AMME2500 Engineering Dynamics
Major Assignment

General Information

• This assignment is due 11:59pm 26th May 2023
• This assignment should take the average student 15 hours to complete.
Assignment Objectives
• Part A focuses on numerical solutions to problems involving kinematics of
a point-mass system.
• In Part B you will study the dynamic behaviour of a system of your choice.
You will use the theoretical principles and analysis techniques developed
in the course to develop and solve the equations of motion from an
example of your selected system. This will test your ability to perform
research, draw relationships between dynamics theory and real systems,
make approximations, and to make realistic predictions about the motion
of the system.
Submission Instructions
• This assignment involves the submission of a single report covering both
Parts A and B
• A single report file should be produced covering both Parts A and B
• Show all of your calculations and working, clearly illustrated diagrams
with relevant variables indicated and working units (use SI units unless
otherwise specified)
• Comment your code thoroughly and perform important steps in the
calculation on separate lines of code
• Graphs and plots must be clearly titled, with correct use of axis labels and
legends, units must be specified
• On completion you will convert it to a pdf file and submit on Canvas via
Turnitin. It is essential the pdf file be readable text, NOT an image. If it is
an image Turnitin cannot check the text for similarity and your report will
not be marked and you will receive zero for the Major Assignment. Only
pictures and sketches may be included as images.
• Any incidence of academic dishonesty or plagiarism will result in the
issue being followed up with the Academic Honesty Coordinator and then
onto the University Registrar, and will result in zero marks for this
assessment, and may result in automatic failure of this unit of study. For
more information on academic honesty, see:
https://sydney.edu.au/students/academic-dishonesty-and-
plagiarism.html
Part A (40 marks)
In this question you will use numerical
simulation techniques to study the
trajectory of a skydiver during their
descent. The skydiver jumps from an
aircraft at an altitude of y=4000m while
the aircraft is flying steady and level
(skydiver has an initial zero vertical
velocity v, and for simplicity, we will
neglect the skydiver’s horizontal
motion). The skydiver initially
experiences freefall (no parachute) until
they reach a height of
yparachute = 500m, at which point a
parachute is deployed to slow the
skydiver’s descent.
During their descent, the skydiver is
subjected to gravitational acceleration
and atmospheric drag. Assuming a
vertical trajectory, an equation of
motion can be developed for the rate of
change of vertical velocity by
considering the balance of weight forces
and drag forces acting on the skydiver
such that:
̇ = −̈ = −
2
where y is the altitude (vertical position) above the ground, v is the downwards
velocity, g is the gravitational acceleration (9.81 m/s2),  is the atmospheric
density (kg/m3) and kdrag is a constant, dimensioned drag coefficient. Before the
parachute is deployed (during freefall), the drag coefficient kdrag = 0.003 m2/kg
and after the parachute is deployed, the drag coefficient increases to kdrag = 0.15
m2/kg.

There are two different models that can be
used to approximate the atmospheric density
(). Using model A (a simplified model), air
density is assumed constant, and at a value of
=1.226 kg/m3. Model B assumes a standard
atmospheric model in which density changes
with altitude according to equations shown to
the right, where T is atmospheric temperature
(oC), P is atmospheric pressure (kPa), y is
altitude (m) and  is the atmospheric density
(kg/m3).
Model B Equations:
= 15.04 − 0.00649
= 101.29[(
273.1 +
288.08
)
5.256
]
=
0.2869( + 273.1)
Question 1 (continued)
i. In a Jupyter Notebook (similar to the examples provided with the lecture
notes) develop a Python function that uses an Euler numerical integration
method to solve the equations of motion shown above for the skydiver’s
descent. Your function should take as an input the initial altitude of the
skydiver, the altitude yparachute at which the parachute should be deployed
(hence changing the drag coefficient of the skydiver). Results should be
obtained for both the constant air density (Model A) and the variable air
density (Model B). Your function should output three arrays/vectors t, y
and v which represent the altitude and velocity profiles vs. time of the
skydiver from the initial time up until the point at which the skydiver
reaches the ground. Start with a timestep of dt = 0.05s but be sure to
demonstrate sensitivity to larger and smaller values. Inside your function
you should iteratively evaluate the velocity and altitude at each timestep
by integrating the equation of motion shown above (15 Marks)
ii. Produce plots of:
a. The velocity of the skydiver vs. time
b. The altitude of the skydiver vs. time
for the two different models A and B. Plot the two trajectory cases (with
different colours/line markers) for each of (a) (first figure) and (b)
(second figure) and include a legend in each plot. Use initial values for the
starting altitude at y0 = 4000m and yparachute = 500m. Remember to
demonstrate sensitivity to the timestep size. (5 Marks)
iii. Use a loop around your developed function to determine the maximum
free fall time possible for the skydiver before hitting the ground as a
function of the jump altitude. Compute the skydiver’s trajectory for a
range of different initial jump altitudes from 1000m to 4000m, while
simulating a “parachute-free” jump (by setting yparachute=0m) and
recording the time taken to hit the ground for each simulation run. Show
two different curves in the one figure, one considering the variable
density model and one considering the approximation of the density as
constant. (7 Marks)
iv. In no more than half a page, discuss the effect of drag and atmospheric
density on the skydiver, considering phenomena such as “terminal
velocity” using you plots as reference, and considering the different
approaches to modelling (8 Marks)
v. Presentation style and elegance of report (5 marks)
Part A Assessment Criteria:
Totally incorrect approach to problem 0%
Correct answer, does not show
working or solution is illegible
0%
Partially correct solution or small
mistakes made, shows all working
including code, free body diagrams
where appropriate
10% to 90%
Correct solution, correct answer,
shows all working including code, free
body diagrams where appropriate,
clear and legible
100%
Part B (100 marks)
Special Instructions:
• Choose one of the four systems suggested below.
• Choose at least one of the following theoretical concepts in dynamics to
describe your system/problem and its dynamic behaviour:
o Work and Energy of particles or rigid bodies
o Linear and/or angular momentum of particles or rigid bodies
o Kinematics and/or kinetics of particles, rigid bodies or machines
• You will perform research into the dynamics of your chosen system and
use the chosen theoretical concepts in dynamics to describe the approach
to predicting the motion of the system.
• You will develop a specific example of your system (with realistic
numbers derived from research) and use principles in dynamics to derive
equations of motion for your example. You will solve the equations of
motion using analytical and/or numerical methods and present and
discuss your results/predictions in relation to the expected system
behaviour uncovered through your own research.
• You will write a formal report in which you discuss the dynamics of the
chosen system and present your example, results and discussion. Your
report must use the following structure:
o Introduction (about one page):
▪ Describe the system under consideration
▪ Describe in your own words your chosen concepts in
dynamics used to describe the dynamics of your system in
general terms, and describe how these concepts can be
used to examine your chosen system.
o System/Problem Example (about two pages):
▪ Provide an example of your system, and use this example to
develop/solve equations of motion for the system
considered.
▪ Your example should include realistic measurements for
the physical structure, mass, inertia, force/torque, velocity,
acceleration etc. that are sourced from your research or
appropriately reasoned.
▪ Draw free body diagrams of your system, detailing and
rationalizing any approximations you have made, then
apply theoretical principles to develop equations of motion.
o Results and Discussion (about two pages)
▪ Solve your equations of motion using analytical and/or
numerical methods and present results illustrating the
behaviour of your system
▪ Discuss the system motion/behaviour using your results
and relate this to the expected system behavior from your
research.
o References
▪ Provide citations for all researched information, figures etc.
o Python code
▪ If you use Python code in Part B, please imbed parts of it in
your report if it is essential to understanding the document
or put it in an Appendix if it is for information only
o The main body of the report (includes the Introduction, Example,
Results and Discussion) has a strict page limit of no more than
five pages including any figures. Your references and appendix
are not included in the page limit.
o An exemplar of Part B is provided on the Canvas site. It is for a
different dynamic system than the four options below but gives
you an idea of the quality required. This particular assignment
scored a credit.
Part B Assessment Criteria:
• Report and Written communication (30%)
o Has the problem/system been clearly described and presented?
o Have the chosen theoretical concepts in dynamics been properly
explained?
o Has research on the system considered been performed and
appropriate reference to external sources made (e.g. textbooks,
websites, journal/conference articles etc.)?
• Application of dynamics principles to your chosen problem/system
(40%)
o Has an appropriate theoretical model for the behaviour of the
system under consideration been developed? Have appropriate
approximations been made?
o Have schematics of the system and freebody diagrams been
developed and clearly presented?
o Are the equations/theory correctly applied to develop equations of
motion for the system, or to evaluate specific motion cases?
• Depth, detail and creativity (30%)
o Has the system been considered in an appropriate level of detail?
o Have numerical modeling techniques/simulation or other physical
experiments been recorded to validate the proposed motion of the
system detailed in the example and results?
Suggested Systems
System 1: Roller Coaster Vertical Loop
During a roller coaster ride, a vertical
loop is a section of the track in which
the car undergoes a complete 360o
turn for which passengers are upside-
down at the top of the loop. Questions
to consider:
• How are these systems
designed from the perspective
of safety and fun for
passengers?
• How is the track shaped and
how are the maximum and
minimum speeds of the car
determined and controlled?
• What are the
accelerations/force vectors
experienced by passengers as
they travel along the track?
System 2: Dynamics of a Medieval Catapult
A catapult is a ballistic device
used to launch a projectile over
large distances without the aid of
explosives. Examples include the
Trebuchet shown here, but you
can choose to analyse different
designs. Questions to consider:
• Mechanically speaking,
how were catapults
designed to transfer forces
and mechanical energy to
a projectile?
• How were the launch
speed and angle calculated
and controlled?
System 3: Kinematics and Forces in a Reciprocating Engine
In a reciprocating engine, forces
induced by pressure on a piston head
are used to drive the rotational motion
of a crank shaft. Questions to consider:
• In a reciprocating engine
consisting of a piston,
connecting rod and crankshaft,
what is the relationship
between the angular and linear
accelerations and velocities?
• How do these vary over the
engine rotation angle, and how
do the size/length of these
components effect the
acceleration of the piston?
• How is vibration calculated and
minimised?
System 4: Rocket Launch into Earth Orbit
A rocket that launches a spacecraft
from the ground into an orbit around
the Earth provides enough velocity to
the spacecraft to achieve a steady
orbit under the influence of gravity.
Questions to consider:
• What are the forces that act
on a rocket during a launch?
• How big must a rocket be and
how much propellant must it
burn to achieve a typical low
earth orbit of 400km above
the surface of the Earth?
• Why do rockets use multiple
stages?