In engineering, it is sometimes difficult to directly measure the velocity or displacement of mechanical equipment components, especially on moving equipment. Even though speed sensors can be arranged in general scenarios, sometimes speed and acceleration need to be monitored at the same time. In order to save costs, the speed signal is often obtained by directly integrating the acceleration signal.
There are two ways to integrate the acceleration signal to obtain the velocity signal. One is to integrate directly in the time domain, which often produces a trend item, and the result of the integration needs to be detrended. The second is to integrate in the frequency domain. The typical frequency domain integration algorithm is the omega algorithm. Here is a brief introduction:
If we have obtained the acceleration signal x''(t), its Fourier transform is X''(f), then:
Record the speed signal as x'(t), which is what we require, let its Fourier transform be X'(f), then:
Because the acceleration can be derived from the velocity, the relationship between them is:
Putting the expression of velocity into the above formula, we have:
Comparing the above formula with the first formula, we can get:
Because the time-domain acceleration signal is obtained by the sensor, its frequency-domain representation can be obtained through calculation, and then the frequency-domain representation of the velocity can be obtained through calculation, which is:
Therefore, the time-domain waveform of the velocity can be obtained through the inverse Fourier transform. It should be noted here that when calculating the frequency domain representation of the speed, the above formula cannot be calculated at the time f=0, so X'(0) = 0 + 0j is directly set, so that the obtained speed signal has no DC value, so The desired velocity waveform can be obtained without detrending. Attached below is the velocity waveform I obtained on matlab through time domain integration (after detrending the least squares algorithm) and frequency domain integration method:
It can be seen that they are basically the same.
Note: The source code is available for a fee.