技術レポート FFTアナライザーについて 7

Acceleration α, velocity v, displacement x, and frequency f are related through calculus.

The symbol ±j and the sign - indicate a phase difference of ±90° and -180°.

The next page will show the displayed waveform of the FFT analyzer while providing a detailed explanation of the Y-axis values (1) to (5) mentioned above.

There are methods to express the amplitude of a sine wave, including total amplitude value, half amplitude value, and effective value.

The data above, Data 7, shows the time-domain waveform of a cosine wave. The width of the + peak and - peak of the amplitude is called the full amplitude value, and half of the full amplitude value is referred to as the half amplitude value. Additionally, the effective value is defined as "the magnitude of alternating current expressed as the direct current value that consumes the same power in the same resistance," which is equal to the square root of the average of the square of the instantaneous values of the alternating current over one cycle. In other words, if the instantaneous value of the alternating current is x(t), then its effective value is;

The fourth segment is represented as the total amplitude value, which can similarly be calculated.

The fifth section presents the effective value in logarithmic form, and according to the previously shown formula, dB Vrms is:

The 6th segment presents the half-amplitude value in logarithmic form, where dBV is;

Data 9 is for explaining the integration of the spectrum. The first row shows the input signal, and the second row displays the spectrum in terms of effective value. (From the spectrum, it can be seen that the input signal has a frequency of f = 1 Hz and a voltage of Vr = 6.924 V.)

The third section shows the spectrum after a single integration. According to the formula presented earlier, it is the second spectrum multiplied by 1/jω. Additionally, the fourth section presents the spectrum after double integration. Since 1/j represents a -90° phase delay, the magnitudes of each integral can be calculated as follows, ignoring the phase.

When the input signal has an acceleration α (m/s ²), the velocity and displacement are obtained by integrating the acceleration once and twice. Therefore, by considering the sensitivity of the acceleration sensor, which is how many volts correspond to 1 m/s ², and converting the voltage to acceleration, it will be possible to determine the velocity and displacement for each frequency from the above equation.

The differential results of the spectrum are shown in Data 10.

The first row is the input signal, and the second row is the effective value display spectrum. From this spectrum, it can be seen that the input signal is 1 Hz and 100.467 V. The third and fourth rows of data are the spectra obtained by taking the first and second derivatives of the second row spectrum, respectively, and according to the earlier section (4) of this item, they are multiplied by jω and (jω) ². Since j represents a phase advance of +90°, the spectrum (magnitude) can be calculated from the following equation.

In contrast to the case of integration, if the input signal is displacement, differentiation can be used to obtain velocity and acceleration.

I will explain about overall (O.A) in Data 11.

Overall power is synthesized from the entire frequency spectrum and corresponds to the power in the time domain (mean square value). Please note that this power is based on a certain number of sample points (for example, 2048 points).

The overall is sought as follows.

In the lower section of Data 11-1, the peak points of each spectrum are listed, but all values other than this peak value are set to zero, and with H _f = 1, the approximate overall value can be calculated as follows.

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

similarly to (1);