If a damped mass-spring oscillator is equivalent to a resonant bandpass filter, then in audio signal terms, what is the input signal for both?

Question

Background

I am working on some personal audio processing and synthesis experiments in the sample domain. I posted here about how a resonant bandpass filter with a given $Q$ and frequency $f_0$ is essentially the same as a damped mass-spring harmonic oscillator set $k$, $m$, and $c$ to get the same $f_0$ and $Q$.

It was commented there my mathematical method (explicit Euler) was likely contributed to instability. I will work on that.

Regardless of stability though, I am trying to understand in this analogy for certain what is analogous in terms of input and output.

1) Input

In audio (resonant bandpass) terms, one passes through an input audio signal sample by sample to the filter. Any audio that is not within the resonant frequency band is filtered out, while those signals within the band are passed through, and resonated based on.

To make the mass spring oscillator perform equally with an input audio signal, what would the audio signal represent? How would it be "inputted" to the damped mass-spring oscillator equation of motion?

Do you just add the input audio sample to the existing position of the mass-spring oscillator?

In my prior post, I thought maybe I need to convert the input into a force (derive twice the input), and apply it as an external force driving the damped mass-spring oscillator ($y = -kx - cx_t + F_i$). But I am not sure if that is legitimate. Maybe that is completely wrong.

I am not sure I am doing this correctly either way.

2) Output

Output of the resonant bandpass filter in usual audio usage is just of course audio yet again (like input), filtered from input.

I presume output of the mass-spring oscillator to match this should just be the amplitude of the mass (ie. $x$ in $y= -kx - cx_t$) at each sample, since its amplitude of motion is directly a harmonic oscillator (and so is the resonant bandpass).

But if I convert input from audio into force in the mass-spring oscillator to get it to behave correctly, do I need to do any conversion on output of the mass-spring oscillator to get it to be audio?

Thanks for any help understanding this analogy and what, if anything, I am getting right or wrong with it.

Answer 1

It depends on how you set up the mechanical equation, but assuming it is a displacement for both, the input will be forced displacement of the base of the spring and the output will be the displacement of the mass at the end of the spring.

These can be set up in labs using precise vibrators of some kind. There are also "real world" examples that cause frequency based forcing. For example, if a roadway has a spatial frequency to its grade, then driving over that causes mechanical forcing at a temporal frequency which also depends on the speed of the car. This is how various damper design problems for an automobile's shock absorbing system can be studied and solved. There are examples of this in almost any book with a title like "Mechanical Vibrations" but that is getting off topic for this forum. Not to dwell on it, but a real roadway will have what amounts to a "spatial noise distribution" to its gradient and then driving over it will imply some sort of temporal frequency driving (that is a whole set of frequencies) that depends on the driving speed. The total output will then be dependent on the frequency response of the shock absorbing system, which will use the same Fourier and convolution and transfer function type math you know from audio circuits.

To be precise, you keep saying "audio" but you likely mean an electrical signal, specifically voltage. You are using the existence of your extremely accurate linear amplifier to treat the voltage as a perfect analog for "audio."