Maneuverability characteristics. Complete system of equations of aircraft motion General vector equations of aircraft motion

The mathematical model of the control object is the basis for the description and study of processes in control loops and the basis for the synthesis of these loops. A mathematical model is constructed to describe a certain group of properties of a real infinitely complex control object.

Equations of spatial motion of an aircraft as a rigid body

In the aerodynamics of the aircraft, the following rectangular right-hand coordinate systems are adopted (Fig. 1.1). The earth's coordinate system, the axis of which is directed vertically, the axes have a constant orientation in the horizontal plane. For ordinary aircraft flight control problems, the influence of the Earth's rotation on the dynamics of motion can be neglected and the system can be considered inertial.

Intermediate (earthly central) coordinate system with

axes parallel to the axes of the earth's system and center O, aligned with the center of mass of the aircraft.

Associated coordinate system. The axes of this coordinate system

usually coincide with the main central axes of inertia of the aircraft. The axis coincides with the longitudinal main axis of inertia, the axis lies in the plane of symmetry, the axis is close to the plane of the wing or coincides with it.

Velocity coordinate system. The axis of this system is oriented along the aircraft's airspeed vector, the axis lies in the plane of symmetry of the aircraft (the lift axis).

The angle formed by the longitudinal axis of the aircraft with the horizontal

plane, is called pitch angle. The angle between the projection of the longitudinal axis onto the horizontal plane and a given direction is called yaw angle, course or track angle. The angle corresponding to the rotation of the aircraft around the longitudinal axis relative to the position at which the transverse axis is horizontal is called roll angle.

The position of the airspeed vector relative to the related axes of the aircraft is characterized by angle of attack b And sliding angle V. The angle of attack is the angle between the projection of the airspeed vector on the plane of symmetry of the aircraft and the longitudinal axis, the angle of slip is the angle formed by the airspeed vector with the plane of symmetry.

Fig.1.1 coordinate systems

Movement of an aircraft as a rigid body in a coupled coordinate system

are described by Euler's equations:

where are the components of the ground speed vector in the associated coordinate system; - components of the angular velocity vector in the associated coordinate system; X 1 ,Y 1 , Z 1, M x1, M y1 , M z1- forces and moments in a related coordinate system; I x ,I y ,I z- moments of inertia about the main axes; m - mass, g - acceleration due to gravity. The mathematical model represented by equations (1.1) - (1.6) corresponds to any rigid body with six degrees of freedom and, in relation to an aircraft, requires further addition.

This specification of the model consists, first of all, in revealing the dependences of forces and moments on aerodynamic and other parameters of motion (coordinates), deviations of controls and disturbing influences, which is the subject of aircraft aerodynamics. Within the framework of stationary aerodynamics, forces and moments acting on an aircraft are expressed as functions of flight parameters and control deflections. Moment of power M y1 expressed as a function of yaw angular velocity, sliding angle V. Angular velocity of roll, rudder deflection, aileron deflection, velocity pressure (- air density, V- airspeed in the absence of wind coinciding with ground speed), Mach number M. Upon closer examination (large angles of attack, in?0) moment M y1 turns out to also depend on the angle of attack b:

M y1= M y1. (1.7)

Forces and moments are not functions, but operators of flight parameters. However, the inertia of the corresponding operators is comparable to the time of movement of air particles relative to the surface, creating a force or moment, and is small. Therefore, the nonstationary nature of aerodynamics in most cases can be approximately taken into account by introducing the first time derivatives. So. The moment about the transverse axis, taking into account the delay of the flow bevel on the stabilizer, is taken as a function of not only the angle of attack, but the derivative of the angle of attack

M z1= M z1 ( 1.8)

Deflection of the elevator or stabilizer.

Detailed consideration of unsteady aerodynamics is necessary when considering some aeroelastic phenomena.

In the future, consideration will be carried out within the framework of stationary aerodynamics.

System of equations (1.1) - (1.6) even in the absence of deviations. The controls are not a closed system.

The direction cosines of the associated coordinate system relative to the earth are expressed through angles according to the formulas given in Table 1.1.

Table 1.1

The velocity components in the earth's coordinate system are related through the direction cosines of Table 1.1 to the quantities V x ,V y ,V z :

On the other hand, according to the data in Table 1.2, the components of ground speed in related axes in the absence of wind are related to the angle of attack and the angle of slip by the formulas

The derivatives of the pitch, roll and yaw angles are described by the expressions

The system of equations (1.1) - (1.6), (1.09), (1.10), (1.11) with the revealed dependences of forces and moments on flight parameters becomes a completely closed system of equations for the aircraft as a control object, if the dependence of air density and sound speed is known A(or temperature) from altitude N=, i.e. the atmospheric model is known. The closedness of the system of equations of an object means that its movement for given deviations of the controls is completely determined by this system of equations.

The mathematical model of the spatial motion of the aircraft as a rigid body, represented by the above equations and the atmospheric model, is asymmetrical and rather cumbersome. However, this model is traditional, at least as a step in the transition to simpler models. The widespread use of this model is due to the fact that it is based on standard angular coordinates: roll, yaw, pitch, slide and attack angles.

If we use direct direction cosines as coordinates of the angular position and express the aerodynamic forces and moments and engine thrust in the form of functions of projections of air speed onto the associated axes and other parameters, then the system of equations of spatial motion of the aircraft takes on a more symmetrical form:

Here is a quantity characterizing the engine thrust control.

If the inertia of traction control (unlimited engine response) is neglected, the value will coincide with the deflection of the engine control handle (motors).

Basic Concepts

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

When studying the stability and controllability of an aircraft, it is represented as a body moving translationally under the influence of external forces and rotating under the influence 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 balance is disturbed, then the center of mass of the aircraft will begin to move unevenly along a curved path, 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 OU axis is perpendicular to the OX axis, and the OZ axis is perpendicular to the XOU plane and is directed
towards the right wing.

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

M x – roll moment or transverse moment;

М Y – yaw moment or traveling moment;

M z – pitching moment or longitudinal moment.

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

Rice. 6.1. Moments acting on an airplane

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

If you look from the origin along the positive direction of the corresponding axis, then the clockwise rotation will be positive.

Thus,

· the moment M z is positive in case of pitching up,

· the moment M x is positive in the case of a roll to the right half-wing,

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

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

· 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 angle called 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 chord of the wing or the axis of the fuselage and the horizon line.

The bank angle is positive if the airplane is in a right bank.

The sliding angle is positive when sliding onto the right half-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 makes uniform linear motion.

From mechanics, 3 types of equilibrium are known:

a) stable b) indifferent c) unstable;

Rice. 6.2. Types of body balance

In the same types of equilibrium there may be
and a plane.

Longitudinal balance- this is a condition in which the aircraft has no desire to change the angle of attack.

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

Transverse balance- the plane has no tendency to change the bank angle.

The balance of the aircraft can be disturbed due to:

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

2) aircraft icing;

3) flying in rough air;

4) non-synchronous deviation of mechanization;

5) destruction of aircraft parts;

6) stall flow around the wing and tail.

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

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

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

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

For flight practice balancing
and the stability of the aircraft are not equivalent. It is impossible to fly on an airplane that is not properly balanced, while flying on an unstable airplane is possible.

The stability of an aircraft's movement is assessed using indicators of static and dynamic stability.

Under static stability refers to its tendency to restore the original equilibrium state after an accidental imbalance. If forces arise when balance is disturbed
and moments tending to restore balance, then the aircraft is statically stable.

When determining dynamic stability It is no longer the initial tendency to eliminate the disturbance that is assessed, but the nature of the disturbed movement of the aircraft. To ensure dynamic stability, the perturbed motion of the aircraft must decay quickly.

Thus, the aircraft is stable if:

· static stability;

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

Quantitative indicators of the static stability of an aircraft include the degree of longitudinal, directional and transverse static stability.

The characteristics of dynamic stability include indicators of the quality of the process of reducing (attenuating) disturbances: the decay time of deviations, the maximum values of deviations, the nature of movement in the process of reducing deviations.

Under aircraft controllability is understood as 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 largely depends on the controllability of the aircraft.

Maneuverability aircraft is its ability to change 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 during 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 has great stability,
then the effort when controlling the aircraft is excessively great and the pilot will quickly
tire. They say about such an aircraft that it is difficult to fly.

Excessively light control is also unacceptable, since it makes it difficult to precisely measure the deflections of the control levers and can cause the aircraft to sway.

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

Lateral stability and controllability are divided into transverse and directional (vane).

Longitudinal stability

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

Longitudinal stability is ensured by:

1) appropriate sizes horizontal tail g.o., the area of which depends on the wing area;

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

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

Rice. 6.3. Determination of the average aerodynamic chord

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

The magnitude and position of the MAR are most often found graphically.

The position of the aircraft’s center of mass, and therefore its alignment, depends on:

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

2) accommodation of passengers and fuel generation.

As centering decreases, stability increases, but controllability decreases.

As centering increases, 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 from the condition of ensuring sufficient stability.

Ensuring longitudinal stability at the angle of attack

Disturbance of longitudinal balance 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 the balance is disturbed, the stability of the aircraft in terms of the angle of attack (in terms of overload) is manifested.

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

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

Fig.6.4. Ensuring longitudinal stability of the aircraft

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

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

The position of the focus during 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 aircraft's layout, and during operation - by refueling or fuel exhaustion, 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 alignments within which the center of mass of the aircraft can be placed.

If the weights on the plane are placed 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 displacement of the center of mass relative to neutral alignment provides the aircraft with longitudinal static stability, and the displacement of the center of mass. backwards makes it statically unstable.

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

In this case, when the angle of attack accidentally changes, a stabilizing moment appears a, returning the aircraft to a given angle of attack (Fig. 6.4).

To shift the focus beyond the center of mass, horizontal tails are used.

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

There is a minimum acceptable margin of stability, which must be equal to at least 3% of the MAR.

The position of the central center at which the minimum permissible centering margin is ensured is called extremely rear centered. With this alignment, the aircraft still has stability, ensuring flight safety. Of course, the back
operational alignment must be less than the maximum permissible.

Permissible center displacement the forward direction of the aircraft is determined by the aircraft balancing conditions.
The worst mode in terms of balancing is the approach mode at low speeds, maximum permissible angles of attack and extended mechanization.
That's why extremely forward alignment is determined from the condition of ensuring the aircraft is balanced during landing mode.

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

When switching from subsonic to supersonic modes, the aircraft's focus shifts back, the balance margin increases several times, and longitudinal static stability increases sharply.

Balancing curves

The magnitude of the longitudinal moment M z that occurs when longitudinal equilibrium is disrupted depends on the change in the angle of attack Δα. This dependency is called balancing curve.

Rice. 6.5. Balancing curves:

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

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

At the trim angle of attack, the aircraft is in a state of longitudinal equilibrium.

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

Aircraft directional stability

Track (weathervane) stability- this is the ability of an aircraft to eliminate slipping without pilot intervention, i.e., to position itself “against the flow”, maintaining a given direction of movement.

Rice. 6.6. Aircraft directional stability

Track stability is ensured by appropriate dimensions vertical tail S v.o.
and the vertical tail arm L v.o, i.e. distance from the center of pressure v.o. to the center of mass of the aircraft.

Under the influence of M, the plane may rotate around the OY axis, but its c.m. by inertia, it still maintains the direction of movement and the aircraft flows around under
sliding angle β. As a result of asymmetrical flow, a lateral force Z appears, applied
in lateral focus. The plane, under the influence of force Z, tends to turn like a weather vane towards the wing on which it is sliding.

In. shifts the lateral focus beyond the central point. airplane. This ensures the creation of a stabilizing traveling moment ΔM Y =Zb.

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

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

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

Changing the angle of attack and releasing the mechanization have little effect on directional stability. In the range of M numbers from 0.2 to 0.9, the degree of directional stability practically does not change.

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

Maneuverability are implemented by the pilot during combat maneuvering, which consists of individual completed or unfinished aerobatic maneuvers, continuously following each other.

Maneuverability is one of the most important qualities combat aircraft any kind of aviation. It allows you to successfully conduct an air battle, overcome enemy air defenses, attack ground targets, build, rebuild and disband the battle formation (formation) of aircraft, bring them 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 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 fire even from a single cannon. On the contrary, if the enemy takes an advantageous position (for example, “hanging 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.

MANEUVERABILITY INDICATORS

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 depicted by vector V1 (Fig. 1), and after one second - by vector V2; then V2=V1+ΔV, where ΔV is the second vector velocity increment.

Rice. 1. Second vector speed increment

In Fig. 2 shown area of possible second vector speed increments for some aircraft during its maneuver 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 be inside the area limited by the line a-b-c-d-e. With the available thrust of the engines Рр, the end of the vector ΔV can only be on the boundary a-b-c-d, on which the following possible maneuvering options can be noted:

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

With zero thrust and brake flaps released, the end of the vector ΔV may appear in a second only at border d-e, for example, at 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 borders a-b-c-d and d-e. Segment g-d corresponds to turns at Sudop with different thrust.

Failure to understand the fact that maneuverability is determined by the second vector increment of speed, i.e., the value of ΔV, sometimes leads to an incorrect assessment of a particular aircraft. For example, before the war of 1941-1945. some pilots believed that our old I-16 fighter had higher maneuverability than the new Yak-1, MiG-3 and LaGG-3 aircraft. However, in maneuverable air battles the Yak-1 performed 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 smaller than those of the Yak-1 (Fig. 3); i.e., in fact, the Yak-1 had higher maneuverability, if the issue is not considered narrowly, from the point of view of “agility” alone. Similarly, it can be shown that, for example, the MiG-21 aircraft is more maneuverable than the MiG-17 aircraft.

The areas of possible increments of ΔV (Fig. 2 and 3) well illustrate the physical meaning of the concept of maneuverability, i.e., they provide a qualitative picture of the phenomenon, but do not allow for quantitative analysis, for which various kinds of particular and general indicators of maneuverability are involved.

The second vector speed increment ΔV is related to overloads by the following relationship:

Due to the earth's acceleration g, all aircraft receive the same speed increase ΔV (9.8 m/s², vertically down). The lateral overload nz is usually not used during maneuvering, so the aircraft’s maneuverability is completely characterized by two overloads - nx and ny (overload is a vector quantity, but in the future the sign of the vector “->” will be omitted).

The overloads nx and nу are thus general maneuverability indicators.

All particular indicators are associated with these overloads:

rg - radius of turn (turn) in the horizontal plane;
wg - angular speed of turn in the horizontal plane;
rв - maneuver radius in the vertical plane;
time to turn at a given angle;
wв - angular velocity of trajectory rotation in the vertical plane;
jx - acceleration in horizontal flight;
Vy - vertical speed at steady ascent;
Vye - speed of gaining energy height, etc.

OVERLOAD

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

Note 1. When moving on the ground, the ground reaction force also participates in the creation of normal overload.

Note 2. SARPP recorders record overloads in a related coordinate system, in which

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

Available normal overload nyр is the highest overload that can be used in flight while maintaining safety conditions.

If we substitute the available lift coefficient Cyr into the last formula, then the resulting overload will be available.

nyр=Cyр*S*q/G (2)

In flight, the value of Cyр, as already agreed, can be limited by stalling, shaking, pick-up (and then Cyр=Cydop) or by controllability (and then Cyр=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 nyе max.

The word “short-term” is sometimes added to the name of the overload nyр.

Using formula (2) and the function Cyr(M), one can obtain the dependence of the available overload nyр on the Mach number and flight altitude, which is shown graphically in Fig. 4 (example). Note that the contents of Figures 4,a and 4,6 are exactly the same. The top graph is commonly used for various calculations. However, for flight personnel it is more convenient to schedule in M-H coordinates(lower), in which the lines of constant available overloads are drawn directly within the range of altitudes and flight speeds of the aircraft. Let's analyze Fig. 4.6.

The line nyр=1 is obviously the boundary of horizontal flight already known to us. The line nyр=7 is the boundary, to the right and below which the maximum operational overload may be exceeded (in our example, nyе max=7).

Lines of permanent available overloads pass in such a way that nyp2/nyp1=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 limit over the range of altitudes and speeds.

Let, for example, it is required to determine nyр at M=1 and H=14 km (at point A in Fig. 4.6). Solution: we 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 = 14,750/5760 = 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 carried out according to the obvious proportion:

Conclusion. Having the level flight boundary (line nyp1=1) constructed for 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=(p2/p1)*(G1/G2) (3)

But in any case, the overload used in flight should not be greater than the maximum operating load. 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 aircraft. From the value of nyp during the most energetic unsteady maneuvers, one can determine such particular characteristics of the aircraft’s maneuverability as the current radii rg and rv, the current angular velocities wg and wv.

Thrust limit normal overload nypr is the greatest overload at which the drag Q becomes equal to the thrust Рр and at the same time nx=0. The word “long-term” is sometimes added to the name of this overload.

The maximum traction overload is calculated as follows:

for a given altitude and Mach number, we find the thrust Рр (according to the altitude-speed characteristics of the engine);
for nypr we have Pр=Q=Cx*S*q, from where we can find Cx;
from the grid of polars using the known M and Cx we find Cy;
calculate the lift force Y=Су*S*q;
We calculate the overload ny=Y/G, which will be the maximum thrust, since in the calculations we proceeded from the equality Рр=Q.

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

we find the thrust Рр;
Let's write Рр = Cр*S*q, where Ср is the thrust coefficient;
by condition we have Рр = Ср*S*q=Q=Cх*Q*S*q+(A*G²n²ypr)/(S*q), from which:

Inductive reactance is proportional to the square of the overload, i.e. Qi=Qi¹*ny² (where Qi¹ is inductive reactance at nу=1). Therefore, based on the equality Рр=Qo+Qи, we can write the expression for the maximum overload in this form:

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

You can notice that the lines nypr=1 in Fig. 5. is the boundary of steady horizontal flight already known to us.

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

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

Based on the value of nypr, one can find such particular characteristics of the aircraft’s maneuverability as radius rg, angular velocity wg, time tf of a steady turn, as well as r, w and t of any maneuver performed at constant energy (prl Pр=Q).

Longitudinal overload nx is the ratio of the difference between the thrust force (assuming Px = P) and drag to the weight of the aircraft

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

If we substitute the available thrust of the engines Рр into the last formula, we obtain the so-called available longitudinal overload:

Rice. 5.5. Thrust overload limits for the F-4C Phantom aircraft; afterburner, weight 17.6 m

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

we find the thrust Рр (according to the altitude-speed characteristics of the engine);
for a given normal overload ny, we calculate the drag as follows:
ny->Y->Сy->Сx->Q;
Using formula (5.7) we calculate nxр.

If the polar is a quadratic parabola, then you can use the expression Q=Q0+Qi¹*ny², as a result of which formula (5.7) takes the form

Recall that when ny=nypr the equality holds

Substituting this expression into the previous one and breaking it apart we get the final formula

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

In Fig. Figure 5.6 shows as an example the dependence of nxр¹ on M and N for the F-4C Phantom aircraft. You can notice that the curves nxр¹(M, Н) on a different scale approximately repeat the course of the curves nyр(М, Н), and the line nxр¹=0 exactly coincides with the line nyр=1. This is understandable, since both of these overloads are related to the thrust-to-weight ratio of the aircraft.

Based on the value of nxр¹, it is possible to determine such particular characteristics of the aircraft maneuverability as acceleration during horizontal acceleration jx, vertical speed of steady ascent Vy, speed of climb to energy altitude Vyе in unsteady linear ascent (descent) with a change in speed.

Fig. 5 6 Available longitudinal overloads in horizontal flight of the F-4C Phantom aircraft; afterburner, weight 17.6 t

8. All considered characteristic overloads (nU9, nupr, R*P> ^lgr1) are often depicted in the form of a graph shown in Fig. 5.7. It is called a graph of generalized aircraft maneuverability characteristics. According to Fig. 5.7 for a given height Hi for any number M, you can find pur (on the line Sur or n^max). %Pr (on the horizontal axis, i.e., for phr = 0), Lhr1 (for pu =) and pX9 (for any overload pu). Generalized characteristics are most convenient for various types of calculations, since any value can be directly taken from them, but they are not visual due to the large number of these graphs and 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 aircraft maneuverability at altitude Hi (example) To get a complete and clear picture of the aircraft’s maneuverability, it is enough to have three graphs p (M, H) - as in Fig. 5.4,6; pupr (M, N) - as in Fig. 5.5,6; nx p1 (M, N) - as in Fig. 5 6.6.

In conclusion, we will consider the question of the influence of operational factors on the available and maximum traction normal overloads and on the available longitudinal overload

Effect of weight

As can be seen from formulas (5.2) and (5.4), the available normal overload pur and the maximum thrust normal overload nypr change in inverse proportion to the weight of the aircraft (at constant M and N).

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

Influence of external suspensions

External suspensions can influence the listed overloads, firstly, through their weight and, secondly, through an additional increase in the non-inductive part of the aircraft's drag.

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

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

The available longitudinal overload lhr also decreases with increasing Cho. The influence of Схо on nxр becomes relatively greater as the overload nу increases during the maneuver.

Influence of atmospheric conditions.

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

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

Turning on the afterburner (or turning it off)

It greatly affects the maximum normal thrust overload nypr, and the available longitudinal overload nхр. Even at speeds and altitudes where Рр >> Qг, an increase in thrust, for example, by 2 times leads to an increase in npr by approximately sqrt(2) times and to an increase in nхр¹ (at nу = 1) by approximately 2 times.

At speeds and altitudes where the difference Рр - Qг is small (for example, near a static ceiling), a change in thrust leads to an even more noticeable change in both npr and nхр¹.

As for the available (according to Сyр) normal overload nyр, the amount of thrust has almost no effect on it (assuming Рy=0). But it should be taken into account that with greater thrust, the aircraft loses energy more slowly during the maneuver and, therefore, can remain at higher speeds for a longer time, at which the available overload nyр has the greatest value.

UDC 629.7333.015
A mathematical model of the spatial motion of a maneuverable aircraft, taking into account the unsteady effects of separated flow at large
angles of attack.
M. A. Zakharov.
Based on a refined model of aerodynamic coefficients longitudinal movement, taking into account the unsteady effects of separated flow at high angles of attack, a mathematical model of the spatial motion of a maneuverable aircraft was constructed, bringing its system of nonlinear differential equations to a canonical form. Initial data have been prepared for entering into the program for solving the specified system on a digital computer. The initial data on aerodynamic coefficients are taken from known ones (covering the ranges 0...900 for angles and -400...400 for angles) and approximately predicted for angles -7200...7200 according to the periodic law. The constructed model is illustrated with solutions for various positions of the aircraft controls.

1 Statement of the problem.
In connection with progress in the field of computer technology, it has become possible to quickly and accurately find a solution to a system of nonlinear differential equations for the spatial motion of aircraft. At the same time, the mathematical apparatus that fully describes this movement is not yet sufficiently developed. There are known works devoted to the consideration of mathematical models of the spatial motion of maneuverable aircraft (for example). At the same time, a mathematical model of aerodynamic coefficients and a motion model (in the form of a system of differential equations) are proposed separately. However, the construction of a general (joint) model for practical use is difficult due to the presence of aerodynamic coefficients of non-stationary components in the model (in particular, components corresponding to the structure of the separated flow around the wing). When substituting aerodynamic coefficients into common system equations, the latter cannot be solved on a digital computer. On the right side of the resulting system there are terms containing the derivatives of the angles of attack and sideslip (,). Another difficulty is that there is practically no information in the press about aerodynamic coefficients for the range of angles and . This paper attempts to overcome these difficulties.
Previously, based on a refined model of aerodynamic coefficients that takes into account the unsteady effects of separation flow at high angles of attack, a mathematical model of the longitudinal motion of a maneuverable aircraft was constructed. The logical conclusion of efforts to implement a refined model of aerodynamic coefficients should be the construction of a model of spatial motion of a maneuverable aircraft, including the specified model of coefficients.
It is also necessary to illustrate the constructed model with solutions when changing the position of the controls.

2 Assumptions, initial equations and construction of a mathematical model.
We assume that a rigid, maneuverable aircraft moves relative to a flat, non-rotating Earth in the absence of wind. The thrust axes of the right and left engines are parallel to the X axis of the associated coordinate system. In this case, the spatial motion of such an aircraft can be expressed by the following system of equations of dynamics and kinematics:
; (1)
; (2)
; (3)
; (4)
; (5)
; (6)
; (7)
; (8)
; (9)
Where:
; (10)
; (11)
; (12)
– linear speed of the center of mass (CM) of the aircraft; , , – its angular speeds of rotation relative to the X, Y, Z axes associated with the aircraft , – wing area;

– wing span; – average aerodynamic chord of the wing; , , – axial moments of inertia relative to the axes OX, OY, OZ;- attack angle; – sliding angle;

– roll angle;

– pitch angle;

Normal terrestrial coordinate system OXgYgZg. This system of coordinate axes has a constant orientation relative to the Earth. The origin of coordinates coincides with the center of mass (CM) of the aircraft. The 0Xg and 0Zg axes lie in the horizontal plane. Their orientation can be taken arbitrarily, depending on the goals of the problem being solved. When solving navigation problems, the 0Xg axis is often directed to the North parallel to the tangent to the meridian, and the 0Zg axis is directed to the East. To analyze the stability and controllability of an aircraft, it is convenient to take the direction of orientation of the 0Xg axis to coincide in direction with the projection of the velocity vector onto the horizontal plane at the initial moment of time of the motion study. In all cases, the 0Yg axis is directed upward along the local vertical, and the 0Zg axis lies in the horizontal plane and, together with the OXg and 0Yg axes, forms a right-handed system of coordinate axes (Fig. 1.1). The XgOYg plane is called the local vertical plane.

Associated coordinate system OXYZ. The origin of coordinates is located at the center of mass of the aircraft. The OX axis lies in the plane of symmetry and is directed along the wing chord line (or parallel to some other direction fixed relative to the aircraft) towards the nose of the aircraft. The 0Y axis lies in the symmetry plane of the aircraft and is directed upward (in horizontal flight), the 0Z axis complements the system to the right.

The angle of attack a is the angle between the longitudinal axis of the aircraft and the projection of airspeed onto the OXY plane. The angle is positive if the projection of the aircraft's airspeed onto the 0Y axis is negative.

The glide angle p is the angle between the aircraft's airspeed and the OXY plane of the associated coordinate system. The angle is positive if the projection of the airspeed onto the transverse axis is positive.

The position of the associated coordinate axes system OXYZ relative to the normal earthly coordinate system OXeYgZg can be completely determined by three angles: φ, #, y, called angles. Euler. Sequentially rotating the connected system

coordinates to each of the Euler angles, one can arrive at any angular position of the associated system relative to the axes of the normal coordinate system.

When studying aircraft dynamics, the following concepts of Euler angles are used.

Yaw angle r]) is the angle between some initial direction (for example, the 0Xg axis of the normal coordinate system) and the projection of the associated axis of the aircraft onto the horizontal plane. The angle is positive if the OX axis is aligned with the projection of the longitudinal axis onto the horizontal plane by turning clockwise around the OYg axis.

Pitch angle # - the angle between the longitudinal# axis of the aircraft OX and the local horizontal plane OXgZg, The angle is positive if the longitudinal axis is above the horizon.

The roll angle y is the angle between the local vertical plane passing through the OX y axis and the associated 0Y axis of the aircraft. The angle is positive if the O K axis of the aircraft is aligned with the local vertical plane by turning clockwise around the OX axis. Euler angles can be obtained by successive rotations of related axes about the normal axes. We will assume that the normal and related coordinate systems are combined at the beginning. The first rotation of the system of connected axes will be made relative to the O axis by the yaw angle r]; (f coincides with the OYgX axis in Fig. 1.2)); the second rotation is relative to the 0ZX axis at an angle Ф (‘& coincides with the OZJ axis and, finally, the third rotation is made relative to the OX axis at an angle y (y coincides with the OX axis). Projecting the vectors Ф, Ф, у, which are the components

vector of the angular velocity of the aircraft relative to the normal coordinate system, onto the related axes, we obtain equations for the relationship between the Euler angles and the angular velocities of rotation of the related axes:

co* = Y + sin *&;

o)^ = i)COS’&cosY+ ftsiny; (1.1)

co2 = φ cos y - φ cos φ sin y.

When deriving the equations of motion for the center of mass of an aircraft, it is necessary to consider the vector equation for the change in momentum

-^- + o>xV)=# + G, (1.2)

where ω is the vector of rotation speed of the axes associated with the aircraft;

R is the main vector of external forces, in the general case aerodynamic

logical forces and traction; G is the vector of gravitational forces.

From equation (1.2) we obtain a system of equations of motion of the aircraft CM in projections onto related axes:

t (gZ?~ + °hVx ~ °ixVz) = Ry + G!!’ (1 -3)

t iy’dt “b U - = Rz + Gz>

where Vx, Vy, Vz are projections of velocity V; Rx, Rz - projections

resultant forces (aerodynamic forces and thrust); Gxi Gyy Gz - projections of gravity onto related axes.

Projections of gravity onto related axes are determined using direction cosines (Table 1.1) and have the form:

Gy = - G cos ft cos y; (1.4)

GZ = G cos d sin y.

When flying in an atmosphere stationary relative to the Earth, projections of flight speed are related to the angles of attack and glide and the magnitude of the speed (V) by the relations

Vx = V cos a cos p;

Vу = - V sin a cos р;

Expressions for the projections of the resulting forces Rx, Rin Rz have the following form:

Rx = - cxqS - f Р cos ([>;

Rty = cyqS p sin (1.6)

where cx, cy, сг - coefficients of projections of aerodynamic forces on the axes of the associated coordinate system; P - gyga of engines (usually P = / (U, #)); Fn - engine stall angle (ff > 0, when the projection of the thrust vector onto the 0Y axis of the aircraft is positive). Further, we will take = 0 everywhere. To determine the density p (H) included in the expression for the velocity pressure q, it is necessary to integrate the equation for the height

Vx sin ft+ Vy cos ft cos y - Vz cos ft sin y. (1.7)

The dependence p (H) can be found from tables of the standard atmosphere or from the approximate formula

where for flight altitudes I s 10,000 m K f 10~4. To obtain a closed system of equations of aircraft motion in related axes, equations (13) must be supplemented with kinematic

relations that make it possible to determine the aircraft orientation angles y, ft, r]1 and can be obtained from equations (1.1):

■ф = Кcos У - sin V):

■fr= “y sin y + cos Vi (1-8)

and the angular velocities cov, co, coz are determined from the equations of motion of the aircraft relative to the CM. The equations of motion of an aircraft relative to the center of mass can be obtained from the law of change in angular momentum

-^-=MR-ZxK.(1.9)

This vector equation uses the following notation: ->■ ->

K is the moment of momentum of the aircraft; MR is the main moment of external forces acting on the aircraft.

Projections of the angular momentum vector K onto the moving axes are generally written in the following form:

K t = I x^X? xy®y I XZ^ZI

К, Iу^х Н[ IУ^У Iyz^zi (1.10)

K7. - IXZ^X Iyz^y Iz®Z*

Equations (1.10) can be simplified for the most common case of analyzing the dynamics of an aircraft having a plane of symmetry. In this case, 1хг = Iyz - 0. From equation (1.9), using relations (1.10), we obtain a system of equations for the motion of the aircraft relative to the CM:

If we take the main axes of inertia as the SY OXYZ, then 1xy = 0. In this regard, we will carry out further analysis of the dynamics of the aircraft using the main axes of inertia of the aircraft as the OXYZ axes.

The moments included in the right-hand sides of equations (1.11) are the sum of aerodynamic moments and moments from engine thrust. Aerodynamic moments are written in the form

where tХ1 ty, mz are the dimensionless coefficients of aerodynamic moments.

The coefficients of aerodynamic forces and moments are generally expressed in the form of functional dependencies on the kinematic parameters of motion and similarity parameters, depending on the flight mode:

y, g mXt = F(a, p, a, P, coXJ coyj co2, be, f, bn, M, Re). (1.12)

The numbers M and Re characterize the initial flight mode, therefore, when analyzing stability or controlled movements, these parameters can be taken as constant values. In the general case of motion, the right side of each of the equations of forces and moments will contain a rather complex function, determined, as a rule, on the basis of approximation of experimental data.

Fig. 1.3 shows the rules of signs for the main parameters of the movement of the aircraft, as well as for the magnitudes of deviations of the controls and control levers.

For small angles of attack and sideslip, the representation of aerodynamic coefficients in the form of Taylor series expansions in terms of motion parameters is usually used, preserving only the first terms of this expansion. This mathematical model of aerodynamic forces and moments for small angles of attack agrees quite well with flight practice and experiments in wind tunnels. Based on materials from works on the aerodynamics of aircraft for various purposes, we will accept the following form of representing the coefficients of aerodynamic forces and moments as a function of motion parameters and deflection angles of controls:

сх ^ схо 4~ сх (°0"

U ^ SU0 4" s^ua 4" S!/F;

сг = cfp + СгН6„;

th - itixi|5 - f - ■b thha>x-(- th -f - /l* (I -|- - J - L2LP6,!

o (0.- (0^- r b b„

tu = myfi + tu ho)x + tu Uyy + r + ga/be + tu bn;

tg = tg(a) + tg zwz/i? f.

When solving specific problems of flight dynamics, the general form of representing aerodynamic forces and moments can be simplified. For small angles of attack, many aerodynamic coefficients of lateral motion are constant, and the longitudinal moment can be represented as

mz(a) = mzo + m£a,

where mz0 is the longitudinal moment coefficient at a = 0.

The components included in expression (1.13), proportional to the angles α, are usually found from static tests of models in wind tunnels or by calculation. To find

Research Institute of Derivatives, twx (y) is required

dynamic testing of models. However, in such tests there is usually a simultaneous change in angular velocities and angles of attack and sliding, and therefore during measurements and processing the following quantities are simultaneously determined:

CO - CO- ,

tg* = t2g -mz;

0), R. Yuu I century.

mx* = mx + mx sin a; tu* = Shuh tu sin a.

CO.. (O.. ft CO-. CO.. ft

ty% = t,/ -|- tiiy cos a; tx% = txy + tx cos a.

The work shows that to analyze the dynamics of an aircraft,

especially at low angles of attack, it is permissible to represent the moment

com in the form of relations (1.13), in which the derivatives mS and m$

taken equal to zero, and under the expressions m®x, etc.

the quantities m“j, m™у are understood [see (1.14)], determined experimentally. Let us show that this is acceptable by limiting our consideration to the problems of analyzing flights with small angles of attack and sideslip at a constant flight speed. Substituting expressions for velocities Vх, Vy, Vz (1.5) into equations (1.3) and making the necessary transformations, we obtain

= % COS a + coA. sina - f -^r )

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