Calculation of the aircraft trajectory according to the equations of motion. Structure of aircraft motion equations
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The movement of an aircraft as a rigid body consists of two movements: the movement of the center of mass and the movement around the center of mass. Since in each of these movements the aircraft has three degrees of freedom, then in general its movement is characterized by six degrees of freedom. To specify motion at any point in time, six coordinates must be specified as functions of time.
To determine the position of the aircraft, we will use the following systems of rectangular coordinates (Fig. 2.1):
the fixed system Ox0y0z0, the origin of which coincides with the center of mass of the aircraft, the axis Oy0 is directed vertically, and the axes Ox0 and Oz0 are horizontal and have a fixed direction with respect to the Earth;
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;
the velocity system Oxyz with origin at the center of mass of the aircraft, the Ox axis of which is directed along the velocity vector V, the Oy axis is in the symmetry plane, and the Oz axis is perpendicular to the symmetry plane;
The position of the coupled system Ox1y1z1 with respect to the stationary system Ox0y0z0 is characterized by the Euler angles: φ is the roll angle, ψ is the yaw angle and J is the pitch angle.
The position of the airspeed vector V relative to the associated system Ox1y1z1 is characterized by the angle of attack α and the angle of slip 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, the lateral deviation from the given flight path Z, and the distance traveled L.
Rice. 2.1 Coordinate systems
Consider a 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 Figure 2.2.
Rice. 2.2 Aircraft in velocity coordinate system
The equations of the longitudinal motion of the center of mass of the aircraft in the projection on the axes OXa and OYa will be written in the form
(2.1)
(2.2)
Where m is the mass;
V is the airspeed of the aircraft;
P is the thrust force of the engine;
a is the angle of attack;
q is the angle of inclination of the velocity vector to the horizon;
Xa is the drag force;
Ya is the aerodynamic lift force;
G is the weight force.
Denote by Mz and Jz, respectively, the total moment of aerodynamic forces acting about the transverse axis passing through the center of mass, and the moment of inertia about 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 the moment of inertia of the elevator about 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), five quantities J, q, a, V and dv are unknown.
As the missing fifth equation, we take the kinematic equation relating the quantities J, q and a (see Fig. 2.2).
The analysis of a nonlinear system of differential equations ((2.1) - (2.7)) and their solution presents certain difficulties. Therefore, the first step towards their study is the linearization of relationships between variables, obtaining a linear mathematical model of an aircraft as a control object, followed by an analysis of the dynamic properties.
To obtain linearized equations of motion, it is necessary to establish the dependence of forces and moments on the quantities and V, as well as on the control factors.
The engine thrust force P depends on internal parameters, as well as on external conditions characterized by flight speed V, pressure p n and temperature T n in the atmosphere.
Aerodynamic forces and moments are usually represented as
where c x and c y are drag and lift coefficients;
m z - pitching moment coefficient;
b A - wing chord length;
S is the area of the wings;
q - velocity head, calculated by the formula:
The coefficients c x and c y are functions of and V, and the coefficient m z is a function of and c.
To linearize equations (2.1) - (2.7), taking into account relations (2.8) - (2.9), we use the well-known method of representing nonlinear dependencies in the form of linear deviations relative to the unperturbed motion (assuming the smallness of these deviations). As the unperturbed motion, we can take horizontal flight at a constant speed. In this case, we will neglect the effect of flow unsteadiness on the aerodynamic characteristics of the aircraft. Let us assume that the unperturbed motion of the aircraft is characterized by parameters V 0 ,H 0 , 0 , 0 , 0 , which do not depend on time. Let at some point in time, due to disturbances acting on the aircraft, we have:
where V, H are small increments.
Consequently, the perturbed motion of the aircraft consists of unperturbed motion and motion characterized by small deviations. This interpretation of the perturbed motion is valid as long as the increments V and H remain small, which is the case for stable systems. Since one of the main purposes of the control system is to ensure the stability of the flight mode, the legality of using linearized equations can be considered secured.
Expanding the forces P, X, Y and the moment M z into Taylor series in small increments and limiting ourselves to the linear terms of the increments, instead of equations (2.1) - (2.5) we get:
where the terms with superscripts denote the partial derivatives with respect to the corresponding variables in the vicinity of the unperturbed motion.
Assume that the undisturbed flight is horizontal, then 0 = 0. For partial derivatives in equations (2.10), taking into account (2.8), we can write:
in these expressions, M is the Mach number.
For the purpose of further transformations, we use the relations:
or, given that
where a is the speed of sound, then
In addition, we will use the relationship between the height H and the parameters of the atmosphere and T H
temperature gradient,
R is the gas constant.
Using expression (2.13), we find:
Consequently
In order to shorten the notation, we introduce dimensionless quantities:
where is the aerodynamic time constant of the aircraft, and instead of increments, and we will write, and, giving the last values the meaning of the same increments.
Using relations (2.11) - (2.16), we bring equations (2.10) to the form:
r is the aircraft's radius of gyration.
The system of differential equations (2.17) is a linear mathematical model of the aircraft longitudinal motion.
The aircraft dynamics in the longitudinal plane is characterized by two components: short-period and long-period. In short-period motion, the parameters and that characterize the motion of the aircraft relative to the center of mass undergo very sharp changes. During long-period motion, the parameters and V, which characterize the position of the center of mass of the aircraft, change. Therefore, in equations (2.17) we can set = 0, assuming that during the change in the angular coordinates and the flight speed practically does not change. In other words, the longitudinal axis of the aircraft can oscillate about the velocity vector of the center of mass.
If we take into account the remarks made and assume that the balance of longitudinal forces is not violated when po and are disturbed, then instead of system (2.17) we obtain for the case of horizontal flight.
In the case of analyzing the dynamics of an aircraft flying at a speed much lower than the orbital one, the equations of motion can be simplified in comparison with the general case of flying an aircraft, in particular, the rotation and sphericity of the Earth can be neglected. In addition, we 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 systems of coordinate axes.
The normal earth coordinate system is 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. Axes 0Xg and 0Zg lie in the horizontal plane. Their orientation can be taken arbitrarily, depending on the objectives 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 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 forms, together with the OXg and 0Yg axes, a right-handed coordinate system (Fig. 1.1). The XgOYg plane is called the local vertical plane.
The associated coordinate system is OXYZ. The origin of coordinates is located at the center of gravity of the aircraft. The OX axis lies in the plane of symmetry and is directed along the line of wing chords (or parallel to some other direction fixed relative to the aircraft) towards the nose of the aircraft. The 0Y axis lies in the plane of symmetry of the aircraft and is directed upwards (in level flight), the 0Z axis completes the system to the right.
The angle of attack a is the angle between the longitudinal axis of the aircraft and the projection of the airspeed onto the OXY plane. The angle is positive if the aircraft airspeed projection on the 0Y axis is negative.
The slip angle p is the angle between the airspeed of the aircraft and the OXY plane of the associated coordinate system. The angle is positive if the airspeed projection on the lateral 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 coupled system
coordinates to each of the Euler angles, one can arrive at any angular position of the coupled system relative to the axes of the normal coordinate system.
When studying aircraft dynamics, the following concepts of Euler angles are used.
The 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 on the horizontal plane by turning around the OYg axis clockwise.
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.
Roll angle y is the angle between the local vertical plane passing through the OX y axis and the related 0Y axis of the aircraft. The angle is positive if the OC 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 the associated axes about the normal axes. We will assume that the normal and associated coordinate systems are aligned at the beginning. The first rotation of the system of coupled axes will be made about the axis O by the angle of yaw r]; (φ coincides with the axis OYgXFig. 1.2)); the second rotation is relative to the 0ZX axis by an angle Ф (‘& coincides with the OZJ axis and, finally, the third rotation is made relative to the OX axis by an angle y (y coincides with the OX axis). Projecting the vectors φ, Φ, y, which are components
vector of the angular velocity of the aircraft relative to the normal coordinate system, on the associated axes, we obtain the equations of connection between the Euler angles and the angular velocities of rotation of the associated axes:
co* = Y + sin *&;
o)^ = i)COS’&cosY+ftsiny; (1.1)
co2 = f cos y - f cos f sin y.
When deriving the equations of motion of the center of mass of the aircraft, it is necessary to consider the vector equation for changing the momentum
-^- + o>xV) = # + G, (1.2)
where w 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
ical forces and traction; G - vector of gravitational forces.
From equation (1.2) we obtain the system of equations of motion of the aircraft CM in projections onto the associated axes:
t (g3?~ + °hVx ~ °ixVz) = Ry + G!!’ (1 -3)
t iy'dt "b Y - \u003d Rz + Gz>
where Vx, Vy, Vz - projections of velocity V; Rx, Rz - projections
resulting forces (aerodynamic forces and thrust); Gxi Gyy Gz - gravity projections on the coupled axes.
The projections of the force of gravity on the associated axes are determined using the 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 that is stationary relative to the Earth, the projections of the flight speed are related to the angles of attack and slip and the magnitude of the speed (V) by the relations
Vx \u003d V cos a cos p;
Vy \u003d - V sin a cos p;
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The expressions for the projections of the resulting forces Rx, Rin Rz have the following form:
Rx \u003d - cxqS - f P cos ([>;
Rty = cyqS p sin (1.6)
where cx, cy, cg are the coefficients of the projections of aerodynamic forces on the axes of the associated coordinate system; P - engine power (usually P = / (Y, #)); Фн - engine jamming angle (φ > 0, when the projection of the thrust vector on the 0Y axis of the aircraft is positive). In what follows, we will take = 0 everywhere.
Vx sin ft+ Vy cos ft cos y - Vz cos ft sin y. (1.7)
Dependence p (H) can be found according to the tables of the standard atmosphere or according to the approximate formula
where for flight altitudes H s 10,000 m K W 10~4. To obtain a closed system of aircraft motion equations in coupled axes, equations (13) must be supplemented with kinematic
relations that allow one to determine the attitude angles of the aircraft y, ft, r]1 and can be obtained from equations (1.1):
■f \u003d Kcos Y - sin V):
■fr\u003d "y sin y + cos Vi (1-8)
Y= co* - tg ft (© y 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 the moment of 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.
The projections of the angular momentum vector K on the moving axles are generally written in the following form:
K t = I x^X? xy®y I XZ^ZI
K, Ixy^x H[ IY^Y 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 with a plane of symmetry. In this case, 1xr = 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 -! *) \u003d MRZ-
If the main axes of inertia are taken as the OXYZ axis, then xy = 0. In this regard, we will perform further analysis of the aircraft dynamics using the main axes of inertia of the aircraft as the OXYZ axes.
The moments included in the right parts of equations (1.11) are the sum of aerodynamic moments and moments from engine thrust. Aerodynamic moments are written as
where mX1 ty, mz - dimensionless coefficients of aerodynamic moments.
The coefficients of aerodynamic forces and moments are generally expressed as functional dependencies on the kinematic motion parameters and similarity parameters depending on the flight mode:
y, r 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 regime, therefore, in the analysis of 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 an 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 deviations of the organs and control levers.
For small angles of attack and slip, the representation of aerodynamic coefficients in the form of expansions in a Taylor series in terms of motion parameters is usually used, while only the first terms of this expansion are preserved. Such a mathematical model of aerodynamic forces and moments for small angles of attack is in good agreement with flight practice and experiments in wind tunnels. Based on the materials of works on the aerodynamics of aircraft for various purposes, we will take the following form of representation of the coefficients of aerodynamic forces and moments as a function of motion parameters and deflection angles of controls:
cx ^ cho 4~ cx (°0"
Y ^ SU0 4 "s ^ ya 4" S! / F;
cg = cfp + CrH6n;
th - itixi|5 - f - ■b mxxa>x-(- mx -f - /l* (І -|- - J - L2LP6,!
o (0.- (0^- p b b „
tu \u003d myfi + tu ho)x + tu Yyy + r + ha / be + tu bn;
tg = tg(a) + tg zwz /i? f.
When solving specific problems of flight dynamics, the general form of representation of aerodynamic forces and moments can be simplified. For small angles of attack, many aerodynamic coefficients of lateral motion are constants, 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 in expression (1.13), which are proportional to the angles air, are usually found from static tests of models in wind tunnels or by calculation. To find-
NIA of derivatives, twx (y) it is necessary to carry out
dynamic testing of models. However, in such tests, a simultaneous change in the angular velocities and angles of attack and slip usually occurs, and therefore, during measurements and processing, the following values are simultaneously determined:
CO - CO-,
r* = r2r -mz;
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0) , R. Yuu I c.
mx* = mx + mx sin a; tu* = Shuh tu sin a.
CO.. (O.. ft CO-. CO.. ft
my% = m, / -|- tiiy cos a; tx% \u003d txy + tx cos a.
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The paper shows that in order to analyze the aircraft dynamics,
especially at low angles of attack, it is permissible to represent the moment
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™y are understood [see. (1.14)] determined experimentally. Let us show that this is admissible by restricting our consideration to the analysis of flight with small angles of attack and slip at a constant flight speed. Substituting into equations (1.3) the expressions for the velocities Vx, Vy, Vz (1.5) and making the necessary transformations, we obtain
= % COS a + coA. sina - f -^r )
