6E4Z1104: Mathematical Methods 1

Mathematical modelling emerged as a powerful, indispensable tool for studying a variety of problems in scientific research, product and process development and manufacturing. It may help you to explain a system and to study the effects of different components, and to make predictions about their behaviour.

Aim

The aim of this assignment is to:

• Use mathematical software such as Excel/MATLAB to investigate the correlation between the variables.

• Use MATLAB to create a mathematical model to investigate the impact velocity of the spacecraft.

Task Description

You are required to write a short report comprising of Tasks 1 and 2. The maximum size of the report should be no longer than 10 pages including Appendices. Save your report as a PDF document and follow the submission guidelines on Moodle.

Task 1

Any Mathematical Software such as Excel or MATLAB can be used for this Task.

A diesel generator is fitted with multiple sensors and instrumentation, which record a sample every second, and downloaded to an Excel spreadsheet (Engine Output_data.xlsx).

1. Calculate the Pearson product-moment correlation coefficient between Fluid Outlet Temperature °C and Gas Valve Setting (%).

2. Draw a scatter plot of the results and comment on the relationship.

3. Calculate the line of best fit using the method of least squares of the form:

Task 2

You are a parachute engineer working for the European Space Agency and you are required to design and implement a piece of software to test the consequences of changing the effective cross-sectional area of the parachute, AParachute, to be used on a spacecraft when it is returning to earth from space.

Your MATLAB program should be able to take user input data of the spacecraft mass, m, spacecraft cross-sectional area, ASpacecraft, the parachute effective cross-sectional area, AParachute, and spacecraft height above earth, H.

Task Details:

MATLAB must be used for this Task. Write a short report comprising of the sections below and use your MATLAB program to answer the questions that are included.

1. INTRODUCTION

· Briefly introduce the problem and the rationale for undertaking the work.

· You do not need to include a literature review.

2. METHOD

The method section should include the following:

· Flow chart illustrating how you have designed your code

· MATLAB code including comments

3. RESULTS

Using your MATLAB code, Answer the following questions for a 900kg spacecraft with an effective cross-sectional area of 7m2 beginning at 140km above earth with an initial velocity of 0m/s.

Question 1: If the spacecraft accelerates towards earth due to the gravitational pull what will be its velocity, V, when it is 100km above earth? Hint: no air resistance (no FD)

Question 2: If the spacecraft continued travelling from the conditions described in Task1 and the parachute failed to open what would be its impact velocity with the earth, V? Also what would be the time of flight from 100km above earth’s surface to impact, T?

Question 3: If a parachute with an effective area of 150m2 was deployed at 4km above the earth’s surface what would be the impact velocity, V? Also what would be its total time of travel from 140km to impact after parachute deployment, T?