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February 11, 2011
AAE 421, Spring 2011
Homework Five
Due: Friday, February 18
Exercise 1 (Your first flight) Consider the model for the Cessna 182 aircraft given in
the notes. Obtain a SIMULINK model of this vehicle with state variables V, α, θ, q, p, h and
“fly” your model to achieve the following trim conditions. (Consider xcm^ = 0.)
(a) Gliding with el = 0.
(b) Horizontal level flight with
th = 100 hp
In this part, you need to vary the elevator position until the desired trim condition is
achieved.
Illustrate your results with plots of the time histories of V , α (deg), γ (deg), and h; also plot
the aircraft trajectory, that is, h versus p.
a) Gliding with el = 0
Simulation Results:
p
h
gamma
alpha
V
S-Function
sfun_Cessna 182
Constant 1
th
Constant
el
0 50 100 150 200 250 300
130
135
140
145
150
155
160
Time (s)
V (ft/s)
Time History of V (Gliding)
0 50 100 150 200 250 300
0
1
2
3
4
Time (s)
alpha (deg)
Time History of alpha (Gliding)
0 50 100 150 200 250 300
0
2
Time (s)
gamma (deg)
Time History of gamma (Gliding)
b) Horizontal Level Flight with th = 100 hp
Simulation Results:
(Note: For these ICs, I used el = 1.592° to achieve horizontal level flight)
0 50 100 150 200 250 300 350 400 450 500
160
165
170
175
180
185
190
195
200
205
Time (s)
V (ft/s)
Time History of Velocity (Horizontal Level Flight)
0 50 100 150 200 250 300 350 400 450 500 0
1
Time (s)
alpha (deg)
Time History of alpha (Horizontal Level Flight)
0 50 100 150 200 250 300 350 400 450 500
0
2
4
6
Time (s)
gamma (deg)
Time History of gamma (Horizontal Level Flight)
S-Function File:
% sfun_pend_drag.m %CHANGE % S-function to describe the dynamics of a % Cessna 182 %CHANGE function [sys,x0,str,ts] = sfun_Cessna182(t,x,u,flag) %CHANGE % t is time % x is state % u is input % flag is a calling argument used by Simulink. % The value of flag determines what Simulink wants to be executed. switch flag case 0 % Initialization [sys,x0,str,ts]=mdlInitializeSizes; case 1 % Compute xdot sys=mdlDerivatives(t,x,u); case 2 % Not needed for continuous-time systems case 3 % Compute output sys = mdlOutputs(t,x,u); case 4 % Not needed for continuous-time systems case 9 % Not needed here end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % mdlInitializeSizes %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % function [sys,x0,str,ts]=mdlInitializeSizes % % Create the sizes structure sizes=simsizes; sizes.NumContStates = 6; %Set number of continuous-time state variables %CHANGE sizes.NumDiscStates = 0; sizes.NumOutputs = 5; %Set number of outputs %CHANGE sizes.NumInputs = 2; %Set number of inputs %CHANGE sizes.DirFeedthrough = 0; sizes.NumSampleTimes = 1; %Need at least one sample time sys = simsizes(sizes); % x0=[160; 0; 0; 0; 0; 3000]; % Set initial state %CHANGE str=[]; % str is always an empty matrix ts=[0 0]; % ts must be a matrix of at least one row and two columns % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % mdlDerivatives %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % function sys = mdlDerivatives(t,x,u) % % Compute xdot based on (t,x,u) and set it equal to sys % sys= Cessna182_ode(t,x,u); %CHANGE % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% mdlOutput %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % function sys = mdlOutputs(t,x,u) % % Compute xdot based on (t,x,u) and set it equal to sys sys = [x(1),x(2)180/pi,(x(3)-x(2))180/pi,x(5),x(6)]; %CHANGE
ODE File:
% Equations of motion for Cessna 182 in HW 5
function xdot = Cessna182_ode(t,x,u) global W S cbar J2 rho m xcm global CL0 CLa CLel global CDm k CLdm global CM0 CMa CMel global CMadot CMq CLadot CLq global epsilon eT eta
% Define some variables el = u(1); th = u(2); V = x(1); alpha = x(2); theta = x(3); q = x(4); qbar = 0.5rhoV^2; gamma = theta - alpha;
% Thrust T = 550theta/V;
% Drag CD = CDm + k(CL0+CLaalpha+CLelel - CLdm)^2; D = qbarS*CD;
% alphadot and Lift alphadot = (-qbarS(CL0 + CLaalpha + CLelel + cbar/(2V)CLqq) + Wcos(gamma) - Tsin(alpha-epsilon) + mVq)/(mV + qbarScbar/(2V)CLadot); CL = CL0 + CLaalpha + CLelel + cbar/(2V)CLadotalphadot + cbar/(2V)CLqq; L = qbarSCL;
% Moments MT = eTT; CM = CM0 + CMaalpha + CMelel + cbar/(2V)(CMadotalphadot + CMqq) - CLxcm/cbar; M = qbarScbar*CM;
% State-dots xdot(1) = (-D-Wsin(gamma)+Tcos(alpha-epsilon))/m; xdot(2) = alphadot;%(-L+Wcos(gamma)-Tsin(alpha-epsilon)+mVq)/(m*V);
