## 4 Experimental Results and Discussion
### The Description of the Beating Phenomenon in Vibration Displacement by Numerical Integration  
The vibration displacement evolution can be obtained from vibration velocity evolution by numerical integration (antiderivative calculus). Based on the approximate definition of velocity (derivative of displacement) \(v=ds/dt\approx\Delta s/\Delta t\), a current sample of velocity \(v_{i}\) is defined using two successive samples of displacement \(s_{i}\), \(s_{i-1}\) (in the displacement interval \(\Delta s=s_{i}-s_{i-1}\)) and the values of time \(t_{i}\), \(t_{i-1}\) for these samples, in the sampling interval \(\Delta t=t_{i}-t_{i-1}\) (usually this is a constant value) as:  
\[v_{i}~{}=~{}\frac{s_{i}-s_{i-1}}{\Delta t} \tag{7}\]  
This is an approximation of the first derivative of displacement as backward finite difference, with \(i>1\)[43]. The current sample of displacement \(s_{i}\) can be simply mathematically extracted from Equation (7) as:  
\[s_{i}~{}=~{}v_{i}\Delta t+s_{i-1} \tag{8}\]  
Figure 4: A zoomed-in detail of Figure 3 in the area labelled as A. Equation (8) describes a sample \(s_{i}\) of displacement related to velocity, this also being our proposal for a description of numerical integration of velocity (antiderivative calculus). According to Equation (8) the sample \(s_{i}\) depends on sampling interval \(\Delta t\) (here \(\Delta t\) is the time \(t_{i}-t_{i-1}\) between two consecutive samples of velocity \(v_{i}\) and \(v_{i-1}\) or two consecutive samples of displacement \(s_{i}\) and \(s_{i-1}\) as well), the velocity sample \(v_{i}\) and the previous sample of displacement \(s_{i-1}\), as the result of a previous step of numerical integration. The numerical integration from Equation (8) is available for \(i>1\). Of course, it is mandatory to know the value of the first sample of displacement \(s_{1}\), this being an indefinite value because \(i>1\). This is exactly the constant \(C\) of integration (usually an arbitrary value). Pure harmonic signals are numerically integrated, in that case, evidently \(C=0\). Figure 5a describes the graphical result of numerical integration of vibration elongation evolution from Figure 3 using Equation (8), with \(C~{}=~{}s_{1}~{}=~{}0\). Certainly this evolution is not strictly related to the vibration displacement from the beating phenomenon. We found that the oscilloscope generates a very small negative constant zero offset. As the theory of integration establishes, the numerical integration of this constant zero offset produces a component with linear evolution, experimentally confirmed in Figure 5a by the evolution with negative slope. The removal of this linear component produces the result from Figure 5b (the evolution emphasized in blue). It is evident that Figure 5b is not the expected evolution of the vibration displacement in beating phenomenon. Surely, there is not a mistake in the numerical integration proposal in Equation (8) because the numerical derivative of the evolution from Figure 5b using Equation (7) produces exactly the evolution of velocity, as Figure 6 indicates (by comparison with Figure 3). Our first attempt to explain this deficiency in this result of numerical integration is related by the constant of integration \(C\). Intuitively it is supposed that somehow the hypothesis that \(C=0\) is wrong. Perhaps the effect of this wrong hypothesis is mirrored in the evolution from Figure 5b and its effect should be removed (as the influence of negative zero offset was removed before). It was discovered by numerical simulation that the numerical integration of a computer-generated beating vibration velocity signal (similarly to those depicted in Figure 3) using Equation (8), with \(C=0\), produces a vertically shifted evolution with a constant nonzero value, which should be mathematically removed. Figure 5: (**a**) The result of numerical integration of velocity evolution from Figure 3; and (**b**) the result of removing the zero-offset influence on numerical integration of velocity from Figure 5a.