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center of gravity of the body and directed downward toward the center of the earth. Opposed to weight is the lift, the highly desirable force produced by the moving airfoil which supports the body and which Is directed perpendicular to the direction of drag.

Acceleration.— In level flight at a constant speed, thrust is exactly balanced by drag, and the lifting force exactly cancels the weight of the body. If any one of these basic forces is changed, the result is acceleration. Acceleration in flight is a change, either in spe'ed or in the direction of motion. It occurs in two ways:

1. The aircraft accelerates as it increases or decreases speed along the line of flight. This kind of acceleration takes place in missile flight during launching and also upon impact with the target.

2. The aircraft is accelerated if it changes the direction in which it is moving, for example in turns, dives, pullouts, and as a result of gusts of wind. During acceleration of this sort while in high-speed flight, the aircraft is subjected to large forces which tend to keep it flying along the line of its previous flight.

The standard unit of acceleration used in aviation is the gravity, abbreviated by the letter “g” A body falling freely in space is pulled downward by a force equal to its weight with the result that it accelerates at a constant rate of about 32 feet per

g-value of the acceleration. The number of g’s which the missile components can withstand is one of the factors which determine the maximum turning rate and the type of launcher suitable for the weapon, since the delicate instruments of the control and guidance systems may be damaged if subjected to accelerations in excess of a certain value.

Production of lift by airfoils.—Lift, the force on which flight depends, is produced by means of pressure differences. One con­ dition and only one is necessary for the lifting action of a wing to occur: the air pressure on the upper surface must be less than the pressure on the underside. The wing, then, is simply a device for creating pressure differences when in motion. The amount of lifting force provided is dependent to a large extent on the shape of the airfoil, or wing. Additional factors which determine lift are the wing area, the angle at which the wing surface is inclined to the airstream, and the density and relative speed of the air passing around it.

The foremost edge of the wing is called the leading edge, and that at the rear is called the trailing edge. A straight line drawn

between the leading and the trailing edges is called the chord; and the maximum distance measured from one wingtip to the

exerted on the various areas of the airfoil. The sum, or resultant,

of lift is directed perpendicular to the direction of the relative wind.

Lift may be considered as resulting from two general causes: one from dynamic pressure, or the pressure of air in motion; and the other from differences in the static pressure of the atmosphere. The dynamic pressure of the relative wind against

by a difference of the static pressures on the upper and lower surfaces. The principal effect is the result of air flowing over the upper wing surfaces with increased velocity and with an accom­

In the form in which it applies to airfoils, ^Bernoulli's principle is as follows: air pressure decreases when air velocity increases. Bernoulli principle says that in the narrow section the pressure exerted laterally against the walls of the tube is less than the lateral pressures on the walls upstream and downstream from this section, and that the difference in pressure results from the increased velocity ok flow.

The same relation exists in the streams of air flowing over the upper and lower surfaces of the wing. During flight, part of the approaching air is forced to flow over the longer path of the curved upper area, and its velocity js thereby increased compared with that of the air passing over the shorter path along the underside. The difference in the flow rate causes a difference in the lateral pressures on the two surfaces, and a net force is then present which is directed upward. This force is the greater part of the total lift supporting the weight of the aircraft, the remain­ der being supplied by the effect of dynamic pressure.

.The lift resulting from dynamic pressure is concentrated near the leading-edge of the wing in normal flight. The contour of the wing and the angle of attack at which the wing is inclined are such that the airstream is split at a point just under the leading edge. Here the air is forcedto change in direction abruptly, and a stagnation point, or high-pressure area is formed. It is important that the design of the wing permits the stagnation

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point to form on the underside of the leading edge rather than

wing. With this in mind, it is easy to understand what takes place when the aircraft goes into, a stall. Up to a point, as the angle of attack is increased, the lift also increases, since the high value of angle causes the air flowing over the upper part of the wing to travel a greater distance. Hence it increases in speed and the pressure difference which produces lift is thereby increased. But if the angle of attack is made too great, lift is destroyed by the formation of turbulence on the upper airfoil surfaces.

At moderately high angles of attack, the flowing air can follow the initial turn of the leading edge but it cannot follow the wing contour completely; and the stream separates from the surface near the trailing edge. Further increase of the angle causes the point at which the separation occurs to move forward. At some value of attack angle, the separation point is placed so near the leading edge that the upper airflow is disrupted, flight characteristics disappear, and the wing is in a stall.

18. Supersonic Flight

Once out of the transonic speed region, the upper limit of which is about Mach 1 .2, the airflow over any area of the aircraft is supersonic in velocity. In this condition, the undesirable effects of mixed supersonic and subsonic flow largely disappear, and the passage of air over the airframe surfaces is without turbulence. The variations in pressure which occur are of two principal kinds: compression waves of the oblique shock wave type, and expansion waves.

Oblique shock waves are those in which the airstream changes in direction upon passing through the transition marked by the

wavefront.

Expansion waves differ from normal and oblique shock waves

in two respects:

1 . The airstream passing through an expansion wave increases in velocity. It undergoes a corresponding decrease in temperature, density and pressure.

2. The increase in velocity of the airstream passing^through the expansion wave is gradual rather than sudden.

The thin symmetrical wings used in supersonic flight deserve further explanation because the sharp leading edges employed at these speeds do not produce the same deviation of airflow as the round-nose counterpart.

The thin wing provides lift by means of pressure differences depending on oblique shock waves and expansion waves. The

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oncoming airstream is deflected by the sharp edge, and then assumes a direction parallel to the wing. On 'the upper wing surface, the air is speeded up by passing through a series of expansion waves with the result that a low-piessure area is formed on the top of the wing, much as in subsonic flow.

Beneath the wing, the force of the airstream (the dynamic pressure), together with the changes occurring in passing through an oblique shock wave, results in the formation of a high-pressure area. As in subsonic flow, the difference in pressures on the upper and lower surfaces of the wing results in an upward lifting force.

cross section and have a small thickness ratio — the ratio of the maximum thickness to the chord length. The double wedge has the least drag for a given thickness ratio, but in certain applica­ tions it is inferior because it lacks the necessary strength. The

manufacture and good overall .performance characteristics make this airfoil the best of presently-known configurations. The bicpnvex has one and one-third greater drag than a double wedge of the same thickness ratio. It is the strongest of the three types shown, but it is difficult to manufacture.

Travel in the transonic and supersonic regions is accompanied by shock waves. With the conventional wing plan, which presents its leading edge perpendicular to the direction of motion, shock waves occur at lower speeds than in other planforms are used. The presence of these shock waves on an airfoil greatly increases the drag and subjects the airfoil to extreme stresses. To reduce the effect of these undesirable features, airfoils for transonic and supersonic flight are built in the shape of an arrow or the Greek letter “delta” (A) and are swept back or forward.

Airfoils are mounted on the airframe in several arrangements. The conventional and cruciform are the most popular tail ar­ rangements; and the high wing and cruciform wings are used for most air-launched missiles. Both the inline and interdigital cruciform arrangements are widely used, especially for superso­ nic missiles.

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a control surface which is deflected so that the force of the airstream turns the missile. In the other method the entire airfoil is deflected. This type requires less movement to produce the necessary turning force, but as a result a very accurate power unit is required to control its motion. Because the airfoils required by subsonic missiles are very large in comparison with

control of most low-speed missiles. In some cases, the movable

Control surfaces are placed on the missile at several locations to provide different types of steering. In the conventional aircraft arrangement movable sections of the tail airfoils control pitch and yaw, and control surfaces on the wings control roll. Move­ ment'of the rudder causes the missile to turn about its yaw axis; the elevators are moved together to make the missile pitch; and the ailerons are moved in opposite directions to make it roll. In the cruciform arrangements pitch is controlled by moving the horizontal surfaces together; yaw is controlled by moving the vertical surfaces together; and roll is accomplished by deflecting either the pitch or yaw surfaces in opposite directions. If the forward set of airfoils is fixed, and steering is accomplished by the tail surfaces, the missile is said to be “tail” controlled. Another type is “canard” control in which the tail fins are fixed and control is provided by the forward surfaces. Other arrange­ ments may also be used.

Airfoil control works efficiently while the missile is in the atmosphere. However, it requires a missile velocity that will create enough air pressure on the surfaces to cause the missile to turn. When the missile moves very slowly or reaches highly rarefied atmosphere, the forces which the control surfaces develop are too low to change the path of the missile. When this happens, it is necessary to use some form of jet steering, such as jet vanes or side jets.

20. Thrust Chamber

The thrust chamber consists of a combustion chamber and an exhaust nozzle. It also has an injector plate. The design-of the thrust chamber is governed by the propellants used, the thrust required, the permissible pressure within the chamber, the altitudes at which the thrust chamber must operate, the combus­ tion temperature, and the method of cooling. After the dimensions of the thrust chamber are determined the throat area, expansion

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The thrust vector control is achieved by gimballing or swivelling the thrust chamber itself as a whole, or by deflecting the jet by jet vanes or paddles, or by swivelling-the nozzle.-

The control system of the ballistic missile rocket engine has to ensure that the engine can be started and shut down at the correct times. It has also to ensure that the thrust is maintained at a predetermined level and that propellants are fed to the combustion chamber at the required pressures and at the correct mixture ratios. The control system must sense any malfunctions and incorrect operations of the start and stop sequences and must shut down the engine if abnormal and dangerous conditions develop. After the shut down of the engine, the control system must arrange for the venting of unused propellants.

The gearbox is used to connect the turbine shaft to the propel­ lant pumps. On each pump is a helical impeller which reduces cavitation,» and a centrifugal pump which gives the high-pressure output. From the pumps the' propellants pass through main control valves to the thrust chamber. They are injected through injectors into the combustion chamber where they burn. Expan­ sion to supersonic velocity takes place through the convergentdivergent expansion nozzle. The thrust chamber is cooled by the passage of fuel through the many tubes which make up its walls.

Rocket thrust chambers used in ballistic missiles have been

tubes in the configuration, of the combustion chamber and expan­ sion nozzle, and brazing or welding them together to form the shaped thrust chamber. Originally steel bands were welded around them to give the necessary hoop strength. A considerable weight reduction was later made in thrust chambers by eliminatJng the steel bands and using untwisted glass filament tape wound around the cylinder and oven cured.

21. The Structure of a Ballistic Missile

The structure of a ballistic missile is an important subsystem. Its purpose is to maintain the correct spatial relationship between the components of the vehicle. In addition the structure protects these components and the other subsystems from exposure to the external conditions. The structure also transmits the thrust from the rocket engine at the rear of the vehicle to the payload at the front of the vehicle and it serves as a container for the propel­ lants.

The structure must be compatible with all other components and subsystems of the vehicle.

The loads which the structure has to carry determine its characteristics. These loads are governed by the type of trajectory

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which the vehicle must follow to inject its payload into a path that will carry it to the distant, target. The loads on the missile structure result from axial acceleration, aerodynamic forces, winds, heating, internal pressure, dynamic effects of launching, staging, propellant sloshing, and control forces.

control system, produce bending loads on the structure of the vehicle. These loads are greatest at the time of maximum dynamic pressure on the structure.

As the vehicle is accelerated rapidly through the atmosphere, the aerodynamic forces also produce heating effects. Temperature differences then lead to thermal loads on the structure. Fortuna­ tely, these are not serious on the upward flight, but they are very important in the design of the re-entry body.

The propellant tanks of a ballistic vehicle are usually pressu­ rized above the pressure of the-surrounding atmosphere at sea level. Internal pressure is used to increase the buckling strength of the structure and to allow lightweight construction. But the use of internal pressure also carries the penalty that it produces a load proportional to the cross-sectional area of the structure and the value of the internal pressure.

Loads produced by propellant sloshing, by the launching impulse and by staging, by the actuation of the control system and by engine vibration, are more rapid in their application than the load previously described.

The ballistic missile is usually designed around two typical structural elements, pressure vessels which house the propellants and certain instrumentation and are designed by reference to their tensile strength, and columns which are subjected to-axial and buckling load that are proportional to the elastic modulus and the moment of inertia and inversely proportional to the square of the length. The weight is accordingly proportional to the square root of the actual buckling load so that the lightest materials give the most efficient columns.

The tanks of the missile are designed by reference to tensile loads, and the weight per unit volume is independent of the size but dependent upon the materials used and the internal pressure required. High strength/weight materials are needed for this type of structure.

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body structure during flight. Later the analysis is extended to include the"bending modes of the structure. The dynamic loads

propellants. When the dynamic considerations are made, we have to regard the structure in terms of its natural oscillation frequ­ encies and the shapes of the corresponding monies. These resultmainly from two motions, bending and longitudinal. Calcula­ tions are made by regarding the missile as being a non-uniform beam.

The propellant within the tanks affects the picture because of its liquid form which allows sloshing. Because the propellants are the main mass of the complete vehicle, their motion plays an important part in applying dynamic forces to the actual structure. The modes of fluid vibration within the tank have to be investi­ gated, and baffling devices must be installed to minimize slosh­ ing. The motion of liquids in mass, even if only small amplitudes are involved, can lead to great pressure forces. These must be avoided in the tanks of large ballistic missiles.

The reduction in dead weight of the modern ballistic missile has been .obtained by the use of thin shell construction for th£ propellant tanks. Appreciable strength can be given to these thin shells by subjecting them to stretching or tensional loads by internally pressurizing the tanks. This type of construction is used in the Atlas missile.

22. Motion of Ballistic Missiles

The ballistic vehicle, even though it may be a large multistage rocket vehicle, has negligible mass compared with the mass of the Earth itself. Because the conditions are those of free flight, the total energy of the ballistic missile remains constant at every point in its path through space when it is moving beyond the appreciable atmosphere.

A body which is moving in space and is acted upon by a central force field, such as that of gravity, possesses a *total energy — known as the total specific mechanical energy — which represents the ability of the body to do work by virtue of its velocity and its position. The crash of a speeding automobile or the falling of a heavy weight give analogies, in destructive work, of these two kinds of energy. The total energy is also equal to the amount of work that would have to be done to get the body to its position and give it its observed velocity. The energy of a body in space is thus made up of two parts, the kinetic energy which it has solely because of its velocity, and the potential

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energy which the body possesses by virtue of its position in the force field.

The work which must be done to bring the body from rest to

the gravitational attraction of the Earth. It is similar in concept to the idea of potential in electrostatics. Note that if the vehicle is moving solely under the influence of the gravity fieldJhere is^ at all times, only one force acting on it, and this force is the centrally-directed force of gravity. Since there is never any force

momentum remains constant at all points in the path through space.

From the military point of view the ballistic missile has the sole objective of carrying a payload from one point to another on the surface of the Earth, this payload being a warhead of some kind. The path followed by the ballistic vehicle can be approxi­ mated by considering two bodies only, the Earth and the payload, the latter being negligible in mass compared with the former. The Earth can be assumed as being fixed in space, and the motion of the payload can then be described in terms of a system of coordinates centred at the centre of the Earth. The entire trajec­ tory from launch to impact takes place in a plane.which is defined by the velocity vector and the point of origin. In the idealized two-body~problem there are no forces to cause the payload to

-move out of this plane. In actual fact there are wind shears which can move the missile from this plane during the powered trajectory and the re-entry trajectory.

A mathematical treatment of the two-body problem shows that if the total specific mechanical energy of the payload, namely the algebraic sum of the potential energy and the kinetic energy, is less than zero, that is, if it is negative, the trajectory is a part of an ellipse. If the total energy is zero the trajectory is a parabola, while if the total energy is positive the trajectory is a hyperbola. The total specific mechanical energy assumes poten­ tial energy as negative and kinetic energy as positive in sign. The specific potential energy is zero when the distance is infi­ nite, and it is a maximum negative value (infinite) when the two bodies are in contact.

For the elliptical trajectory the specific energy is always less than zero, which means that the kinetic energy must be less than the. potential energy of the body at all points of the trajectory. The body is unable, therefore, to exchange kinetic energy for potential energy at infinity and it is accordingly bound in an

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left even though its potential energy is zero. The path is then hyperbolic when referred to the Earth. Elliptical paths are used for ballistic missiles and Earth satellites, hyperbolic paths are used for space probe orbits.

Ellipses; parabolas, and hyperbolas are conic sections. The path of the. ballistic vehicle is thus part of a conic section which can have an eccentricity between zero and less than one. If the eccentricity were zero the path would be a circle about the Earth’s centre. If it were less than unity but not zero, the path would be an ellipse. The path becomes parabolic for an eccentri­ city equal to unity, and hyperbolic for eccentricities greater than unity. Both the circular path and the parabolic path' represent critical conditions. The slightest deviation from the exact condi­ tions produces either ellipses or hyperbolas. For. the circular p’ath the kinetic and potential energies remain constant. For the parabolic path the kinetic energy is always equal to the potential energy.

to be discussed in viewing the ballistic missile’s trajectory. They correspond to the three sections of the trajectory, namely, the range achieved during powered flight, the range achieved in free flight, and the re-entry range.

The free-flight trajectory of a ballistic, missile, disregarding the oblateness and the rotation of the Earth, can be regarded as the path traced out by a body moving under the influence of the

vector is directed correctly in azimuth the target will also be in this same plane.

When the launch point and the target of the ballistic missile have been established the trajectory curve still cannot be deter­ mined until some other parametersare also defined. Assuming that the range and the length of the radius vector are known at the burnout point, it is also necessary to know the magnitude and the direction of the payload’s velocity at this point that are required to carry the payload along the elliptical path to the target.. We would, of course, like to know if there is a direction that will minimize the magnitude of the velocity for a given range. There are, in fact., an infinite number of elliptical trajecto­ ries which could, be used to carry the payload from the burnout point to the re-entry point, but until other limitations are

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