Guidelines:

Failure to follow this guidance might result in a penalty of up to 10% on your marks.

I.      Submit a single PDF document with questions in ascending order. This can be produced for example in Word, LaTeX or MATLAB Live Script. Explain in detail your reasoning for every mathematical step taken. Include units for final answers where possible.

II.      Do not write down your name, or student number, or any information that might help identify you in any part of the coursework. Do not write your name or student number in the title of your coursework  document  file.  Do  not  copy  and  paste  the  coursework  questions  into  your submission - simply rewrite information where necessary for the sake of your argument.

III.      Insert  relevant  graphs  or figures, and describe any figures or tables  in your document. All

figures must be labelled, with their axes showing relevant parameters and units.

IV.      You will need MATLAB coding to solve some questions. Include all code as pasted text in an Appendix at the end of your document. Remember to comment on your code, explaining your steps.

This coursework is worth a total of 100 marks and counts towards 20% of your final ENGF0004 grades.

Coversheet

Please complete this coversheet to declare you have followed good academic practice. Please tick the boxes that apply.

1. I have carefully read and understood Section 9 of the academic manual: Student Academic

Misconduct Procedurehttps://www.ucl.ac.uk/academic-manual/chapters/chapter-6-student-casework-framework/section-9-student-academic-misconduct-procedure.

Yes               No 

2. I have made sure to correctly reference external resources incl. any teaching material.

Yes               No 

3. Have you used any AI tools to support your assignment?

Yes               No  

4. If so, which one(s)? <open ended>

Answer:                                                      

5. I have made sure to correctly reference the use of any AI tools in the entirety of my report.

Yes               No  

Declaration

I declare that all the information provided is true and understand that failure to comply with section 9 of the academic manual may result in penalties as outlined. <tickbox to sign>

Learning Objectives

Skills in Mathematical Modelling and Analysis

Figure 1. Key stages in mathematical modelling.

As part of this module, throughout the core learning activities, workshops and courseworks, we aim to equip you with the essential skills in mathematical modelling and analysis. The process of mathematical modelling can be, for instance, broken down in three key stages, which we hope become part of your skill set in your studies and career going forward.

The three stages are shown in Figure 1. During the first stage, the analytical model is set up. This includes employing the relevant simplifying assumptions, identifying main variables and parameters. To do this, the relevant literature needs to be researched to understand the main theory used to explain the phenomenon under consideration and what physical laws govern it.

In the second stage, this model is implemented computationally. In this stage, the model can be allowed additional complexity if numerical solution methods are going to be employed. Important in this stage is making sure the model is behaving as expected, through sensitivity analysis and validation. The sensitivity analysis also allows us to identify the most important parameters that determine the behaviour of the system.

After this we are ready to use our model to obtain new and useful information. We can change parameters and operating (boundary or initial) conditions to predict the way the system will behave in new conditions. We can use this simulation to design the properties of the system for a required application, or understand how to operate the system to make it more efficient or durable.

Keep this framework of skills in mind for the duration of this module as one of the main learning outcomes you want to obtain at the end.

Detailed Learning Objective

Model 1: Vibrations in a Circular Membrane [100%]

y

Figure 2. Diagram of a circular membrane with centre at the origin and radius R.

Problem Description

Circular membranes are an important fundamental engineering model - they are essential parts of mechanical pumps, microphones, telephones, and even micro-electromechanical systems (used in numerous sensors). They are so common that they are of relevance to most engineering disciplines. For example, circular membranes are also an essential part of the ear and studied in models of the hearing system in biomedical engineering. Membranes are also important for water treatment to clear out oil from contaminated water, relevant in chemical and environmental engineering. Future satellites are looking to use circular membrane mirrors to improve imaging and communication capabilities.

Understanding how these circular membranes vibrate, either to prevent mechanical failure, or control the vibration to ensure stability or improve efficiency and longevity of the membranes, is of interest in all of these applications.

Having been introduced to the wave equation - a partial differential equation which relates the deflection of the membrane (u(x, t)) as a function of time (t) and space (x, y), you are now capable of modelling these vibrations, which you will demonstrate in this coursework.

The one-dimensional wave equation is given by

?2u      2 ?2u

  =  C                  .

?t2           ?x2

As such, it can be used to only model a one-dimensional problem, such as the vibrations of a string, for example. Modelling the two-dimensional membrane, whereby finding a solution as a function of three variables: x, y and t, is more difficult.

However, we will use an ingenious simplification - we are only interested in solutions which are symmetrical around the centre. This means that the deflection along the radius repeats at any radial line around the circle of the membrane, starting from the centre and ending at the edge. It we sweep the solution we obtain for the deflection along the radius in a 360。arc, we

are able to reconstruct what happens on the entire circular membrane. Hence, we will model the vibration along the radius only (one dimension) and have a two-variable solution u(r, t).

We say that such a model is given in polar coordinates: r specifies the position along the radius, and θ specifies the angle around the circle (we assume the solution is the same for all θ).

The model is built on the following assumptions:

1.   The membrane is homogeneous - it has a constant mass per unit area (p = const. ), it is perfectly flexible and does not resist bending.

2.   The membrane is stretched and then fixed along its entire boundary in the rθ-plane. This means the tension per unit length, T, caused by stretching the membrane is constant  and  homogenous  (does  not  change  during  motion  and  is  the  same everywhere).

3.   The deflection, u (r, θ, t), of the membrane during the motion is small in comparison to the size of the membrane and all angles of inclination are small. (This is important for the derivation of the wave equation as the governing equation in the model. By using the wave equation, we assume this is met and use this information no further.)

4.  We will only consider solutions which are radially symmetric. To reiterate, we will model

the vibration along the radius only  = 0).

Hence, the deflection, u (r, t), of the circular membrane with a radius R can then be modelled by the wave equation in polar coordinates given by

 = c2  +  ,                                           (E1)

where

c2 =  .

The boundary and initial conditions are given by

u(R, t) = 0 for all time t ≥ 0 (the membrane is fixed along the boundary circle to the support),

u(r, 0) = f(r) (initial deflection),

 (r, 0) = g(r) (initial velocity).

We summarise the model parameters in Table 1.

Table 1. Model variables and parameters.

Useful Resources

While this is a new problem that you are faced with, this is a fundamental engineering problem that is very well studied in literature. There are many resources such as textbooks, for example “Advanced Engineering Mathematics” by Erwin Kreyszig, and numerous resources on the internet, that you are encouraged to research and consult while working on this coursework. Please do ensure to reference all such resources that you have used.

STAGE 1: Analytical Model [65 marks]

The following questions will help you set up and solve the analytical model, hence they will be more prescriptive in direction.

Question 1 [10 marks]

Demonstrate the separation  of variables technique  applied to  equation  (E1)  in  order  to separate the wave equation into two ordinary differential equations (ODEs).

Question 2 [15 marks]

Write the possible solutions to the time-dependent ODE for G(t) for

.    positive,

.    zero,

.    negative

values of the constant, μ = k2 . Discuss why only the negative values of the constant can produce physically viable solutions.

Find the general solution for G(t), if it is known there are infinitely many k values that satisfy the solution.

Question 3 [10 marks]

Now that you have established that the two ODEs found in Question 1 can produce physically meaningful solutions only in the case of a negative constant, μ = −k2, turn to the second ODE. The ODE which describes the solution w(T), depending on the space variable T, then has the form

 +  + k2 w = 0,                                              (E2)

Use the substitution q = √μT = kT and standard differentiation rules to transform E2 to the Bessel equation (E3)

q  +  + qw = 0.                                                (E3)

Question 4 [10 marks]

Solve the Bessel equation using Laplace transforms, taking the initial conditions

w(0) = c,

d0) = d,

where c and d are constants.

You should make use of the formulae in the standard formula handbook on p.30 and p.32:

ℒ{tf(t)} = −  = − ,

ℒ−1   = J0 (t),         where J0 is the Bessel function of first kind.

After applying the Laplace transforms, use the standard methods of solving ODEs taught in first year in order to solve the resulting ODE in terms of s.

Question 5 [20 marks]

The solution w(T), depending on the space variable T , found in Question 3 should have the form

w(T) = PJ0 (KT),                                                               (E4)

where P isa constant and K  = √μ  is the separation of variables constant.

J0   is the  Bessel function of first  kind and zero order. The Bessel function Jn   is  in-built  in MATLAB through the functionbesselj(nu,Z), where nu = 0 for the zero order function, and z is the function domain, in this case, z  = q.

a)  [5 marks] After reading the information about this function in MATLAB, plot J0 (q)  in MATLAB and find the locations for q at which J0  is 0. This will be useful in finding a solution which matches the boundary condition.

b)  [15 marks] Back in the analytical model, apply the boundary and initial conditions set out in the problem description to find the general form of the solution for the deflection u(T, t) in (E1) .

STAGE 2: Computational Model [35 marks]

The following questions are asking you to set up a computational solution and perform a sensitivity  analysis  of  the   model.  They   resemble  project-style  questions  and  are   less prescriptive.

Question 6 [15 marks]

Having  completed  the  stage  1  questions,  you  should  have  found  that  solutions  (or eigenfunctions)  representing  the  membrane  deflection  which  satisfy  the  given  boundary

condition, have the general form

um(T, t) = (Dmcos(ckmt) + Emsin(ckmt)) J0 (kmT).         (E5)

The coefficients Dm  and Em can be evaluate numerically from E6 and E7

Dm =  RTf(T)J0 (kmT)dT,              (E6)

Em =  RTg(T)J0 (kmT)dT .         (E7)

Note that you can choose any functions for the initial conditions f(T) and g(T) - these can be, for example, linear, quadratic, sinusoidal functions.

The solution um describes themth normal mode of vibration. Note that J1 is the Bessel function of first kind and first order.

Use MATLAB to plot solutions for the first three normal modes of membrane vibration given by (E5) form = 1, 2 and 3, at a single time point chosen by you.

Discuss the shapes obtained, paying attention to where the membrane remains stationary (u(T, t) = 0).

Question 7 [20 marks]

Applying the principle of superposition, the complete solution for the deflection of the circular membrane is given by

u(T, t) = (Dmcos(ckmt) + Emsin(ckmt)) J0 (kmT).                       (E8)

As the last part of your analysis in this coursework, we ask you to consider the effect of model parameters on the amplitude and frequency of the membrane vibrations.

To make it easier to see these effects, use the fast Fourier transform in MATLAB as the main tool in your analysis.

Now explore how different values of the

.    parameter c, and