Optimum Rocket Trajectories.. by N. Coburn Approved by G. E. Hay, Supervisor; Mathematics Group, ~ '..,. Project Wizard-MX-794 (USAF Contract No. W33-038-ac-14222) External Memorandum UMM-48, May 1, 1950

br AERONAUTICAL RESESARCH CENTER- UNWIERS1ITY OF MICHIGAN U vuMM-48 This report was prepared with the active participation of the following members of the Basic Xa-the'matics. Group: R. E. Phinney D. R, Roberts F. S. Spencer....... Page i i

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-48 TABLE OF CONTENTS Section Page I Introduction 1 TI Summary 3 III Notation 5 IV The General Equations of Motion and the Variational Equations 7 V Zero Lift and Drag 14 VI Zero Lift and Linear Drag 24 VII Zero Drag and Linear Lift 31 VIII Other Variational Problems of the Same Type 43 Appendix A: The Problem of Zero Drag and Lift in Unmodified Terms; Burning and Coasting 46 Page iii

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMNM-4 8 Page iv

A ERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-4 8 INTRODUCTION For the purpose of this paper, the flight path of a rocket will be divided into three sections: (a) launching, possibly vertically; (b) the flight path after the launching stage; (c) the homing phase. The present work is concerned only with the second portion. It is assumed that the craft can be considered as a particle of variable mass, and that the flight path lies in a plane. Our problem can be stated as follows. At the end of the launching stage the craft possesses a velocity vector —that is, a speed and a direction of motion can be assigned to the craft. These data furnish the initial conditions of the problem. The forces that will be considered as acting on the craft are: (a) thrust due to the burning of fuel; (b) lift; (c) resistance; (d) gravity. In order to furnish an analytic treatment of the problem, we shall here assume that lift and drag forces can be approximated by expressions that depend on characteristic constants and the first power of the velocity. Thrust will be considered constant in magnitude —that is, the mass of the craft is assumed to decrease linearly with time because of the uniform rate of burning. However, the direction of thrust is to be varied continuously if necessary, so as to obtain the trajectory that is optimum in the sense defined below. WVe shall now define the optimum trajectory. First, we note that two types of motion will be considered: either the craft burns fuel throughout the motion or it burns fuel Page 1

AE R ONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-48 for a specified time and then coasts with fuel jets turned off. We shall fix the horizontal distance attained by the craft and require that the vertical distance at this fixed horizontal distance be stationary, or, alternatively, we shall fix the vertical distance attained and require that the horizontal distance at this fixed vertical distance be stationary. If a trajectory satisfies this requirement, it shall be called an optimum trajectory. The above approach is to be considered as a first attack on the problem of optimum trajectories. The essential idealization is the use of linear drag and lift. One could use quadratic drag or some more complicated drag function as well as a non-linear lift function. The introduction of such functions would lead to problems of the Bolza type in the Calculus of Variations. Because of the non-linear character of the resulting system of differential equations, very little insight into the theory of optimum trajectories could be obtained. It is for this repson that we have here attacked the simplified problem. Further, if the optimum trajectories for a realistic drag function do not depart too widely from the optimum trajectories for linear drag, then our results will prove valuable. With this in mind, we have developed the theory of optimum trajectories in two directions: (1) extensions of the idealized theory to various other criteria for optimum trajectories, such as minimum boost velocity, or maximum terminal velocity; (2) attempts to study the departure of the idealized linear theory from the actual non-linear theory, by perturbations or by stability studies. These will be reported in later papers. Page 2

AERONAUTIGAL RE S EARCH CENTER - UNIVER SITY OF MICHIGAN UMM-48 II SUMMARY The general equations of motion are stated in Section IV, and general formulas for the solution of these equations, to be verified in succeeding sections, are furnished. With the aid of these formulas, the variational equations for the optimum trajectory are determined. In Section V, the case of flight with neither lift nor drag is examined, and solutions of the differential equations of motion obtained. By use of the variational equations, it is shown that for either of the two types of motion mentioned previously —that is, continuous consumption of fuel or consumption of fuel for a specified time followed by a coasting phase —the direction of thrust is fixed in the plane, of motion for an optimum trajectory. The relation of thrust direction to burning time for the first type of motion and the relation of thrust to total flight time for.various burning times in the second type are shown by figures in the text. Section VI considers the case of flight with linear resistance and no lift. Again, it is seen that the direction of thrust is fixed in the plane of motion for an optimum trajectory, and graphs similar to those of the preceding section are given. For the drag coefficients examined, the effect of drag on the desired thrust direction is negligible. The case of flight with linear lift and no resistance is treated in Section VII. For an optimum trajectory the thrust direction is no longer fixed in the plane of motion. Page 3

AAE RONAUTIGAL RESEARCH CENTER - UNLVERSITY OF MICHIGAN -UMM-48 In fact, the thrust direction at various points of the trajectory is obtained by solving a particular equation. Finally, in Section VIII another type of optimum trajectory is discussed. We seek to determine the thrust schedule so that in a fixed burning time and coasting time the horizontal or vertical distance attained by the craft is a maximum, and in maximizing this horizontal or vertical distance we place no restriction on the vertical or horizontal distance, respectively. The thrust schedule for such optimum trajectories is immediately apparent when a linear drpg law and no lift are considered. However, when a linear lift law and no drag are assumed, the thrust schedule is not obvious. The introduction of modified variables is found to simplify the approach to the several phases of the problem, and to permit easier handling of the relations involved. However,'one may not be inclined to follow each modification through, and as a result some difficulty may be encountered. To avoid this possibility, the analysis of one case —that of zero drag, zero lift, with coasting following. the burning stage —is carried through in the original variables in Appendix A. Page 4

AERONA2UTICAL RESEIARCH CENTER - UNLVERSITY OF MICHIGAN UMM-48 TIII NOTA TION c the horizontal component P cos of the modified craft thrust. the acceleration of gravity: 32 ft/sec2. g a modified gravitation coefficient g-/2, a constant. n a modified time variable, 1 - rt. nl the value of n at time t = ti. n2 the value of n at time t = t2. q an independent variable used in solving the equations of motion. r specific burning rate, a constant: time'-. s the vertical component P sin ' of the modified craft thrust. t time, measured from the end of the launching stage. t1 fuel burning time. t2 total time of flight, burning time plus coasting time. x horizontal distance of the craft at any modified time n; a subscript on x denotes a value of n. Page 5

ALER ONAUTICAL RE S ESARCH CENTER - UNIVERSITY OF MICHIGAN UMMI-48 y vertical distance of the craft at any modified time n; a subscript on y denotes a value of n. K a characteristic constant in the linear resistance law. K the modified drag coefficient - a constant. M0rE L a characteristic constant in the linear lift law. L the modified lift coefficient - a constant. Mor MO the initial mass of the craft, including fuel. P the modified craft thrust M0 T the magnitude of the craft thrust acting during the burning time, a constant. / the angle of inclination of thrust to the horizontal axis at any time. dx the modified horizontal component of craft velocity; a dn subscript denotes a value of n. dy the modified vertical component of craft velocity; a subdn dn script denotes a value of n. Note: At time t = 0, n = no = 1. The expressions x)no, (d) no refer to the values of the expressions at modified time 1; the expressions x)nl (dx\ refer to the values of the expres1~ \dn/ n1 sions at modified time n = nj, and so forth. We assume that x)no = Y)n = 0. - - 0~Page 6

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM —48 IV THE GENERAL EQUATIONS OF MOTION AND THE VARIATIONAL EQUATIONS Let the plane of the flight path be specified, and let us introduce a co-ordinate set in this plane in such a way that the rocket is initially at the origin. Then the equations of motion are Mo(1 - rt)d2x dx_ - dy + T cos at2 dt dt dt2 dt dt which can be simplified by introducing the new variable n = 1 - it. Computations show that dx = _ r d2x -2 d2x dt dn dt2 dn2 and that similar expressions are valid for d, d2. Combining the above relations, we may write dt dt2 (4.1) d2x dx + L dy = c dn2 n dn n dn n (4.2) a2r +_K __L dx= s _ l (4.2) d2y9KT Lci g dn2 n dn n dn n In order to discuss the variational equations, it is necessary that we know the form of the solutions of the last Page 7

AERONAUTICAL RESE ARCH CENTER - UNIVERSITY OF MICHIGAN UMM-48 expressions. In the case of burning for a specified time and then coasting, the values of x end y at the end of the total flight time will be shown by integration to be of the form nl (4.3) x(n2,/) = f(nl,n2) + j(nl,n2,n) c dn + nj + h(nl,n2,n) s dn n1 (4.4) y(n2, ) = f(nl,n) + nJ(nl,n2,n) c dn + nl + | h(nl,n2,n) s dn where f, j, h, and so forth, are known functions of their respective arguments: f and f are solutions of equations (4.1) and (4.2) when s = c = 0; and j, j, h, E are integrating factors of the left-hand sides of those equations. It should be noted that x and y depend on n2, when n1 is specified, and also on the thrust direction ~ which is arbitrarily chosen. In the case of continuous burning the values of x and y at the end of the burning time, which is also the flight time, will be shown to be of the form (4.5) x(nl,') = u(nl) + v(nl,n) c dn + nj + |w(ni,n) s dn, nj (4.6) y(n1,') = u(nl) + v(n,n) c dn + nj + w(ni,n) s dn. Page 8

AE R ONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-48 The quantities u and u are solutions of equations (4.1) and (4.2) when s = c = 0, and v, v, w, w are integrating factors of the left-hand sides of those equations. It will be shown that these functions have the property (4.7) v(nl,nl) = v(nl,nl) = w(nl,nl) = w(nl,nl) = 0 and it is of importance to note that 0 < n1 < 1, and also that - 00co < n2 < n1 < 1. We shall first form the variational equations for the case of burning and coasting. Using relations (4.3) and (4.4) we find that (4.8) ~x - +fn2 + (jntC + hn2s)dn] n2 - n2._ (js - hc)6S dn, (4.9) = [n + (jn + hn2s)dn] n2 - fn| (Jis- hc)6 dn, where the subscripts n2 denote partial derivatives with respect to n2. Equating 6x, 8y to zero in equations (4.8) and (4.9), and solving the expression (4.8) for ~n2, we obtain n, (js - hc)6' dn 6n2... nl fn2+| (jn2C + hn2s)dn Page 9

AE R ONAUTICAL RE S EARCH CE NTER - UNIVERSITY OF MICHIGAN UMM-48 Combining this last relation with equation (4.9), we may write nj fnz + (jnC + hn2s)dn r (4.10) - (js - hc)6 dnfnp +I ((in2 + hn2s)dnf J 1 *i2 +n -f (Js -hc)a dn = 0. If we assume that there is an optimum path, then along this path ' is known as a function of nj, n2, and n. Thus, the fractional expression in equation (4.10) is a function of n1 and n2. We denote, this expression by X(nl,n2) so that (4.11) [n2+ ( n2C + Ens)dn - [in2 + (jn2c + hn2S)dn = 0, and expression (4.10) becomes ynl )8 P-rn, (4.12) f (s - hc)8 dn - A (js - hc)8g dn = 0 J1 11 It is to be noted that the derivation of equations (4.11) and (4.12) from (4.8) and (4.9) could have been made directly by use of the Lagrange multiplier X. Since X is a function of nl and n2, it may be introduced under the integral sign in the second expression in equation (4.12) to give (4.13) f [Ss - hc) - X(js - hc3j dn =0 Page 10

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-48 Further, 6' is an arbitrary but integrable function of n. From this it follows that the bracketed expression in the integrand of equation (4.13) vanishes, that is, (4.14) = - h) (: - xi) From this we readily obtain + P(7 - Xj) (j - Xj)2 + (h - Xh)2 -+ P( - Xh) ( - Xj)2 + (h - Xh)2 the meanings of the symbols are given in the section on notation. By use of the above equations, we obtain the expression for X: (4.15) fn2 - Xfn2 + + (Jn2 - Xjn2 ) (J - X j) + (hn2 - hn2) (h - Ah)dn 0 / (; - Xj)2 + (E - Xh)2 Fortunately, in most of our work the fractional expression in equation (4.14) is independent of n, that is, the thrust direction is fixed in the plane of motion. Hence (4.16) X = s - hc Js - hc where X, c and s are independent of n. So, introducing into equation (4.11) this expression for X, we get an algebraic relation between s and c: (4.17) Ac2 + Bs2 + Csc + Dc + Es = 0, Page 11

AE RONAUTICAL RE-SEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-48 where A = h jn2dn - h jn2dn, nl nrl B = - id hndn - C = if jn2 dn - f jn2dn - -hf h ndn + h hn2dn, D = h fn2 - h rn2, E = i fn2- i fn2. By use of equation (4.17) and the relation s2 + c2 p2, we can obtain c, s and ~. The variational equations for the case of continuous burning can be formed in exactly the same manner. Because of the relations (4.7), the theory is exactly the same as in the previous case. For convenience in future use, we include the formulas obtained. Corresponding to equations (4.14) and (4.16) we have (4.18) s - cW v - Xv and = vs - VS - WC corresponding to (4.17), we have (4.19) A'c2 + B's2 + C'sc + D'c + E's = 0, Page 12

AERONAUTICAL RESEIARCH CENTER - UNIVERSITY OF MICHIGAN UMM-48 where n n A' = w J vnldj - w JVnldn, n, rn, B' = vJ wn, ldn vJ nwdn B' ~fin, n, CT = v vfnldn - vJ vnldn C' ~IfnJ1 n, n, - wJ Wnldn + wF wnln, D w un - w un, ' E = v unl -i v Un, Note: in this case we very burning time nl and thrust inclination 1. Page 13

AERONAUTICAL RESEARCH CENTER — UNIVERSITY OF MICHIGAN UMM-48 V ZERO LIFT AND DRAG C ontinuous Burning When both lift and drag are absent, equations (4.1) and (4.2) become d2x c dn2 n d2_ = s dn2 n The integrals of these equations are respectively Hn (5.1) x(nl,e') = (nl - n) ~ dn, x~~ ~ (nj nn- n) ) dn Io n Comparing these with expressions (4.5) and (4.6), we see that for the present case u = (n1 - l) \dn/ n - nw = 0 = Pag n e 14 Page 14

AERONAUTICAL RESE-ARCH CENTER - UNIVERSITY OF MICHIGAN UMM-4 8 u=(n - 1) ( - (nl - 1)2, V = 0 \dn no 2 - nl - n n Thus, equation (4.18) becomes S = -, indicating that thrust is fixed in the plane of motion. Equation (4.19) is reduced to the form P2ln n- + s (n + = 0 and since by definition s = P sin ~ and c = P cos p/, we may write this as (5.3) sin (j + a) =- P n nj + - g - no0 dnn n g(n where (5.4) cos a= d n. dx2no+ - g(n -i 2 /(dnno dn no (5.5) sin a = (n., d no dno n g(no -1)2 Evidently, / will exist only if (5 6) (P in n1)2 <1 2 g(n - 1 Pedn n n Page 15

AERONAUTICAL RESEARCH CENTER -- UNIVERSITY OF MICHIGAN UMM-48 22 20 16 o. xc ~ ~ ~ Pae1,, 12 8 4 ~ 0 ___ _. Figure I Page 16

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-48 -120 -100 -80 o,-60 -40 -20 0 2 4 6 8 10 12 Burning Time - seconds Thrust Angle For Optimum Path. Figure r Page 17

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-4 8 The left-hand side of equation (5.6), when multiplied by the mass of the craft at time t = t1, represents the kinetic energy at time t = t1 of a craft, of initial velocity zero, moving in a vacuum under a fixed thrust. Similarly, the righthand side when multiplied by the mass of the craft at time t = t1, represents the kinetic energy at time t t1 of a craft whose initial velocity vector has components d, (and ~dto o atd which moves in vacuum under the force of gravity. Hence, equation (5.6) implies that an optimum trajectory exists only when the latter kinetic energy is larger than the former. This is clearly a restriction, in the form of a lower bound, on the initial components of velocity. For the case of continuous burning of fuel, some examples of optimum results are shown in Figure I. These assume neither drag nor lift, and give maximum horizontal distance for fixed vertical distance. In the examples we assume five to be the ratio of thrust to total craft weight, inclusive of fuel, and assume a burning rate r of 0.02/sec. The co-ordinates of craft position at the end of the burning time are given as functions of burning time for these values of the craft velocity vector as existing at the end of the launching stage: Case A: xo = 2000 ft/se6, Yo = 500 ft/sec. Case B: Xo = 500 ft/sec, Yo = 2000 ft/sec. Case C: xo = 750 ft/sec, yo = 3000 ft/sec. Case D: xo = yo = 1460 ft/sec. As we can see from the figure at y the left, these values imply for Case A that the craft velocity _l~ vector is initially inclined at an angle of about 140 from the horizontal. Corresponding to _, y| lthe values of the co-ordinates as shown, the necessary angle e' of thrust inclination to the x horizontal is given as given as a function 0o go ~~~of burning time in Figure II. Page 18

AE]R ONAUTICAL RE S E;ARCH CENT ER - UNIVERSITY OF MICHIGAN UMM-4 8 Burning and Coasting During the coasting phase n2 < n < nl, the equations of motion are.7) d2x = 0 (5,8) O dn2 (5.8) d2y dn2 The conditions for x and y at modified time n = n1 are to be obtained from equations (5.1) and (5.2), and by differentiating these equations with respect to nl we obtain the velocity components c (5.9) + — dn.(dnl n, no (5.10) ()- g(nl - 1) + s dn n, n no n Integration of equations (5.7) and (5.8) yields x(n2,g) = x(nj,') + )(n - n1) (n, - (2 -nj) y (n2,) = y (nj,~' + -A (n2 - nj) -g (2-n) (dn\__ _ __ _ 2dn)n1 2 and by substituting into these the co-ordinates of position at the end of burning from equations (5.1) and (5.2), and velocity from equations (5.9) and (5.10), we obtain x~ ~ n,,~~~ > J ~ ~ ~ nn ~x(n2,~) = (n - 1) (d) +f (n2 - n) c dn Pag 19 n - ~Page 19

AERONAUTICAL RE-SELARCH CENTER - UNIVERSITY OF MICHIGAN UMM-48 y(n2, ) = (n2- 1) ( - (n2 - 1)2 + dn no 2 (n2 - n) s dn Comparison of these latter equations with the relations (4.3) and (4.4) shows that f = (n2 - 1) d j = h = 0 \dn no n f = (n2 - 1) d-z - g(nm - 1)2, - 0, dn no 2 -h = n2 - n n Consequently, equation (4.14) becomes s = - C from which we conclude that again thrust direction is fixed in the plane of motion. Equation (4.18) furnishes the relation p2 ln n + s [dn) - (n2 - 1) + c () = 0. It is clear that for this case formulas (5,3) through (5.6) are valid with n1 replaced by n2 in the various denominators and in the right-hand side of equation (5.6). Examples of optimum results for vertical paths are given in Figure III for cases of burning followed by coasting. Values of the co-ordinates are shown as functions of total flight time. Again, lift and drag are assumed negligible, and the conditions at the end of the launching stage are the same Page 20

AERONAUTICAL RESEARICH CENTER - UNIVERSITY OF MICHIGAN UMM- 481 as those used for the examples of Figures I and II. Burning times are 10, 20 and 30 seconds, as indicated on the graphs; the label xA_20, for instance, defines the curve representing the variation of the x co-ordinate with flight time for conditions A —that is, for xo = 2000 ft/sec, yO = 500 ft/sec — and for a burning time of 20 seconds. Corresponding angles of necessary thrust inclination, measured from the horizontal, are indicated in Figure IV. The figures have been drawn so that some indication of maximum vertical distances attained by the rocket may be seen. Actually, at very large vertical or horizontal distances, the effect of the variation in gravity, curvature of earth, etc. should be taken into consideration. In order to make the methods employed more readily understandable, the process of optimization in the case of zero lift and zero drag, burning followed by coasting, is discussed in unmodified terms in Appendix A. Page 21

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-48 3000 2500 2000 1500 I000 500 Terminal Points For Some Optimum Paths. Figure mlI Page 22 APAB

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN UVJMM-48 10I A -30 100 60 40 301 0 100 200 300 400 500 Coasting Time - seconds Thrust Angle For Optimum Path. Figure = Page 23

AERONAUTICALL RES EARCH CENTER -UNIVERSITY OF MICHIGAN UMM-48 VI ZERO LIFT AND LINEAR DRAG Continuous Burning When lift is negligible and a linear drag is-assumed, the equations of motion (4.1) and (4,2) become (6.1) d2x + K dx c dn2 n dn n (6.2) + _K = dn2 n dn n In order to integrate these relations we introduce the function G = G(n,q) which satisfies the adjoint differential equation d2G K dG+ KG (6.3) + 0 dn2 n dn n2 It is easily verified that the value (6.4) G(n,q) K 1 nKqlK K 0,1 satisfies equation (6.3) and has the useful properties G(q,q) = 0, (dG)q 1 The case K = 0 has been covered in the previous sections. If K = 1, then the function G is Page 24

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-48 G(n,q) = n In (n/q) Because of the fact that the case K = 1 is rarely encountered in practice, we omit the discussion of this case and treat equation (6.4). We shall now apply the standard method for treating a differential equation and its adj.oint. Combining equation (6.3) successively with equations (6.1) and (6.2), we find that (6.5) d dx xdG + d /KGx = cG (6.5) A( dn \dn dn d+ n / n and (6.6) dAnt (G ydAK +(d g G dn dn dn dn n/ On integrating these equations between the limits q _ n _ 1 and replacing q by nj, we obtain (6.7) x(nl,) 1 (nllK 1) (+ I - K -n no 1+ 1 nK-lnll-K - c dn 1 K (6.8) y(nl,) =(n 1K - 1) + 1 K \dnno g_ _l-K 1K-iE K- 1 2 K +n 1 2 K +- 1 +1 -K [nlnKilK - s dn Note that K f 0, 1 or -1 in relation (6.8); in future work, we will omit these special values of K. Comparing these latter expressions with equations (4.5) and (4.6), we find Page 25

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-48 1 1-K fdx U = 1 K(nl 1) dnn nK-lnl-K _ 1 V = w = 0, = (n 1-K 1) (d ) + 1 - Kno i-K K-_ l__1 n - K-1 2n, K -1 2 K + 1 K+ 1 V, _ nK-lnll-K - 1 1 - K Again, thrust direction is fixed in the plane of motion. By use of relation (4.19), we find that p2 (nlK - 1) +s + 2g nK+ (6.9) n s 9_nj K nno K+ 1 K+ 1 + c d= \dn/ no The formulas corresponding to expressions (5.3) through (5.5) are (6.10) sin($ + a) = - P (niK - 1) K / - + + g n K+1 \dn no no K + 1 K + K + 1 Page 26 I

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-4 8 (6.11) cos c = ei_ + g __ 2g n l +1 + [no( f K+ 1 K + 1 (612nsn K+1 Kcl dX ~ +r+ g _ 2g nlK+ /(dn no dnno K + K + 1 E+ (6.12) sin a = dx)2 + g _2g nK+ dno dn no K + 1 K+ 1 (6.13) <,(nmK _ 2 2 K2 1 _2 + )+ _2g nK+ 2 no dn no K + 1 K + 1 must be satisfied; this can be interpreted physically in a manner analogous to relation (5.6). Burning and Coasting During the coasting phase n2 < n < nj, the equations of motion are d2x K dx= dn2 n1 dn d+ K d g, dn2 n1 dn Page 27

AER OENAUTICAL RES EARCH CENTER - UNIlVERSITY OF MICHIGAN IUMIM-48 which can be immediately integrated to give (6.14) x(n2) = - nl [eK(l n - 1 (n) + x(n, (6.15) y(n2,) = [K K n, + y(n,) + ~ gnn3- ni + [ - n2 + n, _ eK(l - jT] K K The conditions at modified time n = nl for x and y and their derivatives are obtained from equations (6.7) and (6.8) and their derivatives. We find that (6.16) =()n n1 Kdn) + n-KnK-l c dn n(6.17) = nj + -g (nlK - nl) + \dn/ n \dn/no K+l. n, + ni-KnK-1 s dn 11 Substituting the results from equations (6.7), (6.8), (6.16) and (6.17) into (6,14) and (6.15), we obtain x(n2,/ = ____ nl l-K ____ - [nilK eK(l - n2) nl K) K-] ()no K'K K(K - 1) K 1 nn fn, + j R c dn Page 281

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN. UMM-48 y(n2z,/) -. 1-K n nl-K 1 1 {@ ) eK(1 - 1K+ K] (dy+ L K E(K - 1) K - dn n -nl1-K eK(1 2 n2 1 -+ [nKeK 1 n, n2eK(1- ) 1 K(K + 1) K2(K + 1) nl1-K + n12(K2 + 2K + 2) K(K + 1)(K - 1) 2K2(K + 1) nln2 + _ 1 + rR s dn, K 2(Kl 1- R = 1 _1lKnKl - 1 nl1-KnK-1 eK(1 -n K(1 -1 K K) n 1 K Comparing these expressions with equations (4.3) and (4.4), we find that LKf = [ eK~e nl- K(K -1) K-i] (dnn) -nl nK eK(1 n 1-K) 1 1 n1 K- K(K - E. nK_1 nK 1 -K -K1 _1 K(K- 1)] 1 - K h = 0, Page 29

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-48 1-K n 1-K_ n, Kn, =f ~ i- eK~ -n __ + I+,K(1K(-l an/n0-K =( K(K -- ) 1 dn n 1-K K() n2 2 K(l n ) + g [n e -, ni K K(K + 1) K2(K + 1) -_ nl1K +nK2 + 2K + 2 l n1n2 + + n, + K(K + 1)(K - 1) 2K2(K + 1) K + 1 2 ( K -1 ) =0,Y h = nK~l [n1l-K eK(n1 - ) 1l-K 1 K KK -_ 1 1 K Again equation (4.14) shows that the thrust direction is fixed in the plane of motion. From the relation (4.17) we have (6.18) P2(n K - 1) + K K+l1 K+1 -~" + s dyn + 9 gni gn, eKl 2 + [(dn)no K + 1 K(K + 1) K n3 +c (dx =0 \dn no Formulas similar to (6.10) through (6.13) may be written. The only difference between equations (6.18) and (6.9) lies in the multiplier of s. We shall not explic itly list these results. Page 30

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-48 VII ZERO DRAG AND LINEAR LIFT Cont inuous Burning When linear lift and negligible drag are assumed, the equations of motion may be written (7.1) d2+ L d = c dn n dn n y(7.2) d _ L dx = s g (7.2) - &L - dn2 n dn n Eliminating y from equation (7.1) and x from equation (7.2), we obtain the relations (7) d3x + 1 d2x L2 dx Ls + 1 dc dn3 n dn2 n2 dn n2 n n dn (7.4).d3 + 1 L2 L Lc + ds dn3 n dn2 n2 dn n2 n n dn The adjoint is d3G d2 (G)+ d L2G 0 dn3 dn2 n/ dn n2 / Simplifying this, we may write (75) d3G 1 d2G + 2 + L2 dG 2 + 2L2 G dn3 n dn2 n2 dn n3 Page 31

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-48 We shall attempt to find solutions of equation (7.5) of the form (7.6) G = na Combining equations (7.5) and (7.6), we find that a must satisfy the characteristic equation a3 - 4Q2 + (5 + L2)a - (2 + 2L2) = which has solutions a = 2, 1 + iL, 1.- iL. Hence, the solutions (7.6) are of the form n2, nl+iL, nl-iL with real parts n2, n cos(L ln n), n sin(L ln n) If we choose n n 2 (7.7) G(nq) =-qn cos(L In q) qn sin(L In ) + n (L2 + 1) L(L2 + 1) L2 + 1 as the solution of equation (7.5), this satisfies the conditions (7.8) G(q,q) = () = ()qq = 1 d qyq n q,q By use of relation (7.8) and the standard technique for solving an equation with the aid of the adjoint —see equations (6.5), (6.6) —we obtain Page 32

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-48 (7.9) x(n1,G) = G(l,nl) - L d + d n=1 + G(l) - () ] (dx) + n Ls + d1, dn 1nl 1 + | (nn ) -Ls + L + 1 dc dn +jIfl1G(nnn) n n J 1nndn (7.10) y(n,) = G(1,nl) s g + L dx + LS - ann=1 1,n +f G(n,n2) Lc - g + 1 ds dn l n n These equations can be simplified. Integration by parts and use of relation (7.8) shows that _G(n,nl) dc c d (7.11) (n ) d- dn G(l dn, and an equation similar to this is valid if c is replaced by s. Use of equations (7.7) and (7.11) shows that relations (7.9) and (7.10) mpy be written Page 33

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN. UMM-4 8 (7.12) x(nl,) = -= L2+[1L cos(L In nl) - nl sin(L in nl) - +o L2 +1 dn no + L2+ 1 [n cos(L In nl) t nL sin(L In n) - gnl sin(L in n1) - n,-nn nl cos(L n nl) - sin(L in n -l) dnL2 +i1 J dn nn 1 H in n.4 dn+-L + f ["'1n, L sin(L In ) - nl c o s (L In n) + n dn and (7.13) y(nl,) = = 2- 1 n L cos(L In nl) - nl sin(L In n) - L + L2+ 1 no + 21 [ n cos(L in nj) + nl sin(L in nl) - 11(/tr +'| L2+ n sin(L n n cos(L in n) +n Pg3n 4 nlg co( n~ nsnLI

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-48 Comparing these with equations (4.5) and (4.6), we find that (7.14) u = L2 1 nlL cos(L in n1) - nl sin(L in nl) - L idn + L2 + 1 L dnno + 21 [n1 cos(L in nl) + n1L sin(L in n1) - (dxnn L2 + 1 LJdnj no g nl sin(L in n1) L2 + 1 n. n ~(7 15) _ _ FnL sin(L in n1 cos(L n ) + n (7.15) V =-___ L2 + 1 n n (716) w = L - nlL cos(L in n,) - n, sin(L in n-) + nL (7.16) Wr-,nn ni nl_ L +1 n (7.17) u = 2 1 [niL cos(L ln n1) - n1 sin(L ln n1) - L (dx) + + nL21 n cos(L in n1) + nL sin(L in n1) - i]() +| L2 + 1 dn no + n, cos(L in n1) - 1 L2 + 1 (7.18) v = [-nj cos(L ln n,) - n1 sin(L in n ) + nL (7L18) ne L2 + 1n w = i [niL sin(L in n~) n, cos(L in +- n] Page 35

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-48 From equation (4.18), we have (7.19) s -nl(L+\)sin(L in n) + nl(l-XL)cos(L In n-) - n(l-L) c -nl(L+X)cos(L in n) - nl(l-XL)sin(L in n) + n(L+X) nj n( Thus, we see that the thrust direction varies from point to point on the optimum trajectory. In order to obtain the value of X, one must solve the algebraic equPtion corresponding to equation (4.15) for the case of continuous burning, which is (7.20) unl - Xunl + nl(vnl-Xvn)(V-Xv) + (Wn w)(w-Xw) Pf nj _ ___ n__ dn = 0 J1 $(v —Xv)2 + (w-Xw)2 Because of the complexity of the formulas for u, u, etc., there seems to be little hope of finding the general solution of this equation. However, if we require that X be constant, then equation (7.20) may be simplified. First, we note that in general (7.21) dn (V —Xv)2 + -w)2 = -E~v E Xvn)) (V-X)+(wnj-Xwn) (w-XwI -Xnj [vv+ww- W(v2+w2 I(-v Xv)2 + (w —Xw)2 Since X is Pssumed to be constant, the expression under the integral sign in expression (7.20) can be simplified by use of equation (7.21). Substituting equations (7.14) through (7.18) Page 36

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-48 and (7.21) into the relation (7.20), we may write (7.22) isin(L In n) - Xcos(L in n,) (d + + [cos(L in nl) + Xsin(L in n)() + + - [(1 + XL) cos(L In nl) + (-X+ L) sin(L In nl + L2 + I + + +X2L2+ d 2+l1-2nlncos(L In ) dn = 0. L2 + I dn~ n1 For arbitrarily chosen constants Xand nl (0 < n -< 1), the expression (7.22) determines a linear relation between the initial components of velocity, so that the craft may follow an optimum trajectory. By substituting this value of X into equation (7.19), one obtains the desired thrust direction schedule for the optimum trajectory. Burning and Coasting During the coasting phase n2 < n < nj, the equations of motion are d2x L d O = 0 dn2 n1 dn d2 L dx = dn2 nl dn The equations corresponding to (7.3) and (7.4) are d3x L2 dx = dn3 n~ dn n1 Page 37

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-48 d3y+L2 = o dn3 n1 dn of which the integrals are easily found to be (7.23) x(n,) = - - cos L (ni-n L si(nl-n2z) (-\ + LL L n Jdn n L n\dn nn + x (n1,) + [g (n2 -ni) + 2 sin-(n(-n, )] L2 ni (7.24) y(n2,') = L- cosL (nl-n2)] (d - n sinL (nl-n2)() + L iL nj n n L nl d n Fn2 2 + y(n1,4) - g 2 - 2L cos L (nl-n2)1 The boundary conditions at modified time n = n1 for x, y and their derivatives are obtained from equations (7.12) and (7.13) and their derivatives. Thus, by differentiating equations Page 38

AERONAUTICAL RE;SE:ARCH CENTER — UNIVERSITY OF MICHIGAN UMM-48 (7.12) and (7.13), we get dx- sin(L in ) + cos(L In nl) dx + ndnd n n h)dndno f~n, J+ s[ sin(L in n + c cos(L Inn dn - - 2g - [sin(L in n1) + L cos(L in ni3 and Hi n +,J [s cos(L In n) _ c sin(L Inn dn + 1 nin n n1 + +2 [cos(L ln n1) - L sin(L In n1)] Substituting these into equations (7.23) and (7.24), we obtain Page 39

AER ONAUTICAL RE SE;ARCH' CENTER ~ UNIVERSITY OF MICHIGAN UMM-48 (7.25) x(n2,') = (dxJ-nl.F -L in 1+ \=(dx/nL sin(L in nl) - sin (nl-n2)-L n n + ndno L + 1 [nl, cos(L in n1) + nl L sin(L In nl) - 1]2 + + (nd/LL cos(L in n1) + n cos[L (n-n2)-L in n + *+ 2 cos(L in nl) + nL cos (nl-n2)-L in n gndl(n-nl) + sin (n1-n2 + S+ sL cos( L n n +n sin(L in n -nL dn,n Q- i~ in L sin(L in n )n cos(L n )+n dn +.... cos ( nz- n2)+ L In n) + n n, nn fn sic{ nL sin(L n —L n ) sin[l (ni-n)+L n dn, and n-nl) P nag 4n L (n0_nA nl ~ n )+4

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-48 (7.26) y(n2,) = = dx n cos (L in n) a cos (n-n2)-L In nlj - dn L L - nn Lz+ 1 [Jnl L cos(L in n,) - nl sin(L in n,) - + \dnn L- L1 -nl n, lnz-LILn + (dn.no sin(L n n) L in (nl-n2)-L in n + + 2+i [nl cos(L in nl) + nl L sin(L in nl) - 1i + pni n1 Lo L" +L + s sin(L in n) - sin[L (nl-n2)+L in nJ dn + fn L n, L n, +JiC L cos(L in n) - l cos L(ni-n2)+L in dn + n, L n + {0_ sin(L in nl) + n, cos[k(nl-nz) - L in n] - sin +1 L - nL sin[L (nl-n2) - i n nl - 1 L 2 n2 n2 L g 2 2cos-(n1-n2) L2 T,2 n, L2+ [ n, L sin(L in n )-n1 cos(L in )+n d +2 +1 ] n n n, L2.+ 1 iJ1 nE-n L cos(L in n )-n1 sin(L in n-n)+nL dn. Page 41

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-48 It is clear from the form of equations (7.25) and (7.26) that the thrust direction varies from point to point on the optimum tra_ ector. The determination of optimum path thrust schedules is very complicated in the general case, for equation (4.15) must be solved. However, particular solutions of equation (4.15) corresponding to a constant X can be obtained. The final formula is slightly more complicated than (7.22); hence, we shall not derive it explicitly. -, Page 42

AERONAUTICAL RESEAIRCH CENTER - UNIVERSITY OF MICHIGAN U hMM-48 VIII OTHIER VARIATIONAL PROBLEMS OF THE SAME TYPE WVe spw in Section VII that the problem of maximizing x for a fixed y, or vice versa, either at the end of the total flight time or at the end of the burning period, with variable burning time and variable thrust direction assumed, is rather complicated when lift is involved. A simpler problem of the same type, involving lift, is that of maximizing x or y at the end of the burning period or at the end of the total flight time when both burning time and coasting time are preassigned and thrust direction is variable. The thrust direction schedule cpn be obtained immediately by applying direct variations to equations (7.12) and (7.25), or (7.13) and (7.26). Maximum x for Continuous Burning, Zero Drpg end Linear Lift From equation (7.12) we find thpt nj~~~~~~~~ nj1 L cos(L in n) + ni sin(L in n Ln s + + n1 L sin L(ln n ) - n1 cos(L in n)+ n C 6 / dn = 0. nj n1 If the integrand of this relation is to vanish for arbitrary 6(, then it follows that n ni L sin(L in ) - n cos(L in ) + n tan n x = n n1 L cos(L In na) + n1 sin(L In nna) - Ln Page 43

AERONAUTICAL RESEUARCH CENTER - UNIVERSITY OF MICHIGAN UMM-48 and this formula furnishes the desired thrust direction schedule for the optimum peth. Maximum y for Continuous Burning, Zero Drag and Linear Lift From equation (7.13) we have the result tan y = - cot x. Maximum x for Burning and Coasting, Zero Drag and Linear Lift From equation (7.25) we see that c nj n j.. cos(L In n) + -l cos (n -n,) + L in n n L nj L + Ln n, J + 1 r L cos (L n n)+nj sin(L In n -nL b]/ dn + L2+ lL njnl + S Q sin(L In n ) + sin [L (n-n2) + L ln + 1 n L sin(L ln n-)-n1 cos(L ln -n)+n] dn o. L2 + 1 sin( In n )+ dn 0 Consequently, Page 44

AERONAUTICAL RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-48 tan /x = - n, cos(L in n )+ cos (nl-n2) + L n n + + 21 Inl L cos(L In ) + n1 sin(L in n ) - nL] {nl sin(L ln n )+ n' sin + Llnn] + L n+ L n1 rlj n + n L2 + 1 n, nj + 1 [ni L sin(L in nnf) +n cos(L in nn)+n]2. Maximum y for Burningp and Coasting, Zero Drag and Linear Lift From equation (7.26) we find that tan = - cot x Page 45

AERONAUTICAL RES EARCH CENTER - UNIVERSITY OF MICHIGAN UMM-48 APPENDIX A THE PROBLEM OF ZERO DRAG AND LIFT IN UTMODIFIED TERMS; BURNING AND COASTING For the sake of simplicity, we shall consider optimization in the case of zero lift and drag, burning and coasting, directly in unmodified terms. We may write the equations of motion in the form Mo(l - rt) d2x = T cos, d2 MO(l - rt) O = T sin -, dt2 and integrate to yield the respective velocity components at time t x = Xo +74 T cos dt, t Y = Y gt + T sin ( dt y O t ' - l t) and the co-ordinates of instpntaneous position t rT X = Xgt~co T cos dcr dr x = xot +f 11ol-r [ ~~~~~~~Page 46-

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM -48 y1- 2 =T sin t do dr Y = Yot - gt + d d Moil - If the rocket burns fuel during the intervpl (O,tj), integration by parts gives, as the position co-ordinates of the rocket at some later time t = t2, tl (A.1) x2 = xot2 + I (t2 t) T cosj dt (Al x it Mo(1 - l t ) tl (A.2) Y2 = Yt2 - t2 + (t2 t) MTin dt 2 (lo - itt) For an optimum path we require that x2 be a maximum for fixed Y2 or Y2 be a maximum for fixed x2. Along the path the thrust angle is some function of time t and flight time t2. Thus, for an optimum path we require that 0 Mo(1 - rt) tl (t2- t) T sin dt = 0, 6Y2= ~o - gt2 +f MT(lin dt 6t2 + + (t2 - t) T cos IT dt = 0 Page 47

AERR ONAUTICAL R E S EARC H CENT ER - UNIVERSITY OF MICHIGAN UMM-48 From these equations we get T sin rs YO gt2 + MO(l- it) Sdt t Yo_________ - t)T sin + d at + XO Moj 0 ~t) MO (l-it) (A.3) + ' (t2 t) T Cos ' + ' dt = 0 0 ~~MO(l - it) The substituti on o - gt T sin dt (A.4) X tj~~~~( 0f MO(l ' i t) where Xis independent of t, permits us to write equation (A.3) in the form f 2 -t) T) [cos + Xsin j dbat 0 Since 60 is an arbitrary function of t the bracketed expression above vanishes, so that cot = - X Consequently, thrust angle / is independent of t, and therefore for an optimum path thrust direction is fixed in the plane of mot ion. Page 48

AERONAUTICAL RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMIM-48 Now equations (A.1), (A.2) and (A.4) become respectively (A.5) x2 = xot2 +Tos f t2 - t dt, (A 6) Y2 2 = sin t (A.6) Y2 = ot2 + -2gt2 + T o 1 -. dt 2_ MO T (A.7) Xo cos ' + (Yo - gt2) sin - M In (1 - rtl) For a given launching velocity end e specified burning time we have three equations in x2,y2,t2 and $. For fixed x2 or fixed Y2 equations (A.5), (A.6) and (A.7) give respectively the values for y2,t2,Y or x2,t2,~ for the optimum path. Page 49

AERONAUTICAL RESEARCH CENTER UNIVERSITY OF MICHIGAN UMM-48 DISTRIBUTION Distribution of this report is made in accordance with ANAF-G/M Mailing List No. 10, dated 15 January 1950, including Part A, Part B and Part C. Page 50

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