Industry 4.1. Группа авторов
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Название: Industry 4.1
Автор: Группа авторов
Издательство: John Wiley & Sons Limited
Жанр: Техническая литература
isbn: 9781119739913
isbn:
2.3.3.2 Frequency Domain
Frequency‐domain SFs can reflect the signal’s power distribution over a range of frequencies. Theoretically, periodic signals are composed of many sinusoidal signals with different frequencies, such as the triangle signal, which is actually composed of infinite sinusoidal signal (fundamental and odd harmonics frequencies).
Thus, the frequency spectrum of the periodic signal can be obtained by the projection of these sine and cosine signals in the frequency axis by the Fourier transform (FT) technique [10], which is probably the most widely used method for signal processing. Since then, a signal can be represented by the spectrum of frequency components in the frequency domain.
As the conversion of time and frequency domain shown in Figure 2.17, one time‐domain signal composed of two different waveforms with frequency is converted into the frequency domain. Two magnitudes of corresponding sine or cosine signals are represented at the specific location on the frequency spectrum.
Figure 2.17 View of the time and frequency domains.
One drawback is that the calculation and execution are very time‐consuming when dealing with a large amount of datasets. Thus, fast Fourier transform (FFT) based on FT is implemented to deal with nonperiodic functions and discrete time‐domain signals [10]. FFT can reduce the complexity of computing FT and rapidly compute the global information of the frequency distribution from any signal. The famous mathematician Gilbert Strang also described that FFT is “the most important numerical algorithm of our lifetime” in 1994 [14].
FFT directly decomposes any discrete signal x[n] into the frequency spectrum by the orthogonal trigonometric basis functions as in (2.13), where l = 1, 2, …, N.
Figure 2.18a shows a vibration signal collected from a practical rotary spindle under the speed of 2,000 rotations per minute (rpm) with the sampling rate being 2,048 Hz. Figure 2.18b illustrates that there are three major peaks at 33.3, 66.6, and 99.9 Hz on the frequency spectrum, which correctly represent the fundamental frequency, the second harmonic frequency, and the third harmonic frequency, respectively. Some unimportant frequency components with relatively small amplitude (usually the noises) among three peaks are hard to observe in the time domain, but they are very clear to be detected and can be ignored.
Figure 2.18 A vibration signal: (a) in time‐domain; and (b) in FFT spectrum.
The set of FFT‐based SFs SFFFT(q) can be extracted from the summation of FFT[n] values close to the qth certain frequency band delimited by a lower frequency and an upper frequency of critical characteristics, as expressed in (2.14).
where q=1, 2, …, Q and
ufqqth upper frequency of the critical characteristics; and
lfqqth lower frequency of the critical characteristics.
For stationary signals, FFT provides a good description in global frequency bandwidth without indicating the happening time of a particular frequency component and whether the resolution scale in both time and frequency domains are enough or not.
However, FFT might be limited to processing stationary signals. A highly non‐stationary signal cannot be adequately described in the frequency domain by FFT, since its frequency characteristics dynamically change over time. Thus, extracting other SFs in the time–frequency domain is necessary.
2.3.3.3 Time–Frequency Domain
The time‐frequency analysis describes a nonstationary signal in both the time and frequency domains simultaneously, using various time‐frequency representations. The advantage is the ability to focus on local details compared to other traditional frequency‐domain techniques.
Although short‐time Fourier transform (STFT) method is proposed to retrieve both frequency and time information from a signal afterward, the deficiency is still yet to be overcome completely. STFT calculates FT components of a fixed time‐length window, which slides over the original signal along the time axis.
STFT adopts an unchanged resolution in both time and frequency domains, as shown in Figure 2.19. Heisenberg uncertainty principle [15] states that it is impractical to use good resolution in both time and frequency axes since the product of the two axes is a constant. A longer window has better time resolution but worse frequency resolution, and vice versa. In general, nonstationary components often appear in high frequency and only happen in a very short period of time, but this unchanged window length makes resolution in high frequency unclear.
Figure 2.19 Unchanged resolution of STFT time‐frequency plane.
One representative technique to solve the FT‐related issues is the wavelet packet transform (WPT) decomposition [10, 11, 16]. WPT not only dynamically changes resolutions both in time and frequency scales but also has more options to change its convolution function depending on characteristics of the signal.
In regards to the resolution of Figure 2.20, it is assumed that low frequencies last for the entire duration of the signal, whereas high frequencies appear from time to time as short bursts. This is often the case in practical applications.