BASIC RELATIONS OF MOTION

FLIGHT PERFORMANCE

4.3. BASIC RELATIONS OF MOTION

For any vehicle that flies within the Earth’s proximity, the gravitational attractions from all other heavenly bodies are neglected. Assume next 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 resulting trajectory is two-dimensional and is contained in a fixed plane. Assume further that the vehicle has wings inclined to the flight path at an angle of attack𝛼 providing lift in a direction normal to the flight path. The direction of flight need not coincide with the direction of thrust as shown schematically in Figure 4–5.

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4.3. BASIC RELATIONS OF MOTION 107

FIGURE 4 – 5. Two-dimensional free-body force diagram for flying vehicle with wings and fins.

Let𝜃 be the angle of the flight path with the horizontal and 𝜓 the angle of the direc-tion of thrust with the horizontal. Along the flight path direcdirec-tion, the product of the mass and the acceleration has to equal the vector sum of the propulsive, aerodynamic, and gravitational forces:

m(du∕dt) = F cos(𝜓 − 𝜃) − D − mg sin𝜃 (4–13) The acceleration perpendicular to the flight path is u(d𝜃∕dt); for a given value of u and at the instantaneous Earth radius R (from the Earth’s center) of the flight path, it becomes u²/R. The equation of motion in a direction normal to the flight velocity is

mu(d𝜃∕dt) = F sin(𝜓 − 𝜃) + L − mg cos𝜃 (4–14) By substituting from Eqs. 4–10 and 4–11, these two basic equations can be solved for the accelerations as follows:

du dt = F

mcos(𝜓 − 𝜃) −C_D

2m𝜌u²A − g sin𝜃 (4–15) ud𝜃

dt = F

msin(𝜓 − 𝜃) +C_L

2m𝜌u²A − g cos𝜃 (4–16) No general solution can be given to these equations, since t_p, u, CD, CL, 𝜌, 𝜃, and/or 𝜓 may vary independently with time, mission profile, and/or altitude. Also, C_Dand C_L are functions of velocity or Mach number. In more sophisticated analyses, other fac-tors may also be considered, such as any propellant used for nonpropulsive purposes (e.g., attitude control or flight stability). See Refs. 4–1, 4–8, 4–11, and 4–12 for background material on flight performance in some of these flight regimes. Because rocket propulsion systems are usually tailored to fit specific flight missions, different flight performance parameters are maximized (or optimized) for different rocket flight missions or flight regimes such as Δu, range, orbit height and shape, time-to-target, or altitude.

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For actual trajectory analyses, navigation computation, space flight path determi-nation, or missile-firing tables, the above two-dimensional simplified theory is not sufficiently accurate; perturbation effects, such as those listed in Section 4.4, must then be considered in addition to drag and gravity, and computer modelling is neces-sary to handle such complex relations. Suitable divisions of the trajectory into small elements and step-by-step or numerical integrations to define a trajectory are usu-ally indicated. More generusu-ally, three-body theories include the gravitational attraction among three masses (for example, the Earth, the moon, and the space vehicle) and this is considered necessary in many space flight problems (see Refs. 4–2 to 4–5). The form and the solution to the given equations become further complicated when pro-pellant flow and thrust are not constant and when the flight path is three dimensional.

For each mission or flight one can obtain actual histories of velocities and distances traveled and thus complete trajectories when integrating Eqs. 4–15 and 4–16. More general cases require six equations: three for translation along each of three perpen-dicular axes and three for rotation about these axes. The choice of coordinate systems of reference points can simplify the mathematical solutions (see Refs. 4–3 and 4–5) but there are always a number of trade-offs in selecting the best trajectory for the flight of a rocket vehicle. For example, for a fixed thrust the trade-off is between burn time, drag, payload, maximum velocity, and maximum altitude (or range). Reference 4–2 describes the trade-offs between payload, maximum altitude, and flight stability for sounding rockets.

Equations 4–15 and 4–16 may be further simplified for various special applica-tions, as shown below; results of such calculations for velocity, altitude, or range using the above two basic equations are often adequate for rough design estimates.

A form of these equations is also useful for determining the actual thrust or actual specific impulse during vehicle flight from accurately observed trajectory data, such as from optical or radar tracking data. Vehicle acceleration (du/dt) is essentially pro-portional to net thrust and, by making assumptions or measurements of propellant flow (which usually varies in a predetermined manner) and from analyses of aero-dynamic forces, it is possible to determine a rocket propulsion system’s actual thrust under flight conditions.

Equations 4–15 and 4–16 simplify for wingless rocket projectiles, space launch vehicles, or missiles with constant thrust and propellant flow. In Fig. 4–6 the flight direction 𝜃 is the same as the thrust direction and any lift forces for a symmet-rical, wingless, stably flying vehicle are neglected at 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 given below, where t_pis the operating or burn time of the propellant and𝜁 is the propellant mass fraction:

du dt =

c𝜁∕tp

1 −𝜁t∕t_p − g sin𝜃 − C_D¹

2𝜌u²A∕m₀

1 −𝜁t∕t_p (4–17)

The force vector diagram in Fig. 4–6 also shows the net force (the addition of thrust, drag, and gravity vectors) to be at an angle to the flight path, which will therefore

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4.3. BASIC RELATIONS OF MOTION 109

FIGURE 4 – 6. Simplified free-body force diagram for a vehicle without wings or fins. The force vector diagram shows the net force on the vehicle. All forces act through the vehicle’s center of gravity.

be curved. These types of diagram form the basis for iterative trajectory numerical solutions.

All further relationships in this section correspond to two-dimensional flight paths, ones that lie in a single plane. If maneuvers out of that plane take place (e.g., due to solar attraction, thrust misalignment, or wind), then another set of equations will be needed—it requires both force and energy to push a vehicle out of its flight plane;

Reference 4–1 describes equations for the motion of rocket projectiles in the atmo-sphere in three dimensions. Trajectories must be calculated accurately in order to reach any intended flight objective and today these are done exclusively with the aid of computers. Several computer programs for analyzing flight trajectories exist (which are maintained by aerospace companies and/or government agencies). Some are two-dimensional, relatively simple, and 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 used for more accurate flight path analy-ses may include some or all significant perturbations, orbit plane changes, or flying at angles of attack. Reference 4–4 explains their nature and complexity.

When the flight trajectory is vertical (as for a sounding rocket), then Eq. 4–17 becomes

du

dt = c𝜁∕tp

1 −𝜁t∕t_p − g − C_D¹

2𝜌u²A∕m₀

1 −𝜁t∕t_p (4–18)

The velocity at the end of burning can be found by integrating between the limits of t = 0 and t = t_p where u = u₀ and then u = u_p. The first two terms can readily

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be integrated. The last term is of significance only if the vehicle spends a consider-able portion of time within the lower atmosphere. It may be integrated graphically or numerically, and its value can be designated by the term BC_DA∕m₀, where

B≡ ∫0^tp 1 2𝜌u² 1 −𝜁t∕t_pdt

The cutoff velocity or velocity at the end of propellant burning u_pthen becomes u_p = −c ln(1 −𝜁) − gt_p− BC_DA

m₀ + u₀ (4–19)

where u₀ is an 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 also depends on altitude.

For nonvertical flight paths, the gravity loss becomes a function of the angle between the flight direction and the local horizontal; more specifically, the gravity loss is then given by the integral∫ (g sin𝜃)dt.

When aerodynamic forces within the atmosphere may be neglected (or for vacuum operation) and when no booster or means for attaining an initial velocity (u₀= 0) are present, the velocity at the end of the burning reached with a vertically ascending trajectory becomes simply

u_p= −c ln(1 −𝜁) − gtp

= −c ln MR − gt_p = c ln(1∕MR) − gt_p (4–20) The first term on the right side is usually the largest and is identical to Eq. 4–6.

It is directly proportional to the effective rocket exhaust velocity and very sensitive to changes in the mass ratio. The second term is related to the Earth’s gravity and is always negative during ascent, but its magnitude can be small when the burn time t_p is short or when flight is taking place at high orbits or in space where g is comparatively small.

For the simplified case given in Eq. 4–20 the net initial acceleration a₀for vertical takeoff at sea level is

a₀= (F₀g₀∕w₀) − g₀ (4–21)

a₀∕g₀= (F₀∕w₀) − 1 (4–22)

where a₀/g₀ is the initial takeoff acceleration in multiples of the sea-level gravita-tional 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 typically 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 a_f of a vehicle in vertical Earth ascent usually occurs just before the rocket engine is shut off and/or before the useable propellant

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4.3. BASIC RELATIONS OF MOTION 111 is completely consumed. If drag is neglected, then the final acceleration a_facting on the final mass m_fbecomes

a_f = (F_f∕m_f) − g (4–23)

This equation applies when the powered flight path traverses a substantial range of altitude with g decreasing according to Eq. 4–12. For rocket propulsion systems with constant propellant flow, the final acceleration is usually also the maximum accelera-tion because the vehicle mass being accelerated is a minimum just before propellant termination, and for ascending rockets thrust usually increases with altitude. When this terminal acceleration is found to be too large (e.g., causes overstressing of the structure, thus necessitating an increase in structure mass), then the thrust must be reduced by redesign to a lower value for the last portion of the burning period. In manned flights, maximum accelerations are limited to the maximum g-loading that can be withstood by the crew.

Example 4–1. A simple single-stage rocket for a rescue flare has the following characteris-tics. Its flight path nomenclature is shown in the accompanying sketch.

Launch weight (w₀)

Useful propellant weight (w_p) Effective specific impulse

Launch angle𝜃 (relative to horizontal) Burn time t_p(with constant thrust) The heavy line in the ascending trajectory designates the powered portion of the flight.

4.0 lbf 0.4 lbf 120 sec 80∘

1.0 sec

Drag may be neglected since flight velocities are low. Assume that the acceleration of grav-ity is unchanged from its sea-level value g₀, which then makes the propellant mass numerically equal to the propellant weight in the EE system, or 0.4 lbm. Also assume that start and stop transients are short and can be neglected.

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

SOLUTION. We will 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:

F = I_sw_p∕t_p= 120 × 0.4∕1.0 = 48 lbf

The initial accelerations along the x-horizontal and y-vertical directions are, from Eq. 4–22, (a₀)_y= g₀[(F sin𝜃∕w) − 1] = 32.17[(48∕4.0)sin 80∘ − 1] = 348 ft∕sec²

(a₀)_x= g₀(F cos𝜃∕w) = 32.17(48∕4.0)cos 80∘ = 67.03 ft∕sec²

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At thrust termination the final flight acceleration becomes a₀=

√

(a₀)²_x+ (a₀)²_y= 354.4 ft∕sec²

The vertical and horizontal components of the velocity u_pat the end of powered flight can be obtained from Eq. 4–20. Note that the vehicle mass has been diminished by the propellant that has been consumed:

(u_p)_y= c ln(m₀∕m_f) sin𝜃 − g₀t_p= 32.17 × 120 × ln(4∕3.6) × 0.984 − 32.17 × 1.0

= 368 m∕sec

(u_p)_x= c ln(m₀∕m_f) cos𝜃 = 32.17 × 120 × ln(4∕3.6) × 0.1736 = 70.6 m∕sec

The effective exhaust velocity c = I_Sg₀(Eq. 2–6). The trajectory angle with the horizontal at rocket cutoff in dragless flight is

tan⁻¹(368∕70.6) = 79.1∘

The final acceleration is found, using Eq. 4–22 with the final mass, as a_f = 400 m∕sec². For the powered flight, the coordinates at propulsion burnout y_p and x_pcan be calculated from the time integration of their respective velocities, using Eq. 2–6. The results are

y_p= ct_p[1 − ln(m₀∕m_f)∕(m₀∕m_f − 1)]sin𝜃 − 1∕2g0t²_p

= 32.17 × 120[1 − ln(4∕3.6)∕(4∕3.6 − 1)] × 0.984 −¹₂ × 32.17 × (1.0)²= 181 ft x_p= ct_p[1 − ln(m₀∕m_f)∕(m₀∕m_f − 1)]cos𝜃

= 32.17 × 120[1 − ln(4∕3.6)∕(4∕3.6 − 1)] × 0.173 = 34.7 ft

The unpowered part of the trajectory reaches zero vertical velocity at its zenith. The height gained in unpowered free flight may be obtained by equating the vertical kinetic energy at power cutoff to its equivalent potential energy, g₀(y_z− y_p) = ¹

2(u_p)²_y so that

(y_z− y_p) = ¹

2(u_p)²_y∕g_o= ¹

2(368)²∕32.17 = 2105 ft

The maximum height or zenith location thus becomes y_z= 2105 + 181 = 2259 ft. What remains now is to solve the free-flight portion of vertical descent. The time for descent from the zenith is t_z=√

2y_z∕ g₀= 11.85 sec and the final vertical or impact vertical velocity (uf)y − g₀t_z= 381 ft∕sec.

The total horizontal range to the zenith is the sum of the powered and free-flight contri-butions. During free flight, the horizontal velocity remains unchanged at 70.6 ft/sec because there are no accelerations (i.e., no drag, wind, or gravity component). We now need to find the free-flight time from burnout to the zenith, which is t = (u_p)y∕g₀= 11.4 sec. The total free-flight time becomes t_ff= 11.4 + 11.85 = 23.25 sec.

Now, the horizontal or total range becomes Δx = 34.7 + 70.6 × 23.25 = 1676ft.

The impact angle would be around 79∘. If drag had been included, solving this problem would have required information on the drag coefficient (C_D) and a numerical solution using Eq. 4–18. All resulting velocities and distances would turn out somewhat lower in value. A set of flight trajectories for sounding rockets is given in Ref. 4–3.

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