# Analog Circuit for Mathematical Pendulum

• bob012345
bob012345
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TL;DR Summary
I need some help understanding this block diagram of an analog pendulum circuit. This circuit has a sine generator which complicates the diagram. I need help understanding what comes out of the integrators.
This is an analog circuit block diagram of a mathematical pendulum which solves the equation $$\ddot{\alpha}= -\frac{g}{l}sin(\alpha)$$.
I need some help following the signals into and out of the blocks. I think the input to integrator #1 must be ##\frac{g}{l}sin(\alpha)## as well as ##\ddot{\alpha}## but what goes into and comes out of integrators #3 and #4? I'm planning on implementing this on The Analog Thing computer. The diagram below comes from Bernd Ulmann's book Analog and Hybrid Computer Programming.

https://the-analog-thing.org/

Thanks for any help!

Can you help further explain the functions of the other blocks in the feedback loop diagram besides blocks 1 to 4 which are supposed to be integrators?

The output of integrator 4 must be sin(a), because it is multiplied by the constant g/l.

The lowest line of 5 elements makes the sine wave generator, which produces cosine and sine waves from integrators 3 and 4 respectively.
The sine output is buffered and fed back across the bottom of the diagram to the input to the cosine integrator 3, that has initial output = +1 at time = 0.

The sine and cosine integrator inputs each have a multiplier, +Π, with common input to the multipliers. I believe the angular frequency is controlled by the +Π multipliers before each integrator.

bob012345
alan123hk said:
Can you help further explain the functions of the other blocks in the feedback loop diagram besides blocks 1 to 4 which are supposed to be integrators?
Sure. The blocks with the plus sign are multipliers which just multiply the two input signals and the single triangle is an inverter which just inverts the sign of the signal.

Baluncore said:
The output of integrator 4 must be sin(a), because it is multiplied by the constant g/l.

The lowest line of 5 elements makes the sine wave generator, which produces cosine and sine waves from integrators 3 and 4 respectively.
The sine output is buffered and fed back across the bottom of the diagram to the input to the cosine integrator 3, that has initial output = +1 at time = 0.

The sine and cosine integrator inputs each have a multiplier, +Π, with common input to the multipliers. I believe the angular frequency is controlled by the +Π multipliers before each integrator.
Thanks. I get how the bottom loop makes sines and cosines but I don't get how it makes ##sin(\alpha)## given the buffering from ##-\dot{\alpha}##. For example, the output of the left multiplier must be ##(-\dot{\alpha} )(-sin(\alpha))## but ##\alpha## itself is not trivially ##\omega t## as it would be without the buffering.

Baluncore said:
The output of integrator 4 must be sin(a), because it is multiplied by the constant g/l.

The lowest line of 5 elements makes the sine wave generator, which produces cosine and sine waves from integrators 3 and 4 respectively.
The sine output is buffered and fed back across the bottom of the diagram to the input to the cosine integrator 3, that has initial output = +1 at time = 0.

The sine and cosine integrator inputs each have a multiplier, +Π, with common input to the multipliers. I believe the angular frequency is controlled by the +Π multipliers before each integrator.
Ok, thanks, I see it now. It is exactly as this simpler case also from Bernd Ulmann's book where the ##\omega## is controlled by potentiometers;

Only now, the multipliers in the diagram below act like the potentiometers (circles) in the diagram above because the instantaneous frequency input is ##\omega=-\dot{\alpha}## which implies ##sin(\omega\tau)=-sin(\alpha)##.

I ended up simulating this and then programming the circuit on The Analog Thing analog computer. Here are the results. First the simulation. This is for a large angle of 3 radians. The dark blue is the solution for the angle ##\alpha## with the small angle approximation. The green is the angle without the small angle approximation. Notice it is not a sine wave and has a much longer period. The turquoise is the velocity for that and the red is the acceleration. Notice the double hump which occurs for initial angles larger than ##\frac{\pi}{2}##.

Here is the circuit running on The Analog Thing (THAT). It's not scaled to the exact same conditions but must be a large amplitude and captures the right behaviour. I still have to understand how to calibrate this to exact specified initial conditions. It's not quite as large an amplitude as the simulation because the double hump is less pronounced. Here yellow is the angle, turquoise is the angular velocity and purple is the acceleration.

dlgoff and sophiecentaur

## What is an analog circuit for a mathematical pendulum?

An analog circuit for a mathematical pendulum is an electronic circuit that simulates the behavior of a mathematical pendulum using analog components such as resistors, capacitors, and operational amplifiers. This type of circuit can model the differential equations governing the pendulum's motion, providing a visual or measurable output that corresponds to the pendulum's position, velocity, or acceleration over time.

## How does an analog circuit simulate a mathematical pendulum?

An analog circuit simulates a mathematical pendulum by using components to create a feedback loop that mimics the pendulum's equations of motion. Typically, operational amplifiers are used to integrate and differentiate signals, while resistors and capacitors set the time constants that correspond to the pendulum's physical parameters like length and gravity. The circuit's output can then be observed on an oscilloscope or other measuring devices to study the pendulum's behavior.

## What are the key components of an analog circuit for a mathematical pendulum?

The key components of an analog circuit for a mathematical pendulum include operational amplifiers (op-amps) for signal processing, resistors and capacitors for setting time constants and scaling factors, and sometimes inductors for more complex simulations. Additionally, power supplies are required to provide the necessary voltages for the op-amps, and sometimes potentiometers are used for fine-tuning the circuit parameters.

## What are the advantages of using an analog circuit to study a mathematical pendulum?

Using an analog circuit to study a mathematical pendulum offers several advantages: it provides real-time simulation of the pendulum's motion, allows for easy adjustment of parameters to study different conditions, and can be a cost-effective and intuitive way to understand the underlying physics without requiring complex digital computation. Additionally, analog circuits can offer continuous signals, which can be beneficial for certain types of analysis.

## Can an analog circuit for a mathematical pendulum model nonlinear behavior?

Yes, an analog circuit for a mathematical pendulum can model nonlinear behavior by incorporating nonlinear components or feedback mechanisms that reflect the nonlinear terms in the pendulum's differential equations. For example, diodes or transistors can be used to introduce nonlinearity, and the circuit design can be adjusted to account for the sine function in the pendulum's equation, allowing for more accurate simulation of real-world pendulum dynamics.

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