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:-

 

1. Understand in depth an appropriate range of mathematical principles and use a range of mathematical techniques.

 

The Brief

 

Part 1

 

1. a) Evaluate the following integrals:

 

(i)    (ii)

                                                                                                                                    (6 marks)

1. b) Find the following integrals

 

(i)          (ii)

                                                                                                                                    (8 marks)

1. 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:

C

B

A

 

 

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