ENG2031 Mathematical Modelling

ENG2031 Mathematical Modelling – Instructions for Assignment

Available: Thursday 28 th March 2024. Submission: 23:59 on Monday 29 th April 2024.

You should already have submitted several worksheets on the various problems discussed in teaching

weeks 3-8 in Semester 2.

For the marked assignment, you should write up one of these exercises as a full report, following the

instructions below. One has been allocated to you – listed in a separate document.

General instructions

- Submit your work via Canvas as a single file (PDF preferred, or JPG, JPEG, PNG, DOC), using your

surname as the filename, and check it’s readable on-screen! Best to put your name at the top of

the report also. (Don’t leave this until the final evening in case of IT problems.)

- Maximum of eight pages.

- Properly set-out, with headings: Description of problem (including diagram); Data/parameters,

Assumptions, Simplifications; Equations and solution for basic problem; Interpretation/Application

of solution; Comments, including which simplifications should be re-examined (including when

applied to other situations); Final conclusions about effectiveness of modelling.

- Few marks will be awarded for work that was covered in class and on the basic worksheet. Most

marks will be given for additional sections, and for a good description of the modelling and of

interpretation and application.

Specific instructions for the case-studies

Crop production (output optimization)

- You should give a formula for the optimum length of life in terms of the chosen parameters a, b, c,

d, and check that this gives sensible results for some simple choices. Note that the final formula is

surprisingly simple (expressed as x = sqrt(………))!

- Also find the average yield over that lifetime, and the output value at that point.

- Using a spreadsheet or otherwise, show a range of results for different choices of a, b, c (with d=25

fixed). Comment on how the results depend on each parameter – is one more critical than the

others?

- How would you turn your results into a practical piece of advice? Consider the assumptions and

simplifications that have been made and how those will affect your conclusions.

- How would the advice be different for (a) a single unit or small number of units, (b) a large area or

large number of units?

- If there is an additional ‘scrappage’ cost (for the removal of the tree/scrapping the plant/site

clearance/ redundancy payment), expressed as an amount of yield, how can you include that in

your analysis (NB you don’t need to re-do all the algebra, it’s a simple change).

- Calculate new results using a quadratic curve for the final section of the output curve (you might

need to solve the final equation numerically). How do your conclusions change, if at all?

- Comment on the general application of this approach to the lifetime productivity of various

economic units, also maintenance schedules, etc.

Car-parking (design of manoeuvring space)

- Explain why we can look at the problem of driving out instead of driving in.

- Give a comprehensive table of Assumptions and Simplifications for the Car, the Driver and the

Environment. Indicate which Simplifications are likely to be most critical.

- For the initial simplified analysis, use real data and check your results are credible, also put them

into a practical form. Also look at a range of other vehicles (smaller, larger, bus, etc.). Ensure you

are using the wall-to-wall radius for this.

- How would your results change if the obstruction were a kerb instead of a wall or car? How does

the kerb-to-kerb radius relate to the wall-to-wall radius?- What simple modifications can you make and how significant are they?

- After clearing the car/kerb in front, there are two principal sequences to move to a parallel

position. Produce careful, labelled diagrams and indicate how you might calculate the final position.

- How would you produce practical advice from these results, not only for parking but also for a

range of other manoeuvring problems?

Machine Chatter

Instructions for this assignment are given on a separate sheet.

Concluding remarks

The aim of this part of ENG2031 was to get you thinking about how a real engineering problem is

formulated, starting with a brief from a client or colleague, who may be quite unclear about what they

want or need. Stages include:

- Understanding and defining the problem carefully, including how the solution is to be applied and

by whom;

- Obtaining all necessary information and data;

- Stating explicitly all Assumptions – aspects that can be completely included or excluded;

- Stating explicitly all Simplifications – deliberate changes to the problem to make it possible to solve,

or to make it clearer to understand or apply, but where some might need to be revisited later;

- Being clear about techniques used and how reliable they are;

- Checking that solutions are both correctly obtained and realistic in practice (by using previous

results or simplified values etc.);

- Revisiting major simplifications, identifying critical dependencies;

- If your work is unfinished, make clear where you have reached and what remains;

- Writing clear conclusions, with limitations etc. and including advice on any further work you’d

advise and/or which you think is feasible.

Your report should aim to describe the problem, method and results in line with this general ‘philosophy’.

The classes and your work on the five examples should help you to do this.

John Appleby, March 2024