## 4 Experimental Results and Discussion
### The Evolution of Frequency for Resultant Vibration in Beating Phenomenon  
An interesting item in the beating phenomenon is the evolution of frequency of the resultant vibration \(f_{c}\) (also known as combination frequency or modulation frequency, \(f_{c}=1/T_{c}\), with \(T_{c}\) highlighted in Figure 8b). In [39] this frequency is defined as the average of both frequencies (\(f_{1}\),\(f_{2}\)) involved in the beating (Equation (6)). A beating phenomenon was simulated using the sum of two harmonic vibrations displacements \(y_{1s}(A_{1}f_{1})\) and \(y_{2s}(A_{2}f_{2})\)--already described in Equations (1) and (2) with different values of amplitudes (\(A_{2}>A_{1}\)) and frequencies \(f_{1}\) and \(f_{2}\), close to those from the experiment described in Figures 3 and 7 (with \(f_{1}=17.3856\) Hz and \(f_{2}=17.3959\) Hz), for a duration equal to \(T_{b}\) (placed between two anti-nodes). For six different values of amplitudes (\(A_{1}\) increases and \(A_{2}\) decreases), the evolution of the combination frequency \(f_{c}\) and its average value was determined on the vibration beating simulated signal, as Figure 9 indicates, using a high accuracy measurement technique from a previous work [42]. Here each peak describes the value of frequency \(f_{c}\) in vibration beating node. It is obvious that \(f_{c}\) is not constant and, in contradiction with Equation (6) and [39], \(f_{c}\
eq(f_{1}+f_{2})/2\). Here, with \(A_{1}>A_{2}\) the average \(f_{c}\) is very close to \(f_{2}\), with \(f_{c}>f_{2}\). Figure 8: (**a**) A detail concerning the evolution of velocity (Figure 3) overlaid on the numerical differentiation of the displacement depicted in Figure 7; and (**b**) a detail of area A of Figure 7 with \(T_{c}\), the period of the resultant vibration. A similar simulation was done in the same conditions, now with \(A_{1}>A_{2}\), as Figure 10 indicates (here \(A_{1}\) decreases and \(A_{2}\) increases). Similar to the simulation given in Figure 9, it is obvious that \(f_{c}\) is not constant and again, in contradiction with Equation (6) and [39],\(f_{c}\
eq(f_{1}+f_{2})/2\). The average frequency \(f_{c}\) is very close to \(f_{1}\), with \(f_{c}<f_{1}\). There are two conclusions here, in contradiction with the literature [39]:  
* The combination frequency \(f_{c}\) is not constant over a period \(T_{b}\) (even if its variation is not significant);
* The average value of the combination frequency \(f_{c}\) over a period \(T_{b}\) is practically the same as the frequency of the input vibration in the beating phenomenon (\(y_{1s}(A_{1}f_{1})\) or \(y_{2s}(A_{2}f_{2})\)), whose amplitudes are higher (e.g., if \(A_{2}>A_{1}\) then the average \(f_{c}\approx f_{2}\)). Some supplementary simulations for many other values of frequencies \(f_{1}\) and \(f_{2}\) (and consequently \(T_{b}\)) completely confirm these conclusions. Figure 10: The evolution of the instantaneous combination frequency on simulated vibration beating during a beat period (the same condition as in Figure 9, except for the amplitudes relationship: \(A_{1}>A_{2}\). Figure 9: The evolution of the instantaneous combination frequency \(f_{c}\) on the simulated vibration beating during a beat period \(T_{b}\) (with \(A_{2}>A_{1}\) and \(f_{2}>f_{1}\)). Figure 11 presents the evolution of the instantaneous combination frequency \(f_{c}\) during the vibration beating phenomenon (displacement of headstock) experimentally described in Figure 7.  
Apparently, this is a very noisy evolution. The dominant component of the signal from Figure 7 is the displacement of resultant vibration \(y_{1}+y_{2}\). The frequency measurement method [42] is based on detection of zero-crossing moments of this signal (a topic discussed later on).