# Evaluation of the Acceleration Vibration Signal for Aggregates of the Horizontal Drilling Stand
## 3 Theoretical Background and Calculation
### Frequency Domain Vibration Analysis  
Frequency domain signal processing, when used correctly, eliminates the shortcomings of a time-domain analysis [39; 40]. The method makes it possible to locate emerging faults vibrations of individual parts of the monitored drilling machine [41; 42; 43]. The amplitude spectrum of the measured vibration signal \(x(t)\) is obtained by frequency analysis.  
The basis of frequency analysis is mainly Discrete Fourier Transform (DFT) and Fast Fourier Transform (FFT). The measured time-varying vibration signal is then processed numerically [44]. The vibration acceleration signal \(x(t)\) is sampled in the time-domain in the analog-to-digital converter (ADC).  
Its values are determined at time points distant by the regular sampling period \(T_{s}\) (i.e., sample time) at the sampling frequency \(f_{s}\). Because in practice, only a finite number, \(N\), of measured signal samples is available, it is necessary to use a discrete Fourier transform.  
Before signal processing by the DFT method or FFT algorithm is necessary to be aware of the basic parameters in the calculations:  
* frequency range represents the baseband from 0 Hz to \(f_{s}/2\);
* \"zoom\" factor, when using a frequency magnifier, indicates how many times the frequency range is smaller;
* the number of spectral lines is usually \(N/2\);
* the sequence number of the spectral line;
* frequency analysis resolution, indicating the spacing between the spectral lines.  
The used Fourier transform \(X(j\omega)\) to obtain the amplitude spectrum of the vibration signal \(x(t)\) from individual parts of the monitored drilling rig is defined by the integral equation:  
\[X(j\omega)=\int_{-\infty}^{\infty}x(t)e^{-|2\pi ft}\,\mathrm{d}t. \tag{4}\]  
If a directly measured signal generated by any part of the drilling rig is processed, a numerical method known as Discrete Fourier Transform (DFT) is used [45]. It is suitable for calculation (DFT) to use an efficient Fast Fourier Transform (FFT) algorithm, which is used to process vibration signals in the frequency domain to obtain the resulting amplitude spectrum. The following equation defines DFT:  
\[X(k)=\sum_{n=0}^{N-1}x(n)e^{-|2\pi k\frac{n}{N}},\quad n=0,\ldots,N-1;\quad k =0,\ldots,N-1. \tag{5}\]  
The discrete value \(X(k)\) represents the amplitude. The values of \(x(n)\) and \(X(k)\) have the same physical dimension.  
In industrial practice, power spectra are also calculated in addition to amplitude spectra [46; 47]. In some complex industrial processes, in which signals are generated, they have a stronger informative value about the current state of the technological process [48; 49]. The following equation was used to calculate the power spectrum:  
\[S_{xx}(\omega)=\frac{1}{2\pi N}\left|\sum_{n=1}^{N}x(n)e^{-|2\pi f_{n}|}\right| ^{2}, \tag{6}\]  
where \(1\leq n\leq N\) applies to a signal in the discrete form \(x(n)=x(n\Delta t)\) with a finite number of samples \(N\).  
The amplitude and power spectra of vibration signals from individual parts of the monitored drilling rig were analyzed. The FFT algorithm was used for the measured signal with a sampling frequency of \(f_{s}\) = 18 kHz to obtain a range from 0 to 9 kHz, a sampling period of \(T_{s}\) = 0.55 \(\upmu\)s a segment length of \(N\) = 2048 samples. The measured sampled signal was used; therefore, mathematical equations in a discrete form are presented.