LARP Drawings and Your Very Own State Space Equation

Sorry About My Lack of Posts...

So I've been pretty bad at posting recently and that's because honours year engineering is intense. I'm constantly trying to stay ahead but constantly lagging behind! I'm sure you can all relate in one way or another although I swear some people in my year have at least 20 more hours in the week than me because they are doing just fine! 

Some LARPing Drawings

Those of you who know me, may be aware that I'm a wee bit of a doodler. Right now I'm on a crucible doodle mission. Here's a sneak peak of some drawings that may make an appearance in your next game. I'm not going to say how or why. You'll just have to find that out yourself! 

Creating A State Space Model (3)

Now this is an important one so if you want to continue following my control systems posts, I would advise you to make sure you understand it.

State space models are a a really convenient way of depicting a real life mathematical system. They involve...

•  some inputs which could vary with time (u(t))
•  some outputs which could vary with time (y(t))
• and some state variables which vary with time (x(t)) These are basically just stuff which happens inside a system.

For instance, if we wanted to model a radio we would have an input of some radio waves, u(t), an output of some sound waves y(t) and some internal processes which work on converting radio waves to sound waves, x(t).

If we want to show this mathematically it looks like this ----------------------------->

This may seem a little but complicated but its really very logical when you think about it. 

Top Line: The x with a dot on its head is the derivative of x, this is kind of like the rate of change in x. So the rate of change of the state variable is caused by a combination of the state variables (timed by some constant matrix A) plus some input (multiplied by some constant matrix B)

Bottom Line: Basically says that the outputs of the system is made up of the inputs and the state variables. Which makes sense really. If you think about the radio example, when you change the radio waves, you are going to hear a different sound. Also if your radio is a little bit broken and the internal system isn't working, you will get a different sound output as well. 

EXAMPLE TIME! 

Say we have the system in the picture to the left. A mass on a spring. Someone applies a force to one end f(t) and we measure the distance it travels q(t). 

The first thing we have to do is form some governing equations which involve both f(t) and q(t). In this case we can do this by summing the forces on the mass to get the following governing equation...

By the way the q with a double dot above it is the second derivative of q. So its like the rate of change of the rate of change of distance - acceleration. 

The second thing we do is isolate our inputs, outputs and state variables. So our input ( u(t) ) is f(t) - the force we add to make the system move and our output ( y(t) ) is the measured distance q(t). As a rule of thumb, the number of state variables needed in an equation is equal to the order of the differential equation. Check out those two dots on top of the q! That means we've got ourselves a second order equation so we need two state variables which in this case are x = [q ; qdot] position and speed. 

The third thing (that I like to do) is form the state space equation with the matrices A,B,C and D left blank...

You may be wondering how I know what the size of matrix A, B, C and D are. I can explain it logically but I also have a general rule for you to use. 

Logic: The right hand side of the top equation has two rows [qdot, qdoubledot]. So I know that I need to end up with two rows on the left hand side. Since the A matrix is timed by the 2x1 state variable matrix, I know that in order to end up with 2 rows after the matrix multiplication that I need a 2x2 matrix. If that seems a bit odd, have a look at matrix multiplication again and you'll get the idea. 

Rule: Number of inputs(# i): 1, Number of outputs(# o): 1, Number of state variables(# s): 2

A = (# s) x (# s) , so 2x2

B = (# s) x (# i), so 2x1

C = (# o) x (# s) so 1x2

D = (# o) x (# i) so 1x1

The Fourth step is to fill A,B,C and D in according to the governing equations. 

qdot = 0q + 1qdot + 0f   

^^ qdot = qdot just like 1=1. 

qdotdot = (-k/m)*q + 0qdotdot + (1/m)f

^^notice how I rearranged the governing equation to find this

q = 1q + 0qdot + 0f

So putting these into matrices...

 AND THERE YOU HAVE IT! YOUR VERY OWN STATE SPACE EQUATION!!

Well done you if you followed! See if you can do the whole thing yourself without looking at my notes. That way you'll know if you understand or not. 

The Weird Ways My Dog Sleeps, ThermoPlastics and ThermoSets

My Cute But Weird Dog (obligatory non-technical post)

So you may or may not know that I have a dog called Alfie. I love him to pieces. I am like a crazy cat lady except instead of spreading my affection over 20 cats I give it all to just one doggie. Alfie is the sweetest thing. But he is very weird sometimes...

Here are two pictures of him in some positions he deems 10/10 for comfort, would sleep in again. He stayed in the first position for at least an hour and in the second for at least half an hour. I don't even know how a dog can bend its leg under itself like that??

ThermoPLASTIC Stuff! ((1-2))

If you want to buy these from the polymer shape they come in the form of "solid matter" (pellot like things) Polymer chains already created!

Crystallization and Alien Geometry Races

In one of my previous blogs I talked about crystallization of a polymer. This occurs when a Thermoplastic crystalline is slow cooled. All of the polymer chains huddle together as if trying to warm themselves as the temperature falls before freezing completely (woah, that's a morbid analogy). The slower you cool it, the more crystalline it will be although it can never be completely crystalline, it sort of maxes out at about 45%.

It doesn't stop there though after the chains fold themselves into little sheets of folded polymer called lamallae, the lamallae will twist themselves into a circular arrangement like the picture below.

Once in this arrangement, we call them Spherulites. This is kind of cool because it makes them sound like an alien race of evil spheres. They also look awesome up close (Left).

So... do we think that this arrangement is more or less dense than an amorphous structure (where the polymer strings just hang out in a random arrangement). Its more dense! Which is great because it means that we can measure how crystalline a polymer is just by weighing it.

Report Card For a Highly Crystalline (or dense) Polymer

• Stiffness = A+ all those polymers are hugging each other tightly within their lamallae and spherulites and will not be easily wrenched apart
• Heat Deflection = A+ this means that the polymer now needs to be heated to higher temperatures before it will melt.
• Chemical Resistance = A+ Not really sure why this is, anyone care to share their thoughts
• Hardness = A+ something something that's what she said something. 
• Tensile Strength = A+ yes this is different to hardness, tensile strength is like how much weight it could hold and hardness its ability to dent other materials.
• Impact Strength = C- being crystalline makes polymers brittle! If you hit it with a hammer, it will shatter!
• Ductility = C- those polymer chains are just holding onto each other too tightly for there to be much give

As a side note, if you were to assess an Amorphous structure (with the random arrangement of polymers) the results above would be exactly the opposite. Have a think as to why...

MOAR CRYSTALLIZATION!

So if you think crystallisation is really cool you can increase it by doing a number of things. 

1. Cooling the polymer slower
2. Cooling the polymer to the right temperature
3. Adding bits of fiber to the polymer - these fibres act as nucleation sites which basically means that crystallization likes to happen there.  

ThermoSET Stuff! ((1-2))

If you want to buy these from the Polymer Shop you will find them as either a liquid or solid resin. You have to process these yourself to get your polymer chains and cross links. This makes sense as they can only be processed once!

How do I process My Thermosets?

Thermosets need a little bit of encouragement in order to start reacting and forming their strong covalent cross links. They need a catalyst such as heat, pressure, UV light and sometimes they just need time. Once you get them going though they produce A LOT OF HEAT! This is because they are exothermic (meaning they release a lot of energy when reacting).

I know what some of you clever sods are thinking, "but they degrade in high temperatures, doesn't that mean that they will kill themselves as they form?" Well yes if left them to their own devices, things don't work out. Which is why heat must be removed and moderated during the reaction. 

Stick to Your Strengths!

Thermosets are way stronger than thermoplastics. They are often used for this reason. So when processing them, they should be given the strongest properties possible by being allowed to cure almost completely.

Here's a graph of their cooling over time where alpha is the degree of cure. Obviously you don't want o be waiting around all year for them to be perfect so you have to stop the cure when the graph starts to level out and not when alpha = 1. 

The more inquisitive of you may wonder how the degree of cure is calculated. Obviously you can't figure out how far along the cure is by counting all the cross links between the chains. What you can do though is look at the amount of heat released (in the form of enthalpy) and compare it to the total amount of heat that could be released. dHa/dHt = alpha 

Sweet Buns, Katanas and Heat Exchanges

Sweet Buns ((1))

I am a youtube hair style enthusiast. It's my guilty pleasure to spend an unnecessary amount of time getting ready for parties so that I can learn a new hairstyle. Often I don't even care about doing my makeup.
Hair is both ridiculous and crazy cool. I've always found it weird that unlike most other animals our hair doesn't seem to stick around to keep us warm at all. The fact that our hair pretty much only has an aesthetic use says a lot about people I think. But isn't it cool that somehow, when we twist it into some knots and smooth it down or whatever, this somehow means that we can make ourselves more "attractive". That's whacked, its not just me, right?? But then the general public does seem to like a lot of whacked things that aren't entirely natural or practical. So although its whacked, its not at all surprising.
I would really like to get into doing hair for balls and weddings and things when I get a bit older so hit me up if you are willing to trade me some baking for a hair style for x event in the future. I love that stuff, weird as it is.

Katana ((1))

I bought my first Larp-safe weappon today!! MUCH EXCITE! It's a katana and its beautiful. I have no idea how I am going to be able to use it but man it is cool. Its got a weighted hilt so it feels like you are wielding the real thing except that you're in no danger of splitting flies in half mid air. I will post photos when i get it :) And then I will have to start learning how to use it.

Heat Exchanges ((2))

So heat exchangers are a pretty common place thing for a mechanical engineer. They are basically a component through which flows two fluids which do not mix (i.e. they are in separate pipes). They will be separated by some conductive metal so that heat is conducted between the two streams. This heats on of the streams and cools the other. 

The reason I am telling you about heat exchangers is because they are relevant to my fourth year project. My fourth year project is all about waste heat recovery cycles. This means that take some hot fluid created in an industrial process e.g. some hot water. Then we put it through a thermodynamic cycle in order to turn the heat into electricity. In order to extract energy from heat you need to transfer the heat from the waste fluid into the fluid that you use to drive a turbine (a device which a flow of fluid rotates in order to generate electricity). In our project, the fluid we are investigating is carbon dioxide. I want to see how well heat can be transferred between my waste fluid and carbon dioxide when the carbon dioxide is at different pressures. 

It gets a wee bit more complicated than that but for the sake of simplicity and the time it takes me to write a blog post just keep all of that in mind. 

((3))

One method of calculating the heat transfer rate (how fast heat energy can be passed between streams) is called the Log Mean Temperature Difference Method or LMTD for shorts.This is what it looks like:

• Qdot is just the heat transfer rate
• U is the overall heat transfer coefficient (which basically says how easy or hard it is to transfer heat between the two fluids)
• A is the area over which heat transfer between the two fluids occurs 
• Triangle Tlm is the log mean temperature difference defined in the second equation
• ln is a natural log (basically an operation which is a little more complex than multiplication and stuff)
• Th,in is the temperature of the hot stream at the inlet of the heat exchanger
• Th, out is the temperature of the hot stream at the outlet of the heat exchanger
• Tc, in and Tc,out are the corresponding cold stream values

Creating a Model 

So the tricky thing about the model that I have to create is that the above equation is only viable for constant specific heat (Cp). But at certain pressures (close to the critical pressure, where unfortunately our project's cycle is operating at) the Cp of carbon dioxide changes a lot! By the way, specific heat is pretty much a measure of how much energy it takes to increase the temperature of a fluid by one degrees. Ever wondered why it takes water about twice as long to boil as it does for oil? Yeah oil has about half the Cp value as water. Try it out yourself if you can spare a pot full of oil! 

This variation of cp means that I need to break the heat exchanger up into many very small parts so that I can assume constant specific heat across these tiny sections. Then I can integrate over all of these sections in order to get the overall heat transfer rate. This is why they teach us integration in high school by the way! Because of applications like this. Your teacher wasn't just doing it to see you suffer! 

But anyhow! I will have to talk more about this later because its getting late and university is tomorrow! Goodnight! 

Hair stuff, Matrix Multiplication, Complex Numbers and Modal Method

HAIR STUFF! 

This is basically my obligatory non mathematical post. I did this using an inverse french braid (meaning that you cross strands of hair underneath rather than over top) which gives a really distinct platted look which follows the curvature of the head. 

The back bit I did using a technique which is a little hard to explain without a demonstration. But its super duper easy, holds really well and doesn't require as many bobby pins as you think. Intrigued? Ask me next time I see you and I'll give you a demonstration! 

SO! ONTO THE MATH!

Matrix Multiplication (difficulty 2 - Give it a try)

So matrix multiplication is relevant to a lot of things, including today's lesson! Previously we have done a 2x2 matrix multiplied by a 2x1 matrix. Today we will be doing something a little more complicated and multiplying a 2x2 matrix by a 2x2 matrix. Below I am going to show you how to do it by using letters. Don't panic when you see letters being used to explain math! Letters simply represent any number and they are easier to track than numbers. Anyway, here is our matrix multiplication... 

So we can see to make the top left element on the final matrix we used the top row of the first matrix and the left column of the second matrix, to get the top right element of the final matrix we used the top row of the first matrix and the right column of the second matrix etc etc. 

Give it a try on these two matricies:

[2,1; 2,2] and [3,1; 1,1]; did you get [7,3; 8,4]? if you didn't, let me know and I will see if I can sort out your problem. If you did, GOOD JOB YOU, YOU ARE AWESOME!

Modal Method (3)

So as mentioned before the modal method can only be used to find e^At...

• If A is diagonalisable 
• I've talked about this in a previous blog but in a nut shell you can check for diagonalablity (I feel like I'm just making these words up) finding the eigenvectors, putting them into a matrix (called P) and checking that that matrix has two linearly independent columns.

If you go through and find that the matrix is not diagonalisable then too bad how sad you have to use the expansion method.

The rest of this method involves using this formula... e^(At) = Pe^(LAMBDA.t).P^(-1) where

• P is our friendly old matrix made out of our eigenvectors (which we have to find to prove we can use this method anyway)
• LAMBDA (would usually be written as capital lambda and looks like this) is a matrix with our eigenvalues (also already found) on the diagonals
• t is just a scalar representing time

Basically you put all your values into the formula and bob's your uncle. 

Inverting a Matrix (2)

There is one part that I want you to take notice of though... NOTICE how I have coloured the P^(-1) in red above? It means that we need to take the inverse of the matrix. Usually when we see ^(-1) on a scalar number e.g. 5^(-1) it simply means that we need to push the number down to the other side of the fraction line i.e. 5^(-1) == 1/5 but for a matrix EVERYTHING CHANGES. We now have to do an operation to it which might seem weird but it works on paper because algebra. If you don't believe me try converting a matrix into a scalar equation and try inverting that linearly. You'll get the same answer :)

What you need to know for a 2x2 matrix is that we swap the numbers on the diagonal that goes this way \ and we change the sign (e.g. from positive to negative) on the numbers on the diagonal this way /. And then you times the new matrix by one divided by the determinant of the original matrix (which I tell you how to do here in my calculation of eigenvalues). Let's see an example of that...

See if you can cover up my example and give it a go yourself!

Modal Method Example (4 just because there are so many steps!)

We're going to try to find e^(At) using the matrix A = [0,1; 1,0]. To solve this I have to introduce the idea of complex numbers...

COMPLEX NUMBERS!

These are also known as imaginary numbers because in real life they don't exist as a number!! 

Complex numbers are literally the square root of -1. Think about it, a square root sign is the opposite of a squared sign. The square of positive two and the square of negative two both work out to be the same number 2^2 = 4 & (-2)^2 = 4 as a negative times a negative is a positive. Therefore it is impossible to reverse this and take the square root of a negative number. Try it on a calculator! We represent these numbers as the letter "i"  or "j" which literally means the square root of -1. From this we can see...

• j^1 = j
• j^2 = -1
• j^3 = -j
• j^4 = 1 and so on and so forth.  

Although we call them imaginary, this does not go to say that they are useless. They are actually used in a lot of applications and can tell us a lot about a system if they arise. 

BACK TO OUR EXAMPLE...

Doesn't that just come out beautifully!

Euler's formula can be found here for anyone who is interested in seeing for themselves how it works :)

Cardboard Boxes, Measure for Measure and Exponential Functions

CARDBOARD BOXES!

We bought some speakers,
my little bro drew my face,
I struggled to breathe,
It was awesome.

 Measure For Measure 

So call back auditions were last night. And WOW, what a fantastic cast we have. The auditions were so much fun as well. Mostly a bunch of theatre sports and some individual monologues. It was a little bit nerves, a little bit because I was surrounded by hilarious people but I was in hysterics most of the night. 10/10 would audition again! I'm so excited to start rehearsals with whatever part I get, even if I only have one line. That is how awesome the cast is! <3

Difficulty Ratings

So I've decided that I'm going to introduce difficulty ratings to different sections in my blog. Though I encourage you to give the harder things a read this means that you can just cruise through the easier parts if you want :D
This is the system I will be using:

5 - ((I Don't Even)): This means it's beyond me and my understanding. There probably won't be much of this because if I don't understand something I will probably not write it so that you all still think I'm smart.

4 - ((Break a Sweat)): This means that I found it difficult but got there in the end

3 - ((So-So): It was relatively easy for me as a person with 3 years of mechanical engineering under my belt to understand when broken down into steps

2 - ((All Good)): Relatively easy for me to understand

1 - ((Easy as Pie)): Easy as Pie!

So this is all based around my experience and there are probably geniuses who will find all of it easy and there will also be others who don't have a big background in math or science who might find it harder. If you spend the time thinking about it critically though, you will come to understand it which is the really cool thing about math and physics!!

Exponential Functions ((2))

So here's a nifty trick, if you're ever out and about and have forgotten your phone or calculator and really really really need to know what the exponential function of some number is (e^x) all you have to do is use the algorithm 

x = some number for examble 2!

e = the exponential function

k = a number which starts at zero and increases in jumps of 1 each additional term

! = means factorial. When a number has a factorial sign beside it you write down all the whole numbers from 1 to your number and put multiplication signs in between i.e. 4! = 1x2x3x4 = 24!!

So the equation says to begin with k = 1 and go to k = infinity but you don't need to do that. Noone does that because that's a really lame thing to do with you life and you're always going to fail. You just need to go until the equation converges. And by converging I mean when you are adding extra terms to the equation and it doesn't seem to be making much difference. Give it a go with x = 1. 

See if you can "converge" to 2.7 :D go on !

Here's just one example of where it might come in handy...

Matrix Exponential Function ((3))

So if you want to give this a go but you're unclear about matrices I explain them here.You also might want to know more about matrix operations (multiplication, dot products, determinants etc) you can find them here  because it will take SOOO much time to explain them all. 

So we are basically just going to take the above equation for a linear equation and magic it to make it work for a matrix as well. The other thing we're going to do is add variation with time by putting in the scalar value t because that will make our equation really useful. A lot of applied math varies with time! 

So we want to work out e^(At) which is the exponential function of a matrix which varies with time. Here's the equation we have to use. See if you can spot the similarities to the equation above...

So when we are expanding this out, instead using a 1 for the first term as we did in the above linear case we now use the Identity matrix [1,0  ; 0,1] for when k = 0 and . We simply expand as above to get our answer using a similar method as above. Let's do an example!! 

In this example A = [0,1;1,0]; We want to find e^At

using the equation above our terms are (just expanding out the equation):

= I + tA + (t^2)(A^2)/2 + (t^3)(A^3)/(2x3) +  (t^4)(A^4)/(2x3x4) ...

Subbing A into our matrix and using matrix formatting we get

Which seems really messy until we do this magical magical thing involving the series definition on this page. Now our matrix simplifies really beautifully to this...

Incidentally this is actually how your calculator calculates the sin and cos function. Think about it, it would be completely unreasonable to have manually tried to program in the sin and cos function. The calculator must be working it out somehow and this is exactly how! 

Modal Method - (the easy way)

So here's a wonderful application for the diagonalisation stuff we learnt about in one of my previous blogs! I love it when stuff is useful :D

We can calculate e^At in an easier way than the one above IF we know that matrix A is diagonalisable. 

BUT I'm going to save that for another night because my bed is calling and I still have a speech to write. 

GOODNIGHT EVERYONE!

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Chitty Chitty Bang Bang and Business Plans

Why Failing at Business Plans Made Mr. Potts Fail at Life (well mostly)

Even with a really good idea, businesses can still be doomed to fail. Businesses don't want ideas, if they did, Mr Potts from Chitty Chitty Bang Bang would have been a billionaire from the get go.Businesses want money. Crazy right? And in order to do that, most of all, a business plan needs to be marketable. Let's be honest, Mr. Potts was an ideas man but not a realist. He got lucky with the flying car but the other stuff, when put onto a market, is worthless.

Mr Potts Doing His Thang

Take this breakfast making contraption for example https://www.youtube.com/watch?v=RBJGpNTP_lY. While pretty much one of the coolest things you will ever see in your life, its not going to catch on. This is because he didn't follow all the basic steps for making a business plan...

1. He didn't start with a problem: If he had gone out and interviewed people who cooked in their own home, they might have been able to tell them that they had no problem cooking simple foods like eggs and sausages in their own home. What they actually wanted was for their food to stop sticking to their fry pans. He also might have discovered that people generally wanted more space in their kitchen and did not want it to be taken up by a bulky, intricate contraption which is only able to cook two types of food! 
2. He didn't Identify Customers: The customer for this product are people who really really really like eggs and sausages, all the time for every meal and can't be bothered spending the time making them. This is an almost nonexistent market (pretty much just the Pott household) and should have been assessed before even beginning to make it.
3. He didn't Justify Competitive Value: This isn't as relevant to Mr. Potts because he failed to address a problem and a customer in the first place. But, this step is really important and involves thinking about what you could offer which would make people choose your product over existing products. 
4.  Ideas: To be honest he doesn't have a problem with ideas. He has one beautiful mind. Dat lateral thinking.
5. I guess he kind of thought about distribution: I'm going to use another example for this because it's my blog and I can do what I want to. I want to talk about the  toot sweets. Great idea to bring the toot sweets to a candy factory. Instant loyal customer base right there. 
6. Revenue: Where does the money come from, how much is needed and when? Be specific Mr. Potts, you're  never going to get anywhere in life if you don't think of all this stuff.
7. Costs: Not once in the interview with the sweet factory owner did he mention price. And the boss blindly accepted this. Some day someone's business is going under...
8. Resources: Did you know that if Mr Potts had a partner that businesses would be much much much more likely to invest in his ideas? If you're alone in thinking an idea is good, investors will question why that is...

You Forgot to do Market Research on any of your Products Mr Potts!!

Once you do have an idea, its important to validate it. Ask around and then answer these questions...

can you find customers?

do they have this problem?

is it important?

will they pay for it?

what are the key features of that solution?

Of course you should never people directly if they will buy your product because they will most likely say yes. People like saying yes. But in reality people are lazy and if they don't have a major problem with something, they won't change it. This is why its so so important to do your research! 

Mr Potts Should have Tried a Lean Set Up 
This is all about doing as little as possible to get the maximum possible output in order to attract in investors attention. Investors do not need to see finished prototypes Mr Potts! All they want to see is that the idea has potential and low risk! Ventures don't usually fail because the invention doesn't work, they fail more often because people don't want to buy it. 

WISE UP MR. POTTS!

Clothes Scrambles, Polymer Stiffness and Why Size Does Matter if You're Plastic

My Fourth Blog! Not bad...

Clothes Scramble! 

Second hand shops are great. Clothes scrambles are pure genius! Thank god for impulse buys and changing tastes! I had forgotten that there was one on today and when I figured it out I was so happy. My little brother just happened to snap a picture of my face at that moment of joy...

This clothes scramble had a small donation fee for two very good causes (SPCA and Sweet Louise). And yay! Clothing!

Polymer Necklaces

So I want to have a look at what gives a polymer different properties and not just whether it is a Thermoset or a Thermoplastic. If you do want to review that and what a polymer and a Mer are you can view my previous blog post here http://math-machines-and-playing-pretend.blogspot.co.nz/2014/07/measure-for-measure-thermosets.html

Remember last time we compared polymer strings to beaded necklaces? Well we're going to use that analogy and develop it a little further to have a look at the properties of a chunk of polymer.

Here's three terms to add to your vocabulary of smart sounding stuff:

Intramer Structure: This is the structure of the Mer itself. This refers to how the Mer structure is made up of chemical elements. Think of this as what the bead is made out of and how it is designed to be connected to the necklace.
Intramollecular Structure: (notice the "a") This is the way Mers combine to form polymer chains. If you want you can think of this like the string or the links between the beads.
Intermollecular Structure: How the chains interact with each other. Remember how previously we talked about Van Der Wal and covalent bonds and related them to tangles in the beads strings and perhaps a splash of superglue.

A really really important property of a polymer is its flexibility or rigidity. Take a look around you right now and find a plastic thing. Now ask yourself how absolutely useless this thing would be if that thing had different flexibility properties. The thing closest to me is one of those dog ball thrower things. Although not disastrous without it, a little bit of flexibility is good for this as it provides a bit of spring when throwing a ball. Even though it's a pretty trivial object, someone would have thought this through and intentionally chosen a polymer to balance price and stiffness.

When we say that a plastic is flexible what we literally mean is that its polymer chains can be easily slid over each other without causing permanent damage to the structure of the object. The three structure types that were introduced enough each have an effect on the flexibility. The Intramer structure has the greatest effect and the Intermollecular structure has the smallest effect. Let's examine them one by one...

Intramer structure: This is mostly determined by the atoms that make up the Mers. All atoms within molecules are joined to the other atoms by strong covalent bonds. Depending on the type of atom there can be single bonds or double bonds (in other cases I'm pretty sure there can be more as well). Bulky side groups can also form, an example of this can be seen below. Along with double bonds they are fantastic at making a polymer more rigid. If you're having trouble picturing how those two things can limit a polymers flexibility, Imagine trying to slide a rigid and spiky surface across another rigid and spiky surface. Yeah.

Intramollecular Structure: Different types of plastics form different lengths of polymer chains. And in this case, size matters. Imagine getting a whole bunch of necklace strings as big as your little finger and stirring them around in a bucket. Now Imagine getting another bucket, filling it with necklace strings a meter long and giving that a super long stir. Which bucket of necklaces is going to be the easiest to detangle? It's clear to see that when the average length of the polymer string goes up (and hence the molecular weight) the stiffness of the material will also increase.

Intermollecular Structure: It's all about the Van Der Wal bonds!! When in liquid form, the polymer strings are able to move around like wriggling worms. By really really quickly cooling the polymer down (or quenching it) you can freeze the wriggling worms mid wriggle. But if you cool the polymer down slowly then the worms will line up to form a crystalline structure which looks a little bit like zebra stripes. Take a look at the difference between these two structures...

Look how much more dense the polymer strings in the crystalline structure are! Because of this density the Van Der Waal bonds (which act between two polymer strings) will have more of an effect, they're a little bit like magnets because they pull more strongly when they are closer to another polymer string. And hence the material is also made stiffer by allowing the plastic to slow cool!!

Crucible Badges, Eigenvalues, Eigenvectors, Diagonalisation and Pretty Hair

Mages in Space! (who do their hair nicely)

So its pretty clear for those of you who have spent enough time with me, that I adore being a girl. Mostly because makeup and clothing are some of the coolest things on the planet and it's more socially acceptable for girls to enjoy them (but let's not open up that can of worms right now). I'm unashamed to say that this is one of the biggest reasons that I love things such as Larping and Theatre. All the dressing up!!

Today I am going to a mages in space larp and I have excitedly done my hair differently. Its pretty simple but I like it looots :D I french platted it back (always a struggle because I am way better at inverse braids) and then created a braid bun and put the flower in the middle of it all. Uber cool and took me 3 minutes to do.

To complete with my outfit I have brought my dirty overalls in the hopes of being some snobby first ship engineer. Its kind of gross that they are still grubby with car grease but it certainly adds to the outfit. 

Crucible Badges!

So special snowflakes and Darkest Past players better watch out because I have the following badges! Brace yourselves! 

Eigenvectors and Eigenvalues - The Sliced Bread of Mathematics 

Eigenvectors and Eigenvalues are simply a property of any matrix which some clever cookie discovered were very useful in real life applications. You may not know them yet but these bad boys have improved your life. They were used to design the suspension springs on your car, they could have been used to design the foundations of the building you are sitting in, they even helped you out that one time on Facebook where you posted some photos and Facebook correctly guessed which of your friends were in the picture. And best of all, they're pretty simple to calculate. I'm introducing them now for your benefit because I will probably be talking about them later on. I'll explain them then I encourage you to give my example a try! 

The Matrix (the real deal not the movie)

For those of you who aren't familiar with matrices (the rest of you can skip to the next paragraph) they are basically just a part of a set of linear equations. In this case because there are two rows, there are two equations which might be -5a + 2b = 4 and 2a -2b = 5. Don't panic! All I've done is taken the first half of both the equations, stacked them on top of each other and then factorized out the a's and b's. (I'll show you visual learners that in a picture). Also from now on (because its easier I will be representing matrices using the following notation [-5, 2; 2, -2] because its easier. ; indicates that the following numbers are on a lower row and , indicates that the following number is on a different column)

Eigenvalues 

Alrightie, the first thing you have to do is recognize that the definition of an eigenvalue is det( I lambda - A) = 0 where lambda is a vector of your eigenvalues. I know that seems super duper weird and complex but I'll break it down for you.

"det" stands for the determinant of a matrix. Its just another useful property so don't freak out! A 2x2 matrix e.g [1, 2; 3, 4] can be found by multiplying the top right and bottom left elements of the matrix and then subtracting the top right element timed by the bottom left element. So for the example matrix I gave above this would be 1x4 - (2x3) = -2! Simple right. Good old primary multiplication and subtraction :)

"I" stands for an Identity matrix. This is one of the simplest matrices you will come across. Its just a matrix filled with zeros with 1's going down on a negative diagonal (\). for example a 2x2 identity matrix looks like this [1,0; 0,1].

"lambda" is just the two eigenvalues you want to find. Lambda is the name of a fancy Greek symbol but blogger doesn't know how to speak Greek so I can't write it in text but I can give you this link to good old wiki which has a picture of it http://en.wikipedia.org/wiki/Lambda

Lastly A is just the matrix that you are trying to find eigenvalues and eigenvectors for. Which in this case is our old friend [-5, 2; 2, -2].

So putting this together we have the following working...

Eigenvectors

Cool, cool cool cool. Now we have our eigenvalues we can use them to get the corresponding eigenvectors. Let's start with our eigenvalue -6.

• The formula to get your eigenvectors is AX = lambdaX. Once again A is your matrix, lambda is a single eigenvalue. X represents your eigenvector that you are going to work out which for a 2x2 matrix looks like [x1 ; x2]. 
• We expand out our equation to get 2 linear equations (can be seen in the working below this massive chunk of text).
  Now I know what you're thinking, by now you will already have tried to solve the two equations for x1 and x2 with no success right :P ?? If you have 2 unknowns and 2 equations you can solve something right?? WRONG! well, its right but in this case it doesn't apply. This is because of the way you worked out your eigenvalues using the A matrix. These two equations are actually the same equation with a couple of things thrown in. So if you try to solve it, the answer will always be zero. You need 2 distinct equations to solve something with 2 unknowns. But never fear the next step will explain all.
• Eigenvectors are special because they don't just have one solution. They are able to have any solution so long as the ratio of X1:X2 remains the same. All we have to do is guess a value for X1 or X2 and using only one equation, work out the other X with respect to that value. To make things easy I am going to guess that X1 =1. 

Give the other eigenvector a go using lambda = -1. See if you can get some scaled version of [1;2] :) YOU CAN DO IT!

Test Yourself! 

Try this matrix out for size [0, 1; -2, -3]. See if you can get to the eigenvalues -1 and -2. Then see if you can get some scaled version of the eigenvectors [1,-1] and [1,-2]. If you do it I will give you a hi five next time I see you.

Diagonalisation Time

So like many concepts in math, diagonalisation has a really complex looking definition. Luckily some smart cookies have come along and said "Hey, do you know what would make this a lot easier? Just *insert brilliant world changing idea here*". 

A matrix A can be diagonalisable if you can find a matrix which is able to be used to transform your matrix into a matrix of the same size where the only non zero values are the ones on the diagonal *takes deep breath*. Mathematically this looks like this:  

Luckily there are a number of very clever people who have figured out that...

• If the matrix A (having dimensions nxn) is diagonalisable, then there are n linearly independent eigenvectors of matrix A.
  In other words if you have a 2x2 matrix, if you can prove that there are 2 linearly independent eigenvectors then you have proved that the matrix is diagonalisable. 

This is AMAZINGLY AWESOME!! because we already know how to figure out both eigenvectors and whether two vectors are linearly independent. here is a link to my previous blog where I talk about linear independence 

Tidy Rooms, Thermosets, Thermoplastics Necklaces for Saturday's Party!

Good Afternoon Beautiful People!

Post number two of my blog and hopefully on the way to a habit. I really appreciated those of you who had a keen enough eye to tell me where I'd slipped up. You guys are awesome because as I said I am using these notes as a study resource so you may have saved my future self some confusion as I revise for all my exams at the end of the year. I hope people learned some stuff (or revised some stuff for all of you smarties who already know all about linear independence). Onto post number two...

Apparently My Carpet is Cream Coloured

Today I got my act together and tidied my room! Dad, if you ever read this blog here is photographic evidence that my floor still exists.

I love tidying my room almost as much as I love game of thrones (and I love game of thrones a lot!). These two passions of mine are slightly connected by the fact that I found game of thrones audio books on youtube... Speaking of game of thrones the coolest cool cat of all is coming over to watch season 3 with me tonight. Tessa, prepare yourself for the continuation of George RR Martin's soul crushing plot! 

Thermoset and Thermoplastic Polymer Challenge

So we're going to do a bit of a murder mystery minus the murder. Most of this is a bit of revision for me because it was pretty much all in my chemmat 101 class. But its still super fun to go over and relevant to my 747 paper. Below is a picture of me holding up two types of pastics  (and no I do not think metal is a plastic, I'm talking about the handle of the steamer pot). One of them is a thermoset and the other is a thermoplastic. See if you can guess which is which when I've finished explaining why they are different at a molecular level. Good luck Sherlock! 

Firstly, I wanted to introduce you to the concept of Mers (something most chem heads will know about). Mers are the building blocks of polymers. They are simply a  repeating chemical structures which are able to link together to form long chains. Think of them like beads on necklaces if you want. Then imagine that your chunk of polymer is a pile of beaded necklaces. And depending on how long the necklaces are and whether the beads are covered in Velcro you will get different properties from this pile of beads. You may be able to add a little bit of force to tear them back apart or else the pile may be so tangled and stuck that trying to separate them will only destroy all of your precious jewelry. Keep this in your head. You're doing awesome :)

Thermosets are made by pouring Resin (which is maybe a sort of plant secretion) into a mold with a catalyst, mixing it all up with maybe a little bit of heat and leaving it to cure or set! At a molecular level this results in a network of criss crossing polymers. A chemical reaction was what caused the plastic to set so there are strong covalent bonds or cross links joining the strings of polymers. You can never reverse this process. And if you try to heat the polymer up its only going to burn away leaving only a wonderful smell and some carbon behind. Imagine taking all your beads and putting them into a bucket and then squirting superglue into the bucket, swishing all the beads around and leaving it to dry. Are you ever going to reclaim your favorite string of beads for the party on Saturday? ONLY IF YOU DESTROY ALL YOUR NECKLACES IN THE PROCESS! 

Thermoplastics have got their own thing going on. Things made out of thermoplastic are formed by taking some itty bitty plastic pellets or solid stock, adding some heat and rolling them out and reshaping to get whatever you want. The strings of polymers are connected through secondary bonds or Van Der Wal bonds. These are much much weaker than covalent bonds and can be broken with a little bit of heat. This means that thermoplastics can be processed over and over and over again as long as you don't dump them in landfill. Lets look at your necklace pile again. This time you've stuck them in a bucket without super glue and given them a stir. Even though they will get mega tangled, you're still able to extract them using a little bit of time and finger power and wear them to the Party on Saturday. 

So have you got it? The milk bottle is recyclable and not as rigid as the handle. Also the handle is used in a high heat application making it a little inappropriate if it changed shape every time you steamed some veges. Bottle = Thermoplastic and handle = Thermoset!  Yay!

More on these later! But for now I have a lecture to go to and two classes to teach then GOT to watch :)

A Hello from Me, Linear Systems and Measure for Measure

Introductions

HELLO! THIS IS MY BLOG!

HERE IS A PHOTO OF ME LOOKING HAPPY WITH MYSELF FOR CREATING THIS BLOG!

I've tried making blogs before and blogged about my thoughts and crazy dreams but I thought that I would do something different. So here are my experiences as an amateur actress, a LARPer and a mechanical engineering student. I'm going to be using this as a bit of a study resource as well as a thought map so if you don't like learning things, best not read the sections with titles like "algebra rules for multivariable control systems" or "thermoplastics processing". So i'll get on with it :D

Another Play!

FANTASTIC news today! I've been shortlisted for a part in Measure for Measure (a play from the wonderful Shakespeare) at shoreside theatre. Recalls are next Tuesday and needless to say I am very very excited. The last time I was in a play was last year in December as a pantomime villain Dan Dini at Mairangi Players (picture top right) and the last time I was in the Shoreside Summer Shakespeare was when I was 15. I played William the simpleton in As You Like It (picture bottom right).

Measure for measure is actually a really interesting play about morality and different views on *cough* fornication. I'm currently making my way through the play with reference to the English interpretation (because Shakespeare is totally a foreign language right? :P ). Here's the link to the text with modern interpretation http://books.google.co.nz/books?id=Vg8wXH-2kvoC&printsec=frontcover#v=onepage&q&f=false

Jordan and Amy and Mark and Jackie have also been called back so needless to say I'm really really excited for Tuesday and the rest of the play!!

Linear Independence

Firstly, here's an introduction to matrices if you have no prior knowledge but really want to challenge yourself by reading on. http://www.purplemath.com/modules/matrices.htm

Basic Concept 

So another exciting thing that happened today is that I understand linear independence! Linear independence is a relationship between vectors (in control systems, these vectors are mostly within matrices). The basic concept is that if two vectors are linearly dependent they can be multiplied by a constant and summed together to cancel each other out. 

Visualization of Concept

A good way to visualize a vector is as a coordinate or an arrow. Here's a picture I drew for you!

Linearly Dependent Arrows

The red and the orange arrow are linearly dependent. Visually we can see this as they are both on the same straight line with the same absolute gradient (somewhat limited by my drawing skills in paint but you get the point). The orange vector can be multiplied by a constant of approximately 2 and added to the red vector to cancel out completely. A mathematical example of linear dependent vectors is [2,4] and [4,8] because 2*[2,4] + -1*[4,8] = [0,0]

Linearly Independent Arrows

My second drawing is of linearly independent vectors. They have different gradients so no matter how much you make the arrows grow or shrink (mathematically speaking, no matter what constants you multiply them by) they will never be able to be summed in order to cancel each other out. A mathematical example of linear independent vectors is [2,4] and [3,1]. the ratio of the first and second numbers is different in each of the vectors so it is impossible to times them by a constant in order for them to cancel each other out. No, you are not allowed to multiply them by zero in order to cancel them you sneaky sneaky thing. Also vectors do not count as constants so don't even think about multiplying one of the vectors by another vector.

Rank Limitations and Magic

The magical thing about vectors (and this is really awesome) is that when you have a matrix, you can never ever ever ever have a matrix with more linearly independent vectors than the minimum dimension of the matrix.

For example: if you have a 4x2 matrix [3,2;  4,5;  6,2:  3,1] you can never have more than 2 linearly independent vectors. Any conceivable vector can be created by forming a linear equation using one or more of the linearly independent vectors multiplied by constants. e.g [3,2] = a[4,5] + b[6,2]. We know it is possible to find out a and b because there are two constants and two unknowns!!! MAGICAL RIGHT??

Determining Rank (Full Rank?)

Determinant Method of Determining Rank

There are two nifty ways to figure out if a matrix is full rank. The first method is to find the determinant of the matrix and if it is not equal to zero, the matrix is full rank (det(A) /= 0 ---> A is full rank) This is best used on equations with an equal number of rows and columns. You can use it on 2xn matrices by going through and testing each column against the other columns for linear dependence. This is illustrated to the left. All combinations of vectors need not be tested. Some vectors can have an implied linear dependence. For example if the vectors in the green square have a determinant = 0 and the vectors in the blue square have a determinant = 0, it is implied that vector [a,b] is linearly dependent to vector [d,e] even though this vector combination was not formally tested. Cool right? 

(or not)

The second method of testing is more suited to matrices with unknowns and such or long 2xm matrices if you can't be bothered finding out ALL the determinants. You can simply look at individual vectors, and figure out if they can be multiplied by a constant to create another set of vectors within the matrix (this is only for the case where a vector is dependent on only one linear dependent vector but in trickier cases a vector can be dependent on two or more). If this is possible then only one of the vectors is counted as linearly independent. All the other vectors are clones if you will. They may have different hair cuts and clothing but each has the same DNA. By using this methodology and using implied dependence techniques a solution can be very quickly reached. Here is an example of a problem suited to this second method of doing things:

Topic For Business Plan 

In 703 we have to find an idea for an innovation to market and I have decided to do something a bit larpy. I want to make a large scale GM app for WOD games. Something that will hopefully make large scale combats a lot quicker and smoother to add to the immersion of the game. It allows players to be alerted when their initiative is reached and sends the success of their rolls directly to the GMs. It can be downloaded onto smart phones and bought from the WOD creators for a small fee. The plan is that it will not only make a profit (about 50c per download) but it will also help grow the game which is somewhat limited by how awkward combats can be. I'm pretty excited about this idea so LARPers watch out! I will be doing market research on you!!! :D

Goodbye!

I doubt anyone will read to the end but congrats if you have! If I am good I will post again tomorrow. Otherwise you may abuse me for clearly not doing my homework! CYA!