# 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.

• we call that a 2nd-order homogeneous differential equation. using Laplace Transform, that can be turned into a 2nd-order polynomial with $X(s)$ and powers of $s$. – robert bristow-johnson Jul 26 '18 at 23:29

## 1 Answer

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

• You mean take advantage of the trasnfer function multiplication property? When you say FRF of "system", and "forcing" FRF, what are you refering to? – user76568 Jul 26 '18 at 15:20
• The equation that you provided represents a linear dynamical system. Let me break it down furthur for you. Probably you have seen the videos of building collapsing during earthquake. Now the equation of motion mentioned above represents the building which also works as a filter. The earthquake is the forcing function or Input Signal. Hence, the input signal will be filtered through the filter e.g. building and will produce an output signal. The output signal will represent the vibration of the building. If the vibration is too much to handle for the structure it will collapse. – Debasish Jana Jul 26 '18 at 15:35
• Now if you give a sinusoidal signal as an input, in frequency domain it will show a peak in the sine frequency. And the FRF of the building/filter is shown in Fig.7 (page 19) of the attached pdf. You can multiply the FRFs and can obtain the output. – Debasish Jana Jul 26 '18 at 15:37
• Can't see the earthquake/building analogy :). Given $y(n)=h(n) \ast x(n)$ and for LTI, after z-transfom $H(z) = \frac{Y(z)}{X(z)}$. FRF is defined as feeding an imaginary $z$ to the transfer function. $H$ is nice in that you can break down filters using multiplication in the $z$. One is sinosuidal, and the other is exponential in this case? Hann't read the entire thing yet. Thank you BTW. – user76568 Jul 26 '18 at 15:53
• No... another one is not exponential. It's almost like bell-shaped (like the figure.7) – Debasish Jana Jul 26 '18 at 15:56