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Oscur
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Hey all, I'm a little confused and just wondered if you could help me out with some chaos theory (don't worry, this isn't homework!). My lab partner and I (second-year undergrads) have been doing an experiment in the lab on chaos theory for an extended project and we started off with an LCR (LRVD technically) circuit where the capacitance varies depending on the driving frequency.
We got some lovely Lissajous figures (voltage across the signal generator against voltage across the diode) and neat bifurcation diagrams (instantaneous voltage across the diode against driving frequency), but then came the time to extend our investigation into something more complicated and we decided to try and build a physical system that showed chaotic behaviour. What we came up with was a forced, damped harmonic oscillator as shown in a diagram I'll attach soon.
Basically it's a suspended mass tied vertically to a drive wheel and damped by a pair of springs perpendicular to the direction of the driving force. I'll post a diagram at some point, but it's basically the same system described in The Physics of Vibrations and Waves (Pain) Chapter 11, and Chapter 5 of Classical Dynamics of Particles and Systems (Marion) We talked to our demonstrator, the lab technician and the lab coordinator, and after going through a few iterations of design and prototyping we ended up with a working system.
What's confusing at the moment is what variables to plot in our phase space diagrams for the mechanical system. In our lectures on non-linear dynamics, the lecturer said that for an oscillator, the appropriate coordinates are velocity of the oscillating mass and its position and he even mentioned the van der Pol oscillator that has the behaviour we're looking for:
[URL]http://upload.wikimedia.org/wikipedia/commons/a/a1/Limitcycle.jpg
But this seems really strange, as it doesn't involve the driving force in any way, unlike with the electronic version, when we had to plot driving voltage against "output" voltage... Can anyone help explain why this difference exists?
We got some lovely Lissajous figures (voltage across the signal generator against voltage across the diode) and neat bifurcation diagrams (instantaneous voltage across the diode against driving frequency), but then came the time to extend our investigation into something more complicated and we decided to try and build a physical system that showed chaotic behaviour. What we came up with was a forced, damped harmonic oscillator as shown in a diagram I'll attach soon.
Basically it's a suspended mass tied vertically to a drive wheel and damped by a pair of springs perpendicular to the direction of the driving force. I'll post a diagram at some point, but it's basically the same system described in The Physics of Vibrations and Waves (Pain) Chapter 11, and Chapter 5 of Classical Dynamics of Particles and Systems (Marion) We talked to our demonstrator, the lab technician and the lab coordinator, and after going through a few iterations of design and prototyping we ended up with a working system.
What's confusing at the moment is what variables to plot in our phase space diagrams for the mechanical system. In our lectures on non-linear dynamics, the lecturer said that for an oscillator, the appropriate coordinates are velocity of the oscillating mass and its position and he even mentioned the van der Pol oscillator that has the behaviour we're looking for:
[URL]http://upload.wikimedia.org/wikipedia/commons/a/a1/Limitcycle.jpg
But this seems really strange, as it doesn't involve the driving force in any way, unlike with the electronic version, when we had to plot driving voltage against "output" voltage... Can anyone help explain why this difference exists?
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