Flight Machines And Aerospace Engineering Engineering Essay

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Flight mechanics, a branch of aerospace engineering, concerns with the motion of aerospace vehicles comprising the three complementary and interrelated technologies of performance, guidance and navigation, and flight dynamics [1]. Flight dynamics relies on an understanding of numerous physical disciplines including aerodynamics, propulsion, structures and engineering mechanics. With the familiarization of the subject it could be realised that the affiliation of flight dynamics to the vibration of mechanical bodies also applies to other multi-degree of freedom physical systems such as ships, submarines automobiles, missiles and spacecraft. In a dynamic system, both stability and control are of great concern [2]. Flight dynamics ensure the full balance of performance, guidance and navigational missions and tasks that can be executed successfully and safely in all atmospheric conditions by a pilot, either human or automatic, through appropriate alterations to engine thrust and deflections of control surfaces. Flight dynamics is influenced by factors such as inertial and aerodynamic moments acting on an aircraft, the control surfaces' aerodynamic effectiveness, and the vehicle's response to commands from the pilot [1].

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Many other parameters such as static equilibrium, stability, manoeuvrability, controllability etc. are involved in dynamics of flight. There occurs a need to understand the concept of static equilibrium to comprehend the idea of stability, which is of prime importance to any aircraft.

"Static equilibrium is the state of an aircraft when the total aerodynamic, propulsive and gravity loads acting on it in a steady flight condition, with the throttle and control surfaces fixed at appropriate settings, satisfy the criterion for static equilibrium of zero resultant force and zero resultant moment acting on the aircraft"[1]. With the notion of static equilibrium comprehended, general meaning of stability can be considered. "A dynamical system is said to possess stability, if, when slightly disturbed from a state of equilibrium, it tends to return to and remain in that state, the disturbance acting only for a finite time" [3].

To completely understand the idea of stability, consider an aircraft in equilibrium. If the aircraft is subjected to a disturbance, its response will be a time varying motion. The aircraft can termed as stable if this disturbance die out with time and the aircraft retains its equilibrium state without pilot intervention. This form of stability can be termed as asymptotic stability. If the aircraft was unstable, the response builds up with time and the aircraft does not return to the state of equilibrium [1].

Stability can be classified into static and dynamic stability. Static stability and dynamic stability together constitute overall stability. The aircraft's response to disturbance is associated with the inherent degree of stability, in each of the three axes, that eventuates without any pilot action. The aircraft stability is expressed in relation to each axis: lateral stability-stability in roll, directional stability-stability in yaw and longitudinal stability-stability in pitch. The latter is the most important stability characteristic, while the other two being interdependent [3].

Unmanned Ariel Vehicle Background

Unmanned aerial vehicles are the unpiloted aircraft which can be controlled by remote or pre-programmed flight plans or more complex dynamic systems. They are often preferred for the missions that are dull, dirty or dangerous for manned aircraft. They are mostly used in military application where there is threat to human life. It is also used in reconnaissance, climate (temperature, humidity etc) and urban (controlling crimes) monitoring. It is quite obvious that future war would be more disastrous and losing human life is inevitable so use of UAVs will become necessary. So research and development to improve UAVs must be undertaken. The problem that currently faced by industry is to have the separate airworthiness criteria for UAVs. If UAVs have their own airworthiness criteria, it will result in industry having something to guide them while developing existing UAV or manufacturing entirely new UAV. At present, certification of UAVs in industry are assigned on individual basis which makes certification process troublesome but it can be easy if there is common certification process. It is quite obvious that airworthiness criteria should be similar or shows equivalence with the currently used by manned aircraft. The airworthiness criteria currently set for commercial aircraft contains the paragraph of passenger's safety so it automatically covers safety goals for the people on the ground. The UAVs airworthiness criteria can be evaluated by omitting all the paragraph of passenger's safety and substituting them by the safety of people and property on the ground. Structural requirement for UAVs also defers from the manned aircraft as there will be no cabin pressurisation cycle in UAVs. So ones airworthiness criteria is evaluated it can be easily implemented to do structure and material optimisation. Currently many different material and structural design are used to give optimum result for strength, weight and long life. So it is important to do structure optimisation to achieve light weight, high strength and long life. The following report shows the technique to evaluate the airworthiness criteria and to implement it in improving the structure and material of given UAV.

Birth of a Concept

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"When a young Serbian immigrant stepped off the boat at Ellis Island in New York Harbour in1884, he is said to have arrived with four cents in his pocket, a book of poems he has written, and his plans for a remotely controlled unmanned aeroplane" (about Nikola Tesla in ref[5] page11).

Tesla, an electrical engineer by passion, developed an idea of a mechanical flying machine in 1876. Despite of all the ridicules faced, Tesla learned about a European office opened by Thomas Alva Edison and got himself an opportunity to be an employee by the same. In an electrical exposition in 1898, Tesla presented a new form of torpedo called the "teleautomaton", which was a four-foot-long boat in a tank of water that could be made to stop or go, turn left or right, and blink its light by sending out different radio frequencies. Tesla's invention was discarded by the press and the military, accusing it to be a trick and of no practical value.

Fig 1 The Teleautomaton

The Concept Takes Flight

Originated in the same era as manned aviation, the limiting factors of unmanned flight were the lack of technology mainly in the areas of automatic stabilization, remote control and autonomous navigation. The first person to attempt to address all three in a single unmanned aircraft design was Elmer Ambrose Sperry. The idea of the automatic flight stabilization popped out from his initial approach to stabilize and improve the safety of manned flight. The original idea was to give a pilot in vertigo, a mechanical awareness of wing level. This was experimented by installing gyros. Later it was further experimented by using smaller gyros in the aircraft's pitch, roll and yaw axes coupling them to aircraft control by servo motors. This having had many draw backs and failures, a development came about in the form of mounting all the gyros into a single platform, providing a horizontal reference to any attitude in flight. With success of this design Sperry gained the Navy approval and funding for an "Aerial Torpedo" [5].

The torpedo development program had two phases- to develop initially a gyrostabilized bomb carrying drone with a fully automatic torpedo, and the addition of radio control for directing the torpedo.

Technical Challenge No. 1: Launch

The order was for six Speed Scouts airframe and the first one was delivered in October 1917. This design was the first purpose built unmanned aircraft. The design modifications were made by mounting the torpedo on a modified Marmon car and running it at about 80 miles per hour, thereby simulating a wind tunnel. Launching the drone without upsetting the stabilizing gyroscope was the next big challenge. To solve this problem the drone was initially released to slide down a wire cable. Later this idea was modified and succeeded by transferring the energy of a 5000lbs concrete block dropping 30 feet through a pulley and cable system to the drone. Later the modified Marmon car was used as a launch platform. In the series of failures sits a singular accomplishment. On 6th March 1918, a Curtiss-Sperry Ariel torpedo catapulted cleanly into the air, flew its planned 1000 yard flight then dived at its preset distance into the water off Copiague, Long Island. True to the definition of the UAV, it was recovered and later reflown. Later designs had a modified launch control system based on flywheel-catapult. Using a flywheel spinning at 2175 rpm, the design was capable of accelerating 1950 lbs to 90 kts within 150 ft [5].

Technical Challenge No. 2: Remote Control

With the technical knowhow of launching the aircraft known to both Navy and Army, attention was turned to radio engineers, who were conducting parallel researches to completely control the aircraft by a remote. The Army proposed an idea of modifying the Messenger biplane into an aerial torpedo. If these planes validated the concept, the Army planned to produce Messenger Aerial Torpedoes (MATs).

The army began flight testing by October 1920, focusing on refining the flight control system. Even though the automatic launching system was successful, problems were encountered with the distance measuring system and the gyroscope that controlled the azimuth. Hitting the target was defined as passing within 2o of it laterally and being within 2% of the total range over or under in distance [5].

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In July 1921, Lieutenant Colonel Gearhart, recommended a new gyroscope, both larger and manufactured to tighter tolerances, to be installed as a cure for recurring problems of maintaining accurate heading. Resumed tests showed promising results and on October in the third flight test, an aircraft fitted with the new gyroscope flew for 75 miles and for 119 minutes, maintaining a relatively straight line. By the end of the year design modifications were as accurate as the current technology allowed and the next step of design was correction for unpredictable atmospheric changes. At this point the possibility of adding the capability for remote control by radio was surfaced. In 1922, The Radio Section of The Army Air Services Engineering Division, headed by Lieutenant Redman, developed and installed the radio equipment in several MAT's. Several flight tests were conducted and on 29th June 1922, a MAT took off from Mitchell Field and flew to a target which was 30 miles. Constant corrections to the flight path were made by utilizing radio waves. It then flew back to Mitchell Field, flew over the airfield, then being commanded to make a 360 degree turn overhead before the pilot took over and laded the aircraft. That flight marked the birth of Remote Controlled flight [5].

Categories of UAV

UAVs fall under two main categories of lethal and non-lethal. The table below shows the types of UAVs in each division.

Lethal

Non Lethal

Cruise missiles

Reconnaissance equipment

Radar decoy support

Short-range anti-radiation UAVs

Passive TV surveillance equipment

Tactical air-launched decoy anti-radar devices

Fire and Forget missile with an "intelligent" guidance system

Passive IR surveillance equipment

High-flying meteorological UAVs

Relay or Data link stations

Civil & Police tasks

Target designation equipment

Border patrol

ECM equipment

200-mile EEZ surveillance

Aerial targets

Drug interdiction

SIGINT and ELINT equipment

Environmental research

Emergency microwave radio relay

SAR--Sea-air-rescue

Table 1.UAV categorisation on the basis of role played by them in the market

UAVs can also be classified in terms of the MTOW and performance. The table below gives the following classification: -

UAV categories

Acronym

Range (km)

Altitude (m)

Endurance (h)

T-O mass (kg)

Micro

Μ

<10

250

1

<5

Miniature

Mini

<10

1502 to 3001

<2

<30 (1502)

Close range

CR

10 to 30

3,000

2 to 4

150

Short range

SR

30 to 70

3,000

3 to 6

200

Medium range

MR

70 to 200

5,000

6 to 10

1,250

Medium range endurance

MRE

>500

8,000

10 to 18

1,250

Low-altitude deep penetration

LADP

>250

50 to 9,000

0.5 to 1

350

Low-altitude long endurance3

LALE

>500

3,000

>24

<30

Medium-altitude long endurance

MALE

>500

14,000

24 to 48

1,500

High-altitude long endurance

HALE

>2,000

20,000

24 to 48

12,000

Special purpose UAVs

Unmanned Combat Aerial Vehicle

UCAV

±1,500

10,000

±2

10,000

Lethal

LETH

300

4,000

3 to 4

250

Decoy

DEC

0 to 500

5,000

<4

250

Stratospheric

STRATO

>2,000

20,000 to 30,000

>48

TBD

Table 2.Suggested UAV type definition

UAVs market trends

Chart 1.Showing UAVs market share in the industries for 2009.

Number

Representing

1

Micro

2

Miniature

3

Close range

4

Short range

5

Medium range

6

Medium range endurance

7

Low-altitude deep penetration

8

Low-altitude long endurance

9

Medium altitude long endurance

10

High altitude long endurance

11

Unmanned Combat Aerial vehicle

12

Stratospheric

Table 3.Numbers representing UAVs categorisation.

The UAVs that fall into category of 'Medium range' and 'Miniature' hold the majority of the percentage in the market out of all other UAVs. The UAVs that fall into category of 'High altitude and long endurance are only of 4.3% out of total UAVs exist in the market.

Airworthiness of UAVs

Currently there is not any specific document representing common safety standard to be followed by UAVs manufacturer or developers. UAVs are certified on the individual basis which makes the process troublesome for the UAVs manufacturers and also airworthiness authority.

The current National and European Regulation are as follow: -

National Regulation

Civil and military are the two regulatory regimes in the United Kingdom. Military aircraft requirements are the matter of Minister of defence. An aircraft use for defence purpose or any aircraft which the Secretary of State for Defence certifies should be treated as military aircraft.

Any aircraft which is not military aircraft should comply with the civil requirement led by United Kingdom aviation safety legislation. There is no special provision for aircraft used in police, customs or other similar services.

The Air Navigation Order 2005 and the Rules of the Air Regulation 2007

The most important civil requirements are set out in the ANO. The requirements set in the ANO and the Rules of the Air are related to operational rules, personnel licensing, equipment requirements, aerodrome regulation and regulation of air traffic services apply to all non- military aircraft, organisations, facilities and individuals. The non-military aircraft should fall into one of the exempt categories listed in Annex II to the EASA regulation, so it should not need to be compiled with EASA regulation and it remains subject of only national airworthiness authority. According to ANO the non-military aircraft which is registered in UK and fall outside the EASA Regulation and Implementing Rules must have a certificate of airworthiness or a permit to fly issued by CAA (or be operating under A or B Conditions) under the ANO, unless it is a "small aircraft" as defined in the ANO.

According to ANO any unmanned aircraft weighting not more than 20 kg is defined as small aircraft. None of the above main regulation is applicable to small aircraft. The following are the requirement that has to be met by small aircraft in order to fly:

Article 98 states that

A person should avoid any article or animal (whether it is attached to a parachute or not) to be fallen off the small aircraft so life of the people and property on the ground are not endangered.

The person in charge of small aircraft which is over 7 kg without its fuel but all other payloads are to be included shall not fly it if: -

the person in charge of the aircraft is unsure that flight can be made safely.

has got no permission from appropriate air traffic control to be operated in class A,C,D or E airspace.

the permission from the air traffic unit is not obtained for the given aerodrome to be watched for specific time.

exceeds height above 400 feet above the surface (exception for those who have condition met in 'b' and 'c'.)

it is use for aerial purpose other than agreement with a permission issued by the CAA.

LIGHT UAS CAA policy

There has been a special provisions made by CAA for the development of 'LIGHT UAS' (UAVs which does not weigh more than 150 kg as if it is more then it will not fall into exempt categories of the EASA regulation).

EASA Regulation

The European Aviation Safety Agency has been set up by EC Regulation 1592/2002 to deal with requirements for implementing rules regarding airworthiness certification and continuing airworthiness. Two Implementing rules are set out complying detailed requirements for airworthiness certification and continuing airworthiness. The EASA regulation and its Implementing rules are not applicable to the aircraft that are engaged in military, customs, police or similar services (State aircraft). However it is responsibility of the EU member state to ensure they comply with the objectives of EASA regulation as far as possible. Few civil aircraft are also exempted from the EASA Regulation and its Implementing rules. These exempt categories are Annex II to the EASA Regulation. The exempt categories that can be related to UAVs are:

Initial design of an aircraft which was intended for military purpose only;

aircraft specifically designed or modified for research, experimental or scientific purpose and likely to be produce in very limited number; and

unmanned aircraft with an operating mass of less than 150 kg.

Aircraft that does not fall into one of the exempt categories listed above has to comply with EASA Regulation and its Implementing rules and it should have an EASA airworthiness certificate. The aircraft that falls in one of the exempt categories listed above or it is a State aircraft remains the subject of national regulation in terms of airworthiness certification and continuing airworthiness are concerned. European regulation does not deal with regulation requirement concerning equipment requirement, operational rules, personnel licensing, aerodrome regulation and regulation of air traffic services so these all remains the subject of national regulation for all categories of aircraft.

The table below represent the CAA policy for UAVs that are to be operated within UK airspace and shows the operating constraint that are more likely to be applied.

UAV Mass

Commercial Use (Aerial work)

< 7 kg

'Small Aircraft' under ANO Art 155

Minimum operational constraint (See Note 1)

No airworthiness standards

7-20 kg

'Small Aircraft' under ANO Art 155

Operational constraints required by ANO Art 98(2)(a) - (d) and additional aerial work constraint (See Note 2)

CAA Permission require under ANO Art(2)(e) which is subject to further constraints as the CAA thinks fit (See Note 3)

No airworthiness standards

20-150 kg

Exemption required including constraint at Notes 2 and 3

Impact kinetic energy must be determined to be not more than 95 KJ (See Note 4)

Airworthiness recommendation from accredited body (See Note 5)

> 150 kg

Existing national operating rules

EASA airworthiness standards

Aircraft Stability and Control

Introduction

Flight dynamics is concerned with the motion of an aeroplane in response to various forces and moments

Inertia forces

Aerodynamic damping forces and moments resulting from the aeroplane's angular motion

Aerodynamic forces and moments resulting from linear motion

Aerodynamic forces and moments due to control movement

Gravitational forces

Propulsive forces

Aeroplanes with longitudinal plane of symmetry are considered in this theory background.

Axes System

The aerodynamic forces and moments acting on an aircraft are always described by its disturbance from the equilibrium position. The equilibrium condition and disturbed condition are elucidated by a system of axes. The following sign conventions are adopted throughout this project. The figure below shows the typical axes system of an aircraft while dealing with any segment of flight dynamics.

Figure 2

From the figure, following can be inferred

Aircraft Centre of Gravity O

Aircraft Mass M

Longitudinal Axis Ox

Lateral Axis Oy

Directional Axis Oz

Velocity component of centre of gravity along Ox U

Velocity component of centre of gravity along Oy V

Velocity component of centre of gravity along Oz W

Angular velocity about Ox p

Angular velocity about Oy q

Angular velocity about Oz r

Moment of Inertia about Ox Ix

Moment of Inertia about Oy Iy

Moment of Inertia about Oz Iz

Product of Inertia with respect to Oxy Ixy

Product of Inertia with respect to Oyz Iyz

Product of Inertia with respect to Ozx Izx

External forces acting in the aircraft along Ox, Oy and Oz are X, Y, Z respectively. These external forces will induce aerodynamic moments on the aircraft. These moments are denoted by L, M and N in roll, pitch and yaw respectively.

The external forces acting on the aircraft constitutes of aerodynamic forces and gravitational forces. Aerodynamic forces are denoted as Xa, Ya and Za and gravitational forces are denoted by Xg, Yg and Zg. So it can be said that the total external force acting on the aircraft will be a sum of aerodynamic and gravitational forces. The situation is different in case of moments. Since the aircraft is considered to be symmetrical the origin of the axes system passed through the centre of gravity of the aircraft. Due to the location of the origin at the CG, there will be no moments due to gravitational forces. The only moments acting on the aircraft will be because of aerodynamic forces.

Equations of Motion Referred to Moving Axes

The motion of an aircraft can be expressed in a set of equations, with each equation referring to the longitudinal, lateral and directional axes. The aircraft motion can be categorized into motion parallel to longitudinal axis, ie, Ox, motion parallel to lateral axis, ie, Oy and motion parallel to directional axis, ie, Oz.

These equations of motion are expressed below.

Parallel to Ox

(3.1)

Parallel to Oy

(3.2)

Parallel to Oz

(33)

Where , ,signifies , , with t being time.

Given below are another set of equations formulated to express the angular motion about the three axes.

Angular motion about Ox

(3.4)

Angular motion about Oy

(3.5)

Angular motion about Oz

(3.6)

Due to the nonlinearity of these equations, some assuptions have to be considered for derivation purposes. The idea is to linearise these equations for ease of manipulation.

Aircraft is initially in trim condition with no degree of bank, yaw or sideslip

All disturbances to the aircraft motion are small

Aircraft has aplnae of symmetry

Aircraft mass is distributed symmetrically

Airflow over the aircraft is quasi-steady

When the aircraft is trimmed the equations of motion becomes

Motion parallel to Ox

(3.7)

Motion parallel to Oy

(3.8)

Motion parallel toOz

(3.9)

Due to the symmetry of mass distribution

(3.10)

Therefore the eqatuons of angular motion becomes

(3.11)

(3.12)

(3.13)

Expression for angular velocities

Pitch angle

Roll angle

Yaw angle

Angular velocity can be expressed expressed as rate of change these angles.

Therefore

Where p, q and r being roll, pitch and yaw rates respectively.

Determination of Aerodynamic Forces and Moments

For the evaluation of the aerodynamic forces and moments, the concept of reference condition must be comprehended. The derivations for aerodynamic derivatives come about due to any disturbances from this referece condition.

The Reference Condition

Consider an aircraft in equilibrium straight and level flight as shown in the following figure

G

M

U

X

W

Z

Figure 4.1

For an aircraft flying at a forward speed U, a reference set of axes can be defined where G is the aircraft CG, is horizontal, lying along the flight path and is vertically down. The forces acting on the aircraft will be as a result of thrust, aerodynamics and gravity. The forces due to thrust and aerodynamics can be defined as X and Z in the and directions respectively, and the pitching moment about the CG as M. Because the aircraft is in equilibrium:

X = 0; Z + W = 0; M = 0 (4.1)

The disturbed condition

Disturbed condition refers to the condition of the aircraft when a pulse disturbance induced to it while in the reference condition. The motion of the aircraft is depicted in the figure below.

U+u

x

G

q







W z

Figure 4.2

Now arises a situation to introduce another set of axes called the body axes, which can be represented as , which will coincide with the reference axes in the equilibrium condition, but will move with the aircraft. The aircraft motions can now be defined in terms of velocity components U + u, w along the Gx, Gz axes, together with a pitch angle  and pitch rate q as shown in Figure. The assumption is that the quantities u, w, , q are assumed to be small. Also the forces and moments due to aerodynamics and thrust will be resolved along the body axes.

Aerodynamic Derivatives

As previously explained, the aircraft eequations of motion are generally formulated in terms of disturbances of the aircraft from some reference condition. Therefore the aerodynamic forces and moments acting on the aircraft depends on its motion which could be expressed by the velocity components U+u, v, w, the angular rates p, q, r and the aileron, elevator and rudder deflections ,  and .

for simplicity of estimation of aerodynamic forces and moments the disturbances are assumed to be small. The area of interest is to determine the increments in forces and moments due to the disturbances.

To do this a mathematical formula could be made use of.

Consider a change in pitching moment M due to a disturbance u, w, q and elevator angle . M will be given by:

(4.2)

The partial derivatives etc are known as aerodynamic derivatives.

These partial derivatives are given a special notation. They could be written as

(4.3)

With the mathematic concept of the aerodynamic derivatieves understood, they can be formally defined as

"The rate of change of any aerodynamic force or aerodynamic moment with restect to one disturbance quatities, all other disturbances being assumed Zero"

The aerodynamic derivatives are partial derivatives of the forces and moments X, Y, Z, L, M, N with respect to the disturbance quantities u, v, w, p, q, r, , , , where these derivatives are determined with respect to the undisturbed flight condition.

By considering the asuumptions that has been earlier established, a table of these derivaties are made and is as shown below.

u

v

w

P

q

r







X

Xu

0

Xw

0

0

0

0

Y

0

Yv

0

0

0

0

Z

Zu

0

Zw

0

0

0

0

L

0

Lv

0

0

Lp

0

Lr

L

0

L

M

0

Mw

0

Mq

0

0

M

0

N

0

Nv

0

0

Np

0

Nr

N

0

N

Motions of the Aircraft

The aircraft motions can be distinguished between that those are longitudinal and those that are lateral.

The longitudinal motions are characterised by the following disturbance quantities:

u, w, q, 

and the forces and moments causing longitudinal motion are:

X, Z, M

The most important longitudinal derivatives are:

Lateral motions are characterised by the following disturbance quantities:

v, p, r, , 

and the forces and moments causing lateral motions are:

Y, L, N

The most important lateral derivatives are:

Caculation of Longitudinal Aerodynamic Derivatives

Since this project focuses only on the longitudinal aerodynamic stability derivatives, the report will only show derivation of the same. Lateral derivatives will not be considered.

Determination of Xu

Consider an aircraft flying straight and level with forward speed U as shown in the figure

L

T D

W

When aircraft is flying straight and level it satisfies the condition or equilibrium, or it can be said that the aircraft is in the reference condition. For an aircraft to be in equilibrium all the aerodynamic forces should balance out. Ie, the lift will balance the weight, so no gain or loose in altitude and the thrust will balance drag, ie, no acceleration or deceleration.

Mathematically it can be expressed as

X = T - D = 0; Z = W - L = 0

In order to determine Xu and Zu, a small perturbation 'u' in the forward speed is introduced. Due to this small disturbance there occurs changes in the forces in longitudinal and lateral sense. Ie, X and Z will change as the aircraft speed increases from U to U+u. Assuming there is no rotation of body axes following expressions are obtained.

(4.4)

(4.5)

Therefore

(4.6)

assuming 'u' to be a small disturbance Xu can be expressed as

(4.7)

Assuming the variation of thrust due to change in forward speed is negligible, the compressibility effect can be neglected. (4.7) can be rewritten as

(4.8)

From (4.5) Zu can be calculated by

(4.9)

Therefore

(4.10)

Neglecting aeroelastic and compressibility effects

(4.11)

Determination of Xw and Zw

Consider an aircraft to be in equilibrium. If a perturbation 'w' in vertical velocity is induced, it will give rise to a rotation in the direction of the lift and drag. This condition of flight is depicted in the figure below.

L

G

D

U

X

Incidence 

Freestream velocity direction

w Z

Figure 4.4

The change in incidence α can be expressed by

(4.12)

Hence the X and X components of force may be written:

(4.13)

These could be rewritten as

(4.14)

From equation (4.12)

(4.15)

However

(4.16)

The aerodynamic derivatives are evaluated by setting w=0 which implies  = 0. Therefore

(4.17)

Thus

(4.18)

Determination of Mu and Mw

For an aircraft in equilibrium, there will not be any pithching moment acting on it.

Consider a pulse disturbance acting on it so that it can induce a pitching moment.

The pitching moment of an aircraft about its Centre of Gravity can be written as

(4.19)

Consider there is a perturbation in forward velocity

Therefore

(4.20)

And so

(4.21)

Before any disturbance acted on the aircraft it was at trim. Therefore Cm = 0. If 'u' is set as zero then

(4.22)

For calculating Mw

(4.23)

Thus

(4.24)

Determination of Xq, Zq and Mq

These are the derivatives due to pitch rate. The figure below shows a conventional aircraft configuration.

Tail-off aerodynamic centre

CL

Cm0

CG

h0

h

Weight

Datum

The most important contribution to these derivatives will come from the tail. If the aircraft experiences a nose-up pitch rate q, then the tail will experience a downwards velocity of:

Hence the tail will experience an increase in incidence of

Therefore

(4.25)

In practice, Xq is negligible. Zq may be determined noting that qlT/U is a change in tail incidence and the change in overall aircraft lift coefficient comes only from the tail.

The tail coefficient of lift can be expressed as

(4.26)

Therefore

(4.27)

Tail lift coefficient is defined with respect to tail area . thus

(4.28)

(4.29)

where is the tail volume ratio.

For calculating Mq pitching moment equation has to be introduced.

(4.30)

Then

(4.31)

Therefore

(4.32)

Static Stability

The atatic stability that is dealt with in this section is longitudinal static stability as the scope of this project doesnot include lateral derivatives.

For an aircraft to be longitudianly stable, any disturbance in angle of attack while in the equilibrium state must produce a response that will tend to restore the aircraft back in equilibrium. Ie, for a nose up angle of attack the aircraft must produce a nose down pitching moment.

Mathematically

or (5.1)

this can be illustrated by

 (+ve)

M (-ve)

Figure 5.1

More correctly (1) can be written as

(5.2)

a plot of Cm against CL can be shown as

Cm

CL

Figure 5.2

The more stable the aircraft, the larger in magnitude is dM/d (or dCm/dCL) and so the greater the restoring moment required to produce trim.