An Ceardcolaiste Reigiunach Sligeach

Regional Technical College Sligo

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School of Engineering

Bachelor of Engineering (Civil Engineering)

Stage 4

Autumn 1996

Subject: Mathematical Modelling

Terminal Examination

College Examiner: Brian Mulligan

External Examiners:

Date: Time:

Duration: 2 hours

Answer 3 out of 4 questions.

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Question 1

(a) Explain what is meant by the term ‘parameter’ in relation to mathematical modelling.

(a) A mathematical model is required to predict the movement of sand along a beach due to waves striking the beach at an angle. Write notes on the development of such a model, stating the important variables and any other assumptions that you think may be realistic.

(c) Explain what would influence your decision in choosing between a 0, 1, 2 or 3 dimensional model for simulation of a dynamic (time varying) system (eg. modelling pollution dispersion and decay in a body of water).

Question 2

(a) Suggest a suitable generalised mathematical equation that could be used to describe the cooling of water down to room temperature.

(b) Show how you could use the Stella software package to implement an Euler approximation of the cooling of water to room temperature.

(c) The following equation is a mathematical model of the displacement of a cantilevered beam:

d4y = q(x)

dx4 EI

where x is the distance from the fixed end, y is the displacement of the beam, E is the modulus of elasticity, I is the moment of inertia of the beam and q(x) is a function that describes the distributed load as a function of the position on the beam

Classify this model under the various schemes you are familiar with.

Question 3

(a) Develop a calculation scheme to determine the consumption of BOD and the growth of micro-organisms in a tank given the following:

dS = G - DdB = 1.5D - 2G

dtdt

G =  ( __B___ )SD = k2S

k1 + B

where S is the concentration of micro-organisms, G is the growth rate of the micro-organisms, D is the decay rate of the micro-organisms, B is the concentration of BOD,  is the maximum specific growth rate of the micro-organisms, k1 is the half growth rate concentration, k2 is the unit death rate.

(b) Sketch and explain how you would implement this model using Stella software.

(c) Describe any experiments that might be useful to you in order to calibrate and verify this model

Question 4

(a) The diagram here represents a Stella model of traffic arriving and departing from a traffic light. Describe in words what you think each individual item in the diagram represents.

(b) Using the Stella modelling package to predict the displacement of an undamped pendelum over time produced the following result: a sinusoidal graph of increasing amplitude. Explain the source of this problem and suggest a way of reducing it.

(c) Describe the type of problem that requires the Euler method of solution.

(d) Describe the type of problem that requires a finite difference method of solution.