Coupled pendulum with external force?

[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 w_p = w_a + w_b, then would the balls just get amplified with w_a and w_b conserved?

(Actually the second question is what I am interested

 

[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.

 

[osilmag]

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

 

[osilmag]

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

 

[BvU]

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

 

[questionner]

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?

 

[questionner]

@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!

 

[Mister T]

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.

 

[osilmag]

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.

 

[zoki85]

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 w_p = w_a + w_b, then would the balls just get amplified with w_a and w_b 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 ω_a ,ω_b and coupling (the expressions are not very short/nice ). If ω_p=ω' or ω_p=ω" than the amplification condition occurs