Damped oscillation

2021-12-08T12:21:27Z

MKoch: /* 2. Derivation of a computer circuit */ Possible error?

Damped oscillation

2021-12-08T12:06:39Z

MKoch: /* 2. Derivation of a computer circuit */ typo + that's unclear

Damped oscillation

2021-12-03T20:06:06Z

MKoch: correct small typo

The modeling of a '''damped oscillation''' is a good starting point for analog programming beginners. This article shall give a detailed explanation how to implement a simulation on '''The Analog Thing''' using the full repatriation method and to get results on an oscilloscope.

== Mathematical description of a damped oscillation ==
[[File:Damped oscillator.png|thumb]]
The first and often the hardest step of modeling any to be simulated system on an analog computer is to give an exact mathematical description of the system in the form of differential equations.

Find all acting forces, three in this case:
- Spring force Fs = k*x; k = spring coefficient, x = deflection
- Inertia force Fm = m*a; m = mass, a = acceleration
- Damper force Fd = d*v; d = damper coefficient, v = velocity
Set the sum of all forces to zero (definition of an isolated system):
Fm + Fd + Fs = 0
m*a + d*v + k*x = 0
Replace the velocity '''v''' with '''ẋ''' and the acceleration '''a''' with '''ẍ'''
v = ẋ (the velocity equals the first derivative of x)
a = ẍ (the acceleration equals the second derivative of x)
m*a + d*v + k*x
= m*ẍ + d*ẋ + k*x = 0
Now we have an exact description the damped oscillation in the form of a differential equation.

Lastly, solve this equation for the highest derivative the finish the full repatriation:
-> ẍ = -(d*ẋ + k*x) / m

Rotation matrix

2021-10-30T10:07:46Z

MKoch:

Only two dimensions x,y can be shown on an oscilloscope screen. If you have a circuit with three outputs x,y,z you can see only two of them. But with a rotation matrix you can rotate the x,y,z space so that you can see a two-dimensional image from an arbitrary direction.

A 24-step switch can be used for angle adjustment in 15° steps. These 4x24 switches are available on Aliexpress for less than 10€. Search for "4*24 Rotary Switch".

[[File:4x24_rotary_switch.jpg]]

The switch must be disassembled to remove the limiter, so that the axis can be rotated over the full 360° range.

[[File:limiter_in_switch.jpg]]

In those planes that make the sin() and -sin() signals, the contact must be removed from the A position and inserted in position B, which is 90° phase shifted from A. For the -sin() signal, the phase shift is in the other direction.

[[File:contact_in_switch.jpg]]

For yaw, pitch and roll rotations you need three switches.

[[File:3_switches.jpg]]

This is the schematic for half of the switch. It has an input A and two outputs A*sin(alpha) and A*cos(alpha).
The schematic for the other half is identical, but the contact in the switch is phase shifted in the other direction so that the outputs are -A*sin(alpha) and A*cos(alpha).

[[File:schematic_rotation_switch.jpg]]

This is the main schematic:

[[File:schematic_rotator.jpg]]

It doesn't matter that the outputs X' and Y' are inverted. Just rotate the switch 180° to invert the signs.

Three rotators can be combined for making yaw, pitch and roll rotations:

[[File:Yaw_pitch_roll.jpg]]

Rotation matrix

2021-10-30T10:07:25Z

MKoch: +new picture

Only two dimensions x,y can be shown on an oscilloscope screen. If you have a circuit with three outputs x,y,z you can see only two of them. But with a rotation matrix you can rotate the x,y,z space so that you can see a two-dimensional image from an arbitrary direction.

A 24-step switch can be used for angle adjustment in 15° steps. These 4x24 switches are available on Aliexpress for less than 10€. Search for "4*24 Rotary Switch".

[[File:4x24_rotary_switch.jpg]]

The switch must be disassembled to remove the limiter, so that the axis can be rotated over the full 360° range.

[[File:limiter_in_switch.jpg]]

In those planes that make the sin() and -sin() signals, the contact must be removed from the A position and inserted in position B, which is 90° phase shifted from A. For the -sin() signal, the phase shift is in the other direction.

[[File:contact_in_switch.jpg]]

For yaw, pitch and roll rotations you need three switches.

[[File:3_switches.jpg]]

This is the schematic for half of the switch. It has an input A and two outputs A*sin(alpha) and A*cos(alpha).
The schematic for the other half is identical, but the contact in the switch is phase shifted in the other direction so that the outputs are -A*sin(alpha) and A*cos(alpha).

[[File:schematic_rotation_switch.jpg]]

This is the main schematic:

[[File:schematic_rotator.jpg]]

It doesn't matter that the outputs X' and Y' are inverted. Just rotate the switch 180° to invert the signs.
Three rotators can be combined for making yaw, pitch and roll rotations:

[[File:Yaw_pitch_roll.jpg]]

File:Yaw pitch roll.jpg

2021-10-30T10:05:30Z

MKoch: Three rotators can be combined for yaw, pitch and roll rotations

== Summary ==
Three rotators can be combined for yaw, pitch and roll rotations
== Licensing ==
{{PD}}

Rotation matrix

2021-10-25T21:19:37Z

MKoch: Inverted outputs explained

2021-10-06T05:45:41Z

MKoch: /* Basic Usage of an Integrator on The Analog Thing */ Remove warning about sum of inputs, see discussion page

'''Integrators''' are the essential building blocks of analog computers. They perform '''integration in time''', as in <code>c = integral_0^t integrand(t') dt'</code>. [[The Analog Thing]] features five integrators. This allows computing five integrals in a program.

Electronic analog computers typically perform integration by charging and discharging capacitors. The key difference between an integrator and a [[summer]] is the capacity in the feedback channel instead of a simple resistance. Accordingly, all integrators are also summing their inputs. For more details, please refer to [[Literature|Analog computing literature]].

== Basic Usage of an Integrator on The Analog Thing ==
[[File:Integrator.png|thumb|A single integrator]]

* Circles represent inputs, triangles represent outputs
* Analog integrators sum their inputs. That is, if you have a sum under your integral, you can save a single summer before the integrator. Otherwise, just put your integrand into the upper left input labeled with <code>1</code>.
* There are two output slots. Just use one of them as you like.
* The <code>IC</code> slot stands for ''initial conditions''. It is an input where you have to provide suitable initial conditions. As shorthands, <code>-1</code> and <code>+1</code> are right to the hand. If this slot is left empty, <code>0</code> is taken as initial condition.
* Beginners may ignore the slots <code>SJ</code> and <code>SLOW</code>.
* The time constant is 1ms (1MOhm * 1nF = 1ms). That means if you connect a <code>-1</code> signal to the <code>1</code> input, the output will ramp from 0 to 1 machine unit in 1ms.

== Extended Usage of an Integrator ==

{{todo|write this section}}

=== Changing the integration speed ===
[[File:Slow-integrator.png|thumb|Internal wiring and effect of slowing down integrator.]]

You can change the integration time scale factor <code>k0</code> of a particular integrator by connecting <code>SLOW</code> to <code>OUT</code>. This effectively decreases the <code>k0</code> by factor 100. The circuit on the right was taken from [[:File:Anathing v1.0 base 3.pdf]].

=== Making use of the Summing junction ===
See also [[XIR]].

== Mathematics and Electronics about analog integration ==
On an electronical level, Integrators work quite similar to [[Summer]]s (and [[Inverter]]s, respectively), given the closed-loop operational amplifier setup. However, they feature a capacitor in the feedback channel which holds the current integration level. This makes an integrator ''stateful'' in contrast to the ''stateless'' Summers.

In the [[THAT]] circuits, you find the integrators on page [[:File:Anathing_v1.0_base_3.pdf]]. As you can see there, they are implemented with ICs called <code>TL074H</code>. See [https://www.ti.com/lit/ds/symlink/tl074h.pdf?ts=1628761245896&ref_url=https%253A%252F%252Fwww.ti.com%252Fproduct%252FTL074H] for the datasheet of this chip.

{{todo|write more}}

[[Category:Components of The Analog Thing]]