I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

## Continuous Assessment Results

You are identified by the last four digits of your student number. The first column is your Test 1 result while the second is your Maple Labs participation. The third is your Maple Test Mark out of 5. The fourth is your continuous assessment mark out of 30. The last is the percentage you must have on the final to pass. If you have any issues with this please email me.

 Student Number Test Maple Labs Maple Test CA Mark Pass 8272 93 10 5 28.95 15.79 4673 90 10 3 26.5 19.29 1054 78 10 4 25.7 20.43 9455 65 10 5 24.75 21.79 0902 70 10 3 23.5 23.57 2344 61 10 3 22.15 25.50 2352 58 10 2 20.7 27.57 4346 28 10 3 17.2 32.57 3152 40 10 1 17 32.86 2343 25 10 3 16.75 33.21 2351 15 8 3 13.25 38.21 2345 28 6 3 13.2 38.29 4674 48 4 0 11.2 41.14 3150 25 6 0 9.75 43.21 1215 0 6 2 8 45.71 8171 30 0 0 4.5 50.71

## Study

Please feel free to ask me questions via email or even better on this webpage — especially those of us who struggled in the test.

Please find a reference for some of the prerequisite material here.

## Week 12

We finished our work on Laplace Methods and looked at the general solution of the damped harmonic oscillator. The following is the correct way to categorise over and underdamping:

### Damped Harmonic Oscillator Analysis

The differential equation $\displaystyle m\frac{d^2x}{dt^2}+\lambda \frac{dx}{dt}+kx(t)=0$,

as discussed on page 154 is the equation of a damped harmonic oscillator. There are three behaviours. One way to analyse these is to define the following parameters $\displaystyle \gamma=\frac{\lambda}{2m}$, $\displaystyle \omega_0=\sqrt{\frac{k}{m}}$

and follow the analysis as per the notes. However last night 8 May we outlined an even easier analysis.

First write the differential equation as it will be on your exam paper $\displaystyle \frac{d^2x}{dt^2}+b\cdot \frac{dx}{dt}+c\cdot x(t)=0$.

Now find the Laplace Transform of the solution. It will look like $\displaystyle X(s)=\frac{As+B}{s^2+bs+c}$.

Now there are three cases depending on whether $s^2+bs+c$ has two distinct real roots, equal real roots, or complex roots. Note in all cases $a=1$.

### Underdamping $b^2-4ac<0$

In this case the roots are complex: no real roots implying no real factors hence we must complete the square $\displaystyle X(s)=\frac{As+B}{(s+a)^2+k^2}$

which is composed of shifted sines and cosines when we transform it back $\displaystyle x(t)=Ce^{-at}\cos kt+De^{-at}\sin kt$

### Overdamping $b^2-4ac>0$

In this case the roots are real and distinct so we have two factors and hence a partial fraction expansion like this: $\displaystyle X(s)=\frac{As+B}{s^2+bs+c}=\frac{As+B}{(s+\alpha)(s+\beta)}=\frac{C}{s+\alpha}+\frac{D}{s+\beta}$,

which is composed of two exponentially decaying terms when we transform back. $\displaystyle x(t)=Ce^{-\alpha t}+De^{-\beta t}$

### Critical Damping $b^2-4ac=0$

In this case the root are real and repeated hence we have repeated real factors and hence a partial fraction expansion like this: $\displaystyle X(s)=\frac{As+B}{s^2+bs+c}=\frac{As+B}{(s+\alpha)^2}=\frac{C}{s+\alpha}+\frac{D}{(s+\alpha)^2}$,

which is composed of an exponentially decaying term and (before transforming) a shifted $\displaystyle \frac{1}{s^2}$ which will need the First Shift Theorem when we transform back: $\displaystyle x(t)=Ce^{-\alpha t}+Dte^{-\alpha t}$

### In Conclusion

If you are asked to analyse a damped harmonic oscillator of the form $m\frac{d^2x}{dt^2}+\lambda \frac{dx}{dt}+kx(t)=0$,

then you have three options:

1. Calculate $b^2-4ac$. Over-zero = Over-damping, Under-zero = Under-damping and Equal Zero = Critical-damping
2. Calculate $\gamma$ and $\omega_0$ as described on page 154 of the notes and compare. It is actually equivalent to method 1. In the Laplace marking scheme handout a damped harmonic oscillator is analysed using this method.
3. Solve the differential equation using Laplace Methods and see which behaviour the solution corresponds to.

## Week 13

We will hold a review tutorial on Wednesday 8 May in the usual room. First off, the layout of your exam is the same as Winter 2012: do question one worth 50/100 and two out of questions two, three, four; each worth 25/100.

I will field any questions ye might have at this time and if there are no questions we will do the exam paper from Autumn 2012.

## Formulae to Learn?

Somebody asked me for a list of formulae that ye might need that are not on the tables. I would put the following on that list:

• The Midpoint Rule Formula Here

One could include

• The Differential; if $z=f(x,y)$ then $\displaystyle\Delta z\approx dz=\frac{\partial f}{\partial x}\Delta x+\frac{\partial f}{\partial y}\Delta y$

There are a number of others such as $\displaystyle \int_a^\infty=\lim_{R\rightarrow\infty}\int_a^R$ for example but these are ideas rather than formulae really. Can you think of any others (the Euler method formula will be given to you)?

## Math.Stack Exchange

If you find yourself stuck and for some reason feel unable to ask me the question you could do worse than go to the excellent site math.stackexchange.com. If you are nice and polite, and show due deference to these principles you will find that your questions are answered promptly. For example this question about completing the square.