Assignment#6
AE-322
Submitted by: N/C Zubair Abid
Pak Number: 182201
Course: 92nd EC
Section: Section A
Date: 02/08/2021
Submitted to: Sir Muzaffar
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Question No 1: Explain pure vertical or plunging motion in the aircraft. Derive a 1 st order equation
to describe this motion under the influence of gust. Plot the response of the system under step or
sinusoidal gust loads
Consider an airplane constrained so that movement is possible only in the vertical direction. This
type of motion could be simulated in the wind tunnel using a model constrained by a vertical rod The
model is free to move up or down along the rod, but no other motion is possible.
The equation of motion for this example is obtained by applying Newton's second law that is,
(1)
Where,
Z = Aerodynamic force in the z direction
W = Weight of the airplane model
If we assume the motion of the airplane will be confined to small perturbations from an initial
unaccelerated flight condition, then the aerodynamic force and vertical velocity can be expressed
as the sum of the reference flight condition plus the perturbation (small disturbance theory),
(2)
Putting 1 in 2 gives,
(3)
This equation can be simplified by recognizing that in unaccelerated flight the condition for
equilibrium is
(4)
Thus now 3 is reduced to,
(5)
Where,
(6)
(7)
To simplify our analysis, we will assume that the lag in lift term, 𝑍𝛼̇ , 𝛼̇ is negligible in comparison to
the 𝑍𝛼 ∆𝛼 term
Substitute 6 and 7 into 5 to get,
(8)
2
(9)
Where,
(10)
Now the response will be plotted for step input and sinusoidal input.
1) Step Input / Sharp Edged Gust
(11)
(12)
2) Sinusoidal Input
(13)
clear
clc
Input Values
u0=125; %Approach Velocity
Cl_alpha=4.44; %Lift curve slope
W=2750; %Weight of aircraft
S=184; %Wing Surface Area
Ag=15; %Magnitued of Wind Gust
rho=0.002378; %Density at sea--level
g=32.2; %Gravitational Accelration
omega=2; %Frequency of Vibration
t=0:0.01:8; %Time Duration
Calculations
a) Step Input
m=W/g; %Mass of Aircraft
Q=0.5*rho*u0^(2); %Dynamic Veocity of Aircraft
Z_alpha=-(Cl_alpha*Q*S)/m;
Tou=-u0/Z_alpha; %Determination of Time Constant
fprintf('Value of Time Constant: \n')
fprintf('%0.3f \n', Tou)
w1=Ag.*(1-exp(-t./Tou)); %Response under Step Input
syms t1
w_symbolic=Ag*(1-exp(-t1/Tou));
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w_t=inline(diff(w_symbolic,t1));
slope=w_t(0);
x_tangent=0;
y_tangent=0;
tangent_equation=slope.*(t-x_tangent) + y_tangent; %Acceleration at t=0
figure(1)
plot(t,w1)
xlabel('Time/ (s)')
ylabel('Response of Aircraft/ w(t)')
title('Response of Aircraft to Sharp Edged Gust')
hold on
plot(t,tangent_equation,'r--')
text(5,12,'Acceleration at t=0')
annotation('arrow',[0.2 0.6],[0.8 0.73])
axis([0 8 0 16])
b) Sinusoidal Input
phi=-atan(Tou*2); %Phase Angle
w2=(Ag.*(1./sqrt(1 + (Tou.*omega).^(2)))).*sin(omega.*t - phi); %Response under Sinusoidal
Input
figure(2)
plot(t,w2)
xlabel('Time/ (s)')
ylabel('Response of Aircraft/ w(t)')
title('Response of Aircraft to Sinusoidal Gust')
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