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MAST20029 Engineering Mathematics
ASSIGNMENT
Assignment 2 Assessment Period:
MONDAY Jan 29, 2024, 4PM – WEDNESDAY Feb 07, 2024, 4PM.
ALL ASPECTS OF THIS ASSIGNMENT ARE SUBJECT TO COPYRIGHT LAWS. DO NOT
SHARE ANY PART OF THIS ASSESSMENT DOCUMENT WITH A THIRD PARTY. DOING SO
WILL RESULT IN MISCONDUCT INVESTIGATIONS.
Students should not discuss any assessment during its designated assessment period in ED.
By submitting work for this assignment, you declare that you understand:
1. the University of Melbourne policy on academic integrity, as outlined in the Plagiarism
Declaration,
2. all the information on this cover page, along with theRules & Policy on Assignments and
Feedback to Students & Schedule, as described in the modules section on Assignments
in the LMS subject website.
This assignment is worth 7.5% of your final MAST20029 mark.
You must answer all 3 questions.
Assignments must be neatly handwritten, digitally handwritten or typeset using LaTeX or equiv-
alent. Submissions must be combined into a single PDF and submitted to gradescope.
Full working must be shown in your analytical solutions and use only the methods and concepts
covered in the subject and its prerequisites.
QUESTION 1
Suppose you are an intern in a company that manufactures industrial washing machines. You are
tasked with assisting a team that will study the interplay between drum geometry and fluid velocity.
The team leader hands you a list of problems ((a)-(d) below) to work through to better understand
the fluid flow through paraboloid-shaped machine components involving S, the open surface
z = 4 x2 y2
that lies above z 0, oriented with outward unit normal. The test fluid velocity v is given by
v(x, y, z) = (y z) i (x+ z) j+ (x+ y) k.
The company’s metal forming department constructs the drum components for the team’s study. This
department has ample supply of a thin perforated metal sheet, which they then bend to any shape
according to your team’s specifications. One component P is already formed in the shape of S and
is given to you. You are to prepare a report to the rest of the team detailing your solutions to the
problems below.
1. Sketch
(a) S and its normal n,
(b) C, the boundary of S,
(c) S⇤, the simplest surface bounded by C,
(d) R⇤, the projection of S⇤ on the xy-plane.
2. Consider BP the region of fluid bounded by S.
(a) Let ⇠ be the divergence of v in the region BP . Is the given information above complete
with respect to Gauss’ divergence theorem? If not, state the missing information that would
enable the computation of ⇠. Assume this information refers to an additional component
that can be made from the same sheet metal used for P .
(b) Find ⇠, subject to the conditions of Gauss’ divergence theorem being met.
(c) Find the volume of fluid in BP .
3. Find the mass of P , given the mass per unit area ⇢ of the sheet metal is
⇢ =
|xy|
x2 + y2
.
4. Consider the rotational flow properties of v across S.
(a) Find the circulation of v along C.
(b) Find and visualise r⇥v. Use MATLAB to check and produce the visualisation and include
this as part of your report.
(c) The team is considering Si, i = 1, 2, ..5, a selection of five alternative open surfaces to S.
Although Si 6= S, each Si is smooth and is bounded by C. Find the flux of r ⇥ v across
each Si.
(d) Verify Stokes’ Theorem for S1 = S⇤.
QUESTION 2
Electronic components are sometimes manufactured using screen printing where a material is dis-
tributed over a surface using a blade. The materials used can exhibit both solid-like and viscous-like
behaviour (known as ‘viscoplastic’ materials). Here we denote the average thickness of the material
by x(t) and the maximum height of the boundary between the solid and viscous regions by y(t) (also
known as the ‘yield surface’). The evolution of these two quantities can be described by the following
system (see this paper for an example):
dx
dt
= 9x y2(6x y), dy
dt
= (xy 1)(y 1),
Note that x and y can be negative and y > x is possible.
(a) Find all the critical points for the nonlinear system.
(b) For the critical point in x < 0, y > 0:
i. find the linearisation of the system with the critical point translated to (0, 0).
ii. Using eigenvalues and eigenvectors, find the general solution of the linearised system in part
(i).
iii. Classify the type and stability of this critical point.
(c) For the critical point in x > 0, y > 0, explain why the Linearisation Theorem cannot be applied.
(d) What happens to a layer with zero initial thickness and no yield surface: x(0) = y(0) = 0? Justify
your answer.
QUESTION 3
Transport is a major contributor to anthropogenic CO2 emissions. This has motivated top engineers
(including Elon Musk) to propose novel low-carbon solutions such as Hyperloop.
You are designing a low energy transport system to take passengers between Melbourne, Australia
and Melbourn, Cambridgeshire, UK, via a tunnel drilled through the centre of the earth. Passengers
are transported in a pod that undergoes simple harmonic motion owing to the graviational field of the
earth. To attempt to improve journey times, you sometimes add a jet to the pod, which leads to the
forcing term on the right-hand side of equation (1). In this question, we will explore the influence of
this jet on the ensuing passenger motion.
(a) Find the Laplace transform of
f(t) = |t ⇡|
(b) Use the convolution theorem to find the inverse Laplace transform, a(t), of
A(s) =
2e⇡s + ⇡s 1
s2(s2 + 9)
[Hint: first write A(s) = F (s)G(s) for choices of F (s) and G(s) that you know how to invert].
(c) Using MATLAB, verify that the Laplace transform of your answer to part (b) gives A(s). [Hint:
you may want to use heaviside functions to help define a(t) in MATLAB]
(d) Given that x(0) = 0 and x0(0) = 0, use Laplace transforms to solve
x00(t) + 9x(t) = |t ⇡| (1)
for t 0.
(e) Use MATLAB to plot x(t) for the range 0 t 3⇡.
(f) Given that y(0) = 1, use Laplace transforms to solve
y0(t)
Z t
0
y(⌧) d⌧ = |t ⇡|
for t 0.