Azimuthal Variation of Rotational Resistance Forces and Reactive Torque

Both the rotational resistance forces and the thrust forces of the blades vary azimuthally as a function of the resultant flow velocity over the blades.

Azimuthal Variation of Rotational Resistance Forces and Reactive Torque

Figure 44. Blade element angle of attack diagram.


Azimuthal Variation of Rotational Resistance Forces and Reactive Torque

Figure 45. Blade rotational resis­tance forces at different azimuths.


The resistance to rotation of a single blade reaches its maximal value at the 90° azimuth and minimal value at the 270° azimuth.

At the 0° and 180° azimuths the rotational drag in the forward flight regime is equal to the drag in the axial flow regime, if the main rotor pitch and flight altitude remain the same (Figure 45).

As a result of this variation of the resistance to rotation, there will /61 be azimuthal variation of the main rotor reactive torque from the maximal value when the blades are at the 90° and 270° azimuths to the minimal value when they are located at the 0°and 180° azimuths.

The variation of the reactive torque causes vibration (torsional oscil­lations) with a frequency equal to the main rotor rpm or some multiple thereof.

For two blades at opposite azimuth angles the rotational resistance forces are directed oppositely relative to the rotor diameter. At the 0° and 180° azimuths their sum is zero; however, at the 90° and 270° azimuths the sum of these forces is not equal to zero and is directed opposite the helicopter

flight direction, since is larger at the 90° azimuth than at the ф = 270° azimuth. This force is the profile drag of the main rotor.

Effect of Number of Blades on Main Rotor Aerodynamic Characteristics

Single-blade main rotors are not used because of the high degree of /60


The primary advantage of the two-blade main rotors is the simplicity of the construction. But the two-blade rotor has low solidity and consequently poor aerodynamic characteristics (low thrust coefficient C^).

Increase of the solidity with increase of the area of each blade (by increasing its width) leads to increase of the profile drag and reduction of the main rotor efficiency.

Moreover, the blades of any rotor cannot be made perfectly identical.

They always differ from one another in their characteristics; therefore the overall blade thrust varies in the forward flight regime. The main rotor resistance to rotation will also vary, i. e., the load on the rotor shaft will vary, and torsional vibrations of the shaft, main rotor vibrations, and vibrations of the entire helicopter will develop.

These problems can be resolved by increasing the number of blades. The larger the number of blades, the smaller the amplitude of the main rotor thrust variations and the smaller the azimuthal variation of the rotor torque, i. e., the rotor becomes more balanced. However, at the same time rotor fabri­cation and blade balancing and adjustment become more difficult. On this basis, main rotors with 4-5 blades are most frequently encountered.

Thrust required for various flight regimes and its variation in the different regimes

Answer 1. The thrust required for a given regime is the force necessary to provide flight along the given trajectory at the required velocity. For constant velocity the thrust required is practically the same for horizontal flight, climbing flight, and descending flight.

Answer 2. The thrust required for a given flight regime is the force required necessary to balance the airplane weight and create the propulsive force. In a climb the thrust required will be considerably greater, and during descent it will be considerably less than in the horizontal flight regime.

Answer 3. The thrust required for a given regime is the force necessary to overcome the parasite drag. At a constant speed the thrust required will be the same for horizontal flight, inclined climb, and inclined descent.

Power and Torque Required to Rotate Main Rotor

In order for the main rotor to turn, the action of the reactive torque must be overcome, i. e., driving torque must be supplied to the rotor.

The torque Mt ^ which must be supplied to the main rotor is termed the /22

required torque. In magnitude, it equals the reactive torque — in direction, it opposes the latter

= пі F £ (toR)2 R tor tor 2

where in is the torque coefficient, tor

The torque coefficient is a composite quantity, i. e.,

m = m + m

tor tor tor.

pr і

Подпись: where m.Подпись:Подпись: m.Подпись:is the part of the torque coefficient due to profile drag forces. This part depends on the condition of the blade surface, the rotor rpm, and the blade shape; is the part of the torque coefficient due to the induced drag forces and depends primarily on the main rotor pitch (Figure 18).

The formula for the required torque, and also the curve of this torque coefficient versus rotor pitch, makes it possible to conclude that the main rotor required torque will increase with increase of the pitch, rpm, and air density.

We recall that power is work per unit time. The concept of the power required to turn the main rotor can be obtained if we examine the work ex­pended in overcoming the forces resisting the rotation of a single blade, and then the work expended in overcoming the reactive torque of the entire rotor (Figure 19).

The work of a single blade during one revolution of the main rotor is


The main rotor work per second, i. e., the power required, is /23

Power and Torque Required to Rotate Main Rotor

Figure 19. Action of rotational resistance forces.


Power and Torque Required to Rotate Main Rotor

Figure 18. Torque coefficient versus main rotor pitch.


N = A, kn req b s

where к is the number of blades;

n is the rps.



N = Q. 2irr_kn req b Q s


Q rnk = and 2Ttn. — «),



Подпись: (IDN = M o) .
req p

Both the power required and the torque required for the main rotor
change with change of the pitch, rpm, and air density. In order to turn the



rotor, engine power equal to the power required must be supplied to the rotor shaft. This equality is the condition for constant rpm

N = N sup req

where Ng Is the power supplied to the rotor from the engine.

If the power supplied NgUp > rotor rPm will increase. However,

if N < N the rotor rpm will decrease, sup req r

Vertical Climb

Helicopter flight along a vertical trajectory with constant velocity is

termed the vertical climb regime. The following forces act on a helicopter

in vertical climb (see Figure 53): the helicopter weight force G, the main

rotor thrust force T, and the tail rotor thrust force T.

t. r

In order to balance the tail rotor thrust force, the main rotor thrust force vector must be inclined at the angle 6, which results in the creation of the vertical thrust force component Y and the horizontal component S^.

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

Since the force X is small and may be neglected and the force YftS T, the equality Y = G + X can be replaced by the equality T = G. Then the vertical climb conditions will be analogous to the hovering conditions. The condition T = G assures constant helicopter speed in the vertical climb regime./82 The equality Tt r = Ss assures rectilinear flight.

Power required for vertical climb. The difference between the hovering

and vertical climb conditions is that, first, the force X in the vertical


climb is larger than in hovering, since it depends on two velocities: the

vertical velocity and the induced velocity V^.

Second, while in hovering the equality T = G assures a state of relative rest; in climb the same equality must assure constant velocity of the vertical motion.

Consequently, the work per unit time of the thrust force in a vertical climb is different from that in a hover; during climb this work is made up of the work expended on creating the thrust force equal to the weight (TV^ =

= GV^), and the work expended in creating the vertical velocity (TV ). During hovering, the work per unit time of the thrust force is expended only in creating the induced flow velocity and is equal to TV_^.

Therefore, while the induced power required for hover is found from the formula

the induced power required for vertical climb is expressed by the formula

For a low climb velocity (2-3 m/sec) the induced velocity differs very little

from the induced velocity in hovering, i. e., V. « V. . But this implies

that the induced power in a climb is greater than the hovering power by the magnitude AN (the excess power required for climb in comparison with the hovering power required). Bearing in mind that the profile power in climb is practically equal to the profile power in hover, we can express the formula for the power required for vertical climb through the hovering power formula

Vertical Climb Vertical Climb

Vertical climb is possible only if excess power is available. To transition from hover to climb, the pilot increases the main rotor pitch with the aid of the "collective-throttle" lever; in this process the main rotor rpm remains nearly constant while the thrust increases. The helicopter transitions from hover to vertical climb. The thrust in a vertical climb can be determined from the formula of ideal rotor momentum theory T = 2pFV^V. In this case

+ V. As the vertical velocity increases the induced velocity will /83 decrease. Therefore, the main rotor thrust again decreases to the value the rotor had in hovering prior to increasing the collective pitch. Thus, in transitioning from hover to climb the pilot actually increases the power supplied to the rotor, but the main rotor thrust force remains nearly unchanged. Therefore,

Подпись:T, , but N = N, + AN. hov cl hov

Vertical climb velocity. The vertical climb velocity is the height through which the helicopter center of gravity displaces in one second.

For vertical displacement of any body it is necessary to perform work equal to the product of the weight of the body by the height change, i. e.,

A = GH. Work performed per second is power. This means that to perform a climb additional power must be supplied to the main rotor, which is expended in creating the vertical velocity. This power is the excess power AN = GV^. Hence we find

Подпись: AA/75


The vertical velocity depends on the excess power and the helicopter weight. If the helicopter is heavily overloaded, there is sufficient engine

Vertical Climb

Figure 56. Aerodynamic characteristics in the climb regime.


power only for hovering in the air cushion zone and in this case vertical climb is not possible.

Vertical Climb Vertical Climb

The excess power used for vertical climb is equal to the difference between the power available and the power required for hover

Usually the helicopter excess power is not large and near sea level does not exceed 10-15% of the total engine power.

The vertical velocity at sea level amounts to 2-3 m/sec. The power available and engine power depend on the flight altitude, and with change of the altitude the vertical climb velocity will also change. The variation of the vertical climb velocity is determined by the altitude characteristics of the engine and is shown graphically (Figure 56a).

For a supercharged reciprocating engine the effective power will increase with increase of altitude from sea level to the critical altitude. The power required also increases. Therefore, the excess power may increase slightly or remain constant up to the critical altitude. After reaching the engine’s critical altitude the excess power decreases rapidly. Since the vertical velocity depends on the excess power, it will also decrease.

Using the graph of the variation of engine power and power required for /84 hovering as a function of altitude, we can use (25) to calculate V for various altitudes. On the basis of these calculations we can plot the vertical velocity as a function of altitude (Figure 56b), from which we see that the vertical velocity reaches its maximal value at the engine’s critical altitude, and then decreases.

The altitude at which the vertical climb velocity equals zero is called the helicopter’s static ceiling. The static ceiling is the highest altitude at which the helicopter can be hovered. At the static ceiling the excess power AN = 0.

However, since both AN and AV approach zero as the helicopter approaches the static ceiling, it is not possible to reach an altitude equal to the theoretical static ceiling. The "practical ceiling" concept has been intro­duced on this basis. The practical ceiling is the altitude at which the vertical climbing velocity equals 0.5 m/sec.

The static ceiling is defined in terms of rated engine power. A specific power required corresponds to each vertical velocity. Therefore, we can plot the power required for climb as a function of altitude for different vertical velocities (Figure 56c). From this graph we can find the vertical climb velocity at various altitudes for different power required. The altitude characteristic can be used to evaluate the possibility of climbing with a given vertical velocity.

Transition From Flight With Engine Operating to Flight. in the Main Rotor Autorotation Regime

Gliding in the autorotative regime is not an emergency flight mode ; rather it is a normal, stable flight mode which is often used even with a normally functioning, sound engine. Gliding in the autorotative regime is

used for working out basic piloting techniques or for rapid loss of altitude. However, in order to assure safety, a definite order and sequence of actions must be followed in transitioning from flight with the engine operating into the autorotative regime. What happens with the helicopter in the case of sudden engine stoppage or in case of rapid decrease of the engine rpm?

The main rotor continues to rotate momentarily, and the rpm does not change, since the freewheeling clutch automatically disengages the engine from the transmission. The rotor continues to turn by inertia, since it has definite angular momentum. The greater the mass or weight of the blades, the larger the moment of inertia, the longer time the rotation will continue by inertia. Therefore, heavy blades have an advantage in the autorotative regime.

Under the action of the reactive moment, the main rotor rpm decreases, and therefore the thrust decreases. If the engine fails in the hovering regime, then as a result of main rotor thrust reduction, the helicopter will transition to vertical descent. However, if the engine fails in horizontal flight, reduction of the thrust and lift will cause the helicopter to descend along an inclined trajectory. In both cases, the air flow will approach the main rotor from below.

The presence of the vertical velocity causes increase of the blade element angles of attack by the magnitude Да and deflection of the force vector AR forward, i. e., a driving torque appears; therefore, there is an increase of the rpm or at least no further reduction of the rpm. Moreover, along with reduction of the main rotor rpm, there is reduction of the centrifugal force of each blade, which leads to increase of the main rotor coning angle, i. e., simultaneous upward flapping of the blades. When the flapping angle increases, there is a reduction of blade pitch under the influence of the flapping compensator, i. e., there is an increase of the main rotor rpm.

Thus, we conclude that in the case of engine failure, there are objective factors which facilitate transition of the main rotor into the autorotative regime. But the pilot must not rely on these conditions and expect the rotor

itself to transition into autorotation. Therefore, in case of engine failure, /135 the pilot must immediately reduce main rotor pitch to the minimal value. To this end, the collective-throttle lever is lowered fully. The main rotor rpm increases, and the circumferential velocity of the blade elements increases.

This leads to reduction of the blade element angles of attack and aft deflection of the force AR. Therefore, the main rotor rpm will increase up to some limit, and then the constant rpm regime is established, i. e., the autorotation becomes steady. However, if the rpm is too high, the pitch must be increased somewhat. During flight with the engine not operating, the rpm should correspond to the engine rated power rpm. In this case, the rotor will develop the maximal thrust force, and the vertical rate of descent will be minimal.

Transition of the main rotor into the autorotative regime is facilitated

by the stabilizer mounted on the tail boom. The stabilizer incidence angle

changes with change of the main rotor pitch: when the pitch is reduced to

the minimal value, the stabilizer incidence angle becomes negative (Figure 85).

If at the time of transition into the autorotative regime the helicopter is

moving with a horizontal velocity, the negative lift force Y develops on

the stabilizer. The moment of this force M = Y L causes helicopter

st st st

nose-up pitch. The main rotor angle of attack becomes positive, and the air flow approaches the rotor from below. The angle Да of each blade element increases, and the rpm increases, i. e., the main rotor transitions into the autorotative regime.

So far, we have discussed the factors which accelerate or decelerate autorotation of the main rotor. We have devoted considerable attention to this factor, since main rotor rpm in autorotation is the primary index of flight safety. If the rpm is less than the minimal permissible value during autorotation, the rotor can come to a stop — which is a problem which cannot be rectified.

However, during transition into the autorotative regime, the pilot must devote some attention to factors other than main rotor rpm. The helicopter behavior at this time differs markedly from the behavior in steady-state flight:

First of all, there is a marked reduction of the main rotor reactive moment. As a result of this, the helicopter tends to turn to the right about the vertical axis. Moreover, if there is a horizontal velocity, there will be flapping motions of the blades, and this means that the main rotor coning axis will tilt to the right.

Подпись:As a result of the main rotor thrust force side component, the helicopter will bank and slip to the right.

Подпись: /136At the moment of transition into the autorotative regime, the pilot must prevent rotation of the helicopter about the vertical and longitudinal axes by reversing the tail rotor thrust force and deflecting the main rotor cone of rotation to the left. The tilt of the helicopter fuselage relative to the horizon depends on the flight speed. At low speed, the tilt reaches 10-15°, i. e., the nose of the helicopter is quite high. This cannot be permitted, as the helicopter tail rotor may come in contact with the ground, and tail rotor failure may occur.

In steady-state autorotation the main rotor blades develop a driving torque. Under the influence of this torque, the blades are rotated forward relative to the vertical hinge to a negative lag angle. During glide, the driving torque depends on the azimuth angle. Therefore, the lag angle will vary, i. e., during glide, the blades will oscillate about the vertical hinges.

Resultant Flow Velocity over Blade Element in the Hub Rotation Plane

It is well known that in the vertical flight regime each blade element is in a stream whose velocity is equal to the circumferential velocity of the element u = шг.

The situation is different in the forward flight regime. If the main /45

rotor angle of attack A = 0° , the resultant velocity with which the stream flows over the blade element depends on the element circumferential velocity, the

Подпись:flight speed, and the azimuth angle p. In this case the resultant velocity will not be equal to the geo­metric sum of the circumferential velocity and the flight velocity, since only the flow directed perpen­dicular to the blade longitudinal axis has an influence on the aero­dynamic forces of the element.

Therefore, we must take as the resultant blade element velocity in the forward flight regime the sum of the vectors of the circumferential velocity of the blade element and the projection of the flight velocity vector on the line of the circumferential velocity vector (Figure 31).

V/ = Vclnb. (16)

Consequently, for a constant flight speed and constant angular velocity the resultant velocity will vary as a function of the azimuth angle.

Let us examine the variation of the resultant velocity as a function of blade azimuth (Figure 32) .

V I/ У

Resultant Flow Velocity over Blade Element in the Hub Rotation Plane

Figure 32. Blade element resultant velocity as a function of azimuth.

It is not difficult to see that for ip = 0° and 180° the resultant velocity equals the circumferential velocity, since the projection of the flight velocity on the circumferential velocity vector equals zero (Figure 32a)

W0 = U+V sin 0° == U,

V/m = u + Vsin 180° = U.

For ф = 90° the resultant velocity is /46

V7fl0 = и + V sin 90° = U – f – V.

For ф = 270° the resultant velocity equals the difference of the veloci­ties (Figure 32b)

y270 — u-i-V sin 270° = 11- V.

If we use (16) to calculate the resultant velocity for several azimuths, we can plot the relation W = f(4>) (Figure 32c).

Figure 32 makes it possible to conclude that:

the maximal blade element velocity will occur at ip = 90°, the minimal will occur at ip = 270°; for ф = 0° and 180° the resultant velocities of a given element are equal to the circumferential velocity of this element. Consequently, the forward flight regime differs from the vertical flight regime in the vari­ation of the blade element velocity. In the vertical flight regime this velocity remains constant W = u and is independent of the azimuth. In this regime the "blade azimuth" concept has no meaning. In the forward flight regime the resultant blade element velocity in the hub rotation plane varies continuously.

Horizontal Flight Endurance and Range

Horizontal flight endurance is the time in the course of which the heli­copter can perform horizontal flight using the available fuel supply. Flight endurance is found from the formula

T = Gfuel
hr" %

where the fuel supply for horizontal flight, liters;

c^ is the fuel consumption per hour, liters/hr.

This formula shows that the endurance depends on the fuel supply and the hourly consumption. The fuel supply for horizontal flight is the

difference between the amount of fuel serviced into the tanks G and the amount of fuel expended in the other flight regimes: taxiing, takeoff, climb,

descent, and landing. The fuel consumption in these flight regimes is indi­cated in the instructions for calculating flight endurance and range, which are prepared for each helicopter type on the basis of calculations and flight tests.

Подпись: г
The hourly fuel consumption is the amount of fuel which the engine consumes per hour of operation. It is found from the formula

= = <29)

where ce is the specific fuel consumption;

N is the effective engine power; e

C, is the power utilization coefficient.

Since c£ and £ change only slightly with variation of the flight speed, their ratio can be assumed constant and (29) takes the form

ch = const /Vh.

This formula shows that the hourly fuel consumption depends on the /99

power required for helicopter horizontal flight, and consequently, on the flight speed.

Using the curve of power versus speed (see Figure 62), we can say that the minimal power required for horizontal flight corresponds to the economical speed; therefore, the minimal hourly fuel consumption corresponds to this speed.

In order for the helicopter to stay in the air for the maximal time, flight must be performed at the economical speed. The economical speed depends on helicopter weight: this speed increases with increase of the weight, and

the flight endurance decreases. Since the economical speed changes very little with altitude, the horizontal flight endurance decreases somewhat with increase of the altitude as a result of the increased fuel consumption in climb and descent.

Helicopter horizontal flight range is the distance which the helicopter can fly while utilizing the fuel supply for horizontal flight


Horizontal Flight Endurance and Rangewhere is the fuel consumption per kilometer, liters/km.

The horizontal flight range is larger, the larger the fuel supply and the lower the consumption per kilometer. The fuel supply is defined as in the flight endurance calculation. The consumption per kilometer is found from the formula

c* ^ if – = const ~r~.

The minimal fuel consumption per kilometer will be achieved with the

minimal ratio N, /V.


Let us examine the power required and available curves for horizontal flight (Figure 66). Any point on the power required curve corresponds to definite values of V and N. For example, the point 1 corresponds to the speed V^, and the power required N^. The ratio of these quantities is equal to tg Y, and therefore, the consumption per kilometer is

= const tg

This means that tg у must be minimal in order to obtain the minimal fuel con­sumption per kilometer. This corresponds to the angle between the tangent to the power required curve and the horizontal axis. The point of tangency will correspond to the optimal helicopter horizontal flight speed.

Thus, the maximal horizontal flight range is achieved at the optimal

speed, which corresponds to the minimal fuel consumption per kilometer only if the engine is carefully adjusted. In this case, the calculation of the maximal range is made using the fuel consumption per kilometer curves plotted

on the basis of experimental helicopter operational data. The speed obtained from these curves and the corresponding minimal consumption will be close to the optimal values.

Подпись: optПодпись: Figure 66. Dependence of N/V on flight speed.Horizontal Flight Endurance and RangeThe consumption per kilometer is the fuel burned per kilometer of air distance (relative to the air). Con­sequently, the flight range calculation made using (30) is valid only if there is no wind. If there is wind, the

flight range will vary, depending on the wind direction and velocity. The so – called navigational fuel supply, amounting to 10-15% of the required fuel, is set aside as reserve in case the weather conditions change. Since the power required for horizontal flight depends on helicopter weight and flight altitude, the fuel consumption per kilometer increases, and the range decreases with increase of the weight. The average flight weight is used for exact range calculations

Horizontal Flight Endurance and Range

G = G av to


where G is the takeoff weight;

Gf і is the fuel supply for horizontal flight.

With increase of the altitude the optimal speed increases somewhat, and the power required decreases, therefore, the fuel consumption per kilometer also decreases. But the climb to a higher altitude requires more fuel. In practice, the longest flight range without account for wind is obtained at an altitude from 1000 to 2000 meters. For the Mi-1, the minimal fuel consumption per kilometer is 0.56 liters/km at an altitude of 1000 meters and an indicated airspeed of 130 km/hr. The flight range with this fuel consumption rate is 370 km.

Helicopter Dynamic Stability

General analysis of dynamic stability. While static stability defines the attitude stability, dynamic stability defines the nature of the helicopter motion after disruption of equilibrium. In equilibrium the helicopter travels in a straight line with constant velocity and without rotation. Such motion is called undisturbed. If equilibrium is disrupted, the helicopter rotates about its axes and the flight velocity and direction change. This motion is called disturbed. Disturbed motion may be either aperiodic or oscillatory.

Aperiodic motion is motion in one direction from the equilibrium position. For example, when equilibrium is disturbed the helicopter center of gravity deviates (Figure 101, solid line). After elimination of the factor causing disruption of the equilibrium, the nature of the disturbed motion may differ.

Helicopter Dynamic Stability

Figure 101. Dynamic stability.

If the center of gravity approaches the line of unperturbed motion (Figure 101a, dashed line), the helicopter has aperiodic stability; if the

helicopter center of gravity continues to deviate further from the equilibrium line (Figure 101b) the helicopter has aperiodic instability. Oscillatory motion is periodic back-and-forth motion relative to the equilibrium line.

If after disruption of equilibrium the helicopter center of gravity travels along a wave-like curvilinear trajectory and this motion is damped, the helicopter has oscillatory dynamic stability (Figure 101c). If the amplitude of the disturbed oscillatory motion increases, the helicopter has dynamic instability or oscillatory instability (Figure lOld).

Most frequently the disturbed motion of the helicopter is oscillatory, and the oscillations will be complex, since the helicopter oscillates simul­taneously about all axes. Moreover, the short and long periodic oscillations are superposed on one another. The short-period helicopter oscillations are those about the center of gravity with account for the influence of the main rotor damping moment; the long-period oscillations are those about a center located at a considerable distance from the helicopter.

Helicopter transverse oscillations in the hovering regime. Let us

assume that the helicopter banks to the angle у in the hovering regime (Figure

102 a). We resolve the helicopter weight force into the components:

acting in the plane of symmetry, and G2 perpendicular to this plane. The

force = G sin у is unbalanced and causes sideslip of the helicopter. As

a result of the increase of the sideslip velocity, the main rotor cone-of-

revolution axis will tilt to the side opposite the slip (Figure 102b). The

force Px is created, which reduces the sideslip velocity and the moment of this

force reduces the bank angle. But the force P is less than the force G„;

о x 2

therefore, the sidelsip velocity will increase and the velocity will be maximal at the moment the helicopter arrives at the position shown in Figure 102 c.

Подпись:The helicopter continues its motion in the same Then the force G2 changes from driving to retarding,

Helicopter Dynamic Stability

Figure 102. Helicopter lateral oscillations.


Подпись:decreases. As a result, the tilt of the cone-of-revolution axis decreases and the moment of the force P^ about the longitudinal axis banks the heli­copter in the opposite direction. When the helicopter reaches its maximal deviation (Figure 102e) further motion terminates. The cone-of-revolution axis coincides with the huh axis and the force P =0. But the force G

X /.

reaches a maximum and causes motion in the reverse direction and the whole cycle repeats. This transverse rocking of the helicopter will increase continuously, and the helicopter will turn over if these oscillations are not terminated in time.

We have examined in this example only the transverse oscillations, but in reality the transverse oscillations are supplemented by longitudinal and directional oscillations; therefore the pattern of the oscillatory motions will be more complex.

Longitudinal oscillations of helicopter in flight with horizontal velocity. If the longitudinal equilibrium of a helicopter is disturbed, longitudinal oscillations develop (Figure 103),i. e., the helicopter will travel along a wave-like trajectory. The existence of such oscillations is confirmed by flight tests in which automatic instruments record the nature of the helicopter oscillations about all axes. The longitudinal oscillations have a considerably longer period (total oscillation time) than the transverse oscillations. The amplitude of the longitudinal oscillations increases in the course of time, although more slowly than the amplitude of the transverse oscillations. Helicopter oscillations about the vertical axis also take place; however, they are performed with a period longer than the transverse oscillations but shorter than the longitudinal oscillations.

Helicopter Dynamic Stability

Figure 103. Helicopter longitudinal oscillations.

From this analysis we can conclude that the helicopter has dynamic instability. Therefore, if the equilibrium of the helicopter is disturbed it will have an oscillatory motion with increasing amplitude and cannot by itself eliminate these oscillations. This means that in every case of equilibrium disruption the pilot must take measures to restore the equilibrium, i. e., he must control the helicopter.

Helicopter Forced Vibrations

There are in the helicopter many sources of exciting forces which cause forced vibrations. Such sources include: main and tail rotors, powerplant,

transmission gearboxes, and transmission shafts.

Each of these sources creates exciting forces with a definite frequency.

The lowest exciting force frequency is that of the main rotor. It may be found from the formula

where n^r is the main rotor exciting force frequency; ng is the main rotor rps; к is the number of main rotor blades.

The frequency of the main rotor exciting forces varies in the range of 8-16 vibrations per second. The tail rotor excites forces with a frequency of 10 – 60 vibrations per second. The transmission shafts and gearboxes create a still higher frequency of the exciting forces: from 50 to several

hundred vibrations per second. The powerplant yields a broad spectrum of exciting forces with frequency of 600 – 1000 vibrations per second.

The primary forced vibration source is the main rotor with hinged blade support. Blade oscillations relative to all the hinges are also the source of many vibrations.


Vibrations from the blades of the main and tail rotors are transmitted through the hubs and the airstream deflected by the blades. This slipstream strikes the tail boom and tail fin in the form of periodic pulses and causes vibrations.

All parts of the helicopter are subjected to forced vibrations, but the amplitude of these vibrations differs. The amplitude magnitude depends on the stiffness of the structure, the closeness of the source of the exciting forces, their magnitude and points of application, and on the degree of close­ness to resonance. The degree of closeness to resonance is determined by the relative frequency v, equal to the ratio of the exciting force frequency to the natural vibration frequency

v = —- •



The amplitude of forced vibrations can be expressed graphically, plotting the structure deformation vertically and the relative frequency horizontally (Figure 110). The relative deformation is the ratio of the deformation caused /180 by the dynamic load to the deformation created by the static load. From this graph we can draw the following conclusions:

The largest deformation or the largest amplitude occurs at resonance (v = 1). Therefore, resonant vibrations are very dangerous: they can lead

to structural failure due to material fatigue;

For v > 0.5 the vibration amplitude increases very rapidly and the structural deformation increases sharply;

For v > 1.5 there is a reduction of the structural deformation in com­parison with the deformation caused by a static load of the same magnitude.

Thus, to reduce the structural deformation it is necessary to reduce the degree of closeness to resonance by altering the natural vibration frequency.

If the exciting force frequency is high, the natural vibration frequency must be reduced. Rubber vibration dampers are used in mounting the engine to the frame to avoid resonance. The use of shock mounts reduces the stiffness of the frame-engine structure, which leads to reduction of the natural vibration frequency and increase of the relative frequency (v > 1.5).

Подпись: ; ^dyn 11 і' Figure 110. Relative vibration amplitude versus relative frequency. 0 0.5 1 1.5 V Another example. The main rotor provides low-frequency exciting forces. The main rotor gearbox is mounted rigidly to the gearbox frame, without shock absorbers. This type of mounting increases the natural vibration frequency, and as a result the relative frequency is considerably less than


Helicopter Forced Vibrations

The helicopter control linkage rods are most frequently subjected to forced vibrations. Therefore, it is particularly important to prevent resonance of the control rods. To this end the natural frequency of the rod is determined. If this frequency is close to the exciting force frequency in the region where the rod is located, the natural frequency must be changed. This frequency can be found from the approximate formula

where D is the rod cross-section diameter; l is the rod length;

E is the longitudinal elastic modulus; у is the specific weight of the material.

We see from this formula that the rod diameter must be increased or its length must be reduced in order to increase the natural vibration frequency.

If the rods are long, roller type supports are used to increase the frequency. When it is not possible to determine exactly the possibility of the occurrence of resonance, use is made of rods with inertial dampers. The inertial damper is a weight located inside the rod close to its midpoint, between two rubber plugs. The presence of the damper leads to rapid decay of the vibrations.

Under normal conditions the forced vibrations of the various parts of the helicopter are small; their amplitudes are measured in hundredths or tenths of a millimeter. However, in certain cases they may become hazardous if the normal operating conditions are exceeded.

Most frequently, magnification of the vibrations is caused by the failure of individual structural elements (stiffness is reduced and resonance occurs), by improper the adjustment of structural parts, and by mass unbalance. The acceptable vibration limit is determined by their effect on the structure and on the human organism. Vibrations are considered acceptable if they do not lead to structural failure and do not cause discomfort to the personnel (Figure 111). The higher the vibration frequency, the lower the vibration amplitude which can be endured by the personnel without pain.