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SCEE08014
Programming Skills for Engineers
Referred Coursework
The following question is to be handed in to the ETO by 1400 on Thursday the
5th of August 2021. This is the referred coursework and to pass the course you
must complete the coursework and obtain a mark of more than 40%.
Figure 1: Single stage rocket with instantaneous mass, m, in flight at a velocity u and an
angle of inclination, θ. The rocket exhaust has an exhaust with gas travelling at velocity
ue and pressure pe in a jet with area Ae . The dashed line indicates the control volume
containing the rocket.
Figure 1 shows a single stage rocket in flight. A large fraction (typically 90%) of the
mass of a rocket is propellant, it is thus important to consider the change in mass of the
rocket as it accelerates. By applying the momentum theorem to a small amount of mass
dm ejected from the rocket during a short period of time dt we can write down the rocket
equation
mv
ue dmv
Av ρu2
du
= (pe − p0 )
−
− Cd
− g cos θ
(1)
dt
A e mv dt
2mv
where p0 is atmospheric pressure, Cd is the drag coefficient and Av is the cross sectional
area of the rocket.
Equation (1) can be simplified by assuming that p0 = pe , ignoring drag and making
ue constant, giving the differential equation
du
ue dmv
=
−g
dt
mv dt
(2)
which can be integrated to give
u(t) = −ue ln
mv (t)
− gt.
mv (0)
(3)
Assuming the burn rate is constant the mass of the rocket,
mv (t) = mv0 (mv0 − mvb )
Page 1 of 4
t
tb
Continued
SCEE08014
Programming Skills for Engineers
Referred Coursework
where mv0 is the mass of the rocket at ignition, mvb is the mass of the rocket at burn out
(i.e. when the fuel is exhausted) and tb is the time at which all the propellant has been
used.
scipy.integrate.solve ivp can be used to solve (2) and can also be used to calculate
the altitude of the rocket by solving the additional equation
dh
= u.
dt
scipy.integrate.solve ivp requires a function which defines the differential equations
to be solved.
import numpy as np
from scipy.integrate import solve_ivp
from matplotlib import pyplot as plt
# Constants
Grav = 9.81
RocketMass = 1000 # Kg
JetArea = 0.1**2*np.pi # m^2
JetVel = 1500.0 # m/s
def rocket(t, state):
'''Simplified rocket equation including height.
t is the current, time, state is an array containing
the values of the variables h, u and m. These are
the height of the rocket, the velocity of the rocket
and the mass of rocket.'''
# unpack the variables
h, u, m = state
# mass loss
if m < RocketMass:
DeltaM = 0.0
else:
DeltaM = -JetArea*JetVel
# acceleration
DeltaV = - JetVel / m * DeltaM - Grav
return [u, DeltaV, DeltaM]
Three additional event functions are defined so we know the times at which the rocket
burns out, achieves apogee (it’s maximum altitude) and crashes.
Page 2 of 4
Continued
SCEE08014
Programming Skills for Engineers
Referred Coursework
# event for when rocket crashes back to earth
def hit_ground(t, y): return y[0]
hit_ground.terminal = True
hit_ground.direction = -1
# event for burnout
def burnout(t, y): return y[2]-RocketMass
burnout.terminal = False
burnout.direction = -1
# event for apogee
def apogee(t, y): return y[1]
apogee.terminal = False
apogee.direction = -1
#Launch a rocket with 1500kg of fuel and see what happens
sol = solve_ivp(rocket,
[0, 3600], # 1 hour of flight maximum
[ 0.0, 0.0, RocketMass + 1500.0], # initial conditions
method = 'LSODA', # stiff ODE solver
dense_output=True,events=(burnout,apogee,hit_ground))
#Interpret results
print('Burn out at t={:.2f}s, maximum velocity is {:.2f} m/s '.format(
sol.t_events[0][0],sol.y_events[0][0][1]))
print('Apogee at t={:.2f}s, maximum altitude is {:.2f} km'.format(
sol.t_events[1][0],sol.y_events[1][0][0]/1000))
print('Impact at t={:.2f}s'.format(sol.t_events[2][0]))
Finally the graph can be plotted
# Plot a graph
t = np.linspace(0.0,sol.t_events[2][0],500)
h = sol.sol(t)
plt.plot(t,h[0]/1000.0)
plt.ylabel('Altitude (km)')
plt.xlabel('Time (s)')
plt.axvline(sol.t_events[0][0],color='red')
plt.show()
1. Use the above python code to solve (2) for a rocket carrying 1500 kg of fuel. Produce
a graph to compare the results obtained against the results obtained using (3 ).
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Continued
[10]
SCEE08014
Programming Skills for Engineers
Referred Coursework
2. Modify the python code so it solves (1), including the aerodynamic drag on the rocket.
The drag force is
A
Fd = Cd ρu2
2
where A is the cross-sectional area of the rocket, ρ is the density of the air and u
is the velocity of the rocket. Compare the results for a simple rocket with a 31 cm
diameter in atmospheric flight with that of the rocket in a vacuum from the previous
question. You may assume that Cd = 0.75.
[30]
3. Using the solution from scipy.integrate.solve ivp write a python program which
simulates the flight of a rocket t in real time. The rocket Your program should display
on the screen
[40]
• a 10 second count down.
• a telemetry report every 5 seconds starting with the tag “T+n seconds” and
reporting the velocity and altitude of the rocket together with the remaining
fuel level (as a percentage). This text should be Green if the engine is running,
amber if the engine has stopped and red if the rocket is descending.
• the telemetry report report should also include the exact time of key events, e.g.
ignition, burnout, apogee, etc.
Your answer to Q3 should be a maximum of three sides of A4. It must discuss
1. the flight dynamics of the rocket and any changes you have made to the model
to make the simulation more realistic,
2. the approach taken to simulate the telemetry from the rocket.
3. the testing of the Python program to ensure the code is behaving as expected,
You should also include an appendix containing your properly documented python
code. This is not included in the page count.
4. Create a video showing your program running with an audio commentary explaining
the flight of a rocket launched from a pad with an 83m tall launch tower, carrying
5000 kg of fuel. The video must have a duration of no more than 3 minutes. To
achieve this you may run your simulation at double speed (i.e. 1 simulated seconds
is equivalent to two seconds of flight time).
Each component of Questions 3 and 4 will be marked against the University common
marking scheme, with equal weighting. You are advised to watch the short video which
explains how this scheme works (https://media.ed.ac.uk/id/1_7d7uwivg).
Page 4 of 4
END
[20]

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Anonymous

Just what I needed…Fantastic!