Modeling of Motion and Flying Charactrstics of an Airplane

7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane

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6 D O F S i m u l a t i o n P a g e | 1

Published by : Manish Tripathi

AMITY UNIVERSITY

MATLAB SIMUATION OF 6 DEGREE OF FREEDOM OF

AIRCRAFT

Submitted by:-Aabid Nabi Khandey, 1

Manish Tripathi, 16

Mushfiq Sarfaraz Yasin, 17

Sadhana Singh 22


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Submitted to:-Dr Sanjay Singh

Table of ContentsLIST OF FIGURES :- ................................................................................................................................... 3

Introduction ............................................................................................................................................ 4

MODELING OF MOTION AND FLYING CHARACTRSTICS OF AN AIRPLANE .......................................... 4

STATIC VS DYNAMIC STABILITY ........................................................................................................... 4

Dynamically Stable Motions............................................................................................................ 4

Dynamically Unstable Motion ......................................................................................................... 5

Dynamically Neutral Motion ........................................................................................................... 5

The Differential 6 Degrees of Freedom Equations of motion used i our model ................................. 6

are :- .................................................................................................................................................... 6

THE AERODYNAMIC MODEL TO FIND FORCE COEFFICIENTS : ............................................................ 6

RungeKutta methods ........................................................................................................................ 7

Common fourth-order RungeKutta method ................................................................................. 7

Longitudinal Modes of Motion ........................................................................................................... 9

Lateral Directional Motion Modes :- ................................................................................................. 10

Dutch roll ....................................................................................................................................... 10

Spiral roll :- .................................................................................................................................... 10

Roll subsidence mode ................................................................................................................... 11

IMPORTANCE OF STABILITY DERIVATIVES ............................................................................................ 12

Results and Discussions : ...................................................................................................................... 16

Case 1 : 6 DOF motion modelling ...................................................................................................... 16

Case 2 : Longitudinal motion modelling ........................................................................................... 21

Case 3 : Lateral-Directional motion modelling................................................................................. 22

Observations ......................................................................................................................................... 24

Variation of Longitudinal derivatives ................................................................................................ 24

Variation in Lateral Derivatives ......................................................................................................... 32

Variation in Directional derivatives: ................................................................................................. 47

APPENDIX I ............................................................................................................................................ 53

MATLAB CODE TO PERFORM OUR 6 DOF PROBLEM COMPUTATIONALLY .......................................... 53

AB(1)=AB(i)/2; ................................................................................................................................... 55


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LIST OF FIGURES :-

Figure 1 :- Dynamically sable response .............................................................. 5

Figure 2 Dynamically unstable (oscillating) Response ....................................... 5

Figure 3 Dynamically neutral motion(undamped) response .............................. 5

Figure 4: long period mode at constant alpha ................................................... 9

Figure 5: Spiral roll mode ................................................................................. 11
http://d/manish/7th%20sem/FD/MODELING%20OF%20MOTION%20AND%20FLYING%20CHARACTRSTICS%20OF%20AN%20AIRPLANE.docx%23_Toc338879705http://d/manish/7th%20sem/FD/MODELING%20OF%20MOTION%20AND%20FLYING%20CHARACTRSTICS%20OF%20AN%20AIRPLANE.docx%23_Toc338879705


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Introduction

MODELING OF MOTION AND FLYING CHARACTRSTICS OF AN AIRPLANE

A 6 DOF flight model provides for a fairly accurate modelling of the motion and flying

characteristics of an airplane. It is generally used when the airplane is to be modelled as a

rigid body , considers both the translational motions and rotational motions as being

cantered around the C.G. of the plane. Since there are three axes to be considered involving

six variables thus it is called the 6 degree of freedom model,

Our problem involves three cases as shown :--

Case 1 : Modelling of the 6 DOF modela) Elevator input 3-2-1-1 with no aileron and rudder input

b) Aileron input 3-2-1-1 with no elevator and rudder

c) Rudder input 3-2-1-1 with no elevator and aileron

d) Rudder , Aileron input 3-2-1-1 with no elevator.Case 2 : Modelling of Longitudinal Motion (3 DOF)

Assuming no variation in the lateral directional variables

Elevator input 3-2-1-1 with no aileron and rudder defection.

Case 3 : Modelling of Lateral-Directional Motion (3 DOF)

Assuming no variation in longitudnal variables

a) Aileron input 3-2-1-1 with no rudder and elevator

b) Rudder input 3-2-1-1 with no elevator and aileron

After coding the model obtain the graphs for alpha ,beta with respect to time for differen

vases and observe the effects of changing derivativs of a plane .

Through this we can also study the dynamics of the plane.

Dynamics is concerned with the time history of the motion of physical systems. An aircraft is

such a system, and its dynamic stability behavior can be predicted through mathematicalanalysis of the aircraft's equations of motion and verified through flight test.

STATIC VS DYNAMIC STABILITY

The static stability of a physical system is concerned with the initial reaction of the system

when displaced from an equilibrium condition. The system could exhibit either:

Positive static stability - initial tendency to return Static instability - initial tendency to

diverge Neutral static stability - remain in displaced position A physical system's dynamic

stability analysis is concerned with the resulting time history motion of the system when

displaced from an equilibrium condition.

Dynamically Stable Motions


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A particular mode of an aircraft's motion is defined to be "dynamically stable" if the

parameters of interest tend toward finite values as time increases without limit.

Figure 1 :- Dynamically sable response

Dynamically Unstable Motion

A mode of motion is defined to be "dynamically unstable" if the parameters of interest

increase without limit as time increases without limit.

Figure 2 Dynamically unstable (oscillating) Response

Dynamically Neutral MotionA mode of motion is said to have "neutral dynamicstability" if theparameters of interest exhibit an undamped sinusoidal oscillation as time

increases without limit.


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The Differential 6 Degrees of Freedom Equations of motion used i our model

are :-

= -qw-rv-gsin+dynpSCx/m +T/m

= -ru+pw+gcossin + dynpSCy/m Force equations

=-pv+qu+gcoscos+ dynpSCz/m

=p+qsintan +rcostan

= qcos-rsin Kinematic equation

= qsinsec+rcossec

= 1/IxIz-I2xz(dynpSs(IzCl+IxzCn)-qr(Ixz

2 +Iz2-IyIz)) Moment

= (dynpScCm-(p2

-r2

)Ixz+pr(Iz-Ix))/Iy equn

= (1/(IxIz-Ixz

2))(dynpSs(IxCn+IxzCl)+pq(Ixz2 + Iz

2-IyIx)-qrIxz(Ix-Iy+Iz))

= usin-vcossin-wcoscos

THE AERODYNAMIC MODEL TO FIND FORCE COEFFICIENTS :

Cl=Cl0+Cl +Clq(qc/2V)+Clee

Cd=Cd0 +Cl2/eAR

Cx=Clsin CdcosCz= -Clcos Cdsin

Cm=Cm0+Cm+Cmq(qc/2V)+CmeeCy=Cy0+Cyp(ps/v)+Cyr(rs/V)+Cy+Cyaa+Cyrr

Cn=Cn0+Cnp(ps/v)+Cnr(rs/V)+Cn+Cnaa+Cnrr

CL=CL0+CLp(ps/v)+CLr(rs/V)+CL+CLaa+CLrr

Total Velocity = (u2+v2+w2)0.5

=sin-1(v/V)

=tan-1(w/u)

S=reference area , s=semi span, c= mean aerodynamic chord, g=local gravity

T=thrust , m=mass of plane, I= Inertia about the underscripted axis

dynp = 0.5V2

h = height

These equations are used in the MATLAB (give in APPENDIX 1) to perform the simulation

using RK4 methodology to solve differential equations.

When we solve the equations we get two matrices for longitudinal and lateral variables.

The longitudinal variables matrix get solved to get the roots for its different modes whereas

the lateral variables matrix can be solved to find the roots corresponding to different modesof lateral motion.


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RungeKutta methods

In numerical analysis, the RungeKutta methods are an important family of implicit and

explicit iterative methods for the approximation of solutions ofordinary differential

equations.

.

Common fourth-order RungeKutta method

One member of the family of RungeKutta methods is so commonly used that it is often

referred to as "RK4", "classical RungeKutta method" or simply as "theRungeKutta

method".

Let an initial value problembe specified as follows.

In words, what this means is that the rate at which y changes is a function ofy itself and

oft(time). At the start, time is andy is . In the equation, y may be a scalar or a vector.

The RK4 method for this problem is given by the following equations:

where is the RK4 approximation of , and
http://en.wikipedia.org/wiki/Numerical_analysishttp://en.wikipedia.org/wiki/Ordinary_differential_equationhttp://en.wikipedia.org/wiki/Ordinary_differential_equationhttp://en.wikipedia.org/wiki/Ordinary_differential_equationhttp://en.wikipedia.org/wiki/Initial_value_problemhttp://en.wikipedia.org/wiki/Initial_value_problemhttp://en.wikipedia.org/wiki/Ordinary_differential_equationhttp://en.wikipedia.org/wiki/Ordinary_differential_equationhttp://en.wikipedia.org/wiki/Numerical_analysis


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Thus, the next value ( ) is determined by the present value ( ) plus the weighted

average of four increments, where each increment is the product of the size of the interval, h,

and an estimated slope specified by function fon the right-hand side of the differential

equation.

is the increment based on the slope at the beginning of the interval, using ,

(Euler's method) ;

is the increment based on the slope at the midpoint of the interval,using ;

is again the increment based on the slope at the midpoint, but now

using ;

is the increment based on the slope at the end of the interval, using .

In averaging the four increments, greater weight is given to the increments at the midpoint.

The RK4 method is a fourth-order method, meaning that the error per step is of the orderof , while the total accumulated error has order .
http://en.wikipedia.org/wiki/Weighted_averagehttp://en.wikipedia.org/wiki/Weighted_averagehttp://en.wikipedia.org/wiki/Euler%27s_methodhttp://en.wikipedia.org/wiki/Big_O_notationhttp://en.wikipedia.org/wiki/Big_O_notationhttp://en.wikipedia.org/wiki/Big_O_notationhttp://en.wikipedia.org/wiki/Big_O_notationhttp://en.wikipedia.org/wiki/Euler%27s_methodhttp://en.wikipedia.org/wiki/Weighted_averagehttp://en.wikipedia.org/wiki/Weighted_average


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Longitudinal Modes of MotionExperience has shown that aircraft exhibit two differenttypes of longitudinal oscillations:

1) One of short period with relatively heavy damping that is called the "short period"mode(sp).The short period is characterized primarily by variations in angle of attack and pitch

angle with very little change in forward speed. Relative to the phugoid, the short periodhas a high frequency and heavy damping.

Table 1 :Motion occurs at constant speed

2) Another of long period with very light damping that is called the phugoid mode

The phugoid is characterized mainly by variations in u and with nearly constant. This

long period oscillation can be thought of as a constant total energy problem with exchanges

between potential and kinetic energy. The aircraft nose drops and airspeed increases as the

aircraft descends below its initial altitude. Then the nose rotates up, causing the aircraft to

climb above its initial altitude with airspeed decreasing until the nose lazily drops below the

horizon at the top of the maneuver. Because of light damping, many cycles are required for

this motion to damp out. However, its long period combined with low damping results in an

oscillation that is easily controlled by the pilot, even for a slightly divergent motion.

Figure 4: long period mode at constant alpha

Phugoid - Small n- Large time constant

- Small damping ratio

Short Period - Large n- Small time constant

- High damping ratio

STANDARD

SOLUTION

These are the longitdnal modes of motion of aircraft.


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Lateral Directional Motion Modes :-

There are three typical asymmetric modes of motion exhibited by aircraft.These modes are

the roll, spiral, and Dutch roll.

Dutch roll is a type ofaircraft motion, consisting of an out-of-phase combination of "tail-wagging" and rocking from side to side. This yaw-roll coupling is one of the basic flightdynamic modes .This motion is normally well damped in most light aircraft, though some

aircraft with well-damped Dutch roll modes can experience a degradation in damping,

as airspeed decreases and altitude increases.

Dutch roll results from relatively weaker positive directional stability as opposed to

positive lateral stability. When an aircraft rolls around the longitudinal axis, a sideslip is

introduced into the relative wind in the direction of the rolling motion. Strong lateral stability

begins to restore the aircraft to level flight. At the same time, somewhat weaker directional

stability attempts to correct the sideslip by aligning the aircraft with the perceived relative

wind. Since directional stability is weaker than lateral stability for the particular aircraft, the

restoring yaw motion lags significantly behind the restoring roll motion. As such, the aircraft

passes through level flight as the yawing motion is continuing in the direction of the original

roll. At that point, the sideslip is introduced in the opposite direction and the process is

reversed.

The Dutch roll mode is a coupled yawing and rolling motion lightly damped,

moderately low frequency oscillation.

Spiral roll :-

If a spirally unstable aircraft, through the action of a gust or other disturbance, gets a smallinitial roll angle to the right, for example, a gentle sideslip to the right is produced. The

sideslip causes a yawing moment to the right. If the dihedral stability is low, and yaw

damping is small, the directional stability keeps turning the aircraft while the continuing bank

angle maintains the sideslip and the yaw angle. This spiral gets continuously steeper and

tighter until finally, if the motion is not checked, a steep, high-speed spiral dive results. The

motion develops so gradually, however that it is usually corrected unconsciously by the pilot,

who may not be aware that spiral instability exists. If the pilot cannot see the horizon, for

instance because of clouds, he might not notice that he is slowly going into the spiral dive,

which can lead into the graveyard spiral.
http://en.wikipedia.org/wiki/Aircrafthttp://en.wikipedia.org/wiki/Phase_(waves)http://en.wikipedia.org/wiki/Flight_dynamicshttp://en.wikipedia.org/wiki/Flight_dynamicshttp://en.wikipedia.org/wiki/Dampinghttp://en.wikipedia.org/wiki/Airspeedhttp://en.wikipedia.org/wiki/Altitudehttp://en.wikipedia.org/wiki/Directional_stabilityhttp://en.wikipedia.org/wiki/Flight_dynamics#Lateral_modeshttp://en.wikipedia.org/wiki/Slip_(aerodynamic)http://en.wikipedia.org/wiki/Relative_windhttp://en.wikipedia.org/wiki/Windhttp://en.wikipedia.org/wiki/Graveyard_spiralhttp://en.wikipedia.org/wiki/Graveyard_spiralhttp://en.wikipedia.org/wiki/Windhttp://en.wikipedia.org/wiki/Relative_windhttp://en.wikipedia.org/wiki/Slip_(aerodynamic)http://en.wikipedia.org/wiki/Flight_dynamics#Lateral_modeshttp://en.wikipedia.org/wiki/Directional_stabilityhttp://en.wikipedia.org/wiki/Altitudehttp://en.wikipedia.org/wiki/Airspeedhttp://en.wikipedia.org/wiki/Dampinghttp://en.wikipedia.org/wiki/Flight_dynamicshttp://en.wikipedia.org/wiki/Flight_dynamicshttp://en.wikipedia.org/wiki/Phase_(waves)http://en.wikipedia.org/wiki/Aircraft


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To be spirally stable, an aircraft must have some combination of a sufficiently large dihedral,

which increases roll stability, and a sufficiently long vertical tail arm, which increases yaw

damping. Increasing the vertical tail area then magnifies the degree of stability or instability.

It is characterized by large directional stability and insufficient lateral stability and positive

real part of roots that is divergence and low damping. Thus leading to spiral divergence andthus tighter spiral as shown.

It can also happen that the aircraft possesses low directional stability and high lateral

stability. Thus leading to directional divergence. In this case the bank angle remains constant

and the sideslip angle goes on increasing.

Figure 5: Spiral roll mode

Roll subsidence mode

Roll subsidence mode is simply the damping of rolling motion. There is no direct

aerodynamic moment created tending to directly restore wings-level, i.e. there is no returning

"spring force/moment" proportional to roll angle. However, there is a damping moment

(proportional to roll rate) created by the slewing-about of long wings. This prevents large roll

rates from building up when roll-control inputs are made or it damps the roll rate (not theangle) to zero when there are no roll-control inputs.

Roll mode can be improved by adding dihedral effects to the aircraft design, such as high

wings, dihedral angles or sweep angles.

These are the various lateral motion modes.
http://en.wikipedia.org/wiki/Dihedral_(aircraft)http://en.wikipedia.org/wiki/Dihedral_(aircraft)


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IMPORTANCE OF STABILITY DERIVATIVES

Some of the stability derivatives are particularly pertinent in the study of the dynamic modes

of aircraft motion, and the more important ones appearing in the functional equations which

characterize the dynamic modes of motion should be understood .

are discussed in the following

paragraphs.

1) :- The stability derivative Cnbeta is the change in yawing moment coefficient with

variation in sideslip angle. It is usually referred to as the static directional derivative or the

"weathercock" derivative. When the airframe sideslips, the relative wind strikes the airframe

obliquely, creating a yawing moment, N, about the center of gravity. The major portion of Cn

comes from the vertical tail, which stabilizes the body of the airframe just as the tail feathers

of an arrow stabilize the arrow shaft. The Cnbeta contribution due to the vertical tail is

positive, signifying static directional stability, whereas the Cn due to body is negative,

signifying static directional instability. There is also a contribution to Cn from the wing, the

value of which is usually positive but very small compared to the body and vertical tail

contributions.

The derivative Cnbeta is very important in determining the dynamic lateral stability andcontrol characteristics. Most of the references concerning full-scale flight tests and free-flight

wind tunnel model tests agree that Cnbeta should be as high as possible for good flying

qualities. A high value of Cnbeta aids the pilot in effecting coordinated turns and prevents

excessive sideslip and yawing motions in extreme flight maneuvers and in rough air. Cnbeta,

primarily determines the natural frequency of the Dutch roll oscillatory mode of the airframe,

and it is also a factor in determining the spiral stability characteristics.

2)

Cnr is the change in yawing moment coefficient with change in yawing velocity. It is known

as the yaw damping derivative. When the airframe is yawing at an angular velocity, r, a

yawing moment is produced which opposes the rotation. Cnr is made up of contributions

from the wing, the fuselage, and the vertical tail, all of which are negative in sign.The

contribution from the vertical tail is by far the largest, usually amounting to about 80% or

90% of the total Cnrof the airframe.

The derivative Cnris very important in lateral dynamics because it is the main contributor r to

the damping of the Dutch roll oscillatory mode. It also is important to the spiral mode. For

each mode, large negative values of Cnrare desired.


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3)

This stability derivative is the change in pitching moment coefficient with varying angle of

attack and is commonly referred to as the longitudinal static stability derivative. When the

angle of attack of the airframe increases from the equilibrium condition, the increased lift on

the horizontal tail causes a negative pitching moment about the center of gravity of the

airframe. Simultaneously, the increased lift of the wing causes a positive or negative pitching

moment, depending on the fore and aft location of the lift vector with respect to the center of

gravity. These contributions together with the pitching moment contribution of the fuselage

are combined to establish the derivative Cm The magnitude and sign of the total CM for a

particular airframe configuration are thus a function of the center of gravity position, and this

fact is very important in longitudinal stability and control. If the center of gravity is ahead of

the neutral point, the value of Cm is negative, and the airframe is said to possess static

longitudinal stability. Conversely, if the center of gravity is aft of the neutral point the valueof CH is positive, and the airframe is then statically unstable. CM is perhaps the most

important derivative as far as longitudinal stability and control are concerned. It primarily

establishes the natural frequency of the short period mode and is a major factor in

determining the response of the airframe to elevator motions and to gusts. In general, a large

negative value of Cm,, (i.e., large static stability) is desirable for good flying qualities.

However, if it is too large, the required elevator effectiveness for satisfactory control may

become unreasonably high. A compromise is thus necessary in selecting a design range for

Cm . Design values of static stability are usually expressed not in terms of Cm but rather in

terms of the derivative CmcL, where the relation is Cm = CmcL. It should be pointed out that

CM in the above expression is actually a partial derivative for which the forward speed

remains constant.

4)

The stability derivative Cmq is the change in pitching moment coefficient with varying pitch

velocity and is commonly referred to as the pitch damping derivative. As the airframe pitches

about its center of gravity, the angle of attack of the horizontal tail changes and lift develops

on the horizontal tail, producing a negative- pitching moment on the airframe and hence a

contribution to the derivative Cmq . There is also a contribution to Cmq because of various

"dead weight" aeroelastic effects. Since the airframe is moving in a curved flight path due to

its pitching, a centrifugal force is developed on all the components of the airframe. The force

can cause the wing to twist as a result of the dead weight moment of overhanging nacelles

and can cause the horizontal tail angle of attack to change as a result of fuselage bending due

to the weight of the tail section. In low speed flight, Cmq comes mostly from the effect of the

curved flight path on the horizontal tail, and its sign is negative. In high speed flight the sign

of Cmq can be positive or negative, depending on the nature of the aeroelastic effects. The

derivative Cmq is very important in longitudinal dynamics because it contributes a major

portion of the damping of the short period mode for conventional aircraft. As pointed out, this

damping effect comes mostly from the horizontal tail. For tailless aircraft, the magnitude of

Cmq is consequently small; this is the main reason for the usually poor damping of this type

of configuration. Cmq is also involved to a certain extent in phugoid damping. In almost allcases, high negative values of Cmq are desired.


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5)

This stability derivative is the change in rolling moment coefficient with variation in sideslip

angle and is usually referred to as the "effective dihedral derivative." When the airframesideslips, a rolling moment is developed because of the dihedral effect of the wing and

because of the usual high position of the vertical tail relative to the equilibrium x-axis. No

general statements can be made concerning the relative magnitude of the contributions to Cl

from the vertical tail and from the wing since these contributions vary considerably from

airframe to airframe and for different angles of attack of the same airframe. Cl is nearly

always negative in sign, signifying a negative rolling moment for a positive sideslip.

Cl is very important in lateral stability and control, and is therefore usually considered in the

preliminary design of an airframe. It is involved in damping both the Dutch roll mode and the

spiral mode. It is also involved in the maneuvering characteristics of an airframe, especially

with regard to lateral control with the rudder alone near stall.

6)

The stability derivative Clp is the change in rolling moment coefficient with change in rolling

velocity and is usually known as the roll damping derivative. When the airframe rolls at an

angular velocity p, a rolling moment is produced as a result of this velocity; this moment

opposes the rotation.

Clp is composed of contributions, negative in sign, from the wing and the horizontal and

vertical tails. However, unless the size of the tail is unusually large in comparison with the

size of the wing, the major portion of the total q comes from the wing.The derivative is quite

important in lateral dynamics because essentially Clp alone determines the damping in rollcharacteristics of the aircraft. Normally, it appears that small negative values of q are more

desirable than large ones because the airframe will respond better to a given aileron input and

will suffer fewer flight perturbations due to gust inputs.


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SIMULATION

CONTROL INPUTS :-

CASE 1 :- 6 DOF MODELLINGCASE 1a : time 0-3 secs : elevator=6, aileron=0, rudder=0

time 3-5 secs : elevator=-6, aileron=0, rudder=0

time 5-6 secs : elevator=6, aileron=0, rudder=0

time 6-7 secs : elevator=-6, aileron=0, rudder=0

CASE 1b : time 0-3 secs : elevator=0, aileron=6, rudder=0

time 3-5 secs : elevator=0, aileron=-6, rudder=0

time 5-6 secs : elevator=0, aileron=6, rudder=0

time 6-7 secs : elevator=0, aileron=-6, rudder= 0

CASE 1c : time 0-3 secs : elevator=0, aileron=0, rudder= 6

time 3-5 secs : elevator=0, aileron=0, rudder= -6

time 5-6 secs : elevator=0, aileron=0, rudder= 6

time 6-7 secs : elevator=0, aileron=0, rudder= -6

CASE 1d : : time 0-3 secs : elevator=0, aileron=6, rudder= 6

time 3-5 secs : elevator=0, aileron=-6, rudder= -6

time 5-6 secs : elevator=0, aileron=6, rudder= 6

time 6-7 secs : elevator=0, aileron=-6, rudder= -6

CASE 2 :- LONGITUDNAL MOTION MODELLING

Inputs same as case 1a

CASE 3 :- LATERAL-DIRECTIONAL MOTION MODELLING

CASE 3a : Inputs same as case 1bCASE 3b : Inputs same as case 1c


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Results and Discussions :

Case 1 : 6 DOF motion modelling

FOR LONGITUDNAL MOTION 6 DOF SELECT ELEVATOR AND THE OUTPUT COMES AS :


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WHEN WE SELECT AILERON AS INPUT THE OUTPUT IS :


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FOR RUDDER WE HAVE :


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FOR RUDDER AILERON DEFLECTION :-


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Case 2 : Longitudinal motion modelling


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In the above plots we can say that the lateral state variables do not change with time in

Longitudnal motion model.

Case 3 : Lateral-Directional motion modelling

We select the aileron and the graphs are :--


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Thus no change in heightWhen selected rudder the output is same as before for the 6 dof modelling for rudder

deflection except that v,w,ht remains constant during the period.


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Observationsby changing the values of stability derivatives:-Variation of Longitudinal derivatives

Elevator deflection

Cmalpha =-0.02;

For very small negative values of Cmalpha,we get a very unstable response.


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Cmalpha = -2;

For higher negative values of Cmalpha, we get stable response.The stability increases withincreasing values of Cmalpha.


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Cmalpha = 2;

For positive values of Cmalpha we have a very unstable response of the aircraft .


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Clalpha = 12;

For very high positive values of Clalpha, we have the following graphs:


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Clalpha= -2;


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For negative value it is unstable that is lift decreases and the plane falls . Thus unstable.

Cmq = -10;

With the increase in Cmq, we have reduced oscillations which results in more stable

response.


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Cmq=2.28 :--

For positive values of Cmq we can observe that the damping has reduced and the oscillations

increase . this is because the tail does not provide negative moment to stabilize the plane and

the oscillations take longer to settle out .


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Cmq=10: as we increase the Cmq more positive the plane becomes more and more

dynamically unstable and the the plane diverges to infinity as shown in the graph. Thus the

more desirable value of Cmq is negative .

r


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Variation in Lateral Derivatives

Cyp = 3.00 (6 DOF Modelling)

1) Case 3: Aileron + Rudder

The maximum height and the final height seems to have decreased.thus decrease in effect on

the longitude variables with increase in Cyp in positive sign.it also leads to increase in

damping ant thus stable condition. Similar occurs for rudder and rudder-aileron case.

AILERON

Cyp = -3

0 1 2 3 4 5 6 71700

1800

1900

2000

2100

2200

2300

2400

2500

2600

2700

time

he

ight

height


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Aileron deflection case

The maximum and minimum height achieved during flight seems to have increased.also the

oscillations increase thus leading to instability. Similarly for the rudder and rudder-aileron

configuration increases. Thus leading to instability.

Cyr = 3 (6 DOF Modelling.)

Case : Aileron Deflection

0 1 2 3 4 5 6 72300

2400

2500

2600

2700

2800

2900

3000

3100

time

he

ight

height


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The maximum height achieved has increased and minimum height decreases. Although

stability increases.

Rudder deflection case also faces same thing i.e. increase in stability but decrease in max.

Height and decrease in minimum height.

Cyr = -3


The damping decrease , instability increases.

0 1 2 3 4 5 6 72200

2300

2400

2500

2600

2700

2800

2900

3000

time

he

ight

height

0 1 2 3 4 5 6 7-0.5

0

0.5

time

alph

a,b

eta

alpha

beta

0 1 2 3 4 5 6 7-500

0

500

time

u,v,w

u

v

w

0 1 2 3 4 5 6 7-5

0

5

time

p,q,r

p

q

r


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Case :. Rudder deflection

Beta takes a little longer to stabilise. The amplitude of oscillations have also increased

slightly after 3 seconds till 5 seconds. As a result, velocity along y axis (v) also oscillates

slightly more. The yaw rate (r) also oscillates relatively more.

0 1 2 3 4 5 6 7-0.2

0

0.2

time

alph

a,b

eta

alpha

beta

0 1 2 3 4 5 6 7-500

0

500

time

u,v,w

u

v

w

0 1 2 3 4 5 6 7-1

0

1

time

p,q,r

p

q

r

0 1 2 3 4 5 6 7-0.3

-0.2

-0.1

0

0.1

time

phi,t

he

ta

,s

hi

phi

theta

shi

0 1 2 3 4 5 6 7-0.2

-0.1

0

0.1

0.2

time

de

le,

de

la,

de

lr

dele

dela

delr


36/60



We also find that yaw angle ( shi ) oscillates relatively more and does not damp out during

the test period.

Thus Cyr is desired to be +ve.

Cybeta = -3 (6 DOF Modelling)

Aileron Deflection case

Roll rate ( r ) is highly stabilised. Velocity along y direction ( v ) is also highly stabilised.

0 1 2 3 4 5 6 7-0.5

0

0.5

time

alph

a,b

eta

alpha

beta

0 1 2 3 4 5 6 7-500

0

500

time

u,v,w

u

v

w

0 1 2 3 4 5 6 7-5

0

5

time

p,q,r

p

q

r

0 1 2 3 4 5 6 7-2

0

2

4

time

phi,theta

,s

hi

phi

theta

shi

0 1 2 3 4 5 6 7-0.2

-0.1

0

0.1

0.2

time

de

le,

de

la,

de

lr

dele

dela

delr


37/60



Roll angle ( phi ) and yaw angle ( shi ) are highly damped and stable.

There is a significant increase in the height achieved at the end of the flight. The

trajectory of flight has also changed.

In the lateral directional modelling this effect on height is neglected.

Cyr = 3 (6 DOF Modelling)

Case :- Aileron Deflection

0 1 2 3 4 5 6 72000

2500

3000

3500

4000

4500

time

he

ight

height


38/60



The amplitude of oscillation of beta, v and r keeps on increasing which will result in an

unstable flight. u also starts decreasing instead of being constant.

0 1 2 3 4 5 6 7-1

0

1

time

alpha

,beta

alpha

beta

0 1 2 3 4 5 6 7-500

0

500

time

u,v,w

u

v

w

0 1 2 3 4 5 6 7-5

0

5

time

p,q,r

p

q

r

0 1 2 3 4 5 6 7-4

-2

0

2

time

phi,t

he

ta

,s

hi

phi

theta

shi

0 1 2 3 4 5 6 7-0.2

-0.1

0

0.1

0.2

time

de

le,

de

la,

de

lr

dele

dela

delr


39/60



The amplitude of phi, theta and shi also keeps increasing and does not damp. Therefore it will

result in an unstable flight.

Similar changes occur for the rudder and rudder-aileron deflection for rudder this change

results in decrease in descent rate .

Clp = -3 (6 DOF modelling)

Case : Aileron deflection

0 1 2 3 4 5 6 70

500

1000

1500

2000

2500

time

he

ight

height

0 1 2 3 4 5 6 7-0.2

0

0.2

time

alph

a,b

eta

alpha

beta

0 1 2 3 4 5 6 7-500

0

500

time

u,v,w

u

v

w

0 1 2 3 4 5 6 7-1

0

1

time

p,q,r

p

q

r


40/60



Beta,v and r are very highly damped which means a very stable flight.

The change in euler angles are highly damped which results in almost linear changes and

stable aircraft.

The height decreases with little perturbations ending in a much lower altitude.

0 1 2 3 4 5 6 7-2

-1

0

1

time

phi,the

ta,s

hi

phi

theta

shi

0 1 2 3 4 5 6 7-0.2

-0.1

0

0.1

0.2

time

de

le,d

ela

,de

lr

dele

dela

delr

0 1 2 3 4 5 6 71200

1400

1600

1800

2000

2200

2400

time

he

ight

height


41/60



0 1 2 3 4 5 6 7-0.1

0

0.1

time

alpha

,beta

alpha

beta

0 1 2 3 4 5 6 7-500

0

500

time

u,v,w

u

v

w

0 1 2 3 4 5 6 7-0.5

0

0.5

time

p,q,r

p

q

r

Case:Rudder deflection

Beta, v and r are very highly damped which means a very stable flight.

Phi and beta are highly damped.

Case : AileronRudder

Similarly



The slope of the curve has increased resulting in a steeper descent .

Clp = 1(Lateral Directional Modelling)


0 1 2 3 4 5 6 7-1

-0.5

0

0.5

time

phi,theta

,shi

phi

theta

shi

0 1 2 3 4 5 6 7-0.2

-0.1

0

0.1

0.2

time

de

le,d

ela

,de

lr

dele

dela

delr


42/60



The following curves show highly unstable flight. Thus negative CLp is desired for stability.

0 1 2 3 4 5 6 7-2

0

2

time

a

lpha

,beta

alpha

beta

0 1 2 3 4 5 6 7-2

0

2x 10

89

time

u,v,w

u

v

w

0 1 2 3 4 5 6 7-1

0

1

x 1089

time

p,q,r

p

q

r

0 1 2 3 4 5 6 7-5

0

5

10

15x 10

43

time

phi,th

et

a,s

hi

phi

theta

shi

0 1 2 3 4 5 6 7-0.2

-0.1

0

0.1

0.2

time

de

le,

de

la,

de

lr

dele

dela

delr


43/60



Clr = 0.9 (Lateral Directional Modelling)

Case :Aileron Deflection

Highly damped beta, v and r.

CLr= 2.418

It is a cross derivative. Increseor decrease in its value cause opposite effects on the lateral and

directional parameters and thus due to their coupling effect each other

Damped oscillations of beta, p and phi but the osscilatons in r and shi has increased due to it thus due

to coupling the lateral variables also fluctuate .though stability prevails.

0 1 2 3 4 5 6 7-0.2

0

0.2

time

alpha

,beta

alpha

beta

0 1 2 3 4 5 6 7-500

0

500

time

u,v,w

u

v

w

0 1 2 3 4 5 6 7-2

0

2

time

p,q,r

p

q

r


44/60



Shi and phi are highly stabilised.

As this value becomes more and more posiive the lateral variables oscilatons decrease but due to crosseffect the directional oscillations increase and thus after a point the motion totaly destabilises.

Case: Rudder deflection

Due to increase in CLr the cross ossicaltions due to rudder increase and after a point it becomes

unstable .

Similar effect occurs in rudder-aileron deflection.

CLr = -1.

The below curves show unstable flight conditions.thus positive value of CLr is desirable.


45/60



0 1 2 3 4 5 6 7-2

0

2

time

alpha

,beta

alpha

beta

0 1 2 3 4 5 6 7-1

0

1x 10

53

time

u,v,w

u

v

w

0 1 2 3 4 5 6 7-2

0

2x 10

53

time

p,q,r

p

q

r

0 1 2 3 4 5 6 70

5

10x 10

25

time

ph

i,t

he

ta

,s

hi

phi

theta

shi

0 1 2 3 4 5 6 7-0.2

-0.1

0

0.1

0.2

time

de

le,

de

la,

de

lr

dele

dela

delr


46/60



CLbeta=-0.14

The static stability increase of the plane as shown by the increase in the frequency of beta

with little effect on the other variables.

For aileron deflection:-

As the negative value increases

We can see the increase in oscillations thus instability . As we decrease the stability (static)

increases whereas as it becomes positive it becomes totally unstable. This is because at

CLbeta positive the roll moment increases as the plane sideslips or rolls thus further goes into

instability .

But as it becomes too much negative it results in instability due to increase in the crossvariable like for aileron deflection as the vaue becomes more negative the yaw becomes high

an the plane enters instability. Thus the value of CLbeta is desired to be a low negative

value. Similar case appears for the other two defections.


47/60



Variation in Directional derivatives:

Cnp is more negative (-6.115) very stable response

Cnp is less negative (-0.015),even more stable to the response


48/60



Height remains constant if we consider effect of lateral on longitudinal variables.

When Cnp is positive (4.115) , very unstable response

Cnr :-

Cnr is more negative (-5.087) ,stable response


49/60



For positive values of it the plane is unstable because of increasing yawing moment with

increase in the yaw rate due to the deflection.

Cnr =3.085 ,unstable response


50/60



Thus the desired value of it is positive for stability.it also effects the damping .

More is the positive value more is the damping.

Cnbeta = -1.281 very stable response


51/60



When Cnbeta is more positive i.e. 4.281:unstable response


52/60



Thus the value should be more in positive or very less of negative value to give a stable

response. When we consider the longitudnal effects the change in the longitudinal variables

will be very less although height reduction is observed due to component of weight acting in

the vertical direction. These are the results for directional derivatives.


53/60



APPENDIX I

MATLAB CODE TO PERFORM OUR 6 DOF PROBLEM

COMPUTATIONALLY

% PROGRAM TO GENERATE THE 6 DOF EQUATIONS OF AN AIRCRAFT

clear allclcgra=9.81; % Gravity

u=296;v=0.0;w=0.0;p=0.0;q=0.0;r=0.0;phi=0.0;theta=0.0;shi=0.0; % Initial reference

flight conditionsht=2400;

rho=1.225; % DensityS=64; % Reference areamass=19633.23;

T=2000; % Thrust

chord=3.159;ar=7.22; % Aspect ratioss=10.75; % Semi Spanws=ss*2; % Wing SpanIxx=189367.2;

Izz=415850.9;Ixz=11442.0;Iyy=252687.0;e=0.9112; % EccentricityV=sqrt(u^2+v^2+w^2); % Resultant

velocity

dynp=0.5*rho*(V*V); % Dynamic Pressurealpha=0.0; % Angle of Attackbeta=0.0; % Side-slip AngleCl0=0; % Coefficient of lift at zero alphaClalpha=2.6307;Clq=4.4134;Cldele=1.0425;Cm0=0.0; % Coeff. of moment at zero alphaCmalpha=-1.66;Cmq=-1.228;Cmdele=-0.3557;

Cd0=0.0323;Cy0=0.0;


54/60



Cyp=0.303;Cyr=0.727;Cybeta=-1.333;Cydela=0.29;Cydelr=0.191;

CL0=0.0;CLp=-0.978;CLr=0.418;CLbeta=-0.126;CLdela=-0.247;CLdelr=0.046;Cn0=0.0;Cnp=-0.115;Cnq=-1.228;Cnr=-0.495;Cnbeta=0.281;

Cndela=0.0;Cndelr=-0.166;

fid=fopen('termpaper.txt','w'); % Open file to store valuesch=menu(' Enter the choice of what you want to model ','6 DOF

modeling','Longitudnal motion Modeling','Lateral-Directional

motion Modeling');switch ch

case 1h=0.01;h1=0.01;disp(' MODELING THE 6 DOF ');

ch1=menu('ENTER THE DEFLECTION TYPE

','ELEVATOR','AILERON','RUDDER','AILERON-

RUDDER');if (ch1==1)

disp(' ELEVATOR DEFLECTION ')delr=0;dela=0;

elseif (ch1==2)

disp(' AILERON DEFLECTION ')dele=0;delr=0;

elseif (ch1==3)disp(' RUDDER DEFLECTION ')dele=0;dela=0;

elseif (ch1==4)disp(' RUDDER-AILERON DEFLECTION ')dele=0;

else

disp(' WRONG CHOICE ENTERED (TRY AGAIN ')end


55/60



case 2disp(' Longitudnal motion Modeling ')h=0.01; % time step for longitudnal variableh1=0; % time step for lateral variable

dela=0;delr=0;

case 3

disp(' Lateral-Directional Motion Modeling ')h1=0.01;h=0;dele=0;ch2=menu(' Rudder or Aileron ','Aileron','Rudder');if (ch2==1)

delr=0;else

dela=0;end

otherwisedisp(' wrong choice ')

endt=0.0;

% START OF THE RUNGE KUTTA 4TH ORDER METHOD

while t


56/60



K(1)=K(i);L(1)=L(i);M(1)=M(i);N(1)=N(i);O(1)=O(i);

s(1)=s(i);AB(1)=AB(i);B(1)=B(i);C(1)=C(i);D(1)=D(i);

endu=u+K(1);v=v+L(1);w=w+M(1);phi=phi+N(1);theta=theta+O(1);

shi=shi+s(1);p=p+AB(1);q=q+B(1);r=r+C(1);ht=ht+D(1);

if t


57/60



elseif (ch==3 && ch2==2)delr=-6/57.3;end

elseif t


58/60



Cy=Cy0+Cyp*(p*ss/V)+Cyr*(r*ss/V)+Cybeta*beta+Cydela*dela+Cydel

r*delr;

Cn=Cn0+Cnp*(p*ss/V)+Cnr*(r*ss/V)+Cnbeta*beta+Cndela*dela+Cndel

r*delr;

CL=CL0+CLp*(p*ss/V)+CLr*(r*ss/V)+CLbeta*beta+CLdela*dela+CLdel

r*delr;

K(i+1)=h*(-q.*w-(r.*v)-

(gra*sin(theta))+(dynp*S*Cx/mass)+(T/mass));L(i+1)=h1*((-

(r.*u))+(p.*w)+(gra*cos(theta)*sin(phi))+(dynp*S*Cy/mass));

M(i+1)=h*((-

(p.*v))+(q.*u)+(gra*cos(theta)*cos(phi))+(dynp*

S*Cz/mass));

N(i+1)=h1*(p+(q*sin(phi)*tan(theta))+(r*cos(phi)*tan(thet

a)));O(i+1)=h*((q*cos(phi))-(r*sin(phi)));

s(i+1)=h1*((q*sin(phi)*sec(theta))+(r*cos(phi)*sec(theta))

);AB(i+1)=h1*(((dynp*S*ss*(Izz*CL+Ixz*Cn))-

(q*r*(Ixz*Ixz+Izz*Izz-Iyy*Izz)))/(Ixx*Izz-Ixz*Ixz));

B(i+1)=h*(((dynp*S*chord*Cm)-(Ixz*(p^2-r^2))+(p*r*(Izz-

Ixx)))/Iyy);C(i+1)=h1*((1/(Ixx*Izz-

Ixz^2))*((dynp*S*ss*(Ixx*Cn+Ixz*CL))-

(q*r*Ixz*(Ixx-Iyy+Izz))+(p*q*(Ixz^2-

Ixx*Iyy+Ixx^2))));D(i+1)=h*((u*sin(theta))-(v*cos(theta)*sin(phi))-

(w*cos(theta)*cos(phi)));

endu=u+((K(2)+2*K(3)+2*K(4)+K(5))/6);v=v+((L(2)+2*L(3)+2*L(4)+L(5))/6);w=w+((M(2)+2*M(3)+2*M(4)+M(5))/6);phi=phi+((N(2)+2*N(3)+2*N(4)+N(5))/6);theta=theta+((O(2)+2*O(3)+2*O(4)+O(5))/6);shi=shi+((s(2)+2*s(3)+2*s(4)+s(5))/6);p=p+((AB(2)+2*AB(3)+2*AB(4)+AB(5))/6);


59/60



q=q+((B(2)+2*B(3)+2*B(4)+B(5))/6);r=r+((C(2)+2*C(3)+2*C(4)+C(5))/6);ht=ht+((D(2)+2*D(3)+2*D(4)+D(5))/6);

fprintf(fid,'%6.9f %6.9f %6.9f %6.9f %6.9f %6.9f %6.9f %6.9f

%6.9f %6.9f %6.9f %6.9f %6.9f %6.9f %6.9f

%6.9f\n',t,alpha,beta,u,v,w,phi,theta,shi,p,q,r,ht,dele,dela,d

elr)

t=t+0.01;end

status=fclose(fid); % END OF WHILE

AC=load('termpaper.txt'); % LOAD FILE TO A NEW MATRIXt=AC(:,1);

alpha=AC(:,2);beta=AC(:,3);u=AC(:,4);v=AC(:,5);w=AC(:,6);phi=AC(:,7);theta=AC(:,8);shi=AC(:,9);p=AC(:,10);q=AC(:,11);r=AC(:,12);

ht=AC(:,13);dele=AC(:,14);dela=AC(:,15);delr=AC(:,16);figure

subplot(3,1,1)plot(t,alpha,t,beta)xlabel('time')ylabel('alpha,beta')legend('alpha','beta')hold on

grid onsubplot(3,1,2)plot(t,u,t,v,t,w)xlabel('time')ylabel('u,v,w')legend('u','v','w')hold ongrid onsubplot(3,1,3)plot(t,p,t,q,t,r)xlabel('time')ylabel('p,q,r')legend('p','q','r')


60/60


hold ongrid onfiguresubplot(2,1,1)plot(t,phi,t,theta,t,shi)

xlabel('time')ylabel('phi,theta,shi')legend('phi','theta','shi')hold ongrid on

subplot(2,1,2)plot(t,dele,t,dela,t,delr)xlabel('time')

ylabel('dele,dela,delr')legend('dele','dela','delr')hold ongrid on

figureplot(t,ht)xlabel('time')ylabel('height')legend('height')hold on

grid on

% END OF FILE

Date post:	14-Apr-2018
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Modeling of Motion and Flying Charactrstics of an Airplane

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