A complete system of equations of aircraft motion. Open Library - open library of educational information

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 in the process of 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 must be taken 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 йi = йу + £а-+ - 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 motions. 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 natural oscillations of the aircraft 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 initial 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.

Size: px

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Transcript

1 1 Areas of training: Avionics Aeronautical systems engineering On-board control systems Discipline: Course, semester, academic year. year: 3, spring, 11/1 Department: 31 SULA Head of training: assistant Kopysov Oleg Eduardovich LECTURE 3 TOPIC: EQUATIONS OF MOTION OF AN AIRCRAFT AS A RIGID BODY. LONGITUDINAL AND LATERAL MOTION The motion of the aircraft, as a rigid body in a coupled coordinate system, is described by Euler's equations (six nonlinear second-order differential equations). The forces and moments included in these equations depend in a complex way on altitude, speed and flight mode and change over time as flight conditions change, for example due to changes in the mass and moment of inertia of the aircraft as a result of fuel consumption or cargo dumping. When analytically studying the control processes of an aircraft, its equations of motion are, as a rule, simplified by considering two movements independent of each other: longitudinal and lateral. Longitudinal motion includes translational motion of the aircraft along the OX and OY axes and rotational motion around the O axis. Lateral motion includes translational motion along the O axis and rotational motion around the OX and OY axes. Longitudinal movement. Generalized mathematical model When the aircraft moves longitudinally, the linear velocity vector V of its center of mass is in the vertical plane. External forces acting on the aircraft: P is the thrust force of the engines, the vector of which is directed along the OX axis: X and the drag force, the vector of which is directed opposite to the vector V, i.e. to the negative side of the OX axis a Y a lift force, the vector of which is perpendicular to the vector V mg weight of the aircraft (m mass of the aircraft, g acceleration of gravity). Rotation of the aircraft in the plane

2 X a Y a is possible under the influence of the moment M acting around the axis O a, which is called the aerodynamic pitch moment. According to Fig. 3.1 there are kinematic relations:, (3.1) where ϑ pitch angle θ angle of inclination of the trajectory of the center of mass (CM) of the aircraft ω angular pitch velocity. Figure 3.1 External forces acting on the aircraft in longitudinal motion The rotational motion of the aircraft around the O axis a is described by the equation: I, (3.) where I is the moment of inertia of the aircraft relative to the O axis and M is the moment of aerodynamic forces, which can be represented as: mba S V, (3.3) where m moment coefficient b a - wing chord ρ air density S wing area. Coefficient m can be represented as consisting of the sum of three terms, two of which depend on the static parameters (α, V, δ in) and determine the static moment, and the third on the dynamic parameters (), and determine the damping moment.

3 3 Let us project the forces acting on the aircraft onto the tangent to the flight path (X axis) and onto the normal to it (Y axis). The sum of the projections of forces on the tangent to the trajectory: dv m mv P cos X a mg sin. dt (3.4) When determining the projections of forces on the normal to the trajectory, it must be borne in mind that when an aircraft moves along a curved trajectory with a radius of curvature r, it is affected by the centrifugal inertial force mv of the trajectory), and ds = Vdt, then / mv mv mv d r. Since r = ds/dθ (s arc length mv mv. r ds / d Vdt / d dt Therefore, the sum of the projections of forces on the normal to the trajectory: mv Y Psin mg cos. a (3.5) The traction force P depends on the engine parameters, on external conditions, characterized by flight speed V, flight altitude H and engine control parameter δ р, i.e. in general form P = Р(V, Н, δ р) Aerodynamic forces X а and Y а depend on the angle of attack α). , flight speed V, air density ρ and rudder deflection angle δ in. Since the angle δ in practically does not affect the values ​​of X a and Y a, this influence is neglected and is usually presented in the form: where X a CxaS V Ya CyaS V, (3.6) C xa, C ya coefficients of drag and lift, depending on the angle of attack and flight speed. System of nonlinear differential equations (3.), (3.4), (3.5) taking into account (3.1), (3.3) , (3.6) is a mathematical model of the longitudinal motion of an aircraft. It is known that for manned aircraft of an aircraft design for almost all configurations and most flight modes, the proper motion of the aircraft consists of two oscillatory movements that differ in frequency and degree of attenuation. These movements are called short-period and long-period or fugo-

4 idnymi. The reason for the occurrence of short-period movements is the imbalance of moments around the O a axis, which leads to rotation of the aircraft relative to the CM and a change in angles α and ϑ. The speed of undisturbed linear motion remains virtually unchanged. The reason for the occurrence of long-period movements is a violation of external forces acting in the longitudinal plane of symmetry of the aircraft, which results in a change in its flight speed. 4 Linearized equations of longitudinal motion of an aircraft By applying the method of small perturbations to equations (3.), (3.4), (3.5), linear equations of longitudinal motion of an aircraft can be obtained. Let us assume that in the flight segment under study, the unperturbed motion of the aircraft is characterized by constant forces X, Y, P, and parameters V, α, ϑ, θ, H and ω z =, and the control parameters δ B, δ p are also constant. If a flight section is being studied in which the motion parameters change significantly, it is divided into several sections in which the motion parameters can be considered constant. The equations of unperturbed motion of the aircraft in a section with constant parameters follow from equations (3.), (3.4), (3.5): P cos X mg sin Y P sin mg cos. From the first two equations of the system follows the relation: P cos X tg, P sin Y (3.7) from which we can conclude that at P cos X the aircraft flies horizontally, at P cos X it gains altitude (), and at P cos X it decreases altitude ( ).

5 If at some point in time the motion and control parameters have changed by values ​​V, then the corresponding parameters P of the perturbed motion take the form: V V V P P P. When studying the longitudinal angular motion of an aircraft in the region of small changes in motion parameters, the first equation of system (3.7) can be excluded from consideration, because it represents the sum of the projections of forces on the OX a axis (Fig. 3.1), which do not affect the angular motion of the aircraft. When linearizing the second equation of system (3.7), it is assumed that the projection of gravity onto the OY a axis does not affect the angular motion of the aircraft, and this component can be neglected. As a result of well-known linearization procedures, the simplest equations for the longitudinal motion of an aircraft can be obtained: mv Y I (3.8), where the constant coefficients correspond to the initial unperturbed motion and are determined as follows: Y Y (Pcos) () () (). 5

6 Let us consider the aerodynamic moments in equations (3.8), which determine the short-period motion of the aircraft. When >, which usually occurs, the moment is called the moment of longitudinal static stability, which is a consequence of the effect of the incoming air flow on the tail horizontal empennage, the size and shape of which mainly depends. When the aircraft moves undisturbed, there is no angle of attack and no aerodynamic torque relative to the transverse axis. Ascending or descending air flows lead to a change in the angle of attack by an amount, for example, a change in the alignment of the aircraft. The value, which may change due to other reasons, leads to an increase in the lifting force of the wings, which results in a change in the flight altitude of the aircraft, and to an increase by Y in the lifting force of the horizontal tail, which is applied in the center of pressure (CP) on the shoulder L of the GO , which creates the moment Y L GO, returning the aircraft to the previous angle of attack, i.e. (Fig. 3.). Thus, the moment ensures the longitudinal stability of the aircraft if the center of pressure of the aerodynamic forces is located behind the center of mass of the aircraft towards the tail. If the CM and the CM coincide, then 6 = (neutral aircraft), if the CM is in front of the CM, then< (неустойчивый ЛА). Момент появляется при вращении вокруг поперечной оси и называется моментом демпфирования тангажа. При вращении вокруг ЦМ с угловой скоростью хвостовое оперение получает линейную скорость L, перпендику- ГО лярную вектору скорости V полѐта (рис. 3.3). В результате угол атаки хвостового оперения изменяется на величину LГО / V и, следовательно, аэродинамическая подъѐмная сила хвостового оперения изменится на величину Y, которая создаст момент Y L, ГО направленный против скорости. Эффективность этого механизма демпфирования существенно зависит от высоты полѐта, а

7 its increase by means of aerodynamics leads to an increase in the impact of aerodynamic disturbances on the aircraft. 7 Figure 3. Determination of the moment of longitudinal static stability Figure 3.3 Determination of the pitch damping moment The control moment appears when the tail elevator is deflected, as a result of which its angle of attack changes. The physical picture of the effect of this moment on the aircraft is similar to the influence of the moment of longitudinal static stability (static pitch stability). The elevator, deflected from the neutral position by an angle, is acted upon by the aerodynamic force Y РВ, directed perpendicular to the incoming air flow and applied to the CD of the control surface (Fig. 3.4), which, as a rule, does not coincide with its axis of rotation (OA). The force Y РВ relative to the axis of rotation creates the so-called hinge moment, which is the main load moment for the drive that rotates the elevator. At the point corresponding to the OB, two oppositely directed forces Y PB, equal in magnitude to Y PB, can be applied.

8 8 Figure 3.4 Determination of the control moment in height Then we can write the equality, Y " L Y " l Y L from which P P P P it follows that the control moment applied to the aircraft consists of the sum of the hinge moment acting relative to the rudder OB and the moment of force Y PB on the arm L relative to the aircraft CM. Let's return to the equations of system (3.8) and rewrite them in variable increments of pitch angles where and attacks: I mv () Y F. Y (3.9), F Y disturbing moment and force, acting respectively relative to the axis O a and along the axis OY a. We rewrite the equations of system (3.9) in the form: where a1 a a3 a a a a a 5 a F, 6 Y Y, a I I I 4 1, a 1 5, a6. I mv mv (3.1) (3.11) The constant coefficients in (3.11), corresponding to unperturbed motion, are determined as follows:

9 m qsb m qsl m qsb Y c, yqs (3.1) where q V / velocity pressure b wing chord. 9 Lateral movement Aerodynamic forces and moments acting on the aircraft Lateral movement of the aircraft includes rotation around the longitudinal axis OX, the normal axis OY and linear movement along the O axis. Let us consider the main aerodynamic forces and moments acting on the aircraft (Fig. 3.5). Let us assume that due to some disturbance of the aircraft relative to the normal coordinate system OX g Y g g it received a roll at an angle γ, after which the disturbance disappeared. Angle γ determines the position of the associated coordinate system OXY, and point O coincides with the center of mass of the aircraft aircraft design. The planes of the wings relative to the X plane are located at an angle φ. With a positive roll (on the right wing), a component mg ​​sin of the weight force of the aircraft appears along the O axis, under the influence of which the aircraft slips at a speed V VXtg (V X is the longitudinal component of the speed V, β is the sliding angle). As a result of sliding, the symmetry of the air flow around the wings is broken. To illustrate this circumstance, triangles of air speeds are constructed at the ends of the right and left wings (V to the component of the velocity V of the oncoming air flow along the wings VI is the component perpendicular to the velocity vector V), from which VI V tg follows. Since the velocities V 1 on the right and left wings are directed in different directions, their angles of attack change, which is illustrated by the construction of velocity triangles on the velocity vectors V X and V I, from which V / V follows. In this case, on the right wing there is a positive increment I X angle of attack (+), and on the other negative ().

10 1 Figure 3.5 Determination of the moments of static stability of the roll and path Accordingly, the lifting force of the right wing will increase by ΔY, and the left one will decrease by ΔY. As a result, relative to the OX axis, a moment of transverse static stability or a moment of static roll stability is formed, the root cause of which is sliding and which is denoted in the form x M where (x) x. Obviously, this moment is greater, the greater the change in angle, the value of which, in accordance with the above relations, can be represented in the form: VI Vtg Vxtgtg, V V V x x x which implies that the greater the angle φ, the greater the moment of lateral stability. The sweep of the wings in plan also leads to the appearance of a moment of lateral stability. Changing the angles of attack leads to a change in the drag forces on the wings: on the right wing this force will increase by the amount ΔХ, and on the left it will decrease.

11 is sewn on ΔХ. With the appearance of the angle β, a force Δ also arises on the vertical tail. The consequence of these forces is the emergence of a feathering moment, or a moment of static stability of the path, which tries to turn the aircraft towards the oncoming air flow. This moment ensures stability along the slip angle, tending to turn the aircraft so that the slip angle that occurred before the disturbance is established. The moment of static stability of the path is denoted in the form where (M y) y y. 11 Using literature sources, find graphical dependences of the longitudinal moment coefficient on the angle of attack and elevator deflection, and the dependence of the coefficients C ha, C ua on the angle of attack. Terms to be included in the thesaurus: longitudinal motion, lateral motion, drag coefficient, lift coefficient, undisturbed motion of the aircraft, moment of static stability, hinge moment.

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Regional competition of creative works of students “Applied and fundamental issues of mathematics” Mathematical modeling Mathematical modeling of aircraft flight Loevets Dmitry, Telkanov Mikhail 11

Topic 5. Mechanical vibrations and waves. 5.1. Harmonic oscillations and their characteristics Oscillations are processes characterized by varying degrees of repeatability. Depending on the physical nature of the repetitive

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UDC 69.735.33.18.7.16 A. N. Narozhny, D. A. Nikonov, G. G. Vysokoglyad, A. I. Sheludko Computer support for the process of determining aircraft performance characteristics Part 7 MODELING REAL

Ministry of Education Russian Federation State educational institution of higher professional education "SAMARA STATE TECHNICAL UNIVERSITY" Department of "MECHANICS" DYNAMICS

ROTATIONAL MOTION (lectures 4-5) LECTURE 4, (section 1) (lecture 7 “CLF, part 1”) Kinematics of rotational motion 1 Translational and rotational motion In previous lectures we got acquainted with the mechanics of material

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LECTURE 11 EQUATIONS OF MOTION OF A RIGID BODY WITH A FIXED POINT Let us write out the dynamic and kinematic Euler equations. Let p, q, r be the projections of the angular velocity of the body onto the main axes of inertia, A, B, C are the main

In the case of analyzing the dynamics of an aircraft flying at a speed significantly lower than the orbital speed, the equations of motion compared to the general case of aircraft flight can be simplified; in particular, the rotation and sphericity of the Earth can be neglected. In addition, we will make a number of simplifying assumptions.

only quasi-statically, for the current value of the velocity head.

When analyzing the stability and controllability of the aircraft, we will use the following rectangular right-handed coordinate axes.

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 system OXYZ relative to the normal earth 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 # is 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 is the number 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 c 10,000 m K f 10~4. For getting closed system 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)

Y= co* - tan ft (©у cos y - sinY),

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:

h -jf — — hy (“4 — ©Ї) + Uy — !*) = MRZ-

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 thxha>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) requires

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 )