# Nonlinear Dynamics of Vibro-Impacting Indenter
## 5 Results and Discussion  
For the case of the metal sample, the experimental and numerical time histories of the acceleration of the indenter for the excitation frequencies of 40 Hz, 60 Hz, 80 Hz and 90 Hz are presented as black and red solid lines in the left column in Figure 3, respectively. A total duration of two consecutive cycles is displayed in the time histories for all the four excitation frequencies. The moments at which impacts occur between the indenter and the sample are observed by noting the sharp decelerations in the time histories. It can be seen in Figure 3 that, when the excitation frequency is 40 Hz, the oscillations are of period-1, and the rig appears to indent the metal sample multiple times, as it is forced towards the metal sample with more than five impacts per forcing cycle and a maximum deceleration of approximately 15.7 g at the first impact followed by a decreasing deceleration in subsequent impacts. As the excitation frequency increases, the oscillations of the indenter are still of period-1. The magnitude of the deceleration increases to 33.8 g at 60 Hz, 49.3 g at 80 Hz and 65.8 g at 90 Hz. Moreover, the first impact tends to occur at a later phase of the increasing input excitation voltage in each forcing cycle as the excitation frequency increases. This is accompanied by a lower number of impacts per forcing cycle, where the numbers are 5, 2 and 1 at 60 Hz, 80 Hz and 90 Hz, respectively. Furthermore, by comparing the acceleration time histories and their corresponding numerical results, it can be noted that there is a qualitative match between them, especially in the cases where a higher excitation amplitude and frequency are applied. The maximum deceleration at the first impact and the number of impacts per forcing cycle are both qualitatively and quantitatively similar in general. The overall dynamical behaviour of the indenter observed experimentally is still predicted numerically, although the deceleration magnitudes of the subsequent impacts are slightly different from those of the predicted responses. The difference between the experimental and numerical acceleration time histories may be attributed to some of the characteristics of the experimental dynamical behaviour that are not yet accounted for in the numerical model and the influence of the electrical circuit inside the electrodynamic shaker that is assumed to be negligible. \begin{table}
\begin{tabular}{l c c c c c c c} \hline Excitation Frequency (Hz) & 30 & 40 & 50 & 60 & 70 & 80 & 90 & 100 \\ \hline Input Voltage Magnitude (V) & 0.09 & 0.14 & 0.16 & 0.22 & 0.27 & 0.32 & 0.42 & 0.48 \\ \hline Forcing Magnitude (N) & 220 & 300 & 370 & 440 & 540 & 650 & 840 & 900 \\ \hline \end{tabular}
\end{table} 
Table 1: Input excitation frequencies and their corresponding input voltage magnitudes and forcing magnitudes. The Fast Fourier Transform (FFT) spectra of the acceleration time histories are presented as black solid lines in the right column in Figure 3 for the range from 0 to 1000 Hz. It can be observed in the frequency spectra that, due to the impact nature of the motion, the fundamental frequency, which is also the excitation frequency, appears along with its harmonics at higher frequencies in the spectra. However, the distribution of the magnitudes in the frequency spectrum is different for all four excitation frequencies. For example, it can be seen in Figure 3e that, when the excitation frequency is 40 Hz, the peak value does not occur at the fundamental frequency, while the harmonics of the acceleration response have four major peaks in the vicinity of 80 Hz, 280 Hz, 520 Hz and 720 Hz, with the highest harmonic peak being at 720 Hz followed by the first peak at 80 Hz, with their ratio being 1.81 to 1. This ratio decreases with the increasing excitation frequency, and when the excitation frequency is 90 Hz, the highest peak value occurs at the fundamental frequency.