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I want to simulate a point that moves with random vibration around a mean position (let's say around the position $[X, Y, Z] = [0,0,0]$). The first solution that I found is to sum a couple of sinusoids for each axis, $$ \sum_{i = 1}^n A_i \sin(\omega_i t+\phi) $$ where $A_i$ is a normal random amplitude, and $\omega_i$ is a normal random frequency. I have not tested the phase yet, so I leave it to zero for now. I generated figures of the expect normal distribution and equation results with the following approach. I tried multiple values of N and I'm not sure that the equation is giving a normally distributed results. Is my approach correct? Is there a better way to generate random vibration?

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