In the previous exercise you used
the 'brute force' method to obtain enough (time,velocity) coordinate
pairs for a suitable plot of the velocity curve. Thus each time
you executed the program you obtained a single coordinate pair.
Furthermore you had to painstakingly plot the velocity curve by
hand, count squares in order to evaluate the cumulative area required
for the height plot, and plot the height curve by hand. This was
a time consuming task however you probably derived a great feeling
of satisfaction in completing it.
In this exercise you will extend your program in two ways. The
function find_velo is only
valid for the first phase of the flight from launch until the
"Brennschluss"(burnout) condition of the booster rockets.
For elapsed times greater than Brennschluss we will extend the
function to follow the booster shells which separate from the
Shuttle Vehicle. After separation, gravity forces reduce the velocity
of the booster shells, and they ultimately fall into the sea for
subsequent recovery (splashdown). The equations follow:
Let
where tb is the time to "Brennschluss" of the booster rockets (s) Mfb is the initial total amount of fuel in the booster rockets (kg) qb is the ejection rate of the exhausting fuel in the booster rockets (kg/s)
thus if the elapsed time t is less than tb, the velocity equation applies as in the previous exercise:
where V is the upward velocity (m/s)
t is the elapsed time (s).
qb is the ejection rate of the exhausting fuel in the booster (kg/s)
Mt is the total initial mass of the Space Shuttle Vehicle (including the solid booster fuel) (kg)
qo is the ejection rate of the exhausting fuel in the orbiter (kg/s)
gm is the effective mean value of gravitational acceleration (m/s/s)
log( ) is the natural logarithm function - in the C language it is available in the math library.
However if t is greater than tb, then the velocity equation
becomes:
where Vc is the so-called 'Characteristic Velocity' of the Shuttle
Vehicle at "Brennschluss" (burnout), given by:
With this new set of velocity equations the velocity plot becomes
typically as is shown below. Note that we have arbitrarily chosen
an elapsed time of 380 seconds to illustrate the effect of elapsed
time t >tb. When the velocity becomes zero then the booster
shell has reached its maximum height and starts falling downwards.
After obtaining the velocity curve as shown above, we can now
consider extending the program to evaluate the height attained
by the booster rockets, until splashdown. The height is obtained
by numerically evaluating the area under the upward velocity curve,
using the 'Trapezoidal' method, which you will develop in Lab 5. Note that in Lab 5
we developed the method by integrating the sine function as follows:
In this exercise we wish to integrate the above upward velocity
function in order to evaluate the height reached by the booster
rockets as follows:
Notice the similarity. Once you have mastered the method in Lab
5 you will be able to easily transpose the code to the current
exercise.
There are three parts to this exercise,
evaluating a sequence of n upward
velocity values of the shuttle from liftoff until a specified
elapsed time, using the trapezoidal method to evaluate the cumulative
height at each of these points, and modifying the "find_velo" function to follow the
booster rockets after "Brennschluss" (burnout). One
approach to developing this program is to first copy your "shuttle.cpp" program to a new file
"shutloop.cpp", and then
modify "shutloop.cpp" using
the vi editor as follows:
cp shuttle.cpp shutloop.cpp vi shutloop.cpp
The first modification should be within the function find_velo to account for "Brennschluss"
(burnout). Once this mod is done it is a
good idea to compile, debug, and test the program over a range
of values of elapsed time before continuing.
Subsequently in the main function introduce a while loop
to evaluate upward velocity over a sequence of values, and finally
when this mod has been debugged and tested, do the final mod in
the while loop to evaluate the height over the range of values
using the trapezoidal method. Use the number of points entered
from the keyboard (minus one) to determine the number of intervals
of the Trapezoidal Method.
A typical program execution output follows:
Evaluate upward velocity as function of elapsed time
for 20,000kg payload, initial total mass is 2,034,681[kg]
enter your initial total mass of shuttle [kg]:
2034681
Shuttle initialised as follows:
total mass at liftoff: 2034681.0[kg]
mass of solid fuel at liftoff: 1004250.0[kg]
booster ejection fuel velocity: 2817.0[m/s],
mass flow rate: 8369.0[kg/s]
orbiter ejection fuel velocity: 3636.0[m/s],
mass flow rate: 1378.0[kg/s]
initial gravity acceleration: 9.8[m/s^2]
enter elapsed time from liftoff [s]:
380
value entered is 380.0 seconds
enter the number of points to evaluate
20
number of points is 20
elapsed time(s) velocity(m/s) height(m)
0.0 0.0 0.0
20.0 95.3 952.7
40.0 214.0 4045.0
60.0 360.8 9792.4
80.0 542.1 18821.8
100.0 766.9 31912.5
120.0 1047.9 50061.1
140.0 851.8 69058.3
160.0 655.7 84132.7
180.0 459.5 95284.3
200.0 263.4 102513.1
220.0 67.2 105819.2
240.0 -128.9 105202.4
260.0 -325.0 100662.8
280.0 -521.2 92200.4
300.0 -717.3 79815.2
320.0 -913.5 63507.3
340.0 -1109.6 43276.5
360.0 -1305.7 19122.9
380.0 -1501.9 -8953.5
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Notice the various items of data entry to this program are written in Italics script. In order to enable this program to be graded automatically, this sequence of data entry should be strictly adhered to.
The source code file shutloop.cpp should be in your home directory by 10:00 a.m. of the due date.