## 4 Mechanical Model of the Mechanical Resonator and a Broadband Effect
We used nodal initial conditions defined by the vertical position of the pendulum for initial angular displacement and two different initial conditions for angular velocity: zero and fairly large 80 rad/s. Two different excitation amplitudes were used, \(A=0.002\) and \(A=0.05\) m.  
In Figure 10a we show the results of standard deviation and mean value for nodal initial velocity and small excitation. We noticed that the pendulum oscillates with small amplitude with visible resonance at the frequency of 6.5 Hz. This frequency corresponds to the natural frequency of the mechanical resonator. Simultaneously, the mean angular velocity is zero. In Figure 10b we present the \begin{table}
\begin{tabular}{c c} \hline \hline
**System Parameters** & **Values [Unit]** \\ \hline \(h\) & 0.03325 [m] \\ \(c\) & 0.000003 [N s/(kg m\({}^{2}\))] \\ \(g\) & 9.81 [m/s\({}^{2}\)] \\ \(m\) & 0.0018 [kg] \\ \(F_{m}\) & 0.03 [N] \\ \(I\) & 0.0000377 [kg m\({}^{2}\)] \\ \(A\) & 0.002 or 0.05 [m] \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Model system parameters used for numerical simulations (as written in Equation (4)). Figure 10: The frequency sweep results for a mechanical resonator. \(\Omega_{std}\) and \(\Omega_{mean}\) denote variability (standard deviation) and mean value (averaged in the time interval [100 s, 200 s]) of the rotational velocity versus excitation frequency. Here the amplitude of excitation was fixed at 2 mm (**a,b**) and 5 cm (**c,d**), while the initial conditions: angle \(\varphi(t=0)=0\), velocity \(\dot{\varphi}(\text{t}=0)=0\) (for **a,c**) and 80 (for **b,d**). results of calculations repeated for large initial velocity and we observe another response in angular velocity. It remains finite up to frequency\(f\) of about 24 Hz and collapses to the solution presented in Figure 10a. During the fairly large angular velocity response of 54 rad/s, the variability of this velocity was fairly small. Note that these results were obtained for the excitation amplitude which can be generated on the shaker. To complete the story, we enlarged the amplitude of excitation and obtained richer responses. In Figure 10b, we can observe a number of higher resonances which multiplies \(f\) = 6.5 Hz. This is a typical behavior for a non-linear system. Non-linearity based higher order resonances manifest with large enough excitation. On the other hand higher value of initial velocity drives the system to rotations. Interestingly, we observe a wide spectrum of rotational solutions separated by rotational instabilities leading to oscillatory vibrations. In addition, various rotational solutions appear. For small frequency \(f\) < 6 Hz we observe a similar dynamical response to the situation presented in Figure 10a. The output velocity is stable with changing frequency raging about 54 rad/s. This solution is related to overcoming the potential barrier (Figure 9c). Referring to Figure 9a,b it is easy to find that a velocity of 54 Hz is sufficient to overcome the potential barrier and obtain a rotational solution. This solution resembles the type of vibration coherence resonance [20], where excitation frequency is less important than the amplitude important to generate a large orbit motion with a minimum of excitation energy necessary to overcome the potential barrier. For higher frequency intervals such as [9 Hz, 15 Hz] and [35 Hz, 45 Hz], the response angular velocity is proportional to excitation frequency. Interestingly, for the first mentioned interval, proportionality is 1:1 while for the next it is 1:1/3. This is evidence that the solution has a sub-harmonic 3 property. In a summary of the results of Figure 10, one can conclude that the system has a great capability toward frequency broad band responses. In spite of the strong model simplifications, the simulations shows the large frequency region of rotational solutions. Such sub-harmonic solutions can be very useful for energy harvesting as proved by [17; 53].