Damped Harmonic Oscillation as an LTI The goal is to create an LTI filter which is exactly, or approximates, damping of harmonic modes.

The equation of course is:

$$\frac{d^2 x}{dt^2} + 2 \xi \omega \frac{dx}{dt}+\omega^2x=0$$

This can be done with very costly convolution, I suppose.

I was thinking that instead of treating the velocity term as a part of a time dependent potential (no longer time invariant), I can treat this as a two dimentional state and therefore eliminate second derivatives and have it a $z^{-1}$ feedback filter. It is working weird.

A continous alternative of something close is:

$$e^{-\xi t} \cos \omega t$$

How is this approximated efficiently?

Links are enough.
This is a classic dynamics problem. I think you do not have to go through the convolution to solve these type of problem. Instead, you can draw the frequency response function (FRF) of the system and multiply with the FRF of the forcing function to obtain the resulting FRF. Then the resulting FRF can be transformed into the time domain using the inverse Fourier transform. You can also solve it by doing the convolution. Then you have to use the Duhamel's integral to solve this type of problem.

The link is attached.

http://people.duke.edu/~hpgavin/cee541/sdof-dyn.pdf