Coordinates from the equation of motion of an aircraft on the ground. Linearization of the equations of longitudinal motion of an aircraft. Safety factors

The plane moves in the air under the influence of aerodynamic force, engine thrust and gravity. We became acquainted with the aerodynamic force and its projections on the axes of various coordinate systems when studying the fundamentals of aerodynamics. Traction force is created power plant airplane. The vector is usually located in the base plane of the aircraft and forms a certain angle with the 0 axis X associated coordinate system, but for simplicity we will assume that this angle is equal to zero, and the vector itself is applied at the center of mass.

An airplane flight can be roughly divided into several stages: takeoff, climb, horizontal flight, descent and landing. The plane can also perform turns and other maneuvers. At some stages of the flight, the movement of the aircraft can be either steady or unsteady. In steady motion, the aircraft flies at a constant speed, with constant angles of attack, roll and sideslip. Below we will consider only steady motion during the stages of horizontal flight, climb and descent.

Steady level flight is straight flight at a constant speed at a constant altitude (see Fig. 39). The equations of motion for the center of mass of the aircraft will be written in this case as follows:

(48)

Since the angle of attack a is small (cos a » 1, and sin a » 0), we can write:

Rice. 39. Diagram of forces acting on an airplane in steady state

horizontal flight

If the first of these equalities is not satisfied, then the speed of the aircraft will either increase or decrease, i.e. the condition of steady motion will not be satisfied. If the lifting force is not equal to the force of gravity, then the plane will either rise or descend, which means that the condition of horizontal flight will not be met. From this equality, knowing the lift force formula (35), we can obtain the speed required to perform horizontal flight V g.p.

Considering that G = mg(Where m is the mass of the aircraft, and g– free fall acceleration), can be written:

, (50)

(51)

From this formula it is clear that the speed of horizontal flight depends on the mass of the aircraft, air density r (which depends on the flight altitude), and wing area S kr and lift coefficient C ya. Because the C ya directly depends on the angle of attack a, then each value of the horizontal flight speed will correspond to a single value of the angle of attack. Therefore, to ensure steady horizontal flight at the required speed, the pilot sets a certain engine thrust and angle of attack.

Steady climb is the straight-line upward movement of the aircraft at a constant speed. The diagram of the forces acting on the aircraft during a steady climb with a trajectory inclination angle q is shown in Fig. 40.

Rice. 40. Diagram of forces acting on the aircraft at steady state

climb (the angle of attack is assumed to be small and is not shown)

In this case, the equations of motion will take the form:

(52)

It should be noted that when climbing, the engine thrust P balances not only the drag force X a, as in horizontal flight, but also the gravity component G sinq. Lifting force Y a in this case less is required, since G cosq< G.

An important characteristic of an aircraft is its rate of climb - vertical speed of climb. V y. From Fig. 40 it is clear that:

. (53)

Steady descent is the straight downward movement of the aircraft at a constant speed. In Fig. Figure 41 shows a diagram of the forces acting on the aircraft during descent.

Rice. 41. Diagram of forces acting on the aircraft at steady state

descent (the angle of attack is assumed to be small and is not shown)

The equations of motion for a steady descent are:

(54)

If we divide the first equation of system (54) by the second, we get:

. (55)

From equation (55) it is clear that a steady descent is possible only if the thrust is less than the drag ( P < X a). Typically, the decrease occurs at low thrust values (at low throttle thrust), so we can assume that P» 0. This flight mode is called planning. In this case:

. (56)

An important characteristic is the planning range L pl from a given height H pl. It's easy to see that:

. (58)

From formula (58) it is clear that the higher the aerodynamic quality of the aircraft, the greater the gliding range will be.

Usually, the flight of an airplane is considered as the movement in space of an absolutely rigid body. When compiling the equations of motion, the laws of mechanics are used, which make it possible to write in the most general form the equations of motion of the center of mass of the aircraft and its rotational motion around the center of mass.

The initial equations of motion are first written in vector form

m - aircraft weight;

The resultant of all forces;

The main moment of the external forces of the aircraft, the vector of the total torque;

Vector of the angular velocity of the coordinate system;

The moment of momentum of the aircraft;

The sign "" denotes a vector product. Next, they move on to the usual scalar notation of equations, projecting vector equations onto a certain system of coordinate axes.

Received general equations turn out to be so complex that they essentially exclude the possibility of conducting a visual analysis. Therefore in aerodynamics aircraft Various simplifying techniques and assumptions are introduced. Very often it turns out to be advisable to divide the total movement of the aircraft into longitudinal and lateral. Longitudinal motion is called motion with zero roll when the gravity vector and the aircraft velocity vector lie in its plane of symmetry. Further we will consider only the longitudinal movement of the aircraft (Fig. 1).

We will conduct this consideration using coupled OXYZ and semi-coupled OX e Y e Z e coordinate systems. The origin of coordinates of both systems is taken to be the point at which the center of gravity of the aircraft is located. The OX axis of the associated coordinate system is parallel to the chord of the wing and is called the longitudinal axis of the aircraft. The normal OY axis is perpendicular to the OX axis and is located in the plane of symmetry of the aircraft. The OZ axis is perpendicular to the OX and OY axes, and therefore to the plane of symmetry of the aircraft. It is called the transverse axis of the aircraft. The OX e axis of the semi-coupled coordinate system lies in the plane of symmetry of the aircraft and is directed along the projection of the velocity vector onto it. The OY e axis is perpendicular to the OX e axis and is located in the plane of symmetry of the aircraft. The OZ e axis is perpendicular to the OX e and OY e axes.

The remaining designations adopted in Fig. 1: - angle of attack, - pitch angle, - trajectory inclination angle, - airspeed vector, - lift force, - engine thrust force, - drag force, - gravity force, - elevator deflection angle, - pitch moment rotating the aircraft around the OZ axis.

Let's write the equation longitudinal movement aircraft center of mass

where is the total vector of external forces. Let's represent the velocity vector using its module V and the angle of its rotation relative to the horizon:

Then the derivative of the velocity vector with respect to time will be written as:

Taking into account this equation for the longitudinal motion of the aircraft’s center of mass in a semi-coupled coordinate system (in projections on the OX e and OY e axes) will take the form:

The equation for the rotation of the aircraft around the associated axis OZ has the form:

where J z is the moment of inertia of the aircraft relative to the OZ axis, M z is the total torque relative to the OZ axis.

The resulting equations completely describe the longitudinal motion of the aircraft. In the course work, only the angular motion of the aircraft is considered, so in what follows we will only take into account equations (4) and (5).

According to Fig. 1, we have:

angular velocity of rotation of the aircraft around the transverse axis OZ (angular velocity of pitch).

When assessing the quality of aircraft controllability, overload is of great importance. It is defined as the ratio of the total force acting on the aircraft (without taking into account weight) to the weight force of the aircraft. In the longitudinal movement of an aircraft, the concept of “normal overload” is used. According to GOST 20058-80, it is defined as the ratio of the projection of the main vector of the system of forces acting on the aircraft, without taking into account inertial and gravitational forces, onto the OY axis of the associated coordinate system to the product of the mass of the aircraft and the acceleration of gravity:

Transient processes in overload and pitch angular velocity determine the pilot's assessment of the quality of controllability of the longitudinal movement of the aircraft.

Department: TAU

CALCULATION OF THE LAW OF CONTROL OF LONGITUDINAL MOTION OF AN AIRCRAFT

Introduction

1. Mathematical description of the longitudinal motion of the aircraft

1.1 General information

1.2 Equations of longitudinal motion of an aircraft

1.3 Forces and moments during longitudinal motion

1.4 Linearized equations of motion

1.5 Mathematical model of the stabilizer drive

1.6 Mathematical models of angular velocity and overload sensors

1.7 Mathematical model of the steering wheel position sensor

2. Terms of reference for the development of an algorithm for manual control of the longitudinal movement of the aircraft

2.1 General provisions

2.2 Requirements for static characteristics

2.3 Dynamic performance requirements

2.4 Requirements for parameter spreads

2.5 Additional requirements

3. Course work plan

3.1 Analysis phase

Introduction

The purpose of the course work is to consolidate the material of the first part of the TAU course and master the modal methodology for calculating control algorithms using the example of the synthesis of the law of control of the longitudinal movement of an aircraft. The guidelines contain the derivation of mathematical models of the longitudinal movement of the aircraft, electro-hydraulic elevator drive, helm position sensors, pitch angular velocity, overload, and also provide numerical data for a hypothetical aircraft.

One of the most crucial and difficult moments when implementing the modal synthesis technique is the choice of the desired eigenvalues. Therefore, recommendations for their selection are given.

Mathematical description of the longitudinal motion of an aircraft

General information

The flight of an aircraft is carried out under the influence of forces and moments acting on it. By deflecting the controls, the pilot can adjust the magnitude and direction of forces and moments, thereby changing the parameters of the aircraft's movement in the desired direction. For straight and uniform flight it is necessary that all forces and moments are balanced. So, for example, in straight horizontal flight at a constant speed, the lift force is equal to the gravity force of the aircraft, and the engine thrust is equal to the drag force. In this case, the balance of moments must be maintained. Otherwise, the plane begins to rotate.

The equilibrium created by the pilot can be disrupted by the influence of some disturbing factor, for example, atmospheric turbulence or gusts of wind. Therefore, when the flight mode is set, it is necessary to ensure motion stability.

Another important characteristic of an aircraft is controllability. The controllability of an aircraft is understood as its ability to respond to movement of the control levers (controls). Pilots say about a well-controlled aircraft that it “follows the handle” well. This means that in order to perform the required maneuvers, the pilot needs to perform simple deflections of the levers and apply small but clearly noticeable forces to them, to which the aircraft responds with corresponding changes in position in space without unnecessary delay. Controllability is the most important characteristic of an aircraft, determining its ability to fly. It is impossible to fly an uncontrollable plane.

It is equally difficult for a pilot to control an airplane when it is necessary to apply large forces to the control levers and perform large movements of the yoke, as well as when the deflections of the yoke and the forces required to deflect them are too small. In the first case, the pilot quickly gets tired when performing maneuvers. Such an aircraft is said to be “difficult to fly.” In the second case, the aircraft reacts to small, sometimes even involuntary movement of the stick, requiring a lot of attention from the pilot, precise and smooth control. They say about such an aircraft that it is “strict in control.”

Based on flight practice and theoretical research, it has been established what the characteristics of stability and controllability should be in order to meet the requirements for convenient and safe piloting. One of the options for formulating these requirements is presented in the terms of reference for the course work.

Equations of longitudinal motion of an aircraft

The initial equations of motion are first written in vector form

m – aircraft weight;

– resultant of all forces;

– the main moment of the external forces of the aircraft, the vector of the total torque;

– vector of angular velocity of the coordinate system;

– moment of momentum of the aircraft;

t – time.

The sign "" denotes a vector product. Next, they move on to the usual scalar notation of equations, projecting vector equations onto a certain system of coordinate axes.

The resulting general equations turn out to be so complex that they essentially exclude the possibility of conducting a visual analysis. Therefore, various simplifying techniques and assumptions are introduced in the aerodynamics of aircraft. Very often it turns out to be advisable to divide the total movement of the aircraft into longitudinal and lateral. Longitudinal motion is called motion with zero roll when the gravity vector and the aircraft velocity vector lie in its plane of symmetry. Further we will consider only the longitudinal movement of the aircraft (Fig. 1).

The remaining designations adopted in Fig. 1: – angle of attack, – pitch angle, – trajectory inclination angle, – airspeed vector, – lift force, – engine thrust force, – drag force, – gravity force, – elevator deflection angle, – pitching moment rotating the aircraft around the OZ axis.

Let us write down the equation for the longitudinal motion of the aircraft's center of mass

, (1)

where is the total vector of external forces. Let's represent the velocity vector using its module V and the angle of its rotation relative to the horizon:

Then the derivative of the velocity vector with respect to time will be written as:

. (2)

Taking into account this equation for the longitudinal motion of the center of mass of the aircraft in a semi-coupled coordinate system (in projections on the OX e and OY e axes) will take the form:

The equation for the rotation of the aircraft around the associated axis OZ has the form:

where J z is the moment of inertia of the aircraft relative to the OZ axis, M z is the total torque relative to the OZ axis.

According to Fig. 1, we have:

angular velocity of rotation of the aircraft around the transverse axis OZ (angular velocity of pitch).

When assessing the quality of aircraft controllability, overload is of great importance. It is defined as the ratio of the total force acting on the aircraft (without taking into account weight) to the weight force of the aircraft. In the longitudinal movement of an aircraft, the concept of “normal overload” is used. According to GOST 20058–80, it is defined as the ratio of the projection of the main vector of the system of forces acting on the aircraft, without taking into account inertial and gravitational forces, onto the OY axis of the associated coordinate system to the product of the mass of the aircraft and the acceleration of gravity:

Transient processes in overload and pitch angular velocity determine the pilot's assessment of the quality of controllability of the longitudinal movement of the aircraft.

Forces and moments during longitudinal motion

The forces and moments acting on the aircraft are complex nonlinear functions that depend on the flight mode and the position of the control elements. Thus, the lift force Y and the drag force Q are written as:

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Safety factors

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In the longitudinal plane, the aircraft is subject to the force of gravity G = mg (Fig. 1.9), directed vertically, the lift force Y, directed perpendicular to the speed of the oncoming flow, the drag force X, directed along the speed of this flow, and the thrust of the engines P, directed towards the flow at an angle close to the angle of attack a (assuming the angle of installation of the engines relative to the Ox i axis equal to zero).

It is most convenient to consider the longitudinal movement of the aircraft in a velocity coordinate system. In this case, the projection of the velocity vector onto the Oy axis is zero. Angular velocity of rotation of the tangent to the trajectory of the center of mass relative to the axis Og

<ог= -В = & - а.

Then the equations of motion of the aircraft’s center of mass in projections on the Ox and Oy axes have the following form:

projections of forces on the Ox axis (tangent to the trajectory):

mV = - X-Osm0-f-/°cosa; (1.2)

projections of forces onto the Oy axis (normal to the trajectory):

mVb = Y - G cos 0 - f~ Z3 sin a. (1.3)

The equations describing the rotation of the aircraft relative to the center of mass are most simply obtained in a coupled system

coordinates, since its axes coincide with the main axes of inertia. Since, when considering isolated longitudinal motion, we assume p = 0 (under this condition, the velocity coordinate system coincides with the semi-coupled one) and, therefore, the Oz axis of the velocity coordinate system coincides with the Ozi axis of the coupled system, then the equation of moments about the Oz axis has the form:

where /2 is the moment of inertia of the aircraft relative to the Og axis;

Mg - aerodynamic pitching moment, longitudinal moment.

To analyze the characteristics of the longitudinal motion of an aircraft relative to its center of mass, it is necessary to add an equation for the relationship between the angles of attack, pitch and inclination of the trajectory:

When considering the dynamics of the longitudinal trajectory motion of an aircraft - the movement of its center of mass relative to the ground - two more kinematic equations are needed:

xg = L*=V COS0; (1.6)

yg - H = V sin b, (1.7)

where H is the flight altitude;

L is the distance traveled along the Oxg axis of the earth's coordinate system, which is assumed to coincide in direction with the Ox axis of the velocity system.

In accordance with the stationarity hypothesis, aerodynamic forces and moments are nonlinear functions of the following parameters:

X=X(*% I7, M, Rya);

G = G(*9 1/, m, Rya);

M2 = Mz(bв.<*» а, V, М, рн),

: (th “speed of sound at flight altitude);

rya - air density at flight altitude; bv - elevator deflection angle.

These forces and moments can be written through aerodynamic coefficients:

where Cx - Cx (a, M) is the drag coefficient;

Su -Su (a, M) - lift coefficient;
mz-mz (bv, a, a, d, M) - longitudinal moment coefficient M%

S is the area of the aircraft wing;

La is the average aerodynamic chord of the MAC.

Engine thrust is also a nonlinear function of a number of parameters:

P = P(8d) M, rn, Tya),

where bl is the movement of the body that controls the thrust of the engines; pi - pressure at flight altitude;

Tya is the absolute air temperature at flight altitude.

We will consider steady rectilinear motion as an unperturbed motion

We believe that the parameters of the perturbed motion can be expressed through their steady-state values and small increments:

a = a0-4-Yes;

Є-VU;

Taking into account (1.15) the linearization of the equations of perturbed motion (1.2-1.7) and taking into account the equations of unperturbed motion (1.9-1.14), we obtain a system of linear differential equations with constant coefficients:

mbV = - XvbV - Xm DM -X“Da- A^p&D yg- G cos 0OD0 - f + COS a0DM - P0 sin a0Da - f P? cos a0ridyg -f P T COS a„Tun^Ue +

cos «0Д8д; (1.16)

mV^b = YVW + KmDM + K“Da - f Kiy Dyg + O sin 0OD6 +

RM sin aoDM + PQ cos a0Da - f P? sin а0р^Дyg +

P T sin *ъТу„лув + P5 sin а0Д5д; (1.17)

Izb = M ® Д8В - f M'M - f МІДа - f AlfbA - f

dx, dx< vrp дХ

U - ‘ L 1 — ——

In these equations, to simplify writing, symbolic notation for partial derivatives has been introduced:

When studying the dynamics of approach and landing of an aircraft, equations (1.16-1.18) can be simplified by neglecting (due to their smallness) terms containing derivatives with respect to parameters p, T, derivatives of aerodynamic forces and their moments with respect to the Mach number. For similar reasons, the derivative Yam can be replaced by the derivative Pv, and the increment DM by the increment XV. In addition, in the moment equation it is necessary to take into account that Mzv = 0 and Mrg = 0, since the moment coefficient mZo = 0. Then equations (1.16-1.18) will take the form:

mAV=-XvAV - X'1Aya - O cos 0OD0 + Pv cos a0DK -

P„ s i P a0D a - f - P5 cos a0D&l; (1.16a)

mV0A

R0 cos a0Da-(-P8 sin a0D8d; (1.17a)

1$ = Ш Д8В + m Yes + M Yes + D 8;

Yv=c!/oSpV0; Ya = cauS ;

The values of the coefficients Cti Cy, Cx, Cy, niz, fflz, fflz, tftz are determined using graphs compiled based on the results of purging aircraft models in wind tunnels and flight tests of the aircraft.

Characteristics Pb are necessary when considering cases when, in a disturbed motion, the body that controls thrust moves, for example, when considering the longitudinal movement of an aircraft simultaneously controlled by the autopilot and autothrottle (automatic speed control). If during the perturbed motion D6d = 0, then the last term in equations (1.16 and 1.17) is equal to zero.

When analyzing the stability of the movement of an uncontrolled aircraft (with the controls clamped), it is necessary to take into account that the stability of such movement does not depend at all on the xx coordinate and practically does not depend, due to neglecting the influence of Рн and Тн, on the yg coordinate. Therefore, when analyzing the stability of an aircraft without an automatic control system, equations (1.19 and 1.20) can be excluded from consideration.

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L, . ". South-^ =M-A. v0 K0

Note that the terms containing the control coordinates 6D and 6B are on the right side of the equations. The characteristic polynomial for the system of equations of motion of an uncontrolled aircraft (with clamped controls) has the following form:

A (p) = P4 -f яjP3 + оР2 + а3р - f d4, (1.24)

where dі = dj + £a-+ - f g - ;

+ - f s. + ^ь+с;)(«vr -60);

Н3 = Г« (rtK ~ + + + ^4)(a6^V ~av b*)>

ai - ca(atbv - avbH).

According to the Hurwitz-Rouse criterion, the movement described by a fourth-order equation is stable when the coefficients ab a2, a3 and a4 are positive and a3(aia2-az)-a4ai2>0.

These conditions are usually satisfied not only for landing modes, but also for all operational flight modes of subsonic civil aircraft. The roots of the characteristic polynomial (1.24) are usually complex conjugate, different in size, and they correspond to two different oscillatory movements. One of these movements (short-period) has a short period with strong attenuation. The other motion (long-period, or phugoid) is a slowly decaying motion with a long period.

As a result, the perturbed longitudinal motion can be considered as a mutual superposition of these two motions. Considering that the periods of these movements are very different and that the short-period oscillation decays relatively quickly (in 2-4 seconds), it turns out to be possible to consider the short-period and long-period movements in isolation from each other.

The occurrence of short-period motion is associated with an imbalance in the moments of forces acting in the longitudinal plane of the aircraft. This violation may be, for example, the result of wind disturbance, leading to a change in the angle of attack of the aircraft, aerodynamic forces and moments. Due to the imbalance of moments, the plane begins to rotate relative to the transverse axis Oz. If the movement is stable, then it will return to the previous value of the angle of attack. If the imbalance of moments occurs due to deflection of the elevator, then the aircraft, as a result of short-period movement, will reach a new angle of attack, at which the equilibrium of the moments acting relative to the transverse axis of the aircraft is restored.

During short-period movement, the speed of the aircraft does not have time to change significantly.

Therefore, when studying such motion, we can assume that it occurs at the speed of undisturbed motion, i.e., we can accept DU-0. Assuming the initial mode to be close to horizontal flight (0«O), we can exclude from consideration the term containing bd.

In this case, the system of equations describing the short-period motion of the aircraft takes the following form:

db - &aDa=0;

D b + e j D& - f sk Yes - f saDa == c5Dyv; Db = D& - Yes.

The characteristic polynomial for this system of equations has the form:

А(/>)k = d(/>2 + аі/> + а. Ф where а=ьЛск+с> Ї

Short-period motion is stable if the coefficients “i and 02 are positive, which is usually the case, since in the field of operating conditions the values b*, cx, z” and are significantly positive.

niya tends to zero. In this case, the value

the frequency of the aircraft’s own oscillations in short-period motion, and the magnitude is their damping. The first value is determined mainly by the coefficient ml, which characterizes the degree of longitudinal static stability of the aircraft. In turn, the coefficient ml depends on the alignment of the aircraft, i.e., on the relative position of the point of application of the aerodynamic force and the center of mass of the aircraft.

The second quantity causing attenuation is determined

to a large extent by the moment coefficients mlz and t% ■ The coefficient t'"gg depends on the area of the horizontal tail and its distance from the center of mass, and the coefficient ml also depends on the delay of the flow bevel at the tail. In practice, due to the large attenuation, the change in the angle of attack has the character , close to aperiodic.

The zero root p3 indicates the neutrality of the aircraft with respect to the angles d and 0. This is a consequence of the simplification made (DE = 0) and the exclusion from consideration of the forces associated with a change in the pitch angle, which is permissible only for the initial period of the disturbed longitudinal motion - short-period *. Changes in angles A# and DO are considered in long-period motion, which can be simplified to begin after the end of short-period motion. At

1 For more details on this issue, see

In this case, La = 0, and the values of the pitch and inclination angles of the trajectory are different from the values that occurred in the original unperturbed motion. As a result, the balance of force projections on the tangent and normal to the trajectory is disrupted, which leads to the emergence of long-period oscillations, during which changes occur not only in the angles O and 0, but also in the flight speed. Provided the movement is stable, the balance of force projections is restored and the oscillations die out.

Thus, for a simplified study of long-period motion, it is sufficient to consider the equations of force projections on the tangent and normal to the trajectory, assuming Yes = 0. Then the system of equations of longitudinal motion takes the form:

(1.28)

The characteristic polynomial for this system of equations has the form:

where ai = av-b^ a2=abbv - avbb.

Stability of movement is ensured under the condition “i >0; d2>0. The damping of oscillations significantly depends on the values of the derivative Pv and the coefficient сХа, and the frequency of natural oscillations also depends on the coefficient су„ since these coefficients determine the magnitude of the projections of forces on the tangent and normal to the trajectory.

It should be noted that for cases of horizontal flight, climb and descent at small angles 0, the coefficient bb has a very small value. When excluding a member containing

from the second equation (1.28) we obtain at = av; a2 = aebv.

Isolation of equations of longitudinal motion from the complete system of equations of longitudinal motion of an aircraft.

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

System of equations for longitudinal motion of an aircraft.

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

The relation α = 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 – nonlinear system, because contains products of variables and trigonometric functions, expressions for aerodynamic forces.

To obtain a simplified linear model of the longitudinal motion of an aircraft, it is extremely important to introduce certain assumptions and carry out a linearization procedure. In order to substantiate additional assumptions, it is extremely important for us to consider the dynamics of the longitudinal movement of the aircraft with stepwise deflection of the elevator.

Aircraft response to stepwise deflection of the elevator. Division of longitudinal motion into long-term and short-term.

With a stepwise deviation δ in, a moment M z (δ in) arises, which rotates relative to the Z axis at a speed ω z. In this case, the pitch and attack angles change. As the angle of attack increases, an increase in lift occurs and a corresponding moment of longitudinal static stability M z (Δα), which counteracts the moment M z (δ in). After the rotation ends, 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 in) stops, but, because the aircraft has certain inertial properties, ᴛ.ᴇ. has a moment of inertia I z relative to the OZ axis, then the establishment of the angle of attack is oscillatory in nature.

Angular oscillations of the aircraft around the OZ axis will be damped using the natural aerodynamic damping moment M z (ω z). The increment 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. Based on the balance of moment loads, the pitch angle continues to change synchronously with the change in the inclination angle of the trajectory. In this case, the angle of attack is constant. Angular movements over a short interval occur with high frequency, ᴛ.ᴇ. have a short period and are called short-period.

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

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

When considering the dynamics of longitudinal motion, we did not take into account the additional lift force created by the deflection of the elevator. This effort is aimed at reducing the total lift force, in connection with this, for heavy aircraft, the phenomenon of subsidence is observed - a qualitative deviation in the angle of inclination of the trajectory with a simultaneous increase in the pitch angle. This occurs until the increment in lift compensates for the lift component due to elevator deflection.

In practice, long-period oscillations do not occur, because are extinguished in a timely manner by the pilot or automatic controls.

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

The transfer function is usually called the image of the output value, based on the image of the input at zero initial conditions.

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

In operator form, the record looks like:

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

(6.11)

(6.12)

However, we can write transfer functions that relate the angle of attack and angular velocity in pitch to the elevator deflection

(6.13)

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

, , , , ,

Taking these relations into account, we rewrite 6.13:

(6.14)

Therefore, the transfer functions for the trajectory inclination angle and pitch angle, depending on the elevator deflection, will have the following form:

(6.17)

One of the most important parameters that characterize the longitudinal movement of an aircraft is normal overload. Overload can be: Normal (along the OU axis), longitudinal (along the OX axis) and lateral (along the OZ axis). It is calculated as the sum of the forces acting on the aircraft in a certain direction, divided by the force of gravity. Projections on the axis allow one to calculate the magnitude and its relationship with g.

- normal overload

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

Using expressions for overload, we rewrite:

For horizontal flight conditions ( :

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

-δ in M ω z ν ν α -

θ θ

The 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 direct and reverse reaction of the aircraft to rudder deflection. If M x (δ n) is greater in magnitude than M x (β), the aircraft will tilt in the opposite direction of the turn.

Taking into account the above, we can construct a block diagram for analyzing the lateral movement of an aircraft when the rudder is deflected.
-δ n M y 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 plane rolls, along with this the lift force tilts, which causes a lateral projection Y a sinγ, which begins to develop a large lateral movement: the plane begins to slide onto the inclined half-wing, while the corresponding aerodynamic forces and moments increase, and this means that the so-called “spiral moments” begin to play a role: M y (ω x) and M y (ω z). It is advisable to consider large lateral movement when the aircraft is already tilted, or using the example of the dynamics of the aircraft when the ailerons are deflected.

Aircraft response to aileron deflection.

When the ailerons deflect, a moment M x (δ e) occurs. The plane begins to rotate around the associated axis OX, and a roll angle γ appears. The damping moment M x (ω x) counteracts the rotation of the aircraft. When the aircraft tilts, due to a change in the roll angle, a lateral force Z g (Ya) arises, which is the result of the weight force and the lift force Y a. This force “unfolds” the velocity vector, and the track angle Ψ 1 begins to change, which leads to the emergence of a sliding angle β and the corresponding force Z a (β), as well as a moment of track static stability M y (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) and therefore, to some extent, it reduces the rate of change in the path angle Ψ 1.

The force Z a (β) is also the cause of the moment of transverse static stability. M x (β), which in turn tries to bring 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), the so-called “spiral instability” occurs, in which the roll angle continues to increase after the ailerons return to the neutral position, which leads to the aircraft turning with increasing angular velocity.

Such a turn is usually called a coordinated turn, in which the bank angle is set by the pilot or using an automatic control system. In this case, during the turn, the disturbing moments of roll M x β and M x ωу are compensated, the rudder compensates for sliding, that is, β, Z a (β), M y (β) = 0, while the moment M y (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 the change in the path 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.

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