Article Optimal trajectory after Dave 2 cor 11 25 03а

Optimal Trajectories of Air and Space Vehicles*

Alexander Bolonkin

Senior Research Associate of the National Research Council at Eglin Air Force Base.

1310 Avenue R, #6-F, Brooklyn, NY 11229 USA

Tel/Fax. 718-339-4563

E-mail: aBolonkin@juno.com, http://Bolonkin.narod.ru

 

Abstract

ааа The author developed a theory of optimal trajectories for air vehicles with variable wing area and with conventional wings.а He applied a new theory of singular optimal solutions and obtained in many cases the optimal flight. The wing drag of a variable area wing does not depend on air speed and air density.а At first glance the results may seem strange however, this is correct and this paper will show how this new theory may be used.а The equations that follow allow computing the optimal control and optimal trajectories of subsonic aircraft with pistons, jets, and rocket engines, supersonic aircraft, winged bombs with and without engines, hypersonic warheads, and missiles with wings.

ааа The main idea of the research is in using the vehicleТs kinetic energy for increasing the range of missiles and projectiles.

ааа The author shows that the range of a ballistic warhead can be increased 3-4 times if an optimal wing is added to the ballistic warhead, especially a wing with variable area. If we do not need increased range, the warhead mass can be increased. The range of big gun shells can also be increased 3 - 9 times. The range of aircraft may be improved 3-15% and more.

ааа The results can be used for the design of aircraft, missiles, flying bombs and shells of big guns.

-------------

Key words: optimal trajectory, singular optimal solution, range, aircraft, missiles, and projectiles.

*Theory presented to AIAA/NASA/USAF/SSMO Symposium on Multidisciplinary Analysis and Optimization, Panama City, Florida, USA, Sept. 7-9, 1994. Full text is published in AEAT, Vol.76, No.2, 2004, pp.193-214.

 

Nomenclature (in metric system)

a = the sound speed,

a₁, b₁, a₂, b₂ are coefficients of an exponential atmosphere,

C_L = lift coefficient,

C_D = drag coefficient,а

C_Do = drag coefficient for C_L=0,

C_DW = wave wing drag coefficient when a = 0,

C_Db ааbody drag coefficient,

c = relative thickness of a wing,

c_b is relative thickness of body,

c₁ = relative thickness of a vehicle body,

c_s = fuel consumption, kg/sec/ kg trust,

а= drag of vehicle,

D = drag of vehicle without a,

D_0W = wave wing drag when a = 0,

D_0b = drag of a vehicle body,

H = Hamiltonian,

h = altitude,

K=C_L/C_D is the wing efficiency coefficient,

аk₁, k₂, k₃ are vehicle average aerodynamic efficiencies for 1, 2, 3 sub distances respectively,

аL = range,

M=V/a is Mach number,

m = mass of vehicle,

p=m/S is a load on a wing square meter,

q=rV²/2 is a dynamic air pressure,

R is aircraft range or R = distance from flight vehicle to Earth center; R = R_o+h, where R_o=6378 km is Earth radius,а

t = time,

T = аis thrust,

V = vehicle speed,

V_e = speed of throw back mass (air for propeller engine, jet for jet and rocket engine),

S = wing area,

s = length of trajectory,

T = engine trust,а

Y = lift force,

a = wing attack angle,

b = fuel consumption,

q = angle between the vehicle velocity and horizon,

w = thrust angle between a thrust and velocity,

w_E = Earth angle speed,

j_E = lesser angle between Earth Pole axis and perpendicular to a flight plate,

r = air density.

Introduction

ааа The topic of the optimal flight of air vehicles is very important. There are numerous articles and books about the optimal trajectories of rockets, missiles, and aircraft. The classical research of this topic is in [1]. Unfortunately, the optimal theory of this problem is very complex. In most cases, the researchers obtained the complex equations, which allow one to compute a single optimal trajectory for a given aircraft and for given conditions, but the structure of optimal flight is not clear and simple formulas of optimal control (which depends only on flight conditions) are absent.

ааа The authorТs new theory of singular optimal solutions, developed earlier in [2]-[10], does not contain unknown coefficients or variables as previous theories have. He found that optimal flight path depend only on flight conditions and the addition of certain variable wing structures.

ааа In conclusion, the author applies his solution to ballistic missiles, warheads, flying bombs, big gun shells, and subsonic, supersonic, and hypersonic aircraft with rocket, turbo-jet, and propeller engines. He shows that the range of these air vehicles can be increased 3-9 times.

1. General equations

ааа Let us consider the movement of an air vehicle with the following conditions:

а 1.The vehicle moves in a plane containing the EarthТs center. 2. The vehicle design alloys to change the wing area (this will prove important in the remainder of this article). 3. We neglect the centrifugal force from Earth rotation (it is less 1%). 4. Earth has a curvature.

а Then the equation of flight vehicle (in system coordinate when a center system is located in a center of gravity of the flight vehicle, the axis x - in a flight direction, the axis y - in a perpendicular direction of the axis x, fig.1-1) are

 

 

 

 

 

аа аааааааа

 

 

 

 

 

Fig.1-1. Vehicle forces and coordinate system.

ааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааа (1-1)-(1-2)

ааааааааа (1-3)-(1-5)

ааа All values are taken in metric system and all angles are taken in radians.

2. Flight with small change of vehicle mass and flight path angle

ааа Most air vehicles fly with angle q in the range ▒15⁰ (q = ▒ 0.2618 rad) and the engine located along the velocity vector. It means

sinq =q,аа ааааааа cosq =1 ,ааааааааа w=0 ,аааааааааааааааааааааааааааааааааааааа (2-1)-(2-3)

because sin15⁰=0.25882, cos15⁰=0.9659.

ааа Let as substitute (2-1)-(2-3) in (1-1)-(1-5)

ааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааа (2-4)-(2-5)

аааааааааааааааааааааааааааааааааааааааааааааааааааааа (2-6)-(2-8)

аааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааа (2-9)

where

.аааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааа (2-10)

ааа A lot of air vehicles fly with the small anglar speed dq/dt. The change of mass is also small in flight. It means m=const, dm/dt @ 0.

ааа dq/dt╗ 0,а ааа dm/dt = 0 .аааааааааааааааааааааааааааааа (2-11)- (2-12)

Let us take a new independent variable s = length of trajectory

dt = ds/V, аааааа ааааааааааааааааааааааааааааааааааааааааааааааа ааааааааа(2-13)

and substituteа (2-10)-(2-13) in (2-4)-(2-9). Then system (2-4)-(2-9) takes the form

аааааааааааааааааааааааааааааааааааааааааааааааааааааааааа (2-14)-(2-17)

Let us to re-write the equation (2-17) in form

аааааааааа ааааааааааааааааааааааааааааааааааа (2-18)

If we neglect the last member, the equation (2-18) takes form

.аааааааааааааааааааааааааааааааааааааааааааааааааааааааааааа (2-18)Т

ааа If the V is not very large (V< 3 km/sec), the two last members in the equation (2-17) is small and they may be neglected. The equations (2-18), (2-18)Т can be used for deleting a from .

ааа Note the new drag without a as

D=D(h,V). ааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааа (2-19)

If we substitute a from (2-18) to Eq. (2-16) the equation system take the form

аааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааа (2-20)-(2-22)

Here the variable q is new control limited by

.аааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааа (2-23)

Statement of problem.

ааа Consider the problem: find the maximum range of an air vehicle described by equation (2-20) - (2-22) for limitation (2-23). This problem may be solved by the conventional method. However, this is a non-linear problem and contains the linear control. That means, this problem has a singular solution. For finding the singular solution, we will use the methods developed in [2] and [4].

ааа Write the Hamiltonian

а,ааааааааааааааааааааааааааааааааааааааа (2-24)

where are unknown multipliers. Application of the conventional method gives

аааааааааааааааааааааааа (2-25)-(2-27)

Where аdenote the first partial derivatives D, T per h, V respectively.

ааа The last equation shows that control q can have only two values ▒q_max. We consider the singular case when

ааааааааааааааааааааааааааааааааааааааааааааааааааааааааааа аA =.ааааааааааааааааааааааааааааааааааааааааааааааааааа (2-28)

This equation has two unknown variables l₁ and l₂ and does not contain information about the control q.ааа

ааа Let us, as in [2], to differentiate equation (2-28) to the independence variable s. After substitution the equations (2-22), (2-25), (2-26), (2-28), we received the relation for l₁ ¹ 0, l₂¹ 0

ааааааааааааааааааааааааааааааааааааааааааааааааааааааа (2-29)

ааа This equation does not contain q either, but it contains the important relation between the variables V and h on the optimal trajectory.

If we know formulas (or graphs)

D = D(h,V), ааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааа (2-30)

T = T(h,V), аааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааа (2-31)

we could find the relation

h = h(V) аааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааа (2-32)

and the optimal trajectory for a given air vehicle.

ааа That also gives the important information about the structure of the optimal solution. The investigation of the equation (2-29) shows that the equation has one solution in each, the subsonic, supersonic, and hypersonic fields. The equation can have two solutions for a transonic field.

ааа It means the optimal trajectory in most cases has three parts (see fig. 2-1):

a)      In a climb and flight: a vehicle moves from the initial point A with the angle ▒up to the optimal curve (2-32), then one moves up along the optimal curve (2-32), and further that moves with the angle ▒ to the point B.аа

b)      In a descent and flight (Fig.2-2): a vehicle moves from the initial point A with the angle ▒(up or down) to the optimal curve (2-32), then one moves down along the optimal curve (2-32), and further that moves with the angle ▒(up or down) to the point B.аа

 

 

 

 

ааааааааааааааааааааааааааааааааааааааааааааааааааааааааа аааааааааааааааа

 

 

Fig. 2-1 (left). Optimal trajectory for air vehicle climb and flight.

Fig.2-2 (right). Optimal trajectory for air vehicle descent and flight.

 

ааа The selection of direction (up or down, with аorа -respectively) depends only on the position of the initial and end points A and B.аааааааааааааааааааа

ааа For air vehicles with rocket engines T=const, the equation (2-29) has very simple form

ааааааааааааааааааааааааааааааааааааааааааааааааааааааааааа а.аааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааа (2-33)

ааа The same form (same curve) is also for a ballistic warhead, which does not have engine thrust (after itТs short initial burn)а (T=0).

If we want to have an equation for the control q, we continue to differentiate equation (2-29) with the independent variable s, and substitute the equations (2-21), (2-22), (2-25), (2-26), (2-28), (2-29). We received the relation for q if l₁ ¹ 0, l₂¹ 0

ааааааааааааааааааааааааааааааааааааааааааааааа аааааааааааааааааааааааааааааааааааааааааааа (2-34)

where

аааааааааааааааа ааааааа (35)-(36)

ааа Here signs in form are the second partial derivates D for h, V.

а .аааааааааааааааааааааааааааааааааааааааааааааааааааааааааа (2-37)

ааа If the thrust does not depend from h, Vа (T=const) or no engine (T = 0), the equation for q became simpler

ааааааааааааааааааааааааааааааааааааааааааааааааа (2-38)

ааа With according [2]-[8] (see, for example, [4], Eq. (4.2)) the necessary condition of optimal trajectory is

аааааааааааааааааааааааааааааааааааааааааааааа (2-39)

where k = 1.

ааа For getting results for different forms of the drags and thrusts, we must take formulas (or graphs) for subsonic, transonic, supersonic, hypersonic speed, specific formulas for the thrust

and substitute them in the equation (2-29), (2-34). Consider two cases: subsonic and hypersonic speeds.

Subsonic speed (V<270 m/sec) and different engines

Lift, drag, and derivative equations for subsonic speed are

а where. Magnitude e╗z²/pl is an induced drag coefficient, l=l²/S, l is a wing span.

ааа It is known in conventional aerodynamics that the coefficient of the flight efficiency k is

ааааа (2-41)

a)      Aircraft with rocket engine. For this aircraft the thrust T is constant or 0. The equation (2-29) has form (2-33). Find the partial derivatives

ааааааааааааааааааааааааааааааааааааааааааааааааааааааа (2-42)

ааа Substitute (2-40)-(2-42) in (2-33) we get the relation between an air density r, altitude h, and aircraft speed V:

ааа ,аааааааааааааааааааааааааа (2-43)

where p=m/S is a load on a wing meter square. For diapason h = 0 ¸ 11 km the coefficients a₁=1.225, b₁=9086.

ааа Results of the computation is presented in fig. 2-3.

b) Aircraft with turbo-jet engine. The trust for this engine is

аааааааааааааааааааааааааааааааааааааааааа (2-44)

Substitute (2-44) in (2-29). We get

,ааааааааааааааааааааааааааа (2-44)Т

substitute (2-40), (2-44) in (2-29), we get

а.аааааааааааааааааааааа (2-45)

Find r, h from (2-45)

ааааа (2-46)

Result of computation for the different p, T=0.8 N/kg, a₁=1.225, b₁=9086 are presented in fig. 2-4.

Fig.2-3 (left). Air vehicle altitude versus speed for the wing load p=400, 500, 600, 700 kg/m² and rocket engine.

Fig.2-4 (right). Air vehicle altitude versus speed for wing load p=400, 500, 600, 700 kg/m², turbo-jet engine, and relative trust 0.8 N/kg vehicle.

c)      Piston and turbo engines with propeller. All current propeller engines have propellers with variable pitch. Propeller coefficient efficiency, h, approximately is constant. The trust of this engine is

аааааааааааааааааааааааааааааааааа (2-47)

where N₀=N_eh, N_e = engine power at h = 0.

ааа Substitute (2-47) in (2-29). We get the equation for trust

а.аааааааааааааааааа (2-47)Т

Substitute (2-40), (2-47) in (2-29). We get

ааа (2-48)

Find r, h from (2-48)

ааааааааааааааааааааааааа (2-49)

Result of computation for C_Do=0.025, l=10, different p, N are presented in fig. 2-5.

Fig.2-5. Air vehicle altitude versus speed for the wing load p = 250, 300, 350, 400 kg/m², piston (propeller) engine, and relative engine power 100 W/kg vehicle.

Hypersonic speed (1 km/s< V<7 km/s).

ааа The lift and drag forces in hypersonic flight approximately equal (see (2-18Т)

ааааааааааааааааааааааа (2-50)

Noteа

аааа (2-51)

The derivatives of D by V, h are

аааааа .аааааааааааааааааааааа (2-52)

a) Rocket engine or hypersonic glider. The derivatives from T=const and T=0 are

ааааааааааааааааааааааааааааааааааааааааааааааааааааааа (2-53)

Substitute (2-51) in (2-52), expressions (2-52), (2-53) in (2-33) and find r, h. We get for h > 11,000 m

а,а (2-54)

where a₂ = 0.365, b₂ = 6997 are coefficients of the exponent atmosphere for stratosphere 11 to 60 km.

ааа If we neglect the small member аfor M > 3 in (2-54), the equations get a form

ааааа

where C_DW ╗4c. If we neglect the member gb₂ (for M > 3), then

а.аааааааааааааааааааааааааааааааааааааааааааааааааа (2-55)

In the limit as Rо ¥ in (2-54), we find

а.ааааааааааааааааааааааааааааааааааааааааааааааааааааааа (2-55)Т

ааа Here аis an optimal (maximum C_L/C_D) wing attack angle of the horizontal flight.

Results of computation (2-54) are presented in fig. 2-6.

Fig. 2-6. Optimal vehicle altitude versus speed for specific body load P_b=3, 5, 7, 10 ton/m², body drag coefficient C_b=0.02, wing drag coefficient C_d = 0.025, wing load p = 600 kg/m².

 

d)      Ramjet engine. The trust of jet engine approximately equals (M < 4)

а,аааааааааааааааааааааааааааааааааа (2-56)

where x is a numerical coefficient, r₂ is the air density in the lower end of the selected atmospheric diapason (in our case 11 km).

ааа Substitute (2-56), (2-52) in our main equation (2-29). By repeat reasoning we can get the equation for the given engine

аа (2-57)

where T₀ is taken in the lower end of the exponent atmospheric diapason (in our case 11 km). The curve of the air density vs. the altitude h is computed similar to (2-54).

Optimal wing area

ааа The lift force and drag of any wing may be written as

.ааааааааааааааааааааааааааааааааааааааааааа (2-58)

Substituteа (2-58) in (2-24) and find minimum H vs. S, we get equation

аааааааааааааааааааааааааааааааааааааааааааааа (2-59)

where a is value found from the first equation (2-58). The equation (2-59) is the general equation of the optimal wing area and optimal specific load p=m/S on a wing area.

a)      Subsonic speed.аа Lift force and drag of the subsonic wing are

а,аааааааа (2-58)Т

where q=rV²/2 is a dynamic air pressure for the subsonic speed.а

ааа Substitute the last equation (2-58) to the first equation (2-59). We get an optimal specific load on the wing area

а.ааааааааааааааааааааааааааааааааааааааааааааааааааааа (2-59)Т

Substitute a from (2-58)Т into the last equation (2-58)Т and divide both sides by a vehicle mass m. We get

.ааааааааааааааааааааааааааааааааааааааааааааааааа (2-60)

Here: D/m is a specific drag (drag per a vehicle weight unit). Substitute (2-59)Т into (2-60). We get the minimum drag of variable wing

,аааааааааааааааааааааааааааааааааааааааааааааааааааааааа (2-60)Т

аwhere the member in the right side is wing drag per a lift of one vehicle weight unit. We discover the important fact: the OPTIMAL wing drag of a variable wing DOES NOT DEPEND on air speed. It depends ONLY on the geometry of a wing!а It may look wrong, however consider the following example. Wing drag equals D=mg/K, where K=C_L/C_Dа is the wing efficiency coefficient. The value D/m does not depend on speed.

ааа If the air vehicle has body, the minimum drag is

.ааааааааааааааааааааааааааааааааа (2-61)

Full vehicle drag depends on a speed because the body drag depends from V.

Substitute (2-59)Т to a into (2-58)Т. We get the optimal attack angle

.ааааааааааааааааааааааааааааааааааааааааааааааааааааааааааа (2-62)

This is the angle of optimal efficiency, but C_DW is the wing drag coefficient ONLY when a = 0 (not full vehicle as in the conventional aerodynamic). Coefficient of flight efficiency

а.аааааааааааааааааааааааааааааааааааа (2-63)

b)      Hypersonic speed. Equation of wing lift force and wing air drag for hypersonic speed are the following:

. аааа (2-64)

Substitute a from (2-64) into . We get

.ааааааааааааааааааааааааааааааааааа (2-64)Т

Substitute the wing load p=m/S to (2-64)Т. We get

.ааааааааааааааааааааааааааааааааааааа (2-65)

Find the minimum of the air drag D for p. Take the derivatives and set them equal to zero. We get

а .ааааааааааааааааааааааааааааааааааааа (2-66)

Substitute (2-66) in (2-65). We get a minimum wing drag

.ааааааааааааааааааааааааааааааааааааааааа

Sum of the minimum vehicle drag plus a body drag is

.аааааааааааааа (2-67)

Substitute (2-66) to a into (2-64). We get the optimal attack angle of vehicle without body

.ааааааааааааааааааааааааааааааааааааааааааааааааааааааа (2-68)

Coefficient of the flight efficiency k=Y/D is

а.

For hypersonic speed the coefficients approximately equal

аааа ,ааааааааааааааааааа (2-69)

ааа In numerical computation the angle q can be found from (2-21) as q =Dh/DR_g.

For the rocket engine or a gliding flight we find the following relation:

When S is optimum (variable), the partial derivatives from (2-67) equal

а.

ааа Substitute them in (2-33). We find the relationship between speed, altitude, and optimal wing load for a hypersonic vehicle with rocket engine and VARIABLE optimal wing:

ааааа .аааааааааааааааааааааааааааааааааа (2-70)

For z=4, e=2 the equation (2-69)Т has the form

ааааааааааааааааааааааааааааааааааааа (2-70)Т

ааа Result of computation by (2-70)Т for z=4, e=2, a₂=0.365, b₂= 6997 and different p_b are presented in figs. 2-6 (dash lines). As you see, the variable area wing saves a kinetic energy, because its curve is located over an invariable (fixed) wing. This is advantageous only at the orbital speed (7.9 km/sec) because no lift force is necessary.

Estimation of flight range.

Air and space vehicles without trust

Aircraft range can be found from Eq. (2-22)

,аааааааааааааааааа (2-71)

Consider a missile having the optimal variable wing in a descent trajectory with thrust T=0.

a) Make a simplest estimation using equation of a kinetic energy from the classical mechanics. Separate the flight in two stages: hypersonic and subsonic. If we have the ratio of vehicle efficiency , where k₁, k₂ ratio of flight efficiency for hypersonic and subsonic stages respectively, we find the following equations for a range in each region:аааа

Or more exactly

а,ааааааааааааааааааааа (2-72)

where R₁ is the hypersonic part of the range, R₂ is the subsonic part of the range, V₁ is an initial (maximum) vehicle hypersonic speed, V₂ is a final hypersonic speed, and h is an altitude in the initial stage of the subsonic part of trajectory.

b) To be more precise. Assume in (2-71) r =const (used average air density).

1. Hypersonic part of the trajectory. Substitute (2-67) to (2-71). We have

.ааааааа (2-73)

2. Subsonic part of the trajectory. Substitute (2-61) in (2-71). We get

а,аааааааааааааааааааааааааааааааааааааа (2-74)

where the values C₁, C₂ are

а.ааааааааааааааааааааааааааааааааа (2-75)

The trajectory (without rocket part of trajectory) is

.ааааааааааааааааааааааааа (2-76)

where R₂ = k₂h computed for altitude h at the end of a kinetic part of the subsonic trajectory.

3. Ballistic trajectory of a wingless missile without atmosphere drag is

,аааааааааа (2-77)

where h is the initial altitude, V₁ is the initial horizontal speed of the wingless missile at altitude h, V_y is an initial (shot) vertical speed at h = 0, V_i is the full initial (shot) speed at h = 0 .

ааа For hypersonic interval 5<V<7.5 km/sec, we can use the more exact equation

а,ааааааааааааааааааааааааааааааааааааааааааааааааааааааааа (2-78)

where R=6378 km is the radius of Earth. The full range of a ballistic rocket plus the range of a winged missile equals

аR_f=R_b+R_a+R_g, аааааааааааааааааааааааааааааааааааааааааааааааааааааааааа (2-79)

where R_g =kh is a vehicle gliding range from the final altitude h₂а (fig.5-1) with aerodynamic efficiency k.

The classical method of the optimal shot ballistic range for spherical Earth without atmosphere is

а,ааааааааааааааааааааааааааааа (2-80)

where b_opt is an optimal shot angle, V_A is a shot projectile speed, and Vc is an orbital speed for a circle orbit at a given altitude.

4. Cannon projectile. We divide the distance in three sub distances: 1) 1.2M < M, 2) 0.9M < M <1.2M, 3) 0 < M < 0.9M. аThe range of the wing cannon projectile may be estimated by equation

а.ааааааааа (2-81)

where k₁, k₂, k₃ are average aerodynamic efficiencies for 1, 2, 3 sub distances respectively. Conventionally, these coefficients have values: subsonic k₃=8-15, near sonic k₂=2-3, supersonic and hypersonic k₁= 4-9. If V > 600 m/sec, the first member in (2-81) has the most value and we can use the more simple equation for range estimation:

ааааааааааааааааааааааааааааааааааааааааааааааааааааааааааа .аааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааа (2-80)Т

ааа At the top of the trajectory, modern projectile can have an additional impulse from small rocket engines. Their weight equals 10-15% of the full mass projectile and increases the maximum range in 7-14 km. In this case we must substitute V=V₁+dV in (2-80)Т, where dV is the additional impulse (150-270 m/sec).

Subsonic aircraft with trust. Horizontal flight

ааа The optimal climb and descent of a subsonic aircraft with a constant mass and fixed wing is described by equations (2-46), (2-43). Any given point in a climb curve may be used for horizontal flight (with different efficiency). We consider in more detail the horizontal flight when the aircraft mass decreases because the fuel is spent. This consumption may reach 40% of the initial aircraft mass. The optimal horizontal flight range may be computed in the following way:

ааааааааа (2-82)

where m is fuel mass, c_s is fuel consumption, kg/sec/ kg trust.

a) For fixed wing, we have (from (2-40))

аааа .аааааааааа (2-83)

ааа Substitute (2-83) in (2-82), we get

, (2-84)

b)      For variable wing we have (from (2-61)

,аа (2-85)

Results of the computation are presented in fig. 2-7. Aircraft have the following parameters: C_DW = 0.02; C_Db =0.08; b₁ =9086; S =120 m²; m =100 tons, m_k =80 tons, c_s = 0.00019 kg/sec/kg trust; wing ratio l =10.

ааа As you see, the specific fuel consumption does not depend on speed and altitude, a good aircraft design reaches the maximum range only at one point, in one flight regime: when the aircraft flies at the maximum speed admissible by critical Mach number, at maximum altitude admissible by engine. The deviation from this point decreases the range in 5-10-15 percent or more. The variable wing increases efficiency of the other regime. That approximately decreases the losses by a half.а

ааа The coefficient of the flight efficiency may be computed by equation k=g/(D/m), where values

, ааааааааааа (2-86)

for fixed and variable wings respectively. Result of computation is presented in fig. 2-8. The curve of the variable wing is the round curve of the fixed wing.

Fig. 2-7 (left). Aircraft range for altitude H = 6, 8, 10, 11, 12 km; maximum range R_m = 4361 km; relative fuel mass M_r = 0.2; body drag coefficient C_b = 0.08; wing drag coefficient C_d = 0.02.

Fig. 2-8 (right). Aerodynamic efficiency of non-variable and variable wings for wing load p = 400, 600, 800, 1000 kg/m², wing drag C_D = 0.02, body drag C_Db = 0.08, wing ratio 10.аа

 

3. Optimal engine control for constant flight pass angle

ааа Let as to consider the equation (1-1)-(1-5) for constant trajectory angle q = const. Substitute

q = constant, thrust T=V_eb , and a new independence variable s = Vt (where s is the length of trajectory) into equation system (1-1)-(1-5). We get equations

аааааааааааааааааааааааааааааааааааа (3-1)-(3-6)

The equation (3-5) is used for substitute a into equation (3-3) and for a change of air drag

ааааааааааааааааааааааа =D(V,h).ааааааааааааааааааааааааааааааааааааааааааааааааааа (3-7)

ааа We received a nonlinear system with a linear fuel control b. It means this system can have a singular solution.

Solution

ааа Consider the maximum range for vehicles described by system (3-1)-(3-6).

Let as to write Hamiltonian H

,аааааааааааааааааааа (3-8)

where are unknown multipliers. Application of conventional method gives

а (3-9)-(3-11)

Where аis the first partial derivates D per V.

ааа The last equation shows that a fuel control b can have only two values, ▒b_max. We consider the singular case when

ааааааааааааааааааааааааааааааааааааааааааааааааааааааааааа аA = .ааааааааааааааааааааааааааааааааааааааааааааааааа (3-12)

ааа This equation has two unknown variable, l₂ and l₃, and does not contain information about fuel control b.

ааа The first two equations (3-1)-(3-2) do not depend on variable and can be integrated

L=s cosq , ааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааа (3-13)

h=s sinq .аа ааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааа (3-14)

ааа According with book [2] let as to differentiate the equation (3-12) to the independent variable s. After substitution the equations (3-3)-(3-5), (3-7), (3-9), (3-10), (3-12), (3-14) we get the relation for l₂ ¹ 0, l₃¹ 0:

а .аааааааааа (3-15)

ааа This equation also does not contain b, however it contains an important relation between variables m, h and V, on an optimal trajectory. This is 3-dimentional surface. If we know

D = D(h,V) ,ааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааа (3-16)

V_e = V_e(h,V) ,ааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааа (3-17)

mass of our apparatus m, and itТs altitude h, we can find the optimal flight speed. It means we can know the needed thrust and the fuel consumption for every points m, h, Vа (Fig.3-1).

ааа If we want to have an equation for a fuel control b, we continue to differentiate the equation (3-15) to the independence variable s and substitute equations (3-1)-(3-14). We calculate the relation for b, if l₂ ¹ 0, l₃¹ 0, V_e=const, then

,аааааааааааааааааааааааааааааааааааааааааааа (3-18)

where

.аааааааааааааааааааааааааааааааааааааааа (3-19)

 

 

 

 

 

 

Fig.3-1. Optimal fuel consumption of flight vehicles.

 

ааа The necessary condition of the optimal trajectory as it is shown in [2]-[8] (see for example, [4], Eq. (4.2)) is

аааааааааааааааааааааааааааааааааааааааааааааа (3-20)

where k = 1.

ааа If the flight is horizontal (q = 0), the expression (3-18) is very simply

а.аааааааааааааааааааааааааааааааааааааааааааааааааааааааа (3-21)

It means, the trust equals the drag. This fact is well known in aerodynamic science.

 

ааа For getting the specific equations for different forms of drag and thrust, we must take formulas (or graphs) for a subsonic, transonic, supersonic and hypersonic speed for a thrust and substitute them in the equation (3-15), (3-18).а

4. Simultaneously optimization of the path angle and fuel consumption

ааа Consider the case when the path angle and the fuel consumption are simultaneously optimized.

In this case the general equations (1-1)-(1-5) have a form:

а аааааааааааааааааааааааааааааааааааа (4-1)-(4-5)

Let us to write the Hamiltonian

ааааааааааааааааааааааааа (4-6)

The necessary conditions of optima give

ааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааа а (4-7)-(4-8)

The lamda equations are

ааааааааааааа а(4-9)-(4-11)

Let us to difference A (4-7). From dA/ds=0, we find the optimal fuel consumption

ааааааааааааааааааааааааааааааааааааааааааааааа а.ааааааааааааааааааааааааааааааааааааааааааааааааа (4-12)

Let us difference B (4-8). From dB/ds=0 we find the optimal path angle

ааааааааааааааааааааааааааааааааааа а.аааааааааааааааааааааааааааааааааааааааааа (4-13)

ааа We used the conventional marks for the partial derivatives in (4-9)-(4-13) as in part 2 and 3 (see for example (2-47)).

ааа If we know from analytical formula or graphical functions Ve, D, Y we can find the optimal trajectory of the air vehicle.

а ааIn the general case, this trajectory includes four parts:

1. Moving between limitations q and b.
2. Moving between one limitation q or b and one optimal control b or q.
3. Moving simultaneously with both optimal controls q and b.
4. Moving on a given point along one limitation and/or both limitations.

5. Application to aircraft, rocket missiles, and cannon projectiles

A) Application to rocket vehicles and missile.

аааа Let us to apply the previous results to the typical current middle and long distance rockets with warhead. We will show: if warhead has wings and uses the optimal trajectory, the range of warhead (or useful load) is increased dramatically in most cases. We will compute the following optimal trajectories: the rocket launched warhead beyond altitude (20 Ц 60 km) and speed (1 Ц 7.5 km/sec). The point B is located on the curve (2-54) for a fixed wing and it is located on curve (2-69)Т for a variable wing (fig.5-1). Further, the winged warhead flights (descent) along optimal trajectory BD (fig.5-1) accorded the equations (2-54)(fixed wing) or the equations (2-69)Т(variable wing) respectively. When speed is decreased a small amount (for example, 1 km/sec) (point D in fig. 5-1), the wing warhead glides (distance DE in fig.5-1).

 

 

 

 

 

Fig.5-1. Trajectory of flight vehicles.

The following equations are used for computation:

1.      The optimal trajectory for space vehicle with FIXED wing.

a)      Eq. (2-54) is used for computing h=h(V) of the warhead optimal trajectory of the non- variable fixed wing in the speed interval 1<V<7.5 km/sec. Result is presented in fig.2-6.

b)      аEq. (2-50) computes the magnitude (D/m).

c)      The equation (2-71) in form

аа (5-1)

is used for computation in the intervals R_a, R_g fig.5-1. Here R_g is the range of a gliding vehicle.

d)      The equation (2-71) is used for computation R_b in the launch interval AB fig. 5-1.

e)      The full range, R, of warhead with a fixed wing and the full ballistic warhead range, R_w,а are

.ааааааааааааааааааааааааааааааааааааааааааааа (5-2)

f)        The equation (2-80) is used for computation the optimal shot BALLISTIC trajectory without air drag (vehicle WITHOUT wing). The range of this trajectory, as it is known, may be significantly more then range in the atmosphere.

аFig.5-2 (left). Range of NON-VARIABLE wing vehicle for body drag coefficient C_b = 0.02, wing drag coefficient C_d = 0.025, wing load p = 600 kg/m².

Fig.5-3 (right). Relative range of NON-variable wing vehicle for the body drag coefficient C_b = 0.02, the wing drag coefficient C_d = 0.025, the wing load p = 600 kg/m², the body load P_b=3 Ц10 ton/m².

а

ааа Result is presented in fig. 5-2. Computation of a relative range (for different p_b) by formulas

аааааааааааааааааааааааааааааааааааааааааааааааааааааааа (5-3)

is presented in fig. 5-3. The optimal range of the winged vehicle is approximately 4.5 times that of the ideal ballistic rocket computed without air drag. In the atmosphere this difference will be significantly more.

 

2. Rockets, missiles and space vehicles with VARIABLE wings.

The computation is same. For computing r, h, D/m we are used the equations (2-69)Т, (2-67) respectively. The results for different body loads are presented in fig.2-6.а The optimal trajectories of vehicles with the variable wing areas have a lesser slope. It means, the vehicle loses less energy when it moves.а It is located over the optimal trajectory of the vehicle with fixed wings.а It means it needs a lot of time (10-20) and more wing area then the fixed wing space vehicle (fig.5-4). The computation of the optimal variable wing area is presented in fig. 5-5. The relative range (Eq. (5-3)) is presented in fig. 5-6.

Fig.5-4 (left). Optimal wing load versus speed for specific body load P_b = 3, 5, 7, 10 ton/m², body drag coefficient C_b = 0.02, wing drag coefficient C_d = 0.025, wing load p = 600 kg/m².

Fig.5-5 (right). Range of VARIABLE wing vehicle for the body drag coefficient C_b = 0.02, the wing drag coefficient C_d = 0.025, the wing load p = 600 kg/m².

 

 

Fig.5-6 (left). Relative range of variable wing vehicle for the body drag coefficient C_b = 0.02, the wing drag coefficient C_d = 0.025, the wing load p = 600 kg/m², the body load P_b=3 Ц10 ton/m².

Fig.5-7 (right). Vehicle efficiency coefficient versus speed for specific body load P_b = 3, 5, 7, 10 ton/m², body drag coefficient C_b = 0.02, wing drag coefficient C_d = 0.025, wing load p = 600 kg/m².

 

The aerodynamic efficiency of vehicles with fixed (for different p_b bodies) and optimal variable wings computed by equations (5-1), (2-63) respectively are presented in fig. 5-7. The difference between the vehicle with fixed and variable wings reaches 0.2¸0.6 . The slope of the trajectory to horizontal is small (fig.5-8).

The range of the fixed wing vehicle computed by equation (5-1) is presented in fig. 5-2. The range of the variable wing vehicle computed by equation (5-2) is presented in fig. 5-5. The curve is practically same (see figs. 5-2, 5-5).

3. Increasing of a rocket payload for same range.а If we do not need to increase the range, the winged vehicle can be used to increase payload, or save a rocket fuel. We can change the mass of fuel or payload. The additional payload my be estimated by equation

,аааааааааааааааааааааааааааааааааааааааааааааааааа (5-4)

where m = m/m_b is a relative mass (the ratio of a rocket mass with wing vehicle to the ballistic rocket), DV=V_b-V is difference between the optimal ballistic rocket speedа (Eq.(2-80)) and the rocket with winged vehicle (Eq.(5-2)) for given range (see fig.5-2). Result of computation is presented in fig. 5-9. The mass of the rocket with winged vehicle may be only 20 Ц 35% from the optimal ballistic rocket flown without air drag.

Fig.5-8 (left). Trajectory angle versus speed for body drag coefficient C_b =0.02, wing drag coefficient C_d = 0.025.

Fig.5-9 (right). Mass ratio of wing rocket to ballistic rocket for specific engine run-out gas speed V_e = 1.8, 2, 2.2, 2.4, 2.6 and 2.8а km/sec.

Conclusion: The winged air-space vehicle increases the range by a minimum of 4.5-5 times compared to a shot optimal ballistic space vehicle.а The variable wing improves the aerodynamic efficiency 3-10% and also improves the range. The optimal variable wing requires a large wing area. If you do not need to increase the range, you may instead, increase payload.

B) Application to cannon wing projectiles.а

The typical current cannon properties are shown in table 1 below.

 

 

 

Table 1.

-------------------------------------------------------------------------------------------------------------

Name ааааааааааааа caliberааааааааааааа Nozzle speedааааааааааааааа Mass projectile Rangeаааааааааааааа RAPаааа

ааааааааааааааааааааааа mmааааааааааааааааа m/secаааааааааааааааааааааааааа kgааааааа ааааааааааааааааааааааа kmаааааааааааааааааа km

------------------------------------------------------------------------------------------------------------

M107аааааааааааааа 175ааааааааааааааааа 509-912аааааааааааааааааааааа 67ааааааааааааааааааааааааааааааа 15-33

SD-203ааааааааааа 203ааааааааааааааааа 960ааааааааааааааааааааааааааааа 110ааааааааааааааааааааааааааааа 37.5

2S19ааааааааааааааа 155ааааааааааааааааа 810ааааааааааааааааааааааааааааа 43.6аааааааааааааааааааааааааааа 24.7

2S1ааааааааааааааааа 122ааааааааааааааааа 690-740аааааааааааааааааааааа 21.6аааааааааааааааааааааааааааа -

S-23ааааааааааааааа 180ааааааааааааааааа -аааааааааааааааааааааааааааааааааа -аааааааааааааааааааааааааааааааааа 30.4аааааааааааааааа 43.8

2A36аааааааааааааа 152ааааааааааааааааа -аааааааааааааааааааааааааааааааааа -аааааааааааааааааааааааааааааааааа 17.1аааааааааааааааа 24

D-20ааааааааааааааа 152ааааааааааааааааа 600-670аааааааааааааааааааааа 43.5-48.8аааааааааааааааааааа 20

---------------------------------------------------------------------------------------------------------------а

Issue: JameТs

 

а The computation by Equation (2-80)Т for different k and RAP with dV=270 m/sec are presented in figs. 5-10, 5-11.

Fig.5-10 (left). Cannon winged projectile range for average aerodynamic efficiency k = 3, 5, 7, 9.

Fig.5-11 (right). Cannon winged projectile relative range for average aerodynamic efficiency k = 3, 5, 7, 9.

ааа Conclusion. As you see (figs.5-10, 5-11), the winged projectile increase range 3 Ц 9 times (from 35 up 360 km, k = 9). The projectile with RAP increase range 5-14 (from 40 up 620 km, k=9) times. The winged shells have another important advantage: they do not need to rotate. We can use a barrel with a smooth internal channel. This allows for the increase of projectile nozzle speed up 2 km/sec and the shell range up to 1000 km (k=5).

а C) Application to current aircraft.

ааа We used the equations (2-84), (2-85) for computation of the typical passenger airplane (Fig.5-12, thru 5-14, and 2-7). When all values are divided by maximum range R_m=4381 km (for a fuel mass 20% from a vehicle mass) at speed V=240 m/sec, altitude H=12 km. The speed is limited by the critical Mach number (V<M=0.82), the altitude is limited by the admissible engine trust, when engine stability is such that it works in cruiser regime. Fig. 5-12 shows the typical aircraft long-range trajectory.

а Conclusion: The best flight regime for a given air vehicle (closed to Boeing 737) is altitude H =12 km, speed V =240 m/sec, specific fuel consumption Cs=0.00019 kg fuel/sec/kg trust. The deviation from this flight regime significantly decreases the maximum range (up to 10-50%). The vehicle with a variable wing area losses 50% less then vehicle with a fixed wing.

 

 

 

Fig.5-12. Optimal trajectory of aircraft.

Fig.5-13 (left). Relative aircraft range for altitude H = 6, 8, 10, 11 and 12 km, maximum range R_m = 4381 km, relative fuel mass M_r = 0.2, body drag coefficient C_b = 0.08, wing drag coefficient C_d = 0.02.

Fig.5-14 (right). Relative aircraft range for speed V = 240 m/sec, maximum range R_m = 4381 km, relative fuel mass M_r = 0.2, body drag coefficient C_b = 0.08, wing drag coefficient C_d = 0.02.

6. General discussion and Conclusion

_{a.)       The current space missiles were designed 30-40 years ago. In the past we did hot have navigation satellites that allowed one to locate a missile (warhead) as close as one meter. Missile designers used inertial navigation systems for ballistic trajectories only. At the present time, we have a satellite navigation system and cheap devices, which allow locating of aircraft, sea ships, cars, any vehicle, and people. If we exchange the conventional warhead by a warhead with a simple fixed WING, having a control and navigation system, we can increase the range of our old rockets 4.5 Ц 5 times (fig.5-3) or significantly increase the useful warhead weight (fig.5-9). We also notably improve the precision of our aiming.}

_{b.)       Current artillery projectiles for big guns and cannons were created many years ago. The designers assumed that the observer could see an aim point and correct the artillery. Now we have the satellite navigation system which allows one to get exact coordinates of targets and we have cheap and light navigation and control devises which can be placed in the cannon projectiles. If we replace our cannon ballistic projectiles by a projectile with a fixed wing and control, navigation system, we increase its range 3-9 times (from 35 km up 360 km, see fig. 5-10, 5-11). We can use a smooth barrel to increase nozzle shell speed up to 2000 m/sec and range up to 1000 km. These systems can guide the WINGED projectiles and significantly improving their aiming.а We can reach this result because we use all of the KINETIC energy of the projectile. The conventional projectile cannot keep itself in the atmosphere and drops with a very high speed. Most of its kinetic energy is uselessly spent. In our case 70-85% of the projectileТs kinetic energy is spent for support of the moving projectile. That way the projectile range increases 3-9 times or more.}

_{c.)       All aircraft are designed for only one optimal flight regime (speed, altitude, and fuel consumption). Any deviation from this regime decreases the aircraft range. For aircraft closed to the Boeing 747 this regime is: altitude H=12 km, speed V=240 m/sec, specific fuel consumption Cs =0.00019 kgf/sec/kg trust.а If the speed is decreased from 240 m/sec to 200 m/sec, the range decreases 15% (fig.5-13).а The application of the variable wing area decreases this loss from 15% to 10%.а If the aircraft decreases the altitude from 12 km to 9 km, it loses 12% of its maximum range (fig.5-14).а If it has a variable wing area, the air vehicle loses only 7.5% of maximum range.а The civil air vehicles must deviate from the optimal condition by weather or a given flight air corridor. The military air vehicles must sometimes have a very large deviation from the optimal condition (for example, when they fly at low altitude out of the enemy radar system). The variable wing area may be very useful for them because it decreases the loss by approximately 50%, improves supersonic flight and taking off and landing lengths.ааа}

 

The author offers some fixed and variable wings for air vehicles (fig.5-15). Variants a, b, c, f for missile and warhead, variants d, e for shells.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig.5-15. Possible variants of variable wing: a, b, c and f, for aircraft; d and e for gun projectiles.

 

Acknowledgement

аThe author wishes to acknowledge scientist Dr. D. Handwerk for correcting the English.

 

References

1. A. Miele, General Variable Theory of the Flight Paths of Rocket-Powered Aircraft, Missile and Satellite Carries, Astronautica acta, Vol.4, pp.256-288, 1958.
2. A.A. Bolonkin, УSpecial Extrema in Optimal Control ProblemsФ, Akademiya Nauk, Izvestia, Theknicheskaya Kibernetika, No.2, March-April, 1969, pp.187-198. See also English translation in Eng. Cybernetics, No.2, March-April 1969, pp.170-183.
3. A.A. Bolonkin, New methods of optimization and their application, Moscow, Technical University named Bauman, 1972, pgs.220 (in Russian).
4. A.A. Bolonkin, УA New Approach to Finding a Global OptimumФ, УNew Americans Collected Scientific ReportsФ, Vol.1, 1991. The Bnai Zion Scientists Division, New York.
5. A. A. Bolonkin, Optimization of Trajectory of Multistage Flight Vehicles, in collection УResearches of Flight DynamicsФ, Masinobilding, Moscow 1965, pp.20-78 (in Russian).
6. A.A. Bolonkin, N.S. Khot, Method for Finding a Global Minimum, AIAA/NASA/USAF/SSMO Symposium on Multidisciplinary Analysis and Optimization, Panama City, Florida, USA. Sept.7-9, 1994.
7. A. A. Bolonkin, Solution Methods for Boundary-Value Problems of Optimal Control Theory. 1973. Consultants Bureau, a division of Plenum Publishing Corporation, NY. Translation from Prikladnaya Mekhanika, Vol.7, No 6, 1971, p.639-650 (English).
8. A. A. Bolonkin, Solution general linear optimal problem with one control. Journal УPricladnaya MechanicaФ, v.4, #4, 1968, pgs. 111-122, Moscow (in Russian).
9. A. A. Bolonkin, N. Khot, Optimal Structural Control Design, IAF-94-1.4.206, 45^th Congress of the International Astronautical Federation, World Space Congress-1994. October 9-14,1994/Jerusalem, Israel.
10. A. A. Bolonkin, N. S. Khot, Optimal Bounded Control Design for Vibration Suppression, Acta Astronautica, Vol.38, No.10, pp.803-813, 1996.
11. A. A. Bolonkin, D. Jeffcoat, Optimal Thrust Angle, 2002, (under publication)
12. A. A. Bolonkin, R. Sierakowski, Design of Optimal Regulators, 2^nd AIAA УUnmanned UnlimitedФ Systems, Technologies, and Operations Ц Aerospace, Land, and sea Conference and Workshop & Exhibit, San Diego, California, 15-18 Sep 2003, AIAA-2003-6638.