General vector equations of aircraft motion. Mathematical model of the spatial motion of a maneuverable aircraft. Influence of atmospheric conditions

The presence of a plane of material symmetry in the aircraft makes it possible to divide its spatial motion into longitudinal and lateral. Longitudinal motion refers to the movement of an aircraft in a vertical plane in the absence of roll and slip, with a neutral position of the rudder and ailerons. In this case, two translational and one rotational movement occur. The translational motion is carried out along the velocity vector and along the normal, rotational - around the Z axis. Longitudinal motion is characterized by the angle of attack α, the angle of inclination of the trajectory θ, the pitch angle, the flight speed, the flight altitude, as well as the position of the elevator and the magnitude and direction in the vertical plane of the thrust DU.

The system of equations for the longitudinal motion of the aircraft.

The closed system describing the longitudinal motion of the aircraft can be separated from the complete system of equations, provided that the lateral motion parameters, as well as the deflection angles of the roll and yaw controls are equal to 0.

The relation α \u003d ν - θ is derived from the first geometric equation after its transformation.

The last equation of system 6.1 does not affect the others and can be solved separately. 6.1 is a nonlinear system, since contains products of variables and trigonometric functions, expressions for aerodynamic forces.

To obtain a simplified linear model of the longitudinal motion of the aircraft, it is necessary to introduce certain assumptions and carry out the linearization procedure. In order to substantiate additional assumptions, we need to consider the dynamics of the longitudinal motion of the aircraft with a stepped deflection of the elevator.

Airplane response to elevator stepped deflection. The division of the longitudinal movement into long-term and short-term.

With a stepped deviation δ at, a moment M z (δ at) arises, which rotates about the Z axis at a speed ω z. This changes the pitch and attack angle. With an increase in the angle of attack, an increase in lift occurs and the corresponding moment of longitudinal static stability M z (Δα), which opposes the moment M z (δ in). After the expiration of the rotation, at a certain angle of attack, it compensates for it.

The change in the angle of attack after balancing the moments M z (Δα) and M z (δ c) stops, but, since the plane has certain inertial properties, i.e. has a moment of inertia I z relative to the OZ axis, then the establishment of the angle of attack is oscillatory.

The angular vibrations of the aircraft around the ОZ axis will be damped using the intrinsic moment of aerodynamic damping M z (ω z). The increase in lift begins to change the direction of the velocity vector. The angle of inclination of the trajectory θ also changes. This, in turn, affects the angle of attack. Proceeding from the balance of moment loads, the pitch angle continues to change synchronously with the change in the angle of inclination of the trajectory. In this case, the angle of attack is constant. Angular movements in a small interval occur with high frequency, i.e. have a short period and are called short-period.

After the short-term fluctuations have died out, the change in flight speed becomes noticeable. Mainly due to the Gsinθ component. The change in speed ΔV affects the increment of lift and, as a consequence, the angle of inclination of the trajectory. The latter changes the flight speed. In this case, fading oscillations of the velocity vector in magnitude and direction arise.

These movements are characterized by a low frequency, fade out slowly, therefore they are called long-period.

When considering the dynamics of the longitudinal motion, we did not take into account the additional lifting force created by the deflection of the elevator. This effort is aimed at reducing the total lift, therefore, for heavy aircraft, there is a drawdown phenomenon - a qualitative deviation of the trajectory inclination angle with a simultaneous increase in the pitch angle. This occurs until the incremental lift compensates for the lift component by deflecting the elevator.

In practice, long-term fluctuations do not occur, because are promptly extinguished by the pilot or automatic controls.

Transfer functions and structural diagrams of the longitudinal motion mathematical model.

The transfer function is called the image of the output magnitude, according to the image of the input at zero initial conditions.

A feature of the transfer functions of the aircraft, as a control object, is that the ratio of the output quantity, in comparison with the input, is taken with a negative sign. This is due to the fact that in aerodynamics it is customary to consider the deviations that create negative increments of the aircraft motion parameters as the positive deflection of the controls.

In the operator form, the record looks like:

System 6.10, which describes the short-term movement of the aircraft, correspond to the following solutions:

(6.11)

(6.12)

Thus, we can write down the transfer functions that relate the angle of attack and the angular velocity in pitch from the deflection of the elevator

(6.13)

In order for the transfer functions to have a standard form, we introduce the following notation:

, , , , ,

Given these ratios, we rewrite 6.13:

(6.14)

Thus, the transfer functions for the angle of inclination of the trajectory and for the pitch angle, depending on the deviation of the elevator, will have the following form:

(6.17)

One of the most important parameters that characterize the longitudinal motion of an aircraft is normal overload. Overload can be: Normal (along the OY axis), longitudinal (along the OX axis) and lateral (along the OZ axis). Calculated as the sum of the forces acting on the aircraft in a specific direction divided by the force of gravity. The projections on the axis allow you to calculate the value and its relation to g.

- normal overload,

From the first equation of forces of the system 6.3 we obtain:

Using overload expressions, we rewrite:

For level flight conditions (:

Let's write a block diagram that corresponds to the transfer function:

-δ in M \u200b\u200bω z ν ν α - θ θ

Lateral force Z a (δ n) creates a roll moment M x (δ n). The ratio of the moments M x (δ n) and M x (β) characterizes the forward and reverse reaction of the aircraft to the deflection of the rudder. In the event that M x (δ n) is greater in modulus than M x (β), the aircraft will tilt in the opposite direction of the turn.

Taking into account the above, we can construct a structural diagram for analyzing the lateral movement of an aircraft when the rudder is deflected.

-δ н М у ω y ψ ψ
β β F z Ψ 1 Mx
ω y ω x

In the so-called flat turn mode, the roll moments are compensated by the pilot or the corresponding control system. It should be noted that with a small lateral movement, the aircraft rolls, along with this, a tilt of the lift occurs, which causes a lateral projection Y a sinγ, which begins to develop a large lateral movement: the aircraft begins to slide on the inclined wing, while the corresponding aerodynamic forces and moments increase, and hence the so-called "spiral moments" begin to play a role: М у (ω х) and М у (ω z). It is advisable to consider a large lateral movement when the aircraft is already tilted, or on the example of the dynamics of the aircraft when the ailerons are deflected.

Aircraft response to aileron deflection.

When the ailerons are deflected, the moment M x (δ e) arises. The aircraft begins to rotate around the bound axis ОХ, and the roll angle γ appears. The damping moment M x (ω x) counteracts the rotation of the aircraft. When the aircraft is tilted due to a change in the roll angle, a lateral force Z g (Ya) arises, which is the result of the weight force and lift Y a. This force "unfolds" the velocity vector, while the track angle Ψ 1 begins to change, which leads to the occurrence of the slip angle β and the corresponding force Z a (β), as well as the moment of the track static stability М у (β), which begins to unfold the longitudinal axis aircraft with angular velocity ω y. As a result of this movement, the yaw angle ψ begins to change. The lateral force Z a (β) is directed in the opposite direction with respect to the force Z g (Ya), therefore, to some extent, it reduces the rate of change of the track angle Ψ 1.

The force Z a (β) is also the cause of the lateral static stability moment. M x (β), which in turn tries to get the plane out of the roll, and the angular velocity ω y and the corresponding spiral aerodynamic moment M x (ω y) try to increase the roll angle. If M x (ω y) is greater than M x (β), a so-called "spiral instability" arises, in which the roll angle continues to increase after the ailerons return to the neutral position, which leads to a turn of the aircraft with increasing angular velocity.

Such a turn is called a coordinated turn, and the roll angle is set by the pilot or by the automatic control system. At the same time, in the process of turning, the disturbing roll moments M x β and M x ωу are compensated, the rudder compensates for the slip, that is, β, Z a (β), М у (β) \u003d 0, while the moment М у (β ), which turned the longitudinal axis of the aircraft, is replaced by the moment from the rudder M y (δ n), and the lateral force Z a (β), which prevented a change in the track angle, is replaced by the force Z a (δ n). In the case of a coordinated turn, the speed (maneuverability) increases, while the longitudinal axis of the aircraft coincides with the airspeed vector and turns synchronously with the change in angle Ψ 1.

Basic concepts

Stability and controllability are among the particularly important physical properties of an aircraft. Flight safety, simplicity and accuracy of piloting and full implementation of the technical capabilities of the aircraft by the pilot largely depend on them.

When studying the stability and controllability of an aircraft, it is represented as a body moving translationally under the action of external forces and rotating under the action of the moments of these forces.

For steady flight, it is necessary that the forces and moments are mutually balanced.

If, for some reason, this equilibrium is violated, then the center of mass of the aircraft will begin to make an uneven movement along a curved trajectory, and the aircraft itself will begin to rotate.

The axes of rotation of the aircraft are considered to be the axes of the associated coordinate system with the origin
at the center of mass of the aircraft. The OX axis is located in the plane of symmetry of the aircraft and is directed along its longitudinal axis. The ОУ axis is perpendicular to the ОХ axis, and the ОZ axis is perpendicular to the ХОУ plane and is directed
towards the right wing.

The moments that rotate the plane around these axes have the following names:

M x - roll moment or lateral moment;

М Y - yaw moment or travel moment;

М z - pitching moment or longitudinal moment.

The moment M z, which increases the angle of attack, is called pitching, and the moment M z, which causes a decrease in the angle of attack, is called diving.

Figure: 6.1. Moments affecting the plane

To determine the positive direction of the moments, the following rule is used:

if the gaze is directed from the origin along the positive direction of the corresponding axis, then the clockwise rotation will be positive.

Thus,

The moment М z is positive in the case of nose-up,

Moment М х is positive in case of roll to the right wing,

· The moment М Y is positive when the aircraft turns to the left.

Positive steering deviation corresponds to negative torque and vice versa. Therefore, the following should be considered as a positive deflection of the rudders:

Elevator - down,

Steering wheel - to the right,

· Right aileron - down.

The position of the aircraft in space is determined by three angles - pitch, roll and yaw.

Roll anglecalled the angle between the horizon line and the OZ axis,

sliding angle- the angle between the velocity vector and the plane of symmetry of the aircraft,

pitch angle- the angle between the wing chord or fuselage axis and the horizon line.

The roll angle is positive if the aircraft is in right bank.

The slip angle is positive when sliding to the right wing.

The pitch angle is considered positive if the nose of the aircraft is raised above the horizon.

Equilibrium is a state of an airplane in which all the forces and moments acting on it are mutually balanced and the airplane performs uniform rectilinear motion.

Three types of equilibrium are known from mechanics:

a) stable b) indifferent c) unstable;

Figure: 6.2. Types of body balance

In the same types of equilibrium, there can be
and the plane.

Longitudinal balance - This is a state in which the aircraft does not have a tendency to change the angle of attack.

Travel balance - the plane has no tendency to change the direction of flight.

Transverse equilibrium - the aircraft has no tendency to change the roll angle.

The aircraft's balance can be disturbed due to:

1) violation of engine operating modes or their failure in flight;

2) aircraft icing;

3) flying in turbulent air;

4) asynchronous deviation of mechanization;

5) destruction of aircraft parts;

6) stall flow around the wing, tail.

Ensuring a certain position of a flying aircraft in relation to the trajectory of movement or in relation to terrestrial objects is called balancing the aircraft.

In flight, aircraft balancing is achieved by deflecting the controls.

Aircraft stabilityits ability to independently restore an accidentally disturbed balance without the intervention of a pilot is called.

According to N.E. Zhukovsky, stability is the strength of movement.

For flight practice, balancing
and the stability of the aircraft is not equal. You cannot fly on an airplane that is not balanced, while an unstable airplane can fly.

Aircraft motion stability is assessed using static and dynamic stability indicators.

Under static stability its tendency to restore the initial equilibrium state after an accidental disturbance of equilibrium is understood. If, when the balance is disturbed, forces
and moments seeking to restore balance, the plane is statically stable.

In determining dynamic stability it is no longer the initial tendency to eliminate the disturbance that is evaluated, but the nature of the flow of the disturbed motion of the aircraft. To ensure dynamic stability, the disturbed motion of the aircraft must be rapidly decaying.

Thus, the aircraft is stable in the presence of:

· Static stability;

· Good damping properties of the aircraft, contributing to intensive damping of its oscillations in disturbed motion.

The quantitative indicators of the static stability of the aircraft include the degree of longitudinal, directional and lateral static stability.

The characteristics of dynamic stability include indicators of the quality of the process of reduction (attenuation) of disturbances: the decay time of deviations, the maximum values \u200b\u200bof deviations, the nature of movement in the process of decreasing deviations.

Under aircraft controllability means its ability to perform, at the will of the pilot, any maneuver provided for by the technical conditions for a given type of aircraft.

Its maneuverability also largely depends on the aircraft's controllability.

Maneuverability airplane is called its ability to change the speed, altitude and direction of flight over a certain period of time.

The controllability of an aircraft is closely related to its stability. Controllability with good stability provides the pilot with ease of control, and, if necessary, allows you to quickly correct an accidental error made in the control process,
and it is also easy to return the aircraft to the specified balancing conditions when exposed to external disturbances.

The stability and controllability of the aircraft must be in a certain ratio.

If the plane is very stable,
then the efforts when controlling the aircraft are excessively large and the pilot will quickly
tire. Such an aircraft is said to be difficult to fly.

Excessively light control is also unacceptable, since it makes it difficult to accurately meter the deviations of the control levers and can cause the aircraft to swing.

Balancing, stability and controllability of the aircraft is divided into longitudinal and lateral.

Lateral stability and controllability are subdivided into lateral and track (weather vane).

Longitudinal stability

Longitudinal stability is the ability of the aircraft to restore the disturbed longitudinal balance without the intervention of the pilot (stability relative to OZ)

Longitudinal stability is provided by:

1) the corresponding dimensions of the horizontal tail tail, the area of \u200b\u200bwhich depends on the wing area;

2) the shoulder of the horizontal tail L, i.e. the distance from the center of mass of the aircraft to the center of pressure

3) Centering, i.e. distance from toe average aerodynamic chord (MAX) to the center of mass of the aircraft, expressed as a percentage of the MAC:

Figure: 6.3. Determination of the mean aerodynamic chord

MAR (b a) is a chord of a certain conventional rectangular wing, which with the same area as a real wing has the same coefficients of aerodynamic forces and moments.

The magnitude and position of MAR is most often found graphically.

The position of the center of mass of the aircraft, which means that its alignment depends on:

1) aircraft loading and changes in this load in flight;

2) passenger accommodation and fuel production.

With a decrease in centering, stability increases, but controllability decreases.

With an increase in centering, stability decreases, but controllability increases.

Therefore, the front centering limit is set from the condition of obtaining a safe landing speed and sufficient controllability, and the rear limit is set from the condition of ensuring sufficient stability.

Ensuring longitudinal stability in the angle of attack

Longitudinal imbalance is expressed
in changing the angle of attack and flight speed, and the angle of attack changes much faster than the speed. Therefore, at the first moment after imbalance, the stability of the aircraft in the angle of attack (overload) appears.

When the longitudinal balance of the aircraft is disturbed, the angle of attack changes by an amount and causes a change in the lift force by an amount that is the sum of the increments of the wing lift and the horizontal tail:

The wing and the aircraft as a whole have an important property, which is that when the angle of attack changes, the aerodynamic load is redistributed so that the resultant of its increase passes through the same point F, which is distant from the nose of the MAR at a distance X f.

Figure 6.4. Ensuring the longitudinal stability of the aircraft

The point of application of the incremental lift caused by a change in the angle of attack at a constant speed is called focus.

Longitudinal static stability
aircraft is determined by the relative position of the center of mass and focus of the aircraft.

The focus position for continuous flow does not depend on the angle of attack.

The position of the center of mass, i.e. aircraft alignment is determined during the design process by the layout of the aircraft, and during operation - by refueling or using fuel, loading, etc. By changing the alignment of the aircraft, you can change the degree of its longitudinal static stability. There is a certain range of center of gravity within which the center of mass of the aircraft can be located.

If weights are placed on the plane so that the center of mass of the plane coincides with its focus, the plane will be indifferent to imbalance. Centering in this case is called neutral.

The forward shift of the center of mass relative to the neutral center provides the aircraft with longitudinal static stability, and the c.m. back makes it statically unstable.

Thus, to ensure the longitudinal stability of the aircraft, its center of mass must be in front of the focus.

In this case, with a random change in the angle of attack, a stabilizing moment appears a, returning the aircraft to a given angle of attack (Figure 6.4).

To shift the focus beyond the center of mass and use the horizontal tail.

The distance between the center of mass and the focus, expressed in fractions of MAR, is called the overload stability margin, or centering margin:

There is a minimum allowable stability margin, which must be at least 3% MAR.

The position of the CM, at which the minimum allowable centering margin is ensured, is called extremely rear centering... With such an alignment, the aircraft still has a stability that ensures flight safety. Of course, the back
the operational alignment should be less than the maximum allowable.

Permissible displacement of c.m. aircraft forward is determined by the balancing conditions of the aircraft.
The worst in terms of balancing is the approach mode at low speeds, maximum permissible angles of attack and released mechanization.
therefore extreme front centering is determined from the condition of ensuring the balancing of the aircraft in the landing mode.

For non-maneuverable aircraft, the balance margin should be 10–12% of the MAR.

During the transition from subsonic to supersonic modes, the focus of the aircraft shifts back, the balance margin increases several times and the longitudinal static stability increases sharply.

Balancing curves

The value of the longitudinal moment M z, which occurs when the longitudinal equilibrium is violated, depends on the change in the angle of attack Δα. This dependence is called balancing curve.

Mz

Figure: 6.5. Balancing curves:

a) stable plane, b) indifferent plane,
c) unstable plane

The angle of attack at which M z \u003d 0 is called the balancing angle of attack α.

At the trim angle of attack, the aircraft is in longitudinal balance.

At the corners a stable plane creates a stabilizing moment - (dive moment), an unstable one - destabilizing +, an indifferent plane does not create, i.e. has many balancing angles of attack.

Aircraft flight stability

Track (weather vane) stability- this is the ability of the aircraft to eliminate slip without the intervention of the pilot, that is, to be installed "upstream", maintaining the given direction of movement.

Figure: 6.6. Aircraft flight stability

The track stability is provided by the corresponding dimensions of the vertical tail S in.
and the shoulder of the vertical tail L v.o, i.e. distance from the center of pressure to the center of mass of the aircraft.

Under the influence of M, the plane rotates around the OY axis, but its CM. by inertia, it still retains the direction of movement and the plane is flown around under
sliding angle β. As a result of asymmetric flow, a lateral force Z arises, applied
in lateral focus. The plane, under the influence of the Z force, tends to turn like a weather vane towards the wing on which it glides.

In. shifts the lateral focus behind the CM. aircraft. This ensures the creation of a stabilizing travel moment ΔM Y \u003d Zb.

The degree of static directional stability is determined by the value the derivative of the yaw moment coefficient with respect to the slip angle m.

Physically, m determines the magnitude of the increase in the yaw moment coefficient if the slip angle changes by 1.

For an aircraft with directional stability, it is negative. Thus, when sliding onto the right wing (positive), a ground moment appears that rotates the plane to the right, i.e. coefficient m is negative.

Changes in the angle of attack, release of mechanization insignificantly affect the directional stability. In the range of M numbers from 0.2 to 0.9, the degree of directional stability practically does not change.

The types of motion are considered, the trajectories of which lie strictly either in the vertical or in the horizontal planes. This, of course, is some kind of schematization, but quite acceptable. However, in the general case, the flight trajectory does not lie in one plane, but is spatial. Such maneuvers include combat turn, spiral, oblique loop, roll, etc. Consider the first of the listed maneuvers.

A combat turn is a maneuver of an aircraft, in which, simultaneously with a change in the direction of flight, an increase in altitude is made. The spatial trajectory of such a maneuver is, as it were, a combination of a bend and a hill (Fig. 7.10). When calculating a combat turn, the influence of the lateral force Za and the overload nza is small,

Figure: 7.11. A typical program for changing the ya roll and overloading the pua during a combat turn

and the maneuver can be considered coordinated, p "0, nza" 0, if the NUBS authorities are not applied.

< Расчет боевого разворота ведется численным интегрированием уравнений (7.10) … (7.14).

To calculate the trajectory of the combat turn, in addition to setting the engine operating mode (usually the maximum mode is taken), it is necessary to have two more control functions, for which it is convenient to take nv о (W) and у а (V).

A typical view of the roll and overload change is shown in Fig. 7.11. The choice of the values \u200b\u200bof the parameters yt »x and putax depends on the task of the combat turn. It can be seen from the equations of motion that the lower the overload, the lower the angular velocity of rotation, and the longer the time it takes to complete the combat turn. An increase in roll at a given g-load leads to a decrease in gain. In the extreme case, you can pick up such a large roll that the combat turn will turn into a turn. At very small bank angles, the trajectory will approach the trajectory of the hill.

If the requirement for a minimum maneuver time is presented to the combat turn, without setting the conditions for maximum climb, then equation (7.11) shows that with an increase in the overload and roll angle, the angular velocity of the trajectory rotation increases. From this point of view, the usual law of variation of these parameters, shown in
fig. 7.11, is unprofitable, because in. at the end of the maneuver, the product pua sin ya turns out to be small and the turn is delayed. It is possible to reduce the time of the combat turn by applying the law of roll change shown in Fig. 7.11 dotted line. In this case, by the end of the turn, the aircraft is almost in an inverted position and it is possible to withstand a constant maximum overload until the very end of the maneuver. Such a combat turn, by analogy with a bend, can be called forced. If the purpose of the turn is to increase the altitude, then a slight overload should be taken, and the roll change law should be taken as usual.

Diagrams of some other spatial maneuvers are given in Fig. 7.12.

The possibilities of performing any maneuver, both flat and spatial, are limited by the available value of the normal overload Pua rasp AND the MINIMAL EVOLUTIONAL VELOCITY of flight at which the maneuver is possible (pua rasp\u003e 1, the effectiveness of the controls remains, no stall occurs, etc.).

The maneuverability can be increased by adopting a wing with a profile variable according to flight modes (speed, angle of attack) for aircraft that require high maneuverability indicators. Thus, by deflecting the slats and flaps in flight when reaching large angles of attack, it is possible to significantly increase the acre and Su-dop, prevent stall and stall, and significantly reduce the limit of the minimum speed during maneuver 114]. Such control of the wing configuration during maneuvering should be performed automatically, since the pilot's attention is overloaded during piloting. The speed of the drives, control elements, ma-. neural mechanization of the wing should be sufficient to flexibly change their position during vigorous maneuvers. However, if such a system can be created, then the maneuverability of the aircraft at low speeds increases significantly.

Further reading, p. 104-114, P01, p. 278-294,, p. 339-390.

test questions

1. What maneuver is called coordinated?

2. Why, with a coordinated maneuver in the horizontal plane, is there an unambiguous connection between Pua and ya?

3. What is the limitation of the available value of pua at low indicated flight speeds? On the big ones?

4. Why does the minimum flight speed Utsch (Yia treb) increase with the growth of pu.1res?

5. Output the formula for i? in. Pr for poausT determined by (7.9). Analyze the RB dependency. up from height.

6. Show the approximate nature of the change in the overload of the PUA when performing the Nesterov loop, roll.

Maneuverability aircraft is called its ability to change the vector of flight speed in magnitude and direction.

Maneuverable properties are implemented by the pilot during combat maneuvering, which consists of individual completed or unfinished aerobatics figures, continuously following each other.

Maneuverability is one of the most important qualities of a combat aircraft of any kind of aviation. It allows you to successfully conduct air combat, overcome enemy air defenses, attack ground targets, build, rebuild and dissolve the battle formation (formation) of aircraft, deploy to an object at a given time, etc.

Maneuverability is of particular and, one might say, decisive importance for a front-line fighter conducting an air battle with an enemy destroyer (fighter-bomber). Indeed, having taken an advantageous tactical position in relation to the enemy, you can shoot him down with one or two missiles or even fire from a single cannon. On the contrary, if the enemy occupies an advantageous position (for example, "hangs on his tail"), then any number of missiles and guns will not help in such a situation. High maneuverability also allows for successful exit from air combat and separation from the enemy.

INDICATORS OF MANEUVERABILITY

In the most general case maneuverability aircraft can be fully characterized second vector increment speed. Let at the initial moment of time the magnitude and direction of the aircraft's speed be represented by the vector V1 (Fig. 1), and after one second - by the vector V2; then V2 \u003d V1 + ΔV, where ΔV is the second vector velocity increment.

Fig. 1. Secondary vector speed increment

In fig. 2 depicts area of \u200b\u200bpossible second vector velocity increments for a certain aircraft when it maneuvers in the horizontal plane. The physical meaning of the graph is that after one second the ends of the vectors ΔV and V2 can only appear inside the area bounded by the line a-b-c-d-e-f. With the available engine thrust Pp, the end of the vector ΔV can only be on the border a-b-c-d, on which the following possible options for maneuvering can be noted:

• a - acceleration in a straight line,
• b - U-turn with acceleration,
• c - steady reversal,
• d - forced turn with braking.

With zero thrust and released brake flaps, the end of the vector ΔV can be in a second only at the border d-e, for example, at the points:

• d - energetic turn with braking,
• e - braking in a straight line.

With intermediate thrust, the end of the vector ΔV can be at any point between the boundaries a-b-c-d and e-f. The segment d-d corresponds to turns at Sydop with different thrust.

Failure to understand the fact that maneuverability is determined by the second vector velocity increment, that is, by the value of ΔV, sometimes leads to an incorrect assessment of one or another aircraft. For example, before the war of 1941-1945. some pilots believed that our old I-16 fighter had better maneuverability than the new Yak-1, MiG-3 and LaGG-3 aircraft. However, in maneuverable air battles, the Yak-1 showed itself better than the I-16. What's the matter? It turns out that the I-16 could quickly "turn", but its second increments ΔV were much less than that of the Yak-1 (Fig. 3); that is, in fact, the Yak-1 had higher maneuverability, if the question is not considered narrowly, from the point of view of only one "agility". Similarly, it can be shown that, for example, the MiG-21 is more maneuverable than the MiG-17.

The areas of possible increments ΔV (Figs. 2 and 3) illustrate well the physical meaning of the concept of maneuverability, that is, they give a qualitative picture of the phenomenon, but do not allow quantitative analysis, for which various kinds of particular and generalized indicators of maneuverability are involved.

The second vector velocity increment ΔV is associated with overloads by the following relationship:

Due to the ground acceleration g, all aircraft receive the same speed increment ΔV (9.8 m / s², vertically downward). Lateral overload nz during maneuvering is usually not used, therefore the maneuverability of the aircraft is fully characterized by two overloads - nx and ny (overload is a vector value, but in the future the sign of the vector "-\u003e" will be omitted).

The nx and ny overloads are thus general indicators of maneuverability.

All private indicators are associated with these overloads:

• rg - radius of turn (bend) in the horizontal plane;
• wg - angular rate of turn in the horizontal plane;
• rв - radius of maneuver in the vertical plane;
• turn time to a given angle;
• wв - angular velocity of the trajectory rotation in the vertical plane;
• jx - acceleration in level flight;
• Vy - vertical speed at steady rise;
• Vye - the rate of energy altitude climb, etc.

OVERLOADS

Normal overload ny is the ratio of the algebraic sum of the lift and the vertical component of the thrust force (in the flow coordinate system) to the weight of the aircraft:

Note 1. When driving on the ground, the reaction force of the ground is also involved in the creation of normal overload.

Note 2. The SARPP recorders register overloads in the associated coordinate system, in which

On airplanes of the usual scheme, the value of Ru is relatively small and is neglected. Then the normal overload will be the ratio of the lift to the weight of the aircraft:

Disposed normal overload nyр is called the greatest overload that can be used in flight with observance of safety conditions.

If the available lift coefficient Cyр is substituted into the last formula, then the resulting overload will be available.

nyр \u003d Cyр * S * q / G (2)

In flight, the value of Cyр, as already agreed, can be limited by stalling, shaking, catching (and then Cyр \u003d Cydop) or by controllability (and then Cyр \u003d Cyf). In addition, the value of nyр can be limited by the strength conditions of the aircraft, i.e., in any case, nyр cannot be greater than the maximum operational overload nye max.

The word "short-term" is sometimes added to the name of the overload nyp.

Using formula (2) and the function Cyр (M), it is possible to obtain the dependence of the available overload nyр on the number M and the flight altitude, which is shown graphically in Fig. 4 (example). Note that the content of Figures 4, a and 4.6 is exactly the same. The top graph is usually used for various calculations. However, for the flight crew, it is more convenient to have a graph in the M-H coordinates (lower), on which the lines of constant located g-forces are drawn directly within the range of altitudes and speeds of the aircraft flight. Let's analyze Fig. 4.6.

The line nyр \u003d 1, obviously, is the boundary of horizontal flight already known to us. Line nyр \u003d 7 is the border to the right and below which the maximum operating overload can be exceeded (in our example, nyэ max \u003d 7).

Lines of permanent disposable overloads pass in such a way that nyp2 / nyp1 \u003d p2 / p1, i.e., between any two lines, the difference in height is such that the pressure ratio is equal to the overload ratio.

Based on this, the available overload can be found by having only one horizontal flight boundary in the range of altitudes and speeds.

Suppose, for example, it is required to determine nyр at M \u003d 1 and H \u003d 14 km (at point A in Fig. 4.6). Solution: find the height of point B (20 km) and the pressure at this height (5760 N / m2), as well as the pressure at a given height of 14 km (14 750 N / m2); the desired overload at point A will be nyр \u003d 14 750/5760 \u003d 2.56.

If it is known that the graph in Fig. 4 is built for the weight of the aircraft G1 and we need the available overload for the weight G2, then the recalculation is made according to the obvious proportion:

Output. Having the horizontal flight boundary (line nyp1 \u003d 1), plotted for the weight G1, it is possible to determine the available overload at any altitude and flight speed for any weight G2 using the proportion

nyp2 / nyp1 \u003d (p2 / p1) * (G1 / G2) (3)

But in any case, the overload used in flight should not exceed the maximum operational. Strictly speaking, for an aircraft subject to large deformations in flight, formula (3) is not always valid. However, this remark usually does not apply to fighter planes. By the value of nyp during the most energetic unsteady maneuvers, it is possible to determine such particular characteristics of the maneuverability of the aircraft as the current radii rg and rv, the current angular velocities wg and ww.

Maximum thrust normal overload nypr is the greatest overload at which the drag Q becomes equal to the thrust Pp and at the same time nx \u003d 0. The word "long-term" is sometimes added to the name of this overload.

The thrust limit overload is calculated as follows:

• for a given height and number M, we find the thrust Pp (according to the altitude and speed characteristics of the engine);
• for nyпр we have Pр \u003d Q \u003d Cx * S * q, whence we can find Cx;
• from the grid of polar by the known M and Cx we find Cy;
• calculate the lifting force Y \u003d Su * S * q;
• we calculate the overload ny \u003d Y / G, which will be the ultimate in thrust, since in the calculations we proceeded from the equality Pp \u003d Q.

The second method of calculation is used when the plane's polars are quadratic parabolas and when instead of these polarities in the plane's description the curves Cx0 (M) and A (M) are given:

• we find the thrust Pp;
• we write down Рр \u003d Ср * S * q, where Ср is a thrust coefficient;
• by condition, we have Pp \u003d Cp * S * q \u003d Q \u003d Cх * Q * S * q + (A * G²n²yпр) / (S * q), whence:

The inductive reactance is proportional to the square of the overload, ie Qi \u003d Qi¹ * ny² (where Qi¹ is the inductive reactance at nу \u003d 1). Therefore, based on the equality Pp \u003d Qo + Qi, it is possible to write an expression for the ultimate overload in the following form:

The dependence of the ultimate overload on the M number and flight altitude is shown graphically in Fig. 5.5 (example taken from the book).

It can be seen that the lines nypr \u003d 1 in Fig. 5. is already known to us the boundary of steady horizontal flight.

In the stratosphere, the air temperature is constant and the thrust is proportional to the atmospheric pressure, i.e., Рp2 / Рp1 \u003d р2 / p1 (here the thrust coefficient Ср \u003d const), therefore, in accordance with formula (5.4), for a given number М in the stratosphere, the proportion takes place:

Consequently, the maximum thrust overload at any height over 11 km can be determined by the pressure p1 on the line of static ceilings, where nypr1 \u003d 1. Below 11 km, the proportion (5.6) is not observed, since the thrust with a decrease in the flight altitude grows more slowly than the pressure (due to an increase in the air temperature), and the value of the thrust coefficient Cp decreases. Therefore, for heights of 0-11 km, the calculation of the maximum thrust overloads has to be done in the usual way, that is, using the altitude and speed characteristics of the engine.

By the value of nypr, one can find such particular characteristics of the maneuverability of the aircraft as the radius rg, the angular velocity wg, the time tf of the steady turn, as well as r, w and t of any maneuver performed at constant energy (prl Pp \u003d Q).

Longitudinal overload nx is the ratio of the difference between the thrust force (assuming Px \u003d P) and frontal resistance to the weight of the aircraft

Note When driving on the ground, the friction of the wheels must be added to the resistance.

If we substitute the available thrust of the engines Pp into the last formula, then we get the so-called available longitudinal overload:

Fig. 5.5. Maximum thrust overloads of the F-4C "Phantom" aircraft; afterburner, mass 17.6 m

Calculation of available longitudinal overload for an arbitrary value of nу, we perform as follows:

• we find the thrust Pp (according to the altitude and speed characteristics of the engine);
• for a given normal overload ny, we calculate the drag in the following way:
  ny-\u003e Y-\u003e Cy-\u003e Cx-\u003e Q;
• by formula (5.7) we calculate nxp.

If the polar is a quadratic parabola, then you can use the expression Q \u003d Q0 + Q and * ny², as a result of which formula (5.7) takes the form

Recall that for ny \u003d nypr the equality

Substituting this expression into the previous one and expanding we get the final formula

If we are interested in the value of the available longitudinal overload for horizontal flight, i.e., for ny \u003d 1, then formula (5.8) takes the form

In fig. 5.6 as an example, the dependence of nxp¹ on M and H is shown for the F-4C "Phantom" aircraft. It can be seen that the curves nxр¹ (M, H) on a different scale approximately repeat the course of the curves nyпр (М, Н), and the line nxр¹ \u003d 0 exactly coincides with the line nyпр \u003d 1. This is understandable, since both of these overloads are associated with the thrust-to-weight ratio of the aircraft.

The value of nxр¹ can be used to determine such particular characteristics of aircraft maneuverability as acceleration during horizontal acceleration jx, vertical speed of steady ascent Vy, rate of energy altitude gain Vye in unsteady rectilinear ascent (decrease) with a change in speed.

Fig 5 6 Positioned longitudinal G-forces in horizontal flight of the F-4C "Phantom" aircraft; afterburner, weight 17.6 t

8. All the considered characteristic overloads (nV9, npr, R * P\u003e ^ ngr1) are often depicted in the form of a graph shown in Fig. 5.7. It is called the Airplane's Generalized Maneuverability Graph. Fig. 5.7 for a given height Hi, for any number M, one can find pur (on the line Sup or n ^ max). % Pr (on the horizontal axis, that is, at nxp \u003d 0), Lxp1 (at ny \u003d) and nX9 (at any overload ny). Generalized characteristics are most convenient for various kinds of calculations, since any value can be directly removed from them, but they are not visual due to the large number of these graphs and the curves on them (for each height, you need to have a separate graph similar to that shown in Fig.5.7). Fig. 5 7 Generalized characteristics of the maneuverability of the aircraft at altitude Hi (example) To get a complete and visual representation of the maneuverability of the aircraft, it is enough to have three p (M, H) plots as in Fig. 5.4.6; pupr (M, H) - as in fig. 5.5.6; nx p1 (M, H) - as in Fig. 5 6.6.

In conclusion, let us consider the question of the influence of operating factors on the available and ultimate thrust normal overloads and on the available longitudinal overload

Influence of weight

As can be seen from formulas (5.2) and (5.4), the available normal overload pur and the maximum thrust normal overload nyпр change inversely proportional to the weight of the aircraft (at constant M and H).

If an overload ny is specified, then with an increase in the weight of the aircraft, the longitudinal available overload nxр decreases in accordance with formula (5.7), but simple inverse proportionality is not observed here, since with an increase in G, the drag Q also increases.

Influence of external suspensions

The above-mentioned overloads can be influenced by external suspensions, firstly, through their weight and, secondly, through an additional increase in the non-inductive part of the aircraft drag.

The available normal overload nyр is not affected by the suspension resistance, since this overload depends only on the value of the available lift of the wing.

The thrust limiting overload nypr, as can be seen from formula (5.4), decreases if Cx increases. The greater the thrust and the greater the difference Cp - Cx, the less the influence of the suspension resistance on the ultimate overload.

The positioned longitudinal overload rxp also decreases with increasing Cx. The influence of Cxo on nxp becomes relatively greater with an increase in the overload ny during the maneuver.

Influence of atmospheric conditions.

For definiteness of reasoning, we will consider an increase in temperature by 1% at a standard pressure p; the air density p will be 1% less than the standard one. Where from:

• at a given airspeed V, the available (by Cyp) normal overload pur will drop by about 1%. But at a given indicator speed Vp or the number M, the overload nur will not change with an increase in temperature;
• the maximum thrust normal overload nypr at a given number M will fall, since an increase in temperature by 1% leads to a drop in thrust Pp and thrust coefficient Cp by about 2%;
• the available longitudinal overload nхр with an increase in the air temperature will also decrease in accordance with the drop in thrust.

Turn on afterburner (or turn it off)

It has a very strong effect on the normal thrust limit nyпр, and the available longitudinal overload nхр. Even at speeds and altitudes, where Рр \u003e\u003e Qg, an increase in thrust, for example, 2 times leads to an increase in npr by about sqrt (2) times and to an increase in nхр¹ (at nу \u003d 1) by about 2 times.

At speeds and altitudes, where the difference Pp - Qg is small (for example, near a static ceiling), a change in thrust leads to an even more tangible change in both ncr and nxp¹.

As for the available (according to Cyr) normal overload nyр, the thrust value has almost no effect on it (assuming Рy \u003d 0). But it should be borne in mind that with greater thrust, the aircraft loses energy more slowly during maneuvering and, therefore, for a longer time can be at higher speeds, at which the available overload nyр has the greatest value.