## 2 A Theoretical Approach  
Assume that the rotary unbalance of each shaft (each of which rotates at angular speeds of \(\omega_{1}\) and \(\omega_{2}\), respectively) is reducible at the asymmetry of mass distribution with unbalance masses \(m_{1}\) and \(m_{2}\), respectively, placed in a single plane at distances \(r_{1}\) and \(r_{2}\), respectively, to the rotation axis. The horizontal projection of the centrifugal forces of each rotary unbalance (\(F_{1}~{}=~{}m_{1}\omega_{1}^{2}r_{1}\) and \(F_{2}~{}=~{}m_{2}\omega_{2}^{2}r_{2}\)) generates vibration displacements \(y_{1}=kD_{af1}F_{1}cos(\theta_{1})\) and \(y_{2}=kD_{af2}F_{2}cos(\theta_{2})\), where \(k\) is the stiffness of the headstock and the lathe foundation, \(D_{af1}\) and \(D_{af2}\) are the dynamic amplification factors and \(\theta_{1}=\omega_{1}t+\varphi_{1}\) and \(\theta_{2}=\omega_{2}t+\varphi_{2}\) are the instantaneous values of the angle of the centrifugal forces with respect to the horizontal direction (\(\varphi_{1}\) and \(\varphi_{2}\) being the instantaneous values of these angles at \(t=0\)). With these considerations, a complete description of \(y_{1}\) and \(y_{2}\) of the vibrations is given below:  
\[y_{1}\ =\ kD_{af1}m_{1}\omega_{1}^{2}r_{1}cos(\omega_{1}t+\varphi_{1})\ =\ A_{1}cos(\omega_{1}t+\varphi_{1}) \tag{1}\]  
\[y_{2}\ =\ kD_{af2}m_{2}\omega_{2}^{2}r_{2}cos(\omega_{2}t+\varphi_{2})\ =\ A_{2}cos(\omega_{2}t+\varphi_{2}) \tag{2}\]  
Here \(A_{1}=kD_{af1}m_{1}\alpha_{1}^{2}r_{1}\) and \(A_{2}=kD_{af2}m_{2}\alpha_{2}^{2}r_{2}\) are the vibration amplitudes of the two shafts, respectively. The headstock and the lathe vibrate as a single body on its foundation (as a mass-spring-damper system) with a vibratory motion which is the result of the addition \(y_{1}+y_{2}\) of these two vibrations, a periodical non-harmonic motion that presents itself as a beating phenomenon [23], with nodes and anti-nodes (as the simulation from Figure 1 proves). According to Figure 1, if the period of vibration \(y_{1}\) is \(T_{1}=2\pi\omega_{1}\) and \(T_{2}=2\pi/\omega_{2}\) is the period of vibration \(y_{2}\), then the period \(T_{b}\) of the beating phenomenon (the beat period being the time between two anti-nodes or between two nodes, as well) and the periods \(T_{1}\), \(T_{2}\) (with \(T_{2}<T_{1}\)) should fulfill this obvious condition:  
\[T_{b}\ =\ nT_{1}\ =\ (n+1)T_{2} \tag{3}\]  
with \(n\) being a natural number, defined from Equation (3) as:  
\[n\ =\ T_{2}/(T_{1}-T_{2}) \tag{4}\]  
In Figure 1\(n=7\). If in Equations (3) and (4) the periods are replaced by frequencies (\(T_{b}=1\!f\!_{b}\), \(T_{1}=1\!f\!_{1}\), \(T_{2}=1\!f\!_{2}\)), then the resulting frequency \(f_{b}\) of the beating phenomenon (beat frequency or the number of nodes per second, as well) is:  
\[f_{b}\ =\ f_{2}-f_{1} \tag{5}\]  
In Figure 1\(f_{2}=8\) Hz and \(f_{1}=7\) Hz; these generate \(f_{b}=1\) Hz with \(T_{b}=1\)s (\(A_{1}=10\), \(A_{2}=8\), \(\varphi_{1}=0\), \(\varphi_{2}=-\pi/2\)).  
Figure 1: A simulation of the beating phenomenon, where: \(y_{1}\) is the vibration of shaft 1; \(y_{2}\) is the vibration of shaft 2, \(T_{b}\) is the beat period; and \(T_{c}\) is the period of the resultant vibration \(y_{1}+y_{2}\).  
According to [39], the resultant waveform of the vibration addition \(y_{1}+y_{2}\) has the frequency \(f_{c}=1/T_{c}\) (as a combination frequency or modulation frequency, with the period \(T_{c}\) highlighted on Figure 1) defined as:  
\[f_{c}\ =\ (f_{1}+f_{2})/2 \tag{6}\]  
This paper will prove by simulations and experiments that this definition is not accurate, especially when \(A_{1}\
eq A_{2}\).  
These theoretical considerations and some other supplementary issues and procedures will be confirmed by experimental approach in this paper.