Introduction
The basis of this brief is to perform the various tasks needed to create an informative assignment report, including gathering information from a range of different identified sources, using the facilities available within the University.
Learning Outcomes
On successful completion of this assignment, the student should be able to:
 Understand in depth an appropriate range of mathematical principles and use a range of mathematical techniques.
The Brief
Part 1
 a) Evaluate the following integrals:
(i) (ii)
(6 marks)
 b) Find the following integrals
(i) (ii)
(8 marks)
 c) Evaluate:
(i) (ii)
(6 marks)
Part 2
A turbine manufacturer uses three mathematical models to estimate temperature distribution across the face of the turbine wheel under running conditions.
The three models used are:
Model 1: = 45T
Model 2:
Model 3:
The turbine wheels are as below:



It is necessary that the temperature drops to sensible levels across the disc face. Consequently, the design specification is that, for a temperature at the blade tips (Point A), of 420°C, the temperature at Point B should not exceed 200°C, and the temperature at the bottom of the blade, Point C, should not exceed 125°C
The total blade length from A to C is 55mm, and the distance from B to C is 40mm.
Solve the three equations above, using the initial values given in the manufacturers specifications.
15 marks
Part 3
Produce a set of graphs to establish which of the three models is the most suitable in terms of the design specification.
You should be able to plot the temperature change across the face of the disc, identifying the temperature at point B in each case.
15 marks
Part 4
The piston acceleration of an engine on test is defined by:
(i) Produce an expression for the velocity, , using appropriate boundary conditions
10 marks
(ii) Prove that an expression for the displacement, s, could be represented by: and consequently that maximum displacement is equal to 2r
10 marks
(iii) Plot the graphs of acceleration, velocity and displacement against crankshaft angle θ using suitable axes, and given that r = 10cm, l = 18cm and ω = 50π rad/s
30 marks
Total 100 marks
Reading Materials
Module lecture and support notes.
See also module reading list.
Note: These sources are guides only to commonly available material. Students will also be expected to consult other relevant source material.
 Stroud, K and Booth, D (2011) Advanced Engineering Mathematics Palgrave Macmillan
 Kreyszig, E (2010) Advanced Engineering Mathematics Wiley
 Bird, J (2010)_Higher Engineering Mathematics Newnes
 Singh, K (2003) Engineering Mathematics through Applications Palgrave Macmillan