Mass-and-spring model of parametric amplification

[Swamp Thing]

I was playing with a toy model ( HTML + JS ) involving a mass, spring and damping that lets you vary those parameters using sliders. In the process, I got interested in the idea of varying the stiffness in real time in order to demonstrate parametric amplification.

This does work pretty well when you vary the stiffness directly at twice the resonant freq, but then I decided to simulate something more like an actual RF / microwave parametric amp. I put in a nonlinear equation for the spring, and added a pump excitation and a signal excitation. The idea is that you don't vary the stiffness directly, but rather you apply a pump input that uses the nonlinearity to modulate the stiffness indirectly.

This version doesn't work as expected and I'm trying to debug it.

But I have this question: in order to work, is it necessary to apply the pump force directly to the spring (for example by pulling up and down on the supporting point)? At the moment I am applying the signal force as well as the pump force to the mass itself. I have begun to get a vague intuitive feeling that this is not the way it would work.

Any thoughts?

 

[BvU]

I was playing with a toy model ( HTML + JS ) involving a mass, spring and damping that lets you vary those parameters using sliders. In the process, I got interested in the idea of varying the stiffness in real time in order to demonstrate parametric amplification.

This does work pretty well when you vary the stiffness directly at twice the resonant freq, but then I decided to simulate something more like an actual RF / microwave parametric amp. I put in a nonlinear equation for the spring, and added a pump excitation and a signal excitation. The idea is that you don't vary the stiffness directly, but rather you apply a pump input that uses the nonlinearity to modulate the stiffness indirectly.

This version doesn't work as expected and I'm trying to debug it.

But I have this question: in order to work, is it necessary to apply the pump force directly to the spring (for example by pulling up and down on the supporting point)? At the moment I am applying the signal force as well as the pump force to the mass itself. I have begun to get a vague intuitive feeling that this is not the way it would work.

Any thoughts?

Hi,
main thought: one needs to be clairvoyant to guess what you do, what comes out and what you expect. Please post a concrete description of the model.

 

[Paul Colby]

At the moment I am applying the signal force as well as the pump force to the mass itself.

For parametric amplification, doesn’t one need to change a parameter? In your case it would be changing the spring constant with time. For electronic examples, one uses a voltage dependent capacitance. A reversed biased diode has a junction capacitance that depends on the bias voltage, for example.

 

[Swamp Thing]

For electronic examples, one uses a voltage dependent capacitance.

For mechanical examples, I thought it might make sense to use an elongation / compression dependent stiffness. (Rather than commanding the stiffness to change magically by changing a variable called "stiffness" or "K")

Please post a concrete description of the model.

I will post the model part of the code if I'm not able to fix it in a day or two.

 

[tech99]

For mechanical examples, I thought it might make sense to use an elongation / compression dependent stiffness. (Rather than commanding the stiffness to change magically by changing a variable called "stiffness" or "K")




I will post the model part of the code if I'm not able to fix it in a day or two.

For mechanical examples, I thought it might make sense to use an elongation / compression dependent stiffness. (Rather than commanding the stiffness to change magically by changing a variable called "stiffness" or "K")

If the signal and pump are both applied to the mass, then the deflections will add. But to produce parametric action the spring stiffness must chnage with deflection. This is what you are proposing, sounds correct. I think the pump needs to be large compared with the signal.

 

[Paul Colby]

The notion that just a nonlinear component, like a non-linear spring, leads to parametric amplification left me wondering. This wiki article https://en.wikipedia.org/wiki/Parametric_oscillator actually gives a simple harmonic oscillator model for a parametric amplifier that paints a very different picture. It’s worth a read,

 

[Swamp Thing]

This wiki article paints a very different picture

In a sense, yes it does. That is, as far as the mechanical models discussed in it go.

However, searching for the word "nonlinearity" in the same article will show passages that explain that direct control of a parameter is a more abstract picture, whereas practical implementations often use nonlinear mixing to modulate the instantaneous value of a parameter.

Edit: Yes there are practical examples where a mechanical input controls an inductance, but those are from the pre-semiconductor era.

 

[Paul Colby]

The issue I have with this is the equation of motion,

$m\frac{d^2x}{dt^2} + k(t)^2x = 0$,

Is quite different than this one,

$m\frac{d^2x}{dt^2} + k(x)^2x = 0$.

[edit] Just to add some clarification, the first equation corresponds (less a dampening term) to that found in the WIKI. The spring constant is a function of time. In the second equation, the spring constant is a function of displacement. In the first equation, $x=0$ is a solution even when $k$ is driven at the pump frequency. In the second equation, $x\ne 0$ when driven at the pump frequency.