Glossary of Basic Terms Related to FFT Analysis

In measuring the dynamic characteristics of structures such as machinery, the impulse response obtained by applying an impulse to the structure with a hammer is typically processed using FFT to determine the system's transfer function.
However, since the transfer function obtained using FFT is discrete data with a finite, equally spaced frequency resolution, there are very few measurement points near natural frequencies where the amplitude curve changes abruptly.
Therefore, the Nyquist plot obtained from this process does not represent an ideal circular locus. To obtain the correct modal parameters such as peak values and natural frequencies, curve fitting is necessary, which involves interpolating between these equally spaced data points.

Curve fitting is a technique that involves assuming an analytical equation for the transfer function and adjusting modal parameters such as natural frequency, damping ratio, and vibration mode in this equation to approximate the measured transfer function with the model's transfer function as closely as possible.
This theoretically determines the dynamic response of a structure in modal analysis.

In actual curve fitting, the real and imaginary parts of the discrete complex transfer function obtained through measurement are first used to plot multiple points on the Nyquist line.
Next, a theoretical Nyquist plot is calculated that minimizes the error with these points. The transfer function is then recalculated from this Nyquist plot and fitted to the measured transfer function.

There are two main methods used for curve fitting of measured transfer functions.
When the peaks of each vibration mode are far apart and do not influence each other, a single-degree-of-freedom curve fit (SDOF) is used.

On the other hand, when the characteristics of adjacent vibration modes overlap, it is necessary to consider the influence of multiple vibration modes, and a computational algorithm is required that simultaneously fits the numerous modal parameters that analytically represent the transfer function to the measured transfer function.
This method is called multi-degree-of-freedom curve fit (MDOF).

The time data used for performing an FFT consists of points that are powers of 2. In Ono Sokki 's FFT analyzers, these points are called the number of sample points (frame length).

Performing an FFT on 64, 128, 256, 512, 1024, 2048, and 4096 time data points yields 25, 50, 100, 200, 400, 800, and 1600 frequency data points.

Thus, the frequency resolution depends on the number of samples in the FFT.

Rotational order ratio analysis is a method used to perform frequency analysis of vibrations and noises from rotating machinery. It involves sampling signals using pulses from a pulse generator attached to the rotating body as an external sampling clock.

In frequency analysis, 1 Hz represents a component that completes one cycle in one second. In contrast, in rotational order ratio analysis, rotational first order refers to a component that completes one cycle for one rotation of the reference rotating body.
The second rotational component completes two cycles per revolution and is twice that of the first rotational component. Thus, in order to perform analysis based on the variation per revolution, sampling synchronized with the rotational speed is necessary.
If the internal sampling clock is used as is, the number of sampling points per revolution will change if the rotation speed changes. However, if a clock synchronized with the rotation pulse is used as the sampling clock, the number of sampling points per revolution will always remain constant.

For example, if we consider a rotating body rotating at 600 r/min, the primary rotational frequency is (600 r/min) / 60 = 10 Hz, and the secondary rotational frequency is 20 Hz.
When the rotational speed increases to 700 r/min, the primary rotational frequency rises to 11.7 Hz and the secondary rotational frequency rises to 23.3 Hz. As shown above, the frequency fluctuates with changes in rotational speed, but by normalizing it as an order, it becomes possible to focus on a particular component without being affected by rotational fluctuations.

One application of rotational order ratio analysis is rotational tracking analysis.

Rotational tracking analysis traces the change in amplitude of a certain order component with rotational speed as the parameter on the horizontal axis. This allows us to determine which components of a rotating machine are resonating at a given rotational speed, or which component (of what order) is resonating at what rotational speed.

The measured open-loop and closed-loop transfer functions can be converted into closed-loop and open-loop transfer functions, respectively, through calculation.

If there is no feedback element

The obtained open-loop transfer functionTherefore, the closed-loop transfer functionteeth

The obtained closed-loop transfer functionTherefore, the open-loop transfer functionteeth

The Fourier transform and the inverse Fourier transform have the following relationship:

Furthermore, the inverse Fourier transform of the cross-spectrum function is the cross-correlation function, and the inverse Fourier transform of the frequency response function is the impulse response.

A signal input to a filter experiences a delay before it is output. The group delay characteristic represents how much delay each frequency of the output signal has relative to the input. Specifically, it is the derivative of the phase characteristic (phase difference between input and output) with respect to angular frequency, and is used to evaluate the characteristics of a filter circuit. This characteristic (delay time) varies depending on the filter circuit and frequency. When using filters with different delay times depending on frequency, waveform distortion occurs in the output signal relative to the input signal.

Two signalsLet X(f) and Y(f) be the Fourier transforms of X(f) and Y(f), and let X*(f) be the complex conjugate of X(f). Then the cross spectrum ​_Wxy (f) is defined by the following equation.

A cross-spectrum is calculated by multiplying the frequency components of the spectra of two signals and then averaging them, with the X-axis representing frequency and the Y-axis representing...It is represented as follows. When the cross spectrum shows a large value at a certain frequency, it means that at that frequency, the correlation between the frequency components of the two signals is large, and the magnitudes of both components are also large. The cross spectrum is also used in the calculation of the cross-correlation function, transfer function, and coherence function.

The cepstrum is obtained by performing a Fourier transform on the logarithm of the power spectrum, which was itself obtained by the Fourier transform. The horizontal axis of the cepstrum takes on a time-dimensional value called the cefrency.

When a signal input to a system has periodicity and its period is long, that period appears as a line cepstrum in the long-ceflancy region, which can be extracted as the fundamental period. Furthermore, the short-ceflancy region contains concentrated information representing the system's transfer characteristics, and by performing an inverse Fourier transform on this region, the envelope of the logarithmic power spectrum can be obtained (lifted envelope). This envelope is system-specific and does not depend on the spectrum of the input signal.

Applications include extracting fundamental frequencies and spectral envelopes from audio waves, bio-waves, and other sources.

Vibration waveforms observed in mechanical vibration systems typically contain various harmonic components in addition to the fundamental wave component. When a sine wave is applied to a transmission system, the output signal exhibits distortion components, which are harmonic components of the applied sine wave, due to the nonlinear characteristics of the transmission system. Therefore, by focusing on this distortion, the harmonic components of the vibration waveform and output signal are analyzed to examine the vibration characteristics and the fidelity of the transmission system.

Currently, the observed waveform, generally the output waveform, consists of the fundamental frequency f1 and the second harmonic. If the effective values of the harmonic components formed by f2, the third harmonic f3, etc. are | E1 |, | E2 |, | E3 |, ... then the overall distortion rate is

While the measured value is read as a voltage, if a reference value is established for the signal being measured, such as acceleration, pressure, or sound, the voltage value can be calibrated to the reference value and read as a physical quantity.

Example 1: Accelerometer sensitivity isIf the voltage is 100 mV, then multiply the obtained voltage by 10 to 0.1 V/EU and change the unit to m/ ^s².

Example 2: When calibrating a microphone, acoustic calibrator, and sound level meter, the power spectrum data should be adjusted so that the overall (dB value) matches the calibration value.

A Coqad plot is created by plotting the real and imaginary parts of a frequency response function separately on the frequency axis and displaying them side by side. It can be used for estimating natural frequencies, among other things.

Coherence function (relevance function)This indicates the degree of causal relationship between the system's input and output.It takes a value between 0 and 1.If the value is 1, it indicates that at that frequency, the entire output of the system is attributable to the measurement input.In this case, the output of the system is completely unrelated to the measurement input in terms of that frequency.If this is the case, it is likely due to signals unrelated to the measurement, noise occurring within the system, system nonlinearity, or system time delay.

​Coherence function of the results between the two measured channelsA small value indicates that the measurement result is inaccurate. These inaccurate parts are not displayed.The coherence blanking function displays only the larger portion of the data.The value can be set arbitrarily.The transfer function will not be displayed at frequencies below this value.

The product of the coherence function and the output autopower spectrum is called the coherent output power (COP).

COP represents the autopower spectrum of the output that is attributable to the measurement input.