Measured values are read as a voltage value. If the reference value of the signal under measurement (acceleration, pressure, sound, etc.) is defined, it can be read as a physical quantity
by calibrating the voltage value to the reference value.
When the input of the acceleration pickup is 1m/s2 and the output 100mV, the physical quantity is 10.0EU/V (10
times the obtained voltage value) and the unit is m/s2.
Example 2: When calibrating a sound level meter using a microphone and
a sound calibrator, the overall value (in dB) is used as the
output sound pressure level of the sound calibrator by means of
the power spectrum data.
The cepstrum is obtained by the Fourier transform of the
logarithmic value of the power spectrum obtained by the Fourier
transform. The horizontal axis of the cepstrum is assigned a
time-dimensional value referred to as quefrency.
If a signal
input to a certain system has a periodicity and the period is
long, the period appears as a linear cepstrum in the long
quefrency portion, and can be extracted as a fundamental period.
The information representing the transfer characteristic of the
system is concentrated in the short quefrency portion. The
envelope of the power spectrum is obtained by performing the
inverse Fourier transform of this portion. The liftered envelope
is specific to the system and does not depend on the spectrum of
the input signal.
Applications of the cepstrum include extraction of the
fundamental frequency and spectrum envelope from sound wave,
organism wave, and so on.
Channel Delay Function
In measurement of the transfer function of a mechanical system,
if the system has a long signal propagation time causing
temporal shift (delay) between the input signal and the output
signal of the system, the transfer function cannot be measured
If there is temporal shift between channels, the
delay function is used to correct the timing by delaying the
sampling start timing of the slave channel with respect to the
sampling start timing of the master channel.
A small value of the coherence function HREF_COH, γ2 (f), between two channels under measurement indicates that the
measurement result is not accurate. The coherence blank function
does not display such inaccurate components but displays only
components with large γ2 (f). The value of γ2 (f) can be set to any value. The transfer function is not
displayed at frequencies when γ2 is equal to or less
than the specified value
The coherence function γ2 indicates the degree of the
correlation between the input and output of a system. It gives a
value from 0 to 1 for each frequency.
When γ2 (f) is 1, it indicates that at frequency f
all outputs of the system are due to the input. When γ2 (f) is 0, it indicates that at frequency f the output of the
system is completely independent of the input.
When the following condition is met,
presence of noise occurred inside the system or
non-linearity or time delay of the system can be assumed.
?2 is obtained by
where Wxy is the cross spectrum and Wxx and Wyy are the power spectrum of x and y,
respectively. The coherence function, γ2, is the
square of the absolute value of the cross spectrum divided by
each power spectrum of the input and output of the system.
The coherence function, in its nature, is not
meaningful unless averaging is performed. When measuring the
coherence function, be sure to perform averaging.
Coherence Output Power
The product of the coherence function and the auto power
spectrum of the output signal is referred to as the coherent
output power (C.O.P.).
The C.O.P represents only the auto power
spectrum caused by the input under measurement out of the auto
power spectrum of the output.
The coincident quadrature diagram is a diagram which plots the
real part and imaginary part of the frequency response function
separately on the frequency axis and displays them with vertical
arrangement. This diagram is used for estimation of inherent
It is plotting the real and imaginary parts of the frequency response function separately against the frequency axis and displays them side by side on the top and bottom. It can be used for estimating natural frequencies, for example.
The crest factor is defined as the ratio of the peak value to
the RMS value (peak value/RMS value). The crest factor of a DC
signal is 1 and that of a sine wave is √2=1.414.
As for the peak value and RMS value, for example, the vibration
value changes relatively with the bearing size. (With large
bearings, the RMS value of vibration is large and the peak
value, under abnormal condition, becomes larger.) Since the
crest factor denotes the ratio of the peak value to the RMS
value, the vibration value is not affected by the bearing size,
allowing accurate judgment of the extent of flaws and other
abnormal conditions. If the value of the measured crest factor
is large, a large extent of abnormal condition is judged.
There are two types of correlation functions: auto-correlation
function and cross-correlation function.
The cross-correlation function is a function of delay amount τ
when the waveform of one of two waveforms is delayed by τ. It is
defined by the following expression:
The cross-correlation function is used for
measurement of correlation and time delay between two signals.
If the two signals are different completely, the
cross-correlation function approaches zero irrespective of τ. If
the two signals correspond to the input and output of a certain
system, the cross-correlation function is used to estimate the
delay time within the system, detect signals embedded in
external noise, and estimate the signal propagation path.
With the FFT analyzer, the cross-correlation
function is obtained by the inverse Fourier transform of the
When the Fourier transform of two signals x (t) and y (t) is X
(f) and Y (f) respectively and the complex conjugate of X (f) is
X*(f), the cross spectrum, Wxy (f), is defined by
The cross spectrum is given by obtaining the product of the same
frequency component of the Fourier spectrum of two signals and
then averaging the result. The X axis is represented by the
frequency and the Y axis by V2. A large value of the
cross spectrum at a certain frequency means that there is
intense correlation between the frequency components of the two
signals at the frequency and that the magnitudes of these
components are large. The cross spectrum is used to calculate
the cross-correlation function, transfer function, and coherence
In dynamic characteristic measurement of machines and other
structures, the transfer function of the system is usually
obtained by applying an impulse to the structure with a hammer
and performing the FFT operation for the impulse response
However, the FFT-based transfer function is discrete data with
limited regular-interval frequency resolution and therefore has
a problem that there are very few measuring points near the
characteristic frequency at which the function changes abruptly.
Therefore, the Nyquist diagram obtained from the transfer
function does not provide the ideal circle tracing. To obtain
correct peak values, characteristic frequency, and other modal
parameters, curve fitting, which interpolates the
regular-interval data during calculation, is necessary.
With the curve fitting method, an analytical expression of the
transfer function is assumed and, by setting the characteristic
frequency, damping factor, vibration mode, and other modal
parameters to an appropriate value, the measured transfer
function is approximated to the transfer function of the model.
This method determines theoretically the dynamic response of a
structure in modal analysis.
In actual curve fitting, multiple points are plotted on the
Nyquist line using the real part and imaginary part of the
complex transfer function of the discrete system which is
obtained as a result of measurement. Then, a theoretical
Nyquist diagram with the minimum error to these points is
calculated, the transfer function is obtained from it, then the
calculated transfer function is fit to the measured transfer
The following two methods are mainly used for curve fitting.
If the peaks of vibration mode are separated and do not have any
mutual influence, SDOF (Single-Degree-Of-Freedom) curve fitting
is used. Conversely, if the adjacent vibration modes
overlap each other, it is necessary to take the influence of
many vibration modes into consideration, and a calculation
algorithm is required which applies a number of modal parameters
representing the transfer function analytically to the measured
transfer function at the same time. This method is
referred to as MDOF (Multi-Degree-Of-Freedom) curve fitting.