UNDERGRADUATE EXAMINATION
Courses that will take this exam;
MEng/BEng (Hons) Automotive Engineering
MEng/BEng (Hons) Biomedical Engineering
MEng/BEng (Hons) Civil Engineering
MEng/BEng (Hons) Electronic Engineering
MEng/BEng (Hons) Mechanical Engineering
Module Code and Title: ENG4125 Mathematical Modelling 2
This assessment has been converted from a closed book examination to an online open book examination
Date: Moodle – 20/MAY/2020 – 0900 British Summer Time (GMT + 1)
Duration: 2 hours + 10 minutes reading time.
You should spend no longer than the scheduled duration of the exam plus any approved additional time that you have been granted in an exam concession (e.g. Student Support Summary)
Latest Submission Deadline – 21/MAY/2020 – 0900 British Summer Time (GMT + 1)
You may complete the exam at any time during the scheduled 24 hour period, ensuring you submit your answers by the above deadline. This takes account of any reasonable adjustments.
Examiner: Jagjit Sehra
| Learning Outcomes Assessed:
· Recall techniques to solve 1st and 2nd order linear differential equations (homogeneous and non-homogeneous). · Recall techniques used to solve arithmetic and/or differential equation problems using matrices. · Convert and analyse signals from the time domain to frequency domain and vice versa
|
| Instructions to Candidates:
· Answer three questions in total · Candidates should ensure that all workings are shown clearly |
| Material Available on Moodle:
· Question Paper including Formulae sheets |
| Summary of Covid – 19 Revisions : The exam will be available on Moodle from 0900 British Summer Time (GMT +1). You will have the duration time to complete the assessment. Additional time has been provided for scanning and uploading your answers. You must ensure you upload your answers to Moodle by 0900 British Summer Time (GMT +1) on the following day. |
- a) Explain, giving example applications the concept of the Fourier analysis technique.
[12 marks]
- b) Calculate the following trigonometric Fourier series coefficients for the periodic, piecewise function shown, where T= 2π and hence ω=1.
- i) The DC average over one period, a0.
[4 marks]
- ii) The amplitude of the cosine terms, an, clearly showing the repeating pattern.
[9 marks]
iii) The amplitude of the sine terms, bn, clearly showing the repeating pattern.
[9 marks]
Q1 Total [34 marks]
- a) Explain and prove using integration by parts the derivative property of Laplace transforms.
[18 marks]
- Determine the solution for the differential equation with initial conditions shown using Laplace transforms. Use Laplace transform table included.
[15 marks]
Q2 Total [33 marks]
- A coupled system of differential equations is expressed in matrix form. Using the matrices technique, determine the following.
- a) The characteristic equation. [5 marks]
- b) The 2 Eigen values. [5 marks]
- c) The 2 Eigen vectors. [6 marks]
- d) The homogeneous solution (complementary function). [3 marks]
- e) The non-homogeneous solution (particular function). [12 marks]
- f) The full solutions expressed for x1and x2. [2 marks]
Q3 Total [33 marks]
END OF PAPER
Appendix A.1 – Formulae Sheet
Laplace Transform used for differentiation:
Derivative property of Laplace transforms
,
Laplace transform table
Partial Fraction Decomposition (Note the numerator has to 1 less order than the denominator)
Trigonometric Fourier series synthesis and coefficient equations
Roots of a quadratic equation:
Second order differential Equation solutions form
Appendix A.2 – Formulae Sheet
Solutions to non-homogeneous particular solutions:
Table of common integrals and derivatives
Function Derivative Function Integral