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(Drawn with a Philips PM 8043 X-Y Recorder)
Sunday, April 20, 2014
Friday, April 18, 2014
The Butterfly Effect
Integrator rack completed. Banana plugs purchased and leads made up.
And now for a first attempt at a serious bit of programming. Following in the footsteps of many others suggests solving the Lorenz equations...three coupled, first-order, nonlinear differential equations which describe the trajectory of a particle through time, originally derived by Edward N. Lorenz (1963) as a model of atmospheric convection (there's even part of MIT named after him). (It's either that or 'hello world'!)
The equations describe the evolution of the spatial variables x, y, and z, given the parameters σ, β, and ρ, through the specification of the time-derivatives of the spatial variables:
dx/dt = σ(y − x)
dy/dt = x(ρ − z) − y
dz/dt = xy − βz .
Now, given I haven't built the potentiometer rack yet (it's next on my list), I am a bit limited as to parameter choice, and I can't readily use the values Mr Lorenz used (σ = 10, β = 8/3 and ρ = 28). After some thought, I've opted for σ = 5, β = 1, and ρ = 7. These I can effect without the potentiometer rack: σ and β are integrator/summer input multiplicands and and ρ = 7 volts (which I can cook up from 2 x 3 V plus 1 V). Also, these values give a solution which (I thought) lies within +/- 10 volts, and also it's a reasonably interesting solution.
To get (minus) x we just integrate the first equation and so on. Two multipliers are used to create the xy and x(ρ − z) parts respectively, and all the bits are added together as required. Here's a sketch of my thinking:
...and here's the plugged up integrator rack:
.JPG)
which yields the plot, below right:
The left hand plot is taken from a very useful online resource written by Andrew Baxter:
http://highfellow.github.io/lorenz-attractor/attractor.html
for the above parameter values, the squares are 2.5 units (i.e. volts) for the left hand plot, and the oscilloscope is set at 2 V per division.
I think the discrepancy on the right side of output from the analog computer is because the thing went outside the -10 V limit on summing amplifier number 2 (this cooks up the xy − βz term). Clearly I have yet to get to grips with the mysterious business of scaling problems on the analog computer.
And now for a first attempt at a serious bit of programming. Following in the footsteps of many others suggests solving the Lorenz equations...three coupled, first-order, nonlinear differential equations which describe the trajectory of a particle through time, originally derived by Edward N. Lorenz (1963) as a model of atmospheric convection (there's even part of MIT named after him). (It's either that or 'hello world'!)
The equations describe the evolution of the spatial variables x, y, and z, given the parameters σ, β, and ρ, through the specification of the time-derivatives of the spatial variables:
dx/dt = σ(y − x)
dy/dt = x(ρ − z) − y
dz/dt = xy − βz .
Now, given I haven't built the potentiometer rack yet (it's next on my list), I am a bit limited as to parameter choice, and I can't readily use the values Mr Lorenz used (σ = 10, β = 8/3 and ρ = 28). After some thought, I've opted for σ = 5, β = 1, and ρ = 7. These I can effect without the potentiometer rack: σ and β are integrator/summer input multiplicands and and ρ = 7 volts (which I can cook up from 2 x 3 V plus 1 V). Also, these values give a solution which (I thought) lies within +/- 10 volts, and also it's a reasonably interesting solution.
To get (minus) x we just integrate the first equation and so on. Two multipliers are used to create the xy and x(ρ − z) parts respectively, and all the bits are added together as required. Here's a sketch of my thinking:
...and here's the plugged up integrator rack:
which yields the plot, below right:
The left hand plot is taken from a very useful online resource written by Andrew Baxter:
http://highfellow.github.io/lorenz-attractor/attractor.html
for the above parameter values, the squares are 2.5 units (i.e. volts) for the left hand plot, and the oscilloscope is set at 2 V per division.
I think the discrepancy on the right side of output from the analog computer is because the thing went outside the -10 V limit on summing amplifier number 2 (this cooks up the xy − βz term). Clearly I have yet to get to grips with the mysterious business of scaling problems on the analog computer.
Sunday, April 6, 2014
Hooking up
In the midst of a lot of wiring...
.JPG)
...but eventually in a position to plug up my first program on the new front panel. This is Programming Example 8.1 from Ulmann's book Analog Computing (this is the example I used to test the integrator/summer boards back in October)...two integrators (left hand side) and one inverter (right hand side). It worked.
It really is beginning to look like an analogue computer!
.JPG)
Next step is to hook up the final four integrators (top left of panel)...trouble is that each integrator has a three way connector (null potentiometer), a nine way connector (toggle switch - selects INT or SUM mode), twelve banana jack connections plus three LED connections..that's a lot of soldering!
...but eventually in a position to plug up my first program on the new front panel. This is Programming Example 8.1 from Ulmann's book Analog Computing (this is the example I used to test the integrator/summer boards back in October)...two integrators (left hand side) and one inverter (right hand side). It worked.
It really is beginning to look like an analogue computer!
Next step is to hook up the final four integrators (top left of panel)...trouble is that each integrator has a three way connector (null potentiometer), a nine way connector (toggle switch - selects INT or SUM mode), twelve banana jack connections plus three LED connections..that's a lot of soldering!
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