Department of Aerospace Sciences Elie Carafoli [617547]

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Department of Aerospace Sciences 'Elie Carafoli'

Mass, Balance and Trim consideration of jet transport
BEng Final Project

Author: Osilagun Sandra Zainab

Supervisor(s): Dr. Ing., Ion Fuiorea

Session: July 2019

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Anti-Plagiarism Declaration

I the undersigned Osilagun Sandra Zainab , student: [anonimizat], Faculty
of Aerospace Engineering declare herewith and certify that this final project is the result of my own,
original, individua l work. All the external sources of information used were quoted and included in the
References. All the figures, diagrams, and tables taken from external sources include a reference to the
source.

Date: _________ Signa ture: __________

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List of figures

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Acronyms

BEW – Basic Empty Weight
DOW – Dry Operating Weight
ZFW – Zero Fuel Weight
MTW – Maximum Takeoff Weight
MRW -Maximum Ramp Weight
MLW – Maximum Landing Weight
MTW – Maximum Taxi Weight
MTOW – Maximum Take‐Off Weight
MLW – Maximum Landing Weight
MZFW- Maximum Zero Fuel Weight
MAC – Mean aerodynamic chord

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Executive Summary

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Rezumat

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Table of Contents
Anti-Plagiarism Declaration ………………………….. ………………………….. ………………………….. ………… 2
List of figures ………………………….. ………………………….. ………………………….. ………………………….. …… 3
Acronyms ………………………….. ………………………….. ………………………….. ………………………….. …………. 4
Executive Summary ………………………….. ………………………….. ………………………….. ……………………… 5
Rezumat ………………………….. ………………………….. ………………………….. ………………………….. ……………. 6
Introduction ………………………….. ………………………….. ………………………….. ………………………….. ……… 9
1. Introduction to weigh t and balance ………………………….. ………………………….. ……………….. 10
1.1 Airplane weight definitions ………………………….. ………………………….. ………………………….. 10
1.2.1 Weight/Mass distinction ………………………….. ………………………….. ………………………….. ……. 10
1.2.2 General aircraft weight definition ………………………….. ………………………….. …………………. 10
1.2.3 Structural Weight Limitation ………………………….. ………………………….. …………………………. 11
1.2.4 Certified and Operational Weights ………………………….. ………………………….. ……………….. 12
1.2 Center of Gravity ………………………….. ………………………….. ………………………….. ……………… 12
1.2.1 Location of the CG ………………………….. ………………………….. ………………………….. ……….. 13
1.2.2 Aircraft CG Range and Limit s ………………………….. ………………………….. ………………….. 14
1.2.3 Terms and Definition ………………………….. ………………………….. ………………………….. ……. 15
1.2.4 CG and Balance in an Airplane ………………………….. ………………………….. ……………….. 16
1.3 Weight control ………………………….. ………………………….. ………………………….. ………………….. 17
1.3.1 Effect of weight ………………………….. ………………………….. ………………………….. …………….. 18
1.4 Weight Changes ………………………….. ………………………….. ………………………….. ………………. 18
1.4.1 Balance Limits ………………………….. ………………………….. ………………………….. ……………… 19
1.4.2 Stability and Balance Control ………………………….. ………………………….. ………………….. 19
1.4.3 Effects of Adverse Balance ………………………….. ………………………….. ……………………… 20
1.4.4 Basic Principles of Weight and Balance Computations ………………………….. …….. 21
1.5 Weight and Balance Restrictions ………………………….. ………………………….. ……………….. 22
1.5.1 Importance of Aircraft Wei ght and Balance ………………………….. ………………………… 23
1.5.2 Factors Affecting Weight and Balance in Aircraft ………………………….. ………………. 23
2. Aircraft Stability and Control ………………………….. ………………………….. ………………………….. …. 29
2.1 Introduction ………………………….. ………………………….. ………………………….. ………………………….. . 29
2.2 Static Stability ………………………….. ………………………….. ………………………….. ………………………. 29
2.2.1 Static Stability in Aircrafts ………………………….. ………………………….. ………………………….. … 30
2.2.2 Equilibrium and the basic trim equation ………………………….. ………………………….. ………. 34
2.3 Dynamic Stability ………………………….. ………………………….. ………………………….. ………………….. 36
2.3.1 Static Forces and M oments on an Aircraft ………………………….. ………………………….. …… 37
2.4 Moment on an Aircraft ………………………….. ………………………….. ………………………….. …………. 38

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2.4.1 Center of Gravity of a Stable Aircraft ………………………….. ………………………….. ……………. 39
3.Trim Considerations in Aircrafts ………………………….. ………………………….. ………………………… 41
3.1 Trim Systems in Aircrafts ………………………….. ………………………….. ………………………….. ……. 41
3.2 Types of Tabs ………………………….. ………………………….. ………………………….. ……………………….. 45
3.3 The Trim Condition of an Aircraft ………………………….. ………………………….. ……………………. 48
References ………………………….. ………………………….. ………………………….. ………………………….. ……… 53

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Introduction
Aviation is one of the most dynamic industries since its beginning. New aircraft are frequently
being developed with enhancement over previous models. Improvements in design have, in many
cases, tended to increase the importance of the proper loading and balancing of today's airplanes.
Weigh t-and-balance calculations are performed in step with the actual rules and specifications and
should be ready when aircraft are manufactured and whenever they’re altered, whether or not the
airplane is massive or small. The dynamic ally changing conditions of modern aircraft operation
present more advanced combinations of cargo, crew, fuel, passengers, and baggage. The necessity of
getting the maximum efficiency for all flights has increased the requirement for a precise system of
controlling the weight and balance of an aircraft.
Most current -day airplanes are designed clean and sleek, which usually ends up in greater range,
speed, payload, and increased efficiency. This type of airplane is mostly used for cross -country
flights. Airplanes used for short term flights and those used for carrying heavy loads, like those
utilized in some certain agricultural operations, are designed differently, However, they still exhibit
good performance for their purpose. Some of the factors that represent good performance ar e short
takeoff and landing distance, exaggerated climb capability, and bigger speeds using less fuel.

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1. Introduction to weight and balance

An aircraft's weight is a n accurately essential performance parameter for every aircraft ever made,
attention is usually given to an airplane's takeoff weight just at the very begin of the design proce dure,
and weight calculations and analyses maintain every step of design and manufactur ing process for every
new airplane . Aircraft weight are registered as part of type certification, and every single aircraft are
weighed before been released . Weight growth during the life of an aircraft is inevitable , proceeding
from repairs , modifications and contamination, paint ing and so on. Therefore , its standard custom for
operators to keep records of each aircraft's empty weight.

1.1 Airplane weight definitions

Definition: Mass is the quantity of matter in a body regardless of its volume or of any forces acting
on it and is measuring its inertial properties.
: Weight is a measurement of the gravitational force acting on an object.
Effect of gravity: Mass is always constant at any place and any time
: The weight of an object depends on the gravity at that place
Unit of Measurement: Mass is expressed in kilogram (kg), grams (g), and milligram (mg) , pounds
(lb).
: Weight is expressed in Newton (N)

1.2.1 Weight/Mass distinction

“The distinction between the terms weight and mass is important in aircraft engineering. To be entirely
consistent and technically correct at all times, However, especially when discussing flight operations,
can be difficult. FAA documents, for example, generally refer to the aircraft weight (e.g FAA AC 120 –
27 [3]), whereas many comparable EASA documents refers to the aircraft’s mass (e.g EASA OPS part –
CAT [5]). Similarly , operators are spilt in their approaches: some use aircraft mass but majority prefer
aircraft weight.” [1]
1.2.2 General aircraft weight definition

Basic Empty Weight (BEW):
The basic empty weight of an airplane is the standard empty weight of the airplane plus optional
equipment installed.
Dry Operating Weight (DOW):
This is the total weight of the airplane ready for a specific type of operation excluding all usable fuel
and traffic load.
Zero Fuel Weight (ZFW) :

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This is the operating weight of the aircraft plus the traffic load (payload). An aircraft's cargo, baggage,
and passengers (including pilots) make up its payload.
Maximum Takeoff Weight (TOW):
The maximum weight limits for an aircraft to begin its takeoff roll is called the max takeoff weight. It
is the zero weight plus the fuel weight at takeoff. It is also occasionally called the brake release weight
(BRW).
Maximum Ramp Weight :
The maximum weight for maneuvering on the ground is called the max ramp weight . It is the takeoff
weight plus the fuel weight used for the auxiliary power unit (APU). Max ramp weight includes the
weight of fuel used for start, tax i, and aircraft run -up procedures.
Maximum Landing Weight :
The maximum landing weight is, as you might imagine, the maximum aircraft weight limit approved
for an aircraft to land. Landing above this weight can cause structural damage. [2]
1.2.3 Structural Weight Limitation

The structural capability governs the absolute weight limits applicable to an aircraft for various stages,
these limits are applicable to individual aircraft. There are mainly four structural limitation which are

Maximum Structural Taxi Weight (MSTW):
This is the structural limitation on the weight of the airplane at the commencement of taxi, It’s
structural limit is imposed by ground load on the landing gear. Usually refer to as the maximum taxi
weight (MTW).
Maximum Structural Take‐Of f Weight (MSTOW):
This is the maximum acceptable total airplane weight from the start of the takeoff its structural limits
are usually dictated by flight load, braking system and landing impact consideration, which are based
on a “vertical impact of 6ft/s (1.83 m/s) (as required on by FAR/CS 25.473 [10, 11])” [1]. And MSTOW
is frequently referred to as Maximum Take‐Off weight (MTOW).
Maximum Structural Landing Weight (MSLW):
This is the maximum acceptable total airplane weight on landing under normal circums tances. This
defines the upper limits to the acceptable landing weight (LW). Its structural limit is governed either by
the braking system limitation or the landing impact which is based on the “vertical impact of 10ft/s
(3.05m/s) (as required by FAR/CS 25 .473 [10,11])”. [1] Usually referred to Maximum landing weight
(MLW).
Maximum Zero Fuel Weight (MZFW):
The maximum acceptable weight of an airplane with no usable fuel. The structural limit is usually
governed by the maximum allowable wing root bending mo ment. Increasing the MZFW increases the
average wing root bending moment, which can lead to a long -term structural damage. The maximum
zero fuel weight is usually referred to as MZFW. [2]

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1.2.4 Certified and Operational Weights

Certified Weight
An oper ator can only legally use the maximum weight of an airplane that has already been recorded in
the Airplane Flight Manual or in the Weight and Balance Manual. This weight boundary is known as
certified weight, which is used for financial and operational pur poses. The certified weight of an
airplane is usually chosen by the operator when the aircraft is acquired instead of the operating costs
been linked to an aircraft certified takeoff and landing weight. Individual aircraft not by the aircraft
type have to have certified weight individually .
Operational weight
An operator ’s maximum takeoff weight in use for service cannot exceed the certified maximum
takeoff weight for the precise aircraft. Takeoff weight can restricted by number of factors which are:
i. The aircraft performance capability under certain weather and atmospheric conditions
ii. Noise restrictions
iii. Runway loading limit
iv. Center of gravity limit
Certified maximum landing weight cannot be exceeded by operators and it can also be limited by
other operatio nal factors. [1]

1.2 Center of Gravity

The CG is defined as the point through which the force of gravity acts on a mass, in aircraft terms, it is
the point in which the aircraft’s entire mass acts in a vertically downward manner. The center of gravity
is als o the point of balance and because of this, it tends to affect the stability of the aircraft both on the
ground and in the air.
Each body of matter in the universe attracts each and every other body with a specific force that is
known as gravitation force. The term gravity is used to refer to the force that tends to draw all bodies
toward the center of the earth. The weight of a body is the result of gravitational force acting on the
body.
Every particle of an object is acted on by the force of gravity. Ho wever, in every object there is one
point in which a single force is equal in magnitude to the weight of the object and is directed upward
which will keep the body at rest, which means that, it will keep it in balance and prevent it from falling.
This poin t is what known as the center of gravity (CG), The CG might be defined as the point at which
all the weight of a body can be considered concentrated. Thus, the CG of a perfect circular ball would
be the precise center of the ball, provided that the ball wa s made of homogenous material and that there
were no air or gas pockets inside (see Figure 1.I).
The CG of a uniform ring would be at the center of the ring but would not be at any point on the ring
itself (see Figure 1.II). The CG of a cube of solid mate rial would be equidistant from the eight comers,

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as shown in Figure 1.III. In airplanes or helicopters, simplicity of control and maneuverability require s
that the placement of the CG should be within specifi c limits.

I II III
Figure 1.1: Center of Gravity of different shapes

1.2.1 Location of the CG

Since the CG of a body is that point at which its weight can be considered to be con centrated, the CG
of a freely suspended body will always be vertically underneath the point of support when the body is
supported at a single point. To locate the CG, therefore, it is necessary only to determine the point of
intersection of vertical lines drawn downward from two separate points of support employed one at a
time.
This technique is demonstrated in Figure 2A, which shows a flat, square feet of material lettered A, B,
C, and D at its four comers, suspended first from point B and then from poi nt c. The lines drawn
vertically downward from the point of suspension in every case meets at the CG.
The CG of an irregular body can be found in the same way. If an irregular object, such as the one
shown in Figure 2B, is suspended from a point P in such a manner that it can turn freely about the
point of suspension, it will come to rest with its CG directly below the point of suspension, P. If a
plumb line is dropped from the same point of suspension, the CG of the object will coincide with
some point al ong the plumb line; a line drawn along the plumb line passes through this point.
If the object is suspended from another point, which will be called A, and another line is drawn in the
direction indicated by the plumb line, the intersection of the two lin es will be at the CG. In order to
verify the results, the operation can be repeated, this time with the object suspended from another
point, called B. No matter how many times the process is repeated, the lines should pass through the
CG; therefore, it can be shown that the CG of the object lies at the point of intersection of these lines
of suspension. Therefore, any object behaves as if all its weight were concentrated at its CG.

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Figure 1. 2: Location of Center of Gravity
1.2.2 Aircraft CG Range and Limits

The first -class lever is in balance only when the CG is at the point. However, an aircraft maybe
balanced in flight anywhere within a certain specified forward and aft limits if the pilot operates the
trim tabs or elevators to exert an aerodynamic force sufficient to overcome any static unbalance. CG
locations outside the specified limits will cause unsatisfactory or even dangerous flight characteristics.
The allowable variation inside the CG are varied carefully and determined by the engineers who
design an airplane. The CG varies sometimes by extends forward and rearward from a point a round
one-fourth the chord of the wing, back from the leading edge, provided that the wing has no
sweepback. The exact location is always shown in the Aircraft Specifications or the Type Certificate
Data Sheet. Heavy loads near the wing location are balanced by much lighter loads at or near the nose
or tail of the airplane. In Figure 5A, a load of 5 Ib [2.268 kg] at A will be balanced by a load of 1lb
[0.4536 kg] at B because the moments of the two loads are equal.
Since the CG limits constitute the range of movement that the aircraft CG can have without making it
unstable or unsafe to fly, the CG of the loaded aircraft must be within these limits at takeoff, in the air
and on landing. In some cases, the takeoff limits and landing limits are not exactly the same and the
differences are given in the specifications for the aircraft.
Figure 3B, shows typical limits for the CG location in an airplane. As previously stated, these limits
establish the CG range. The CG of the airplane must fall within this range if the airplane is to fly
safely; that is, the CG must be to the rear of the forward limit and forward of the aft limit.

A
B

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Figure 1. 3: Aircraft CG Range and Limits
1.2.3 Terms and Definition

Centre of Gravity Limits
The CG is not a n immovable point it has a series of mo tion between a maximum forward position and
a maximum back ward position which is established by the aircraft manufacturer and cannot be
surpassed . The CG must be on or within the limit range at all phases . The restriction are usually given
in the flight manual and are explained relativ ely to the datum. They may also be assumed as a
percentage of the mean chord of the wing. (The wing mean chord was called the Standard Mean
Chord but is now known as the Mean Aerodynamic Chord or more simply, the MAC.)
Datum
A point along the longitudinal axis ( center line) of the a irplane designated by the manufacturer as the
zero o r reference point from which all balance arms (distances) commences. By taking moments
concerning the datum the CG position of the aircraft will be determined. For the purposes of this
phase of study the lateral displacement of the CG from the longitudinal axis is presumed to be zero.
Balance Arm
The distance from the aircraft’s datum to the CG position or centroid of a body or mass. For example,
the centroid of a square or rectangle is the exact center of the square or rectangle and, in such cases,
the ba lance arm is the distance from the datum to the exact center of the square or rectangle.
Unfortunately, cargo bays are seldom exact squares or rectangles and so the centroid (the point the
total weight acts through) is given by the manufacturer.
Delta
This is a Greek letter expressed by the symbol D to indicate a change of values. For example, D CG
indicates a change (or movement) of the CG.
Fuel load
This is the expendable part of the load in the airplane. It includes solely usable fuel, not fuel requir ed
to fill the lines or that which remains trapped in the tank sumps.

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Moment
This is the product of the weight of an item multiplied by its arm. Moments are expressed in pound –
inches (lb -in). Total moment is that the weight of the airplane is multiplied b y the distance between
the datum and the CG .
Moment index (or index)
This is a moment divided by a constant such as 100, 1,000, or 10,000. The us e of a moment index is
to simplify weight and balance computations of airplanes where ver heavy items and long arms result
in massive , unmanageable numbers.
Mean aerodynamic chord (MAC)
This is the average distance from the leading edge to the trailing edge of the wing
1.2.4 CG and Balance in an Airplane

The CG of an airplane is also defined, for the aim of balance co mputations, as an imaginary point
regarding the nose -heavy ( -) moments and tail -heavy (+) moments are specifically equal in
magnitude. Thus, the aircraft, if suspended from that point (CG), would have no tendency to rotate in
either direction (nose -up or nose-down).
This condition is illustrated in Figure 4A. As expressed previously, the weight of the aircraft can be
assumed to be concentrated at its CG. The CG with the aircraft loaded is allowed to vary fore and aft
within a certain limit that are deter mined throughout the flight tests for type certification. These limits
are the most forward and rearward loaded CG positions at which the aircraft will meet the
performance and flight characteristics required by the authorities.
These limits are also expressed in terms of a percentage of the mean aerodynamic chord (MAC) or in
inches forward or to the rear of the datum line. The relative positions of the CG and also the center of
lift of the wing have critical effects on the flight characteristics of the a ircraft. Consequently, relating
the CG location of the chord of the wing is convenient from a design and operations standpoint.
Usually, an aircraft will have satisfactory flight characteristics if the CG is located somewhere near
the 25% average chord po int. This means the CG is located at one -fourth of the total distance back
from the leading edge of the average wing section (see Figure 4B). Such a location can place the CG
forward of the aerodynamic center for most airfoils. The mean aerodynamic chord (MAC) is
established by the manufacturer. If the wing has a constant chord, the straight line distance from the
leading edge to the trailing edge (the chord) would also be the MAC.
However, if the wing is tapered, the mean aerodynamic chord is more compl icated to define. The
MAC is the chord o f an imaginary airfoil, which has the same aerodynamic characteristics as the
actual airfoil. The MAC established by the manufacturer defines its leading edge (LEMAC) and
trailing edge (TEMAC) in terms of inches fro m the datum.
The CG location and various limits are then expressed in percentages of the chord. The MAC is
usually given in the aircraft's Type Certificate Data Sheet when it is required for weight -and-balance
computations; therefore the person working on the airplane is expected to have only a general
understanding of its meaning. For simplicity purposes, most light -aircraft manufacturers express the
CG range in inches from the datum, while transport -category aircraft are expressed in terms of
percentages of the MAC.

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Figure 1.4:CG and Balance in an Airplane
1.3 Weight control

Weight is a major factor in airplane construction and operation, and it demands respect from all pilots
and particular attentiveness by all A&P mechanics and repairment. Excessive weight reduces the
potency of an aircraft and also the safety margin available if an emergency condition should arise.
When an aircraft is designed, it is designed as light -weight because the essential structural strengt h
can permit , and also the wings or rotors are designed to support the maximum allowable weight.
When the weight of an aircraft is exaggerated , the wings or rotors should produce extra lift and also
the structure should support not only the additional stat ic loads, but also the dynamic loads imposed
by flight movements. Simple uncoordinated action/maneuver of flight into turbulence can impose
dynamic loads on the aircraft great enough to cause failure. In accordance with Title 14 of the Code of
Federal Regu lations (14 CFR) part 23, the structure of a standard category airplane must be strong
enough to sustain a load factor of 3.8 times its weight. Meaning, every pound of weight added to an
aircraft requires that the structure be strong enough to support an a dditional 3.8 pounds.
An aircraft operated within the effective category should sustain a load factor of 4.4, and acrobatic
category aircraft must be strong /sturdy enough to resist 6.0 times their weight. The lift created by a
wing is determined by its air foil shape, angle of attack, speed through the air, and the air density.
When an aircraft takes off from an ai r field with a high -density altitude, it should accelerate with a
speed quicker than that, that would be need ed at sea level to provide enough lif t to permit takeoff;
Thus , an extended takeoff run is critical . The distance required could be longer than the available
runway. When in operati on from a high -density altitude airport, the Pilot’s Operating Handbook
(POH) or Airplane Flight Manual (AFM) should be consulted to determine the maximum weight
allowed for the aircraft under the conditions of altitude, temperature, wind, and runway conditions. [3]

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The flight characteristics of an airplane at gross weight with the center of gravity very near its mo st
aft limits are very different from those of the same airplane lightly loaded. For lift and weight to be in
equilibrium as to keep up the desired attitude of flight, more lift should be creat ed to balance the
heavy weight. To obtain this, the air craft must be flown at an increased angle of attack. As a result,
the wing will stall sooner (i.e. at a higher airspeed) when the airplane is fully loaded than when it was
light-weight . Stalling speed in turns (that is, at increased load factors) will also be high er. In fact,
everything connected with lift are going to be affected. Take -off runs are going to be longer, angle of
climb and rate of climb will be reduce d and because of the increased drag generated by the higher
angle of attack, fuel consumption will be higher than normal for any given airspeed. Severe g -forces
are a lot likely to cause stress to the airframe supporting a heavy payload.
An aft center of gravity makes the airplane less stable, making recovery from manoeuvres more
difficult. The airplane i s more easily upset gusts. However, with an aft center of gravity, the airplane
stalls at a slightly lower airspeed. To counteract the tail heaviness of the aft center of gravity, the
elevator must be trimmed for an up load. The horizontal stabilizer, as a result, produces additional lift
and also the wings, correspondingly, hold a slightly lower angle of attack. An airplane with a forward
center of gravity, being nose heavy, is more stable but more pressure on the elevator controls will be
necessary to rai se the nose – a fact to remember on the landing flare. The forward center of gravity
means a somewhat higher stalling speed another fact to remember during take -offs and landings. [4]

1.3.1 Effect of weight

Most modern aircraft are so designed that if all seat s are occupied, all baggage allowed by the
baggage compartment is carried, and every one of the fuel tanks are full, the aircraft will be grossly
overloaded. This type of design need s the pilot to g rant great consideration to the requirements of the
trip. If maximum range is need ed, occupants or baggage should be left behind, or if the maximum
load should be carried, the range, d etermined by the amount of fuel on board, must be reduced. Some
of the issues caused by overloading an aircraft are:
• The aircraf t will need a higher takeoff speed, which results in a longer takeoff run.
• Both the rate and angle of climb will be reduced.
• The service ceiling will be lowered.
• The cruising speed will be reduced.
• The cruising range will be shortened.
• Maneuverab ility will be decreased.
• A longer landing roll will be required because the landing speed will be higher.
• Excessive loads will be imposed on the structure, especially the landing gear. [6]
1.4 Weight Changes

The weight of the airplane can be altered by alt ering the fuel load. Gasoline has considerable weight,
6 pounds per gallon of 30 gallons may weigh more than one passenger. But it should be recalled that

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if weight is decreased by reducing fuel, the range of the airplane is reduced. During flight, fuel bu rn is
often the sole weight modification that takes place. As fuel is used, the airplane becomes lighter and
performance is better. Changes of fixed equipment have a serious impact on the weight of the
airplane. An airplane can be burdened by the installat ion of extra radios or instruments. Repairs or
modifications usually have an e ffect on the weight of the airplane.

1.4.1 Balance Limits

The position of the center of gravity along its longitudinal axis affects the stability of the airplane.
There are forward a nd aft limits established by the aircraft design engineers beyond which the center
of gravity should not be located for flight. These limits are set to assure that sufficient elevator
deflection is available for all phases of flight. If the center of gravi ty is too far forward, the airplane
will be nose heavy, if too far aft, tail heavy. An airplane whose center of gravity is too far aft may be
dangerously unstable and will possess abnormal stall and spin characteristics. Recovery could also be
difficult if not impossible as a result of the pilot is running out of elevator control. It is, therefore, the
pilot's responsibility when loading an airplane to see that the center of gravity lies within the
recommended limits.
If the center of gravity is too far for ward, the airplane will be nose heavy, if too far aft, tail heavy. An
airplane whose center of gravity is too far aft may be dangerously unstable and will possess abnormal
stall and spin characteristics. Recovery may be difficult if not impossible because the pilot is running
out of elevator control. It is, therefore, the pilot’s responsibility when loading an airplane to see that
the center of gravity lies within the recommended limits.
Usually the Airplane Owner's Manual lists a separate weight limitation for the baggage compartment
additionally to the gross weight limitation of the total aircraft. This is an element to which the pilot
should pay close attention, for overloading the baggage compartment (even if the plane itself is not
overloaded) may move the center of gravity too far aft and affect longitudinal control. The Airplane
Owner's Manual may specify such things as the seat to be occupied in solo flight (in a tandem seating
arrangement) or which fuel tank is to be emptied first. Such instructions should be carefully complied
with. As the flight of the aircraft progresses and fuel is consumed, the weight of the airplane
decreases. Its distribution of weight also changes and hence the center of gravity changes. The pilot
should take into consideratio n the situation and calculate the weight and balance not just for the
beginning of the flight but also for the end of it.
1.4.2 Stability and Balance Control

Balance refers to the location of the center of gravity (CG) of an air craft and is vital to air craft
stability and safety in flight. The center of gravity is a point in which the airplane would balance if it
were suspended at that point. The prime concern of airplane balancing is the fore and aft location of
the center of gravity along the longitudinal axis . Location of the center of gravity with re spect to the
lateral axis is also vital. For each item of weight existing to the left of the fuselage centerline, there is
an equal weight existing at a corresponding location on the right. This may be upset, howe ver, by
unbalanced lateral loading. The position of the lateral center of gravity is not computed, but the pilot
must be aware that adverse effects will certainly arise as a result of a laterally unbalanced condition.
Lateral unbalance can occur if the fue l load is mismanaged by supplying the engine(s) inconsistent ly
from tanks on one aspect of the air craft. The pilot can make amends for the resulting wing -heavy
condition by adjusting the aileron trim tab or by holding a constant aileron control pressure. H owever,

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this places the air craft controls in an d out-of-streamline condition, increases drag, and end up in
decreased operating efficiency.

Figure 1. 5: Lateral or longitudinal unbalance
The center of gravity is not essential ly a fixed point; its location depends on the distribution of weight
in the air craft. As variable load items are shifted or expended, there is a resultant shift in center of
gravity location. The pilot should realize that if the center of gravity of an airplane is displaced too far
forward on the longitudinal axis, a nose -heavy condition will result. Conversely, if the center of gravity
is displaced too far aft on the longitudinal axis, a tail -heavy condition will result. It is possible that an
unfavorable locat ion of the center of gravity could produce such an unstable condition that the pilot
could not control the airplane.
When in event, flying an airplane which is out of balance can increase pilot fatigue with obvious effects
on the safety and efficiency of f light. The pilot’s natural correction for longitudinal unbalance could be
an amendment of trim to get rid of the excessive control pressure. Excessive trim, however, has the
effect of not only reducing aerodynamic efficiency but also reducing primary contr ol travel distance in
the direction the trim is applied.
1.4.3 Effects of Adverse Balance

Adverse balance conditions affect airplane flight characteristics the same manner as those mentioned
for an excess weight condition. In addition, there are two essential a irplane characteristics which may
be seriously affected by improper balance; these are stability and control. Loading in a n exceedingly
nose-heavy condition causes issue s in controlling and raising the nose, especially during takeoff and
landing. Loading i n an exceedingly tail-heavy condition includes a mo re serious effect on the
longitudinal stability and may cut back the air craft’s capability to recover from stalls and spins.
Another undesirable characteristic produced from tail -heavy loading is that it p roduces very light
control forces. This makes it easier for the pilot to unknowing ly overstress the air craft.
Limits for the location of the airplane’s center of gravity are established by the manufacturer. These
are the fore and aft limits beyond which th e center of gravity should not be located for flight. These
limits are p rinted for every aircraft in the Type Certification Data Sheet or Aircraft Specification. If,
after loading, the center of gravity is not within the allowable limits, it will be necess ary to relocate
some items within the airplane before flight is attempted. The forward center of gravity limit is
commonly established at a location that is determined by the landing characteristics of the air craft. It

21
may be possible to maintain stable an d safe cruising flight if the center of gravity is located ahead of
the prescribed forward limit; but during landing which is one of the most critical phases of flight,
exceeding the forward center of gravity limit may cause problems. Manufacturers purpose ly place the
forward center of gravity limit as far rearward as possible to aid pilots in avoiding damage to the
airplane when landing.
A restricted forward center of gravity limit is also specified to assure that sufficient elevator deflection
is availabl e at minimum airspeed. When structural limitations or large stick forces do not limit the
forward center of gravity position, it is located at the position where full -up elevator is required to
obtain a high angle of attack for landing.
The aft center of g ravity limit is the most rearward position at which the center of gravity can be
located for the most critical maneuver or operation. As the center of gravity moves aft, a less stable
condition occurs which decreases the ability of the airplane to right it self after maneuvering or after
disturbances by gusts. For some airplanes the center of gravity limits, both fore and aft, may be
specified to vary as gross weight changes. They may also be modifi ed for certain operations such as
acrobatic flight, retracti on of the landing gear, or the installation of special loads and devices which
change the flight characteristics. The actual location of the center of gravity can be altered by many
variable factors and is usually controlled by the pilot. Placement of bagg age and cargo items
determines the center of gravity location. The assignment of seats to passengers can also be used as a
means of obtaining a favorable balance. If the air craft is tail -heavy, it is only logical to place heavy
passengers in forward seats. [6]
1.4.4 Basic Principles of Weight and Balance Computations

By determining the weight of the empty air craft and also adding the weight of everything loaded on
the air craft, a total weight can be determined. But to distribute this weight in such a way that the entire
mass of the loaded airplane is balanced around a point (center of gravity) which must be located
within specified limits.
The point where the airplane will balance can be determined by locating the center of gravity, which
is mentioned in the defi nitions of terms, the imaginary point where all the weight is concentrated. To
provide the mandatory balance between longitudinal stability and elevator control, the center of
gravity is usually located slightly forward of the center of lift. This loading condition causes a nose –
down tendency in flight, which is desirable during flight at a high angle of attack and slow speeds. A
safe zone within which the balance point (center of gravity) must fall is called the center of gravity
range. The extremities of the range are called the forward center of gravity limits and aft center of
gravity limits. These limits are usually specified in inches, along the longitudinal axis of the airplane,
measured from a datum reference. The datum is a random point, established by air craft designers,
which may vary in location between completely different airplanes.

22

Figure 1. 6: Weight and balance illustrated
The distance from the datum to any component part of the airplane, or any object loaded on th e
airplane, is called the arm. When the object or component is located aft of the datum, it is measured in
positive inches; if located forward of the datum, it is measured as negative inches, or minus inches.
The location of the object or part is commonly referred to as the station. If the weight of any object or
component is multiplied by the distance from the datum (arm), the product is the moment. The
moment is the measurement of the gravitational force which causes a tendency of the weight to rotate
about a point or axis and is expressed in pound -inches. [Figure 4]
This is sometimes called torque. The metric equivalent, used commonly in engineering, is the
Newton -meter (N -m). The pound -inch unit is used here only for the reason that most aircraft in
service were designed with the older engineering units, as much as 50 or more years ago. Using the
older units for weight and balance calculations cause s no difficulty for m any pilots as long as the
aircraft will be balanced properly for flight.
1.5 Weight and Bal ance Restrictions

The airplane’s weight and balance restrictions ought to be closely followed. The loading conditions
and empty weight of a specific aircraft may differ from that found in the Airplane Flight Manual or
Pilot’s Operating Handbook because mo difications or equipment changes may have been made.
Sample loading issues in the Airplane Flight Manual or Pilot’s Operating Handbook are meant for
guidance only; therefore, each airplane must be treated separately. Although an air craft is certified for
a fixed maximum gross takeoff weight, it will not safely take off with this load under all conditions.
Conditions which have an effect on takeoff and climb performance such as high elevations, high
temperatures, and high humidity (high density altitudes) ma y require a reduction in weight before
flight is attempted. Other factors to consider before takeoff are runway length, runway surface,
runway slope, surface wind, and the presence of obstacles. These factors m ight require a reduction in
weight before flight.
Some airplanes are designed so that it is impossible to load them in a manner that will place the center
of gravity out of limits. These are usually small airplanes with the seats, fuel, and baggage areas
located near the center of gravity limit. These airplanes, however, can be overloaded in weight. Other
airplanes can be loaded in such a manner that they will be out of center of gravity limits even though
the useful load has not been exceeded. Because of the effects of an out -of-balance or overweight
condition, a pilot should always be sure that an airplane is properly loaded. [3]

23
1.5.1 Importance of Aircraft Weight and Balance

An aircraft is a dynamic device that need s a careful balance between all of its forces to maintain safe
and e conomical flight. The lift creat ed by the wing is focused at a point approximately one -third ["that
is, the CG must be to the rear of the forward limit and" of the way back from the leading edge, and to
provide stability, the center of gravity, or that point at which all of the aircraft weight can be
considered to be concentrated, is located slightly before the center of lift. This location results in a
moment that tries to pitch the nose of the aircraft down, but this nose -down moment is balanced by a
tail m oment, which pulls the nose up.
The magnitude of tail moment is determined by the airspeed, and it drops off when the airplane slows
down. The weight remains constant and ahead of the center of lift, so it pulls the nose down and the
airplane will automa tically regain the speed it has lost. If the center of gravity falls outside of the
rather narrow limits allowed by the aircraft designer, serious control problems can result. If it is
allowed to get too far aft, the stall characteristics will be adversely affected, and if it is too far
forward, there will be difficulty in slowing the airplane for landing.
The structure of the aircraft is designed to safely accept certain loads, but in flight through rough air
and on the impact of a hard landing, the force s due to acceleration could overload the structure and
cause it to fail. When an aircraft is designed , limits are put on its maximum weight, and restrictions
are set up regarding the range within which the center of this weight is allowed to va ry. A part of
the certification procedur e for an airplane is to determine that its weight and balance are within the
allowable limits, and this information is furnished with the aircraft as part of its operations manual.
It is the responsibility of t he pilot to know before each flight that his aircraft is properly loaded, that it
does not exceed the allowable gross weight, and that the center of gravity is within the allowable
range. The weight of an aircraft changes throughout its operational lif e as equipment is added or
removed and as repairs are made. All of these changes should be monitored and also the weight and
balance data utilized by the pilot must be kept up -to-date. This is the responsibility of the aviation
maintenance technician.
Very close track must be kept of the weight and balance of aircraft used to carry passengers or cargo
for hire, and they must be reweighed periodically and have their center of gravity recompute ["that
is, the CG must be to the rear of the forward limit an d"]. Large aircraft have several rows of seats,
some of which are ahead of the center of gravity range and some behind it and there are often
both forward and aft baggage compartments. This wide range of loading possibilities makes the us e
of charts or alternative aids to loading a necessity for the pilot to make sure that the center of gravity
is within the allowable range.
1.5.2 Factors Affecting Weight and Balance in Aircraft

Thrust and Drag: The forces are formed so that lift acts behind we ight and thrust acts below drag.
Preferably, the pitching moments should terminate each other out, but in practice, a secondary
technique of balancing is used. It is usually done by the tail -plane.

24

Tail-plane : It provides the force required to balance an y remaining pitching moment. Long Moment
Arm necessitates smaller aerodynamic surface. It generates either a down force or an up force. At
slow speed the force is inadequate to balance. In such cases, the elevator is deflected to increase or
reduce the for ces. It creates drag, which is also known as Trim Drag.

CG Constraints : To ensure that the aircraft is accurately controllable and stable, the manufacturer
places front and rear constraint or limits for the location of CG, both on the ground and in flig ht. If the
aircraft’s CG falls on or within these lines, the CG is in limits. [5]
Forward Limit : The forward limit is decided by the authority that the tail -plane has to trim out the
increased nose down pitching moment established by the location of the CG . This authority is decided
by the range of the elevator, the airspeed of the aircraft, and the size of the tail -plane and the lever
arm.

25

Aft Limit: As the CG is positioned backward, there is a reduction in the nose down pitching moment.
This reduces dow n force demand and large elevator deflections. It permits additional control surface
deflection to be used to alter pitch a ngle of the aircraft. There is additionally a reduction in trim drag
and stick fixed force making it easier for the pilot to maneuver the controls.

Neutral Point: When the CG moves further aft, the ability to control increases with the decrease in
stability. The rear limit is set to keep up sufficient stability. The rear limit is forward of the neutral point.
The neutral point occurs at point called the aerodynamic center, which in subsonic aircraft is normally
located at the ¼ chord point behind the leading edge.

26

Aerodynamic Center : For this calculation, the AC is considered to be located at 25% chord. Which in
reality it moves be tween 23 -27% of the chord for a subsonic aircraft. If the CG is located on the AC,
there is no net pitching moment (increased controllability, neutral stability) causing it to not be
desirable.
Since it is not advis able in an air transport aircraft, The c g rear limit is set forward of this point. This
gives the aircraft longitudinal static stability. The distance between the neutral point and the CG is
therefore termed the static margin or CG margin. Therefore, in r eality the real limit is forward of the
neutral point, and the CG is not permitted to get this far aft.

27

Fuel Consumption : In flight, the mass of the airplane reduces through consumption of fuel. If the
aircraft has fuel tanks with varying arms, as the trip fuel is burnt off, the CG position v aries due to this
consumption and the drop s in aircraft mass.

Fowler Flaps : Transport aircraft oftentimes uses Fowler flaps as trailing edge flaps. These flaps
translate rearward on extension, moving rearward furthermore lowering the trailing edge. This leads to
the CG moving rearward with the flaps’ extension and forward with the flaps’ retraction .

28
Landing Gear Design : Most aircraft have main gears that retract laterally (no effect on the longitudinal
CG). However, the raising of a forward retracting nose gear moves the CG forward and the other way
around .
Cargo : When an aircraft is loaded, The CG and mass must be within prescribed limits of the aircraft.
If the cargo should shift in flight, the aircraft will become either too stable or uncontrollable .
Additionally, cabin crew and passenger movements have an effect on the trim of the aircraft. [5]

29
2. Aircraft Stability and Control

2.1 Introduction

Among the numerous however, often -overlooked obstacles to power driven flight overcame by the
Wright brothers was the question of the way to build an air craft that was stable enough to be controlled
and maneuvered by a pilot. It has been shown that the Wright’s first power driven aircraft in 1903 was
thus unstable that solely the Wrights themselves flew it, because of the intensive self -training on their
previous glider versions in 1902 (Abzug, 2002). As they and other different aviation pioneers took steps
to unravel the stability and controls problem s, the capabilities and performance of airplanes increa sed
significantly. In the early days of flight, it was observed that certain designs of airplanes were more
stable and controllable than others, but it was not until the 1930s that a large amount theory existed to
explain why. Much of the modern stability and control theory and specifications were not developed
until the 1960s or later (Abzug, 2002).
Airplanes of all sizes should be capable of stable, trimmed flight in order to be controllable by a human
pilot and useful for various applications. Stable fli ght by a n individual’s pilot is achieved only if the
aircraft possesses static stability, a characteristic that requires aerodynamic forces on the airplane to act
in a direction that restores the plane to a trimmed condition after a disturbance. Dynamic st ability
requires that any oscillations in aircraft motion that result from disturbances away from equilibrium
flight conditions must eventually dampen out and return to an equilibrium or “trimmed” condition.
Certain dynamic instabilities may be tolerated by the pilot, depending mostly upon pilot skill and
experience. If computer -augmented feedback control is used even statically unstable aircraft can be
flown successfully (Abzug, 2002). Both static and dynamic stability characteristics can be expect ed
when an air craft is still with in the design stage of development. Static stability can be foreseen using
information about the air craft aerodynamic center and the center of gravity in addition to alternative
geometric parameters. Dynamic stability is foresee n using the airframe geometric and inertial properties
to calculate the natural frequencies, damping ratios and time constants of the characteristic dynamic
modes of the six degree -of freedom aircraft model. The handling qualities of an ai rcraft are stated to be
a measure of how well an air craft is able to perform its designated mission. They are typically evaluated
using flight test data and the pilot feedback on performance . [7]
2.2 Static Stability

Stability is the tendency of an aircraft to return to a steady state of flight without any help from the pilot,
after being disturbed by an external force.
The following qualities must be met by an Aircraft:
• Adequate stability to maintain a uniform flight condition.
• The ability to recover from different disturbing influences.
• Sufficient stability to reduce the workload of the pilot.
• Appropriate response to the controls so that it may achieve its design performance with suitable
manoeuvrability.
There are two broad categories of stability, static and dynamic stability.

30
Static stability can be envisioned by a ball (or any object) on a surface. At the beginning, the ball is at
equilibrium. The ball is then displaced from the equilibrium position and its initial presence is observed.
There are three form s of static stability –
i. Statically stable form (Positive)
ii. Statically unstable form (Negative)
iii. Neutrally stable form. (Neutral)
The Statically stable form takes place if the forces and moments on the body caused by a disturbance
tend initially to return the body towards its equilibrium position, the body is statically said to be stable.
The Statically unstable form occurs if the forces and moments are such that the body continues to move
away or continue in the direction from its equilibrium posit ion after being disturbed.
The Neutrally stable form occurs if the body is disturbed but the moments remain zero, the body stays
in equilibrium
Figure 2.2: Below shows the diagrammatic representation of the different forms of static stability.
.

Figure 2.2A: Statically stable ball

Figure 2.1B: Statically unstable ball

Figure 2.1C: Neutrally stable ball
The importance to this study, are the first two cases (Statically stable and unstable) as neutral stability
occurs very rarely. A very important poi nt is that static stability deals only with the initial tendency
for an aircraft to return to the equilibrium.
2.2.1 Static Stability in Aircrafts

An air craft possesses static stability if the aerodynamic forces and moments introduced on the
airframe as a result of it being disturbed from equilibrium tend to act in a direction that will return the
aircraft to an equilibrium condition. Static stability is compar able to a marble in a bowl. If the marble
is disturbed from its equilibrium position at the b ase of the bowl, gravitational forces at all other

31
positions will tend to pull it back towards the b ase. Aerodynamic forces and moments on a statically
unstable aircraft will tend to m aneuver it away from a trimmed flight condition once it is disturbed
from eq uilibrium. This condition is comparable to a marble on the top of a smooth hill / surface or
balancing a pendulum bulb upside down. This condition would be nearly impossible for a human pilot
to control, But it could also be possible if some form of feedbac k control is used. Static stability can
be thought of as a special case (steady -state) of the aircraft dynamics. It is exhibited in both the
decoupled longitudinal and lateral -directional axes. It will a dditionally become clear that both
longitudinal and l ateral -directional static stability are a prerequisite for dynamic stability.

a. Longitudinal Static Stability
Longitudinal stability is motion about the lateral axis. It is the axis about which a particular type of
stability takes place. Thus, lateral s tability is about the longitudinal axis (rolling), directional stability
is about the normal axis (yawing) and longitudinal stability is about the lateral axis (pitching).
Longitudinal static stability is crucial to make sure that pilot s may successfully f ly an air craft without
any stability augmentation. It depends principally on a parameter referred to as the static margin,
which is defined as the distance between the aircraft center of gravity and the neutral point of the
aircraft, normalized by the mean geometric chord, of the wing. An air craft with longitudinal static
stability should initially possess a positive (nose -up) pitching moment from the combination of the
aerodynamic forces and moments on the wing and tail. For flying wings, an airfoil with a natural
positive pitching moment must be chosen or washout and wing sweep must be combined to give the
airplane a natural positive pitching moment. If this condition is met, a positive static margin, defined
as the center of gravity in front of the neutra l point, will ensure static stability. For an aircraft to
exhibit a Longitudinal Static Stability it will tend towards the trim angle of attack when displaced.
Static stability can be simply represented by plotting the pitching moment of the aircraft about its
center of gravity versus angle -of-attack as shown in Figure 2.2.
Longitudinal Static Stability is the measure ment of the reaction of the aircraft once pitch is disturbed.
In other words, it describes the rate of change of pitching moment in response t o a change in angle of
attack. For example , if the nose is raised by some means, increasing the lift coefficient, is there a
natural tendency for the aircraft to lower the nose and return its initial condition . [6]

Figure 2.2: A moment coefficient curve for an airplane possessing longitudinal static stability. Any
disturbance away from trim will result in aerodynamic forces and moments which will act in a direction
that will tend to return the plane to equilibrium .

32
A small static margin (center of gravity near the neutral point) will provide marginal static stability and
will be represented by a nearly flat line on the graph in Figure 2.2. A large static margin can provide a
steep line and might make the aircraft feel “nose -heavy.” It may cause the plane t o be less controllable
because it doesn’t respond to control inputs. These constraints on the static margin are presented in
Figure 2.3. It is also vital that the static margin be chosen because that will enable the plane to be
trimmed at a reasonable angl e of attack.

Figure 2.3: Aircraft center of gravity envelope. The C.G must fall within these limits set by the
stability and controllability of the aircraft. (Kimberlin, 206)
To consider longitudinal static stability in mathematical terms, we first have to determine the
aerodynamic center, It is usually the longitudinal location along the centerline of the aircraft measured
from the leading edge of the wing about which the pitching moment is constant over a range of
angles -of-attack. It is also the point in which the lift effectively acts. Equation 2.1 shows the
mathematical definition of the aerodynamic center of an aircraft:
0 /=d dMac
at
acx 2.1

Figure 2.4: Graphical definitions of the various parameters affecting air plane static stability.

33

b. Longitudinal Static Stability Margin
An aircraft is said to be statically stable if a disturbance from equilibrium generates a restoring force
that tries to return it to equilibrium. An aircraft is considered dynamically stable is it really does return
to equilibrium. A wing by itself is naturally unstable. In reality , both the wing and the fuselage have
negative, or destabilizing, moment coefficients. It is the tail that provides a stabilizing component. In
the normal flight regim e, these values are always constant, or nearly so. The total curve is normally a
sum of all the contributing components, fuselage, tail, nacelle, etc. The final curve of pitching
moment with respect to lift coefficient must have a negative slope to posse ss natural stability. The
slope of this curve is what the pilot recognize as the stability of the aircraft. A steep slope is felt as
very stable, a flatter slope is recognized as a less stable aircraft. (Figure 4). Note that in general this
should not be c onfused with responsiveness to control inputs. Stability refers to the natural response
of an aircraft when disturbed from equilibrium. While it actually requires less force to maneuver a less
stable aircraft, controls are usually created to be more or les s effective through other design aspects.
[1]

Figure 2.5 Longitudinal stability

Criteria for Longitudinal Static Stability

34
𝐶𝑀,0 must be positive and 𝜕𝐶𝑀,𝑐𝑔
𝜕𝛼𝑎 must be negati ve for an aircraft can be trimmed. Thus, from equation
3.4 for a longitudinal static stability to be achieved in an aircraft ,
𝐶𝑀,𝑎𝑐𝑤𝑏 <0,𝑉𝐻>0,𝛼𝑡>0 ⟹ 𝑖𝑡>0 𝑓𝑜𝑟 𝐶𝑀,0>0 2.2

Neutral Point and Static Margin
The neutral point is usually determined both analytically and by flight test. For this case, the analytical
approach will be the focus of the researcher. Many details of the aircraft geometry are required for the
determination of the neutral point and they include: Aircraft plan form, incidence angles, airfoil data,
etc. The neutral point is defined by Equation below;
𝛿𝐶𝑀,𝑐𝑔
𝛿𝛼𝑎=𝑎[(ℎ−ℎ𝑎𝑐)−𝑉𝐻𝑎𝑡
𝑎(1−𝛿𝜀
𝛿𝛼)] 2.3
𝑊ℎ𝑒𝑟𝑒 ℎ𝑛=ℎ𝑎𝑐+𝑉𝐻𝑎𝑡
𝑎(1−𝛿𝜀
𝛿𝛼)
Thus equation 2.4 is the static margin,
𝛿𝐶𝑀,𝑐𝑔
𝛿𝛼𝑎=𝑎(ℎ−ℎ𝑛)=−𝑎(ℎ𝑛−ℎ)=−𝑎×𝑠𝑡𝑎𝑡𝑖𝑐 𝑚𝑎𝑟𝑔𝑖𝑛 2.4

c. Lateral -Directional Static Stability
Similarly, to the longitudinal case, an ai rcraft possesses directional static stability if there is a slight
increase in sideslip results in a restoring yawing moment as well as a restoring side force. This is
sometimes called weathervane stability. Lateral (about the x -axis or roll axis) static stability is expressed
in terms of dihedral effect, or the stability derivatives for rolling moment due to sideslip or If they are
negative, then the ai rcraft will possess a positive lateral stability and will exhibit a negative rolling
moment (left wing down) for a positive sideslip (nose left).
2.2.2 Equilibr ium and the basic trim equation

An airplane is in a state of equilibrium when opposing forces are balan ced. The sum of forces acting
on an airplane is zero and the sum of the moments acting about the center of gravity is also zero.
Equilibrium in pitch can be achieved through various combinations of elevator and horizontal
stabilizer positions. An airplane is in a state of trim if equilibrium is achieved by positioning the
stabilizer such that the elevator is in null (mid) position. The stabilizer position required to balance the
airplane also depends on (1) Elevator position, (2) Flight speed, (3) Air densi ty, and (4) Airplane
configuration.
The pitching moment coefficient:
𝐶𝑀=𝑀
1
2𝜌𝑈2𝑆𝐶̅̅ 2.5
Pitching moment coefficient is significant to the definition of aerodynamic center of an airfoil. The
aerodynamic center is described to be the point on the chord line of the airfoil at which the pitching
moment coefficient does not vary with angle of attack . The pitchin g moment is considered to be
positive when it acts to pitch the airfoil in the nose -up direction. Therefore, airfoils cambers supported
at the aerodynamic center pitch nose -down of the aircraft, the pitching moment coefficient of the
airfoils is negative. [1]

35
The tailplane volume coefficient
𝑉̅=𝑙𝑇1𝑆𝑇
𝑐̅̅ 2.6
For an airplane to be trimmed, the sum of th e moments about the center of gravity must be zero as
shown in Equation 2.7. Referring to Figure 2.3 .
𝑀𝑐𝑔=𝑀𝑎𝑐+(𝑥𝑐𝑔−𝑥𝑎𝑐)𝑐̄𝐿−𝑙𝑡𝐿𝑡=0 at trim 2.7
Equation 2. 7 can be written in non -dimensional form as shown in Equation 2. 8.
𝐶𝑚𝑐𝑔=𝐶𝑚𝑎𝑐+(𝑥̄𝑐𝑔−𝑥̄𝑎𝑐)𝐶𝐿−𝑉̄ℎ𝐶𝐿𝑡=0 at trim 2.8
Where: 𝑥̄𝑐𝑔=𝑥𝑐𝑔√𝑐
𝑥̄𝑎𝑐=𝑥𝑎𝑐√𝑐

𝑉̄ℎ=𝑙𝑡𝑆𝑡√𝑐̄𝑆
The static margin, σ, is then defin ed in Equation 2. 9 as the non -dimensional distance:
𝜎=𝑥̄𝑐𝑔−𝑥̄𝑎𝑐𝐴 2.9
Aacx
is the aerodynamic center of the complete aircraft, or the point about which the pitching moment
is constant with angle of attack . For t ailless aircraft, it is also indicated as the neutral point. A positive
static margin is a qualification for longitudinal static stability. The magnitude of a static margin is the
most i mportant aircraft parameter on the longitudinal dynamic stability of an aircraft.
Mean Aerodynamic Center and Mean Aerodynamic Chord
The mean aerodynamic chord is the length parameter that is the mathematical equivalent of the
complete wing planform reduced to a single chord length, longitudinal position and angle of attack .
The entire wing is approximately replaced by the mean aerodynamic chord for all calculations relating
to stability. The mean aerodynamic center describes a point on the wing where the pitching moment is
not any longer a function of angle of attack. For a subsonic airfoil, this point is located at 25% of the
chord. The ability to treat both the mean aerodynamic center and also the pitching moment coefficient
as constants makes subsequent calculation much easier.

Figure 2.5 Mean aerodynamic center of a n airfoil
Mean Aerodynamic Center of an Airfoil
For big aircrafts like Boeing 737, the mean aerodynamic center moves aft by the addition of the
horizontal stabilizer. This new location is also referred to as the aircraft aerodynamic center or neutral
point. In order to satisfy the criteria for natural stability the aircraft CG should stay ahead of the
neutral point. The distance between the neutral point and the aircraft CG is referred to as the static

36
margin and is prese nted as a percentage of the mean aerodynamic chord. In the figure below, the final
summation of lift and moments acting on an aircraft is demonstrated.

Figure 6:Total Lift and Moment acting on aircraft
h = CG location as perce ntage of mean aerodynamic chord
hn = neutral point location as percentage of mean aerodynamic chord
NP = neutral point
CL = total life coefficient, CL = 2𝑊
𝑆𝜌𝜈2
Static Margin = hn – h
Cm = Total Moment
Static Margin = h n – h

2.3 Dynamic Stability

The analysis of static stability provides some measurement of the air craft dynamics, however only a
rather crude one. Of larger relevance, especially f or lateral motion, is th at the dynamic response of the
aircraft. it is possible for an airplane to be statically stable, yet dynamically unstable, resulting in
unacceptable characteristics. An aircraft is considered dynamically stable if it actually does r eturn to
equilibrium.
An airplane possesses dynamic stability if the amplitudes of any oscillatory motions induced by
disturbances eventually decrease to zero relative to a steady -state flight condition. This means that if
an air craft experiences a little disturbance from trimmed flight, it will eventually return to trim on its
own. This is comparable to a marble in a bowl eventually returning to rest at the b ase of the bowl. If
the amplitude of oscillatory motion instead tends to extend with time, the airp lane is said to be
dynamically unstable. Dynamic instabilities are clear ly undesirable, but certain mild dynamic
instabilities can be tolerated by a human pilot. If automatic controls are used, more severe dynamic
instabilities can also be tolerated. Graph ical representations of dynamic stability and instability are
shown in Figures 2.6 and 2.7.
To study dynamic stability, it is necessary to analyze the well -known differential equations of aircraft
motion. For small perturbations, these equations can be dec oupled into longitudinal and lateral –
directional portions, with 3 degrees of freedom in each. Small perturbation theory also allows us to

37
approximate the actual non -linear equations as linear differential equations with constant coefficients
while ignoring any less significant non -linear aerodynamic effects. This greatly simplifies the analysis
of the dynamic modes of aircraft motion.

Figure 2. 7: A graphical example of dynamically stable aircraft motion relative to a steady -state
condition

Figure 2. 8: A graphical example of dynamically unstable aircraft motion.

2.3.1 Static Forces and Moments on an Aircraft

There is an aerodynamic force created by the pressure (and shear stress i.e. Pressure is the force
perpendicular to the surface per unit area, whi le shear stress is the force along the surface per unit
area. Both have units of pascals) distribution over the wing surface. The resultant (net) force R can be
resolved into two components; the life L (perpendicular to the relative wind) and the drag (in the
direction of the relative wind).
If we consider only the pressure on the top surface of the wing. The net force due to that pressure
distribution, called F 1, points downward and is acting through point 1 on the chord line. The pressure
distribution on the bottom surface results in a net force F 2, pointing upward and acting through point 2
on the chord line. The total aerodynamic force on the wing is of course a summation of F 1 and F 2. If

38
F2 > F 1, there is life. Since the two forces do not act through an equivalent point there will be a net
moment on the wing.

Figure 2.9: The origin of the moment acting on an airfoil
The magnitude of the moment depends upon the point of reference regarding where the moment is
taken. If the moment is taken with respect to the leading edge, it is denoted by MLE. For subsonic
wings, it is often customary to take the moment about the quarter -chord point (i.e. the point that is a
distance c/4 away from the leading edge). This moment is denoted by Mc/4. Both MLE and Mc/4
vary with the angle of attack. However, a special point exists about which the moment essentially
does not vary with α. This point is called the aerodynamic center (ac). For that point;
Mac = constant (independent of angle of attack) 2.10
The moment coeffici ent about the aerodynamic center is defined as;
𝐶𝑀,𝑎𝑐≡𝑀𝑎𝑐
𝑞∞𝑆𝑐 2.11
Where
q is the dynamic pressure, S the wing area and c the chord length. The value of
acMC, is zero
for symmetric airfoils an d varies from -0.02 to -0.3 or so for cambered airfoils. [6]
2.4 Moment on an Aircraft
Considering looking at a wing only, we can now consider the complete aircraft, as shown in figure
2.7. In examining a n entire aircraft, the pitching moment about the cen ter of gravity (center of mass)
is of interest. The moment coefficient a round cg is defined comparably to the moment coefficient
about the ac;

Figure 2. 10: Contributions to the moment acting about the center of gravity

39
An airplane is in pitch equilibrium when the net moment about the center of gravity is zero.
𝑀𝑐𝑔=𝐶𝑀,𝑐𝑔=0 airplane is trimmed 2.12
Note that while drag plays an essential part in performance determination, its role is small for stability
and control. Its value is much less tha n that of the lift, and its acts not too far from the center of gravity,
so its effects are often neglected.
2.4.1 Center of Gravity of a Stable Aircraft
As with any other vehicle, the position of the center of gravity is of great importance. For an aircra ft,
it will determine its pitch stability. For some studies which analyzes the outer parts of the aircraft it
can be hard to determine a stable center of gravity as little is known about the placement of the inner
components such as avionics. A way to appr oach this is by looking at the stable case at set the center
of gravity accordingly as a requirement. Pitch stability is often described in terms of the stability
margin which must be positive according to the stability criteria and usually ranges from 5 – 10%.
Stability margin =𝑙𝑛.𝑝.−𝑙𝑐.𝑔.
𝑐̄ 2.13
Here,
..gcl is the center of gravity’s position away from the main wings MAC as shown in figure 2.9.
..pnl
is the position of the point on the aircraft which gives neutral pitch stabilit y which means it is
neither stable nor unstable. This point also is the aerodynamic center of the whole aircraft.

Figure 2.1 1: Center of gravity’s position
The neutral point position
..pnl can be shown with several approximations to beco me;
𝑙𝑛.𝑝. =𝑐̄×𝑐ℎ𝑡×𝑎ℎ𝑡
𝑎𝑤(1−𝑑∈ℎ𝑡
𝑑𝛼) 2.14
The yet unknown terms in equation 2. 14 can for unswept wing be estimated from;
𝑎𝑤=𝐴𝑅
2+𝐴𝑅×2𝜋 2.15
𝑎ℎ𝑡=𝐴𝑅ℎ𝑡
2+𝐴𝑅ℎ𝑡×2𝜋 2.16
𝑑∈ℎ𝑡
𝑑𝛼=(1+√𝜉2+1
𝜉)×𝑎𝑤
2𝜋𝐴𝑅 2.17

40
𝜉=2𝑙ℎ𝑡
𝑏 2.18
The higher the stability margin, the more stable is the aircraft considered to be. So if a stability margin
of 10% is desired,
..gcl must = 0.111m with
..pnl = 0.165. the center of gravity for the aircraft can be
seen in figure 2.10.

Figure 2.1 2: Displaying the position of the aircrafts center of gravity

41
3 Trim Considerations in Aircrafts
Establishing aircraf t trimmed states is of primary importance in a variety of engineering studies.
According to Duke and Berndt (2007), trim defines conditions for both design and analysis based on
aircraft models. The analysis point is broadly based on the conditions which is normall y associated
with trim conditions to facilitate the analysis or design. In simulation, the analysis points establish
initial conditions comparable to flight conditions. It is further added that, based on aerodynamic and
propulsion systems models of an aircraft, trim analysis can be used to provide e the data needed to
define the operating envelope or the performance characteristics. A trim state defines the initial
condition that the dynamics of interest is also studied. Trimming aims to bring the forces and
mome nts acting on the aircraft into equilibrium so that the three axes forces , and moments are all at
zero.

3.1 Simplification and Assumption of Trim Equation
Simplification and assumption are made in the elementary analysis of an airplane’s longitudinal stat ic
stability to aid the derivation of the basic trim equation:
• The airplane is in steady level flight or at an angle of climb or decent small enough for total lift to be
approximately equal to the aircraft weight.
• The aerodynamic force and moment deriv atives are constant.
• Compressibility effect are neglected, so that the aerodynamic coefficients are independent of Mach
number.
• The influence of aeroelastic effects is negligible.
• The CG is located close to the mean aerodynamic chord (MAC) of the win g.
• The lift forces act in a direction approximately perpendicular to the MAC of the wing.
• The thrust and drag forces are assumed to have a negligible influence on the static stability of the
airplane.
3.2 Trim Systems in Aircrafts

Although an aircraft can be operated through a wide range of attitudes, airspeeds, and power settings,
it has been designed to fly hands free within a very limited combination of these variables. Trim
systems are used to relieve the pilot requi rement to keep up a constant pres sure on the flight controls,
and usually consist of flight deck controls and small hinged devices connected to the trailing edge of
one or more of the primary flight control surfaces that are designed to minimize a pilot’s workload,
trim systems aerodynamicall y assist movement and position of the flight control surface to which they
are connected . Common form of trim systems include s trim tabs, balance tabs, antiservo tabs, ground
adjustable tabs, and an adjustable stabilizer.
1. Trim Tabs
The trim tab is one of th e most common types of tabs used in small single -engine airplanes. A trim tab
is attached to the trailing edge of an elevator, and it's controlled by moving a small control wheel in
the cockpit. The most common mechanism on small aircraft is a single trim tab attached to the trailing
edge of the elevator. Most trim tabs are manually controll ed by a small, vertically mounted control
wheel. However, a trim crank can be found in some aircraft. The flight deck control includes a trim
tab position indicator. Pla cing the trim control in the full nose -down position moves the trim tab to its
full up position. With the trim tab up and into the airstream, the airflow over the horizontal tail surface
tends to force the trailing edge of the elevator down. This causes th e tail of the air craft to mane uver

42
upward, and the nose to move down ward . If the trim tab is set to the full nose -up position, the tab
moves to its full down position. In this case, the air flowing under the horizontal tail surface hits the
tab and forces the trailing edge of the elevator up, reducing the elevator’s AOA. This causes the tail of
the airplane to maneuver down ward , and the nose to move up ward .
In spite of the opposing directional movement of the trim tab and also the elevator, control of trim is
natural to a pilot. If the pilot needs to apply constant back pressure on a control column, the demand
for nose -up trim is indicated. The normal trim procedure is to continue trimming until the aircraft is
balanced and the nose -heavy condition is no t any longer apparent. Pilots usually establish the
speci fied power, pitch attitude, and configuration first, and then trim the aircraft to ease the control
pressures that may occur for that flight condition. Whenever power, pitch attitude, or configuration is
changed, expect that retrimming will be essen tial to relieve the control pressures for the new flight
condition . The advantage of the trim tab is the complexity is reduced while the disadvantage is that
the drag is increased.

a

b
Figure 3.1: Trim Tab Working Principle

43
From figure 3.1b, the operation involves the rolling the wheel in the nose an upward position , that
makes the tab to move down and when the wheel is rolled in the nose down direction, the tab moves
up. When the trim tab is moved up or down, it sticks out into the free air stream, and deflects the
elevator in the opposite direction.
2. Adjustable Stabilizer
Some aircrafts have a movable tab on the trailing edge of the elevator, while others have an
adjustable stabilizer. With this structure , linkages pivot the horizontal stabilizer about its rear spar.
This is done by use of a jackscrew mounted on the leading edge of the stabilator. On small aircraft,
the jackscrew is operated with a trim whe el or crank. On larger aircraft, it is motor driven. The
trimming effect and flight deck indications for an adjustable stabilizer are related to those of a trim
tab. The advantage of the adjustable stabilizer is that the drag is lower with respect to the t rim tab
while the disadvantage is that the constructive complexity is increased.

Figure 3.2: Adjustable Stabilizer Working Principle
3. Balance Tabs
The control forces may be enormous ly high in some aircraft, and in order to decrease them,
manufacturer s of the aircraft use balance tabs. They are alike to trim tabs and are hinged in
approximately the same places as trim tabs. The important difference between the two is that the
balan ce tab is coupled to the control surface rod so that when the primary cont rol surface is moved in
any direction, the tab automatically moves in the o ther direction. The airflow striking the tab
counterbalances some of the air pressure against the primary control surface and enables the pilot to
move more easily and hold the c ontrol surface in position. If the linkage between the balance tab and
the fixed surface is adjustable from the flight deck, the tab acts as a combination trim and balance tab
that can be adjusted to any desired deflection. Some aircraft have very heavy co ntrol loads, especially
at high speeds. Th is is where balance tabs come in handy. As seen in figure 3.3 below, by moving the
balance tab in the opposite direction, the control load on the yoke is significantly reduced, making the
aircraft easier to fly.

44

Figure 3.3: Balance Tab Working Principle

4. Antiservo Tab
Antiservo tabs are sim ilar to the balance tabs, but they move in the opposite direction. For instance,
when the elevator or stabilator moves up wards , the antiservo tab moves in the same direction.
Antiservo tabs work s in the same method as balance tabs except, instead of moving in the opposite
direction, they move in the same direction as the trailing edge of the stabilator. In addition to
decreasing the sensitivity of the stabil ator, an antiservo tab also functions as a trim device to relieve
control pressure and maintain the stabilator in the desired position. The f astened end of the linkage is
on the opposite side of the surface from the horn on the tab; once the trailing edge of the stabilator
moves up, the linkage forces the trailing edge of the tab up. When the stabilator moves down, the tab
will also moves down. Equal ly, trim tabs on elevators move opposite of the control surface.

a

45

b
Figure 3.4: An Antiservo tab Working Principle

In small aircraft, it increases the control feel, and helps prevent you from over -controlling your
aircraft's pitch. One of the most popular examples of the antiservo tab is on the Piper Cherokee.
Without it, the plane would be much easier to pitch up and down, but it would also be easy to over –
control, and possibly overstress the airframe.
5. Ground Adjustable Tab
Alot of small aircraft have a non -movabl e metal trim tab on the rudder. This tab is bent in one
direction or the o pposite whereas on the ground to apply a trim force to the rudder. The correct
displacement is determined by trial and error. Usually, small change s are necessary until the aircraft
no longer skids left or right throughout normal cruising flight. The ground tab is used to keep the
plane flying in a coordinated level flight.

Figure 3.5: A Ground Adjustable Tab
3.3 Types of Tabs

Trim Tab s: Trim tabs trim the aircraft in flight. To trim means to correct any tendency of the aircraft
to move toward an un favorable flight attitude. Trim tabs control the balance of an aircraft so that it

46
maintains a straight and level flight without pressure on the control column, control wheel, or rudder
pedals.

Figure 3.6: Trim tab
Working principles : The tab has a change able linkage which is adjustable from the cockpit.
Movement of the tab in one direct ion causes a deflection of the control surface in the opposite
direction. Most of the trim tabs fixed on aircraft are mechanically operated from the cockpit through
an individual cable system. However, some aircraft have trim tabs that are operated by an electrical
actuator. Trim tabs are either controlled from the cockpit or adjusted on the ground before taking off.
Trim tabs are installed on elevators, rudders, and ailerons.

Figure 3.7: Trim tab working p rinciples
Servo Tabs : Servo tabs are very similar in operation and appearance to the trim tabs just discussed.
Servo tabs, sometimes referred to as Sight tabs, are used primarily on the large main control surfaces.
They aid in moving the control surface an d holding it in the desired position. Only the servo tab
moves in response to movement of the cockpit control. The servo tab horn is free to pivot to the main
control surface hinge axis. The force of the airflow on the servo tab then moves the primary cont rol
surface. With the use of a servo tab less force is needed to move the main control surface.

Figure 3.8: Servo tab
Balance Tabs : The linkage is designed in such a way that when the main control surface is m oved,
the tab moves in the opposite direction. Thus, aerodynamic forces, acting on the tab, assist in moving
the main control surface. Thus, reducing the effort the pilot needs to apply, to move the control
surface.

47

Figure 3.9: Balance tab
Anti -balance Tab : This works in the opposite direction to a servo tab. It deploys in the same
direction as the control surface, making the movement of the control surface more difficult and
requires more force appli ed to the controls by the pilot.

Figure 3.10: Anti -balance tab
Spring Tabs : Spring tabs are similar in appearance to trim tabs, but serve an entirely different
purpose. Spring tabs are used f or the same purpose as hydraulic actuators, that is, to aid in moving a
primary control surface.

Figure 3.11: Spring tab
Working principles: There are various spring arrangements used in the linkage of the spring tab. On
some aircraft, a spring tab is h inged to the trailing edge of each aileron and is actuated by a spring –
loaded push -pull rod assembly which is also linked to the aileron control linkage. The linkage is
connected in such a way that movement of the aileron in one direction causes the spring tab to be
deflected in the opposite direction. This provides a balanced condition, thus reducing the amount of
force required to move the ailerons. The deflection of the spring tabs is directly proportional to the
aerodynamic load imposed upon the aileron .

48

Figure 3.12: Spring tab working principles
At low speeds the spring tab remains in a neutral position and the aileron is a direct manually
controlled surface. At high speeds, however, where the aerodynamic load is great, the t ab functions as
an aid in moving the primary control surface. To lessen the force required to operate the control
surfaces they are usually balanced statically and aerodynamically.
3.4 The Trim Condition of an Aircraft

A second condition must be met in co njunction with stability for straight and level flight, is the trim
condition. To this end the aircraft must have a positive moment (nose up) at zero lift coefficient.
Combined with a negative pitching moment slope, the aircraft must seek equilibrium at a positive lift
coefficient. The trim condition is a summation of the pitching moments produced by the wing,
fuselage tail, engine and CG etc. when the net moment goes to zero.
The pilot has a few options for changing the trim condition. Most convenient and common is the
elevator. This changes the tail contribution. The flaps can also be used. Flaps change the camber of
the wing and thus its pitching moment at any given angle of attack. The CG can be moved, although
this is not easily achieved on demand in mo st small aircraft. Fuel burn does change the contribution of
CG and thus moves the trim point. The figure below shows how elevator deflection shifts the total
Cmα curve vertically. The aircraft then finds a new angle of attack and a new trim air speed.

Figure 3.13: Effect of Elevator Deflection on Trim Speed

49
Neutral Point
The neutral point can be determined both analytically and by flight test. For the analytical approach,
many details of the aircraft geometry are required: Aircraft plan fo rm, incidence angles, airfoil data,
etc. The neutral point is defined by Equation below;
ℎ𝑛=ℎ𝑛𝑤−𝑎𝑓
𝑎𝑤+𝑉ℎ𝑎𝑡
𝑎𝑤(1−𝑑𝜀
𝑑𝛼)
Where
𝑉ℎ=𝑙𝑡𝑆𝑡
𝑆𝑤𝐶
Vh is the tail volume ratio
Lt is the distance between mean aerodynamic center of the tail and the aircraft CG
St is the effective area of the tail
Sw is the area of the wing
C is the mean aerodynamic chord
hnw is the aerodynamic center of the wing
at is the life curve slope of the fuselage
aw is the life curve slope of the wing
at is the life curve slope of the tail
𝑑𝜀
𝑑𝛼 is the effect of down wash on the tail with respect to angle of attack

3.4.1 Experimental Determination of Neutral Point of an Aircraft

The neutral point can be determined experimentally with minimal instrumentation. For a given
trimmed flight condition and CG location there is a nearly linear relationship between change in
elevator position and change in lift coefficient. This relationship can be exploited to find the neutral
point in flight.
The process involves conducting at least two flight tests with the aircraft l oaded to two different CG
positions, preferably near both ends of the envelope. During each test flight the aircraft is trimmed at
moderate power for hands -free level flight. Then, without re -trimming, the aircraft is manually held
off trim speed long enou gh to capture steady state data for airspeed and elevator position. The altitude
must remain in a reasonable band for each test point. A 1,000’ window is sufficient. It is convenient to
alternate one point above the trim speed and one point below to keep t he data points at nearly the
same altitude. If for example, the trim speed was 168 KIAS, the test points would be: 175, 160,
185,155, 190, 150, 195, 145, 200, 140, 205 KIAS, etc.
Since each aircraft speed is converted to a lift coefficient, weight must al so be known. Scales and a
fuel totalizer can be used to obtain instantaneous weight during the flight. Elevator position needs to
be captured to a resolution of better than 1/10th of a degree. This fine resolution is needed since each
test flight will see the total elevator position vary by less than one degree. All data can be recorded to
a micro SD card via a PIC micro controller. Aircraft weight and CG location were measured prior to
each flight.

50
Each CG position will produce a curve similar to charts be low. It is the slope of these curves that is of
interest. As the CG moves aft and approaches the neutral point, the slope reduces. At the neutral point
it would become a horizontal line as lift coefficient would become independent of elevator input. The
slopes obtained from these two curves are plotted in the next chart.

Figure 14B:Stability Test, Aft CG
Figure 14A :Stability Test, Forward C G

A line is then drawn through these points and extrapolated until crossing zero on the x -axis. This
intersection represents the CG location where pitch is no longer a function of elevator position, in
other words, the neutra l point. See the figure below.

Figure 14C:Neutral Point Result by Flight Test

Trim Tabs
Trim tabs can be used by the pilot to trim the vehicle at zero control force for any desired speed. Trim
tabs are small control surfaces mounted at the trailing edges of primary control surfaces. A linkage is
provided that allows the pilot to set the angle of the trim tab, relative to the primary control surface, in
a way that is independent of the deflection of the primary control surface. Deflection of the trim tab
creates a hinge moment that causes the elevator to float at the angle desired for trim. The geometry of
a typical trim tab arrangement is shown in Fig. 3.15.

51

Figure 3.15: (a) Typical location of trim tab on horizontal control (elevator), and (b) sc hematic
illustration of aerodynamic forces responsible for hinge moment due to trim tab deflection.
Here, zero control force corresponds to zero hinge moment, or

𝐶ℎ𝑒=0=𝐶ℎ𝑒0+𝐶ℎ𝛼𝛼+𝐶ℎ𝛿𝑒𝛿𝑒+𝐶ℎ𝛿𝑡𝛿𝑡 3.1

and the trim tab deflection that achieves this for arbitrary angle of attack and control deflection is

𝛿𝑡=−1
𝐶ℎ𝑒0(𝐶ℎ𝑒0+𝐶ℎ𝛼𝛼+𝐶ℎ𝛿𝑡𝛿𝑡) 3.2

so the tab setting required for zero control force at trim is

𝛿𝑡𝑡𝑟𝑖𝑚=−1
𝐶ℎ𝛿𝑡(𝐶ℎ𝑒0+𝐶ℎ𝛼𝛼𝑡𝑟𝑖𝑚 +𝐶ℎ𝛿𝑒𝛿𝑒𝑡𝑟𝑖𝑚 ) 3.3
Where
𝛼𝑡𝑟𝑖𝑚 =−𝐶𝐿𝛿𝑒𝐶𝑚0−𝐶𝑚𝛿 𝑒𝐶𝐿𝑡𝑟𝑖𝑚
∆
And
𝛿𝑒𝑡𝑟𝑖𝑚 =𝐶𝐿𝛼𝐶𝑚0+𝐶𝑚𝛼𝐶𝐿𝑡𝑟𝑖𝑚
∆

Figure 3.16: Variation in trim tab setting as function of velocity for stable aft tai l

52

Substituting 𝛼𝑡𝑟𝑖𝑚 and 𝛿𝑒𝑡𝑟𝑖𝑚 values into Equation 3.3 gives the required trim tab setting as

𝛿𝑡𝑡𝑟𝑖𝑚=−1
𝐶ℎ𝛿𝑡(𝐶ℎ𝑒0+𝐶𝑚0
∆(−𝐶ℎ𝛼𝐶𝐿𝛿𝑒+𝐶ℎ𝛿𝑒𝐶𝐿𝛼)+1
∆(−𝐶ℎ𝛼𝐶𝑚𝛿 𝑒+𝐶ℎ𝛿𝑒𝐶𝑚𝛼)𝐶𝐿𝑡𝑟𝑖𝑚 ) 3.4

Note that the coefficient of C Ltrim in this equa tion – which gives the sensitivity of the trim tab
setting to the trim lift coefficient, can be written as;

𝑑𝛿𝑡
𝑑𝐶𝐿=−𝐶ℎ𝛿𝑒
𝐶ℎ𝛿𝑡∆(𝐶𝑚𝛼−𝐶ℎ𝛼𝐶𝑚𝛿𝑒
𝐶ℎ𝛿𝑒)=−𝐶ℎ𝛿𝑒
𝐶ℎ𝛿𝑡∆(𝑥!𝑁𝑃
𝑐̅−𝑥𝑐𝑔
𝑐̅)𝐶𝐿!𝛼 3.5

And the equation can be re -written as

𝛿𝑡𝑡𝑟𝑖𝑚=−1
𝐶ℎ𝛿𝑡[𝐶ℎ𝑒0+𝐶𝑚0
∆(−𝐶ℎ𝛼𝐶𝐿𝛿𝑒+𝐶ℎ𝛿𝑒𝐶𝐿𝛼)+𝐶ℎ𝛿𝑒
∆𝐶𝐿!𝛼(𝑥!𝑁𝑃
𝑐̅−𝑥𝑐𝑔
𝑐̅)𝐶𝐿𝑡𝑟𝑖𝑚] 3.6

Thus, the tab setting for trim is a linear function of trimmed lift coefficient whose slope is proportional
to the control free static margin. This variation is shown schematically for a conventional (aft tail)
config uration in Fig. 3.3.

Moment Around cg
𝑀𝑐𝑔=𝑀𝑎𝑐𝑤𝑏+𝐿𝑤𝑏(ℎ𝑐−ℎ𝑎𝑐𝑐)−𝑙𝑡𝐿𝑡 3.1
Divide by 𝑞∞𝑆𝑐 𝑎𝑛𝑑 𝑛𝑜𝑡𝑒 𝑡ℎ𝑎𝑡 𝐶𝐿.𝑡= 𝐿𝑡
𝑞∞𝑆𝑡
𝐶𝑀,𝑐𝑔=𝐶𝑀,𝑎𝑐𝑤𝑏+𝐶𝐿𝑤𝑏(ℎ−ℎ𝑎𝑐)−𝑙𝑡𝑆𝑡
𝑐𝑆𝐶𝐿,𝑡 3.2
or it can be re -written as
𝑪𝑴,𝒄𝒈=𝑪𝑴,𝒂𝒄𝒘𝒃+𝑪𝑳𝒘𝒃(𝒉−𝒉𝒂𝒄)−𝑽𝑯𝑪𝑳,𝒕 3.3
Where
𝑉𝐻=𝑙𝑡𝑆𝑡
𝑐𝑆
𝐶𝐿𝑤𝑏=𝑑𝐶𝐿𝑤𝑏
𝑑𝛼𝛼𝑎,𝑤𝑏=𝑎𝑤𝑏𝛼𝑎,𝑤𝑏
𝐶𝑙,𝑡=𝑎𝑡𝛼𝑡=𝑎𝑡(𝛼𝑤𝑏−𝑖𝑡−𝜀)
Where 𝜀 is the downwash at the tail due to the life on the wing
𝜀=𝜀0+(𝛿𝜀
𝛿𝛼)𝛼𝑎,𝑤𝑏
𝐶𝐿,𝑡=𝑎𝑡𝛼𝑎,𝑤𝑏(1−𝛿𝜀
𝛿𝛼)−𝑎𝑡(𝑖𝑡+𝜀0)
𝐶𝑀,𝑐𝑔=𝐶𝑀,𝑎𝑐𝑤𝑏+𝑎𝛼𝑎[(ℎ−ℎ𝑎𝑐)−𝑉𝐻𝑎𝑡
𝑎(1−𝛿𝜀
𝛿𝛼)]+𝑉𝐻𝑎𝑡(𝑖𝑡+𝜀0) 3.4

53
References

[1] T. M. Young, *Performance of the Jet Transport Airplane: Analysis Methods, Flight Operations
and Regulations, First Ed ition.

[2] FAA “Aircraft Weight & Balance Handbook”, FAA -H-8083 -1A
[3] Pilot’s Operating Handbook (POH) and Airplane Flight Manual (AFM)
[4] CAE Oxford Aviation Academy Mass and Balance
[5] Ion fuiorea *Mass and balance of an aircraft
[6] CAE oxford Aviation Acedemy Principles of Flight