## 4 Experimental Results and Discussion  
### A Beating Phenomenon Described in Vibration Velocity  
Figure 3 presents the evolution of the headstock vibration velocity during a time interval of 200 s, described with 1 MS (or 1,000,000 samples as well) so a sampling interval of \(\Delta t\) = 200 \(\upmu\)s, when the main spindle (shaft 1) rotates (and shaft 2, as well) in the steady-state regime, with constant IAS, with an average value of \(\omega_{1}\) = 109.2369 rad/s (for \(f_{1}\) = 17.3856 Hz average rotation frequency, or 1043.1 revolutions per minute on average).  
It is obvious that Figure 3 depicts a vibration beating phenomenon with nodes and anti-nodes, with a very high value of the period \(T_{b}\) (96.6 s) and consequently with a very small value of beat frequency \(f_{b}\) = 1/\(T_{b}\) = 1/96.6 Hz. The beating phenomenon proves that the IAS \(\omega_{1}\) and \(\omega_{2}\) and also rotation frequencies \(f_{1}\) and \(f_{2}\) as well, are slightly different because the diameters of pulleys 4 and 5 involved in belt transmission are not strictly the same. With \(T_{1}\) = 1/\(f_{1}\) and the relationship between \(T_{1}\) and \(T_{b}\) from Equation (3) there are \(n\approx T_{b}f_{1}\approx\) 1679 periods \(T_{1}\) between nodes (and between anti-nodes as well). This is an approximated value of \(n\) because the frequency \(f_{1}\) is not rigorously constant (as is proved, later in this paper). According to Equation (3), at each \(n\) complete rotations of shaft 1, the shaft 2 makes \(n+1\) or \(n-1\) rotations (one rotation difference), which means that the experimentally revealed value of the belt transmission speed ratio is \(\omega_{2}/\omega_{1}\) = \(T_{1}/T_{2}\) = \(n/(n\pm 1)\approx\) 1679/(1679 \(\pm\) 1). This is also the ratio between pulleys diameters: the diameter of pulley 4 divided by the diameter of pulley 5 (assuming that there is no slipping between the belt and the pulleys).  
The additional mass 6 (Figure 2a) was placed in a certain angular position on pulley 5, in order to obtain a maximum value of the amplitude \(A_{2}\), and a maximum difference between amplitudes in anti-nodes and nodes as well. It is obvious that the main spindle (shaft 1) is also unbalanced; otherwise the beating phenomenon does not occur.  
Figure 3: The evolution of the velocity of headstock vibrations with a beating phenomenon due to rotary unbalanced shafts; here \(T_{b}\) is the beat period, A is a label for a future comment on signal evolution.  
Figure 4 presents a zoomed-in detail in the area labelled A in Figure 3. Here the dominant component (\(\approx\)6 mm/s amplitude) is the sum of two vibrations created by rotary unbalances; the other low amplitude (and high frequency) components are related by vibrations generated by some other headstock rotary components.  
With the values for \(f_{1}\) and \(f_{b}\) revealed before, the frequency \(f_{2}\) is a result of Equation (5), with two possible values (\(f_{2}=f_{1}-f_{b}=17.3752\) Hz or \(f_{2}=f_{1}+f_{b}=17.3959\) Hz). Because in Equation (5) the notations \(f_{1}\) and \(f_{2}\) are arbitrary, this equation should be reconsidered as \(f_{b}~{}=~{}\big{|}f_{2}-f_{1}\big{|}\).  
As a consequence, the angular speed \(\omega_{2}=2\pi f_{2}\) has two possible values (\(\omega_{2}=109.1716\) rad/s or \(\omega_{2}=109.3016\) rad/s), as does the speed ratio (\(\omega_{2}/\omega_{1}\)) of the driving belt (with \(\omega_{1}=109.2369\) rad/s). To find the right value of \(f_{2}\) (and \(\omega_{2}\) as well) the technique described in [42] should be used (with an IAS sensor placed on shaft 2).  
The evolution from Figure 3 is an addition of vibrations velocities (\(v=dy_{1}/dt+dy_{2}/dt\)) generated by both of the unbalanced shafts (1 and 2). It is expected that the beating phenomenon keeps the main characteristics (e.g., \(T_{b}\), \(T_{c}\) values or \(f_{b}\), \(f_{c}\) values, as well) if it is described using the addition of vibration displacements (\(s=y_{1}+y_{2}\)), except for the amplitudes in nodes and anti-nodes which significantly decreases.