# Eliminating inertia effect from the acceleration signal

I am struggling with a problem, and would be happy if you could assist me. I put an accelerometer sensor inside a spherical enclosure, and release it to the water. While it was translating in the flow, I recognized that the enclosure was also wobbling (like a kicked pendulum), which I think that it is likely caused by the inertia of the enclosure. I then added some weight inside the enclosure and observed that the peak was more pronounced. Left panel is the spectra of the acceleration magnitude for three different tests without additional weight (inertia is only described by the sensor + enclosure + battery).

Since the sensor is like a black box (I do not know the mass distribution inside so that I can try to match it with the geometrical center of the enclosure), I would like to find a way to exclude the effect of the wobbling in the signal without disturbing it because the wobbling might also affect the lower frequency part of the signal, as seen in green line in page 118.

Based on the comment below, I have added the spectra after notch filter. Notch filter does its work very well, and attenuated the signal in the region (2 - 5 Hz), without affecting the rest. However, the oscillations (around 3.5 Hz) likely influences the lower frequency range of the spectra. Is there any quantity that I can derive to see the effect of these oscillations in the lower frequency range or any function that I can try to model the effect of inertia? Any ideas and suggestions would be appreciated.

It sounds like you want to remove the frequency component at about 2.5Hz in the plot, but leave the rest of the plot as untouched as possible.

The usual first approach to that problem is to just use a notch filter:

$$H(z) = K \displaystyle\frac{1 - 2 \cos \omega_0 z^{-1} + z^{-2}}{1 - 2r \cos\omega_0 z^{-1} + r^2 z^{-2}}$$

where $\omega_0$ is the (normalized) frequency you want to remove and $r \lt 1$.

There are more complex approaches that use, for example, Kalman filtering... but that might require more assumptions about the dynamics of your system that you are prepared to make given your "black box" comments.

• Thanks a lot for your very well explanation! Notch filter worked quite well in attenuating signal. I have slightly edited my question, considering your suggestion. – Tethys Sep 15 '15 at 9:39
• Any other approach (e.g. Kalman Filtering) will require a model of what you're doing (and what you're measuring). I don't have the time right now, but may in a couple of days (too many work deadlines!). – Peter K. Sep 15 '15 at 11:34
• Thank you very much for your comment and kind assistance! I am measuring time-series of accelerations in a water channel using a spherical enclosure fitted with an accelerometer, and am interested in translation of the particle (accelerations due to rotation are negligible). I also work with the norm of the accelerations. However, I am not sure i) if there is a way to see the effect of above mentioned wobbling on lower frequencies, and ii) how I can keep track of accelerations measured by the wobbling with Kalman filter? – Tethys Sep 16 '15 at 11:12