# Tag Archives: Learning

## Dynamic Arches…

Tinkering about with SystemModeler a little further, I’ve managed to finally build a sprung arch, complete with dampers on the revolute joints.  I’m intending on using this principle in my research to create folded structures, so it’s interesting to see what effect the spring stiffness will have on the behaviour of the arch during the unpacking process – specifically looking at the accelerations on the masses at key points.

The thing that I was struggling with was creating a structure that had a set of equations that could be solved, the key concept I was initially missing was the closing of the structure with the special type of revolute joint to complete the chain.  Without this special revolute chain the equations are essentially unsolvable, so it’s important that one of these joints sits in the system somewhere.

Another concept is that the structure in the video has 3 straight segments, each 1m long; but the supports are only 2m apart…. forcing the arch to pop into a stable shape that balances the weights at each of the joints.  This is essentially what makes the arch wobble when solving the initial set of equations.  Next step is applying external forces and measurement points along the structure for displacement etc…

Teaching

## Mentoring…

Ever since I graduated I’ve always wanted to learn new things and for that reason I’ve pretty much always had at least three different types books on the go at any given time.  Typically I have a fiction book to read when my brain is really tired, a technical book linked to engineering, maths, or programming, and some sort of self-improvement book.   I want to be the best I can be, to reach my full potential and not be limited by my lack of understanding.

Over the past few years I’ve been spending quite a bit of time thinking about how mentoring (and what I believe to be the lack of) is affecting the environment for modern graduates in industry.  As margins become tighter and tighter in industry, the experienced engineers have less and less wriggle room to spend mentoring graduates.  Instead the experienced engineers merely push the less-experienced engineers to a quick solution so that deadlines are met and fee structures are not blown – resulting in a shallow learning experience.

I’ve always tried to make time to help less experienced engineers understand new concepts, even if I think they should know them already from University.  This has meant frequently giving up my dinner hour and time after work to help someone understand a new concept.  I’ve not always been successful and that’s a limitation of my skill and something that I’m getting better at with time as I improve as a mentor, but as time goes by I wonder if my approach is mentoring or coaching?

Most of the time I spend on a 1 to 1 basis with students takes the form of conversations and specific questioning to change the way they think about things.  To change or reinforce their perceptions about how something can be improved in the future.  This is the commonest form of mentoring that I’ve taken over the years, but through what I’ve read of Starr (2010) this approach isn’t mentoring at all, it’s more like coaching.  It’s essentially a series of conversations to help change a future outcome.  One of the favourite activities I teach is the supervision of dissertations, this is the module I can see the biggest leap in a student’s abilities, particularly with regards critical thinking and it presents the biggest opportunity that I get to mentor and build a connection with the students.  It’s the one area that probably sucks a disproportionate amount of time out of my week but it’s definitely where my mojo (Goldsmith, 2010, p17) lies when I’m teaching.

It’s this connection with the students and their topics I think that I enjoy the most, especially as I watch them grow and I still get emails and calls from some of the students that I supervised a few years ago simply so they can let me know what they’re up to.  Of course quite a few students only get in touch when they’re chasing a reference or contacting me because they want something from me but when someone drops you a note to say hi or let you know what they’re up to now without wanting anything from you it’s personally very rewarding and it’s this type of contact that I treasure as it’s then that you know you’ve made a real impact on someone’s perspective on life.

Dissertations are also one of the key activities that is keeping me in academia, as it’s giving me an opportunity to continually learn and grow on a personal level although the longer I stay in academia the lower my career progression opportunities and the lower my earning potential become…. Perhaps Robert Greene (2012) is right, when looking back on my life nobody will remember the wonderful report I sacrificed weekends and evenings to complete, but perhaps they’ll remember the time I gave them to help improve their critical thinking skills and understanding of structural behaviour and I know my kids will hopefully appreciate the time we have together now I’m no longer being continually shipped out all over the place to design buildings.

After all I’ve changed quite a few skylines around the world when I had a proper job, maybe there’s more satisfaction to be had creating brilliant engineers to create even more radical designs.  Or perhaps there’s a compromise to be had by spending some of my time teaching and another chunk of my time working with brilliant engineers and architects to design really radical and life-changing designs… I think this could be where my future lies in all honesty, it’s about time I seized the steering wheel again as these things don’t happen by themselves.

References.

Goldsmith, M. (2010). Mojo. London: Profile Books.

Greene, R. (2012). Mastery. London: Profile Books.

Starr, J. (2010). The Coaching Manual (Third ed.). London: Pearson Business.

## Critical thinking…

One of the biggest challenges that I face as a lecturer is helping the students to develop their critical thinking skills, especially with regards problem solving as this is one of the key skills an engineer should possess.

It’s a fundamental skill for a practicing engineer to not only problem solve, but to critically review the problem and reconstruct it so that only the important parts are focused upon.  Essentially honing in on the nub of the problem.

I’m a fan of using stories in lectures and I used to work with an Italian engineer who unfortunately died last year and he had a plethora of stories and anecdotes that he paraded out whenever the occasion arose.  One of his numerous stories that sticks in my mind and that I think I’ll start using in my lectures about critical thinking involves three construction professionals on a train ride to Scotland for the first time.

There is an Architect, a Project Manager, and an Engineer on a train together and as they cross the border into Scotland for the first time on a train racing through the countryside they spot a brown cow in a field.

The Architect, being the first to spot the cow, declares that clearly all Scottish cows are brown, the Project Manager who feels that he clearly has the organisational skills to rationalise this observation and thinks that the architect hasn’t been critical enough, decides to correct the Architect and notes that in reality what the Architect has actually observed is that in Scotland there is a field which has brown cows.

The Engineer meanwhile has been listening intently to this discussion and decides to correct them both by sharing his observation, that in fact there is in Scotland, a field, that contains a cow, one side of which is brown.

General

## 10,000…

I’ve been doing my PhD for a while now and in all honesty it goes through fits and starts with regards its progress, but on the whole I think that it’s getting there more or less.  I follow quite a few different people of on Twitter who are on a similar journey and often take a quick run through the #phdchat channel although because of a commitment on a Weds night with the British Red Cross I can never join the conversation.

I’ve seen a few comments over the past few months that a PhD is about 10% inspiration and 90% perspiration… but it surely can’t be that straightforward, can it?  Since I graduated all those years ago I’ve usually kept a trio of books on the go at any one time, each one from a different genre: fiction, self improvement, and a technical book.  The self improvement book that I’m reading at the minute is called ‘Outliers’ and is all about those over achievers that sit far and beyond the datum of us mere mortals, but interestingly there are a few things in the book that have really got me thinking about this 90:10 statement.

One of the points raised in the book is that on average it takes about 10,000 hours to get really good at something, whether it’s playing the piano, ice hockey, or other activities.  This got me thinking about how much effort it is going to take me to finish my PhD, given that it takes about 4 years on average for a student to complete their PhD and most students work insanely hard at the end which I guess makes for a 50 hour week averaged out over the duration (including time for noodling stuff over and reflection) and that most people take a couple of weeks off a year for a holiday this gives a rough idea of how long it should take a typical PhD student.  After all gaining your PhD is essentially demonstrating that you know an awful lot about a very focused topic.

$\Large \underbrace {50}_{Hours/Week} \times \underbrace {50}_{Weeks} \times \underbrace 4_{Years} = 10,000{\text{ }}hours$

So the amount of effort seems to match pretty neatly on my guesstimated figures, but what about the 10% inspiration part?  It’s argued by Gladwell that once a persons IQ is over a certain point, say 130 or so then they’re deemed to be ‘capable enough’ to be a contended for a Nobel prize or a reasonable University education, in fact just as many people win with an IQ of 130 or so as compared to the ultra intelligent folk who have IQ’s of 200+.  Now clearly if you have a huge IQ, then the chances are you going to find it easier to grasp particle physics than say someone with a lower IQ, but this doesn’t necessarily mean that you’re going to come up with that 10% inspiration easier.  Now I’m never going to aim as high as a Nobel prize, but certainly completing my PhD would be great!

As part of my PGCAP course I was interested to see how different people learned and one thing that I thought was interesting was a test for multiple intelligences as I really connected with the idea that different people will excel at different types of activities, but how can you argue that a musical genius is any less intelligent than a physics genius? Surely they have similar genius qualities, but they’re subtly different… My results for the multiple intelligences test are below which show I’m spread over a few different strands, but clearly I can barely hum a decent tune.  If you would like to see how you’re intelligences are distributed then you could take the test here.

Personally I think that the 10% inspiration part is going to be easier for the sort of person who likes to think of new uses for existing things, or indeed someone who can noodle over and think creatively and abstractly and I’m remaining hopeful that my experience designing buildings and other structures will be useful in thinking creatively on my PhD.

There is an interesting test that can be taken called the divergence test, this test asks candidates to think of different uses for a common everyday item such as a brick for example.  Creative sorts should be able to come up with all sorts of examples that are beyond the every day uses of this item… some people will list that they could build a house with it and a BBQ and then run out of ideas, a creative sort would be able to list all sorts of madcap ideas from weighing down the corners of your duvet, to using it in a smash and grab, to leaving a car supported whilst you steal the wheels.  I tend to fare pretty well on these sorts of tests and I’m one of the few that’s still writing ideas down as the time runs out and I’m hopeful that it’s this creative thinking that is going to help me draw upon the 10% inspiration part of my PhD.  The downside is though that I frequently go off on madcap related and unrelated tangents whilst I’m in this kind of thought process, so the biggest risk for me completing my PhD will be focusing on the task in hand I think.

References:

Gladwell, M. (2009). Outliers: The story of success. London: Penguin.

Teaching

## Golden ratio…

A quick link through to my first ever screen case written for a Maths MOOC that’s hopefully going live soon… this was much more hard work than I gave it credit for when I agreed to write it.

The slides are available from below as a slideshare link.

## Cable-Chains…

Part of the work that I’ve been undertaking on Mathematica is to create a series of sheets that will calculate the geometry of a cable-chain arch with a set of given parameters.  In part I’m interested in how the cable-chain arch can behave as a deployable structure and looking to build on the work of (Li, Vu, & Richard, 2011) to see how practical a cable-chain structure can be made with regards economy, efficiency, and robustness.  Essentially a cable-chain structure is a curved arch that is sub-divided into straight sections, with cables spanning across the base of two struts as can be seen in the figure below.  Simple versions of these types of structures are widely used for the likes of temporary and deployable aircraft hangers to create large open spans.

Now that I’ve got the makings of a simple Mathematica sheet up and running and I’ve taught myself some rudimentary programming and graphics manipulation skills I’ve managed to start to knock together what I feel are some high quality illustrations for my thesis.

I’ve done this with a mixture of Mathematica to create the base diagram, which I save as a PDF.  I then import the PDF into OmniGraffle to annotate the diagram and then export to a PNG file to maintain the transparent background, this figure I can then host for linking into blogs etc.  Below is a sample figure which shows how the number of segments (nSeg) affects the internal area available for habitation within a typical parabolic arch.

Given that both of my brothers are colour blind and I’ve never done the test, I’m not convinced on my choice of colour schemes, but the good news is that it won’t take long to change if it turns out I’ve made my figures look like something off the set of Austin Powers.

So far I’m finding OmniGraffle quite limited compared to Visio that I’ve been using for my diagrams for perhaps 20 years or so.  I decided to use OmniGraffle though as most of my writing work is done on a Mac, although I also have a PC so I can always create the more complex diagrams on Visio if need be, especially as I’ve managed to get a legitimate copy from work for £12.

I’d love to hear how other engineers and academics approach creating technical figures and sketches on their Macs though, I’ve a feeling that I’m really missing out on something and there’s got to be a much slicker workflow out there.

References:

Li, Y., Vu, K. K., & Richard, J. Y. (2011). Deployable Cable-Chain Structures: Morphology, Structural Response And Robustness Study. Journal for the International Association for Shell and Spatial Structures, 52(168), 83-96.

General

## Geometry…

I’ve posted a few times that I feel my mathematics skills have slowly atrophied having been out in industry and that this I feel will be my achilles heel when it comes through to my research.  One of the areas that I’ve been investigating is the relationship between cable-chain structures and their behaviour.

Depending on how many segments you divide the arc into will determine the geometry and efficiency of the cable-chain arch, now if it’s a semi-circle that’s not too hard to calculate the geometry using simple triangles.  In degrees the internal angle of each triangle will be given by the following formula, where nseg is the number of segments you’re dividing the arc by:-

$Internal{\rm{{ \,\,}}}Angle = \frac{{180}}{{{n_{seg}}}}$

This then can allow you to calculate the length of the segment along the arc as set out below using the Cosine rule which I haven’t used for probably about 20 years, this though is only half the problem as you still have to determine the length of the cables.

$Length = \sqrt {2{R^2} – 2{R^2}Cos\theta }$

$Length = \sqrt {2{R^2} – 2{R^2}Cos\left( {\frac{{180}}{{{n_{seg}}}}} \right)}$

Next to determine the length of the cable, you will need to combine two of the triangles previously calculated and which gives double the internal angle and repeat the process for the cable that is shown highlighted in red below.

$Length = \sqrt {2{R^2} – 2{R^2}Cos\left( {2\theta } \right)}$

$Length = \sqrt {2{R^2} – 2{R^2}Cos\left[ {2\left( {\frac{{180}}{{{n_{seg}}}}} \right)} \right]}$

This all seems fine and well for simple circles, but how would you know that a semi-circle gives the most efficient arrangement for a cable-chain?  What about if it’s a parabola?  Well fortunately for the mathematically out of practice determining the equation for a parabolic curve is pretty easy for us Mathematica users…  Let’s say I know I have a draped cable that spans 4100m horizontally between supports and has a 500m drape at the middle.  We can define three points on this cable easy enough as can be seen below:

This is where Mathematica starts to flex its muscles for me and why I feel I get good value from the student version that I’ve purchased.  I don’t know what the Mathematica code for determining a line through the points is, but if I press the = key twice I get access to the Wolfram Alpha Servers and try and describe using every day English what it is that I’m trying to do…  Following a blog post from Mathematica on determining arc lengths using calculus I tried the following term which doesn’t use any mathematical code at all, simply describes what it is that I’m trying to do:-

Parabola through the points (0,500), (2050,0), (4100,500)

Which gives the me the following output, which is pretty helpful to the maths novice who is trying to explore and understand the relationship between the data, the expression, and the graph…

Which makes my life a lot easier as I can start to tinker with geometry very quickly now using this approach.  One of the cool things about working this way is that you can then start to dissect the code and determine what the correct format should be when creating Mathematica sheets and it makes learning Mathematica much much easier than say MatLab or Maple.

Whilst creating code like this would not be difficult for someone with a little rudimentary programming experience, it isn’t that intuitive for a newb like myself, but now I can start to adapt this syntax to suit what I need…  The next step is to try and work out how to subdivide this into nseg number of equal length links… but that’s a job for another day.  Maybe I’ll need to start using polar geometry to make my life easier? Who knows… but at least I’m not being held to ransom by my elderly fuzzled brain anymore and I can explore and play with the maths rather than being pummelled by them.

I’d love to hear from other Mathematica users, especially to hear what their experiences are and particularly to hear if I’m missing a trick as I’m going through and learning the syntax…

General

## Patterns…

One of the challenges when researching in an engineering field is to determine what patterns your data may present you with.  This can be an unfamiliar skill for those of us that have come from industry as we’re used to dealing with certainty when designing buildings, not uncertainty.  One of things I’ve been messing about with lately is patterns and series of numbers as I’ve been learning Mathematica, using Roozbeh Hazrat’s book to help.

This has let me calculate the palindromic prime numbers less than 10,000 and other long winded sums in a single line of code.  It has had me thinking about other series of numbers such as the Fibonacci series which is present in nature and Pascal’s triangle… (below)  I’ve been seeking these types of patterns and puzzles out to try and sharpen my powers of observation and help with rebuilding my maths skills which have atrophied over the years.

$\begin{array}{c} {\rm{1}}\\ {\rm{1\, 1}}\\ {\rm{1\, 2\, 1}}\\ {\rm{1\, 3\, 3\, 1}}\\ {\rm{1\, 4\, 6\, 4\, 1}} \end{array}$

Pascal’s triangle is quite a simple pattern to determine, you simply work through the line above and it helps you create the next line below in the series.  Take the first number, it’s always 1, then add the first two numbers together on the line above to get your next number, rinse repeat to see what you get…

$\begin{array}{c} {\rm{1}}\\ {\rm{1 1}}\\ {\rm{1 }}\underbrace {{\rm{(1 + 1)}}}_2{\rm{ 1}} \end{array}$

In the example above when you add together the two numbers on the second row, you get the number 2…. I’ve highlighted this with a bracket on the example above to illustrate how you get the number 2 on the third line…  try working through this process to see if you can get the pattern above to repeat and see if you can follow the logic.

Now this is all fine and well, but Pascal’s triangle is well known and it isn’t going to win you a bet down the pub… the Aha! moment that goes with puzzles is the key to a successful puzzle (Badger, Sangwin, Ventura-Medina, & Thomas, 2012)  the simpler the explanation, the more readily the solution will be accepted (Michalewicz & Michalewicz, 2008).

So following the pattern theme of this post, let’s see what we can make of the following pattern and see if it can help sharpen up the old grey stuff… and if you crack the puzzle, maybe you can try it down your local and see if it earns you a free pint…

$\begin{array}{c} 1\\ 1{\rm{ 1}}\\ {\rm{2 1}}\\ {\rm{1 2 1 1}}\\ {\rm{1 1 1 2 2 1}} \end{array}$

What is the next line for this pyramid?  If you’re struggling and would like a hint, then try saying the series aloud.

The solution is 3 1 2 2 1 1 because there are THREE ONES, TWO TWOs, and ONE ONE on the line above…

If you’re struggling to solve this puzzle, then you can highlight the text from this line to the line above to reveal the answer.

References:

Badger, M., Sangwin, C. J., Ventura-Medina, E., & Thomas, C. R. (2012). A guide to puzzle-based learning in STEM subjects. Birmingham: University of Birmingham.

Michalewicz, Z., & Michalewicz, M. (2008). Puzzle-based learning: An introduction to critical thinking, mathematics, and problem solving. Melbourne: Hybrid Publishers.

## Mathematica…

One of the reasons that I started this blog was so that I could mess about with embedding some Mathematica files to help with testing out some ideas.  For this to make sense it’s easiest if I embed a few simple examples in this blog post.  Now if you want to interact with these examples, I’m afraid you’re going to have to download the Wolfram CDF player, which is completely free and works on PC’s and Mac’s alike.  Imagine it as a sort of PDF viewer but it lets you interact with the files as opposed to a PDF which is typically just a static and lifeless document.

Consider the following equation:

$Sin\left( {2x} \right)$

Most text books would draw the graph for this over whichever range they deemed to be suitable and then students would try and learn from these dull and boring diagrams.

[WolframCDF source=”https://dl.dropboxusercontent.com/u/22612196/Wordpress_Test5.cdf” width=”653″ height=”405″ altimage=”” altimagewidth=”https://dl.dropboxusercontent.com/u/22612196/Wordpress_Test5.png” altimageheight=””]

Now this is how I was taught maths and in fairness, it’s pretty dull and it’s difficult to gain any form of intuition as to how it might behave if the 2 became a 3 for example, this is where Mathematica’s CDF files come in handy because it has some nice tricks for letting you explore maths in an interactive fashion… let’s consider the following equation, from the previous graph most people wouldn’t really know how it would affect the graph.

$Sin\left( {a.{\rm{ }}x} \right)$

But if we crank this through Mathematica we can create a really nice interactive widget that can be shared with anyone for free!  As you change the slider, the graph updates in real time, and if you want to know what number you’re changing ‘a’ to be then simply click the little + sign next to the slider itself to expand the input values beneath it.  In fact if you think that messsing with sliders is far too much like hard work, then simply click the little play button in the top right and the widget will work the sliders for you… sit back and watch the pattern.

[WolframCDF source=”http://dl.dropbox.com/u/22612196/Wordpress_Test4.cdf” CDFwidth=”752″ CDFheight=”574″]

If you’re not familiar with Mathematica, you may be concerned that this sort of widget is really difficult to create, but actually I’m still on Chapter 3 on the text that I’m working through and the code is incredibly simple to create this kind of interactive learning tool and I’ve replicated it below to show how few lines of text can create this level of interaction.

Manipulate[ Plot[ Sin[a x], {x, -10, 10}], {a, 1, 5}]

Essentially this code starts with “I want a slider widget”, “Plot me a graph of Sin(a.x) over a range of values for x from -10 to 10”, then “make the slider vary a from 1 to 5”.

Now this seems ok, but the Manipulate command is actually incredibly powerful and with a little more twiddling, high quality interactive 3D plots can be created, so let’s consider the following expression.

$f{\rm{ }}Sin\left( x \right) + g{\rm{ }}Sin\left( y \right)$

This expression has four variables: f,g,x, and y.  Of course, I bet you’re dying to know what the graph looks like for this function so you can boost your maths skills…

[WolframCDF source=”http://dl.dropbox.com/u/22612196/Wordpress_Test3.cdf” CDFwidth=”752″ CDFheight=”758″]

This is where the CDF player starts to flex its muscles a little, not only can you mess around with the sliders to change the values of f and g… but you can click and rotate the 3D graph itself to get a better view of how you think it’s working.  For me this level of interaction is a real opportunity for playing with the maths to help build up a level of intuition and feeling of how the maths will behave.  And once again the code to get it to work is fairly straight forward even for a novice such as myself.

Manipulate[Plot3D[(f ) Sin[ x] + (g)  Sin [y], {x, 1, 10}, {y, 1, 10}], {f, -10, 10}, {g, -10, 10}]

Now here’s the rub, a full Mathematica licence is the best part of £1,000 for a lecturer to use, in these hard times that’s a lot of money.  But because I carry ‘dual’ status as I’m studying 2 degrees as well as working full time as a lecturer I was able to pick up a student licence for roughly £80.  Normally the cost for a student licence is a shade over £100 but it is possible to reduce the normal student price by 15% by using the discount code PD1637 at the Wolfram store checkout and I still retain the full functionality of sharing my CDF files via export.

I hope this helps someone, if you’ve any feedback on this post or would like to ask any questions, please get in touch or leave a comment below.