# Evaluation of the Acceleration Vibration Signal for Aggregates of the Horizontal Drilling Stand
## 3 Theoretical Background and Calculation  
### Time Domain Vibration Analysis  
Values of oscillating motion parameters the vibration signal respectively generally changes over time [28; 29; 30]. For evaluation purposes, it is usually necessary to have a peak, peak-to-peak, and the total energy content of the signal, which is given by the average and effective value [31; 32].  
The following are considered essential time characteristics for the vibration acceleration signal:  
* amplitude \(x(t)\), which is the instantaneous value of the monitored signal parameter in time \(t\);  
Figure 4: Result of practical drilling in laboratory conditions ((**a**) block-shaped rock sample with boreholes, (**b**) drilling rock core).  
peak \(x_{p}(t)\), that represents the maximum distance of the peak of the wave from the reference value, usually the \(x\)-axis;
* peak-to-peak \(x_{pp}(t)\), which is the maximum distance of opposite peaks of the wave;
* the average value \(x_{avg}(t)\) that represents the average value of the amplitude during the wave according to the equation: \[x_{avg}(t)=\frac{1}{N}\sum_{i=1}^{N}|x_{i}(t)|;\] (1)
* effective value (Root Mean Square--RMS) which is the objective value used in the diagnostic guidelines, determined by the equation [33]: \[\text{RMS}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}x_{i}^{2}(t)}.\] (2)  
The established statistical characteristics evaluate the vibration signal only statically at a given time and do not provide information about the dynamic properties of the signal [34; 35; 36; 37]. They do not contain information on the statistical dependence of the signal realization values at two different time points \(t_{1}\) and \(t_{2}\).  
The dynamic properties of changes over time are characterized by an autocorrelation function in evaluating a single signal. The autocorrelation function can be evaluated from a single implementation of the measured signal and the mutual time shift \(\tau=t_{2}-t_{1}\). Although, in general, the autocorrelation function depends on the time points \(t_{1}\), \(t_{2}\) resp. \(t_{2}-t_{1}\), in the case of a stationary signal, it depends only on \(\tau\).  
An essential statistical characteristic in time analysis is the signal's autocorrelation, resp. autocorrelation function \(R_{xx}(\tau)\) of the measured vibration signal \(x(t)\). The autocorrelation function \(R_{xx}(\tau)\) is determined from one implementation and represents a generalization of the mean value of the product of time-shifted values. The following equation defines it:  
\[R_{xx}(\tau)=\frac{1}{N}\sum_{i=1}^{N}x_{i}(t)(t+\tau). \tag{3}\]  
The periodic parts of the vibrational signal \(x(t)\) remain in the autocorrelation function, and the non-periodic ones quickly disappear. From the autocorrelation function, it can be determined the period of the periodic part of the measured signal. The autocorrelation function provides an information about the dependence of the values of the function \(x(t)\) at time \(t\) from the values at time \(t+\tau\).  
From the time waveform of the autocorrelation function, it can be said for practice that the faster the autocorrelation function decreases for increasing \(|\tau|\)[38], the smaller the statistical dependence of the signal's successive values \(x(t)\) and the signal will also cover a broader frequency spectrum. The slower time waveform of the autocorrelation function is characterized by more significant statistical dependencies of successive signal values and greater system inertia. If the signal contains a periodic component, the autocorrelation function has the same period as the signal.