 The magazine of the Melbourne PC User Group Saturn V - Apollo 11 Ken Holmes |
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Everything has been said about this event, but it is very suitable subject
for a programming exercise in QBasic. I hope these articles might illustrate the main allure of programming,
in that it provides a spur to find interesting subjects and, to produce credible programmes, it is necessary
to collect hard data. Hence you find the subject even more exciting. In 1988, I collected all the data
available in local libraries and in a few professional papers to reconstruct the 8 day timetable of this
journey to within a few minutes accuracy and to show the whole path graphically. That program will be
rehashed in this series, making a few simplifying short cuts and assumptions to save magazine space and give
shorter, less abstruse code. It should be acceptable to all except those with a detailed knowledge of the
project and the dynamics governing it.
Timings are well recorded but some of the spot data on positions, thrusts and fuel flows were hazy. A degree
of "creativity" was needed to get a result conforming generally with the data. Naturally, units were pounds
for mass, lbs.wt for thrust and miles for distance, but we will convert to kgm for mass, kgf for force
(thrust) and km for distance. This will make it a little harder for you to check up on me.
Let's launch
For example, the launch mass was loosely quoted as 6 M pounds, but totting up components gave a figure of
6468706 which converts to 2940321 kgm; the thrust was 7653854 lb.wt or 3479020 kgf. Now, a kgf is a
gravitational unit and will accelerate one kgm at g, which is 980 cm/s/s or 9.8 m/s/s or 0.0098 km/s/s. We
want accelerations in km/s/s, so we will multiply the number of kgf by 0.0098 to get the number (34094) of
kga, my absolute unit with dimensions kgm.km/s/s. If this figure is divided by the current mass of the rocket
in kgm, we will get the thrust acceleration, at, in km/s/s. At launch, this was 0.0116, which overcame the
gravity by a margin of 0.0018 km/s/s and the rocket left with this acceleration (0.184g).
The acceleration due to gravity is g at the earth's surface, radius 6378 km (erd) so we use a gravity
constant gc (= g * erd * erd) which can be divided by the square of any radius to get local ag. We will try
to get into a circular orbit at 180 km height, or radius 6558 (orbh).
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Stage 1
Stage 1 fuel flow was essentially constant so we will adopt a value based on the known total fuel amount and
the burning time (150 secs). The fuel/oxidiser is kerosene/LOX and dividing quoted launch thrust (kgf) by
quoted fuel flow (kg/sec) indicates a Specific Impulse (kg thrust per kg/sec) of 242 which is consistent with
other information for this fuel at sea level. However, other info. indicates this should increase in vacuo;
the percentage increase is influenced by the combustion chamber pressure, which I don't know, and is due to
the better expansion ratio in vacuo. We'll assume a 20% (?) increase and a linear relationship between thrust
and ambient pressure; the atmosphere's pressure/ height profile was used to get some working values for
thrust. If it's wrong, you can sue me. The curve is very curved so it is approximated to by three linear
segments for Stage 1 thrusts at different heights. See Listing 1a.
With no information on the pitch-over rate towards horizontal, we assumed a constant rate and used the
honourable method of trial and error to choose a value which gets us into orbit with least fuel consumption.
Figure 1 shows the variation of the significant factors. Velocity starts at 0.41, the same as Florida, and
increases slowly at first; the shape of the trajectory at the top does not correspond to the vertical
take-off as seen by observers. Also, the height profile below, being against time, is not what you would get
if it were plotted against distance from launch. The Specific Energy curve is similar in shape to the
velocity curve. The mass follows a straight line due to the constant fuel consumption, as does our arbitrary
angle to the horizon. In Figure 2, we see the velocity vector in green and a force polygon. The vertical
black line shows the decreasing weight and the small cyan line shows the build-up of "centrifugal" force to
relieve the weight - very slowly at this stage. The red line shows the constant thrust as it swings towards
horizontal. The resultant force direction (not drawn) is from the top of the weight line to the end of the
thrust line. This diagram is constantly redrawn (on Figure 1) as the program runs but Figure 2 was prepared
to convey the information here, sampled at regular intervals through all three stages.
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Stage 2
At Stage 1 burnout, the spent unit and interstage structures were discarded and, in a 4 second interlude, not
shown, Stage 2 was pitched up so that its much smaller thrust could maintain the upward momentum of the fully
fuelled
Stage 2. As fuel was consumed it pitched down to provide
more horizontal acceleration; to achieve a smooth levelling out at 180 km orbital height it is necessary to
institute a control on incidence so that the vertical thrust component will cause a critically
velocity-damped, simple harmonic motion. The vertical acceleration needs to be proportional to the height
deficit, reduced by an amount proportional to the vertical velocity. T&E was used to determine the
coefficients required to deliver Stage 3 to just below orbit height; without delay loops the program runs in
a flash which greatly assisted the process. In a separate program, the whole program was run in a repeated
loop, varying one parameter at a time and printing and plotting results; the optimum values soon became
evident. This Stage was aimed at 6550 km (aimh) radius (172 km height), with less than critical damping, to
overshoot and crest just below orbit height so that Stage 3 would not need to pitch too markedly to correct
upward velocity. In Figure 2, we see the "centrifugal" force increasing its support as the velocity increases
and the local vertical is now more noticeably inclined to our fixed plotting axes. The fuel is now
Hydrogen/LOX with a Specific Impulse of 420 which increases by about 3% over the Stage 2 height range; we
adopt a single linear relationship for this.
Note that I don't like using the words "centrifugal force" since there is no such thing; centripetal force
causes an object to curve around a centre - if it is insufficient or absent, the curvature is less or the
object goes in a straight line, which is interpreted as "fleeing the centre". However, it is peculiarly
difficult to write simple sentences using "centripetal".
Stage 3
We are approaching orbital speed now with the weight exceeding the centripetal force, appropriate for a
circular orbit, by about 15%; of course, we have the vertical component of the thrust to compensate. Our aim
now is to reach orbital velocity at 180 km altitude, moving perfectly horizontally at correct orbital speed
for a circular orbit. We need a new pair of coefficients for the vertical acceleration control; you will note
that the transition was reasonably smooth by the angle plot in Figure 1 and the thrust directions in Figure
2. It pitched up and down a few degrees and settled into orbit. Having calculated the Specific Energy (Eh)
required to orbit at 180 km, the engine was cut when this was achieved. Fuel is also H2/LOX for this
stage.
Arrays of programmes were run to optimise the trajectory to achieve minimum fuel consumption or maximum
weight into orbit. In practice each Saturn-Apollo mission provided real performance data to improve the
technique and, during each flight, with any variation from expected performance, the control parameters could
be rapidly recalculated to optimise the rest of the mission.
Next month
Note that we still have 88 tonnes of fuel, in Stage 3, which will be used to correct the orbit and blast off
to the moon - or TLI (Trans Lunar Injection), as NASA preferred to call it.
Reprinted from the October 2000 issue of PC Update, the
magazine of Melbourne PC User Group, Australia
