For a vehicle that flies within the proximity of the earth, the gravitational attraction of all other heavenly bodies may usually be neglected. Let it be assumed that the vehicle is moving in rectilinear equilibrium flight and that all control forces, lateral forces, and moments that tend to turn the vehicle are zero. The trajectory is two-dimensional and is contained in a fixed plane. The vehicle has wings that are inclined to the flight path at an angle of attack a and that give a lift in a direction normal to the flight path. The direction of flight does not coincide with the direction of thrust. Figure 4-4 shows these conditions schematically.

Let 0 be the angle of the flight path with the horizontal and r/r the angle of the direction of thrust with the horizontal. In the direction of the flight path the product of the mass and the acceleration has to equal the sum of all forces, namely the propulsive, aerodynamic, and gravitational forces:

The acceleration perpendicular to the flight path is u(d6/dt)-, for a constant value of u and the instantaneous radius R of the flight path it is u2/R. The equation of motion in a direction normal to the flight velocity is mu(d6/dt) = F sin(^ - 6) + L - mg cos 0 (4-14)

By substituting from Equations 4-10 and 4-11, these two basic equations can be solved for the accelerations as

dt m 2m

dt m 2m

No general solution can be given to these equations, since tp, u, CD, CL,p, 6, or ijr can vary independently with time, mission profile, or altitude. Also, CD and CL are functions of velocity or Mach number. In a more sophisticated analysis other factors may be considered, such as the propellant used for nonpropulsive purposes (e.g., attitude control or flight stability). See Refs. 4-1 to 4-5 for a

background of flight performance in some of the flight regimes. Different flight performance parameters are maximized or optimized for different rocket flight missions or flight regimes, such as Au, range, time-to-target, or altitude. Rocket propulsion systems are usually tailored to fit specific flight missions.

Equations 4-15 and 4-16 are general and can be further simplified for various special applications, as shown in subsequent sections. Results of such iterative calculations of velocity, altitude, or range using the above two basic equations often are adequate for rough design estimates. For actual trajectory analyses, navigation computation, space-flight path determination, or missile-firing tables, this two-dimensional simplified theory does not permit sufficiently accurate results. The perturbation effects, such as those listed in Section 4.6 of this chapter, must then be considered in addition to drag and gravity, and digital computers are necessary to handle the complex relations. An arbitrary division of the trajectory into small elements and a step-by-step or numerical integration to define a trajectory are usually indicated. The more generalized three-body theory includes the gravitational attraction among three masses (for example, the earth, the moon, and the space vehicle) and is considered necessary for many space-flight problems (see Refs. 4-2 and 4-3). When the propellant flow and the thrust are not constant, the form and the solution to the equations above become more complex.

A form of Eqs. 4-15 and 4-16 can also be used to determine the actual thrust or actual specific impulse during actual vehicle flights from accurately observed trajectory data, such as from optical or radar tracking data. The vehicle acceleration (du/dt) is essentially proportional to the net thrust and, by making an assumption or measurement on the propellant flow (which usually varies in a predetermined manner) and an analysis of aerodynamic forces, it is possible to determine the rocket propulsion system's actual thrust under flight conditions.

When integrating Eqs. 4—15 and 4—16 one can obtain actual histories of velocities and distances traveled and thus complete trajectories. The more general case requires six equations; three for translation along each of three perpendicular axes and three for rotation about these axes. The choice of coordinate systems and the reference points can simplify the mathematical solutions (see Refs. 4-2 and 4-4).

For a wingless rocket projectile, a space launch vehicle, or a missile with constant thrust and propellant flow, these equations can be simplified. In Fig. 4-5 the flight direction 6 is the same as the thrust direction and lift forces for a symmetrical, wingless, stably flying vehicle can be assumed to be zero of zero angle of attack. For a two-dimensional trajectory in a single plane (no wind forces) and a stationary earth, the acceleration in the direction of flight is as follows:

du _ c$/tp _ sin0_ CD\pu2A/mo dt~\-St/tp 8Sm l-V/tp

A force vector diagram in Fig. 4-5 shows the net force (by adding thrust, drag and gravity vectors) to be at an angle to the flight path, which will be curved. These types of diagram form the basis for iterative trajectory numerical solutions.

The relationships in this Section 4.3 are for a two-dimensional flight path, one that lies in a single plane. If maneuvers out of that plane are also made (e.g., due to solar attraction, thrust misalignment, or wind) then the flight paths become three-dimensional and another set of equations will be needed to describe these flights. Reference 4—1 describes equations for the motion of rocket projectiles in the atmosphere in three dimensions. It requires energy and forces to push a vehicle out of its flight plane. Trajectories have to be calculated accurately in order to reach the intended flight objective and today almost all are done with the aid of a computer. A good number of computer programs for analyzing flight trajectories exit and are maintained by aerospace companies or Government agencies. Some are two-dimensional, relatively simple, and are used for making preliminary estimates or comparisons of alternative flight paths, alternative vehicle designs, or alternative propulsion schemes. Several use a stationary flat earth, while others use a rotating curved earth. Three-dimensional programs also exit, are used for more accurate flight path analyses, include some or all perturbations, orbit plane changes, or flying at angles of attack. As explained in Ref. 4—3, they are more complex.

If the flight trajectory is vertical (as for a sounding rocket), Eq. 4—17 is the same, except that sin 8 = 1.0, namely du ct;/tp CD\pu2 A/m$

The velocity at the end of burning can be found by integrating between the limits of / = 0 and t = tp when u = u0 and u = up. The first two terms can readily be integrated. The last term is of significance only if the vehicle spends a considerable portion of its time within the atmosphere. It can be integrated graphically or by numerical methods, and its value can be designated as BC£,A/m0 such that

The cutoff velocity or velocity at the end of propellant burning up is then where w0 is the initial velocity, such as may be given by a booster, g is an average gravitational attraction evaluated with respect to time and altitude from Eq. 4-12, and c is a time average of the effective exhaust velocity, which is a function of altitude.

There are always a number of trade-offs in selecting the best trajectory for a rocket projectile. For example, there is a trade-off between burning time, drag, payload, maximum velocity, and maximum altitude (or range). Reference 4-6 describes the trade-offs between payload, maximum altitude, and flight stability for a sounding rocket.

If aerodynamic forces outside the earth's atmosphere are neglected (operate in a vacuum) and no booster or means for attaining an initial velocity (w0 = 0) is assumed, the velocity at the end of the burning reached in a vertically ascending trajectory will be

The first term is usually the largest and is identical to Eq. 4-6. It is directly proportional to the effective rocket exhaust velocity and is very sensitive to changes in the mass ratio. The second term is always negative during ascent, but its magnitude is small if the burning time tp is short or if the flight takes place in high orbits or in space where g is comparatively small.

For a flight that is not following a vertical path, the gravity loss is a function of the angle between the flight direction and the local horizontal; more specifically, the gravity loss is the integral of g sin 0 dt, as shown by Eq. 4-15.

Kp = -cln(l -£)-gtp = —c In 1VR — gtp = cln(l/]V«)-g/,

For the simplified two-dimensional case the net acceleration a for vertical takeoff at sea level is where a/g0 is the initial takeoff acceleration in multiples of the sea level gravitational acceleration g0, and F0/w0 is the thrust-to-weight ratio at takeoff. For large surface-launched vehicles, this initial-thrust-to-initial-weight ratio has values between 1.2 and 2.2; for small missiles (air-to-air, air-to-surface, and surface-to-air types) this ratio is usually larger, sometimes even as high as 50 or 100. The final or terminal acceleration aj of a vehicle in vertical ascent usually occurs just before the rocket engine is shut off and before the propellant is completely consumed.

In a gravity-free environment this equation becomes ay/go = Ff/wf. In rockets with constant propellant flow the final acceleration is usually also the maximum acceleration, because the vehicle mass to be accelerated has its minimum value just before propellant exhaustion, and for ascending rockets the thrust usually increases with altitude. If this terminal acceleration is too large (and causes overstressing of the structure, thus necessitating an increase in structure mass), then the thrust can be designed to a lower value for the last portion of the burning period.

Example 4-1. A simple single-stage rocket for a rescue flare has the following characteristics and its flight path nomenclature is shown in the sketch.

Launch weight 4.0 lbf

Useful propellant mass 0.4 lbm

Effective specific impulse 120 sec

Launch angle (relative to horizontal) 80°

Burn time (with constant thrust) 1.0 sec a = (F0g0/w0) - g0 a/go = (F0/w0) - 1

Drag is to be neglected, since the flight velocities are low. Assume no wind. Assume the local acceleration of gravity to be equal to the sea level go and invariant throughout the flight.

Solve for the initial and final acceleration of powered flight, the maximum trajectory height, the time to reach maximum height, the range or distance to impact, and the angle at propulsion cutoff and at impact.

SOLUTION. Divide the flight path into three portions: the powered flight for 1 sec, the unpowered ascent after cutoff, and the free-fall descent. The thrust is obtained from Eq. 2-5:

The initial accelerations along the x and y directions are, from Eq. 4.22,

(a0)y = g0[(Fsin6>/w) - 1] = 32.2[(48/4)sin80° - 1] = 348 ft/sec2 (flo)x = go(F/w)cos9 = 32.2(48/4) cos 80° = 67.1 ft/sec2

The initial acceleration in the flight direction is a0 = y(a0)2x + (ao)v = 354-4 ft/ sec2

The direction of thrust and the flight path are the same. The vertical and horizontal components of the velocity up at the end of powered flight is obtained from Eq. 4-20. The vehicle mass has been diminished by the propellant that has been consumed.

(up)y = cln(wQ/wf)smO - g0tp = 32.2 x 1201n(4/3.6)0.984 - 32.2 = 375 ft/sec (up)x = cln(w0/wy)cos<9 = 32.2 x 1201n(4/3.6)0.1736 = 70.7 ft/sec

The trajectory angle with the horizontal at rocket cutoff for a dragless flight is tan-1 (375/70.7) = 79.3°

Final acceleration is af = Fg^/w = 48 x 32.2/3.6 = 429 ft/sec2. For the short duration of the powered flight the coordinates at propulsion burnout yp and xp can be calculated approximately by using an average velocity (50% of maximum) for the powered flight.

yp=Wp)ytp= ^x 375 x 1.0 = 187.5 ft xp = \(up)xtp = \ x 70.7 x 1.0 = 35.3 ft

The unpowered part of the trajectory has a zero vertical velocity at its zenith. The initial velocities, the x and y values for this parabolic trajectory segment, are those of propulsion termination (F = 0, u = up, x = xp, y = yp); at the zenith (uy)z = 0.

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