## 4 Experimental Results and Discussion
### The Influence of the Lathe Suspension Dynamics on Beating Vibrations Amplitude  
The relative high vibration displacement amplitude of the headstock during the beating phenomenon (as shown in Figure 7) has an evident explanation: the vibration frequencies \(f_{1},f_{2}\) and \(f_{c}\) as well, are close to the first resonant frequency (vibration mode) of the headstock and lathe on its foundation (as a single body mass-spring-damper vibratory system). This means that the dynamic amplification factors \(D_{af1}\) and \(D_{af2}\), (involved in Equations (1) and (2)) are significantly higher than 1 (because of resonant amplification). In order to prove that, the resources of a very simple experiment performed with the same experimental setup are available: the evolution of headstock vibration velocity after an impulse excitation produced with a rubber mallet (hammer) in the same direction with \(y_{1}\) and \(y_{2}\) vibrations (as Figure 17 describes). Figure 16: The evolution of the frequency measurement error \(\varepsilon_{f}\) (for a simulated harmonic signal with combination frequency \(f_{c}=17.383\) Hz) versus the evolution of the first sampling time (\(t_{1}=0\div\Delta t\), or \(t_{1}=0\div 200\)\(\upmu\)s) involved in the first zero-crossing time (\(t_{zc1}\)) calculus. Here the blue curve partially depicts the free damped vibration velocity \(v_{fd}\) (acquired with the geophone sensor); the red coloured one depicts the best fitting curve of a part of the free response (with 25,000 samples and 500 ns sampling time). The curve fitting [44] was done in Matlab, with an adequate computer program specially designed for this paper, based on a known theoretical model of free viscous damped vibration velocity response [45]:  
\[v_{fd}(t)\ =\ a\cdot e^{-bt}sin(p_{1}t+\alpha) \tag{9}\]  
The best fitting curve (in red in Figure 17) is described with \(a\) = 1.170\(\cdot\)10\({}^{-3}\) m/s, \(b\) = 5.899 s\({}^{-1}\) (as damping constant), \(p_{1}\) = 117.952 rad/s (as angular frequency of damped harmonic vibration) and \(\alpha\) = 4.986 rad (as phase angle at the origin of time \(t_{0}\) on Figure 17). The angular natural frequency (\(p\ =\ \sqrt{p_{1}^{2}+n^{2}}\) = 118.099 rad/s) and the damping constant \(b\) are useful in the definition [45] of dimensionless dynamic amplification factor \(D_{df}\) from forced vibrations of harmonic excitation (as happens during the beating phenomenon, assuming that the combination frequency is approximately constant):  
\[D_{df}\ =\ \frac{1}{\sqrt{\left[1-\left(\frac{\omega}{p}\right)^{2}\right]^{ 2}+\left(2\frac{\omega}{p}\cdot\frac{b}{p}\right)^{2}}} \tag{10}\]  
Here \(\omega\) = 2\(\pi\)\(f\) is the angular frequency of harmonic excitation on frequency \(f\). Based on previous experimental results of curve fitting (with \(b\) and \(p\) values in Equation (10)) Figure 18 presents the simulated evolution of \(D_{df}\) related to the frequency of excitation (1 \(\div\) 35 Hz range). Because of a low damping constant \(b\), the system presents resonant amplification, with a maximum value \(D_{df}\) = 10.01 on \(f\) = 18.749 Hz frequency. Based on the previous experimentally determined frequencies \(f_{1}\) and \(f_{2}\), with \(f\) = \(f_{1}\) = 17.3856 Hz gives the result \(D_{df1}\) = 5.831 and with \(f\) = \(f_{2}\) = 17.3752 Hz (or \(f\) = \(f_{2}\) = 17.3959 Hz) the result is \(D_{df2}\) = 5.803 (or \(D_{df2}\) = 5.859). For \(f\) = \(f_{c}\) = 17.383 Hz (Figure 12) the result is \(D_{dfc}\) = 5.824 (the coordinates of point A on Figure 18). This means that, because of mechanical resonance, the vibration amplitude generated by the beating phenomenon of the headstock and the lathe on its foundation (already revealed in Figure 7) is amplified on average by 5.824 times. Figure 17: Some experimental results on signal processing related to free damped vibration of headstock after an impulse excitation (with a rubber mallet).