Assignment#5
AE-322
Submitted by: N/C Zubair Abid
Pak Number: 182201
Course: 92nd EC
Section: Section A
Date: 15/07/2021
Submitted to: Sir Muzaffar
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Question No 1:
Write a MATLAB code to find the complete longitudinal response of the Boeing 747 aircraft
while flying at Mach 0.25 near to the ground using equation (4.51). Take geometric and
aerodynamic parameters from Table B-27 of your textbook.
a) Construct the A matrix
clear
clc
Data Input for Geometric Properties
Iyy=33.1e6; %Moment of Inertia about the y-axis
W=636600; %Weight of a/c
S=5500; %Surface Area of Wing
b=195.68; %Span of Wing
c=27.31; %Mean Chord of Wing
M1=0.25; %Mac Number of a/c
rho=0.002378; %Density of ai in slugs/ft^3
gamma=1.4; %Specfic Heat Ratio
R=287; %Gas Constant
T=288.15; %Atmospheric Temperature in K
g=32.2; %Gravitational Acceleration in ft/s^2
Data Input for calculating Stability Derivatives
CDu=0;
CLu=-0.81;
CMu=0.27;
CD0=0.102;
CL0=1.11;
CDalpha=0.66;
CLalpha=5.70;
CMalpha=-1.26;
CZalphadot=0;
CMalphadot=-3.2;
CZq=0;
CMq=-20.8;
CZdel_e=-0.338;
CLdel_e=0.338;
CMdel_e=-1.34;
Determination of Basic Quantities
m=W/32.2; %Mass of the a/c
u0=(M1*(gamma*R*T)^(1/2))*3.28084; %Forward Velocity in ft/s
Q=0.5*rho*(u0)^2; %Dynamic Pressure
Equations for Longitunal Stability Derivatives
Xu=(-(CDu + 2*CD0)*Q*S)/(m*u0);
Xw=(-(CDalpha - CL0)*Q*S)/(m*u0);
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Zu=(-(CLu + 2*CL0)*Q*S)/(m*u0);
Zw=(-(CLalpha + CD0)*Q*S)/(m*u0);
Zwdot=-CZalphadot*((c)/(2*u0))*((Q*S)/(m*u0));
Zalpha=u0*Zw;
Zalphadot=u0*Zwdot;
Zq=CZq*((c)/(2*u0))*((Q*S)/(m));
Zdel_e=CZdel_e*((Q*S)/(m));
Mu=CMu*((Q*S*c)/(u0*Iyy));
Mw=CMalpha*((Q*S*c)/(u0*Iyy));
Mwdot=CMalphadot*((c)/(2*u0))*((Q*S*c)/(u0*Iyy));
Malpha=u0*Mw;
Malphadot=u0*Mwdot;
Mq=CMq*((c)/(2*u0))*((Q*S*c)/(Iyy));
Mdel_e=CMdel_e*((Q*S*c)/(Iyy));
Generation of A Matrix
A=[Xu Xw 0 -g;Zu Zw u0 0;Mu+(Mwdot*Zu) Mw+(Mwdot*Zw) Mq+(Mwdot*u0) 0;0 0 1 0] %Generation
of Stability Matrix
b) Find the Damping ratio and frequency using the complete longitudinal response.
Calculation of Natural Frequency and Damping Ratio for the complete Longitunal Response
sympref('FloatingPointOutput',true);
syms x
I=[1 0 0 0;0 1 0 0;0 0 1 0;0 0 0 1]; % Identity Matrix
D=det(x.*I - A)==0; %Taking Determinant of Matrix and putting it equal to zero and
assigning that to a variable
Characteristic_Equation=vpa(D) %Simplifying the expression for the Characteristic Equation
Roots=vpa(solve(D,x)) %Obtaining roots of the Characteristic Equation
%Phugoid Mode
PhugoidRoots=[Roots(1);Roots(2)] %Roots for Phugoid Mode
Phugoid_t_half=0.69/abs(real(Roots(1))); %Time to half amplitude for Phugoid Mode
Phugoid_Period=(2*pi)/abs(imag(Roots(1))); %Time Period for Phugoid Mode
Phugoid_Cycles=((Phugoid_t_half)/(Phugoid_Period)); %Number of Cycles to half amplitude
for Phugoid Mode
syms zeta wn
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[NaturalFrequency_Phugoid,DampingRatio_Phugoid]=solve(zeta*wn==abs(real(Roots(1))), wn*(1(zeta)^(2))^(1/2)==abs(imag(Roots(1))));
fprintf('Natural Frequency for Phugoid Mode: %0.2f rad/s\n',NaturalFrequency_Phugoid )
fprintf('Damping Ratio for Phugoid Mode: %0.3f \n',DampingRatio_Phugoid )
fprintf('Time to half amplitude for Phugoid Mode: %0.2f s\n ',Phugoid_t_half)
fprintf('Time Peirod for Phugoid Mode: %0.2f s\n ',Phugoid_Period)
fprintf('Number of cycles to half amplitude for Phugoid Mode: %0.2f \n ',Phugoid_Cycles)
%Short Period Mode
ShortPeriodRoots=[Roots(3);Roots(4)] %Roots for Short Period Mode
ShortPeriod_t_half=0.69/abs(real(Roots(3))); %Time to half amplitude for Short Period Mode
ShortPeriod_Period=(2*pi)/abs(imag(Roots(3))); %Time Period for Short Period Mode
ShortPeriod_Cycles=((ShortPeriod_t_half)/(ShortPeriod_Period)); %Number of Cycles to half
amplitude for Short Period Mode
[NaturalFrequency_ShortPeriod,DampingRatio_ShortPeriod]=solve(zeta*wn==abs(real(Roots(3)))
, wn*(1-(zeta)^(2))^(1/2)==abs(imag(Roots(3))));
fprintf('Natural Frequency for Short Period Mode: %0.2f
rad/s\n',NaturalFrequency_ShortPeriod )
fprintf('Damping Ratio for Short Period Mode: %0.3f \n',DampingRatio_ShortPeriod )
fprintf('Time to half amplitude for Short Period Mode: %0.2f s\n ',ShortPeriod_t_half)
fprintf('Time Peirod for Short Period Mode: %0.2f s\n ',ShortPeriod_Period)
fprintf('Number of cycles to half amplitude for Short Period Mode: %0.2f \n
',ShortPeriod_Cycles)
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c) Apply short period and phugoid approximations to find the approximate solution.
Phugoid Approximation
wn_p=((-Zu*g)/(u0))^(1/2);%Natural Frequency obtained from Phugoid Approximation
zeta_p=(-Xu)/(2*wn_p); %Damping ratio obtained from Phugoid Approximation
w_d_p=wn_p*(1-(zeta_p)^(2))^(1/2); %Damped Frequency obtained from Phugoid Approximation
Phugoid_Approx_Roots=[-zeta_p*wn_p - 1i*w_d_p; -zeta_p*wn_p + 1i*w_d_p] %Roots obtaiend
from Phugoid Approximation
Phugoid_Approx_t_half=0.69/abs(real(Phugoid_Approx_Roots(1))); %Time to half amplitude
from Phugoid Approximation
Phugoid_Approx_Period=(2*pi)/(w_d_p); %Time Period for Phugoid Approximation
Phugoid_Approx_Cycles=(Phugoid_Approx_t_half)/(Phugoid_Approx_Period); %Number of Cycles
to half amplitude for Phugoid Approximation
fprintf('Time to half amplitude for Phugoid Mode Approximation: %0.2f s\n
',Phugoid_Approx_t_half)
fprintf('Time Peirod for Phugoid Mode Approximation: %0.2f s\n ',Phugoid_Approx_Period)
fprintf('Number of cycles to half amplitude for Phugoid Mode Approximation: %0.2f \n
',Phugoid_Approx_Cycles)
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Short Period Approximation
wn_sp=(((Zalpha*Mq)/u0) - Malpha)^(1/2); %Natural Frequency obtained from Short Period
Approximation
zeta_sp=-(Mq + Malphadot + ((Zalpha)/u0))/(2*wn_sp); %Damping ratio obtained from Short
Period Approximation
w_d_sp=wn_sp*(1-(zeta_sp)^(2))^(1/2); %Damped Frequency obtained from Short Period
Approximation
ShortPeriod_Approx_Roots=[-zeta_sp*wn_sp - 1i*w_d_sp; -zeta_sp*wn_sp + 1i*w_d_sp] %Roots
obtaiend from Short Period Approximation
ShortPeriod_Approx_t_half=0.69/abs(real(ShortPeriod_Approx_Roots(1))); %Time to half
amplitude from Short Period Approximation
ShortPeriod_Approx_Period=(2*pi)/(w_d_sp); %Time Period for Short Period Approximation
ShortPeriod_Approx_Cycles=(ShortPeriod_Approx_t_half)/(ShortPeriod_Approx_Period); %Number
of Cycles to half amplitude for Short Period Approximation
fprintf('Time to half amplitude for Short Period Mode Approxiamtion: %0.2f s\n
',ShortPeriod_Approx_t_half)
fprintf('Time Peirod for Short Period Mode Approxiamtion: %0.2f s\n
',ShortPeriod_Approx_Period)
fprintf('Number of cycles to half amplitude for Short Period Mode Approximation: %0.2f \n
',ShortPeriod_Approx_Cycles)
d) Compare the exact and approximate solution and construct table like Table 4.4 of
Example 4.3
Difference between the complete Longitunal Response and the Phugoid and Longitunal Approximations
%Errors in Phugoid Mode
Error_t_half_Phugoid=((abs(Phugoid_t_half - Phugoid_Approx_t_half))/(Phugoid_t_half))*100;
%Error in time to half amplitude due to Phugoid Approximation
Error_Period_Phugoid=((abs(Phugoid_Period - Phugoid_Approx_Period))/(Phugoid_Period))*100;
%Error in time period due to Phugoid Approximation
Error_Cycles_Phugoid=((abs(Phugoid_Cycles - Phugoid_Approx_Cycles))/(Phugoid_Cycles))*100;
%Error in number of cycles to half amplitude due to Phugoid Approximation
fprintf('Error in time to half amplitude due to Phugoid Approximation: %0.2f %%
\n',Error_t_half_Phugoid)
fprintf('Error in time period due to Phugoid Approximation: %0.2f
%%\n',Error_Period_Phugoid)
fprintf('Error in number of cycles to half amplitude due to Phugoid Approximation: %0.2f
%% \n',Error_Cycles_Phugoid)
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%Errors in Short Period Mode
Error_t_half_ShortPeriod=((abs(ShortPeriod_t_half ShortPeriod_Approx_t_half))/(ShortPeriod_t_half))*100; %Error in time to half amplitude
due to Short Period Approximation
Error_Period_ShortPeriod=((abs(ShortPeriod_Period ShortPeriod_Approx_Period))/(ShortPeriod_Period))*100; %Error in time period due to Short
Period Approximation
Error_Cycles_ShortPeriod=((abs(ShortPeriod_Cycles ShortPeriod_Approx_Cycles))/(ShortPeriod_Cycles))*100; %Error in number of cycles to half
amplitude due to Short Period Approximation
fprintf('Error in time to half amplitude due to Short Period Approximation: %0.2f %%
\n',Error_t_half_ShortPeriod)
fprintf('Error in time period due to Short Period Approximation: %0.2f
%%\n',Error_Period_ShortPeriod)
fprintf('Error in number of cycles to half amplitude due to Shoet Period Approximation:
%0.2f %% \n',Error_Cycles_ShortPeriod)
Exact
Method
Approximate Method
Difference
t1/2
(s)
194.40
73.28
62.30 %
P
(s)
44.68
51.42
15.09%
t1/2
(s)
1.33
1.34
1.14%
P
(s)
9.02
8.94
0.86%
Modes and Properties
Phugoid
Short Period
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e) Comment on the flying qualities of the aircraft in both modes and suggest design level
changes to get better flying and handling qualities.
 The Short Period mode is much more accurate as compared with the Phugoid Mode. The
preferable mode of operation is the short period mode as the amplitude decreases much
more quickly and the period is much shorter.
 In order to improve the Phugoid Mod, the stability derivatives have to be changed to get
lesser t1/2 and smaller period.
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