Weight and Balance Control

There are many factors that lead to efficient and safe
operation of aircraft. Among these vital factors is proper
weight and balance control. The weight and balance
system commonly employed among aircraft consists of
three equally important elements: the weighing of the
aircraft, the maintaining of the weight and balance records,
and the proper loading of the aircraft. An inaccuracy in any
one of these elements nullifies the purpose of the whole
system. The final loading calculations will be meaningless
if either the aircraft has been improperly weighed or the
records contain an error.

Improper loading cuts down the efficiency of an aircraft
from the standpoint of altitude, maneuverability, rate
of climb, and speed. It may even be the cause of failure
to complete the flight, or for that matter, failure to start
the flight. Because of abnormal stresses placed upon the
structure of an improperly loaded aircraft, or because of
changed flying characteristics of the aircraft, loss of life
and destruction of valuable equipment may result.
The responsibility for proper weight and balance control
begins with the engineers and designers, and extends to the
aircraft mechanics that maintain the aircraft and the pilots
who operate them.

Modern aircraft are engineered utilizing state-of-the-art
technology and materials to achieve maximum reliability
and performance for the intended category. As much
care and expertise must be exercised in operating and
maintaining these efficient aircraft as was taken in their
design and manufacturing.

The designers of an aircraft have set the maximum weight,
based on the amount of lift the wings or rotors can provide
under the operation conditions for which the aircraft
is designed. The structural strength of the aircraft also
limits the maximum weight the aircraft can safely carry.
The ideal location of the center of gravity (CG) was very
carefully determined by the designers, and the maximum
deviation allowed from this specific location has been
calculated.

The manufacturer provides the aircraft operator with the
empty weight of the aircraft and the location of its emptyweight
center of gravity (EWCG) at the time the certified
aircraft leaves the factory. Amateur-built aircraft must have
this information determined and available at the time of
certification.

The airframe and powerplant (A&P) mechanic or
repairman who maintains the aircraft keeps the weight and
balance records current, recording any changes that have
been made because of repairs or alterations.
The pilot in command of the aircraft has the responsibility
on every flight to know the maximum allowable weight
of the aircraft and its CG limits. This allows the pilot to
determine on the preflight inspection that the aircraft is
loaded in such a way that the CG is within the allowable
limits.

Weight Control

Weight is a major factor in airplane construction and
operation, and it demands respect from all pilots and
particular diligence by all A&P mechanics and repairmen.
Excessive weight reduces the efficiency of an aircraft
and the safety margin available if an emergency
condition should arise.
When an aircraft is designed, it is made as light as the
required structural strength will allow, and the wings or
rotors are designed to support the maximum allowable
weight. When the weight of an aircraft is increased,
the wings or rotors must produce additional lift and the
structure must support not only the additional static loads,
but also the dynamic loads imposed by flight maneuvers.
For example, the wings of a 3,000-pound airplane must
support 3,000 pounds in level flight, but when the airplane
is turned smoothly and sharply using a bank angle of 60°,
the dynamic load requires the wings to support twice this,
or 6,000 pounds.
Severe uncoordinated maneuvers or flight into turbulence
can impose dynamic loads on the structure great enough
1–
to cause failure. In accordance with Title 14 of the Code
of Federal Regulations (14 CFR) part 23, the structure of a
normal category airplane must be strong enough to sustain
a load factor of 3.8 times its weight. That is, every pound
of weight added to an aircraft requires that the structure
be strong enough to support an additional 3.8 pounds.
An aircraft operated in the utility category must sustain a
load factor of 4.4, and acrobatic category aircraft must be
strong enough to withstand 6.0 times their weight.

The lift produced by a wing is determined by its airfoil
shape, angle of attack, speed through the air, and the air
density. When an aircraft takes off from an airport with a
high density altitude, it must accelerate to a speed faster
than would be required at sea level to produce enough
lift to allow takeoff; therefore, a longer takeoff run is
necessary. The distance needed may be longer than the
available runway. When operating from a high-density
altitude airport, the Pilot’s Operating Handbook (POH)
or Airplane Flight Manual (AFM) must be consulted to
determine the maximum weight allowed for the aircraft
under the conditions of altitude, temperature, wind, and
runway conditions.

Relation between Pressure, Density, and Temperature of a Gas

 Relation between Pressure, Density, and Temperature of a Gas

AIR AT REST, THE ATMOSPHERE AND STATIC LIFT 13

ii. Relation between Pressure, Density, and Temperature of a Gas

By the experimental laws of Boyle and Charles, for constant

temperature the pressure of a gas is proportional to its density ; for

constant volume the pressure of a gas is proportional to its absolute

temperature. The absolute temperature is denoted by T and, if 6 is

the temperature on the centigrade scale, is given by

T = + 273.

Combining these laws, we have, for a given mass of a particular

gas:

pV^Bt (8)

where V is the volume, or, if V is the volume of 1 lb.,

P/9=gBi: (9)

B is a constant which is made characteristic of a particular gas by

treating 1 lb. of the gas ; it is then evaluated from measurements of

pressure and volume at a known temperature. It follows that B

will vary from one gas to another in inverse proportion to the

density under standard conditions of pressure and temperature.

If N is the ixumber of molecules in V, N will, by Avogadro's law,

be the same for all gases at constant p and T. Hence, writing pV/N

= B'T, B' is an absolute constant having the same value for all

gases. Equation (9) is more convenient, however, and the variation

of B is at once determined from a table of molecular weights.

Some useful data are given in Table II. It will be noticed that, if p

is kept constant, B measures the work done by the volume of gas in

expanding in consequence of being heated through unit temperature

change. The units of B are thus ft.-lb. per lb, per degree centigrade,

or ft. per C.

 

Isothermal Atmosphere

Isothermal Atmosphere

We now examine the static equilibrium of a bulk of gas under

gravity, taking into account its compressibility. Equation (2)

14 AERODYNAMICS [CH.

applies, but specification is needed of the relationship between p

and p. The simple assumption made in the present article is that

appropriate to Boyle's law, viz. constant temperature TO, so that />/p

remains constant. From (2) :

*.-*. 9g From (9) : 1 _ J5r 9g P ' Hence : ,. BiQ = - dh. P

Integrating between levels Ax and h2 , where p = pt and p2 respectively,

BTO log (pjpj = h, - h, . . . (10)

The logarithm in this expression is to base e. Throughout this

book Napierian logarithms will be intended, unless it is stated

otherwise. The result (10) states that the pressure and therefore

the density of a bulk of gas which is everywhere at the same temperature

vary exponentially with altitude.

The result, although accurately true only for a single gas, applies

with negligible error to a mass of air under isothermal conditions,

provided great altitude changes are excluded. The stratosphere is

in conductive equilibrium, the uniform temperature being about

- 55 C. The constitution of the air at its lowest levels is as given

in Article 1. As altitude increases, the constitution is subject to

Dalton's law : a mixture of gases in isothermal equilibrium may be

regarded as the aggregate of a number of atmospheres, one for each

constituent gas, the law of density variation in each atmosphere

being the same as if it constituted the whole. Hence argon and

other heavy gases and subsequently oxygen, nitrogen, and neon will

become rarer at higher levels. The value of B for the atmosphere

will consequently increase with altitude, although we have assumed

it constant in order to obtain (10). The variation of B for several

miles into the stratosphere will, however, be small. At greater

altitudes still the temperature increases again.

The Troposphere

The Troposphere

The atmosphere beneath the stratified region is perpetually in

process of being mechanically mixed by wind and storm. When a

bulk of air is displaced vertically, its temperature, unlike its pressure,

has insufficient time for adjustment to the conditions obtaining at the

new level before it is moved away again. The properties of this

part of the atmosphere, to which most regular flying so far has been

restricted, are subject to considerable variations with time and place,

excepting that B varies only slightly, depending upon the humidity.

There exists a temperature gradient with respect to altitude, and on

the average this is linear, until the merge into the stratosphere is

approached.

Centre of Pressure

Centre of Pressure

Th point on a surface exposed to pressure through which the
resultant force acts is called the centre of pressure. The centres of
pressure with which we are concerned relate to the pressure difference,
often called the gas pressure, unevenly spread over part of an
envelope separating gas from the atmosphere. Gas pressures are
small at the bottom of an envelope and reach a maximum at the top,
as illustrated in Fig. 6, and positions of the centres of pressure are
usually high.

The high centres of the total gas pressures exerted on walls which
restrain a gas-bag, as in the case of the wire bulkheads or transverse
frames of a rigid airship, lead to moments internal to the structure.
BCDE (Fig. 8) is a (full) gas-bag of an airship which is pitched at
angle a from a level keel. The longitudinal thrusts P, P' from the

* gas pressure

' are supported by bulkheads EC and DE of areas A,

A', assumed plane, B and E being lowest and C and D highest points.

The gas is assumed to be at rest, so that pressure is constant over

horizontal planes, and its pressure at B, the bottom of the bag, is

taken as equal to that of the atmosphere. Let p be the excess

pressure at height h above the level of B. Then from (3) p = pigA,

where p x is the difference in the densities of the gas and the surrounding

air.

Lower Bulkhead BC. Let 8A be the area of a narrow horizontal

strip of BC distant y from a horizontal axis in its plane through B.

Then h = y cos a, and the total thrust on BC is given by :

re re P = p dA = pig cos a y dA

JB JB = P!# cos a . AyQ

Balloons and Airships

Balloons and Airships

In balloons and airships the gas is contained within envelopes of
cotton fabric lined with gold-beaters

1 skins or rubber impregnated.
Diffusion occurs through these comparatively impervious materials,

and, together with leakage, contaminates the enclosed gas, so that
densities greater than those given in the preceding article must be
assumed. Practical values for lift per thousand cubic feet are 68 Ib. for
hydrogen and 62 Ib. for helium, at low altitude. Thus the envelope of a
balloon weighing 1 ton would, in the taut state at sea-level, have a diameter
of 39-8 ft. for hydrogen and 41-1 ft. for helium ; actually it would be
made larger, filling only at altitude and being limp at sea-level.\

the variation of atmospheric pressure from the level of the top of

the open filling sleeve S to that of the crest of the balloon, OH the

corresponding variation of pressure through the bulk of helium

filling the envelope.

 The difference between these external arid

internal pressures acts radially outward on the fabric as shown to

the right. The upward resultant force and part of the force of

expansion are supported by the net N, from which is suspended the

basket or gondola B, carrying ballast and the useful load.

Balloons drift with the wind and cannot be steered horizontally.

Airships, on the other hand, can maintain relative horizontal velocities

by means of engines and airscrews, and are shaped to streamline

form for economy of power. Three classes may be distinguished.

The small non-rigid airship, or dirigible balloon (Fig. l(a)} has a

faired envelope whose shape is conserved by excess gas pressure

maintained by internal ballonets which can be inflated by an air

scoop exposed behind the airscrew. Some stiffening is necessary,

especially at the nose, which tends to blow in at speed. A gondola,

carrying the power unit, fuel, and other loads, is suspended on cables

from hand-shaped strengthening patches on the envelope. (Only

a few of the wires are shown in the sketch.)

In the semi-rigid type (b) some form of keel is interposed between

the envelope and gondola, or gondolas, enabling excess gas pressure

to be minimised. Several internal staying systems spread the load

carried by the girder over the envelope, the section of which is not as

a rule circular.

Buoyancy of Gas-filled Envelope

Buoyancy of Gas-filled Envelope

The maximum change of height within a balloon or a gas-bag of an
airship is usually sufficiently small for variation of density to be
neglected. Draw a vertical cylinder of small cross-sectional area A
completely through the envelope E (Fig. 5), which is filled with a
light gas of density p', and is at rest relative to the surrounding
atmosphere of density p. Let the cylinder cut the envelope at a
lower altitude-level ht and at an upper one Aa , the curves of inter-

section enclosing small areas Slt Sa , the normals to which (they are

not necessarily in the same plane) make angles oc lf a8 with the

vertical. On these areas pressures />',, p'2t act outwardly due to the

gas, and^>lf p9 act inwardly due to the atmosphere.

There arises at h2 an upward force on the cylinder equal to

(pi ^a)S2 cos a,.

The similar force arising at h^ may be upward or downward, depending

on the position of Sl and whether an airship or a balloon is

considered, but in any case its upward value is

(Pi P()SI cos i- Since Sa cos oc 8 = A == St cos al, 

the resultant upward force on the

cylinder due to the pressures is

Substituting from (3),

Measurement of Small Pressure Differences

Measurement of Small Pressure Differences

Accurate measurement of small differences of air pressure is often
required in experimental aerodynamics. A convenient instrument
is the Chattock gauge (Fig. 4). The rigid glasswork AB forms a
U-tube, and up to the levels L contains water, which also fills the
central tube T. But above L and the open mouth of T the closed
vessel surrounding this tube is filled with castor oil. Excess of air
pressure in A above that in B tends to transfer water from A to B
by bubbling through the castor oil. But this is prevented by tilting
the heavy frame F, carrying the U-tube, about its pivots P by means
of the micrometer screw S, the water-oil meniscus M being observed
for accuracy through a microscope attached to F. Thus the excess
air pressure in A is compensated by raising the water level in B
above that in A, although no fluid passes. The wheel W fixed to S
is graduated, and a pressure difference of O0005 in. of water is easily

FIG. 4. CHATTOCK GAUGE.

detected. By employing wide and accurately made bulbs set close

together, constantly removing slight wear, protecting the liquids

against appreciable temperature changes and plotting the zero

against time to allow for those that remain, the sensitivity

* may be

increased five or ten times. These gauges are usually constructed

for a maximum pressure head of about 1 in. of water. Longer forms

extend this range, but other types are used for considerably greater

heads.

At 15 C. 1 cu. ft. of water weighs 62-37 Ib. Saturation with air

decreases this weight by about 0'05 Ib. The decrease of density

from 10 to 20 C. is 0-15 per cent. A 6 or 7 pet cent, saline solution

is commonly used instead of pure water in Chattock gauges, however,

since the meniscus then remains clean for a longer period.

The Hydrostatic Equation

The Hydrostatic Equation

We now approach the problem of the equilibrium of a bulk of air
at rest under the external force of gravity, g has the dimensions of
an acceleration, L/T*. Its value depends slightly on latitude and
altitude, increasing by 0-5 per cent, from the equator to the poles
and decreasing by 0-5 per cent, from sea-level to 10 miles altitude,
At sea-level and 45 latitude its value is 32173
in ft.-sec. units. The value 32-2 ft./sec.

2 is sufficiently accurate for most purposes.
Since no horizontal component of external force
acts anywhere on the bulk of air, the pressure
in every horizontal plane is constant, as otherwise
motion would ensue. Let h represent altitude, so
that it increases upward. Consider an elementcylinder
of the fluid with axis vertical, of length
SA and cross-sectional area A .

Pressure

Pressure

Consider a small rigid surface suspended in a bulk of air at rest.
The molecular motion causes molecules continually to strike, or
tend to strike, the immersed surface, so that a rate of change of
molecular momentum occurs there. This cannot have a component
parallel to the surface, or the condition of rest would be disturbed.
Thus, when the gas is apparently at rest, the aggregate rate of
change of momentum is normal to the surface ; it can be
represented by a force which is everywhere directed at right
angles towards the surface. The intensity of the force per unit
area is the pressure pt sometimes called the hydrostatic or static
pressure.

It is important to note that the lack of a tangential component to
p depends upon the condition of stationary equilibrium. The
converse statement, that fluids at rest cannot withstand a tangential
or shearing force, however small, serves to distinguish liquids from
solids. For gases we must add that a given quantity can expand to
fill a volume, however great.

It will now be shown that the pressure at a point in a fluid at rest is
uniform in all directions. Draw the small tetrahedron ABCO, of

* In this system, the units of length and time are the foot and the second, whilst
forces are in pounds weight. It is usual in Engineering, however, to omit the word ' weight/ writing

*Ib.' for 'lb.-wt.,' and this convention is followed. The
appropriate unit of mass is the 'slug,' viz. the mass of a body weighing g Ib.
Velocities are consistently measured in ft. per sec., and so on. This system being
understood, specification of units will often be omitted from calculations for brevity.
For example, when a particular value of the kinematic viscosity is given as a number,
sq. ft. per sec. will be implied. It will be desirable occasionally to introduce special
units. Thus the size and speed of aircraft are more easily visualised when weights
are expressed in tons and velocities in miles per hour. The special units will be
duly indicated in such cases. Non-dimeasional coefficients are employed wherever
convenient.