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ECMM171P Programming for Engineers
创建日期：2024-04-23 15:58:34
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Programming for Engineers

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ECMM171P Programming for Engineers

Assignment 3: Projectiles
ˆ This assignment will count for 50% of your total grade in the course.
ˆ Remember that all work you submit should be your own work. This assignment is not
meant to be a team exercise. The University treats plagiarism very seriously.
ˆ Ensure that your programs run without any error on the Harrison computers - Matlab
version R2019a and later. Programs with errors may get zero marks. Additionally, ensure
that you pay careful attention to the instructions and marking scheme below, particularly
in regards to defining specific functions, commenting your code appropriately and ensuring
its readability. Programs with no comments will get 20 % less than what would normally
be awarded.
ˆ You should submit your source code by 16:00 Monday 13th December 2021
ˆ If you have more general or administrative problems please e-mail me. Always include
the course number (ECMM171P) in the subject of your e-mail.
The path of a projectile is an interesting, important, and sometimes challenging problem. It’s
very likely that you will have studied the path of a projectile in the absence of atmospheric
drag in basic mechanics, since this can be derived analytically. In the real world, however,
drag can significantly alter the trajectory. Unfortunately, in introducing this effect, we lose the
ability to analytically obtain solutions for the projectile’s trajectory: simulating the trajectory is
therefore required. In this assignment, you will therefore write a Matlab program that computes
trajectories under the influence of drag, and use this to solve a simple targeting problem.
1 Theory
We will neglect any crosswind effects, so that the projectile travels in a two-dimensional plane
with coordinates (x, y). The projectile’s position, velocity and acceleration can therefore be rep-
resented as vector functions of time, which we denote by r(t) = [rx(t), ry(t)], v(t) = [vx(t), vy(t)]
and a(t) respectively. As the projectile travels through space, we will assume that it experiences
two forces: acceleration due to gravity, and a drag force Fd acting in the direction opposite to
its trajectory. A free body diagram of this scenario can be found in Figure 1. The drag force
can be roughly approximated by the drag coefficient cd through the relationship
Fd =
1
2
ρcdA|v|v,
which also involves the velocity v, air density ρ and the frontal area A (i.e. the two-dimensional
cross-section exposed to the drag force). cd is dimensionless and is used to quantify the drag
of an object in a fluid: the lower the value, the lower the drag force.
To simulate the trajectory of the projectile, we can use Newton’s second law:
F = ma⇒ a(t) = 1
m
(−1
2
ρcdA|v|v −mg
)
,
1
2 YOUR TASKS
velocity v
mg
Fd
Figure 1: Free body diagram of the projectile.
where m is the mass of the projectile and g = (0, g) is acceleration due to gravity with g =
9.81 ms−2. Since we are interested in the projectile’s trajectory r, we can then utilise the fact
that
a(t) =
dv
dt
= v˙(t) and v(t) =
dr
dt
= r˙(t)
to give us a ordinary differential equation
r¨(t) =
1
m
(−1
2
ρcdA|r˙|r˙ −mg
)
(1)
Notice that this is a non-linear equation that cannot be solved analytically – we therefore
need a computer to simulate it. It is also second-order: that is, it depends on the second
derivative of r.
All that is left to do is equip the above with an appropriate set of initial conditions at time
t = 0: we will imagine that the projectile is fired from an initial starting height h at an angle
of θ degrees and initial speed (velocity magnitude) v0, which gives:
r(0) = (0, h) and r˙(0) = v(0) = [v0 cos(θ), v0 sin(θ)].
2 Your tasks
Your program will be structured in two parts. In the first part, you will write a function to solve
the ODE system defined above. In the second part, you will use these functions to create a
simple targeting computer. In all of the below, you should assume that the projectile is a sphere
of diameter 0.1 m being fired through air of density ρ = 1.225 kg m−3, with corresponding drag
coefficient cd = 0.479 and mass m = 5.5 kg.
2.1 Getting a trajectory
In the first part of this assignment, you will write a solver for the ODE in equation (1).
1. Using the techniques discussed in the lectures, first transform equation (1) as a system of
first-order equations for the two components of velocity and two components of position,
respectively. Mathematically, this system will be expressed as
y˙ = F (t,y(t)),
where y(t) = [vx(t), vy(t), rx(t), ry(t)]. Write a function drag ode, which takes as ar-
guments the time t and the vector y(t), and returns the function F (t,y) as a column
vector.
2
2.1 Getting a trajectory 2 YOUR TASKS
0 100 200 300 400 500 600 700 800 900
Downrange dustance [m]
0
5
10
15
20
25
30
35
40
45
H
ei
gh
t [m
]
Without drag
With drag
(a) Comparison of trajectories with and without drag
for initial conditions v0 = 150, h = 10 and α = 10.
0 200 400 600 800 1000
Downrange distance [m]
0
20
40
60
80
100
120
H
ei
gh
t [m
]
Trajectory
Target
(b) Example of target being hit in target.m with
(tx, ty) = (1000, 50)
Figure 2: Graphs that your codes should produce.
2. Using the lecture notes, write a function called solve ode euler, which uses the forward
Euler time integration method and your drag ode function to solve the ODE in equa-
tion (1). solve ode euler should take as arguments the initial speed v0, the angle of
attack θ, the initial height h and a timestep size ∆t. You should also include a maximum
integration time as a parameter. You should stop integrating once either the maximum
integration time has been reached or once the projectile reaches the ground.
The function should return: the solution time, the two components of v and the two
components of r. Include all outputs into a single variable with 5 columns. Each row
should contain these values at a given timestep (i.e. the first row is their values at t = 0,
the second at t = ∆t, and so on).
3. To validate your implementation, write a function called solve ode45, which instead
uses Matlab’s built-in function ode45 and your drag ode function to solve the ODE in
equation (1). solve ode45 should take the same arguments and return the same array as
solve ode euler.
4. You should include a script drag which will prompt the user for initial speed v0, the angle
of attack θ, the initial height h and will call the solve ode euler function to solve the
ODE system and produce a plot that compares this to the analytic solution of the particle
trajectory without drag. You may use the following formulae,
Distance travelled : d =
v cos θ
g
(
v sin θ +
√
(v sin θ)2 + 2gh
)
Height at position x : y = h+ x tan θ − gx
2
2(v cos θ)2
Travel time : t =
d
v cos θ
Think carefully about the choice of ∆t. Your script should output a graph that shows
the two trajectories with and without drag, as in Figure 2, and will print: initial speed,
angle of attack, initial height, travel time, distance travelled and maximum height of the
projectile.
3
2.2 A targeting computer 2 YOUR TASKS
2.2 A targeting computer
In your second part, you should use the Matlab’s optimisation package to build a simple tar-
geting system. The idea here is that you are given the position (tx, ty) of a target which is
downrange from your projectile. Given the constraints that:
ˆ the projectile’s initial velocity can be between 100ms−1 and 800ms−1;
ˆ the projectile’s initial angle can be between 10◦ and 80◦.
your program should determine what the initial velocity and angle of the projectile should be
in order to hit this target. In all of the below, you can assume that the initial height is h = 0.
1. Write a function called objective, which defines the objective function to be minimised.
objective should take as arguments: the two variables being optimised (i.e. the initial
velocity and angle), and the target coordinates tx and ty. You may combine the arguments
into two variables for convenience. It should call solve ode45 with the initial conditions
and return the shortest distance between the computed trajectory and (tx, ty).
2. Write a function called objective image. The purpose of this function will be to produce
an image, along the same lines as discussed in lectures, that shows the value of the
objective function across the range of constraints, as shown in Figure 3. objective image
should take as arguments the target (preferably in a single variable as in objective), and
two integers nx and ny, which are the number of pixels to be used to represent the range
of initial velocities (in the x-direction of the plot) and the initial angle (in the y-value of
the plot). Your function should then plot the objective function across these ranges and
produce an output file called objective.pdf.
3. Write a function called target, which takes as two arguments the target’s position tx and
ty. Use routines from Matlab’s optimisation toolbox to minimise objective, and return
True if the optimisation succeeds, along with v0, θ and the minimum distance; otherwise
return False and three zeros.
Some hints:
ˆ This aspect is the most open-ended in the assignment. The function can be quite
hard to optimise and is sensitive to initial guesses.
ˆ Use your images from the objective image as a guide: think about how the opti-
misation (minimisation of the objective function) process works and how it might
work for some of these images to guide your choices of e.g. initial guesses.
ˆ You can use any combination of functions from the optimisation toolbox, but you
may explore fmincon and ga in particular: note the former is much faster than the
latter, but is more prone to becoming stuck during optimization.
ˆ You should consider in particular the bounds and args options in these functions
and you can consult Matlab’s documentation to determine appropriate arguments
to call.
Note that there is no unique solution to this problem: full marks will be awarded for
creativity and devising a relatively robust optimisation strategy that works for most cases,
which demonstrates knowledge of the programming and numerical techniques under the
hood.
4. Finally, write a script called test target that prompts the user for the coordinates (tx, ty),
calls your target function, and prints a message target was hit! along with some
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MAS316/414 ACCT2522 MSIN0181 ATMS 201 COMP3425 MATH7861 SOLA2060 MA 325 7CCEMCTH ECON 432 INFT2051 SOLA 2060 ECOS2002 Stats141XP BIOA02H3 S2022 MA4601 K5 COMP5329 EG 284 IB3A70 WECO1020 COMP3027 Energy ACS61011 MAST20026 STAT 533 MATA31 ENVS257 BEM2031 EC 486 COMP0172 CS3214 EECE5644 IB2B20 COSC2536 COSC1147 COMP6214 5CCM242A COMP0130 AMME2000 MTMG50: EG238 ACF6006 QBUS2810 COSC2391 ECON7560 MATH377 STA305 CS 486 COMP1117B EL648 ELEC3909 ECON 241 COMP 4901W BUSI4428 ENEN20002 ENGN4524 ELEC3115 CSC 413 PSYC 100B ECON1002 MATH256 ANSC20005 MTHS2008 MATH40082 COSC 2406 FIT3174 STA 238 CMT202 Differential Equations MTHM017 GARCH 101 MATH2069 CSC336S ACSC12 ETF2121 CEME 2001 DB B474F ARHT1001 EECS 3311 3220 STATS 786 CS 484 COMP20008 MATH4091 ECE 455 CSCI 204 BUS B200F BA 875 BU5562 COSC 1147 X11 GEOG 101 IMM250F CSE 584A MATH38172 LAB 4 ELEC5882M FINS5523 EE 430 COMP 465 6CCS3PRE 7CCSMNSE FINANCE ACMB02H3 ENGG2112 ELEC5616 SCM B371 CHEM3120 CS22 ECMT2130 MAST30011 ADSPBF706 CENG0013 ACCOUNTING BUSS1040 CHEM 181 MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 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