Microsoft Word - Mini-Project 2_Spr2023.docx
AE 4802 CON Mini-Project 2 Spring 2023
Complete the following problems. Please write or type your discussion and results neatly, organize
your work in a logical way, and describe your assumptions and steps. Include any needed plots
from computer analyses, along with titles, axis labels, and line types/colors as needed for clarity.
You may discuss solution techniques with other students, but copying results or code is prohibited.
Please write your code in either MATLAB or Python but note that the provided code snippets and
examples are written in MATLAB. Your assignment must be submitted electronically as a PDF
on Canvas along with any supporting code by Tuesday, March 28, at 11:59 PM.
We will model a small commercial transport airplane and produce some skymaps and a payload-
ange diagram. Presume the following characteristics about the airplane.
MTOGW: 110,000 lbs
MFW: 25,000 lbs
OEW: 60,000 lbs
MPW: 35,000 lbs
Fuel required for takeoff and climb: 2000 lbs
Fuel remaining after cruise (descent, reserve): 4500 lbs
Span: 92 ft
Wing area: 1,100 ft2
Zero-lift drag coefficient (at low speed; without wave drag): 0.021
Oswald efficiency: 0.87
Critical Mach number: 0.74
JT8D engine (deck provided)
Two engines, both operating
Now complete the following problems:
1. An engine deck that is roughly cali
ated to a Pratt and Whitney JT8D is provided with
this assignment in the form of a MATLAB function ([thrust, fuelflow, tsfc] =
jt8d(M,h,pc)), along with a function for the standard atmosphere model ([temp, pres,
ho, mu, a] = atmosphere(h)).
a. Plot maximum thrust (in lbs) vs. Mach number over a range of 0 ≤ M ≤ 0.80. Plot
two curves on the same plot axes: (1) sea level and (2) 35,000 ft altitude.
. Plot maximum thrust (in lbs) vs. altitude over a range of 0 ft ≤ h ≤ 35,000 ft. Plot
two curves on the same plot axes: (1) M = 0 and (2) M = 0.80.
c. Repeat (a), but this time for TSFC (in lbs/lbs-hr).
d. Repeat (b), but this time for TSFC (in lbs/lbs-hr).
Use a legend to identify the curves on each plot and include axis labels. Discuss how the
engine deck behavior illustrated in your plots compares qualitatively to the “simple”
models we discussed in class for thrust and TSFC variation with Mach number and altitude.
2. Make a drag polar function for the airplane of the form CD = dragpolar(CL, Mach). The
drag polar should include effects of wave drag using Lock’s 4th Power Method (just
presume that the wing ? , is equal to the airfoil ? , predicted from Lock’s
method). Use your drag polar to make two plots:
a. A plot of CD vs. Mach with different lines co
esponding to CL=0.1, CL=0.2, and
CL=0.3. Plot over the domain 0 ≤ Mach ≤ 0.86.
. A plot of CL vs. CD (a typical drag polar plot) with different curves co
esponding
to M=0.70, M=0.75, M=0.8, and M=0.85. Plot over the domain 0 ≤ CL ≤ 0.4.
Discuss how transonic drag rise affects the shapes of these plots compared to a simple 2-
parameter drag polar with constant coefficients.
3. Enforce steady level flight conditions and make a skymap for specific excess power (in
ft/min). Plot over the ranges 0 ≤ M ≤ 0.86 and 0 ≤ h ≤ 35,000 ft. Use “hatch” marks to
indicate the boundary of where steady level flight can occur with respect to specific excess
power considerations. You might find the following hatched line and hatched contour
functions for MATLAB or Python to be helpful:
https:
www.mathworks.com/matlabcentral/fileexchange/29121-hatched-lines-and-contours
https:
matplotlib.org/stable/gallery/images_contours_and_fields/contours_in_optimization_demo
.html#sphx-glr-gallery-images-contours-and-fields-contours-in-optimization-demo-py
4. Enforce steady level flight conditions and make a skymap for CL. Plot over the ranges 0 ≤
M ≤ 0.86 and 0 ≤ h ≤ 35,000 ft. Overlay a boundary of the buffet CL using “hatch”
marks. Presume that buffet occurs at the following values of CL vs. Mach number (you
can interpolate the numbers for making your plot):
Mach XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX
CLb XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX
5. Enforce steady level flight conditions and make the following skymaps of aerodynamic
performance. For both figures, plot over the ranges 0 ≤ M ≤ 0.86 and 0 ≤ h ≤ 35,000 ft.
Overlay both the specific excess power limit and the buffet limits from Problems 3 and 4
and trim data outside these constraints. See the note and code snippets at the end of this
assignment for some help in trimming the data.
c. L/D
d. ML/D
For each of these plots, discuss “where” you would like to fly (in terms of M and h) for
est L/D and ML/D, respectively, and why.
6. Enforce steady level flight conditions and make the following skymaps for range and
endurance performance. For both figures, plot over the ranges 0 ≤ M ≤ 0.86 and 0 ≤ h ≤
35,000 ft. Overlay both the specific excess power limit and the buffet limits from Problems
3 and 4 and trim data outside these constraints. See the note and code snippets at the end
of this assignment for some help in trimming the data.
a. Specific endurance (in sec/lbs)
. Specific range (in nm/lbs)
For each of these plots, discuss “where” you would like to fly (in terms of M and h) for
est endurance and range, respectively, and why. Compare/contrast these optimal flight
conditions with those you found in Problem 5, being sure to note the physical
similarities
elationships between the aerodynamic metrics in Problem 5 and specific range
and specific endurance.
7. Compute and list the following weights (in lbs) at each “corner point” of the payload-range
diagram, i.e. points 1, 2, and 3 labeled on slide 8 of “8_Payload_Range.pptx” on Canvas:
(a) takeoff gross weight, (b) payload weight, (c) zero-fuel weight, (d) fuel weight. Present
these values in a table with rows co
esponding to each of the weights above and columns
co
esponding to the particular corner points. Show your work. Can the airplane fly with
max payload and max fuel simultaneously? Why or why not?
8. Compute the range of the airplane at each “corner point” from Problem 7. To do this,
presume flight at 30,000 ft at Mach 0.76. Use the Breguet range equation for a jet airplane
in the following form:
? =
? (?/?)
????
ln
?
?
where ? is the weight of the airplane at the beginning of cruise (subscript “i” for “initial”)
and ? is the weight of the airplane at the end of cruise (subscript “f” for “final”). Use the
drag polar and engine deck to find the needed information at the given altitude and Mach
number. “Fly the airplane” by enforcing the equations of motion for steady level flight, and
presume that the airplane flies at constant CL and constant V at the conditions
co
esponding to the initial cruise weight, ? . Be sure to use the weights co
ectly from
Problem 7 and also to consider the climb fuel and reserve fuel values provided in the
airplane data at the beginning of the assignment. Show your work.
9. Plot the bounding envelope of the payload-range diagram using the information from
Problems 7 and 8. Use OEW+PW (which equals ZFW) as the y-axis variable (instead of
just PW). Could this aircraft fly from New York to London with 27,000 lbs of passengers
and cargo? Why or why not?
Note on Trimming Data Outside Constraint Boundaries
To trim parts of the skymaps that fall outside of the two constraint curves, a handy trick is to note
that MATLAB will not plot “not a number” (NaN) values. We can use this to our advantage. You
can define a “thrust margin” as Tmargin=Tmax-D and a “CLmargin” as CLmargin= CLbuffet-CL.
(Note that the specific excess power > 0 limit is effectively nothing more than a steady level flight
thrust limit.) Then, you can trim your specific range (SR) data as follow:
SR(Tmargin<0)=NaN;
SR(CLmargin<0)=NaN;
You can also trim the buffet and maximum thrust boundary curves as in the example below, where
you can adjust the value to the right of the < to make things work nicely for your contour spacing:
Tmargin(CLmargin<-0.05)=NaN;
CLmargin(Tmargin<-0.05)=NaN;
As an example implementation, here’s how I did it. If you learn nothing more from this homework,
then learn just how to make nice MATLAB plots by controlling everything in the figure and then
exporting to a .png:
% Compute SR and/or SE and all of the other needed intermediate
variables (e.g. CL, CD, etc.) here; I recommend using the MATLAB
meshgrid function to create matrices of Mach number and altitude
points that you then use to make the contour plots by running each of
the resulting points through your computations.
% Now compute the skymap limits and trim the data
[Tmaxeng,~,~]=jt8d(M,h,50*ones(size(M))); % PC = 50 for max thrust
Tmax=2*Tmaxeng; % For 2 engines operating
Tmargin=Tmax-Treq;
CLmargin=CLbuffet-CL;
% Preceding the line above, you’ll need to compute CLbuffet as a
% function of Mach number by interpolating the numbers given in
% Problem 4
Tmargin(CLmargin<-0.05)=NaN;
CLmargin(Tmargin<-0.05)=NaN;
SR(Tmargin<0)=NaN;
SR(CLmargin<0)=NaN;
% Make the figure
[c,handle]=contour(M,hplot,SR,'LineColor','k');
% M, hplot, and SR are in meshgrid format
clabel(c,handle);
title('Specific Range (nm/lb)')
xlabel('Mach Number')
ylabel('Altitude (1000 ft)')
axis([ XXXXXXXXXX])
hold on
oC = ocontourc(Mvect',hvect', Tmargin, [0,0]);
%Note that Mvect and hvect are the vectors that were used to define
%the meshgrid for the skymap contour plot
handle2=hatchedcontours(oC);
oC = ocontourc(Mvect',hvect', CLmargin, [0,0]);
handle2=hatchedcontours(oC,'r');
hold off
% Export the figure. Use the stencil below for exporting all of your
% MATLAB graphics!
ez=600; %resolution (dpi) of final graphic. 600 is a good number.
f=gcf; %handle of figure to be exported
figpos=getpixelposition(f);
esolution=get(0,'ScreenPixelsPerInch');
set(f,'paperunits','inches','papersize',figpos(3:4)
esolution,...
'paperposition',[0 0 figpos(3:4)
esolution]);
path=
; %Where should it go? C:\\ etc.
name='SR.png'; %What should it be called?
print(f,fullfile(path,name),'-dpng',['-r',num2str(rez)],'-opengl')
10.