AE 426 Homework Assignment #2
Out: 9/30/14
Due: 10/7/14
Problem 1 (Teams 1 and 2): In this problem we design a telecommunications link,
using S-band (2 GHz) from the Moon to Earth, assuming two scenarios, as follows.
Part a: The first scenario assumes an overall architecture based on central command and
control from Earth. The resulting average data rate is equivalent to 10 color television
channels, i.e. 925 Mbps. We use BPSK modulation and require a bit error rate of 10 6 ,
which implies a required Eb N0 of 5 dB. Take the required beam width of the transmit
antenna equal to the angular size of the Earth from the Moon. Use autotracking so that
pointing error is 10% of the beam width. Assume a 5 m diameter receive antenna with
pointing error 10% of its beam width. Let both antennas be circular aperture, parabolic
antennas with efficiency = 0.55. Assume a system noise temperature of 25.7 dB-K.
If we require a link margin of 3 db, what is the diameter of the transmit antenna
and what is the transmitter power?
Part b: In this scenario, there is a substantial degree of local control with the highest
volume communications being handled by local, shorter range comm. Links. The link
with Earth is used for high-level communications with an average rate equivalent to ten
command messages, or 640 bps. With all other assumptions as in Part a, what is the
diameter of the transmit antenna and what is the transmitter power?
Problem 2 (Teams 3, 4, and 5): Next, we study the dynamics and structural mechanics
of an air bag landing system for final touchdown of a cargo package on the lunar surface..
Although the landing package would be encased in at least 12 air bags, we need consider
only the one airbag that finds itself between the landing package and the ground at the
instant of touchdown. As illustrated below, we are concerned with the performance of the
air bag from the time of first contact through the end of the first bounce, given that the
landing package approaches the ground with initial speed, 0 . How well does the air bag
shield the landing package from the deceleration loads? How large a payload mass and/or
touchdown speed can the airbags handle without breaking? How do these characteristics
vary with air bag pressure, skin thickness, and size?
Let us now put together a simple model of this system. First, to get a conservative
estimate of deceleration loads, assume that the Lunar surface is rigid. Regarding the air
bag, suppose this consists of an initially spherical skin of wall thickness w, and initial
radius r0 . The bag contains a gas with initial pressure P0 . We model the air bag skin as a
membrane that is flexible in bending but relatively stiff against stretching deformations.
Therefore, we approximate the deformed shape of the airbag as shown in the next
diagram. During the landing event, the air bag is compressed between the landing
package and the ground such that there are two, planar surfaces of circular form in touch
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Landing Package
Mass  M L
Landing Package
Mass  M L
0
Air bag
Landing Package
Mass  M L
Air bag
with the landing package and with the ground. The edges of these form an angle  with
the vertical axis. The surface between these two planar end caps is a portion of a sphere
of radius r  r0 . Thus, while the area and volume of the undeformed air bag are
4 r02 and 43  r03 , one can readily show that the area and volume of the deformed air bag
are:
Area of deformed shape  4 r 2  cos   21 sin 2  
(1.a,b)
Volume of deformed shape  43  r 3 1  21 sin 2   cos 
P
d

r0
Air bag
deformed
shape
r
P=
pressure
in the
deformed
shape
P
2
Part a: Assume the air bag skin cannot be stretched so that the surface area remains the
same when the air bag is deformed. With this constraint, use (1.a) to show that the radius
of the spherical section after deformation is related to the initial radius by:
r0
r
(2)
2
1
1
2  cos   2 cos 
Next, use this expression and (1.b) to find that the volume, V, of the deformed air bag is
given by:
 32  12 cos2   cos 
V

V0  1  cos   1 cos 2  3 2
2
2
(3)
1  rd0
2
 d
cos  
1  r0  2  2 1  rd0 
2 
d

2  1 r 

0



Where V0  4 r is the volume of the undeformed air bag and d  r0  r cos  is one half
the reduction of the air bag’s vertical height (see the above figure) relative to its
undeformed diameter.
2
0
Part b: Recall the ideal gas law ( PV  nRT ) to argue that if the temperature of the gas
does not change appreciably during the landing event, the gas pressure, P , in the
deformed air bag is :
V
P  P0 0
(4)
V
Next suppose the airbag skin in contact with the ground or the lander on the flat contact
segments carries no load. Thus, as we tried to sketch in the diagram above, the pressure P
acts directly on the payload. Show that the total force, FB , that opposes the downward
motion of the payload is:
2
V r
FB   P r 0   1  cos 2  
V  r0 
2
0 0
(5)
Plot FB 2 P0 r02 as a function of d r0 for d r0  0 to 0.9 . Lots of extra credit if you can
show that (5) with (2) and (3) can be simplified to read:
d r0
FB  2 P0 r02
(6)
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1   d r0 
As you should expect, as d r0 approaches unity, the pressure and force blow up.
Part c: Using (6), determine the equation of motion of the landing package in terms of d,
which is half the displacement of the landing package from its position at the instant the
air bag makes contact with the ground. Neglect the effect of the Lunar gravity. For
d  t0   v0 2, and d  t0   0 , where t 0 is the time of contact, solve this equation of motion
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and determine the maximum value of d , the maximum landing package deceleration,
2d , and the maximum gas pressure within the air bag.
Part d: Since the portion of the air bag skin in contact with either the landing package or
the ground supports little in-plane membrane forces, the maximum stresses occur in the
spherically – shaped segment of the skin. These principal stresses are applied uniformly
along the cross-section and their maximum value is given approximately by:
1
rP
(7)
2w
Where w is the wall thickness of the skin. Show that the minimum w such that the
maximum stress given by (7) is less than one third the yield stress,  yield , of the airbag
material is:
 max 
w
r  d max 
3 PV
0 0
2  yield V  d max 
(8)
Where d max is the maximum value of d found in part c, and r and V are determined from
(2) and (3).
Part e: Assume the following values:
v0  15 m s
r0  3 m
P0  0.02 Bar 1 Bar  105 N m 2 
M L  1500 kg
 yield  1.1109 N m 2
 ab
g max
mass density of air bag skin  2000 kg m3
maximum tolerable deceleration load on lander  10 gs
Now find out if the air bag can satisfy all of the following constraints:
 i  d max  0.7r0 (air bag deformation not excessive)
 ii  2d max  g max (lander deceleration is tolerable)
 iii   max  13  yield (air bag stresses well below yield stress)
 iv  4 r02 wab  0.1M L (air bag mass a small fraction of the total)
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