How is an Acceleration Value from an Acceleration Sensor Converted to Velocity and Displacement?

Acceleration, velocity, and displacement are parameters indicating the status of vibrating objects. As is widely known, these parameters are mutually related in differential and integral term. Acceleration sensors are commonly used for the measurement of vibration. Therefore, the FFT analyzer is equipped with integration functions for the conversion of acceleration values measured to velocity and displacement. Furthermore, when calibrating EU (What is EC calibration?) with acceleration values in m/s2 units, velocity units are automatically read directly in m/s, and displacement units are automatically read directly in m. The differential and integral functions of the FFT analyzer support both differentiation and integration on the frequency and time axes.


For power spectrums subject to narrow band analysis, since power at each frequency can be seen as being equivalent to the square of the amplitude of the sine wave, frequency-axis differentiation and integration in the FFT is implemented by multiplication and division of angular frequency ω(i.e. 2π f, f: frequency). In the integration example, the power spectrum value is found by dividing by ω2 with the single integral, and by ω4 with the double integral (equivalent to ω and ω2 respectively for the linear spectrum). Differentiation and integration of the frequency is commonly used if only the amplitude value on the frequency axis is required.


Time axis differentiation and integration involves numerical pre-processing of the direct time waveform, followed by an FFT to obtain the power spectrum. By using the time axis integral, the velocity and displacement time waveforms can be found from the acceleration waveform, thus permitting reading of instantaneous amplitude of shock amplitude waveforms.


The point you should notice is about time axis integration. Since integration (with the double integral in particular) requires a large numerical dynamic range, even integration of a time waveform with a wide frequency band results is implemented, the waveform with only the low frequency band results, and the waveform for the high frequency band cannot be observed in practice. As a practical method, a high-pass filter (analog or digital) is used to limit to the targeted frequency band when implementing the time-integration of a time waveform.


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