# Coupled pendulum with external force?

• questionner
In summary, the conversation discusses a picture of a mechanical analog of a non-degenerate parametric amplifier, specifically two coupled oscillators with an external force. The question asks about how the frequencies of the oscillators would be determined and if they would be amplified. The expert responds by explaining that the motion of the oscillators is more complicated and that there are two resonant frequencies that determine if amplification occurs.
questionner

I got this picture from a superconducting parametric amplifier text I was reading.
(The picture is a mechanical analog of a non-degenerate parametric amplifier.)

If the balls(red and blue) were oscillating at their own natural frequencies, and an external force is driven(purple), how would the balls' frequencies be determined?
If the problem is too sophisticated, then assuming that wp = wa + wb, then would the balls just get amplified with wa and wb conserved?

(Actually the second question is what I am interested

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osilmag
The period, and thus the frequency of a pendulum, depends upon length and gravity (##T=2pi \sqrt{L/g}##). So, I would state that ##dT/dt## would be proportional to ##dL/dt##, the length being the parameter the force is changing.

I worked it out that it could be ##dT/dL = (2pi^2)/(Tg)##.

Then your force would change the length ##dL/dt=-k*dy/dt##.

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@questionner: the picture you show is of three coupled oscillators, not two with an independent external force !

osilmag said:
The period, and thus the frequency of a pendulum, depends upon length and gravity (##T=2pi \sqrt{L/g}##). So, I would state that ##dT/dt## would be proportional to ##dL/dt##, the length being the parameter the force is changing.
Okay thank you for the reply. So is the answer yes? or no?

BvU said:
@questionner: the picture you show is of three coupled oscillators, not two with an independent external force !
Actually the right most one is not a pendulum. Sorry for the misunderstanding, but it's a two coupled oscillator!

questionner said:
I got this picture from a superconducting parametric amplifier text I was reading.

Does it contain a differential equation that describes the behavior of the amplifier?

(The picture is a mechanical analog of a non-degenerate parametric amplifier.)

Then there should be an analogous differential equation that describes its motion.

questionner said:
Okay thank you for the reply. So is the answer yes? or no?

You asked how the pendulum frequencies would be changed and I put forward a possible way to find how they're changing. I was speculating how the contraption works since I have limited info on the driver. To me it looks like a driven spring.

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questionner said:
Summary:: Here's an interesting situation(picture): when the two pendulums(red and blue) oscillate and there is an external force(purple), how would their frequencies change?

View attachment 254134
I got this picture from a superconducting parametric amplifier text I was reading.
(The picture is a mechanical analog of a non-degenerate parametric amplifier.)

If the balls(red and blue) were oscillating at their own natural frequencies, and an external force is driven(purple), how would the balls' frequencies be determined?
If the problem is too sophisticated, then assuming that wp = wa + wb, then would the balls just get amplified with wa and wb conserved?

(Actually the second question is what I am interested
No. Motion of harmonically driven 2 coupled oscillators is more complicated. There are generally two resonant frequencies of the system ω',ω" depending on ωab and coupling (the expressions are not very short/nice ). If ωp=ω' or ωp=ω" than the amplification condition occurs

## 1. What is a coupled pendulum with external force?

A coupled pendulum with external force is a system of two pendulums that are connected to each other and are also influenced by an external force. This external force can be in the form of a push or pull, and it affects the motion of both pendulums simultaneously.

## 2. How does an external force affect the motion of a coupled pendulum?

The external force applied to one pendulum in a coupled pendulum system causes both pendulums to move in a synchronized manner. This means that the pendulums will have the same period of oscillation, and the amplitude of their motion will also be affected by the external force.

## 3. What is the difference between a coupled pendulum with external force and a simple pendulum?

The main difference between a coupled pendulum with external force and a simple pendulum is that in the former, the motion of one pendulum affects the motion of the other due to their connection. In a simple pendulum, the motion of the pendulum is only affected by gravity and its initial conditions.

## 4. How can the motion of a coupled pendulum with external force be described mathematically?

The motion of a coupled pendulum with external force can be described using the equations of motion for each pendulum, taking into account the external force and the coupling between the two pendulums. This results in a system of differential equations that can be solved to determine the motion of the pendulums over time.

## 5. What are some real-life applications of coupled pendulums with external force?

Coupled pendulums with external force can be seen in various real-life applications, such as in clocks and metronomes. They are also used in seismometers to measure earthquakes and in engineering to study the stability of structures under external forces.

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