Mathematical Modelling and Analysis II 2022/2023 Coursework 2
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Mathematical Modelling and Analysis II
2022/2023
Coursework 2
Learning Objectives
Model 1 |
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Apply a technique: Line integrals |
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Apply a technique: Surface integrals |
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Apply a technique: Polar coordinate systems |
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Apply a technique: Eigenvalues and eigenvectors |
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Model computationally: Implement computationally solutions for system behaviour. |
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Show an understanding the meaning of eigenvalues and eigenvectors. |
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Show an understanding of the implications of mathematical results about the behaviour of the system. |
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Model 2 |
Apply a technique: Hypothesis test on two means. |
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Apply a technique: Hypothesis formulation, hypothesis tests on means or proportion. |
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Show an understanding of the implications of hypothesis tests. |
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Data analysis: Review and sort data appropriately. Show an ability to understand and use raw data from independent sources. |
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Communicate technical information: Present clearly and concisely data, data analysis results and conclusions based on this information. |
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Show independent and critical thinking. |
Model 1: Magnetic Levitation Trains [50 marks]
Figure 1. Schematic of the principle of operation of magnetic levitation trains. Magnetic repulsive forces keep the train
levitating above the track. Magnetic forces are used for propulsion in a push-pull manner, where like poles repel, followed
by attraction between opposite poles pulling the train forward.
Introduction
Magnetic levitation trains or maglev trains, are vehicles which float in air, supported by either electromagnetic attraction or repulsion. Maglevs make use of a fundamental property of magnetic forces – like magnetic poles repel each other, and opposite magnetic poles attract each other – to lift, propel, and guide a vehicle over a track (Figure 1). Most commonly superconductors or electromagnets are used to create the magnetic forces required in the maglev technology.
One of the main benefits of maglevs is the fact they eliminate a key source of friction – that of train wheels on the rails. However, similarly to conventional trains, they must still overcome air resistance. This lack of friction means that they can reach higher speeds than conventional trains. At present maglev technology has produced trains that can travel in excess of 500 km per hour. This speed is twice as fast as conventional commuter trains.
Reaching higher speeds is not the only advantage of maglevs. They are less expensive to operate and maintain, because the absence of rolling friction reduces the wear and tear of parts such as railcar wheels. Crucially, their design also makes derailment highly unlikely.
Maglev railcars can also be wider than conventional railcars, allowing for a more comfortable journey, which is already smoother as a result of the levitation. Maglevs produce little to no air pollution during operation, because no fuel is being burned, and are very quiet due to the lack of friction. Finally, due to the absence of friction, maglev systems can operate on higher ascending grades (up to 10 percent) than traditional railroads (limited to about 4 percent or less), reducing the need to excavate tunnels or level the landscape to accommodate the tracks.
The greatest obstacle to the development of maglev systems is that they require entirely new infrastructure that cannot be integrated with existing railroads and that would, however, compete with existing highways, railroads, and air routes.
The most common element of the design of maglev systems is the electromagnet. In this model, you will explore a number of considerations that contribute to the design of electromagnets for maglev systems.
Question 1. Magnetic Field Force [8 marks]
The magnetic force acting on a charge Q moving with velocity v in a uniform magnetic field B is defined as
F = Q(v × B). (E1)
Find the work done by the magnetic force in moving the charge a distance d s = v dt. Comment on what your findings imply about the work done by the magnetic force in general.
Question 2. Principle of Magnetic Levitation [16 marks]
y
Figure 2. Schematic of the set-up in Question 2: a mass levitating due to magnetic force.
This question will demonstrate a very simple example of magnetic force levitation , similar to that occurring in maglev trains.
In Question 1, we considered the magnetic force on a moving charge. Now let us consider what happens when there are more than one charge moving, i.e. if there is a current flowing through a wire. The current flowing through a wire is
I = 入v, (E2)
where 入 is the moving charge density. The magnetic force acting on a segment dl of this wire when in a uniform magnetic field B is
F = ∫(I × B)dl. (E3)
Now consider the case where a square loop of wire (with a side a), supporting a mass m, hangs vertically with one end in a uniform magnetic field B, which points into the page in the shaded region as shown in Figure 2.
a) [8 marks] For what current magnitude I, in the loop, would the magnetic force upward balance the gravitational force downward? Explain what would happen if the current is increased from this value.
b) [8 marks] Explain how this principle fits in the maglev technology. Using this as the foundation of your reasoning, what considerations and elements would you include in a prototype for a magnetic levitation system.
Question 3. Current and Current Density [10 marks]
Figure 3. Schematic of the set-up in Question 3. Cartesian coordinate system centred at the middle of the wire is defined as
shown on the right.
The current flowing through a conductor is not always uniform . The current density J at each point in a cross-section of the conductor is a measure of the current per unit area and similarly to density, can be integrated over the conductor area to determine the total current flowing through the conductor.
Suppose we take the wire shown in Figure 3 which has a radius a, in which the current density is described by the following function with relation to its position
J = X + y + X2 + y2.
Find the current flowing through the wire.
Hint: use polar coordinates.
Question 4. Future Innovation of Maglev Technology [16 marks]
Maglev trains are not the only competitive technology aiming to increase the share of short to medium distance travel done by rail rather than flight, thus reducing carbon emissions. Hydrogen powered trains and electric high-speed trains are just as promising. A key to wider- spread availability of such new technologies is a reduction in the cost of adoption driven by research and innovation, which is very dependent on the level of funding invested in each.
Recent trends in the funding rates for research into these three technologies indicate that the likelihood that the three main research funding bodies will continue to fund one of the three technologies or switch to another after each year is given by the probability matrix A:
From T1 From T2 From T3
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A = [ 0.2 |
0.9 |
0.2] |
To T2 |
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Given this, the funding distribution for each technology after, for example, two years can be found by y2 = A A y0 for an initial state y0. Note the following notational convention is used:
T1: Maglev trains,
T2: Hydrogen trains,
T3: Electric-power high speed trains.
a) [6 marks] Find the eigenvalues and eigenvectors of this matrix analytically (without the use of in-built software tools).
b) [10 marks] Using your knowledge of the characteristic equation
A v = 入v,
and the meaning of eigenvalues and eigenvectors, determine if the funding distribution will reach a steady-state and if so, what that steady-state funding allocation is. What are the implications for adoption of maglev trains?
Model 2: Data-driven Social Policies [50 marks]
The field of big data is taking increasingly more prominent part in shaping public policy. There is a growing interest in making use of the vast amounts of data collected by public institutions to make better informed decisions in areas ranging from medicine to social care. Enterprises
such as the CHIMERA centre are at the forefront of research-based data science. But
there are also initiatives such as the Health Equity DataJam in the USA which aims to harness the public and the powerful analytical tools available to them to drive data-based innovation and enact real-life change.
In this model, you will analyse a real-world dataset obtained by the Australian Capital Territory government as part of their general health annual survey. The data was obtained in 2019 and there were 2002 participants, as detailed in Table 1.
Table 1. Details of the number of respondents in the Australian Capital Territory’s General Health
Survey, 2019.
Please refer also to the accompanying documents to help you in your analysis:
1) The raw data from the survey for the years from 2011 to 2020 as documented in the csv file “ACT_General_Health_Survey_2011-2020” (can be opened in EXCEL).
2) The summary of findings from the 2019 survey in the pdf file “2019 General Health Survey Summary of Results” .
Question 1. Overall Personal Wellbeing [25 marks]
Table 2 summarises the average overall personal wellbeing index, with the mean for male respondents at 76.8 and for female respondents at 75.3. The corresponding standard error of the measurement (also known as the estimated standard deviation) is 0.5 for males and 0.6 for females.
Determine if there is a statistically significant difference between the average overall personal wellbeing index reported by males and females at 1% significance level.
Table 2. Personal Wellbeing Index in the Australian Capital Territory’s General Health Survey, 2019.
Question 2. Overall Personal Wellbeing [25 marks]
The full survey results for years 2012 to 2019 are available in the accompanying Excel spreadsheet file “ACT_General_Health_Survey_2011-2020” . Many valuable insights can be gained from a survey of this scale. In this part of the question we want you to demonstrate your self-guided ability to
• [6 marks] select a research question,
• [10 marks] analyse the data pertaining to it, performing any necessary hypothesis tests,
• [9 marks] make recommendations on the basis of your findings.
To answer this question, start by reviewing the survey data and formulating a research question. This can be, for example, investigating if there is an unmet need in a particular population group, e.g. females or individuals aged 18-24. You may find it helpful to read the summary of the survey findings in the accompanying pdf file “2019 General Health Survey Summary of Results” .
State the data from the survey you are going to use as evidence for assessing your research question. Perform no more than two hypothesis tests as part of your analysis.
Comment on your findings and their implications to your research question. Make short recommendations for further steps, either in policy or further investigation. Extensive outside research is not expected in answering this question.
Please limit your analysis to no more than 800 words and 7 pages.
2023-03-15