Vehicle Design and Performance
Master Solution - CONFIDENTIAL
- January 2002
This paper contains the master solution for the vehicle design and performance question
for the doctoral qualifying examination in the Department of Aeronautics and Astronautics at MIT. This paper is to be used for grading purposes only.
(a) Spacecraft Design Problem
(al) Assuming 45 degree incidence of sunlight onto the solar panel, find Pt = f (A).
Compute a lower bound Aminfor Pt = 0.
The power generated by the solar panel is the solar constant times the area A times the
efficiency times the cosine of the incidence angle, thus:
In order to cor~lputethe transmission power as a furlctiorl of A, we have to subtract the
bus power and substitute the constants:
The lower bound Aminis found by setting Pt = 0 as:
(a2) The two subsystem teams suggest an initial design x, = [D AIT = [I 5IT. Compute
the performance R(x,). Does this design meet the requirement? Explain.
We substitute the three last equations from the problem formulation into the link budget
equation for the data rate R and obtain:
where p = 93703.3 [J-1m-2] is a constant. Thus, the performance goes linearly with solar
panel area A and quadratically with antenna diameter D. Substituting the values lor x,
in the above equation we obtain R(xo) = 25.9 [Mbps]. This design does not meet the
requirement (80-120 [Mbps]) and is significantly underdesigned. We need to increase the
data rate R by increasing D, A or both.
(a3) If design x, meets the requirement, set XL = x, and continue directly with (a4).
Otherwise, use x, as a starting point for finding an acceptable design xl. Plot your
path from x, to x1 in the (D,A)-tradespace. Note that the tradespace is bounded by
Dmin5 D 5 Dmaxand Ami, 5 A 5 A,,,
respectively. Hint: Plot the (D,A)-tradespace
as large as you can on a separate sheet.
Starting from x, we can find xl by increasing the antenna diameter D. The question is
by how much? A sensitivity analysis can help determine a good step size:
Substituting the values for x,, we obtain a sensitivity dR/dD = 51.5 [Mbps/m]. Thus
the step size is found as:
Thus, we attempt the new design point XI = [2.45 5IT and substitute into the performance
equation R = f ( D , A ) found above. We obtain R(xL) = 154.7 [Mbps]. It appears
that we overshot somewhat. We recompute the sensitivity at this point and obtain
i3Rli3D = 126.3 [Mbps/m], which results in a step size A D 1:-0.45 [m]. The new design
point is x1 = [2 5IT resulting in a performance R(xl) = 103 [Mps]. This design meets the
requirement. The path from x, to x1 is shown in Figure 1.
(a4) Ma.na,gementlikes your design xl, but points out tha,t it is not unique. Find a. second
acceptable design, x2, at some distance from XI. Sketch the line that corresponds to the
requirement &, = 100 [Mbps] in (D,A)-space (isoperformance contour).
The design is not unique, since we could have also increased the solar panel area A to
meet performance, while holding D fixed. The sensitivity of R with respect to A is:
This sensitivity is independent of A and is evaluated as 8R/8A = 10.8 [Mbps/m2] at x,.
We compute the increment as:
The performance at this point x2 = [I 12IT is equal to R(x2) = 101.2 [Mbps] and meets
The fie,= 100
the requirements. Note that xz is pegged at the upper bound A,.
[Mbps] isoperformance contour is shown in Figure I. We see that for an increasing
diameter D we may decrease A and vice-versa.
(a5) The accounting department gives you the cost estimation relationships for the antenna CD= 2500 - D2 [$] and solar panel CA = 12000 - (A 1) [$], respectively. Assuming
that there are no other cost drivers, what is your final recommended design x3?
+
We want to find x3 such that is satisfies the equation
while minimizing the cost
We substitute D2 from the first equation into the cost equation and obtain:
13
Satellite Design (D,A) trade space - contours in [Mbps]
x2
12
Solar Panel Area A [m^2]
11
10
20
0
25
8
0
40
9
0
10
7
20
6
0
5
x1 100
xo
x1*
4
25
3
2
0.5
1
x3
100
25
1.5
2
2.5
Antenna Diameter D [m]
3
(b) Unmanned Aerial Vehicle (UAV) Design Problem
( b l ) First compute the constant a used in the endurance equation: a = 2yCL/(clf).
Then compute the wing area S [m2],assuming that f = 0.25 and A? = 6. Hint: The lift
is L = (p/2)1T2SCLand L = g . m, during straight and level flight.
The value of a is simply obtained by substituting the appropriate constants:
a=- 2vCL - 2 - 0.8 1.2 = 48000 [sec]
cT-'
10-6 40
Next we compute the takeoff mass m, as:
+
+
m, = ma (1 &/18) (1 f ) = 220 (413) - (514) = 367 [kg]
(14)
We can solve for the wing area S from the lift equation and substitute:
(b2) Compute the performance E(x,) for the initial design vector x, = [f A] = [0.25 61.
Does this design meet the performance requirement E,.,,?
First compute the total drag coefficient for this configuration:
c; = 0.05 + 1.22 = 0.126
CD = CD, + TA?
x.6
(16)
The then endurance E(x,) is found by substituting into Brkguet's equation:
E(x,) =
CD
(dm
-
&-%
1) =
-
1 = 44965 [sec] E 12.5 [hr]
(17)
The design x, does not meet the requirement E,.,, = 18 - 22 [hr], since the loiter (=endurance) is too short.
(b3) If design x, meets the requirement, set X L = x, and continue directly with (b4).
Otherwise, use x, as a starting point for finding an acceptable design XL. Note that the
design space is bounded by 0.1 5 f 5 0.5 and 5 5 A 5 15, respectively. Hint: Try
adjusting the fuel mass fraction, f .
We can increase E by adding more fuel, i.e. by increasing f . The question is by how
much? Computing the sensitivity of endurance E with respect to f will help:
dE
--
af
-
-.-a
I
ac, dl+f
-
48000
a .o.las
I
.-
= 170367 [sec] = 47.32
The fuel mass fraction increment A f is then obtained as:
The new design point
XL =
[J AIT= [0.41 6IT yields an endurance of
E(xL)= 48000 0.126
(a
1
-
= 71403.5
[sec] 2: 19.8 [hr]
[hr]
(18)
This design xl meets the requirement E,,,.
(b4) Management likes your design xl, but points out that it is not unique. Find a
second acceptable design, x2, at some distance from xl. Hint: This time, try adjusting
the aspect ratio A, starting from design x,.
This time we will try to change A. We see from the drag coefficient equation that
increasing the aspect ratio A will reduce the total drag and thus increase the endurance
E for a constant f . The derivative of CDwith respect to A is:
The sensitivity of E with respect to A is obtained via the chain rule as:
-a(dm
c;
-
1) ,--C;
-
48000,
7
1
B
2
(m 1)- 1 . 2 ~= 4543.8 [sec] = 1.26 [hr]
-
0.1262 - i7 62
The A increment is obtained as:
The endurance for the new design point x2 = [0.25 12IT is computed as E(xz) = 64238.2
[sec] = 17.8 [hr], which still does not meet the requirement. Recomputing the sensitivity
at this new point results in another increment AA=3.4. The new design point x2 =
[0.25 15.4IT yields E(x2) = 70820 [sec] -. 19.6 [hr], which satisfies the requirement.
(b5) Plot your path from x, to xl and x2 in the (f ,A)design space. Sketch the line that
corresponds to the requirement E,,, = 20 [hours], i.e. the isrrendurance contour.
Figure 2 shows the design points x,, xl and x2 in the (f,&) design space along with
the iso-endurance curves at 10, 20 and 30 [hr]. As expected one may achieve the same
endurance with a large fuel mass fraction f and smaller A, i.e. carry lots of fuel and
have shorter wing span or with long slender wings (large A) and less fuel. The larger
aspect ratio desiin will be heavier due to the larger wing bending moment and the need
for thicker skins and bulkier spars.
(b6) The UAV lifecycle cost is the sun1 of the production cost C, = 1000, ml and the
total fuel cost Cf = 5000. y , f . ml. Which design, xl or x2, has the lower lifecycle cost
C, Cf, given that y = 4 [$/kg]? Briefly explain the tradeoff between both designs.
+
The best way to do this comparison is in the format of a table, see Table 1.
It turns out that the production cost of design xl are lower, since it is a lighter aircraft
(and tooling costs are lower for low A designs). This advantage, however, is lost when
considering the lifecycle cost. Design x2 has lower life cycle cost because it is more
fuel efficient. It is likely that a cost optimal design could be found somewhere on the
imendurance curve.
Figure 2: (f ,At) design space with bounds and performance contours
Table I: Design comparison XL and
51
r n ~
[kg]
52
293.3 415.5
52
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16.842 Fundamentals of Systems Engineering
Fall 2009
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