Equations of motion for the center of mass of an aircraft. The structure of the equations of motion of an aircraft The concept of the components of the longitudinal motion of an aircraft

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 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 (р) = Р4 -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 relative 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.

Page 1

The motion of an airplane as a rigid body consists of two motions: the motion of the center of mass and the motion around the center of mass. Since in each of these movements the aircraft has three degrees of freedom, its overall movement is characterized by six degrees of freedom. To specify movement at any time, it is necessary to specify six coordinates as functions of time.

To determine the position of the aircraft we will use the following rectangular coordinate systems (Fig. 2.1):

a stationary system Ox0y0z0, the beginning of which coincides with the center of mass of the aircraft, the Oy0 axis is directed vertically, and the Ox0 and Oz0 axes are horizontal and have a fixed direction relative to the Earth;

a coupled system Ox1y1z1 with the origin at the center of mass of the aircraft, the axes of which are directed along the main axes of inertia of the aircraft: the Ox1 axis is along the longitudinal axis, the Oy1 axis is in the symmetry plane, the Oz1 axis is perpendicular to the symmetry plane;

velocity system Oxyz with the origin at the center of mass of the aircraft, the Ox axis of which is directed along the velocity vector V, the Oy axis in the symmetry plane, the Oz axis perpendicular to the symmetry plane;

The position of the coupled system Ox1y1z1 in relation to the stationary system Ox0y0z0 is characterized by Euler angles: φ – roll angle, ψ – yaw angle and J – pitch angle.

The position of the airspeed vector V relative to the coupled system Ox1y1z1 is characterized by the angle of attack α and the gliding angle b.

Often, instead of an inertial coordinate system, a system associated with the Earth is chosen. The position of the center of mass of the aircraft in this coordinate system can be characterized by the flight altitude H, lateral deviation from the given flight path Z and the distance traveled L.

Rice. 2.1 Coordinate systems

Let us consider the plane motion of an aircraft in which the velocity vector of the center of mass coincides with the plane of symmetry. The aircraft in the high-speed coordinate system is shown in Fig. 2.2.

Rice. 2.2 Aircraft in a high-speed coordinate system

We write the equations of longitudinal motion of the aircraft’s center of mass in projection onto the OXa and OYa axes in the form

(2.1)

(2.2)

Where m is mass;

V – aircraft airspeed;

P – engine traction force;

a – angle of attack;

q – angle of inclination of the velocity vector to the horizon;

Xa – drag force;

Ya – aerodynamic lift force;

G – weight force.

Let us denote by Mz and Jz, respectively, the total moment of aerodynamic forces acting relative to the transverse axis passing through the center of mass, and the moment of inertia relative to the same axis. The equation of moments about the transverse axis of the aircraft will be:

(2.3)

If Mshv and Jv are the hinge moment and moment of inertia of the elevator relative to its axis of rotation, Mv is the control moment created by the control system, then the equation of motion of the elevator will be:

(2.4)

In four equations (2.1) – (2.4), the unknowns are five quantities J, q, a, V and dв.

As the missing fifth equation, we take the kinematic equation connecting the quantities J, q and a (see Fig. 2.2).

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 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 # 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 - 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)

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) 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 )