Equations of spatial maneuver of an aircraft. Maneuverability characteristics

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1 Electronic journal “Proceedings of MAI”. Issue 78 UDC 57.95: Solution of the boundary value problem of forming the trajectory of an aircraft when performing a spatial maneuver Tang Thanh Lam Moscow Institute of Physics and Technology (State University) MIPT st. Gagarina Zhukovsky Moscow region 484 Russia e-mal: Abstract The problem of planning the trajectory of an aircraft when performing a spatial maneuver is considered. To obtain a trajectory in compliance with the specified boundary conditions, two approaches are used based on the concepts of the inverse problem of dynamics and the representation of the trajectory in a parameterized form. In the first case, the simplest parameterization is considered, ensuring only the fulfillment of boundary conditions. In the second case, parameterization provides for additional optimization of some quality criterion, which corresponds to some implementation of the direct variational method. Specific examples are used to compare these two approaches. Key words: spatial maneuver of an aircraft, trajectory planning, boundary value problem, inverse dynamics, direct variational method. Introduction One of the main tasks of flight dynamics is to determine the trajectory and controls that ensure the transfer of the aircraft from a given starting point to

2 a given end point in space. If a control quality criterion is additionally specified, then the problem can be solved by methods of optimal control theory. But in any case, the formation of a flight path is essentially a boundary task. To date, many methods have been developed for solving problems of this type. Among them, the methods of targeting finite differences of finite elements, the Galerkin-Ritz method, methods of reduction to Fredholm integral equations, etc. are well known. Among the promising directions proposed recently are solution methods based on trajectory parameterization and the application of the concept of inverse problems of dynamics. Parameterization of the trajectory allows you to reduce the problem to finding the required values ​​of a finite number of parameters, and the concept of inverse dynamics makes it possible to easily determine the controls necessary to carry out movement along the required trajectory. If it is additionally necessary to optimize the quality of control according to any criterion, then this approach corresponds to one of the possible implementations of the direct variational method. The main advantage of this direction is the comparative simplicity and efficiency of calculation algorithms. In the future, this will allow generating trajectories in real time, which is attractive for on-board applications. This article discusses two characteristic ways of forming a trajectory based on specifying it in a parameterized form. In the first method, the coordination of boundary conditions is carried out through the appropriate choice of coefficients [ 3 4 5] and in the second method - through a special choice

3 basic functions. Free coefficients of parameterized dependencies in the second method are determined based on the optimality condition of a given quality criterion and restrictions on controls, which makes this method significantly more flexible. However, calculating the trajectory requires a fairly large amount of calculations. Using specific examples, the article shows that the first method, despite its attractive simplicity, can hardly be used for autonomous generation of an aircraft trajectory. Equations of motion and inverse problem The movement of the center of mass of an aircraft in space is described by the following system of equations: V g na sn gn a cos γ cos Ψ gn a sn γ/ V cos V cos cos V sn V cos sn /V () n a cosα X mg a n a snα Y mg a () Here the coordinates of the aircraft’s center of mass in the normal earthly coordinate system V flight speed trajectory inclination angle heading angle angle of attack angle of roll engine thrust X a aerodynamic drag Y a aerodynamic lift m aircraft mass g gravitational acceleration na - longitudinal overload na - transverse 3

4 overload. Aerodynamic forces X a and Y a depend on the speed V and on the density of the atmosphere at the flight altitude X a c V Y c V a where c c () and c c () are the aerodynamic drag and lift coefficients, the magnitude of which depends on the angle of attack (the angle between the longitudinal aircraft axis and flight speed vector). For trajectory motion described by the model (), the control variables are engine thrust (), angle of attack () and roll angle (). However, in problems of trajectory formation, the overloads n a and n a can be considered instead of and as variables. The attractiveness of this approach is due to the fact that the values ​​n a n a and are directly determined by the dependencies () () and () without any additional parameters and variables. To apply the inverse problem methodology, it is required that the control forces can be uniquely determined along given trajectories. System () allows this, which is easy to verify. Let the dependences of the aircraft coordinates on time () () and () be given. Directly from () it follows: sn V cos sn cos (3) V. V By differentiating these relations we find V V cos Ψ Ψ V. (4) V V cos 4

5 Directly from () it is also easy to obtain expressions for determining overloads and roll angle cos g g cos/ V n a V sn g n a V g cos g cos. (5) On the other hand, differentiating the last three equations of the system (), taking into account the first three equations of this system, we obtain the following relations: n a g n n g cos cos n a a g sn n a a g cos sn n g cos sn cos n g cos cos a g cos sn sn n a a g sn sn g sn cos ( ) This result allows us to write: n n a a g sn cos sn g cos cos sn arcg g g cos sn cos cos sn g cos cos sn. sn (7) Formulas (7) together with formulas (3) will determine the control variables na na and γ in the form of functions of coordinates () () () and their first and second derivatives with respect to time. Engine thrust and angle of attack can be determined from the relations (). Thus, system () can be used to solve inverse dynamics problems. It should be noted that by now there are already a number of methods for generating a trajectory based on the concept of the inverse problem of dynamics. This article discusses the two most typical approaches: simple trajectory planning and trajectory formation based on the principle of optimality. 5

6. Simple trajectory planning It is assumed that the given initial state = T and final state = T of the aircraft, as well as the initial and final time of the maneuver. The initial and final control vectors u= T u = T can also be specified. It is required to construct a flight trajectory and control that satisfies all these boundary conditions. When considering the trajectory () () (), we replace physical time with relative time τ in accordance with the transformation formula. (8) Here Δ = - so that τ = at = and τ = at =. The result should be the dependencies ((τ)) = (τ) ((τ)) = (τ) ((τ)) = (τ). The trajectory planning procedure involves specifying functions (τ) (τ) (τ) in the form of parameterized dependencies using basis functions. For example, polynomials of the form h w (9) can be taken as (τ) (τ) (τ) where h w are constant coefficients and... basis functions with the property of linear independence. To simplify calculations, the structure of the basis functions is assumed to be sufficiently

7 7 simple only requires that the functions (τ) (τ) (τ) be continuous and at least twice differentiable. In particular, power relations of the form are convenient for use. Options with trigonometric functions can be used, as well as combinations of power and harmonic functions, for example. cos sn Differentiating dependences (9) with respect to τ we obtain the derivatives w h. w h Polynomials (τ) (τ) (τ) and their derivatives must satisfy the given boundary conditions: Based on these relations, we will compose three systems of equations:

8 8 w w w w w w h h h h h h () In () the values ​​Δ na na γ na na γ s s s s s s= =.. are known. Values ​​of quantities are determined by equations () and values ​​by relations (). System () represents 3=8 equations for 3=8 unknown coefficients (...) (h h...h) and (w w...w). The task of calculating coefficients from the system () is made easier by the fact that this system is divided into 3 independent subsystems. Getting a solution is easy. For example, for the first subsystem using vector-matrix notation T T B

9 A we can write A = B and thus the required formula for calculating the coefficients will take the form =A - B. Because the basis functions used have the property of linear independence, then the matrix A is not singular, therefore the inverse matrix A exists and there is a unique solution. The solutions of system () for the remaining coefficients (h h...h) and (w w...w) are determined in a similar way. 3. Trajectory planning using the direct variational method. In formulas (9) of the previous section, the fulfillment of the boundary conditions was ensured by a special choice of coefficients for given arbitrary basis functions. However, the boundary value problem can be solved in another way by means of a special choice of basis functions for arbitrarily given coefficients. In this case, the presence of freedom in the choice of coefficients allows you to combine the trajectory planning procedure with the optimization of any quality criterion and also take into account restrictions on phase and control variables. Apparently, such an approach for flight dynamics problems was first proposed by Taranenko in the context of optimization of direct control 9

10 by variation method. Taranenko's method involves replacing the argument of physical time with some generalized argument τ in accordance with the equation where λ is an unknown function. The trajectory is given by the relations d d (τ) = (τ) (τ) = (τ) (τ) = 3(τ) V(τ) = 4(τ). Here the functions (τ) = 4 must be continuous, single-valued and differentiable over the entire interval of values ​​of the argument τ. Functions (τ) are sought as a combination of known a priori specified basis functions: where j j j = 4 j = n basis functions j unknown n j coefficients. Functions and j are selected to satisfy inhomogeneous and homogeneous boundary conditions, respectively: For example, according to recommendations j. j

11 j j sn j or j j. It is easy to see that this choice of basis functions guarantees for (τ) the satisfaction of the boundary conditions for any values ​​of the parameters j. On the other hand, the functions (τ) depend on the coefficients j and therefore by choosing these coefficients one can influence the trajectory, ensuring optimization of a given quality criterion and fulfillment of control restrictions without worrying about boundary conditions. Let's transform the system () to a new argument τ: V g na sn / g a Ψ gna snγ/ V cos V coscos / V sn/ V cossn / / n cosγ cos/ V () Proceeding in the same way as described in the section from equations () is not difficult obtain the following kinematic relations: V sn V g V cos V 3/ 3/ cos. For control variables, the following formulas are obtained:

12 cos arcg g cos/ V n a V sn g n a V g cos. g cos The above formulas show that all control and state variables are expressed through (τ) (τ) (τ) V(τ) and their derivatives, but unlike the formulas in the section, a scaling function is additionally present here. The choice of free coefficients j will be subordinated to the optimization of the functional J p which depends on the goal of the problem (here p is the vector of coefficients j). Thus, the formation of an optimal trajectory that satisfies the given boundary conditions is reduced to a nonlinear programming problem: mn J (p) or pc ma J (p) () pc where C is the region of permissible values ​​of the parameters p ensuring the fulfillment of the required restrictions on controls and state variables. Recommendations regarding ways to solve this problem are given in. 4. Calculation examples The trajectory planning options discussed above were tested by numerical calculations for a number of typical maneuvers. The calculation results for two examples are presented in graphs in Figure 4. Graphs of simple trajectory planning (option) are displayed with dashed lines, and graphs of trajectory planning using the direct variational method (option) with optimization according to the performance criterion are displayed with solid lines. In both cases the boundary conditions are the same.

13 Example (turn by 8 with climb) Boundary conditions: - start of maneuver = V = 35 m/s Θ = rad Ψ = rad = m = 5 m = m na = na = γ = rad. - end of maneuver = 4.5 s V = 35 m/s Θ = rad Ψ = π rad = m = 8 m = -7 m na = na = γ = rad. In the calculations of the option, restrictions on controls and state variables are taken into account: 35 m/s V 8 m/s Θ -9 Ψ 7 -. na. -. na γ. 3

14 Fig.. Aircraft trajectories (Example). 4

15 Fig. Behavior of control and state variables (Example). In this example, the turn occurs with a fairly large radius. The curvature of the trajectory is small, so changes in control and state variables are slow and smooth. The graphs show that the results of the two options differ, but they are not too large. We can conclude that both options provide practical solutions. Example (turn by 8 with return to original altitude) Boundary conditions: - start of maneuver = 5

16 V = 35 m/s Θ = rad Ψ = rad = m = 5 m = m na = na = γ = rad. - end of maneuver =.5 s V = 35 m/s Θ = rad Ψ = π rad = m = 5 m = -8 m na = na = γ = rad. In the calculations of the option, restrictions on control and state variables are taken into account: 35 m/s V 8 m/s Θ -9 Ψ 7 -. na. -. na γ. Rice. 3. Aircraft trajectories (Example).

17 Fig. 4. Behavior of control and state variables (Example). In this example, the option produces a turning path with a very small radius. The curvature of the trajectory is large; therefore, changes in control and state variables occurred faster and more sharply than in the first example. The results of the options differ greatly. Analysis of the behavior of the dependences V() and na() for the variant (Fig. 4) shows that the overload na remains at the level of ~ under conditions of very low speeds V, which is completely unrealistic for a conventional aircraft. The minimum speed reaches ~7 m/s (at the th second), which is significantly less than the stall speed and is unacceptable under flight safety conditions. In the vicinity of this point, the graph of the dependence Ψ() (Fig. 4) 7

18 shows a sharp increase in the rotation angle. But this is quite natural because... in accordance with the kinematics of movement (see 3rd equation ()), the situation V in conditions n leads to the receipt. a Thus, in this example, the option produced a trajectory that was unacceptable for use. The result is quite predictable because This option does not take into account restrictions important for the practical implementation of the generated trajectory. At the same time, a formal check of the resulting solution for consistency between control variables and state variables does not provide any information about the unacceptability of the solution. In Fig. (5) shows graphs of the behavior of state variables for the approximating solution (9) and for the results of numerical integration of the original system of equations of motion () (4th order Runge-Kutta method) using controls calculated by formulas (7) for the generated trajectory. The graphs of both types coincide, which indicates the consistency of the approximating solution with the dynamics of the system under consideration. This one example alone demonstrates the insufficiency of simply planning an aircraft flight path without taking into account the restrictions associated with the implementation of this trajectory. The considered method of trajectory planning with optimization (option) in this example generated a completely realizable trajectory since this method takes into account the necessary restrictions. However, the volume of calculations by this method turns out to be very large because getting 8

19 solutions require the use of iterative nonlinear programming procedures. Rice. 5. Consistency check (markers o solution to the trajectory planning problem; solid lines; result of integration). Conclusion The article examines and analyzes with numerical examples two methods for planning the trajectory of an aircraft's spatial maneuver based on parametrization of the trajectory and the use of the concept of the inverse problem of dynamics. From the given calculation examples it follows that the simplest method is 9

20 planning that does not take into account restrictions on phase variables and controls can lead to unrealistic results. And despite its attractiveness due to its simplicity, this method is hardly acceptable for onboard use (we are talking about conventional aircraft). To more reliably solve the problem of generating a maneuver trajectory, you can use more complex methods that allow you to take into account at least the most important restrictions. The method of direct solution of the variational problem proposed by Taranenko, discussed in the article, in principle allows one to take into account such restrictions and at the same time perform optimization of the maneuver according to any given criterion. The main disadvantage of this method is the large amount of calculations caused by the need to perform nonlinear conditional optimization using iterative procedures. It should be noted that even a very complex method of generating a trajectory is not immune from obtaining unrealizable solutions, so the results obtained must be analyzed and verified. For onboard applications this poses a challenge. Bibliographic list. Taranenko V.T. Momdzhi V.G. Direct variational method in boundary value problems of flight dynamics. - M.: Mechanical Engineering s.. Nonlinear dynamics and control: Collection of articles / Ed. S.V. Emelyanova S.K. Korovina. - M.: FIZMATLIT. - 4 s.

21 3. Velishchansky M.A. Synthesis of a quasi-optimal trajectory of an unmanned aerial vehicle // Electronic journal “Science and Education” 3: hp://echnomag.bmsu.ru/doc/447.hml (publication date.3). 4. Kanatnikov A.N. Construction of trajectories of aircraft with a non-monotonic change in energy // Electronic journal “Science and Education” 3 4: hp://echnomag.bmsu.ru/doc/554.hml (publication date 4.3). 5. Kanatnikov A.N. Krischenko A.P. Tkachev S.B. Acceptable spatial trajectories of an unmanned aerial vehicle in the vertical plane // Electronic journal “Science and Education” 3: hp://echnomag.bmsu.ru/doc/3774.hml (publication date 3.).

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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;

Related

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 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 . Such control of the wing configuration during a maneuver must be performed automatically, since the pilot’s attention is overloaded when piloting. Speed ​​of drives that control elements, ma-. The neural mechanization of the wing should be sufficient to flexibly change their position during vigorous maneuvers. However, if such a system can be created, then the maneuverability of the aircraft at low speeds increases significantly.

Further reading, p. 104-114, P01, p. 278-294, , p. 339-390.

Control questions

1. What maneuver is called coordinated?

2. Why is there an unambiguous connection between Pua and ua during a coordinated maneuver in the horizontal plane?

3. What is the limitation of the available pua value at low indicated flight speeds? On the big ones?

4. Why does the minimum flight speed Utsh (Yaia req) increase with increasing pua.1res?

5. Derive the formula for i? V. Pr at pauT determined by (7.9). Analyze the RB relationship. up from height.

6. Show the approximate nature of the change in the overload of the pua when performing the Nesterov loop, barrel.