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Теоретическая механика
UDC 533.6.013
DOI: 10.22363/2312-9735-2017-25-2-161-169
General Integral for a Class of Non-Steady Atmospheric Flights and Applications to Trajectory Analysis
D. M. Azimov*, R. G. MukharlyamoV
* Department of Mechanical Engineering The University of Hawaii at Manoa 2540 Dole St., Holmes 202A, Honolulu, HI, 96822, USA ^ Department of Theoretical Physics and Mechanics Peoples' Friendship University of Russia (RUDN University) 6 Miklukho-Maklaya St., Moscow, 117198, Russian Federation
A complete analytical integration of the aircraft kinematic and dynamic equations of motion is presented. Different applications of defined integrals to trajectory analysis are considered. The dynamic equations are obtained under the assumptions, that acceleration due to aerodynamic lift, the difference between the accelerations due to propulsive thrust and aerodynamic drag are not changed, the aircraft body rate about the velocity axis is zero and the sideslip angle is zero. The general integral of these equations consists of six independent first integrals of motion and describes a class of non-steady flight trajectories in a maneuver plane. It will be shown that the dynamic equations can be derived and completely integrated in a closed-form for more general assumptions. The problem of computing thrust for a given trajectory has been considered. The trajectory is defined by constraint equation. Constraints stabilization equations, which have asymptotically stable trivial solution, are constructed. Explicitness can make the integrals applicable to modeling the trajectories of spacecraft, re-entry vehicles and missiles, and to the design of on-board targeting and guidance. An illustrative example is presented.
Key words and phrases: analytical integration, general integral, analytical solutions, aircraft nonlinear model, programmed constraints
1. Introduction
This paper presents a complete analytical integration of the aircraft kinematic and dynamic equations obtained under the following assumptions: (a) acceleration due to aerodynamic lift, and the difference between the accelerations due to propulsive thrust and aerodynamic drag are not changed; (b) the aircraft body rate about the velocity axis is zero; (c) the sideslip angle is zero. It will be shown that the general integral of these equations consists of six independent first integrals which lead to the closed-form analytical solutions. The studies of the existing literature show that the aircraft equations can be integrated in a closed-form for some specific cases of quasi-steady and non-steady flights, including the cases of climb and cruise with constant altitude, velocity or lift acceleration, negligible flight path angle or small angle of attack [1,2]. In some cases of optimal quasi-steady cruise trajectories, the equations of motion have been implicitly integrated or reduced to quadratures [2]. It should be noted that the studies presented in this paper were initiated with the purpose of integration of the 3rd order differential equation,
dr r x r
6t = wx (r - g), (1)
obtained for the nonlinear model of the aircraft tracking problem under the following assumptions: (1) acceleration due to aerodynamic lift, and the difference between the
Received 7th October, 2016.
The work is performed under support of RFBR, grant No 16-08-00558 A.
accelerations due to propulsive thrust and aerodynamic drag are not changed; (2) the aircraft body rate about the roll axis is zero; (3) the angle of attack and the sideslip angle are zero [3-5]. Note that the left hand side of Eq. (1) represents the jerk vector, and its expression does not explicitly depend on the accelerations due to thrust, drag and lift. Analysis show, however, that as the drag is a function of the square of the velocity, it would be very difficult to hold the lift, and the thrust-drag accelerations constant with zero angle of attack. In this paper, it will be shown that Eq. (1) can also be derived and completely integrated in a closed-form for a more general assumptions (a-c) with non-zero and variable angles of attack. It is demonstrated that the assumptions (a-b) can significantly extend the applicability of Eq. (1). Explicitness can make the integrals applicable to modeling the trajectories of spacecraft, re-entry vehicles and missiles, and to the design of on-board targeting and guidance [4].
2. Equations and Integrals for Non-steady Flight
Consider the F-frame formed by the triad of orthogonal unit vectors ef, e,f, e;f and with the origin O at the aircraft center of mass (COM): the unit vector ef is aligned with the velocity vector, e;f forms the angle ^ with lift and e,f completes the right handed system (see Fig. 1). The angle 0 is measured in the Oefef-plane. It is assumed that the non-steady flight trajectory lies in a vertical plane xz containing ef and ef. Then if P is the sum of external forces acting on the aircraft, that is thrust, weight, drag and lift, then [1]
Figure 1. To the nonlinear aircraft model
P = W + T + D + L
can be rewritten in the form:
P = (T cos a — D — go sin 0)ef + [(T sin a + L) cos 0 — g0 cos 0]ef,
where a = a + ar is the angle between the thrust vector and the velocity vector. If a = v ef + vd ef, then with the assumptions given above, the Newton's second law yields the following equations valid in the maneuver ¡rz-plane [3]:
v = —go sin 6 + c\, v0 = —go cos 0 + c2, c\ = 0, ¿2 = 0, (2)
where
c\ = I0 (T cos a — D), c2 = T0 (T sin a + L) cos 6 (3)
W W
with 6 = const. Complete analytical integration of Eqs. (2) and application of the resulting solutions to trajectory analysis is the main purpose of this paper. As will be shown below, the complete integration of Eqs. (2) reveals a general integral which consists of six independent first integrals with their constants. These constants will be denoted below by r]i, (i = 1,..., 6), and one can accept that r]\ = c\ = const and r]2 = c2 = const. Eqs. (2) are valid for a flight with the assumptions (a-c) in the maneuver plane. The first integrals of Eqs. (2) for r]i and r]2 represent the relationships between the velocity magnitude, flight path angle, the propulsive and aerodynamic accelerations.
3. Integrals for Velocity Vector, Time and Position Vector
Integrals for magnitude of velocity vector
In this subsection, it will be shown that the first two equations of Eqs. (2) can be explicitly integrated in elementary and transcendental functions in terms of the angle . By considering 0 as an independent variable instead of time, t, we have v = d v/dt = d v/dddd/dt. Then by eliminating d d/dt from Eqs. (2), one can obtain
d v _1 — go sin d + d d 6 — g0 cos 6 + c2
(4)
which can be integrated in the form [6]:
v( 0) = r]3(a + 6sin^)-1 exp
a tan x + d3
2 A a tan x + b
—— arctan---
d1 d1
v( 0) = rj3(a + bsinx)
-1
a tan x + d4
(A/d2)
v( 0) = rj3(a + 6sinx) exp
A
— tan (x — ^
[a2 > b2], [a2 < b2], [a2 = b2],
(5)
where r]3 is the integration constant, x = 6 + ^,x = x/2 and a + frsinx = 0, and the following constants are used:
A = c i, a = c2, b= —B = — g0, (6)
^ = Va2 — b2, d2 = \/ b2 — a2, d3 = b — \/ b2 — a2, d4 = b + b2 — a2. (7)
Note that in a particular case when a + 6sinx = 0, the system of equations in Eqs. (2) describes a motion with constant and . This case is of a very limited theoretical and practical interest, and not considered in this paper.
Integrals for time
Once v = v(6) is determined, the second equation of Eqs. (2) can be integrated as:
x — ^
=
v(x)dx a + sin x
+ '14,
(8)
xq—j
which can be reduced to the following final forms:
m
t =
t =
A2 + a2 - b2 m
A2 + a2 - b2
exp
exp
2A a tan x + b
—— arctan---
d\ d\
A a tan x + d3
— ln-=--
d2 a tan x + d4
A + b cos x a N a + b sin x A,
+ '14,
A + b cos x a N a + b sin x A,
+ ^4,
, m
t = ^ exp
A ( x ■k\] / â ta\2 - 4 J. {
A + b cos x a a(1 + sin x) A
+ V4,
[a2 >b2], [a2 < b2], (9) [a2 = 62],
where is the new integration constant, sin x = —1. As Eq. (1) is a 3rd-order vector differential equation, which describes the motion in the maneuver plane, its complete integration would require to find six independent first integrals with six scalar integration constants of motion in the maneuver plane. So far, four independent first integrals and four new constants have been found above, that is 'qi and ^2 in Eq. (2), in Eqs. (5) and in Eqs. (9). Eqs. (5) and Eqs. (9) also represent the general solution of Eqs. (2) with constants ^3, ^4, cl and c2.
Integrals for position vector components
It can be shown that the magnitude of the velocity vector and angle between the velocity vector and the local horizon are not enough to uniquely determine the position of the aircraft in the maneuver plane. If v = v cos d \x — v sin d \z and r = p \x — h iz, where p and h are the aircraft horizontal and vertical coordinates (crossrange and downrange respectively), then the equation v = r written in terms of its components yields
p = v cos 6, = v sin 0.
(10)
Noting that 6 = x — k/2, p = dp/dx dx/dt and h = dh/dx dx/dt, one can rewrite Eqs. (10) as
dp di . dh di
v— sin x, — = —V— cos X, (11)
dx
dx
dx
Integration of Eqs. (11) yields the aircraft coordinates p and h:
and
p(x) = P exp p(x) = P exp p(x) = P exp
h(x) = Q exp h(x) = Q exp h(x) = Q exp
4A a tan x + b
—— arctan---
d\ d\
2 A a tan x + d3 —— ln
d2 a tan x + d4 x K
2 - 4
2^ ( x ■k \
— tan----
a V 2 4)
+ [a2 >b2], + 'n5, [a2 <b2] + V5, [a2 = b2],
4 A a tan x + b
— arctan---
d\ d\
2 A a tan x + d3 —— ln
d2 a tan x + d4 f x k \ U - 4 )
+ >q6, [a2 >b2], + rie, [a2 <b2],
2A (x n
— tan----
a V 2 4
+ Ve,
[a2 = b2],
(12)
where r]5 and r/e are the new integration constants, x = x/2, and P and Q are known functions of x. Eqs. (12) and (13) represent the first integrals of Eqs. (10) (and Eqs. (2)), and allow us to determine the aircraft's horizontal and vertical cartesian coordinates (crossrange and downrange) in the maneuver plane. The first integrals presented in Eqs. (2), (5), (9), (12) and (13) with constants rji, i = 1,...,6 represent the general integral of Eq. (2). Any point on the trajectory can be considered as a target point and the constants can be chosen to achieve this point. Consequently, the targeting problem can be solved at any desired point thereby providing a foundation for the development and design of the targeting and guidance schemes.
4. Expressions for Thrust and Angle of Attack
As mentioned above, the assumptions (a)-(c) can be justified and validated by analyzing the thrust, drag and list accelerations using the solutions for altitude and velocity, and by comparing the results for the angle of attack and the thrust to existing ranges of these quantities [2,7,8]. The assumptions (a) and (b) mean that
T cos a — D (T sin a+ L) cos 6
-= ci = const, -= c2 = const, (14)
m m
where [1]
6 = const, m =—, D = -CDpaS v2, L = -CLpaS v2, go 2 2
and it is assumed that | = n/2 + kn, k = 0, 1, 2 ..., the drag and lift coefficients, Cd and Cl can be computed according to Ref. [2] and the air density, pa is changed according to the exponential law. From Eq. (14) one can obtain
(mci + D) sin a — I m°2 — L ) cos <5 = 0. (15)
\cos| )
Eq. (15) is a transcendental equation and solvable for a = a(h, v) only by numerical schemes. Once a = a(h, v) is determined, then the thrust can be computed as
T =\/<m 1 +D>2 + (— li ■ <16>
5. Illustrative Example
Consider the example of a flight simulation using the analytical solutions presented above for a2 > b2. One can compute magnitude of velocity vector, time, altitude and downrange in terms of the angle 6 which is assumed to satisfy the inequality: d0 < 0 < di. The following values have been accepted: t0 = 0.0, v0 = 250, g0 = 9.8, b = —9.8, po = 28500, ho = 7000, W = 5000, a = 10.0, A e [0.1, 0.5], do = —10°, Ad = 65°. The results of the simulations for this case are illustrated in figures 2-4. The plots of the angle of attack and the thrust vs magnitude of velocity are presented in figures 2 and 3. These figures show that when the velocity magnitude is decreased in the beginning of the simulation for ~ 10 ft/s, a is increased for ~ 0.5° and T is decreased for ~ 45 lbs. Analysis has also shown that an increase in values of W leads to the higher values for a and T, and also T is increased together with altitude and Mach number, computed as M = v/va, where va is the speed of sound [2]. From Fig. 4 it can also be seen that each additional value 0.1 to A would yield an increase of the final altitude by only 100 m.
a2=100; b2=96.04; W=5000
14 13.5 13
_ 12.5
S 12
1 11.5
■¡Î 11 <
10.5 10 9.5
620 640 660 680 700 Magnitude of velocity vector (ft/s)
a2=100; b2=96.04; W=5000
1500 1450 1400 1350 ) 1300 1250 1200 1150 1100
1050
580 600 620 640 660 680 700 720 740 Magnitude of velocity vector (ft/s)
600
720
740
Figure 2. Angle of attack vs magnitude Figure 3. Propulsive thrust vs of velocity magnitude of velocity
a=10; v=250; e=-0.17453; p0=28500; h=7000
9500
2.45 2.5 2.55 2.6 2.65
Range on x-axis (m) x 104
Figure 4. Downrange vs altitude
6. Computation of Thrust with Stabilization of Constraints
Let the aircraft trajectory be given in the form h = f (p). Then the equations of a programmed constraint and its derivative are y = h — f (p) and ^ = vy with [9]
uy = —^ (sin 6 + f '(p)cos e), f(p) = , (17)
The equations of the perturbed constraint can be given as
^ = vy, ^ = K (y,uy ,p,h,v,6), K (0, 0,p,h,v,6) = 0, (18)
and in particular, ^ = —cy — kvy. Eqs. (2), (17) and (18) yield the equation with respect to the thrust:
£ = — dH (sin 0 + f Mcos D) + * (/'(p)sin e — cos 0) ^ — ^/"(p)cos 9
TO) = ^
After substitution of these expressions into Eqs. (18), which provides
^ = — d- (sin e + f'(p)cos 0) + ^ (/'(p)sin 0 — cos 6) d- — uf ''(p)cos 0d,
one can obtain the equation for determination of T: T = A/B, where
A = mK — (sin 0 + f'(p) cos 0) (mg sin 0 + D) — — {f'(p) sin 0 — cos 0) (—mg cos 6 + L cos (p)+ + mv2f "(p) cos2 0, B = — (sin 0 + f'(p) cos 0) cos a + (/'(p) sin 0 — cos 0) cos sin a.
Assume now that K = —cy — kvy = c(f (p) — h) + fc^(sin 0 + f'(p) cos 0), with c > 0, k > 0. Then one can show that
A = —mch + mkv sin 0 — mg + D sin 0 + L cos cos 6+
+ mc/ (p) + (mkv cos 6 — D cos 6 — L cos sin d)f' (p)+
+ mv 2f'(p) cos2 d,
B = — cos a sin 0 — sin a cos cos 6 + (cos a cos 6 + sin a cos sin d)f' (p).
7. Conclusions
The general integral of the aircraft's kinematic and dynamic equations of motion in the non-steady flight conditions has been obtained. These equations represent the 3rd order vector differential equation, the general integral of which consists of six independent first integrals with six corresponding constants. All integrals are expressed in elementary and transcendental functions in terms of the flight path angle. The applications may also include the flight trajectories in the transonic, low and high supersonic conditions. These results can find potential applications in the design of on-board targeting and guidance schemes.
References
1. N. X. Vinh, Optimal Trajectories in Atmospheric Flight, Elsevier Scientific Publishing Company, Amsterdam, 1981.
2. M. Asselin, An Introduction to Aircraft Performance. AIAA Education Series, Write-Patterson Air Force Base, Ohio, 1965.
3. R. H. Bishop, A. C. Antoulas, Nonlinear Approach to Aircraft Tracking Problem, Journal of Guidance, Control and Dynamics. AIAA 17 (5) (1994) 1124-1130.
4. D. M. Azimov, R. H. Bishop, One Class of Nonlinear Model Solutions for Flight Vehicles and Applications to Targeting and Guidance Schemes, in: Advanced Maui Optical
and Space Surveillance Technologies Conference (AMOS). Maui, Hawaii. September 11-14, 2012, 2012.
5. D. M. Azimov, On One Case of Integrability of Atmospheric Flight Equations, AIAA Journal of Aircraft 48 (5) (2011) 1722-1732.
6. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products, Academic Press, New York, 1965, fourth Edition, 1965. See formulas TI(344), 2.552(2), TI(250), 2.555(3).
7. A. I. Matyuhin, Control of Motion of Mechanical Systems, Fizmatlit, Moscow, 2009, in Russian.
8. A. A. Burov, I. I. Kosenko, The Lagrange Differential-Algebraic Equations, Journal of Applied Mathematics and Mechanics 78 (6) (2014) 587-598, in Russian.
9. R. G. Mukharlyamov, Simulation of Control Processes, Stability and Stabilization of Systems with Program Constraints, Journal of Computer and Systems Sciences International, 2015, Vol. 54, No. 1, pp. 13-26. 54 (1) (2015) 13-26, in Russian.
УДК 533.6.013
DOI: 10.22363/2312-9735-2017-25-2-161-169
Общий интеграл для одного класса нестационарных атмосферных летательных аппаратов и приложения для
анализа траекторий
Д. М. Азимов*, Р. Г. Мухарлямов1"
* Кафедра машиностроения Гавайский университет в Маноа ул. Доул, 2540, Холмс 202A, Гонолулу, США, 96822 ^ Кафедра теоретической физики и механики Российский университет дружбы народов ул. Миклухо-Маклая, д. 6, Москва, Россия, 117198
Представлено полное интегрирование уравнений кинематики и динамики движения самолёта. Рассмотрены различные применения полученных интегралов к анализу траекторий. Уравнения динамики получены в предположении, что разница между ускорением, вызванным аэродинамической подъёмной силой, и ускорением тяги не меняется, направление курса самолёта относительно продольной оси остаётся постоянным, угол атаки и угол скольжения равны нулю. Общее решение состоит из шести первых интегралов уравнений движения и описывает множество траекторий в вертикальной плоскости. Показано, что уравнения динамики могут быть получены и проинтегрированы в замкнутой форме при более общих предположениях. Рассматривается задача определения величины тяги, соответствующей данной траектории, заданной уравнением связи. Строится уравнение возмущений связи, имеющее асимптотически устойчивое тривиальное решение. Предлагаемый метод построения интегралов может быть использован в задачах построения траекторий космических аппаратов, ракет и спускаемых аппаратов, а также при проектировании бортовых систем целеуказания и наведения. Приводится иллюстрационный пример.
Ключевые слова: аналитическое интегрирование, общий интеграл, аналитические решения, нелинейная модель самолёта, программные связи
Литература
1. Vinh N. Optimal Trajectories in Atmospheric Flight. — Amsterdam: Elsevier Scientific Publishing Company, 1981. — 402 p.
2. Asselin M. An Introduction to Aircraft Performance. AIAA Education Series / Ed. by J. Przemieniecki. — Ohio: Write-Patterson Air Force Base, 1965. — 334 p.
3. Bishop R., Antoulas A. Nonlinear approach to aircraft Tracking Problem // Journal of Guidance, Control and Dynamics. AIAA. — 1994. — Vol. 17, No 5. — Pp. 1124-1130.
4. Azimov D., Bishop R. One Class of Nonlinear Model Solutions for Flight Vehicles and Applications to Targeting and Guidance Schemes // Advanced Maui Optical and Space Surveillance Technologies Conference (AMOS). Maui, Hawaii. September 11-14, 2012. — 2012.
5. Azimov D. On One Case of Integrability of Atmospheric Flight Equations // AIAA Journal of Aircraft. — 2011. — Vol. 48, No 5. — Pp. 1722-1732.
6. Градштейн И., Рыжик И. Таблицы интегралов, сумм, рядов и произведений. — М.: Физматлит, 1963. — 1100 с.
7. Матюхин В. Управление механическими системами. — М.: Физматлит, 2009. — 320 с.
8. Буров А., Косенко И. О дифференциально-алгебраических уравнениях Лагран-жа // Прикладная математика и механика. — 2014. — Т. 78, № 6. — С. 818-832.
9. Мухарлямов Р. Моделирование процессов управления, устойчивость и стабилизация систем с программными связями // Известия РАН. Теория и системы управления. — 2015. — № 1. — С. 15-28.
© Azimov D.M., Mukharlyamov R. G., 2017
