In real-time analysis, FFT operation for sampled data is
performed continuously without intervals between windows.

In usual FFT analysis, when sampling for the analysis data
length (1024 or 2048 points) of signal has been performed, FFT
operation for the data is performed. The next data is sampled
during FFT operation and, when FFT operation for that data is
completed, FFT operation for the next data starts. If the
duration of FFT operation (including the time necessary for
display) is shorter than the sampling time, real-time analysis
can be performed. However, if the sampling time is shorter than
the duration of FFT operation, a new signal for one ore more
frames is sampled during operation resulting in missing signal.
If the sampling time is longer than the duration of FFT
operation, part of the window can be overlapped with the
previous window (overlap processing).

Rotational Order Ratio
Analysis

With the rotational order ratio analysis, signal sampling is
performed using the pulse of the pulse generator attached to the
body of rotation as an external sampling clock for the purpose
of frequency analysis of vibration and noise of rotating
machinery.

With frequency analysis, 1 Hz is a component which completes one
cycle in one second. With rotational order ratio analysis, on
the other hand, the rotational 1st order is a component which
completes one cycle for each rotation of the reference body of
rotation.
The rotational 2nd order is a component which completes two
cycles for each rotation of the body of rotation, which is twice
the case of the rotational 1st order. Thus, in order to perform
analysis based on the variation for each rotation, sampling must
be performed in synchronization with the revolution speed.
If only the internal sampling clock is used, the number of
samples per rotation changes with the variation of the
rotational speed. However, if a clock synchronizing with the
rotation pulse is used as a sampling clock, the number of
samples per rotation becomes constant.

For example, suppose a body of rotation with a rotational speed
of 600 r/min. The rotational 1st order is (600 r/min)/60 = 10 Hz
and the rotational 2nd order 20 Hz. If the revolution
speed reaches 700 r/min, the rotational 1st order increases to
11.7 Hz and the rotational 2nd order to 23.3 Hz.
Thus, the frequency changes with the variation of the rotational
speed, However, normalizing the frequency as an order makes it
easier to focus a certain component without being affected by
the variation of the rotational speed.

Rotational Tracking
Analysis

The rotational tracking analysis is an application of the
rotational order ratio analysis.

The rotational tracking
analysis is an analysis method which is used to determine which
component of the rotating machine resonates or what time (what
order) component of the rotational speed resonates, with respect
to a certain rotational speed, by tracing the level variation of
a certain order component as a horizontal-axis parameter.

Root Mean Square (RMS)

The root mean square denotes the mean of squares of the signal,
which is calculated by

With a sine wave, the RMS value is 1/√2 times the peak value. The data of each line of the power spectrum equals to the RMS
value of the signal in the relevant frequency band.

RMS value

Take an electric heater as an example. When DC voltage is applied to the electric heater, it generates heat. Since the
nichrome wire of the electric heater (or ceramic with recent ceramic heaters) is a resistive element (resistor), the circuit is as shown in Fig. 1.

Figure 1

Suppose the energy (heat) generated when the voltage is 100 V
and the resistance is 100 ohms. From Ohm's law,

The current flowing in the resistor is I=E/R=100/100=1(A), and
the power W is W=E x I=100x1=100(W). Since the calculation of
current is troublesome, the following expression is generally
used from I=E/R:

When 100 V is applied to a 100-ohm electric heater, heat for 100
W is generated. In the Kanto area, the 100 V rms domestic power
voltage is supplied, not as DC voltage but as 50 Hz AC voltage
(sine wave). When a 100-ohm electric heater is connected to the
100 V power outlet, what amount of heat (in watts) is generated?
The answer is 100 W, i.e., the RMS value is the equivalent DC
voltage that performs the same work. So when the same electric
heater is connected to a 200-V power outlet, what amount of heat
(in watts) is generated? (Actually, the heater will be damaged.)
Since 200 VAC has the same power as 200 VDC, the power is 200^{2}/100=400W,
not 200W. When the voltage is doubled, the current is also
doubled and the power is quadrupled (2^{2}). What is the
RMS value for the following case in Fig. 2?

Figure 2

The simple mean of the absolute value is 1.5 V. Let us
consider the power actually generated. Since
calculation is troublesome with a resistance of 100Ω, a
resistance of 1Ω is assumed. The power generated
with 1V and 2V is given by the following expressions. Even
if the resistance (1Ω) is omitted, the result remains unchanged.

In this case, the square of the voltage becomes the power
generated at the resistor. If the voltage or current is
multiplied by itself, you may recognize the expression as the
power. (In the case of 1Ω, the voltage and current are equal and
therefore the square of the current also becomes the power from
expression (2).)

The mean power is 2.5W because the 1W section is 1/2 and the 4W
section is 1/2. as for mean power, the section of 1W is set to
2.5W. That is, this waveform has the same power as the DC
voltage that generates 2.5W power in the 1Ω resistor.

The RMS value is the square root of the mean of the power.
In other words, it is the square root of the mean of the square
value. This is referred to as "root mean square." If you
want to emphasize that the value is the RMS value, it is
described as "rms."

Since the RMS value of a sine wave equals 1/√2=0.707 times the
peak value, the waveform of the domestic 100 V power has a peak
value of 141V which equals √2 times the RMS value. (The mean is
2/π=0.637.)

Figure 3

Let us connect the 100-ohm electric heater to this power. The
maximum power is

Since it is apparent that the minimum power is 0 W, you can
easily imagine that the power swings between 0 and 200W.
This phenomenon is shown in Fig. 4. It is apparent that the mean
at this time is 100W.