Glossary of Basic Terms Related to FFT Analysis

In the measurement of the dynamic characteristics of structures such as machines, the impulse response obtained by applying an impulse to the structure with a hammer is usually processed with FFT to determine the transfer function of the system.
However, since the transfer function using FFT is discrete data with finite equidistant frequency resolution, there are very few measurement points near the natural frequencies where the amplitude curve changes sharply.
Therefore, the Nyquist diagram obtained from this will not be an ideal circular trajectory, so to obtain correct peak values, natural frequencies, and other modal parameters, it is necessary to perform curve fitting while interpolating between this equidistant data.

The method called curve fitting assumes an analytical expression of the transfer function and aims to approximate the measured transfer function and the model's transfer function as closely as possible by adjusting the modal parameters such as natural frequency, damping ratio, and vibration modes to appropriate values.
This is what theoretically determines the dynamic response of the structure in modal analysis.

In actual curve fitting, first, using the real and imaginary parts of the complex transfer function of the discrete system obtained from measurements, multiple points are plotted on the Nyquist plot.
Next, a theoretical Nyquist plot is calculated such that the error with these points is minimized, and from this Nyquist plot, the transfer function is recalculated and obtained, fitting this to the measured transfer function.

Two main methods are used for curve fitting of the measured transfer function.
When the peaks of each vibration mode are separated and do not affect 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 effects of multiple vibration modes, and a computational algorithm is required that simultaneously fits numerous modal parameters, which analytically express the transfer function, to the measured transfer function.
This method is called Multi-Degree-of-Freedom curve fit (MDOF).

The time data for performing an FFT must consist of a number of points that is a power of two. In Ono Sokki's FFT analyzer, this number of points is referred to as the sample points (frame length).

By performing an FFT on time data of sizes 64, 128, 256, 512, 1024, 2048, and 4096, frequency data points of 25, 50, 100, 200, 400, 800, and 1600 are obtained.

In this way, frequency resolution depends on the number of FFT sample points.

Rotational frequency ratio analysis is a method used for frequency analysis of vibrations and noise in rotating machinery, where the pulses from a pulse generator attached to the rotating body are used as an external sampling clock for signal sampling.

In frequency analysis, 1 Hz is a component that completes 1 cycle in 1 second. In contrast, in rotational harmonic analysis, the first harmonic refers to a component that completes 1 cycle for one rotation of the reference rotating body.
The second harmonic is a component that completes 2 cycles for one rotation, which is twice that of the first harmonic. Thus, to perform an analysis based on the variations per rotation, it is necessary to conduct sampling synchronized with the rotational speed.
If the internal sampling clock is used as is, the number of sampling points per rotation will change if the rotational speed varies; however, if a clock synchronized with the rotational pulse is used as the sampling clock, the number of sampling points per rotation will remain constant.

For example, considering a rotating body spinning at 600 r/min, the primary rotation is (600 r/min) / 60 = 10 Hz, and the secondary rotation is 20 Hz.
When the rotational speed increases to 700 r/min, the primary rotation rises to 11.7 Hz, and the secondary rotation increases to 23.3 Hz. In this way, the frequency fluctuates with changes in rotational speed, but if normalized as orders, it becomes easier to focus on certain components without being affected by rotational variations.

As an application of rotational frequency ratio analysis, there is rotational tracking analysis.

Rotational tracking analysis involves tracing the changes in amplitude of certain frequency components with rotational speed as the parameter on the horizontal axis. This allows us to determine which components of the rotating equipment are resonating at a specific rotational speed, or which multiples (or orders) of frequency components are resonating.

The measured open-loop and closed-loop transfer functions can be converted into their respective forms through calculations.

If there are no feedback elements.

The obtained open-loop transfer function.Then, the closed-loop transfer function.is

The obtained closed-loop transfer function isThen, the open-loop transfer function.is

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

Furthermore, the inverse Fourier transform of the cross-spectrum yields the mutual correlation function, and the inverse Fourier transform of the frequency response function results in the impulse response.

The signal input into the filter experiences a delay when it is output. The group delay characteristics refer to the amount of delay time each frequency of the output signal has relative to the input. Specifically, it is derived from the phase characteristics (the phase difference between input and output) differentiated with respect to angular frequency, and is used to evaluate the characteristics of filter circuits. This characteristic varies in value (delay time) depending on the filter circuit and frequency. When using filters that have different delay times depending on frequency, waveform distortion occurs in the output signal relative to the input signal.

two signalsLet the Fourier transforms be X (f ) and Y (f ), and let the conjugate complex of X (f ) be X* (f ). The cross-spectrum W _xy (f ) is defined by the following equation.

The cross-spectrum is the average of the products of the spectral components of two signals at certain frequency components, where the X-axis represents frequency, and the Y-axis is...It is represented by the cross-spectrum. A large value in the cross-spectrum at a certain frequency indicates that there is a strong correlation between the frequency components of the two signals at that frequency, as well as a large magnitude of both components. Furthermore, the cross-spectrum is used in the calculation of mutual correlation functions, transfer functions, and coherence functions.

Cepstrum is the result of applying a Fourier transform to the logarithm of the power spectrum obtained through a Fourier transform. The horizontal axis of the cepstrum takes values in a dimension of time called kefrequency.

When a signal with periodicity is input into a system and the period is long, it appears in the long frequency part of the cepstrum and can be extracted as the fundamental period. Additionally, the short frequency part concentrates information that represents the system's transfer characteristics, and by performing an inverse Fourier transform on this part, the envelope of the logarithmic power spectrum (referred to as the lifted envelope) can be obtained. This envelope is specific to the system and does not depend on the spectrum of the input signal.

As an application, there is the extraction of fundamental frequencies and spectral envelopes from sound waves and biological waves.

Vibration waveforms observed in mechanical vibration systems typically contain various harmonic components in addition to the fundamental frequency component. When a sine wave is added to the transmission system, distortion components, known as harmonic components of the added sine wave, appear in the output signal due to the nonlinear characteristics of the transmission system. Therefore, by focusing on this distortion, analyses of the harmonic components of the vibration waveforms and output signals are conducted to examine the characteristics of the vibrations and the fidelity of the transmission system.

Now, if the observed waveform, generally the output waveform, is composed of the fundamental frequency f ₁ and harmonic components such as the second harmonic f ₂, the third harmonic f ₃, …, and their respective RMS values are | E ₁ |, | E ₂ |, | E ₃ |, …, then the total distortion factor is

The measured value is read as a voltage value, but if the reference value for signals such as acceleration, pressure, or sound is set, the voltage value can be calibration to the reference value to be read as a physical quantity.

Example 1: The sensitivity of the acceleration pickup isWhen it is 100 mV, it becomes 0.1 V/EU (the obtained voltage value multiplied by 10) and the unit is m/s ².

Example 2: In the case of calibrating a microphone, an acoustic calibrator, and a sound level meter, the calibration should be done using power spectrum data to ensure that the overall (dB value) matches the calibration value.

The real and imaginary parts of the frequency response function are plotted separately against the frequency axis and displayed one above the other, which is called a Cochrane diagram. It can be used for estimating natural frequencies.

Coherence function (relevance function)indicates the degree of causal relationship between the input and output of a system.takes a value between 0 and 1.In the case where it is 1, it indicates that the output of the system at that frequency is entirely due to the measured input.In that case, regarding that frequency, the output of the system is completely unrelated to the measurement input.In such cases, it is considered that there are signals unrelated to the measurement, noise generated within the system, non-linearity of the system, or time delays in the system.

The coherence function of the results measured between the two channelsA small measurement indicates that the measurement results are inaccurate. Such inaccuracies should not be displayed.The feature that displays only the large parts is the coherence ranking function.The value can be set arbitrarily,The transfer function is not displayed for frequencies below that value.

The product of the coherence function and the output side's auto power spectrum is called coherent output power (C.O.P.).

C.O.P represents the auto power spectrum attributed to the measurement input among the output's auto power spectra.