Technical Report: About FFT Analyzers 7

If acceleration is α, velocity is v, displacement is x, and frequency is f, then acceleration, velocity, and displacement are related by differential and integral calculus.

The ±j and - signs indicate phase differences of ±90° and -180°, respectively.

On the next page, we will explain in detail the Y-axis values (1) to (5) mentioned above, while showing the waveform displayed by the FFT analyzer.

There are three ways to express the amplitude of a sine wave: total amplitude, partial amplitude, and RMS value.

Data 7 above shows the time-domain waveform of a cosine wave. The width between the positive and negative peaks of the amplitude is called the total amplitude value, and half of the total amplitude value is called the single-amplitude value. Furthermore, the RMS value is defined as "the magnitude of the AC expressed as the DC value at which the power consumed by the same resistance is the same," and is therefore equal to the square root of the average value obtained by averaging the square of the instantaneous AC value over one period. That is, if the instantaneous AC value is x(t), then its RMS value is:

The fourth stage represents the total amplitude value, which can be calculated in the same way.

The fifth row shows the RMS value in logarithmic form, and according to the formula shown earlier, dB Vrms is:

The sixth row shows the single-amplitude values in logarithmic scale, and the dBV is:

Data 9 is used to illustrate the integration of the spectrum. The first stage shows the input signal, and the second stage shows the spectrum in RMS value. (From the spectrum, we can see that the input signal has f = 1 Hz and Vr = 6.924 V.)

The third stage shows the single integral spectrum. Using the formula shown earlier, it is obtained by multiplying the second stage spectrum by 1/jω. The fourth stage shows the double integral spectrum. Since 1/j represents a phase lag of -90°, the integral values (magnitudes) can be calculated as follows, ignoring the phase.

When the input signal is acceleration α (m/ ^s²), the velocity and displacement are obtained by the single and double integrals of the acceleration, respectively. Therefore, by considering the sensitivity of Accelerometer and the voltage per 1 m/ ^s², and converting the voltage back to acceleration, the velocity and displacement for each frequency can be determined from the above formula.

Data 10 shows the differential results of the spectrum.

The first stage shows the input signal, and the second stage shows the spectrum in RMS value. From this spectrum, we can see that the input signal is 1 Hz and 100.467 V. The third and fourth stages of the data are the spectra obtained by differentiating the second stage spectrum once and twice, respectively, from the calculus formula in section (4) above, multiplied by jω and (jω) ^². Since j represents a phase lead of +90°, it can be calculated from the following formula when considering the spectrum (magnitude).

Conversely to integration, if the input signal is displacement, you can find velocity and acceleration by differentiating it.

Data 11 explains the concept of overall (O.A.).

The overall power is the combined power of the entire frequency spectrum and equals the time-domain power (mean squared). Please remember that this power is calculated based on the number of sample points (e.g., 2048 points).

The overalls are calculated as follows:

The peak points of each spectrum are listed in the lower section of Data 11-1. If we assume that all values other than these peaks are zero and that _Hf = 1, the approximate overall value can be calculated as follows.

The overall value is displayed in the upper right corner of the data.

(1) as in the same case;