English translation
Demonstration Example No. 5 — Ball in a Box
This file translates the original German-language document “Demonstrationsbeispiel Nr. 5: Ball im Kasten” published by AEG-Telefunken, Informationstechnik, Analogrechner division.
Cover (Page 1)
AEG / TELEFUNKEN — Informationstechnik Analogrechner (Analog Computer)
Demonstration Example No. 5 Ball in a Box
Problem Description (Page 2)
An elastic ball is thrown with a specific initial velocity v₀ into a box. The process is treated as a two-dimensional (planar) problem.
Figure 1: Shape of the Box
The box dimensions are normalized to 1 (10 V). The initial velocity is composed of the components v_ox and v_oy. Within the box, a constant gravitational acceleration g acts on the ball in the negative y-direction; therefore the ball tends to fall downward in the box. When the ball reaches the floor or a side wall of the box, it bounces back elastically. If the initial velocity is large enough that the ball touches the lid of the box, it also bounces back there. Due to air friction, a damping force acts on the ball that opposes the current direction of motion and causes the body to eventually come to rest.
Circuit Structure
The motions in the x- and y-directions are generated separately. To display the ball trajectory, a rapid circular motion is superimposed, so that a stationary image produces a circle. The x- and y-components of the circular motion are superimposed on the respective current x-y position of the body. This causes the circle to shift its position according to the motion of the body described above.
Circuit Structure (Pages 2–3)
Figure 2: Circuit Structure
The circuit generates the ball motion in y-direction and x-direction separately. A sine-cosine generator (for the ball display) produces a rapid circular orbit (ball shape), which is superimposed on the actual x-y position signals to create a moving dot display on the oscilloscope.
Motion in the y-Direction
The motion in the y-direction is described by the following equations:
- Position: y = ∫v_y dt + y₀
- Velocity: v_y = ∫b_y dt + v_oy
- With acceleration: b_y = −g + d·v_y + (c/m)·(|y| − h₂) for y < h₂
- or: b_y = −g + d·v_y − (c/m)·(y − h₁) for y > h₁
where the terms represent gravitation, damping, and elastic bounce respectively.
For simplicity in the circuit, the damping is approximated as proportional to velocity.
Motion in the y-Direction — Continued (Page 4)
The acceleration function for the elastic bounce is modeled using a dead zone (Tote Zone): no restoring force acts while the ball is within the box walls; only when it touches or passes a wall boundary does the elastic restoring force engage. For y < h₂ (floor), the force is proportional to the penetration depth |y| − h₂ with spring constant c/m. For y > h₁ (ceiling), the force is proportional to y − h₁.
Motion in the x-Direction
The motion in the x-direction follows similarly: no gravity acts, but elastic bounce occurs at the left wall (x = −1) and the right wall (x = +1). A dead zone element is again used for the bounce logic. Damping also acts in the x-direction.
Generation of a Circle (Ball Display)
The y-components of the circle are generated by a sine-cosine oscillator running at high frequency. These circle signals are superimposed on the actual x- and y-position signals. If the oscillation tends to decay, an additional amplitude control (v_y term) keeps the circle amplitude constant.
Circuit Diagrams (Pages 5–7)
Figure 3: Circuit for RAT 700 / RA 742
Complete analog circuit diagram for the RAT 700, RA 741, and RA 742 machines, showing the full y-direction motion subsystem (with dead zones for ceiling h₁ and floor h₂), the x-direction motion subsystem (with comparator/relay-based bounce logic), and the sine-cosine generator circle subsystem. Component numbering and potentiometer settings are indicated for these machines.
Figure 4: Circuit for RA 741 / RA 742 (variant)
A variant circuit diagram for the RA 741 and RA 742 machines, showing the same three subsections with labeling specific to those amplifier numbering conventions.
Figure 5: Circuit for RA 770
Complete circuit diagram adapted for the RA 770 machine, with potentiometer and amplifier numbering specific to that model. The three subsections (y-motion, x-motion, circle generator) are shown.
Figure 6: Circuit for RA 800 HYBRID
Complete circuit diagram for the RA 800 HYBRID machine, with numbering convention for that platform.
Potentiometer List (Page 9)
| Coefficient | RAT 700 / RA 741 / RA 742 Pot. No. | RA 770 Pot. No. | RA 800 H Pot. No. | Value | Remark |
|---|---|---|---|---|---|
| g | 1 | 00 | 00 | 0.1 … 0.5 | Gravitational acceleration constant |
| d | 2 | 01 | 06 | 0 … 0.2 | Damping |
| v_oy | 5 | 05 | 04 | −0.5 … +0.5 | Initial velocity of ball, y-direction |
| y₀ | 10 | 15 | 05 | −1.0 … +1.0 | Initial y-position (note: h₂ ≤ y₀ ≤ h₁) |
| h₁ | 6 | 06 | 02 | 0.5 … 1.0 | Box height above y = 0 |
| h₂ | 7 | 16 | 01 | −(0.5 … 1.0) | Box depth below y = 0 |
| c/m | 8; 9 | 03; 04 | 10; 20 | 0.5 | Spring constant c per mass m at elastic bounce |
| v_ox | 18 | 22 | 30 | 0.2 … 0.8 | Initial velocity of ball, x-direction |
| ω | 11; 16 | — | — | 0.5 | Frequency of the sine-cosine generator |
| a | 4 | 30 | 33 | 0.002 | Damping for sine-cosine generator |
| r_x | 13 | 31 | 38 | 0.1 | Radius of ball (x) |
| r_y | 12 | 32 | 39 | 0.1 | Radius of ball (y) |
Operating mode: Repetitive (Repetierend Rechnen) Repetition period T_rep: ≈ 30 sec.