We all knew already how helpful LEGO Technic might be to demonstrate some mathematical properties.
Yet, with this infinitely simple construction we get also a demonstration from the chaotic behavior of the Double Pendulum.
In mathematics a double pendulum is a pendulum with another pendulum attached to its end, and is a simple physical system that exhibits rich dynamic behavior with strong sensitivity to the initial conditions.
The motion of a double pendulum is governed by a set of coupled ordinary differential equations. For small angles, the double pendulum behaves like a linear system. However for large angles or certain energies, the double pendulum is non-linear and its motion turns chaotic.
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8 comments:
what is your education background may i ask??
Engineering Physics (EP). Why?
There used to be a small toy that worked on that principle..
In the Electronics department of my old university (York, UK) there is a chaotic pendulum with a 'T' shape. The first pendulum has three arms of the 'T', with similar single-arm second pendulums on the ends of each arm. If you wind it up fast enough it goes on for 20 minutes or so. Not so easy to create a pendulum with low enough friction from LEGO parts though.
It is possible to make a LEGO chaotic system that includes electrical power inputs and motors, maybe with NXT feedback too, to overcome friction in the system. The use of a "-x^2" term in an equation in the NXT would invite chaos - that term is in the population curve and Mandelbrot chaotic equations. This is known in real control systems and such chaos often has to be damped out.
A Power Functions system could do it with motors controlling the handsets for each other's IR receivers, like this: http://www.brickshelf.com/cgi-bin/gallery.cgi?f=241795
It might be possible in pneumatics too. My pneumatic servo http://www.brickshelf.com/cgi-bin/gallery.cgi?f=405269 could be made chaotic, given more gain. Since it uses a non-regulated pressure and a compressible medium, the valve movement is in proportion to the acceleration of the cylinder movement, representing a double integration in the control loop. With a regulated pressure of a non-compressible fluid (hydraulics) it would be proportional to the velocity of the cylinder, a single integration. Both are capable of chaotic behaviour.
>>Engineering Physics (EP). Why?
I was just wondering, cause your blog has an interesting mix of engineering and maths, thanks!
I am not sure if I understand the point of this?
http://www.youtube.com/watch?v=C-rJm3E6wtM
;-)
Incredible! :)
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