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

SPACE MISSION

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Lubricity Lubricity is the ability to reduce friction between solid surfaces in

relative motion, so it is a measure of a material’s effectiveness as a lubricant.

Jet fuel must possess a certain degree of lubricity because jet engines rely on

the fuel to lubricate some moving parts in fuel pumps and flow control units.

The lubrication mechanism is a combination of hydrodynamic lubrication

and boundary lubrication. In hydrodynamic lubrication, a layer of the liquid

lubricant prevents the opposing moving surfaces from contacting each other.

Higher viscosity liquids provide more hydrodynamic lubrication than lower

viscosity liquids. While jet fuel specifications do not include an explicit lower

limit on viscosity, the distillation specification serves as a surrogate limit. Jet

engines are designed to work with jet fuels within the normal viscosity range,

and therefore, typical jet fuels provide adequate hydrodynamic lubrication.

When close tolerances squeeze out most of the liquid layer that provides

hydrodynamic lubrication, boundary lubrication becomes important. Now,

small areas of the opposing surfaces are in contact. Boundary lubricants

are compounds that form a protective anti-wear layer by adhering to the

metal surfaces.

Straight-run jet fuels (see page 32) are good boundary lubricants. This is not

due to the hydrocarbons that constitute the bulk of the fuel, but is attributed

to trace amounts of certain oxygen-, nitrogen-, and sulfur-containing compounds.

Evidence for the role of trace quantities is the fact that adding as

little as 10 ppm of a lubricity enhancing additive to a poor lubricity fuel can

make it acceptable

Stability and control studies are concerned with motion of the center
of gravity (cg) relative to the ground and motion of the airplane about
the cg. Hence, stability and control studies involve the use of the six
degree of freedom equations of motion. These studies are divided into
two major categories: (a) static stability and control and (b) dynamic
stability and control. Because of the nature of the solution process, each
of the categories is subdivided into longitudinal motion (pitching motion)
and lateral-directional motion (combined rolling and yawing motion).
While trajectory analyses are performed in terms of force coefficients
with control surface deflections either neglected (untrimmed drag polar)
or eliminated (trimmed drag polar), stability and control analyses are in
terms of the orientation angles (angle of attack and sideslip angle) and
the control surface deflections.
The six degree of freedom model for flight in a vertical plane
is presented in Chap. 8. First, the equations of motion are derived in
the wind axes system. Second, formulas for calculating subsonic aero-
dynamics are developed for an airplane with a straight, tapered, swept
wing. The aerodynamics associated with lift and pitching moment are
shown to be linear in the angle of attack, the elevator angle, the pitch
rate, and the angle of attack rate. The aerodynamics associated with
drag is shown to be quadratic in angle of attack. Each coefficient in
these relationships is a function of Mach number.
Chap. 9 is concerned with static stability and control. Static
stability and control for quasi-steady flight is concerned primarily with
four topics: trim conditions, static stability, center of gravity effects, and
control force and handling qualities. The trim conditions are the orienta-
tion angles and control surface deflections required for a particular flight
condition. Given a disturbance from a steady flight condition, static
stability investigates the tendency of the airplane to reduce the distur-
bance. This is done by looking at the signs of the forces and moments.
Fore and aft limits are imposed on allowable cg locations by maximum
allowable control surface deflections and by stability considerations, the
aft cg limit being known as the neutral point because it indicates neu-
tral stability. Handling qualities studies are concerned with pilot-related
quantities such as control force and how control force changes with flight
speed. These quantities are derived from aerodynamic moments about
control surface hinge lines. Trim tabs have been introduced to allow the
1.5. Stability and Control 11
pilot to zero out the control forces associated with a particular flight con-
dition. However, if after trimming the stick force the pilot flies hands-off,
the stability characteristics of the airplane are reduced.
To investigate static stability and control for accelerated flight,
use is made of a pull-up. Of interest is the elevator angle required to
make an n-g turn or pull-up. There is a cg position where the elevator
angle per g goes to zero, making the airplane too easy to maneuver. This
cg position is called the maneuver point. There is another maneuver
point associated with the stick force required to make an n-g pull-up.
While dynamic stability and control studies can be conducted
using wind axes, it is the convention to use body axes. Hence, in Chap.
10, the equations of motion are derived in the body axes. The aerody-
namics need for body axes is the same as that used in wind axes. A
particular set of body axes is called stability axes. The equations of
motion are also developed for stability axes.
Dynamic stability and control is concerned with the motion of
an airplane following a disturbance such as a wind gust (which changes
the speed, the angle of attack and/or the sideslip angle) or a control
input. While these studies can and are performed using detailed com-
puter simulations, it is difficult to determine cause and effect. As a con-
sequence, it is desirable to develop an approximate analytical approach.
This is done in Chap. 11 by starting with the airplane in a quasi-steady
flight condition (given altitude, Mach number, weight, power setting)
and introducing a small disturbance. By assuming that the changes in
the variables are small, the equations of motion can be linearized about
the steady flight condition. This process leads to a system of linear,
ordinary differential equations with constant coefficients. As is known
from linear system theory, the response of an airplane to a disturbance is
the sum of a number of motions called modes. While it is not necessary
for each mode to be stable, it is necessary to know for each mode the
stability characteristics and response characteristics. A mode can be un-
stable providing its response characteristics are such that the pilot can
easily control the airplane. On the other hand, even if a mode is stable,
its response characteristics must be such that the airplane handles well
(handling qualities). The design problem is to ensure that an aircraft
has desirable stability and response characteristics thoughout the flight
envelope and for all allowable cg positions. During this part of the de-
sign process, it may no longer be possible to modify the configuration,
and automatic control solutions may have to be used