# Differences between FFT from acceleration, velocity and displacement data of a dynamic system

I am new to vibration mechanics and signal processing. Recently, I’ve performed an experimental test with an accelerometer mounted on a plate (shaker). The plate vibrates with exact frequencies 10 and 20 Hz. The purpose of the experiment is to extract the acceleration vs. time and FFT diagrams. I have a basic and fundamental question.

What are the differences between applying FFT to acceleration, velocity or displacement diagram of a dynamic system? Does FFT give the same peaks for the three mentioned diagrams so that I assume that the peaks are natural frequencies of the vibrating system (10 and 20 Hz in this problem)?

I would be appreciated if you could help me with these questions.

• That's three questions, and even just the first one is really broad. I'd strongly recommend reducing to one question, and that would be the first, and to reduce that to a very specific question. Typically, that should be something that you encountered during researching about the topic that you couldn't answer yourself. Such questions are much better to answer! Work from there. You show some misconceptions, and it's hard to clear up multiple complex questions with misunderstandings in one go. Sep 3, 2017 at 19:41
• @MarcusMüller: Thank you for helping me. I'll do that and edit my post. Sep 3, 2017 at 21:04

I have the same problem and I made some simulation and calculations to understand this phenomenon.

Facts:

• velocity is an integral of acceleration
• displacement is an integral of velocity
• sine wave having frequency f is sin( 2*pi * f * t ) where t is time in seconds
• integral of sin( n*t ) is -1/n * cos( n*t )
• double integral of sin( n*t ) is -1/n^2 * sin( n*t )
• To make equations cleaner, I'll use k = 2*pi

Calculations:

If you have hypothetical acceleration measurement signal having frequencies 1Hz, 10Hz, 100Hz with equal amplitudes, then:

• acceleration: sin(k*t) + sin(k*10*t) + sin(k*100*t)
• velocity: -1/(k) * cos(k*t) - 1/(k*10) * cos(k*10*t) - 1/(k*100) * cos(k*100*t)
• displacement: -1/(k^2) * sin(k*t) - 1/(k^2*10*10) * sin(k*10*t) - 1/(k^2*100*100) * sin(k*100*t)

So, it seems that:

• Both acceleration, velocity and displacement signals have the peaks at the same place: 1Hz, 10Hz, 100Hz
• the 10Hz and 100Hz peaks of velocity are lower and the peaks of displacement are MUCH lower that the peaks of acceleration
• the 100Hz peak of displacement is 10,000 times lower that 100Hz peak of acceleration

Simulations:

I made short R script which created the acceleration signal above. Then I integrated it (cumulative sum) to get velocity and displacement signals and finally took fft of them. The results verified the calculations