical axis).

If we draw through the polar a secant parallel to the horizontal axis, the points of intersection 1 and 2 will correspond to two gliding regimes, for which the vertical rate of descent will be the same and the velocities along the trajectory will be different. The point M on the polar corresponds to the minimal velocity along the trajectory.

## Main Rotor Thrust as a Function of Flight Speed

The thrust of a particular rotor at constant air density depends on the flight speed and the induced velocity. With increase of the flight speed there is an increase of the resultant velocity, which leads to increase of

 Figure 28. Main rotor induced velocity and thrust versus speed flight.

the mass flowrate of the air deflected by the main rotor. Consequently, the higher the velocity in the forward flight regime, the larger the air mass flowrate and the greater the thrust developed by the rotor. But thrust increase is possible only up to some limit. This is associated with the change of the induced velocity which, in turn, depends on the flight speed. However, this relation is complicated by the variation of the main rotor angle of attack (Figure 28a). This figure makes it possible to draw some important conclusions:

the induced velocity decreases with increase of the flight speed;

with increase of the main rotor angle of attack the induced velocity increases and vice versa;

for negative angles of attack the induced velocity decreases with increase of the flight speed;

for A > 0° the induced velocity first increases with increase of the flight speed up to 15-20 km/hr and then decreases;

for flight speeds up to 50-60 km/hr the induced velocity depends to a considerable degree on the main rotor angle of attack, while at higher

flight speeds this dependence becomes less significant;

the induced velocity decreases very rapidly with flight speed in the range from 0 to 60-70 km/hr.

With further increase of the flight speed, the reduction of the induced velocity becomes more gradual.

These conclusions are necessary for understanding the nature of main rotor thrust variation in the forward flight regime, and also for under­standing the nature of helicopter motion in horizontal flight, climb, and descent along an inclined trajectory. If we take into account the nature /43 of the induced velocity variation, then the variation of main rotor thrust with change of the flight speed becomes clear (Figure 28b). This figure shows that main rotor thrust increases with increase of the flight speed and reaches the maximal value for a speed of about 100 km/hr. All the conclu­sions drawn on the variation of the induced velocity and thrust relate to operation of a main rotor with constant power expended in turning the rotor.

The thrust increase with increase of the flight speed is explained by the fact that, as the flight speed increases, a larger amount of air approaches the rotor, i. e., the mass flowrate of the air interacting with the rotor in­creases. The rotor deflects the large air mass downward and, thus, force impulse increases, i. e., the main rotor thrust increases.

Upon further increase of the flight speed, the time of interaction of the rotor with the air diminishes. The rotor "fails to" deflect the air markedly downward, which means a decrease of the induced velocity and, therefore, of the force impulse. Moreover, the energy received by the rotor from the shaft is expended not only in creating the induced velocity, but also in overcoming frictional drag forces, and with increase of the flight velocity these forces increase.

## Factors Limiting Maximal Horizontal Flight Speed. and Ways to Increase This Speed

It was established above that to increase the flight speed we must increase the angle у of deflection of the thrust force vector (see Figure 60) and the magnitude of the main rotor thrust force. The thrust force can be increased in two ways: by increasing the angular velocity CO; and by increasing the

main rotor pitch, since this leads to increase of the thrust coefficient c^

(see Figure 16).

Increase of the main rotor thrust force by increasing the rpm involves increase of the blade drag profile, and consequently increase of the fuel consumption per unit rotor thrust developed. This approach is not advisable. Moreover, increase of the thrust by increasing the rotor rpm is possible only up to a definite limit.

It is well-known that increase of the angular velocity increases the circumferential velocity u = U)R. In the forward flight regime the resultant

blade element velocity is W = u + V sin ip, i. e., at the azimuth 90° W = u + V.

With increase of the rpm there is an increase of the circumferential velocity and the flight velocity; consequently, the resulting velocity W will increase. Increase cf the resultant velocity of the flow past the blade elements is permissible only until the velocity reaches the critical value, i. e., until the appearance on the blade of a local flow velocity equal to the speed of sound (compression shocks develop, and blade shock stall manifests itself).

Thus, the first technique for increasing main rotor thrust by increasing the rpm is limited by the appearance of shock stall at the tip of the blade when the blade is located at the 90° azimuth.

Let us examine the second approach. Increase of main rotor thrust by increasing the pitch and the coefficient c^ involves increase of the blade element angles of attack. It is well-known that in the forward flight regime the blade element angles of attack vary in azimuth: the angles of attack are

smallest at the 90° azimuth, and they are largest at the 270° azimuth. The higher the flight speed, the larger the angles of attack of the blade tip elements at the 270° azimuth. Increase of the pitch leads to further increase /97 of the angles of attack. If the angles of attack at the tip elements approach the critical value, flow separation analogous to the separation from an airplane wing is formed at the end of the blade (Figure 64).

Moreover, with increase of the flight velocity the reverse flow zone expands; in accordance with the formula d = yR, the diameter of this zone increases with increase of the flight speed at constant rpm. Expansion of the stall separation and reverse flow zones leads to reduction of the main rotor thrust force and causes severe roughness. Flight cannot be continued under separated flow conditions, therefore, the flight speed can be increased only until the angles of attack of the blade tip elements become close to the critical value. This flight speed limitation is called the 270°-azimuth blade stall limitation. Thus, the maximal speed for the Mi-1 helicopter is limited to 170 km/hr, while the Mi-4 is limited to 175 km/hr.

 Figure 64. Flow separation from blades with increase of flight speed.

If these limitations did not exist, the power available would permit a maximal speed at altitude of 205 km/hr for the Mi-1 and a speed of 225 km/hr at an altitude of 1500 m for the Mi-4.

## Helicopter Equilibrium in the Hovering Regime

The conditions of helicopter equilibrium in the hovering regime can be applied (with some additions) to any flight regime.

Hub horizontal hinge offset. Most modern helicopters have offset of the main rotor hub horizontal hinges, i. e., separation between the hub axis and the horizontal hinge axis, which is denoted by (Figure 95a). Horizontal

hinge offset has an effect on helicopter equilibrium, stability, and controllability conditions.

The centrifugal forces acting on the rotor blades are transmitted to the horizontal hinges. When the main rotor plane of rotation is parallel to the hub plane (no tilt of the coning axis), the blade centrifugal forces are in

a single plane and their moment relative to the center of the hub equals zero.

If the main rotor coning axis is deflected from the hub axis, the main rotor plane of rotation will not be parallel to the hub rotation plane. The blade centrifugal forces act in a plane parallel to the main rotor plane (Figure 95b).

If the horizontal hinges are offset, there will be an arm c between the centrifugal forces; therefore, these forces create the moment = Nc relative to the center of the hub. This moment rotates the main rotor hub and consequently the entire helicopter so as to make the hub axis approach the coning axis.

Equilibrium of the helicopter relative to the principal axes of rotation can be subdivided into longitudinal, transverse, and directional. Common to all these modes is the first equilibrium characteristic: uniform rectilinear

motion or, as a particular case, relative rest in the hovering regime (V = 0). Therefore, in the definitions of the equilibrium modes we omit the first characteristic, assuming that it holds.

Helicopter longitudinal equilibrium is that state of the helicopter in which it does not rotate about the transverse axis. Since the velocity in the hovering regime equals zero, there will be no forces parallel to the helicopter longitudinal axis. Then the first equilibrium characteristic is

expressed by the two equalities

The second equilibrium condition is that the sum of the longitudinal moments must equal zero: EM^ = 0.

Longitudinal moments are created by (Figure 96):

main rotor thrust force (M^ = Та);

stabilizer lift force (M. = Y L.);

st st st

horizontal hinge moment (M^ = NC);

tail rotor reactive moment де.

r

t. r

If the eg location is far forward the helicopter will hover with the nose down (Figure 96a). In this case the main rotor thrust force moment will be nose-down; the moments of the horizontal hinges, stabilizer, and the tail rotor reactive moment are nose-up. Therefore, the second longitudinal °quilibrium characteristic is expressed by the equation

or

NC + Y. L. 4 M – Та = 0. st st r

t. r

To satisfy this condition the main rotor coning axis must be tilted aft. The more forward the eg, the larger the coning axis deflection angle л must be.

If the eg moves forward beyond the limiting forward position, hovering longi­tudinal equilibrium cannot be achieved.

 Figure 96. Helicopter longitudinal equilibrium in hover.

If the helicopter eg is aft or slightly forward, the helicopter will hover with the nose high (Figure 96b). In this case the main rotor coning axis must be deflected forward relative to the hub axis through the angle n. The moment of the main rotor thrust force may be positive, negative, or zero. The moment of the stabilizer and tail rotor, as in the first case, will be nose-up. The second equilibrium characteristic will be expressed by the equality

Mst+Mr ±“1-^ = °.

t. r

If the helicopter eg moves aft beyond the permissible limit, the main rotor coning axis is deflected full forward. In this case the helicopter cannot be transitioned into the horizontal flight regime, and if this cannot be done the flight speed cannot be increased to the maximal value.

Helicopter hover in the horizontal attitude is possible with a slightly forward eg location. In this case the main rotor coning axis will coincide

with the hub axis. The moment of the horizontal hinges will be zero. The main rotor thrust force will be nose-down (Figure 96c). The longitudinal equilibrium condition is expressed by the equality

Longitudinal equilibrium of the helicopter in other flight regimes as a function of eg location can be expressed by one of the equalities discussed above. However, to the terms of these equalities we must also add the helicopter parasite drag force moment, which will generally be a climbing moment.

Helicopter transverse equilibrium is that state of the helicopter in which there is no rotation about the longitudinal axis. The transverse equilibrium conditions are expressed in general form by the following equalities: = 0, no side displacement, this equality expresses the first

transverse equilibrium characteristic; £Mx = 0, no rotation about the longi­tudinal axis, this equality expresses the second transverse equilibrium characteristic.

In order to express the transverse equilibrium conditions through the forces acting on the helicopter during hovering and through the transverse moments of these forces, we must examine the conditions for equilibrium of the single-rotor helicopter without an aft fin and the single-rotor helicopter with an aft fin.

In the single-rotor helicopter without an aft fin, the tail rotor is located right on the tail boom (Figure 97a).

This helicopter can have transverse equilibrium in the hover regime only in the presence of a bank in the direction opposite the tail rotor thrust (Figure 97c). A side component of the weight force = G sin у (у is the

 Figure 97. Helicopter lateral equilibrium.

bank angle) is formed when the helicopter banks. This component balances the tail rotor thrust force. Then the equality expressing the first transverse equilibrium characteristic is written as follows

T = G = G sin у or T – G„ = 0 . t. r 2 ‘ t. r 2 •

Hence we find the magnitude of the bank angle

T_

. t. r

у = arcsm—–

G

The thrust force vector is not deflected from the helicopter symmetry

plane, therefore the thrust force moment about the helicopter longitudinal

axis will be zero. The moment of the horizontal hinges will also be zero.

The tail rotor thrust force is applied to the longitudinal axis of the

helicopter, and its moment will also be zero. Therefore, the second helicopter

equilibrium characteristic will be expressed in general form by the equality

EM = 0. x

The transverse equilibrium of this type of helicopter in the other flight regimes can be achieved in the same way as in the hover regime, i. e., by banking; or it can be obtained by sideslipping the helicopter in the direction of the tail rotor thrust (Figure 97d)• During the sideslip there is formed the side air pressure force on the fuselage, and this force balances the tail rotor thrust force

the tail boom, since there is no twisting moment of the tail rotor thrust force, and lower helicopter weight in view of absence of the vertical fin.

The disadvantages of the helicopter include:

(1) large bank angle during hover, which creates discomfort for passen­gers and crew and makes helicopter control difficult;

(2) large bank angle or large sideslip in translational flight, which increases helicopter parasite drag;

(3) probability of damage to the tail rotor in view of its very low position;

(4) danger to servicing personnel because of the low tail rotor location.

Therefore, helicopters without a vertical fin are seldom encountered.

The presence of a vertical fin on which the tail rotor is mounted elimi­nates these drawbacks, since the tail rotor is raised above the ground and its axis approaches the main rotor plane of rotation (Figure 97b). Let us examine the transverse equilibrium of a helicopter with vertical fin (Figure 97d). This helicopter, like the helicopter without a vertical fin, hovers with a bank angle, but the bank angle will be small, about 1°, and is not noticeable

in practice. The necessity for the bank angle follows from the transverse equilibrium conditions. The bank angle leads to the side component G^ – G sin у of the weight force. The first transverse equilibrium criterion will be expressed by the equatior

S + G_ or T

S 2 t.

(S= + G_) = 0. s 2

 To satisfy this condition the main rotor thrust force vector (coning axis) must be deflected by some angle in the direction opposite the tail rotor thrust direction; then the side component Sg is created, which together with the force G2 balances the tail rotor thrust.

 Since the tail rotor is raised above the helicopter longitudinal axis, the transverse moment rolling Mr = T b is developed. This moment is t « Г t • г balanced by the side force moment Mg = Sgh and the moment of the horizontal hinges relative to the longitudinal axis. Consequently, the second transverse equilibrium criterion will have the form

 /158

 Tt. rb = Ssh + h.

 Why does the helicopter require a bank angle in hover? If there were no bank angle, the first transverse equilibrium criterion could be written as

 T = S or T – S = 0. t. r s t. r s

 t. r

 S + s

But T = S ; therefore contradictory requirements are obtained: the

t • IT s

tail rotor thrust must at the same time be larger than and equal to the side force, which is not possible; therefore a small bank angle and the side weight force G2 are required for equilibrium.

During flight with translational velocity, transverse equilibrium is achieved either as a result of a bank angle, as in hovering, or as the result of sideslip, which results in side pressure force on the fuselage. The transverse equilibrium condition will be expressed by the equalities

Ф

Heliconter directional equilibrium is the state of the helicopter in which it does not rotate about the vertical axis. The directional equilibrium conditions are

To ensure the latter condition it is necessary that the reactive moment be balanced by the moments of tail rotor thrust and main rotor thrust side force. Then

Here the following circumstance must be noted. With change of the main rotor thrust there will also be a change of its reactive moment, i. e., the directional equilibrium will be disturbed. Therefore, when the main rotor thrust is changed, it is necessary to change the tail rotor thrust to preserve directional equilibrium. But this requirement complicates the control helicopter.

In the translational flight regimes with horizontal velocity, directional /159 equilibrium is achieved in the same way as in the hovering regime

where b is the distance from the point of application of the force Z^ to the helicopter vertical axis.

## Blade Element Angle of Attack Change Owing to Flapping Motions

The blade angle of attack change +Да depends on the vertical flapping velocity on шг> and on V sin ip, i. e., on the azimuth angle, which we

see from the formula

The sign of the vertical flapping velocity is determined by the direction of the flapping motion: a minus sign for upward blade flapping, a plus sign for

downward flapping. Since the maximal upward blade flapping velocity occurs at the 90° azimuth, the negative angle of attack increment will be greatest at this azimuth and the angle of attack of a given blade element will be minimal. The highest downward vertical flapping velocity occurs at the 270° azimuth, and the positive angle of attack increment Да will be maximal at this azimuth. This means that a given blade element has its maximal angle of attack at the 270° azimuth (Figure 42). Moreover, in analyzing the curve we see that the maximal magnitude of the negative angle of attack increment at the 90° azimuth is less than the maximal magnitude of the positive angle of attack increment at the 270° azimuth.

This variation of the angle of attack increment is explained by the fact that in (17a) for ф = 90° the second term of the denominator is positive, and tg Да will decrease as a result of increase of the resultant flow velocity over the blade.

 Figure 42. Azimuthal variation of blade element angle of attack.

0°.. S0° 130а 270а 350° V*

1) with rigid mounting;

2) with hinged support.

For ф = 270° the second term of the denominator is negative, and this means that the angle of attack increment Aa will increase as a result of reduction of the resultant velocity of the blade element. Moreover, at the 90° azimuth the vertical upward flapping velocity Vq will be less than at the 270° azimuth, when the blade flaps downward. But the blade element angle of attack does not change only in azimuth. It also varies along the main rotor radius (Figure 43). We see from the figure that the angles of attack will be highest for the tip elements at an azimuth close to 270°, and lowest at the 90° azimuth, with the angles of attack being nearly the same for elements at different radii.

The following azimuthal variation of the angle of attack is characteristic: from the 0° azimuth the angles of attack, remaining nearly constant along the length of the blade, decrease up to about the 110° azimuth and then begin to increase.

The following angle of attack variation along the radius is characteristic of the retreating blade: from the root to the tip of the blade the blade

element angles of attack increase by 4-5°, with the variation being less at the root elements than at the tip. The angle of attack variation equalizes the blade thrust force azimuthally (Figure 44), and the blade flapping motions are reduced.

## Helicopter Rate of Descent With Operating Engine

During inclined descent the helicopter speed along the trajectory and the

vertical rate of descent may vary from zero to the limiting permissible values.

In accordance with the second descent condition, the speed along the trajectory

will be constant if = X r + px* The flight speed can be altered by varying

the magnitude and direction of P. Since P depends on the position of the

X x

main rotor cone of rotation axis, there is a change of the helicopter speed along the trajectory when this position is changed. The helicopter motion velocity along the trajectory is connected with the vertical rate of descent as follows (Figure 70b)

 V, sin 0. des^

 (38)

The vertical rate of descent is measured with the aid of a special instrument — a variometer. We see from the figure and Formula (38) that this rate depends on the velocity along the trajectory and the descent angle. The descent angle depends on the magnitude of the lift force Y, while the rate of descent depends on the force P, and therefore on the magnitude of the power supplied to the rotor. The helicopter rate of descent can be found from the Zhukovskiy grid.

If we draw on the Zhukovskiy grid, the lines showing power available N ; N

«X c*2»

for the given altitude, the points of their intersection with the

power required curves for the different regimes will correspond to definite

 а) ‘ Ъ)

 Figure 71. Power required versus flight speed for different climb and descent angles.

flight speeds along the trajectory. Thus, from the intersection of the line N& with the power required curves we can find the maximal helicopter hori­zontal flight velocity V, the vertical rate of climb V, for 0 = 10°, the ° max 1

vertical rate of climb V„ for 0 = 30°, and so on. Drawing the line N , we

L a2

find the vertical rate of descent for 0 ■ -10°. Other power available lines

must be drawn, i. e., lower engine powers must be used, to find the rates of

descent with larger descent angles.

The descent angles and velocities along the trajectory are often deter­mined for some average altitude and some definite flight weight in the aero­dynamic analysis. Such calculations are made for different values of main rotor pitch, for example, ф^; ; ф^. From the results of this calculation

we plot a graph — the polar curve of helicopter descent trajectories with operating engine (Figure 71b).

In this figure each curve corresponds to a definite velocity along the trajectory, if this velocity is plotted in the form of a vector from the coordinate origin. The horizontal axis will be the horizontal velocity

component V = V, cos 0, and the ordinate axis will be the vertical rate x des•t

of descent V, = V, ^ sin 0. The vertical rate of descent is measured in des des. t

m/sec, and the horizontal velocity is in km/hr. The velocity along the tra­jectory can be found if this velocity vector is transferred to the horizontal axis of the figure. Let us examine some characteristic points on the polar.

The point A lies at the intersection of the characteristic curve (for = 8°) with the ordinate axis. This point corresponds to helicopter vertical descent rate with the engine operating. If this rate of descent is more than 3 m/sec, transition to the vortex ring regime is possible and this cannot be permitted. This means that the pitch ф and the engine power must be increased.

The point В corresponds to descent with minimal permissible velocity along the trajectory, and the point C corresponds to descent with the minimal vertical rate of descent. In this case, the velocity along the trajectory corresponds to the economical velocity for horizontal flight of the helicopter.

The point D corresponds to flight with the minimal descent angle 0 , and in

this case, the velocity along the trajectory will be equal to the optimal helicopter horizontal flight speed.

## Bending and Bending-Torsion Vibrations of. Rigidly Restrained Blade

Bending vibrations are those in which the blade chord displaces parallel to itself, i. e., there is not twisting of the blade. These vibrations occur when the line of blade section centers-of-gravity coincides with the elastic /184

axis. If the tip of such a blade is bent down, an elastic force P ^ appears, applied at the elastic center and directed upward (Figure 113). When the deflected blade is released, it begins to travel upward with an acceleration under the action of the elastic force. As a result of the acceleration there is formed the inertia force P^ , directed downward and applied at the center of gravity. We are examining a blade for which these points coincide; there­fore the twisting moment of the inertia force will be zero, and there will be no twisting of the blade.

As the blade element moves upward, the elastic and inertia forces will decrease, while the bending motion velocity will increase. When the blade element reaches the neutral position, the elastic and inertia forces will be equal to zero while the velocity will be maximal. With further upward

movement of the blade, the elastic force changes its direction and will cause reduction of the velocity. The element motion stops in the extreme upper position. After passage through the line of equilibrium, the inertia force will be directed upward. This motion is shown in Figure 113, positions 1,

2. 3. The blade element will travel downward from position 3 (positions 3,

4, 5). The forces, acceleration, and velocity will vary just as in the upward motion. It follows from this analysis that the largest elastic forces and the largest accelerations of the bending motion will occur in the extreme positions. The inertia forces will be maximal for the largest accelerations, i. e., also in the extreme positions. If the blade section passes through the neutral position, the forces and acceleration change their signs and the velocity reaches its maximal magnitude. The variation of the velocity and!

acceleration is shown on the graphs of the vibratory motion. If the line of centers-of-gravity does not coincide with the elastic axis, the inertia force of the given blade element creates a torsional moment, which twists the blade relative to the elastic axis. The bending vibrations will be supplemented by torsional vibrations.

For most blades the line of centers-of-gravity lies behind the elastic axis. The nature of the bending-torsion vibrations of such a blade is shown in Figure 114. The direction and variation of the forces, velocity, and acceleration will be the same as for bending vibrations. As a result of the aft shift of the center of gravity relative to the elastic center, in the case of upward bending motion of the blade section there are created the additional positive twist angles (+ ф), which increase up to the neutral /185

position and then decrease (positions 1 – 5).

During downward bending motion of the blade, negative twist angles are created, which first increase in absolute magnitude and then decrease (positions 5 – 9).

This nature of the natural bending-torsion vibrations is determined by the mutual position of the elastic axis and the line of centers-of-gravity, and occurs for all blades.

 і

We have examined bending and bending-torsion vibrations of the blade in stationary air, i. e., when the rotor is not turning. If the rotor turns, the aerodynamic forces must be added to the elastic and inertia forces. The effect of these forces on the blade leads to flutter under certain conditions.

Essence of bending-torsion blade flutter. We imagine that bending – torsion vibrations of a blade take place while the main rotor turns. We resolve these vibrations along the blade line of motion with the air flow approaching at the velocity u (Figure 115). The elastic and inertia forces which arise during bending-torsion vibrations are not shown in Figure 115, but they do act. To these forces we add the additional lift force AY. As the blade flexes upward the twist angles will be positive, and this means that /186

there will be a positive increment (Да = ф) of the blade element angle of attack.

The additional lift force will be directed upward. When the flexing motion is downward, the twist angles will be negative, which means there will be a negative angle of attack increment (- Aa = – ф). The additional lift force will be directed downward. It follows from this analysis that the additional lift force coincides with the direction of the bending motion, i. e., it will be an exciting force.

 м=о. А а-9 = О Act = (f-0 АУ=0

 Acx = cPsO £4=0

Under the influence of the exciting force the bending vibration amplitude will increase, which leads to increase of the elastic forces, bending vibra­tion accelerations, inertia forces, and their torsional moment. The twist angles and the additional lift forces increase. The conclusion is that bending vibrations cause twisting of the blade. Increase of the twist leads to increase of the bending vibrations, which in turn leads to increase of the torsional vibrations.

As a result the amplitude of the vibrations increases so rapidly that blade failure may occur.

Flutter occurrence conditions. Both exciting forces and damping forces arise in the case of bending-torsion vibrations. Flutter will be possible only if the exciting forces are larger than the damping forces. We have already shown that the exciting forces arise as a result of torsional vibra­tions, which lead to increase of the angle of attack.

The lift force

27k

We see from the figure that the increment ДС

У

depends on the angle of

attack increment Да and on

the slope of the curve to

the abscissa axis, which can

ДС

be defined as frr 3 — . 1 – v

Да ~a’

Hence

Thus, the exciting forces are proportional to the twist angle and the square of the blade tip circumferential velocity.

The damping forces arise as a result of the bending vibrations. During these vibrations there is a change of the blade element angle of attack as a result of the vertical velocity V of the bending motions (Figure 116b).

In the case of downward bending motion, the angle of attack increment is positive and the additional lift force is directed upward. In the case of downward bending motion, Да is negative and the additional lift force is directed downward. Thus, in all cases of bending motion the additional lift force is directed opposite this motion, i. e., it is a damping force

The lift force coefficient increase is proportional to the angle of aLtack, і. e. , ДСу — СЛя.

The conclusion is that the damping force is proportional to the velocity of the bending motion and the blade tip circumferential velocity. This is shown graphically in Figure 117, from which we see that the damping forces equal the exciting forces at a definite circumferential velocity. This velocity is called the critical flutter speed. Since u = 2irRns, the critical speed corresponds to the critical flutter rpm.

At speeds below the critical flutter speed (for rpm below the critical value) the damping forces are larger than the exciting forces, the bending-torsion oscillations will be damped, and flutter is not possible. At rpm above the critical value the damping forces are smaller than the exciting forces. The bending-torsion

vibrations will be increasing and flutter is inevitable. Consequently, flutter develops if the blade section eg line is behind the elastic axis and the main rotor rpm is above the critical flutter rpm. The first of these conditions holds for all blades; therefore the second condition is sufficient for the onset of flutter.

Dependence of critical flutter rpm on various conditions. The critical flutter rpm depends on the blade stiffness, mutual positioning of the eg line and the elastic axis, mutual positioning of the center-of-pressure line

and the elastic axis. The higher the blade stiffness in bending and torsion, the larger the exciting forces must be in order to deform the blade.

High velocity is necessary in order to obtain large forces. This means that the critical flutter rpm increases with increase of the stiffness.

Flutter of a metal blade is possible at a considerably higher rpm than for a blade of composite construction of the same dimensions. This is one of the advantages of metal blades.

If the blade eg axis coincides with the elastic axis, the natural vibrations will be flexural; without the torsional vibrations there will not be any exciting forces, and flutter will not occur.

If the eg axis is ahead of the elastic axis, the natural vibrations will be the bending-torsion type. However, the signs of the twist angles will be opposite to those for which flutter occurs, i. e., when the blade flexes up­ward the twist angles will be negative, and when the blade bends down the angles will be positive. With these vibrations associated with twisting, additional damping forces are created and flutter is impossible.

If the eg axis is behind the elastic axis, then the larger the distance between these axes, the larger the twisting moment of the inertia forces, and the lower the critical flutter rpm. This means that the blade eg axis must be moved forward in order to increase the critical rpm. To this end special weights in the form of metal bars, mass balance weights, are installed in the leading edge of the blade.

The mutual influence of the center-of-pressure axis and the elastic axis amounts to the fact that if the center-of-pressure axis is ahead of the elastic axis, an additional twisting moment on the blade is created from the action of the aerodynamic forces. The sign of this moment is the same as the sign of the inertia force moment. Consequently, in this case the critical flutter rpm will decrease.

Flutter characteristics of blade with hinged support. The majority of main rotors have blades with hinged support. The presence of the horizontal and axial hinges leads to the so-called flapping flutter, i. e., a combination of flapping motions with oscillations of the blade relative to the axial /189

hinge. The amplitude of such oscillations will increase. These oscillations are possible even for an absolutely rigid blade, in the absence of bending and twist.

During the flapping motions there is an increase of the amplitude of the vertical displacements of the blade element. The amplitude is made up of the flapping and bending displacements. In this case the critical flutter rpm decreases. Consequently, the hub horizontal hinges contribute to the onset of flutter.

The axial hinges permit the blade to rotate about its longitudinal axis.

This is equivalent to reduction of the blade torsional stiffness. Rotation about the axial hinge is possible as a result of elastic deformation of the blade pitch control linkage elements. The lower the stiffness of these elements, the lower the critical flutter rpm. The hub axial hinges also pro­mote the onset of flutter.

The flapping compensator causes reduction of blade pitch as they flap upward, and increase of the pitch as they flap downward. This pitch change is analogous to the bending-torsion oscillations during flutter. Consequently, the flapping compensator also promotes the onset of flutter. The larger the compensation coefficient, the lower the critical flutter rpm. For most hubs the compensation coefficient К = 0.5. In the case of a large value of this coefficient, the critical flutter coefficient decreases to rpm values within the operational range.

The critical flutter rpm depends on the flight speed: with increase of

the speed there is an increase of the resultant velocity of the flow over the blade at the 90° azimuth, and this promotes the onset of flutter. This means that the critical flutter rpm decreases with increase of the flight speed.

278

In addition to the factors listed above, blade flutter is affected by centrifugal forces. As a result of the centrifugal forces the flapping motions and bending of the blade are reduced, and the blade becomes effectively stiffer. Consequently, the critical flutter rpm increases under the influence of the centrifugal forces.

In view of the lower blade stiffness, flutter of the main rotor blades does not develop as violently as does the flutter of the airplane wing, and therefore rotor blade flutter can be detected in time and measures can be taken to stop the flutter.

Operational sources of flutter. Flutter is avoided in the design of airplanes and helicopters. This means that in calculating the critical flutter rpm the blade stiffness and eg location are selected so that the critical flutter rpm is made considerably higher than the maximal permissible main rotor rpm. However, flutter can arise from operational causes, as a result of mass unbalance and reduction of the structural stiffness. Disruption of /190 the mass balance is particularly characteristic for blades of composite skeleton construction. In these blades the wooden ribs absorb moisture from the air markedly. With increase of the moisture content, the eg line shifts aft, which leads to reduction of the critical flutter rpm, and flutter becomes possible at operational rpm.

The mass balance may be disrupted as a result of careless overhaul of the blade, which also leads to the onset of flutter^ Reduction of the structural stiffness, which leads to reduction of the critical flutter rpm, occurs if there is a failure of the individual structural elements or in case of damage to the blade skin.

Under operational conditions flutter may occur with the main rotor operating on the ground and in flight. Flutter is detected by heavy vibration of the helicopter and from "blurring" of the main rotor cone of rotation. If there are no blade vibrations, the blades of a properly adjusted rotor will

travel along a single track and will form a definite cone, which is visible from the cockpit* If vibrations are present, the blades travel along different trajectories and the cone will be "blurred," vague.

When flutter is detected, the main rotor rpm must be immediately reduced to the minimal permissible value. After landing the reason for the flutter must be investigated. It must be kept in mind that "blurring" of the cone occurs not only in the case of flutter, but also if the blades are not in track, i. e., in case of improper adjustment of the main rotor. However, in the latter case the "blurring" is independent of the rpm.

Measures to prevent vibrations of all types. During helicopter operation, special attention is devoted to preventing all forms of vibration. These measures concern, first of all, strict adherence to all the operating conditions and performance of all the instructions and scheduled maintenance procedures for the particular helicopter type. These operations include the adjustment and alignment of all parts of the helicopter, verifying the proper adjustment of the vertical hinge dampers, proper charging of the landing gear shock absorbers and tires, verifying main rotor tracking, and checking the main rotor for flutter.

The essence of the flutter check amounts to the following: a definite

weight is attached to the trailing edge of each blade and the main rotor rpm is increased (helicopter parked) to the rpm indicated in the instructions.

Then the rpm is increased J>y 1 – 2%,and the rotor is operated in this condi­tion for 1-2 minutes. Then the rpm is again increased and brought up to the maximal value indicated in the instructions. If flutter does not occur with the weights installed, there will be just that much more margin with the weights removed, since the critical flutter rpm increases as the blade eg moves forward.

[1] Motion power, i. e., the work expended per unit time in displacing

the helicopter, N = PV.

mot

[2] Induced power, i. e., the work expended per unit time in obtaining the lift force equal to the helicopter weight

(27)

[3] Power required for overcoming the main rotor blade profile drag

[4] V

gl sin 0 + fl

u – V, cos 0 gl