modeling of motion and flying charactrstics of an airplane
TRANSCRIPT
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
1/60
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
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
2/60
6 D O F S i m u l a t i o n P a g e | 2
Published by : Manish Tripathi
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
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
3/60
6 D O F S i m u l a t i o n P a g e | 3
Published by : Manish Tripathi
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 -
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
4/60
6 D O F S i m u l a t i o n P a g e | 4
Published by : Manish Tripathi
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
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
5/60
6 D O F S i m u l a t i o n P a g e | 5
Published by : Manish Tripathi
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.
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
6/60
6 D O F S i m u l a t i o n P a g e | 6
Published by : Manish Tripathi
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.
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
7/60
6 D O F S i m u l a t i o n P a g e | 7
Published by : Manish Tripathi
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 -
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
8/60
6 D O F S i m u l a t i o n P a g e | 8
Published by : Manish Tripathi
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 -
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
9/60
6 D O F S i m u l a t i o n P a g e | 9
Published by : Manish Tripathi
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.
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
10/60
6 D O F S i m u l a t i o n P a g e | 10
Published by : Manish Tripathi
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 -
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
11/60
6 D O F S i m u l a t i o n P a g e | 11
Published by : Manish Tripathi
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) -
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
12/60
6 D O F S i m u l a t i o n P a g e | 12
Published by : Manish Tripathi
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.
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
13/60
6 D O F S i m u l a t i o n P a g e | 13
Published by : Manish Tripathi
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.
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
14/60
6 D O F S i m u l a t i o n P a g e | 14
Published by : Manish Tripathi
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.
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
15/60
6 D O F S i m u l a t i o n P a g e | 15
Published by : Manish Tripathi
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
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
16/60
6 D O F S i m u l a t i o n P a g e | 16
Published by : Manish Tripathi
Results and Discussions :
Case 1 : 6 DOF motion modelling
FOR LONGITUDNAL MOTION 6 DOF SELECT ELEVATOR AND THE OUTPUT COMES AS :
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
17/60
6 D O F S i m u l a t i o n P a g e | 17
Published by : Manish Tripathi
WHEN WE SELECT AILERON AS INPUT THE OUTPUT IS :
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
18/60
6 D O F S i m u l a t i o n P a g e | 18
Published by : Manish Tripathi
FOR RUDDER WE HAVE :
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
19/60
6 D O F S i m u l a t i o n P a g e | 19
Published by : Manish Tripathi
FOR RUDDER AILERON DEFLECTION :-
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
20/60
6 D O F S i m u l a t i o n P a g e | 20
Published by : Manish Tripathi
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
21/60
6 D O F S i m u l a t i o n P a g e | 21
Published by : Manish Tripathi
Case 2 : Longitudinal motion modelling
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
22/60
6 D O F S i m u l a t i o n P a g e | 22
Published by : Manish Tripathi
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 :--
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
23/60
6 D O F S i m u l a t i o n P a g e | 23
Published by : Manish Tripathi
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.
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
24/60
6 D O F S i m u l a t i o n P a g e | 24
Published by : Manish Tripathi
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.
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
25/60
6 D O F S i m u l a t i o n P a g e | 25
Published by : Manish Tripathi
Cmalpha = -2;
For higher negative values of Cmalpha, we get stable response.The stability increases withincreasing values of Cmalpha.
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
26/60
6 D O F S i m u l a t i o n P a g e | 26
Published by : Manish Tripathi
Cmalpha = 2;
For positive values of Cmalpha we have a very unstable response of the aircraft .
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
27/60
6 D O F S i m u l a t i o n P a g e | 27
Published by : Manish Tripathi
Clalpha = 12;
For very high positive values of Clalpha, we have the following graphs:
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
28/60
6 D O F S i m u l a t i o n P a g e | 28
Published by : Manish Tripathi
Clalpha= -2;
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
29/60
6 D O F S i m u l a t i o n P a g e | 29
Published by : Manish Tripathi
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.
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
30/60
6 D O F S i m u l a t i o n P a g e | 30
Published by : Manish Tripathi
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 .
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
31/60
6 D O F S i m u l a t i o n P a g e | 31
Published by : Manish Tripathi
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
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
32/60
6 D O F S i m u l a t i o n P a g e | 32
Published by : Manish Tripathi
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
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
33/60
6 D O F S i m u l a t i o n P a g e | 33
Published by : Manish Tripathi
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
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
34/60
6 D O F S i m u l a t i o n P a g e | 34
Published by : Manish Tripathi
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
Case : Aileron Deflection
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
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
35/60
6 D O F S i m u l a t i o n P a g e | 35
Published by : Manish Tripathi
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
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
36/60
6 D O F S i m u l a t i o n P a g e | 36
Published by : Manish Tripathi
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
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
37/60
6 D O F S i m u l a t i o n P a g e | 37
Published by : Manish Tripathi
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
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
38/60
6 D O F S i m u l a t i o n P a g e | 38
Published by : Manish Tripathi
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
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
39/60
6 D O F S i m u l a t i o n P a g e | 39
Published by : Manish Tripathi
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
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
40/60
6 D O F S i m u l a t i o n P a g e | 40
Published by : Manish Tripathi
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
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
41/60
6 D O F S i m u l a t i o n P a g e | 41
Published by : Manish Tripathi
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
Beta, v and r are very highly damped which means a very stable flight.
Beta, v and r are very highly damped which means a very stable flight.
The slope of the curve has increased resulting in a steeper descent .
Clp = 1(Lateral Directional Modelling)
Case : Aileron Deflection
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
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
42/60
6 D O F S i m u l a t i o n P a g e | 42
Published by : Manish Tripathi
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
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
43/60
6 D O F S i m u l a t i o n P a g e | 43
Published by : Manish Tripathi
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
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
44/60
6 D O F S i m u l a t i o n P a g e | 44
Published by : Manish Tripathi
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.
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
45/60
6 D O F S i m u l a t i o n P a g e | 45
Published by : Manish Tripathi
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
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
46/60
6 D O F S i m u l a t i o n P a g e | 46
Published by : Manish Tripathi
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.
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
47/60
6 D O F S i m u l a t i o n P a g e | 47
Published by : Manish Tripathi
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
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
48/60
6 D O F S i m u l a t i o n P a g e | 48
Published by : Manish Tripathi
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
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
49/60
6 D O F S i m u l a t i o n P a g e | 49
Published by : Manish Tripathi
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
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
50/60
6 D O F S i m u l a t i o n P a g e | 50
Published by : Manish Tripathi
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
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
51/60
6 D O F S i m u l a t i o n P a g e | 51
Published by : Manish Tripathi
When Cnbeta is more positive i.e. 4.281:unstable response
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
52/60
6 D O F S i m u l a t i o n P a g e | 52
Published by : Manish Tripathi
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.
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
53/60
6 D O F S i m u l a t i o n P a g e | 53
Published by : Manish Tripathi
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;
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
54/60
6 D O F S i m u l a t i o n P a g e | 54
Published by : Manish Tripathi
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
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
55/60
6 D O F S i m u l a t i o n P a g e | 55
Published by : Manish Tripathi
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
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
56/60
6 D O F S i m u l a t i o n P a g e | 56
Published by : Manish Tripathi
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
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
57/60
6 D O F S i m u l a t i o n P a g e | 57
Published by : Manish Tripathi
elseif (ch==3 && ch2==2)delr=-6/57.3;end
elseif t
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
58/60
6 D O F S i m u l a t i o n P a g e | 58
Published by : Manish Tripathi
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);
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
59/60
6 D O F S i m u l a t i o n P a g e | 59
Published by : Manish Tripathi
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')
-
7/29/2019 Modeling of Motion and Flying Charactrstics of an Airplane
60/60
6 D O F S i m u l a t i o n P a g e | 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