Reference / Paper · 2006
Workshop – Analog Computing
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Slide deck from the 'Analog Computing' workshop held at VCFE-2006 in Munich, presented by Bernd Ulmann. Covers hands-on analog computer programming through three worked examples of increasing complexity: a mass-spring-damper system, Volterra predator-prey (rabbit-lynx) coupled differential equations, and a bouncing ball in a box with elastic reflections, friction, and gravity. All examples were patched and run on real Telefunken RA741/RA742 hardware; circuit diagrams, scaling considerations, and oscilloscope output photos are included throughout.
- Manufacturer
- Telefunken
- System
- Telefunken RA741 / RA742
- Author
- Bernd Ulmann
- Year
- 2006
- Type
- Reference / Paper
- Language
- English
- Learning track
- introduction
- Pages
- 40
- Telefunken RA741 / RA742
- Telefunken
- analog computing
- differential equations
- simulation
- programming examples
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Workshop – Analog Computing
Workshop – Analog Computing
Bernd Ulmann
19-APR-2006
Commercial use prohibited.
Workshop – Analog Computing
19-APR-2006
ulmann@vaxman.de
http://www.vaxman.de
1
Structure of this workshop
The following slides are part of the workshop ”Analog Computing” which was held at
the VCFE-2006 in Munich.
This workshop gives an introduction to the art of analog computing by outlining
some examples ranging from a mass-spring-damper system to a bouncing ball in a
box.
All examples have been programmed and executed on real analog computers like
the Telefunken RA742, etc.
Workshop – Analog Computing
19-APR-2006
ulmann@vaxman.de
http://www.vaxman.de
2
Simulating a mass-spring-damper system
The first example shows how to simulate the behaviour of a rather simple
mechanical system consisting of a mass, a spring and a damper. The basic
elements of this system are shown below with the mathematical representation of
the forces belonging to each:
'$
&%
Fm = ma = mÿ
Workshop – Analog Computing
19-APR-2006
........
........
HH
HH
...
.........
.....
Fs = sy
ulmann@vaxman.de
Fd = dẏ
http://www.vaxman.de
3
Putting all elements together
Connecting these three elements together yields (thanks to nature and Newton) the
following setup and equations:
.........
..
H
Fm + Fd + Fs = 0
H
.........
..
Workshop – Analog Computing
19-APR-2006
ulmann@vaxman.de
mÿ + dẏ + sy = 0
http://www.vaxman.de
4
Rearranging the equation
To solve the equation
mÿ + dẏ + sy = 0
on an analog computer it is rearranged in a way that yields the highest derivative of
y on the left hand side:
1
ÿ = − (dẏ + sy) .
m
For setting up the computer assume that ÿ is known and generate the remaining
terms incorporating lower derivatives of y by successive integration, multiplication
and summing of terms.
Note that each summer and each integrator will change the sign!
Workshop – Analog Computing
19-APR-2006
ulmann@vaxman.de
http://www.vaxman.de
5
Generating −ẏ
Assuming that ÿ is known, its next lower derivative, −ẏ , can be generated by using
an integrator. The initial condition input of this integrator is used to set the initial
value ẏ0 as shown in the picture below:
ÿ
Workshop – Analog Computing
19-APR-2006
ny˙0
@ −ẏ
@
ulmann@vaxman.de
http://www.vaxman.de
6
Generating Fs = sy
In the following step the force Fs exerted by the spring will be generated:
ÿ
Workshop – Analog Computing
19-APR-2006
ny˙0
@ −ẏ
@
ulmann@vaxman.de
n−y0
@ y
@
s
n
http://www.vaxman.de
7
Generating Fd and the sum of forces
The force generated by the damper, Fd , can be generated accordingly using the
already knwon value −ẏ . The setup shown below then creates the sum of Fs and
Fd with a negative sign:
ÿ
ny˙0
@ −ẏ
@
n−y0
@ y
@
@
@
Workshop – Analog Computing
19-APR-2006
ulmann@vaxman.de
ẏ
s
n
@ − (dẏ + sy)
@
d
n
http://www.vaxman.de
8
Closing the loop
1
The sum − (Fs + Fd ) can now be multiplied by the constant m
yielding ÿ which is
exactly what we expected at the input of the circuit. So closing the loop will result in
a computer setup solving the initial differential equation readily:
ÿ
ny˙0
@ −ẏ
@
n−y0
@ y
@
− m1 (dẏ + sy)
Workshop – Analog Computing
19-APR-2006
n
ulmann@vaxman.de
n
@ − (dẏ + sy)
@
d
ẏ
@
@
s
n
1
m
http://www.vaxman.de
9
Setting up the computer
Workshop – Analog Computing
19-APR-2006
ulmann@vaxman.de
http://www.vaxman.de
10
Simulation run with s = 0.2 and d = 0.8
Here and in the following m = 1 is assumed.
Workshop – Analog Computing
19-APR-2006
ulmann@vaxman.de
http://www.vaxman.de
11
Simulation run with s = 0.4 and d = 0.8
Workshop – Analog Computing
19-APR-2006
ulmann@vaxman.de
http://www.vaxman.de
12
Simulation run with s = 0.6 and d = 0.8
Workshop – Analog Computing
19-APR-2006
ulmann@vaxman.de
http://www.vaxman.de
13
Simulation run with s = 0.8 and d = 0.6
Workshop – Analog Computing
19-APR-2006
ulmann@vaxman.de
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14
Simulation run with s = 0.8 and d = 1
Workshop – Analog Computing
19-APR-2006
ulmann@vaxman.de
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15
Solving two coupled differential equations
The following example is more complicated than the simple mass-spring-damper
system shown before. The goal is to simulate the changes in population numbers in
a two species ecosystem populated by rabbits r and lynxes l. Such a system is
readily described by Volterra’s differential equations:
ṙ = α1 r − α2 rl
l˙ = −β1 l + β2 rl
The parameters are as follows:
Workshop – Analog Computing
α1
Rabbit birth rate
α2
Rate of Rabbits killed by lynxes
β1
Lynx mortality rate
β2
Lynx population growth due to killed rabbits
19-APR-2006
ulmann@vaxman.de
http://www.vaxman.de
16
Partial circuit for ṙ = α1 r − α2 rl
First of all, let us solve ṙ = α1 r − α2 rl assuming that there is a value rl already
known. This leads to the following program:
nr0
+1
ṙ
@
@
AKA @@
A
nα1
−r
n rl
α2
−(−α1 r + α2 rl) = α1 r − α2 rl
Workshop – Analog Computing
19-APR-2006
ulmann@vaxman.de
http://www.vaxman.de
17
Partial circuit for l˙ = −β1 l + β2 rl
Next, let us solve l˙ = −β1 l + β2 rl – again under the assumption that there
already exists a term rl:
nl0
+1
l˙
@
@
nβ1
β1 l − β2 rl
O @@
...
...
...
...
...
−β1 l + β2 rl
Workshop – Analog Computing
19-APR-2006
ulmann@vaxman.de
@
@
−l
n rl
β2
http://www.vaxman.de
18
Partial circuit for l˙ = −β1 l + β2 rl
Obviously we can do better and save two summers:
n
rl
+1
nl0
β2
Workshop – Analog Computing
19-APR-2006
ulmann@vaxman.de
@
@
nβ1
−l
http://www.vaxman.de
19
Coupling both differential equations
+1
lr0
@
@
lα1
l
l
@
@
α2
@
@
+1
ll0
@
@
Workshop – Analog Computing
19-APR-2006
β1
@
@
........
.........
-r
-l
@
@
lβ2
ulmann@vaxman.de
http://www.vaxman.de
20
Scaling the equations
Due to the finite range of values which can be processed by an analog computer, it
is necessary to scale the equations to be solved in order to avoid overloading the
operational amplifiers and thus introducing erroneous terms.
Coupled differential equations like the example above are normally quite difficult to
scale since it is challenging to estimate maximum values for the variables.
If a direct scaling is not possible (or if the programmer is too lazy which may be the
case much more often) it is possible to run the calculation with a guessed scaling
and check for overloads. Then use the values at the moment the overload occurred
to determine the next ”guess” and so on.
The values used for the following run were:
α1 = 0.17, α2 = 0.4, β1 = 0.1, β2 = 0.27, r0 = 0.2, l0 = 0.8 (quite
unrealistic number of initial lynxes to be honest).
Workshop – Analog Computing
19-APR-2006
ulmann@vaxman.de
http://www.vaxman.de
21
The completed program
The following picture shows the program as patched for a Telefunken RA741 analog
computer:
Workshop – Analog Computing
19-APR-2006
ulmann@vaxman.de
http://www.vaxman.de
22
The overall setup
The next picture shows the overall setup featuring a two channel storage
oscilloscope:
Workshop – Analog Computing
19-APR-2006
ulmann@vaxman.de
http://www.vaxman.de
23
Running the simulation
The picture below shows the results of the running simulation:
Workshop – Analog Computing
19-APR-2006
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24
Simulating a ball in a box
The following example is yet a bit more complicated – the simulation of a ball
bouncing in a box (cf. [1]) as shown below:
1
v0
y
(x, y)
−1
1
x
−1
Workshop – Analog Computing
19-APR-2006
ulmann@vaxman.de
http://www.vaxman.de
25
Overall setup of the simulation
The ball is thrown into the left upper corner of the box with an initial velocity of v0 .
Whenever it hits a wall of the box it will be reflected elastically.
The ball is influenced by a gravitational force pointing downwards and it looses
energy by air friction (which is assumed to be proportional to the speed of the ball).
The simulation setup consists of essentially four parts:
1. A (sin(ωt), cos(ωt))-generator to create a real ball instead of a single moving
point,
2. a circuit to generate the y -component of the ball’s movement in the box,
3. a circuit to generate the x-component and, finally,
4. a summing circuit to overlay these signals in a proper manner.
Workshop – Analog Computing
19-APR-2006
ulmann@vaxman.de
http://www.vaxman.de
26
Generating the ball itself
This is the easiest part of the simulation. A simple sine/cosine-generator with a
rather high output frequency is necessary to create the impression of a ball (circle).
These two values are generated by solving the well known differential equation
ÿ = −αy as shown below:
HH
H
k
=
100
0.02
H
H0H
S
+1
0.5
k0 = 100
1 HH
H
S
0.1
0.5
?
−rx sin (ωt)
Workshop – Analog Computing
19-APR-2006
ulmann@vaxman.de
....
.... .......
. ....
....
....
.
...
−rx cos (ωt)
0.1
10V
http://www.vaxman.de
27
Generating the ball itself
At the heart of this circuit is the simple sine/cosine-generator constisting of two
integrators and a summer.
The first thing to note is that the summing junctions of the integrators are used as
the main inputs, thus allowing the use of variable input resistances by means of
coefficient potentiometers. This is necessary to obtain the desired high output
frequency (large ω ).
The feedback path from the summer output to the 1-input of the rightmost integrator
is used to ”heat up” the oscillation avoiding excessive decay.
The two Zener-diodes are used to avoid overloading the integrator. They will clip the
output signal once it reaches one machine unit. This, indeed, will result in a
distorted output signal but this distortion is negligible for this application.
Workshop – Analog Computing
19-APR-2006
ulmann@vaxman.de
http://www.vaxman.de
28
Calculating the y -position
The next step towards a complete simulation is the calculation of the y -position of
the bouncing ball. Drawing y(t) with t as the free variable results in a graph as
shown below:
y(t)
t
Workshop – Analog Computing
19-APR-2006
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29
Calculating y(t)
Three terms constitute ÿ : The (constant) gravitation, the damping proportional to ẏ
and the elastic rebound when the ball hits the floor (y < −1) or the ceiling of the
box:
ÿ = −g + dẏ
+ c (|y| + 1) if y < −1
m
c
−m
(y − 1) if y > 1
From ÿ the velocity ẏ and position y can be easily derived:
ẏ
=
ZT
ÿ dt + ẏ0
0
y
=
ZT
ẏ dt + y0
0
Workshop – Analog Computing
19-APR-2006
ulmann@vaxman.de
http://www.vaxman.de
30
Computer setup to calculate y(t)
-1
0.1 ẏ0
g
+1
+1 -1
0.27
b k0 = 10
bb
S
"
"
"
0.03
0.5
Workshop – Analog Computing
19-APR-2006
ulmann@vaxman.de
0.52
+1
y0
−y
b k0 = 10 b
bb
bb
"
"
"
"
"
"
....
....
....
. .......
.
.
....
..
..
....
10V
http://www.vaxman.de
31
Tricks
There are some tricks used in this computer setup:
1. The condition of hitting the floor or the ceiling of the surrounding box is detected
by the two 10V-Zener-diodes instead of a classical backlash setup. This has the
disadvantage that box heights different from ±1 are not covered as would be
possible by using a backlash. The advantage is that two backlashes would
require two amplifiers, two potentiometers and four diodes which are saved this
way.
2. The slower the ball gets, the smaller the acceleration of the elastic rebounds will
be. This is a bit unrealistic and will be partly compensated for by using the
summing junction of the first integrator as the input from the simplified backlash
instead of using a weighted input.
Workshop – Analog Computing
19-APR-2006
ulmann@vaxman.de
http://www.vaxman.de
32
Calculating the x-position
The calculation of x(t) assumes that the velocity diminishes with time t, eventually
reaching zero (at this point the computer should enter the halt or initial condition
mode).
x(t)
+1
t
−1
Changing the direction of the ball when it hits the left or right wall is a bit tricky as the
following computer setup will show.
Workshop – Analog Computing
19-APR-2006
ulmann@vaxman.de
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33
Computer setup to calculate x(t)
m
0.05
+1
Workshop – Analog Computing
HH k0 = 1
H
19-APR-2006
+1
X
@
-1
HH
H
ulmann@vaxman.de
-1
@
m
0.7
http://www.vaxman.de
-1
HHk0 = 10
−x
H
34
Putting everything together
Now having calculated y(t) as well as x(t) all that is left to do is to superimpose
those values with the (sin (ωt) , cos (ωt))-pair generated previously to display a
real ball at a particular position vector (x(t), y(t)):
−y(t)
−ry cos(ωt)
−x(t)
−rx sin(ωt)
Workshop – Analog Computing
19-APR-2006
HH
HH
-y
HH
HH
-x
ulmann@vaxman.de
http://www.vaxman.de
35
The final computer setup
Workshop – Analog Computing
19-APR-2006
ulmann@vaxman.de
http://www.vaxman.de
36
Bouncing ball
Workshop – Analog Computing
19-APR-2006
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37
The end
I hope you enjoyed the lecture and maybe you got a bit of the feeling of thinking the
”analog way” as Dr. Giloi once said.
Analog computers are more than just fascinating relics – they are the last
reminescences of a wonderful technology and (way more important) they teach one
to think in a way completely different from the way one with a background in digital
processing is trained to follow.
Thinking the analog way will result in solutions which might never have been thought
of in a conventional digital environment.
Thank you for your interest and your patience.
Workshop – Analog Computing
19-APR-2006
ulmann@vaxman.de
http://www.vaxman.de
38
Help
As you may have noticed, analog computing is my passion! Therefore I would like to
ask you for help:
• I am trying to save analog computers from scrap whereever I can. If you happen
to know about a system looking for a good home, please let me know. (I am not
afraid of large systems and I would really love to get an RA770 or an RA800(H),
for example! I will pay for all expenses to save the machine from scrap and I will
arrange shipping, etc.)
• I am interested in trivia, documentation, computer setups, sales brochures, etc.
• Please spread the word and help to save these machines from getting lost and
forgotten.
You can reach me always at ulmann@vaxman.de or by mobile phone at
0177/5633531 (in Germany) in case of an emergency. Thank you very much!
Workshop – Analog Computing
19-APR-2006
ulmann@vaxman.de
http://www.vaxman.de
39
References
[1] AEG Telefunken, ”Demonstrationsbeispiel Nr.5, Ball im Kasten”.
Workshop – Analog Computing
19-APR-2006
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40