So that's why they are 'circular functions'...

It just occurred to me that if I connect the output (y) and its derivative (dy/dt) from the analog computer's solution of the second order DE (see previous post) to X and Y inputs of an oscilloscope, I should get a circle (with respect to time, t). Pretty obvious really...but pleasing nonetheless:

First Program - Solving a second order DE - part the second

Having set up Figure 8.2 from Ulmann's book (integrator capacitors 1 uF and resistors 1 M ohm), switched on the analog computer, pressed COMPUTE and it just worked! Sine and cosine clearly shown on the two meters (10 V peak to peak since the initial conditions were set to + 10 V for the first integrator and 0 V for the second integrator).



The time constant of the integrators was 1 second and so the expected time period of the sine /cosine wave is 2 time pi seconds = 6.28 seconds. And it was!

Pressing the FAST button speeds the integrators up by a factor of 1000 (i.e. the 1 uF capacitors get replaced by 1 nF capacitors). Obviously too fast for the meters, but on an oscilloscope the output looks like this:

What's interesting is that the output stays like this for several minutes - typically growing slowly in amplitude until it starts clipping (FAST mode); in the normal (6.28 second time period) mode the output tends to decay very slowly (i.e. several minutes). In both cases such longevity of the solution suggests very little leakage in those Russian military capacitors!

So there you have it - not Hello World, but the solution of a second order DE! It's early days, but I can see that programming an analog computer is completely different (and a lot more exciting) than programming a digital one! I like the way the analog computer is so direct - if you want to solve a DE then you solve a DE - not mess about with some numerical scheme which may or may not be stable.

A final thought for the moment: I feel I am entering an exclusive club of analog computer programmers :) And how cool is that?..

Next steps: finish off three more integrator boards; sort out hardware for the integrator 19 inch rack; make a proper front panel for the (now finished) control unit 19 inch rack.

First Program - Solving a second order DE - part the first

(1) Have bought a copy of the recently published  Analog Computing by Bernd Ulmann. Excellent book - I particularly like the extensive foot-noting. Essential reading!

(2) Have completed the first of the (eight) integrator boards - each of which has two integrator circuits.

Messy and not yet assembled into a 19 inch rack of integrators/summers.

Anyway, I've now got three integrator/summer units to play with - enough to solve a second order differential equation of the form d2y/dx2 + Ay = 0. This is the first programming example in Ulmann's book (page 126). 

As I say, all a bit messy, but my implementation of Figure 8.2 of Ulmann's book looks like this:

(The top board is a power supply/control board for the integrators; the middle board is my original single integrator/summer and the lower one the latest pair of integrators/summers.) As mentioned, all the integrators boards etc will end up in their own 19 inch 4U rack - I just wanted to test things out!

The boards are bussed up to the main control unit (which is pretty much completed) via a 25 way D connector, via at the moment, a 25 way break out board.