4-1. INTRODUCTION

In this chapter, we shall analyze the trajectory of the vehicle from the launching pad, point A, to the burnout position, point B . During the thrusting phase, the energy provided by the propellant is transformed into potential energy through the increase in the altitude of the vehicle, and kinetic energy through its increase in speed. Also, a part of the energy provided by the propulsion system is dissipated in the form of heat by action of the aerodynamic drag. The powered phase is the phase during which it is possible to have a guidance system to control the trajectory such that at the end of the thrusting program, the vehicle reaches a prescribed position B, specified by the position vector rg , and a prescribed velocity Vg . We have seen in the preceding chapter that the trajectory required by the mission may be completely specified by these conditions at burnout.

The guidance is achieved by the following modes of control, a/ Control of the thrusting force T . This control is performed by the direction of the vector thrust. Its magnitude can also be controlled by the variation of the mass flow rate.

b/ Besides the main engine, the vehicle can be equipped with several small rockets providing lateral thrusting forces for its guidance. We shall assume that the resultant thrusting force of all the engines is represented by the vector thrust T .

c/ Control of the aerodynamic force A . This control is performed by varying the angle of attack of the vehicle and possibly by varying its aerodynamic configuration. In three-dimensional flight, the aerodynamic force is also a function of the bank angle.

To write the equations of motion, we shall assume that the trajectory lies in the plane of the great circle containing the launch point A and the burnout position B . Hence, it is necessary that all the forces involved be contained in that plane. This leads to the assumption that the vehicle has a plane of symmetry and that the velocity vector V , the aerodynamic force A and the thrusting force T are all contained in that plane. The duration of the powered phase is generally short and it is convenient for a first-order approximation to assume that the Earth is an adequate inertial reference and in this reference system the atmosphere is at rest.

The center of mass M of the vehicle is defined by its coordinates x and z in a ground coordinate system Axz , where the axis Ax is the horizontal at the launching point A taken as the origin of the coordinates, with positive x in the direction of motion, and the axis Az is the vertical at the point A, taken positively up (Fig. 4-1).

At any point along its trajectory, the flight path angle of the vehicle is defined as the angle between the local horizontal (the plane perpendicular to the gravitational force mg ), and the velocity vector V . The angle <(> between the local horizontal and the line Mx' drawn parallel to the horizontal Ax of the launch point is precisely the range angle as defined in the preceding chapter.

At each instant t , the vehicle is subject to three forces (Fig.

a/ The gravitational force W = mg applied at the center of mass M .

b/ The aerodynamic force A applied at the aerodynamic center P . The aerodynamic force can be decomposed into a drag force D in the opposite direction to the velocity V , and a lift force L orthogonal to it.

c/ A propulsive force represented by the thrust vector T , applied at a point Q . To simplify the force diagram we shall assume that the three points M , P and Q are aligned and constitute a body axis, fixed with respect to the vehicle. Then the angle of attack a can be conveniently measured from this body axis to the velocity vector V . The thrust angle e is defined as the angle between the body axis and the direction of the thrust.

Fig. 4-2. Forces acting on the vehicle.

Using Newton's second law, we can write the equation of motion in vector form dV - - -

dt where m is the mass of the vehicle. By projecting this equation into the tangent and the normal to the trajectory of the vehicle we have dV

These equations are the dynamical equations. They can be obtained directly from the general equations for flight over a spherical Earth derived in Chapter 2.

The lift and the drag forces are assumed to have the form

where p is the atmospheric mass density, and S a reference area.

The coefficients C^ and C^ are lift and drag coefficients. They are functions of the angle of attack a , the Mach number M and the Reynolds number Re

The longitudinal range x , and the altitude z are obtained from the kinematic relations x t

Finally we have the pitching moment equation, describing the motion of the vehicle about the center of mass d2

B —— ( y + o< - 4> ) = L i pcost( +Dip sina - Ti Q sin t dt

q dt where B is the moment of inertial of the vehicle about an axis passing through the center of mass and perpendicular to the plane of symmetry, I the distance between P and M and £q the distance between Q and M . The term M is the aerodynamic pitching moment, and the term Kd(y + a- — 4> )/dt represents the moment due to the mass flow of the gas ejected from the propulsion system. The thrust can be written as

re e o e e where p = - dm/ dt is the overall mass flow, Vre the average relative velocity, pe the average pressure and A the area over the exit of the engine. It is assumed that the tangential stress over the exit area is negligible. The unit vector ng normal to the area Ag is positively directed inward. Finally, p is the average free stream pressure. For simplicity, we may assume that the vectors in Eq. (4-9) are all collinear and write the one dimensional equation as

We define the effective exhaust velocity c as

re ' lro re' p Hence, the expression for the thrust magnitude is simply dm

This equation gives the thrust in terms of the mass flow rate and the parameter c which can be characterized as a function of the propellant used in the propulsion system on board the vehicle. In engineering practice, we may use the specific impulse I as an alternate parameter which specifies the thrust performance. It is defined as the thrust impulse per unit mass of propellant or

sp dm p

From the last two equations, it is seen that the specific impulse I

may be alternatively defined as the thrust obtained per unit mass ffow which is precisely the same as the effective exhaust velocity. But it is a common practice to use different units for I and c through the

sp where g is the acceleration of gravity. Therefore, while c is given in meters per second, I is given in seconds. Using Eqs. (4-11) IBd (4- 14) we have

The mass flow rate can be computed from p = CpPcAc (4-16)

where

= mass flow coefficient, function of the propellant p = average pressure in the combustion chamber. This c pressure is also called the operating pressure.

A = area of the throat of the nozzle c

Hence

sp g gCpPc Ac

The ratio of the areas A / A can be expressed in terms of the expansion factor P /p as c e

e c where k is the ratio of the specific heats.

From these relations, we see that the specific impulse is a function of the following four factors

2. The operating pressure p

4. The altitude of flight (g and p^ are functions of the altitude).

For a given type of propellant, we can evaluate its specific impulse under some reference conditions. These conditions are:

For a solid propellant, p =70 atm. , p / p = 70/ 1 at sea level.

c ce

For a liquid propellant, p =25 atm. , p / p = 25/ 1 at sea level.

c ce

Let the specific impulse of the given propellant evaluated at these reference conditions be denoted (I ) . Then we define the coefficient sp o

sp I

sp o

This dimensionless coefficient characterizing the propellant under the actual operating condition is now a function of four parameters--the ratio of the specific heats k , the altitude z , the pressure in the combustion chamber pc and the expansion ratio p / p . If we assume that the ratio of the specific heats is the same for all propellants, then the function i = f(z,p , p /p ) can be tabulated for practical reference.

For an anlytical integration of the equations of motion, we shall assume that, under normal operational conditions, the specific impulse Igp , or equivalently the effective exhaust velocity c , is constant.

Finally, if R is the radius of the Earth, then the range angle § is seen to be given by tan 4> = (4-20)

Now we see that, for each stage of the rocket vehicle, the dynamical equations, Eqs. (4-2) and (4-3), the kinematic equations, Eqs. (4-6) and (4-7), the moment equation, Eq. (4-8), and the mass flow program equation, Eq. (4-12), constitute a system of six equations for the following eight unknowns:

coordinates of the center of mass components of the velocity vector mass of the vehicle angle of attack angle of the thrust magnitude of the thrust

Therefore, to specify the flight trajectory, we have at our disposal two control variables. They may be taken to be the thrust magnitude T, and the thrust direction t . On the other hand, for a fully controlled flight, the angle of attack a has to be adjusted constantly to render the moment equation, Eq. (4-8), identically satisfied. Consequently, with a flight program fully controlled, the remaining equations constitute a system of five equations--the Eqs. (4-2) and (4-3), (4-6) and (4-7), and the Eq. (4-12)--which provide the solution for the variables x , z, V , y and m as functions of the time t .

If the time history of the thrust magnitude T(t) is prescribed in advance, then by integrating the mass flow equation, Eq. (4-12), we have the variation of the mass as a function of the time. If we assume a constant mass flow rate, then m is a decreasing linear function of the time. In general, the mass of the vehicle is a decreasing function with respect to time as shown in Fig. 4-3.

The figure represents the variation of the mass of a multi-stage rocket. The first stage of the rocket is operating between the initial time and the time t^ . At tj the first stage is released providing a discontinuity in the mass of the vehicle. If a lapse of time Atj exists before the engine in the second stage is ignited, during that time the vehicle is in coast flight with constant mass. Next m continues to decrease between the time (t^ + At^) and t^ and so on. In the subsequent analysis, we shall assume that all the At^ are zero.

The remaining control variable can be selected either as the angle of attack a , or the thrust angle e or a combination of both by specifying a relation between these variables and possible other variables also. With this selection the ascending program is completely specified.

In general, in considering an ascending program, we are trying to obtain the optimum of some performance criterion. For example, for a preprogrammed motor, we would like to select a time history for the thrust orientation such that the range achieved is a maximum. Problems like these involve the calculus of variations or the equivalent modern control theory and will not be discussed here. From an engineering standpoint, the selection of a best flight control program is severely restricted by other technical constraints. For example, for a thrusting flight giving the maximum range, it can be shown, upon using not unrealistic assumptions, that the flight must be at maximum lift-to-drag ratio, with the thrust directed orthogonally to the aerodynamic force, and hence making a constant angle with the velocity vector. But the thrust angle e , due to the technical construction of the propulsion system, cannot deviate at a large angle from the axis of the vehicle. In general e is constrained by a maximum angle e max a degrees from the main thrusting line.

Another factor to be considered is the normal acceleration. In general, due to structural constraints, this acceleration is severely limited. Hence, for practical purposes, we are led to adopt some simple ascending program which is satisfactory for the analysis during the preliminary stage of the design project. The simplifying hypotheses will provide an analytical solution to the problem considered. The analytical solution has the advantage that it displays explicitly the many relationships among the different variables allowing a global analysis. For example, the solution will give the approximate size of the engine, and the weight of the propellant required to launch a certain given pay-load (final weight of the vehicle) into a prescribed final orbit. From these approximate data, with the aid of high speed computers, we may update the numerical results to obtain the exact solution to the problem.

4-3. ASCENDING TRAJECTORY AT CONSTANT FLIGHT PATH ANGLE

The equations of motion derived in section 4-2 cannot be integrated analytically. For a prescribed initial condition, and a specified thrusting program, numerical integration using high speed computers has to be performed in order to obtain the variables describing the dynamical system as functions of the time.

For advanced planning purposes, it is useful to adopt some simplifying assumptions in order to obtain an analytical solution of the ascen ding powered flight. Such a solution will give explicit relationships among the different variables and permit a preliminary selection of the size of the vehicle, its aerodynamic characteristics, the propulsion system required to perform a given mission. With these data we can then use numerical integration to readjust the different characteristic values.

There exists a simple ascending law which can be used to approximate the real powered flight trajectory. Using this program, as a first approximation, we can assume that, after lift-off, the vehicle essentially follows a straight line trajectory having a constant angle of inclination with respect to the local horizontal. In reality, if the flight path angle is constant, the trajectory will be a logarithmic spiral in the plane of the motion, but since we shall assume a flat Earth model for the gravitational field, the trajectory with constant flight path is essentially a straight line.

More specifically, we shall use the following assumptions to simplify the equations in section 4-2:

a/ The powered flight trajectory involves short longitudinal range and a relatively small altitude compared to the radius of the Earth, Hence, from Eq. (4-20)

Therefore, we can use 4> = 0 in the equations in section 4-2. This assumption is usually called the flat Earth assumption.

b/ For the same reason, the acceleration of gravity g can be considered constant for the altitude range considered, c/ We shall neglect the aerodynamic force.

With these assumptions, the dynamical equations, Eqs. (4-2) and (4-3) become, as can be seen from the simplifying force diagram in Fig. 4-4

Since we assume that y = constant, then dy/ dt = 0 . Hence, from Eq. (4-22) we have the relation between the thrust and the weight

In general, for rocket flight the thrust is large as compared to the weight. Hence from this equation we see that the angle ( a - e ) is necessarily small and we can take cos ( e - a) ~ 1 in Eq. (4-21). Then we have the simplified equation m = T - W sin y (4-24)

Using Eq. (4-12) for the thrust we rewrite this equation, using the relation W = mg dV = - c ^ - g sin y dt (4-25)

To integrate this equation for a multi-stage rocket vehicle, we refer to the Fig. 4-3 and assume that all the time intervals At. between the separation of the ith stage and the engine ignition of the (i+ 1 )th stage are zero. Then by integrating Eq. (4-25) starting from the time t. of the separation of the (i - l)th stage, we have during the operation of the ith stage m (t)

l where

V(t) = instantaneous speed at the time t

V\ j = speed at the initial time of burning of the ith stage c^ = effective exhaust velocity of the ith stage m.(t) = instantaneous mass at the time t m. = mass at the initial time of burning of the ith stage t' = t - t. j , time interval from the initial time of burning of the ith stage

Now, consider the operation of one single stage. For example, let us assume that the vehicle is a single-stage rocket. At the burnout time tj of this stage, the change in the speed is m

where c is the effective exhaust velocity of the stage considered, mo = mi 1

vehicle. If v in the speed m = m is the initial mass and m = m (t ) is the final mass of the 0 111 vehicle. If we neglect the gravitational force, we have for the change m

We see that, in this case, AV can be used as a measure of the fuel consumption. This quantity is called the characteristic velocity of the maneuver. From this simple formula, we can see that AV must have a certain upper limit. The exhaust velocity has an upper limit which depends on the propulsion system used. For example, ordinary chemical propulsion systems currently provide exhaust velocities up to 3000 m/ sec, with a theoretical maximum in the neighborhood of 4000 m/ sec. On the other hand, the ratio of the masses mQ/ mj also cannot be made arbitrarily large. Let Am = m^ - m^ be tne mass of the fuel spent. Then we define the fuel ratio m t s = -L (4-29)

It is obvious that f can never approach unity, since any amount of fuel always requires a certain provision of structure for its operation. Therefore, the characteristic velocity for a single stage is limited due to technological constraints. Some optimistic predictions advance a figure in the neighborhood of 9000 m/ sec for its ultimate value.

Equation (4-30) gives the performance of a single stage rocket in the hypothetical situation of gravity-free, vacuum space. If we include the gravitational force, the increase in the speed during a thrusting phase of a stage is given by Eq. (4-27). The term gt^ siny characterizes the losses due to the gravitational force. Because of this component, the performance of a single stage rocket is further limited. Therefore, to obtain high final speed, one must use a multi-stage rocket.

Let t be the total burning time for a rocket vehicle having n stages. By repeated application of Eq. (4-26), we have the final speed at burnout, assuming a zero initial speed.

where (j.. is the ratio of the masses of the ith stage, defined as m il _ mass of the vehicle at burnout of ith stage itage (4-32)

1 mi0 mass of the vehicle at initial time of ith stage

For the range and the altitude at the end of the powered phase, we use the Eqs. (4-6) and (4-7) with § = 0 , and y = constant. We have x t

sin y 0

Using V(t) as given by Eq. (4-26) to evaluate the integral, we have, for the case of constant mass flow |3 , (|3 = - dm/dt) ,

{ n r |J'' ii xB cos y J £ |vi_iTi + c.Ti(l + T-i- logji.)J-IgT siny 1 i = 1 i

sin y

where t. is the burning time of the ith stage. The final altitude is simply

4-4. OPTIMUM STAGING

The final speed of a rocket vehicle, having a prescribed number of stages is given by Eq. (4-31). This expression for V is a function id of the characteristic parameters c. and (j.. of the different stages, of the constant flight path angle y aniJ the total burning time t of the powered phase. By these considerations one may ask the following question:

"Is there an optimum distribution of the masses of different stages such that, for a prescribed burnout speed Vg , the ratio m^ / m of the initial mass at launching to the final mass at the end of the powered phase is a minimum? "

If such a solution exists, it therefore gives the lightest rocket for a prescribed payload (final mass m ) for a prescribed final speed

In solving this problem, we write the ratio of the masses m m m m

where to ease the notation, in this section, we have used the subscripts as follows m. - total mass of the vehicle at the initial burning time of the ith stage. This mass is also referred to as the gross mass of the ith stage.

From Eq. (4-36) we see that we have defined the ratio s. , called the staging ratio, as m.

We notice that s. is the ratio of the gross mass of the (i + l)th stage to the gross mass of the ith stage, the masses are all evaluated at the initial burning time of the corresponding stage. Also it is seen that m.j = m^ is the initial gross mass of the rocket vehicle, while m^j = m is the resulting payload of the operation.

There is a basic difference between the ratio [j.. as defined by Eq. (4-32) and the staging ratio s. as defined by Eq. (4-37). This is illustrated by Fig. 4-5 showing the mass distribution in the ith stage of a rocket vehicle

Fig. 4-5. Distribution of the masses in the ith stage of a rocket vehicle.

Fig. 4-5. Distribution of the masses in the ith stage of a rocket vehicle.

The total mass shown is the gross mass m. of the ith stage. The mass m^ denotes the mass of the fuel used during the operation of the ith stage, while the mass m. denotes the mass of the structural components of the propulsion system used in the operation of the ith stage. This mass is to be discarded leaving the mass m. as the initial mass for the operation of the (i + l)th stage. Hence, the mass of the vehicle at burnout of the ith stage is m = m- m = m. + m (4-38)

Since we have used m. = m.^ to denote the mass of the rocket vehicle at the initial time of the ith stage, the ratio [ji. , as defined by Eq. (4-32) now becomes m - m „ - i if _ , p.. = - = 1

On the other hand, the staging ratio s. , as defined by Eq. (4-37) is s = 1

We define the structural ratio to. for the propulsion system used in the operation of the ith stage as m.

Using this relation in Eq. (4-40) we have

By eliminating (m^/m.) between the Eqs. (4-39) and (4-42), we have the relation

We can now formulate the optimization problem as follows. The final speed V , the climb angle y and the total burning time t are given. That is, we have from Eq. (4-31)

i=l where V^ is therefore prescribed. We write this equation as a constraining relation n f^) = Eci lo8 M-i + V0 = 0 (4-44)

The number of stages n , the different propellant characteristics c. , and the different structural ratios co . are also given. Find a mass distribution u , fi (j. such that the following function

1=1 1=1 i is a maximum. This is equivalent to minimizing the product of 1/ s. .

In solving this problem, we introduce a Lagrange multiplier £ to form the augmented function

The solution to the problem is obtained by solving the system of (n + 1) equations

9m.. i l for the (n+1) unknowns u. , u. . . . , u. and X . Explicitly, we

12 n write the first n equations air i TT (h1--"-) c-

For each of the n equations (4-48), we have c.^.-c.)

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