# Numerical integration and filtering of acceleration experimental data

I have a vector containing acceleration measurements and the corresponding vector of times in which measurements are taken.

To obtain velocity and displacement I used the cumtrapz() function already implemented in MATLAB. This should be fine, as I read here Numerical integration of non-uniform acceleration samples and here https://stackoverflow.com/questions/9881430/numerical-integration-using-simpsons-rule-on-discrete-data.

My doubts are related to the disturbances introduced by the integration algorithm as I found in a book a chapter in which the author says that the trapezium rule introduces low-frequency disturbances.

I tried to remove these disturbances and unwanted 0-frequency components (the acceleration was not 0-mean therefore I had a linear trend in the velocity) applying a 2nd order, highpass, butterworht filter with cut-off frequency at 5 Hz (in both directions to null the phase shift) to the velocity and the displacement but I don't know if this is enough or too much because I need very accurate values for velocity and displacement since I have to use them to plot a surface in the state-space (EDIT: phase-space).

How can I choose the correct cut-off frequency in order not to loose any important information knowing that the experimental structure is excited in a range of frequencies from 5 to 100 Hz?

EDIT Since I have to integrate noisy data, is it advisable to use an higher order method if my goal is accuracy?

• Did you try a higher order method? Something like this? – nicoguaro Nov 2 '14 at 2:23
• I tried that method but nothing changed – Rhei Nov 2 '14 at 13:47
• I presume from your comments regarding frequencies that the acceleration data is oscillatory? – Geoff Oxberry Nov 3 '14 at 21:06
• My acceleration is a measurement of the structure vibration due to a logarithmic sine-sweep base excitation. – Rhei Nov 3 '14 at 21:09
• If you have acceleration data and initial conditions for velocity and position, plus knowledge of the allowable frequency range, you might try using a Fourier representation of your acceleration data and integrate that twice in time. – Geoff Oxberry Nov 29 '14 at 3:03