Further Maths and Simulation (SC2153)


Course: B.Eng/M.Eng Engineering Year 2
Module: Further Maths and Simulation (SC2153)
Topic: Assignment 1
Assignment submission date: 28th February 2018
Assignment feedback: 21st March 2018
You must attempt all questions and show all working.
Please state if you are rounding numbers, and include units where applicable.
1: The admittance, π‘Œ, in a circuit is given by
π‘Œ =
1
𝑍
where 𝑍 is the impedance of the circuit. If Z = 150 βˆ’ j22, find π‘Œ.
5 marks
2: By applying Kirchoff’s laws on a circuit, the following mesh equations were obtained:
18𝐼1 + 𝑗12𝐼1 βˆ’ 𝑗25𝐼2 = 200
20𝐼2 + 𝑗40𝐼2 + 15𝐼2 βˆ’ 𝑗25𝐼1 = 0
Determine, in polar form, the primary and secondary currents, 𝐼1and 𝐼2 respectively,
through the circuit.
12 marks
3: If 6π‘₯
2 + 3𝑦
2 + 𝑧 = 1, find the rate at which 𝑧 is changing with respect to 𝑦 at the
point (3, 5, 2)
4 marks
Course: B.Eng/M.Eng Engineering Year 2
Module: Further Maths and Simulation (SC2153)
Topic: Assignment 1
4: The stream function, πœ“ (π‘₯, 𝑦),has a circular
function shape as shown, and is related to the
velocity components 𝑒 and 𝑣 of the fluid flow by
𝑒 =
πœ•πœ“
πœ•π‘₯ and 𝑣 =
πœ•πœ“
πœ•π‘¦
a) If πœ“ = π‘™π‘›βˆšπ‘₯
2 + 𝑦
2, find 𝑒 and 𝑣.
b) The flow is irrotational if πœ“ satisfies Laplace’s equation,
𝛿
2 πœ“
𝛿π‘₯2 +
𝛿
2πœ“
𝛿𝑦2 = 0
Determine if the flow is irrotational.
15 marks
5: Determine the force, F, which has a magnitude of 64kN in the direction of the vector
𝐴𝐡 where A = (2, 4, 6) and B = (6, 3, 4).
8 marks
6: An object is dropped and its position vector, r, is given by
π‘Ÿ = (𝑑
5 βˆ’ 2𝑑
2
)𝑖 + (7𝑑
3 βˆ’ 6𝑑
2
)𝑗
a) Find the velocity, 𝑣, and the acceleration, π‘Ž.
b) What is the angle between 𝑣 and π‘Ž when 𝑑 = 2?
12 marks
Course: B.Eng/M.Eng Engineering Year 2
Module: Further Maths and Simulation (SC2153)
Topic: Assignment 1
7: If 𝛷 = π‘₯𝑧
2 + 3π‘₯𝑦
2 + 𝑦𝑧
2

a) Determine grad 𝛷 at the point (3, 5, 1).
b) Find the direction from the point (1, 1, 0) which gives the greatest rate of
increase of the function of πœ™.
9 marks
8: Consider a two-storey building subject to earthquake oscillations as shown:

Oscillations
The period, T, of natural vibrations is given by 𝑇 =
2πœ‹
√(βˆ’πœ†)
where πœ† is the eigenvalue of the matrix A.
Find the period(s) if A = (
βˆ’30 10
10 βˆ’20)
10 marks
The equations of motion are
expressed as
π‘₯̈= 𝐴π‘₯
π‘€β„Žπ‘’π‘Ÿπ‘’ π‘₯ = (
π‘₯1
π‘₯2
)
π‘₯1
π‘₯2
Course: B.Eng/M.Eng Engineering Year 2
Module: Further Maths and Simulation (SC2153)
Topic: Assignment 1
9: Below is a screenshot of a mathematical function written in MATLAB, along with its
corresponding vector diagram.
What does this code do? Explain with respect to each line in the code and the figure
shown.
10 marks
Course: B.Eng/M.Eng Engineering Year 2
Module: Further Maths and Simulation (SC2153)
Topic: Assignment 1
10:
a) Model Q6 using MATLAB. Include a screenshot of the code and explain what
each line of code is doing.
b) Give two advantages of MATLAB over manual calculations and support your
answers with examples
15 marks
Total available marks – 100

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