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Homework 1 AERE355 Fall 2015 Due 9/5(F)
Name __________________________________________
NOTE: Answer (no work required) ; Solution (You need to show work used to arrive at the answer)
Problem 1(20pts) In the altitude region 0  h  10km , we have the following atmospheric equations:
T  T0   h .
P T 
 
P0  T0 
 T 
 
 0  T0 
 (1
go
)
R
 go
R
(1.9*)
.
or  
(1.55)
P
RT
(1.57)
Write a Matlab code that will generate a table with columns h T P   for the array of altitude values
h  0 : 1000 : 10000 , give the table, and comment on how it compares to Table A.1 on p.395 in APPENDIX 1.
Solution:
Standard Atmosphere Table
HG
Comment:
h
T
P

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PROBLEM 2 (35pts) [Related to book Problem 1.2] An airplane’s altimeter reading is 5,000m.
(a)(15pts) If the outside ambient temperature is -20oC and the plane’s true airspeed is 300m/s, find the plane’s indicated
airspeed.
Solution:
(b)(5pts) Suppose that the altimeter reading fluctuates between roughly 4,990m and 5,010m, and that the mean reading is
h  5,000 m . Assume that the fluctuation h has a normal distribution with   3m . Use the Matlab command ‘normpdf’
to arrive at a plot of the probability density function (pdf) for the altimeter reading, h.
Solution: The Matlab code in the Appendix gives the plot below:
Figure 1.2.1 Altimeter reading pdf.
(c)(5pts) On p.17 we have equation (1.54): ln( P / P1 ) 
 T  h 

detailed steps to show that P  Po  o
 To 
 g o  T1   (h  h1 ) 
 . Beginning with this equation, carry out
ln 
R 
T1

 g o / R
.
Solution:
(d)(5pts) To arrive at a theoretical expression for the pdf of P from that of h is beyond the scope of this course. For this
reason, the pdf will be arrived at using simulations. Specifically, use the Matlab command ‘normrnd’ to generate
N=10,000 simulations of h. Next, use these in the expression in (c) to obtain simulated measurements of P. Then: (i) use
the command ‘hist’ to arrive at a plot of an estimate of the pdf of P, (ii) use the commands ‘mean’ and ‘std’ to estimate the
mean and standard deviation of P, (iii) overlay a normal pdf model on your plot, and (iv) comment on whether you think
the normal pdf model is accurate. NOTE: Recall that R  287 m2 /o K  s 2 and   .0065 o K / m .
Solution: (i) & (iii): The code in the Appendix gave the plot below:
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Figure 1.2.2 Plots of the histogram-based and normal pdfs for P
(ii) The estimated mean and standard deviation are:
(iv) COMMENT:
 R 
 1

g o 
 P 
(e)(5pts) Show that the relation between density and pressure is:    o  
 Po 
Solution:
(f)(5pts) EXTRA CREDIT Repeat part (d) for the density.
Solution: (i) & (iii): The code in the Appendix gave the plot below:
Figure 1.2.3 Plots of the histogram-based and normal pdfs for  .
(ii) The estimated mean and standard deviation are:
(iv) COMMENT:
.
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(g)(5pts) EXTRA CREDIT Recall that VIAS  VTAS   300  . Repeat part (e) for VIAS .
Solution: (i) & (iii): The code in the Appendix gave the plot below:
Figure 1.2.3 Plots of the histogram-based and normal pdfs for VIAS .
(ii) The estimated mean and standard deviation are:
(iv) COMMENT:
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PROBLEM 3 (25pts) [Related to book problem 2.1] A given wing moment equation is: Cmcg  0.08  0.15C Lw
w
(a)(5pts) Recall from the discussion below Figure 2.5 on p.43 that the trim condition is defined as that condition such that
Cmcg  0 . (i.e. the trim moment coefficient equals zero). Give the corresponding trim lift coefficient.
Answer:

(b)(5pts) The (scaled) wing center of gravity of the wing is hcg w 

point h N w 
xNw
c
x cg w
c
 0.3 . Use (2.6) on p.45 to find the wing neutral
.Assume that the neutral point equals the aerodynamic center point.
Solution:
(c)(10pts) In (2.7) we see that C Lw  C Lw  C L  w . Suppose that the wing lift equals zero when
0
w
 w  iw  2o ( / 180o )  .0349rad , and it is at trim when  w   wtrim  5o ( / 180 o )  .0873 rad . Find C Lw0 and C Lw . Then plot
CLw as a function of  w over the range 0o   w  10o .
Solution:
Figure 2.1 Plot of CLw versus α.
(d)(5pts) For a wing-alone configuration as shown in Figure 2.7, explain why it is not statically stable.
Explanation:
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PROBLEM 4 (10pts) Consider a thin, symmetric airfoil.
From Fundamentals of Aerodynamics by J. D. Anderson, we have the following the following:
(i) hn = 1/4 . p.328 Fig. 4.27
(ii) CL ( )  2  . p.325 eqn. (4.33)
(iii) CmAC = 0. p.327 eqn. (4.41)
Use the above results to obtain the expression for Cm ( ) .
(a)(6pts) Use your results in (a) to obtain the expression for Cm ( ) .
Solution:
(b)(4pts) From your equation identify  trim . If it does not exist, explain why.
Answer:
PROBLEM 5 (10pts) Proper understanding of notation is of utmost importance in this course. Give the definitions of the
following sets of variables: (i) ( x, y, z) ; (ii) ( X ,Y , Z ) ; (iii) (u, v, w) ; (iv) ( L, M , N ) ; (v) (V ,  ,  ) .
Answers:
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Appendix Matlab Code