aerospace World

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Jet propulsion can be traced back to the 1st century B.C. when an Egyptian,

Hero, is credited with inventing a toy that used jets of steam to spin a sphere.

Sixteen centuries later, Leonardo da Vinci sketched a device that used a flux

of hot gas to do mechanical work. By the 17th century, inventors were beginning

to develop simple turbine systems to operate machinery.

The development of a turbine engine for aircraft began independently in

Germany and Britain in the 1930s. In Germany, Hans von Ohain designed

the engine that powered the first jet flight in 1939. Germany deployed the

jet-powered Messerschmitt 262 late in World War II.

In Britain, Frank Whittle obtained a patent for a turbine engine in 1930. An

aircraft powered by an engine he designed first flew in 1941. The first British

jet fighter, the Gloster Meteor, also flew late in World War II.

S A F E T Y P R O P E R T I E S

Jet fuel can be hazardous if not handled properly. First, and foremost, it is

easy to ignite and it burns rapidly. Second, exposure to jet fuel liquid or

vapor should be limited. Anyone planning to handle jet fuel should obtain

and read the Material Safety Data Sheet (MSDS) issued by the supplier.

Liquid doesn’t burn; only vapor burns. And vapor doesn’t always burn – the

mixture of vapor and air must be within the flammable3 range. Mixtures with

insufficient vapor (below the lower flammability limit) or too much vapor

(above the upper flammability limit) will not burn. For kerosene-type jet fuel,

the lower and upper flammability limits4 are 0.6 volume percent vapor in air

and 4.7 volume percent vapor in air, respectively. For wide-cut jet fuel, the

lower and upper flammability limits are 1.3 volume percent vapor in air and

8.0 volume percent vapor in air, respectively.

In most circumstances, the hydrocarbon vapor-air mixture in an enclosed

space over kerosene-type jet fuel will not be in the flammable range; the

mixture will be below the lower flammability limit. However, high ambient

temperature can heat the fuel enough to bring the vapor space into the flammable

range. The flash point of a fuel is the lower flammability temperature

of the fuel under the specific test conditions. However, this is not necessarily

the lower flammability temperature under other conditions, such as in an

aircraft fuel tank.

For the more volatile wide-cut fuel, the hydrocarbon vapor-air mixture in an

enclosed space may be in the flammable range. The upper flammability temperature

limit depends on the vapor pressure of the fuel. A fuel with a vapor

pressure of 18 kPa (2.6 psi) will have an upper flammability temperature limit

of approximately 18°C (64°F).

However, in the absence of specific information to the contrary, any jet fuel

handling situation should be considered potentially hazardous and the appropriate

safety measures observed.

1.4 Trajectory Analysis

Most trajectory analysis problems involve small aircraft rotation rates

and are studied through the use of the three degree of freedom (3DOF)

equations of motion, that is, the translational equations. These equa-

tions are uncoupled from the rotational equations by assuming negligi-

ble rotation rates and neglecting the effect of control surface deflections

on aerodynamic forces. For example, consider an airplane in cruise.

To maintain a given speed an elevator deflection is required to make

the pitching moment zero. This elevator defection contributes to the

lift and the drag of the airplane. By neglecting the contribution of

the elevator deflection to the lift and drag (untrimmed aerodynamics),

the translational and rotational equations uncouple. Another approach,

called trimmed aerodynamics, is to compute the control surface angles

required for zero aerodynamic moments and eliminate them from the

aerodynamic forces. For example, in cruise the elevator angle for zero

aerodynamic pitching moment can be derived and eliminated from the

drag and the lift. In this way, the extra aerodynamic force due to control

surface deflection can be taken into account.

Trajectory analysis takes one of two forms. First, given an

aircraft, find its performance characteristics, that is, maximum speed,

ceiling, range, etc. Second, given certain performance characteristics,

what is the airplane which produces them. The latter is called aircraft

sizing, and the missions used to size commercial and military aircraft

are presented here to motivate the discussion of trajectory analysis. The

mission or flight profile for sizing a commercial aircraft (including busi-

ness jets) is shown in Fig. 1.6. It is composed of take-off, climb, cruise,

descent, and landing segments, where the descent segment is replaced

by an extended cruise because the fuel consumed is approximately the

same. In each segment, the distance traveled, the time elapsed, and the

fuel consumed must be computed to determine the corresponding quan-

tities for the whole mission. The development of formulas or algorithms

for computing these performance quantities is the charge of trajectory

analysis. The military mission (Fig. 1.7) adds three performance com-

8 Chapter 1. Introduction to Airplane Flight Mechanics

putations: a constant-altitude acceleration (supersonic dash), constant-

altitude turns, and specific excess power (PS). The low-altitude dash

gives the airplane the ability to approach the target within the radar

ground clutter, and the speed of the approach gives the airplane the

ability to avoid detection until it nears the target. The number of turns

is specified to ensure that the airplane has enough fuel for air combat in

the neighborhood of the target. Specific excess power is a measure of the

ability of the airplane to change its energy, and it is used to ensure that

the aircraft being designed has superior maneuver capabilities relative

to enemy aircraft protecting the target. Note that, with the exception

of the turns, each segment takes place in a plane perpendicular to the

surface of the earth (vertical plane). The turns take place in a horizontal

plane.

An airplane operates near the surface of the earth which moves about the

sun. Suppose that the equations of motion (F = ma and M = I) are

derived for an accurate inertial reference frame and that approximations

characteristic of airplane flight (altitude and speed) are introduced into

these equations. What results is a set of equations which can be obtained

by assuming that the earth is flat, nonrotating, and an approximate

inertial reference frame, that is, the flat earth model.

The equations of motion are composed of translational (force)

equations (F = ma) and rotational (moment) equations (M = I)

and are called the six degree of freedom (6DOF) equations of motion.

For trajectory analysis (performance), the translational equations are

uncoupled from the rotational equations by assuming that the airplane

rotational rates are small and that control surface deflections do not

affect forces. The translational equations are referred to as the three

degree of freedom (3DOF) equations of motion.

As discussed in Chap. 1, two important legs of the commercial

and military airplane missions are the climb and the cruise which occur

in a vertical plane (a plane perpendicular to the surface of the earth).

The purpose of this chapter is to derive the 3DOF equations of motion

for flight in a vertical plane over a flat earth. First, the physical model is

defined; several reference frames are defined; and the angular positions

and rates of these frames relative to each other are determined. Then,

the kinematic, dynamic, and weight equations are derived and discussed

for nonsteady and quasi-steady flight. Next, the equations of motion for

flight over a spherical earth are examined to find out how good the flat

2.1. Assumptions and Coordinate Systems 17

earth model really is. Finally, motivated by such problems as flight in

a headwind, flight in the downwash of a tanker, and flight through a

downburst, the equations of motion for flight in a moving atmosphere

are derived