### ENGF0004 Mathematical Modelling and Analysis II

matlab | lab代写 – 本题是一个利用matlab进行练习的代做, 对matlab的流程进行训练解析, 是比较有代表性的matlab等代写方向, 这是值得参考的lab代写的题目

### Late Summer Assessment 2021/

### Coursework 1

#### Standard Formula Handbook: Assessment section on MMA II Moodle site

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Topics Covered: Topics 1 - 4
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Expected Time on Task: 10 hours
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### Guidelines

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Failure to follow this guidance might result in a penalty of up to 10% on your marks.
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I. Submit a single PDF document with questions in ascending order. This can be produced
for example in Word, LaTeX or mat lab Live Script. Explain in detail your reasoning for
every mathematical step taken.
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
(for the purposes of plagiarism checks) in an Appendix at the end of your document.
Remember to comment on your code, explaining your steps.
```

**LONDONS GLOBAL UNIVERSITY**

This coursework counts towards 12.5% of your final ENGF000 4 grades and comprises one

question, referred to as model, worth 100% of your coursework grade.

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On Academic Integrity (Read more about it here)
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Academic integrity means being transparent about our work.
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- Research: You are encouraged to research books and the internet. You can also include and paraphrase any solution steps accessible in the literature and online content if you reference them.
- Acknowledge others: We are happy when you acknowledge someone else’s work. You are encouraged to point out if you found inspiration or part of your answers in a book, article or teaching resource. Read more about how to reference someone else’s work here and how to avoid plagiarism here.
- Do not share and do not copy: We expect students not to share and not to copy assessment solutions or MATLAB code from their peers, even if partially.
- Do not publish ENGF000 4 assessment material: We expect students not to share ENGF000 4 assessment materials on external online forums, including tutoring or "homework" help websites.

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Students found in misconduct can receive a 0 mark in that assessment component and
have a record of misconduct in their UCL student register. In some extreme cases,
academic misconduct will result in the termination of your student status at UCL.
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### Model 1 : Vibrations in a Drumhead [ 10 0%]

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Figure 1. Image of a drumhead the circular membrane at the top of a drum.
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To model the vibrations of a drumhead, which is the circular membrane at the top of a drum, we will use the two-dimensional wave equation in polar coordinates (,). This will allow us to expand our knowledge of a well-known theory such as the one-dimensional wave equation to a wider range of applications. Similar circular membranes are also important parts of pumps, microphones, and telephones to name a few, making them an important fundamental engineering model.

The model is built on the following assumptions:

- The membrane is homogeneous it has a constant mass per unit area (=.), it is perfectly flexible and does not resist bending.
- The membrane is stretched and then fixed along its entire boundary in the -plane. This means the tension per unit length, , caused by stretching the membrane is constant and homogenous (does not change during motion and is the same everywhere).
- The deflection, (,,), 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.)
- We will only consider solutions which are radially symmetric (

(^2) ^2 =^0 ).^ The deflection, (,), of the drumhead membrane with a radius can then be modelled by ^2 ^2

##### =^2 (

##### ^2

##### ^2

##### +

##### 1

##### ),

where

##### ^2 =

and the boundary and initial conditions are

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(,)= 0 for all 0 (the membrane is fixed along the boundary circle to the drum),
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(, 0 )=() (initial deflection),
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```
(, 0 )=() (initial velocity).
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Question 1 [ 15 marks]

Use the separation of variables method to separate the wave equation into two linear ordinary differential equations (ODEs).

Question 2 [ 15 marks]

The two ODEs you found in Question 1 can produce physically meaningful solution in this case only if the constant they are equal to is negative (.=^2 ). The ODE which describes the solution (), depending on the space variable , then has the form

```
^2
^2
```

##### +

##### 1

##### +^2 = 0.

Use the substitution = and standard differentiation rules to transform this equation to the Bessel equation:

##### ^2

##### ^2

##### +

##### += 0.

Question 3 [2 5 marks]

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

```
( 0 )=,
( 0 )
```

##### =.

You should make use of the formulae in the standard formula handbook on p.30 and p.3 2 :

##### {()}=

##### {()}

##### =

##### ()

##### ,

##### ^1 {

##### 1

##### ^2 + 1

```
}= 0 (), where 0 is the Bessel function of first kind,
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and standard methods of solving ODEs taught in first year to solve the ODE in terms of after applying the Laplace transforms.

Question 4 [2 5 marks]

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

```
()= 0 (),
```

where is a constant.

0 is the Bessel function of first kind of order = 0 , which can be found for any from the general form for ()

##### ()=

##### ( 1 )^2

##### 22 +!(+)!

```
```

```
= 0
```

##### .

The Bessel function is in-built in MATLAB through the function besselj(nu,Z).

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a) [10 marks] After reading the information about this function available in MATLAB, plot
0 () and find the locations for at which 0 is 0. This will be useful in finding a solution
which matches the boundary condition.
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b) [15 marks] Apply the boundary and initial conditions set out at the start of the problem
to find the solution for the deflection (,).
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Question 5 [20 marks]

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a) Solutions (or eigenfunctions) representing the drumhead deflection, which satisfy the
given boundary condition, have the general form
```

```
(,)=(cos+sin) 0 (
```

##### ),

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where you can evaluate numerically the coefficients and from their definition below
for chosen by you initial conditions () and () these can be, for example, constants
or linear, parabolic, or sinusoidal functions.
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##### =

##### 2

##### ^2 12 ()

##### () 0 (

##### ),

```
```

```
0
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```
=
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##### 2

##### ^2 12 ()

##### () 0 (

##### )

```
```

```
0
```

##### .

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The solution describes the th normal mode of vibration.
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Plot surface plots of the first three normal modes of drumhead vibration for = 1 , 2 and 3
at a set time point chosen by you. Discuss the shapes obtained, paying attention to lines
where the drum membrane remains stationary ((,)= 0 ).
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b) The complete solution for the deflection of the drumhead is given by
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```
(,)=(cos+sin) 0 (
```

##### )

```
```

```
= 1
```

##### .

Find in MATLAB the fast Fourier transform of this solution over some period of time using either = 1 , 2 and 3 terms in the infinite sum and plot the resulting frequency spectra in the three cases. Discuss differences you can observe in the three spectra due to the different number of terms and compare to what you expected based on the mathematical expression for the solution. Discuss how the frequencies obtained compare to the frequency of musical notes and discuss ways specific notes could be produced.