Application of calculus of variationsto optimize range for gliding flight

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Authors Waible, Leo Charles, 1932-

Publisher The University of Arizona.

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APPLICATION OF CALCULUS OF VARIATIONS TO OPTIMIZE RANGE FOR GLIDING FLIGHT

byLeo Charles Waible, Jr.

A Thesis Submitted to the Faculty of theDEPARTMENT OF AEROSPACE ENGINEERING.

In Partial Fulfillment of the Requirements For the Degree ofMASTER OF SCIENCE

In the Graduate CollegeTHE UNIVERSITY OF ARIZONA

1 9 6 5

STATEMENT BY AUTHOR

This thesis has been submitted in partial fulfill­ment of requirements for an advanced degree at The Univer­sity of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library.

without special permission, provided that accurate acknowledgement of source is made. Requests for permis­sion for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author.

This thesis has been approved on the date shown below:

Brief quotations from this thesis are allowable

SIGNED: ^f

APPROVAL BY THESIS DIRECTOR

(J, l/lvlTHOMAS L. VINCENT

Assistant Professor of Aerospace Engineering

ACKNOWLEDGEMENT

I wish to express my appreciation to Dr. Thomas L« Vincent for his interest and guidance in this paper. With­out his valuable assistance this thesis would not have been possible.

I am also deeply grateful to my loving wife, Jane, and my children, Leo, Edward, and Stephen, for their patience and understanding during the long hours of prep­aration.

TABLE OF CONTENTS

LIST OF ILLUSTRATIONS SYMBOLS ABSTRACT

o » e © o o o o o o o © o o o

O & Q & O O O O Q O Q Q Q O O G

CHAPTER 1CHAPTER 2

CHAPTER 3

CHAPTER 5

INTRODUCTIONTECHNIQUES IN THE CALCULUS OF VARIATIONS Mayer Formulation . . . . . . . . . . . .DEVELOPMENT OF THE CONSTRAINT EQUATIONS ,The Equations of Motion Kinematic Relationships O Q O O O O O O ©

CHAPTER 4 FORMULATION OF THE PROBLEMGeneral Three Dimensional Problem Solution for Negligible Induced Drag Induced Drag in the Vertical Plane Quasi-steady SolutionCOMPLETE SOLUTIONS FOR VARIOUS DRAG FUNCTIONSThe Analogue Computer Drag Proportional to Velocity Squared Density Effects . .Inverse Drag Law . .Combination of Velocity Effects

0 o o o © o o o

CHAPTER 6 REFERENCES

CONCLUSIONS .0 0 0 & 0 0 0 0 9

iv

Page

viix1471010141818212931

3333354041425760

LIST OF ILLUSTRATIONS

Figure Page3-1 General Vector G in Fixed and Rotating

Coordinate Systems ......................... 11

v> fO Three Dimensional Representation of Forces and Kinematic Relations ................... 13

3.3 Rotation of a Coordinate System About z Axis 153.4 Rotation of a Coordinate System About y^ Axis 154.1 Range as a Function of Final x and y

Coordinates ................................ 23

5.1 Drag Versus Velocity for D/m = 0.5

445.2 Velocity Versus Altitude for Various Initial

Flight Path Angles (D/m - ku2 ) ........... 485.3 Range Versus Altitude for Various Initial

Flight Path Angles (D/m = ku2 ) .......... 495.4 Velocity Versus Altitude for Various Ini­

tial Flight Path Angles (D/m = ku2expBz). . 505.5 Range Versus Altitude for Various Initial

Flight Path Angles (D/m = ku^expBz) . . . . 515.6 Velocity Versus Altitude for Various Ini­

tial Flight Path Angles (D/m = kAi) . . . . 525.7 Range Versus Altitude for Various Initial

Flight Path Angles (D/m = k/u) .......... 535.8 Velocity Versus Altitude for Various Initial

Flight Path Angles (D/m = k^u^ + kg/u^) . . 54

(_a_j2 + m100' ( — )

V

vi

Figure Page5.9 Range Versus Altitude for Various Initial

Flight Path Angles (D/m = k^u^ + k^/u^) . . . 545.10 Velocity Versus Altitude for Various Initial

Flight Path Angles (D/m = k^u^ + kg/u^) . . . 555.11 Range Versus Altitude for Various Initial

Flight Path Angles (D/m = k^u^ + ^ 2 * ' * 555.12 Velocity Versus Altitude for Various Initial

Flight Path Angles (D/m = k^u^ + kgcos^a/u^) 565.13 Range Versus Altitude for Various Initial

Flight Path Angles (D/m = k^u^ + kgCOs^a/u^) 56

SYMBOLS

constant defined on page 40 constant defined on page 21 lift coefficient drag coefficient dragfunction defined on page 4functional form defined on pages 6 and 8partial derivative of F with respect to

partial derivative of F with respect to time acceleration of gravity 32.2 ft/sec^ general vector defined on page 10 general function defined on page 7 unit vector defined on page 12 and in Figure 3 general integralunit vector defined on page 12 and in Figure 3unit vector defined on page 12 and in Figure 3drag constantliftmassangular velocity of vehicle in i direction

viiiq •angular velocity of vehicle in j direction

% variables defined on page 4r Aangular velocity of vehicle in k directiont time

Rm range as defined in Figure 4.1 ort page 23u velocity component of vehicle in i directionAUT unit tangent vectorV velocity component of vehicle in j' directionV velocity vectorw velocity component of vehicle in Ak directionX range coordinatey lateral range coordinatea altitude coordinate

flight path angle

x. Lagrange multiplierroll angle

5 constraint function as defined on page 6

V azimuth angle

V function defined on page 7u> rotation vector

SUBSCRIPTS1 initial conditions2 final conditions

denotes partial differentiation with respect to q1

A dot denotes differentiation with respect to time

ABSTRACT

The optimization of range for a glide vehicle in three dimensional space is discussed in this paper. Investigation reveals there are two degrees of freedom in this problems, hence, two optimizing equations exist, one each for the two control variables.

When drag is considered a function of lift, vel­ocity, and altitude, the problem is sufficiently compli­cated to prevent obtaining closed form solutions for the optimizing equations.

If the effect of induced drag is considered negligible, then the solution reduces to one in the verti­cal plane. In this case, the control variables are the azimuth angle y , and the flight path angle x . Specific solutions for various postulated drag functions are obtained from the analogue computer and the results are presented graphically.

The effects of induced drag on the optimizing equation are investigated in the vertical plane only. Even though three dimensional problems including induced drag do not reduce to vertical plane solutions, information on the general effects of induced drag on the control vari­able are obtained.

■x

CHAPTER 1

INTRODUCTION

It is the natural tendency for man to pursue his interests in the best way possible. Given a choice of solutions to a problem, man tends to make his decisions so that the results appear "best" to him. However l o g i c a l

the approach, this tendency exists. The optimal solution to a physical problem is often intuitively manifest; however, much of the guess work in obtaining the "best" solution can be eliminated by applying proven mathemati­cal techniques.

Elementary differential calculus provides a strong tool for obtaining extremal solutions to certain classes of problems. Namely, for a given function, y = y(x^), one seeks to find those particular values of x^, the inde­pendent variables, which will optimize y.

There exists another class of problems in which the dependent variable is a function of a set of func­tions, z - z(yj(xi ) ) • In this instance, z is referred to as a functional. That is, for each function from a set of functions y^.(x^) there exists a value for z. For optimization, one now seeks that particular function

1

2which will extremise s. Non-conservative work is an exam­ple of a functional. The work done in moving from one point to another is clearly a function of the distance traveled which in turn is a function of the path.

The ext realization of functionals is beyond the realm of elementary differential calculus. Solutions to problems involving functionals are given by such methods as the Calculus of Variations, Method of Gradients, and Dynamic Programming (Lietmann, (1) )*. The techniques of the Calculus of Variations are employed in this paper.

For another example of the difference between the two concepts, consider the motion of an-uncontrolled ballistic projectile and compare it to that of an aircraft in a vertical plane. In the former case, the trajectory is completely determined by the equations of motion within two constants, the initial velocity v^, and the flight path angle Range is a function of the initialconditions and the maximum range can be determined by application of the principles of ordinary differential calculus.

Now, consider the flight path of an aircraft in the vertical plane. By its very nature, a controllable vehicle can follow an infinite number of paths in

* Numbers in parentheses refer to REFERENCES.

traveling from one point to another physically attainable point in this plane regardless of the initial conditions. Because of the drag forces acting, the aircraft is in a non-conservative field; and, thus, there are, also, an infinite number of paths connecting two points in the velocity altitude plane. The range attained is different for each path. Each point in this plane represents a specific energy level; and, therefore, in the case of the controllable aircraft we must talk about maximum range for a specific change in this energy level.

The object of this thesis is to find that particu­lar path which will maximize range for an aircraft travel­ing between two fixed points in the velocity altitude plane. Clearly, range is a functional, and, hence, the problem is a variational one.

CHAPTER 2

TECHNIQUES IN THE CALCULUS OF VARIATIONS

One of the simplest functionals that may be consid­ered for optimization by the calculus of variations is one of the form

Bliss (2), Elsgoc (3), et al. show that the so-called Euler-Lagrange equation,

is a necessary condition along an extremal arc. This equa­tion is a second order differential equation, and, hence, its solution is a family of extremals containing two con­stants of integration. The extremum arc must either satisfy given end conditions or special conditions predi­cated by the theory if the endpoints are left free.

set of n dependent variables qi# and the independent variable, time, we then have a functional of the form

(2.1)

(2.2)

If we consider the integrand as a function of a

(2.3)

4

In this case, the necessary conditions become a set of n equations,

i-1

There is a constant associated with equation (2.4). The equation

(2.5)

is a mathematical consequence of the Euler-Lagrange equa­tions and may be used as an alternate to one of them.This may be demonstrated by performing the indicated opera­tion on equation (2.5) and reducing it to equation (2.4).In that particular instance when the integrand, f, is not an explicit function of time, equation (2.5) becomes the most useful first integral expression:

There are 2n+2 end conditions associated with the problem, two each for the n dependent variables and two for the independent variable. These end conditions may be fixed.

variables, respectively. The end point relationships

constant i=l,...n. (2.6 )

free or have some functional interrelationship such as q ^ = ^j2^k2 * w^ere anc* q^ are the jth and kth dependent

6are provided for by the transversality condition:

2(f - f*. )dt + f • dq . = 0 i=l, . . .n

1 qi qi 1 1i=l (2.7)

(Lietmann (1) et al.).Another consideration that must be introduced is

that of constraints, i.e., equations of the form

which express relationships between variables as dictated by the physics of the problem. The original functional plus the constraint equations now form a system of linearly independent equations with n - m degrees of freedom, where m must be less than n for an optimization to take place.It is important that all applicable constraints be applied.

ily accomplished by introducing a new set of variables Xj(t), referred to as Lagrange multipliers, and forming

the augmented function:

If the quantity f is replaced by F in the Euler-Lagrange equations, the first integral expression, and the trans- versality condition, the solution may be effected as before. A problem formulated in this manner is called a problem of Lagrange.

(2.8)

The solution to a problem with constraints is read-

(2.9)

7Mayer Formulation

Another and theoretically equivalent problem is the problem of Mayer. For a Mayer formulation, the quantity to be optimized is considered in the form

2 (2.10)

This problem can be put into the Lagrange form by simply considering the following equivalent form for G(q^,t):

2C2 — Gdt (2.11)

where the quantity G is related to the other dependent variables of the problem by the functional form

G = g(qi#t ). (2.12)

In Lagrangian form, the problem becomes one of optimizing

I = / Gdt V 1

subject to the constraints

1j(qi,qi,t ) = 0 j=l,...m and

G - g(qi,t) = 0 i=l,...n.

8The augmented function becomes

F = G + X (G - g(q. ,t)) + X.3.. (2.13)o i J J

The Euler-Lagrange equations are

Fq + FG " 5i<Fq + F6 ) = 0 (2.14)i i

By performing the indicated operations, equation (2.14) reduces to

= °- (2a5)qi qi

Note that G is considered as a dependent variable, and, hence, the additional constraint equation does not remove a degree of freedom.

Using the conditions for the problem of Lagrange, the necessary conditions for optimizing ig-in equation (2.10) subject to the constraint equations (2.8) are obtained by setting

F = X.$j (2.16)

and solving the Euler-Lagrange equations

F - i-(F. ) = 0. (2.17)q. at qi

9Equation (2.5) becomes

(2.18)

and, the first integral expression (2.6) reduces to

-F. q = constant. 4. i

(2.19)i

Finally, the transversality condition becomes

-F. q.dt + dG + F. dq. = 0. (2.20)1

The Mayer formulation is used exclusively in this thesis.

must be extracted from a set of m + n non-linear differen­tial equations. This set is composed of m constraint and n Euler-Lagrange equations in n dependent variables and m Lagrange multipliers, and, hence, the set can have a unique solution. However, the Euler-Lagrange equations are uncoupled from the constraint equations in X ; there­fore, the Xj 8 s may be eliminated separately. The result­ing n - m optimizing equations are then solved in conjunc­tion with the constraint equations. Except in simpler cases, this set of highly non-linear equations must be solved with the aid of a computer.

The final solution to the problem, q^ = q.(t),

CHAPTER 3

DEVELOPMENT OF THE CONSTRAINT EQUATIONS

The Equations of MotionThe aircraft model chosen for this analysis is an

aerodynamic glider considered as a particle flying over a flat earth. The motion of such an aircraft may be described using Newton's second law of motion:

The velocity vector is defined as

the magnitude of the velocity. The first time derivative becomes

vector G and its relationship with respect to a rotating and a fixed coordinate.system. The time rate of change of the vector G in the inertial system is

V = V u 1 T (3.2)

where u^ is a nonstationary unit tangent vector and Vj is

(3-3)

where the quantity u , can be evaluated by considering a

dG,dt'inertial - d )dt'rotating

10,+ cj x G. (3.4)

11where u defines the rotation of the prime system as shown in Figure 3.1. Proof of this is found in Etkin (4), Appendix A.

y Gy

E

Figure 3.1 General Vector G in Fixed and Rotating Coordinate Systems

By simply substituting u^ for G in equation (3.4) and noting that the unit vector u^ is attached to the rotating system, u , becomes

Equation (3.1) then becomes

— = V u + w x V m I T (3.5)

Let

gj = pi + qj + rk andA. A AV = ui + vj + wk

(3.6)

(3.7)

so that

V u? = ui + »j + wk.

12Then equation (3.5) may be taken as the following three scalar equations:

Fi • ,— = u + qw -rv, (3.8a)m

F i • ,—u = v + ru -pw, and (3.8b)

Fk— = w + pv -qu. (3.8c)

The center of mass of the glider is placed at the origin ofthe rotating coordinate system. The orientation of this system is shown in Figure 3*2 and is further described as follows:

1. i is in the forward direction tangent to the flight path,

2. j is always in the horizontal plane, and3. k extends downward and lies in the vertical

plane.Using this coordinate system, the following quantities vanish:

* *v = v = w = w = 0 .

The forces - lift, drag, and gravity - acting on the glider (Figure 5.2) are defined as follows:

-D t %

Figure 3.2 Three Dimensional Representation of Forces and Kinematic Relations

141. drag acts in the -i direction,2. lift acts in the j k plane only, and3. gravity acts in the i k plane only.

The equations of motion (3.8) become

- - gsinY = u, m (3.9a)

—c o s = ru, and m (3.9b)

(3.9c)m

Kinematic RelationshipsSince the frame of reference used for the equations

of motion is fixed to the glider and moves with it, the position and orientation of the glider cannot be described relative to it. The one exception to this is the angular

Amotion in the i direction. Let us introduce a space fixed system Oxyz with Oz defined downward and Ox defining the direction of the initial velocity vector. The origins are assumed to coincide at t = 0. The orientation of the glider is then obtained by a series of two consecutive rotations whose order is important. Three independent quantities are needed to completely fix a body in space; however, one quantity, the roll angle <j> , has already been incorporated into the equations of motion.

Consider the initial orientation of the aircraft as Cxyz where these axes are parallel to the inertial set. The following rotations are then applied:

1. A rotation y about the body z axis to the axes^X1^1Z1 * k0dy x axis is at its finalazimuth. (Figure 3.3)

2. A rotation y about the body y^ axis to theCx y z . The body y axis is at its final2 2 2 1elevation. (Figure 3.4)

Figure 3.3 Rotation of a Figure 3*4 Rotation of aCoordinate System CoordinateAbout z Axis System About

y Axis

The transformation for the first rotation is given by

16and the transformation for the second rotation is given by

If the matrix equations (3.10) and (3.11) are combined to

vector quantities considered are the velocity components in the respective systems, then we arrive at the result

This set of equations establishes the kinematic relation­ships. T^e reader is referred to Goldstein (5), Chapter 4, for a complete derivation and discussion of these relationships.

Writing equations (3.13) in scalar form and remem­bering that v = w = 0, we get

sin*

eliminate the intermediate vector (x ,y , z ), and if theV ' V 1

cosvf/coss -siny cosysin*

sinycosx cosy sinysinx

x = ucosxcosy.y = ucosxsiny, andz = usinx.

(3.14a)(3.14b)(3.14c)

17Returning to the equations of motion (3.9), we are

left with expressing the angular velocities q and r in terms of ^ and x . The appropriate transformation, which is completely derived in Goldstein (5), Chapter 4, is given by

cos* 0

0 1

sins 0

-sin if

0

cos*

0

(3.15)

In scalar form (3.15) becomes

p = -y/sinv,

q = y , and

r = 4>cos*.

(3•16a)

(3.16b)

(3.16c)

Substituting these results into equation (3.9), we arrive at the final form of our equations of motion, i.e..

D •- — - gsiny = u, m—sincf) = yucosa, andmgeosy - iicos^ = -yu. m

(3.17a)

(3.17b)

(3.17c)

Equations (3.14) and (3.17) represent the physical constraints placed on this problem.

CHAPTER 4

FORMULATION OF THE PROBLEM

General Three Dimensional ProblemThe problem is to determine the particular flight

path a glide vehicle should follow to maximize range sub­ject to certain specified end conditions. Since the prob­lem is analyzed using the Mayer formulation, we will use equations (2.10) and (2.16) through (2.20) and note that the quantity to be optimized, = G(q^,t), only appears in the transversality condition. Therefore, the Euler- Lagrange equations (2.17), the necessary conditions for optimizing the function-y"* may be set down irrespective of the specific functional form of-y. A generaly function which may be specialized to range will be considered here. Mathematically, the problem may be stated as follows: Determine, among that class of functions q^ = q^(t) which satisfy the constraint equations, that special set which extremizes

y = G(x,y,z,u) (4.1)

subject to the constraints$]_ = x - ucosscosiy = 0, (4.2a)

18

3 2 = y - ucosYsin y = 0,

I3 = z + usins = 0,

^4 858 m + Ssina + u = 0,

= sin(|> - vj/ucosy = 0, and

$5 = gcosy - cos<j? + yu = 0.

19(4.2b)

(4.2c)

(4.2d)

(4* 2e)

(4.2f)

There are eight dependent variables included in these equa­tions, i.e.,

x,y,z,u,y , , L•

Drag is in general a functional of the form

D = D(L,u,z).

Since there are six constraint equations, two degrees of freedom exist. The resulting solution will, therefore, produce two optimizing conditions for the control variables L and <J>.Substituting the constraint equations into equation (2.16), F = Xjlj, we obtain

F = X^(x - ucosycos^) + /^(y - ucosysiny) (4.3 )

+ X^tz + usiny) + + gsiny + u)

+ X^t^sin^ - yucosy) + A^(gcosy - —cost)) + %u).

20The eight necessary conditions (2.17) for optimization are

Fq. ' dt(Fqi ) = ° 1=1'* * *81where

^ " x> q2 = y, q3 = 2,

q4 = U, q5 = f , q6 = y .

q7 = 41 , and qQ = L.

The indicated operations are performed to obtain

x: - = °» (4.4a)

y : - --( X2 ) = 0, (4.4b)

- 3t(V ~ °' U -kc)

u : - X^cosifcosy - Agcosysiny + A^sinx (4«4d)

+ \ - x5 V cos* + V " = °»

ty: A^ucosysinv - Agucosfcosy + — •(a ucos ) = 0, (4«4e)

r: x^usinxcosy + Agusinasiny + A^ucosy + X^gcosx (4.4f )

+ A^uysintf - X^gsinx - A^u) = 0,

21■#>: X5Ecos<fl + x 6Esin* = 0> and (4.4g)

L: + AjSin^/m - A^cos^/m = 0. (4.4h)

The first integral equation (2.19) becomes

X^x + XgY + + A^u - X^ucos<f^ + X^ux = -C. (4.5)

Consider the form of equations (4.4). It may be seen that there are three equations containing derivatives of the Lagrange multipliers even if the first integral is utilized. Thus, the multipliers cannot be readily deter­mined.

Solution for Negligible Induced DragIn an effort to simplify the solution, let us

assume that the drag is not a function of lift, that is, induced drag is assumed negligible. One immediate result of this assumption is that equations (4.2e) and (4«2f) become uncoupled from the remaining equations (4.2). These two equations no longer act as constraints on the problem, but, rather, they become a definition of lift and roll angle. It might be noted, however, that, if these equations are included in the analysis with the assumption that D = D(u,z) only, they would have no effect oil the solution as their associated Lagrange multipliers would be found

22to be equal to zero ( = 0). This can be readily shown by solving equation (4.4g) and the reduced form of equation (4.4h) for Xc and X^.

There are still two degrees of freedom associatedwith the problem; however, the control variables are now yand y. The optimizing condition for the control variable y may be immediately determined. From equations (4.4a) and (4.4b) we obtain the results

\ 1 = constant, and

Xp = constant.

Equation (4»4e) can be put in the form

x^/Xg = cot f ; (4.6 )

and, therefore,

y = constant. (4*7)

Thus, motion of the glider is limited to a given vertical plane.

In order to determine the optimizing condition for Y, the transversality condition must be investigated and the quantity to be optimized specified. Let us choose to maximize the range, R^, in the vertical plane where is a function of the final values of x and y as shown in Figure 4.1.

23

x

y

Figure 4*1 Range as a Function ofFinal x and y Coordinates

If we set ~ Yi ^ °» then equation (4.1) becomes2 2 iY = G = (x2 + y2 ) (4.8)

The transversality condition (2.20) becomes

-Cdt + cosydx + sinydy + X- dx + Agdy + A^dz + A^du = 0 .

(4.9)The initial conditions are all fixed so that

dtj, = dx^ = dy^ = dz^ = dz1 = 0;

and, therefore, the initial condition terms in equation(4.9) are all zero. All the final conditions are indepen­dent, and, therefore, to satisfy equation (4.9), we may conclude that

-Cdtg “ 0, (4.10a)

( + cos^)dx |2 “ 0,24

(4.10b)

(X2 + sin^Jdy |2 = 0, (4.10c)

X^dz I2 = 0, and (4.10d)

X^du Ig = 0. (4.10e)

Because X^ and X2 are related by equation (4.6 ), equation (4.10c) may be written in an alternate form

(X^ + cosy)dy I2 = 0. (4.10f)

For the present, let us consider only equation (4.10a); the remaining conditions (4.10) will be discussed after the Lagrange multipliers and the optimizing equation for Y have been determined. If the final time is left free, dt2 ^ 0, then

C = 0. (4.11)

The kinematic relations, equations (4.2a,b,c), may be substituted into the first integral equation (4.5) to obtain

Xj(ucostcosy) + X2(ucostfsin4/) - X^fusins) + X^u = 0. (4.12)

If equation (4.6) is used to eliminate Xg from equations (4.12) and (4.4f), they become

25X^ucos* - X^usinrfcos^ + ^^ucosy = 0, and (4.13)

X^usintf + A^ucos^cosy + X^gcosycosy = 0. (4.14)

Solving this pair of equations for X^ and X^ and noting that from (4.2d)

- D/m = u + gsin*,

we obtain

x3 = " X1 cos^ (tan* + eoftlT/m> (4-15)

^4 ^1 cos*cosyD/m (4.16)

Finally, if we use equation (4.12) to eliminate Xi, X2, and X^ from equation (4*4d), it becomes

uiP/g + u - u = 0 (4.17)au \

X^ may be calculated from equation (4.16) and isv •

X, = -A(u + utany i + u l^Ell) (4.18)^ u D /mwhere

/mau dz

Thus, equation (4.17) becomes

26

u + utan«s + 2S82X. (u + z = 0.6u u m du dz (4.19)

Substituting for u and z from equations (4.2c) and (4.2d), and solving for * , we obtain the optimizing condition

tions of the transversality conditions (4.9) for various end conditions. Because we are optimizing range, Xg and or yg must necessarily be left free. Noting that the coefficients are the same in both equations (4.10b) and (4.10f) and remembering that y = constant, we obtain the result

For the case where either dxg or dyg is fixed, the specific value of y cannot be determined until the solution to the problem is obtained. The problem is a two dimensional one, that is, maximum range is not a function of the azimuth angley, and, therefore, it can be simplified by choosing either the x-z or the y-z plane to compute the solution. Having determined the Maximum range, f^, the angle y and the free coordinate x% or yg , may be calculated from the relations (Figure 4.1)

cos 4, jD/m . JD/m," , (gT- + U - ) . (4.20)

We are now in a position to discuss the implica-

X - cosy . (4.21)

For the case where both and are free, equations (4.22)and (4.23) have an infinite number of solutions, that is,yremains unspecified.

Next, consider equation (4.10d)

x3dz I2 = 0

If Zg is left free, dzg^ 0; and, thus, = 0. This con­dition can be evaluated by applying the result of equation (4.21) to equation (4.15) to obtain

- a : g f o * *■

The condition X? = 0 can be met only if->2sin's D/m = -g. (4.24)

Before discussing this result, let us consider the final transversality condition (4*10e). For U2 free, du2 ^ 0, and thus X. =0. Examining (4.16)

28we see that the condition X , = 0 can only be satisfied if

2u2 - 0. (4.25)

If drag is a function of velocity such that it will have the value zero for zero velocity, then equations (4.24) and (4.25) cannot be simultaneously satisfied. That is, we cannot leave both Ug and Zg free and meet the transver- sality condition.

For the case where z2 is fixed and u2 is free, the glider would attain maximum possible range by expending all its kinetic energy. However, several difficulties imme­diately arise if the final velocity is left free. One is that an aircraft has some realistic stalling speed and another is that for certain drag functions the aircraft can never slow to zero velocity along the optimum path, as there are some optimized flight regimes (Chapter 5) where i remains negative and the aircraft simply goes into a terminal dive. Furthermore, if we fix the final velocity, we would still be at liberty to choose u% = 0 as a specific end condition; and, thus, we could determine the correspon­ding maximum range in those instances when the solution permits the velocity to go to zero. For this reason, Ugwas considered fixed.

For the case where Z2 is left free, equation (4.24) must be satisfied.

Complete solutions to the two dimensional problem are found in Chapter 5• In all instances, both Ug and Zg are considered fixed.

Induced Drag in the Vertical PlaneAn investigation was made to determine the effects

induced drag would have on the optimizing equation. Note that Y appears in equation (4.4d), thus, the three dimen­sional induced drag problem does not reduce to a vertical plane solution. However, for simplification, the problem was limited to the x-z plane with the additional assumption that altitude effects were negligible.

The problem restated becomes one of finding among that class of functions qi(t) satisfying the constraints, that particular set which extremizes

Y = x2 -x1

subject to the constraint equations (4«2a,c,d,f), with V = = 0.

The optimizing conditions are given by equations (4.4a,c,d,e,h) with the one alternative equation, (4.5).The transversality condition becomes

2-Cdt + dx + A^dx + Ajdz +A^du

so that with z and u fixed and t and x unspecified,

30C = 0, and

X — —1.

Substituting the first integral expression (4.12) and equa­tion (4•4h) into equation (4-4d) yields

u + u + 2 u s' ^2 - u = 0. (4.26)au dL X^

Equations (4.12), (4.4f), and (4•4h) may be solved to obtain

= u ^cosy(du A2 + u) + sina(gcos* - gsins (4,27)

- i_(u — )) - ^0/^ siny(u 0!m + u + 2 u* AS) ~] dt dL Tu du TO J

The optimizing equation is then given by equation (4.26) where X^ is known and X^ can be calculated from equation (4.27). It is apparent that the complexity of the optimi­zing equations makes the solution to the problem quite difficult to attain. In this case, the optimizing equation is well beyond the capabilities of the computer utilized. (See Chapter 5). The main reason for the increased diffi­culty is that the Lagrange multipliers appear as deriva­tives.

In order to simplify the optimizing equation, an assumption was made that the normal acceleration in equa­tion (4.2f) is small. As a result of this assumption, equation (4.4f) reduces to an algebraic equation, and.

31thus, the Lagrange multipliers are more easily eliminated. The optimizing equation is of reduced complexity and is of the form

However, even this equation is unmanageable and cannot be programmed on the analogue computer.

cannot be found until the drag function is specified. Itis the purpose of the following chapter to give a detaileddiscussion of the results for various drag functions.Further discussion of equation (4.28) will be delayed until that chapter.

Quasi-steady SolutionThe standard approach to the glide problem in

flight mechanics text books (Miele (6), et al.) is toassume quasi-steady flight and a shallow glide angle. Under these conditions equations (4*2) simplify to

The final time program for the dependent variables

x - u = 0 (4.29a)

z + u* = 0 (4.29b)

jj + < g ” 0, and (4.29c)

32

(4.29d)

Miele (6) shows the condition for the flattest glide, i.e., maximum range, is

This solution is verified by using the calculus of varia­tions procedures. Interestingly enough, the result is obtained by applying only the quasi-steady flight condition, that is, no small angle assumption is needed. In the final analysis, we see that this is a very special case where the final velocity is approximately equal to the initial velocity. Work done by the drag force comes almost entire­ly from potential energy. Further discussion of this con­dition is found in the next chapter.

aD/m _ 3u (4.30)

CHAPTER 5

COMPLETE SOLUTIONS FOR VARIOUS DRAG FUNCTIONS

The Analogue ComputerThe high degree of non-linearity of the differen­

tial equations to be solved dictated the necessity to seek computer solutions. The computer utilized was a Donner model 3400 analogue computer plus ancillary equipment„This computer is available within the Aerospace and Mech­anical Engineering Department of The University of Arizona.

The analogue computer is able to perform the oper­ations outlined below.

1. changing of sign of a function2. addition and subtraction of functions3. four quadrant multiplication of functions4. two quadrant division of functions5. integration of functions

Differential equations are programmed into the computer along with initial conditions for each integrat­ing network. The machine computes the problem continu­ously and, unless otherwise scaled, solves the problem in real time.

33

34In solving the optimizing equation, we note that

the initial flight path angle,cannot be arbitrarily speci­fied. This is because the flight path angle X is a control variable. It does not appear as a time derivative in any of the constraint equations, and, thus, is not introduced into the transversality condition. The two dimensional problem has one degree of freedom which allows the 'k pro­gram to be provided for by the optimizing equation. The solution to this equation, X == x(t) + X f i x e s the flight path the aircraft must follow to obtain maximum range. The.complete optimal solution to the problem is obtained within a constant,X

Since the solution to the problem of optimum range is obtained on arriving at a final altitude velocity com­bination, it is apparent that the initial flight path angle must be varied until the solution passes through the desired end conditions^ The initial condition depends on the final conditions. One method of finding the opti­mal trajectory is to generate a family of extremal curves in the velocity altitude plane and determine the result by interpolation.

Data are obtained directly from the computer with the aid of an x-y recorder. Any two dependent variables in the problem may be plotted. The ones of major interest

are velocity versus altitude and range versus altitude. It is also possible to plot any of the dependent variables as functions of time.

ment, it was necessary to rerun the problem several times to obtain different families of curves. In most instances this could be done with a high assurance that the solu­tions were the same. In some instances$ however, y could change sign, and, thus, the solution became less accurate in the region near i = 0 . The reason for this was that the solution depended on the difference of two large num­bers going to zero, and, thus, small errors introduced because of amplifier drift become magnified. It was difficult to reproduce the results in this region.

Drag Proportional to Velocity SquaredFor the first solution to the two dimensional prob­

lem drag was assumed to be of the form

This simple drag function is based on the assumption that the drag coefficient, CD, and the density, f, remain con­stant throughout the flight regime. For this condition the optimizing equation (4.10) reduces to

Because of the lack of additional recording equip-

(5.1)

36Note that x is not explicitly a function of the drag con­stant k= The set of equations which were programmed into the computer reduce to

B. + gsiny + u = 0S

x - ucos# - 0,

z 4- usin& = 0,

i f = o, and

D % t u >2m * 1100) - 0e

The following initial conditions were imposed:

u^ = 500 feet per second,

x^ - 0, and

= 0-

The drag constant was chosen at a nominal value of 0.5.On the basis of a maximum velocity of 1000 feet per second for the aircraft, the maximum value of D/m is 50 pounds per slug. By considering a 1^100 pound aircraft flying at 500feet per second, we see that the drag is equal to 6250pounds.

(5.3a)

(5.3b)

(5.3c)

(5.3d)

37For the special case where tij = + 90°, the prob­

lem may be solved in closed form and the set of equations (5.3) reduce to

k(i m |2 + 8 " u S ' (5'4)

Separating the variables, integrating and evaluating theconstant of integration yields the equation

uM ^ ) 2 + g - k(Y~)2 + g exp 2kz°10~^, (5.5)

If we apply the previously assigned values of k = 0.5 and Uj = 500 feet per second, and solve this equation for maxi­mum altitude (u - 0), we find that z - - 3340 feet. The velocity that the glider attains as it passes through z=0 was calculated by re-solving equations (5«3) with the condition - - 90°, U]_ = 0, and zg - 3340 feet. This velocity is 430 feet per second. Finally, the terminal velocity was computed by considering u = 0 and solving equation (5.3a)

k(™B_)2 = g. (5.6)

The terminal velocity for this problem is 804 feet per second. The above result and computer solutions to equa­tions (5.3) for y, other than ± 90® are given in Figures 5.2 and 5.3.

Explanation of Figure 5®2 The area bounded by the curves generated for ^ ~

+ 9 0 ° represents an envelope of points which are reached with a finite range. Note that the upper curve should be extended to the point u = 0, z = -3340 feet. The left boundary of the envelope can also be considered a velocity versus altitude plot for the conditions tf-j. = -90® and u-j_ = 0. All points to the right of and above this bounded area are at a higher energy level and thus are excluded.All points to the left and below the bounded area may be reached and have infinite range. Since the drag is pro­portional to the velocity squared, infinite range may be attained by flying straight and level until the velocity decreases to zero. (Certainly we may talk about infinite range from an academic standpoint only.) However, the fixed end conditions must be met, and, therefore, a scheme must be found to allow solutions to pass through a speci­fied point in the velocity altitude plane. This is done by first expending all the kinetic energy to arrive at infinity and then utilizing the aircrafts* potential energy to arrive at the final velocity altitude condition.We can determine from the left boundary the amount of poten­tial energy, i.e., the change in altitude, needed to arrive at the correct final velocity. By comparing this change

39in altitude with the altitude differential specified by the end conditions, we can determine the value of z at which the aircraft must travel to infinity.

Consider the example where we seek a final value of z - -1000 feet and u = 200 feet per second. From the left boundary curve of Figure 5*2, a velocity of 200 feet per second corresponds to a change in altitude of 500 feet (3340 feet - 2840 feet). Thus, the flight program would consist of climbing to the point z = -1500 feet, flying straight and level to zero velocity and, finally, diving straight down through the predetermined point.

All the curves in Figure 5.2 converge to the terminal velocity at z equal infinity, and, therefore, an upper bound is placed on the velocity attainable. To determine the final range for a point located inside the envelope, first determine the initial flight path angle of the optimum curve associated with that particular point (Figure 5.1) and then find the range corresponding to the specified initial flight path angle from Figure 5.2. Choosing values of 520 feet per second and 2000 feet for the final velocity and change in altitude respectively, we determine that *-j_ = 0®; and, therefore, from Figure 5.2 we see that the maximum range is 3350 feet.

A comparison was made with a non-optimum * program, specifically, y = 0, and it was found that the range

attained was 3300 feet, or less than the range obtained in the optimized case.

There is one other region of interest. This is defined as that area between the left boundary in Figure 5.1 and the extension of the extremal curves after they cross the boundary. These extended curves, although not shown, also converge to the same terminal velocity after passing through their respective maximum altitudes. Thus, there is an apparent maximum range for the points located in this region. However, these solutions are just sta­tionary solutions since in this region infinite range can be obtained by more than one path.

Density EffectsA more realistic drag function can be postulated

by introducing an exponential term to account for the vari­ation in density with altitude. This drag function is

” = k(Ig^)2expBz. (5.7)

A reference density is included in the constant k and corresponds to the actual initial altitude. Recall that the initial altitude is designated as z = 0 and that z is defined positive downward. A value of B = 0.308xl0~^feet ^ gives a very good approximation to the NAGA standard atmos­phere from sea level to fifty thousand feet. The problem

41is similar to the first case discussed with the one excep­tion that the optimizing equation (4.20) now becomes

2g + 0.308(-H_)2 _u 100 (5.8)

The solutions to this problem are presented in Figures 5.4 and 5.5. When these curves are compared with the constant density curves» one sees that the range is decreased for the same initial flight path angles due to the increased magnitude of y . However, in order to pass through a given final point, for example, Ug = 520 feet per second and zg ~ 2000 feet, we must have an initial flight path angle of 30®. From Figure 5.5 the optimal range is found to be 4150 feet which is an increase of 500 feet over the constant density range. In effect, the optimal solution takes advantage of the less dense higher atmosphere to minimize energy loss along the flight path.

Inverse Drag LawThe next drag function that was investigated was

of the form& = (100)2 - c (5.9)m u

For k chosen such that 0.1< k<1.0, D/m is of the same order of magnitude as in the first problem discussed. The opti­mizing equation becomes

42

(5.10)

Again, we are able to solve the problem in closed form for the case ^ + 90®, Uj - 500 feet per second, and 3 ^ - 0with the results shown in Figure 5.6. The computer results are shown in Figures 5.6 and 5.7. Points lying to the right of the ± 90® solutions, again, represent higher energy points and are physically unattainable. All other points in the velocity altitude plane have a unique extremal curve passing through them and a finite range associated with them. The quantity v remains greater than zero for all solutions. The curvature of the range altitude curves is reversed from the previous cases. Thus, the flight paths are such that they take advantage of the increased velocities at lower altitudes to minimize the energy loss.

Combination of Velocity Effects

where CDq is the "zero lift" drag coefficient and is

If we consider the coefficient of drag to be givenby

(5.11)

the induced drag coefficient, and if we further consider

(5.12a)

CT = k, iZs, and L 5 «2u(5.12b)

then (5.12c) can be written in the form

2 i nnv T,2(5.13)

This relationship can be simplified by assuming that lift is a constant. Drag is then a function of velocity only and equation (4.20) is the applicable optimizing equation.

If and are considered in the range 0. 1<k^< 1.0,the drag will be of the same order of magnitude as in the previously discussed cases. For ease of computation, the values k^ = kg = 0.5 were chosen. Figure 5.1 is a plot of drag versus velocity under these assumed conditions. Minimum drag occurs at a velocity of 316 feet per second. The optimizing equation (4.20), now, takes on the form

100k(5.14)

Note that y may be either positive or negative, and it takes on a value of zero at the critical velocity corre­sponding to a minimum value of D/m.

44

fH

o a.o

o §00 1000ovelocity - feet per second

Figure 5°1 Drag Versus Velocity for D/m =0.5■ * A " ,

100

The maximum altitude gain computed from the closed form solution for # " 90* and u- = 500 feet per second is z = - 2750 feet. The terminal velocity is 793 feet per second. Comparing this with 804 feet per second from the first case discussed, we see what little effect the inverse velocity term has on the terminal velocity.

Computer solutions to this problem are given graphically in Figures 5.8 and 5.9 for an initial velocity of 500 feet per second and in Figures 5=10 and 5=11 for an initial velocity of 200 feet per second. The solution

becomes quite sensitive as it passes through the critical velocity. When the velocity is below the critical velocity, the second term in equation (5.14) is the dominant factor on the drag and y is positive. For a velocity above the critical value, the first term in equation (5.14) is dom­inant and X is negative.

When V goes through zero, equation (4.20) goesthrough the solution = 0 which is the optimizing con-dudition for the quasi-steady solution as given by equation (4=30). Thus, the solution passes through the quasi­steady solution. If D/m = -gsinx at the same instant y becomes zero, then both u and y will remain zero and the solution will remain quasi-steady. Note that the extremal curve does not become linear until the critical velocity is reached. If the initial velocity is equal to the cri­tical velocity, then one curve in the family of extremal curves would correspond to the quasi-steady solution. If the initial velocity is other than the critical velocity, there are no quasi-steady solutions starting from the initial condition. Equation (5.14) has introduced the effect of induced drag into the problem. Let us consider the validity of the constant lift assumption. For that portion of the trajectory when X and y are small, lift is approximately equal to weight and, thus, the assumption is satisfied. Secondly, for those curves where y is negative

46

and the velocity is much higher than the critical value, the induced drag term has a small effect in general and, thus, the error introduced by considering lift a constant is small. For those solutions which lie below the critical velocity, X is positive and the induced drag is the predom­inant term. Note from Figures 5-9 and 5.11, however, that s remains small until the final portion of the curves.In this region, the approximation is invalid.

As a further refinement to equation (5 <>14)# we can remove the small angle assumption by considering

L/m = gcosx. (5=15)

Equation (5*13) becomes

o 100kocos*>D/m = k-, ( ~ ) + ----— — (5.16)

100 r u v2l100J

This is precisely the case discussed on page 31 where (4.28) is the optimizing equation. Unfortunately, this equation is very sensitive in the neighborhood of the cri­tical velocity and cannot be programmed in this region on the computer. However, in the region where the velocity is always greater than the critical velocity, the effects of induced drag become appreciably smaller and the optimizing equation is given by equation (5.2). Furthermore, if ki = 0.5 and k^ = 0.1,.the critical velocity is 150 feet

47per second. Equations (5.2) and (5.16) were programmed on the computer with these values of k^. The results are given in Figures (5.13) and (5.14) and are quite similar to the first case discussed.

Further investigation with the aid of a digital computer is warranted not only for this case but also for the complete three dimensional problem on page 20.

jjooGl

±&z !T rT e l6 c It' CTerSuSLA: .tltM e d S E C ? * #

1D&Q

5.3 Range'T-ti-stig^Aititiiae: for Ta’tons: Intt:atLthL'hi

t i± _: 1 L l^ lIL i-L lii LLx n x ^ g - .^xUuihix b r :-t :

aoflo

r luQG

IflCO

ilfllic

aoflfi

T1

. i ?a(►city-feet per eeconc

-— —L-*—t- j--1ragars-JjrH Fit Acglee

n:i

•UDOQ exp; j a

r- iQgO5 0 0 £pa -

IflQO

aooQ

t

-x rrangedefor^V

le$nirtt.;Range InitialFigure:5.

' ' h m w r-1: feet::

JSftQ

JAee

titude ;rat. esTart.Qu^velocity

InitialFigure 5

rrrrni'"tlij-i

rtrm-t

:kj=:vi5LL.<ic? :; * 1 7 1 T ‘ i{ ■ « ? * ! H -4- t

Hi I! Li-zsi

44-+ ! | H-+-H-. i * . t -1" ’ t ■

' | ;"+ t *

-i’itHji-l! Hr!

U-14 : I |-x: rang for; VM M :

H | ' n Figure: 5:♦ ♦ *t : t 4 I - | 4 " i H ; t* : ; "t r -| I

;U1; Fllghrsus: AltiT . i I l [

1 1 ■ 1 : i i

ariouBi I M! 4 Rankfe

. r 1 , ♦ .: i U'i-: •;♦ 1

— rtr1 +1 t

t r t ■ 1 1 n ~ !Ll±;r

m m- u . 4 ; I i-T-

1:%:%H-

P:1

u'H

Ii-

E

- - * . «. -4-♦ ee • "I / XV*V .—*-. ♦. . . • * *• - - $-* —-4-- •--- ♦ — • -9r- -«- - p • e... - ♦---.- * —1 ■ *- i —+ — ■*" * • - -» —» » - y -IX'

;>00:fp&feet

tlio.....- . . 3po-. . , .>.«* . . velocity -. feet* par" second

?imire: 5.10T- Velocity- Versus: Altitude^foi»i Various: ^ x: :x:;:{:: initial; Flight: Path: Angles:..:.::;:: ;x:

■CO

Initial

7d:so:- i —

Ttl"

y. -17 oaooo

41 >Q

.-T-v ol'Ooity - rf-©©t - -jS Qfooiwi* + •,2 * Velocxt t Versus A l "t Various Initial F

Ititmde fo r-; 'light feth - -

ook^eas24; + 7o^ 4 133 2»- -- ■ ■ ■ i—•- I/' "U'v *■ *

" 100 ~ .

:Q.:i ry:r- -i

- f t

■lltOCX—i <l£M.

E m s "Alt

CHAPTER 6

CONCLUSIONS

The object- of this thesis was to determine paths a powerless aircraft should follow to optimize range under the influence of various postulated drag functions» No attempt was made to choose characteristics of a particular aircraft, but, rather, the initial conditions and constants were selected such that the solutions would be represen­tative of aircraft in general. The intention was to per­form a qualitative analysis and the objective was obtained with the aid of a computer. An analogue computer was found to be especially useful because of its flexibility and ease of operation. A further advantage is acquired through the visual observation of the results.

In many instances, the solution to the variational problem is quite easily attainable provided one is careful to fully investigate the implications of the transversal- ity condition. The' results from this expression are all germane to the problem.

57

The following conclusions are made for the various drag functions considered:

lo The solution to the three dimensional problem is not easily attainable if drag is considered a function of lift and velocity,

2, The three dimensional problem reduces to a solution in the vertical plane if drag is considered a function of velocity only,

3, For D/m = ku there is a limited region in which the maximum range is finite. The majority of the points in the velocity alti­tude plane are attainable by an infinite number of paths and have infinite range.

4, If the effect of density variation is consid­ered, the initial flight path angle and maximum range are increased over a compara­tive constant density situation,

5° For D/m = k/u, all the physically attainable points in the velocity altitude plane have finite maximum range solutions. As Y is positive, the final point is reached from below.

6. A finite maximum range solution is attainable for U2 = 0, if the drag contains an induced drag term.

7. The solution to the induced drag problem is similar to the solution for drag propor­tional to velocity squared for the region where the velocity is greater than the cri­tical velocity.

Certainly, there are many areas of future study and investigation left to pursue, a most interesting endeavor being the complete solution to the induced drag problem. Actual evaluation of a particular aircraft could be made and possibly verified through physical test ing.

Beyond that, optimization of a manned re-entry vehicle with aerodynamic characteristics could be inves­tigated. Certainly, in a problem of this nature many physical constraints must be introduced. Typical con­straints to be considered include maximum allowable heat load of the vehicle and maximum allowable acceleration load of a human. Other considerations for this problem would include the shape of the earth and the density structure of the atmosphere. One might sum up the possi­bilities of the future by saying, “the sky is the limit.“

REFERENCES

1. Leitmann. G., "Optimization TechniquesAcademicPress, 1962. ‘

2. Bliss, G. A., "Lectures on the Calculus of Varia­tions," University oT~Chicago Press,~T946.

3. Elsgolc, L. E., "Calculus of Variations, Addison-Wesley, 1961.” "" — —

4. Etkin, B., "Dynamics of Flight," John Wiley and Sons,Inc., 19591

5. . Goldstein, H., "Classical Mechanics," Addison-Wesley,1950.

6. Miele, A., "Flight Mechanics-I. Theory ofPaths," Addison-Wesley, 19o2.

60