Water Rocket 4 - Program "rockfile"
In the previous exercise we extended the water
rocket analysis to include evaluation of the upward velocity of
the rocket throughout the flight. In this exercise we complete
the analysis by evaluating the height attained by the rocket throughout
the flight, from liftoff and until it crashes back to earth. Once
more we summarize the relevant equations presented in the previous exercise as follows:
The basic rocket force equation:
The thrust force in terms of the water expelled
through the nozzle:
Bernoulli's equation, relating the pressure
applied by the compressd air to the velocity of the exhausting
water:
Substituting equation 3 into equation 2 and
simplifying:
The adiabatic expansion process relating the
pressure to the volume of the compressed air, which was developed
in the section on Adiabatic Expansion
Analysis:
The volume variation of the compressed air
due to the water escaping through the nozzle is given by:
The total mass of the rocket (ignoring the
mass of the compressed air) is given by:
The frictional drag force is given by:
The acceleration a (and thus the velocity u) is obtained by rewriting equation 1 above as follows:
We now complete the analysis by adding the
equation for the height as follows:
where h is the height attained by the rocket [m]
Unlike the nonlinear differential equations
for volume V and velocity u (equations 7 and 10), equation 11
can be directly integrated using the Trapezoidal method which
we developed in Lab 5.
This leads to the following numerical solution for height attained:
This leads to the algorithm for evaluating
both the velocity and the height attained as functions of time
as the basis for the new method findvelht as follows:
Thus the only change that we find is in the
last three lines. Note that u1 refers to the velocity u(t+
)
The Computer Program "rockfile"
This program relies on Chapters 6 through 8
of our text, including the concepts of passing arguments by reference
(Sections 6.1, 6.2) and the reading and writing of text data files
(Section 8.2).
There are three distinct parts to this exercise,
all of which are extensions of the previous exercise (rockfor),
and they should be solved and debugged separately in the following
sequence:
1. Using the vi editor, create a text
datafile containing your personal rocket input data values
(based on measurements done on your team rocket), and named yourname.data, thus
for example the file which your instructor is using follows
urieli.data
43.0 591.0 0.4 0.3 4.0 0.3 8.0
################################################
The small urieli root beer water rocket -10/16/06
The above data items are (in sequence):
mass [gm] - solid mass (rocket + payload)
vtotal [cc] - total volume (water + air)
dnoz [cm] - diameter of nozzle
fill - fill ratio (water/total volume)
pgage [atm] - charge gage pressure [atmospheres]
cdrag - Coefficient of drag
dbottle [cm] - outside diameter of bottle
################################################
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Notice that the file has the data items followed
by lines of explanatory information. These should include your
name as well as the date when you created the data file. You will
need to write a function getrocket which should both get the data from the file and assign
them to the respective variables in the main function, and then
read and display all the remaining lines of information.
2. Develop a new method findvelht to replace
the method findvelo of the previous exercise in order to evaluate both
the upward velocity and the height attained as functions
of time in accordance with the above algorithm. The basic structure
of this method is shown in the the following structure flow diagram:
3. Notice that the program also creates a text
datafile of (time, velocity,
height) values in your home directory which
will subsequently be used for off-line plotting. Thus the
main function should create, write-to, and ultimately close a
data file named rocket.out using the various file commands that we learn about
in lab7. This third part
of the exercise will give you experience in transferring files
from condor to your local computer using the FTP
(File Transfer Protocol) program, editing local textfiles with
the Wordpad program, and plotting data using the MATLAB application program.
A typical execution is shown below together
with the output data file rocket.out. Note that since the input data defining the rocket
is read from the input data file, only two data items are input
by the user, as shown in italics. Please be sure to enter your data in the same order
as this solution shown below (which is available to test in the
programs directory of condor) since the exercises
are checked automatically using a data file for input. As in the
previous exercise, your program will be separated into two files,
the program file rockfile.cpp and the header file rockfile.h. Both
of these files together with the input data file should all be
in your home directory before 10:00 am of the due date.
get rocket data from the "urieli.data" file
data read:
43, 591, 0.4, 0.3
4, 0.3, 8
################################################
The small urieli root beer water rocket -10/16/06
The above data items are (in sequence):
mass [gm] - solid mass (rocket + payload)
vtotal [cc] - total volume (water + air)
dnoz [cm] - diameter of nozzle
fill - fill ratio (water/total volume)
pgage [atm] - charge gage pressure [atmospheres]
cdrag - Coefficient of drag
dbottle [cm] - outside diameter of bottle
################################################
Water rocket initialized as follows:
Solid mass[gm]: 43, total volume[liters]: 0.591
absolute charge pressure[kPa]: 500
initial air charge volume[liters]: 0.4137
Nozzle free flow area [sq.cm]: 0.125664
Bottle cross sect. area [sq.cm]: 50.2655
Density - water: 1000kg/m^3, outside air: 1.2kg/m^3
Ratio of specific heat capacities for air (k): 1.4
Drag coefficient: 0.3
gravity acceleration: 9.807m/s^2
enter the max time to test (sec)
6
value entered is 6(s)
enter the number of points to evaluate
21
21 points entered
time(s) velocity(m/s) height(m)
0.00 0.00 0.00
0.30 11.66 1.68
0.60 29.60 7.61
0.90 22.44 15.36
1.20 17.02 21.23
1.50 12.72 25.71
1.80 9.05 29.00
2.10 5.70 31.09
2.40 2.69 32.46
2.70 -0.30 32.71
3.00 -3.21 32.21
3.30 -6.01 30.85
3.60 -8.60 28.69
3.90 -10.97 25.59
4.20 -13.01 21.92
4.50 -14.69 18.13
4.80 -16.10 13.77
5.10 -17.33 8.09
5.40 -18.24 3.27
5.70 -19.00 -2.52
6.00 -19.58 -8.02
|
rocket.out
0.0000 0.0000 0.0000
0.3000 11.6613 1.6850
0.6000 29.5986 7.6092
0.9000 22.4417 15.3604
1.2000 17.0247 21.2308
1.5000 12.7197 25.7057
1.8000 9.0538 29.0012
2.1000 5.7049 31.0881
2.4000 2.6946 32.4647
2.7000 -0.3017 32.7064
3.0000 -3.2112 32.2077
3.3000 -6.0067 30.8535
3.6000 -8.6003 28.6913
3.9000 -10.9662 25.5877
4.2000 -13.0083 21.9183
4.5000 -14.6883 18.1251
4.8000 -16.1030 13.7688
5.1000 -17.3331 8.0896
5.4000 -18.2419 3.2690
5.7000 -19.0034 -2.5212
6.0000 -19.5830 -8.0214
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Once the output datafile is available in your
home directory then you should use the FTP program to fetch
the file and save it on the local computer disk. At this stage
you should be able to examine the file, and any out-of-range data
lines can be edited out. Finally you should read the file into
the MATLAB application and plot both the upward velocity
and the height attained vs elapsed time, as shown below. The final
graph can be printed on the Laserprinter in the lab.
A few words about the graph. When I tried to plot
the above file with 21 points it gave very unsatisfactory results.
After experimenting a while I decided on 500 points, which resulted
in the above plot. The very jagged cumulative height plot is a
result of using a very unconventional numerical quadrature technique
to solve the set of three differential equations, when in fact
we should be using a standard differential equation solving method.
Experiment with your own rocket graph until you are satisfied
with the results.
A printout of your graph should be submitted
in the box outside Room 265 (Stocker) by 10:00 am of the due date.
Wow - 0.6 seconds
thrust for 6 seconds flight attaining a maximum height of 33 meters
(100 ft) - isn't this exciting?!