## 4 Mechanical Model of the Mechanical Resonator and a Broadband Effect  
Research on small portions of mechanical energy conversion into electrical energy was started from mechanical vibrations. Results obtained were adjusted to needs of specific energy receiver paying special attention for output power, resonance frequency or energy conversion. Continuous development favors usage of new materials and systems managing conversion and flow of energy to the receiver. Nowadays, in the beginning of work on the design of new device, mathematical model is derived. Few papers have been dedicated only to mathematical modeling of different types of EH and provide crucial and useful information on this topic [47; 48; 49]. The mechanical model of the energy harvester set with a rotational pendulum can be represented by the equation of motion [50; 51; 52]. Into the equation of motion (4) for a pendulum the equation of motion correcting factor was introduced in order to reflect real situation from the experiment. To keep the rotational movement of the pendulum, the entire system with a vibration exciter was tilted, and for this angle \(\varphi\) is added to the equation of motion. We did not consider the equation for the magnetic  
Figure 8: Results of (**a**) root mean square (RMS) voltage in the operating frequency domain for different values of additional resistance, (**b**) output power in the operating frequency domain for different values of additional resistance, (**c**) RMS voltage in the operating frequency domain for open loop, (**d**) output power in operating frequency domain for open loop. circuit. In our simplified model we took into account all the variables having an impact on pendulum's movement and they are contained in the following simplified equation of motion:  
\[I\ddot{\varphi}+c\ddot{\varphi}+m\omega^{2}Ah\cos(\omega t)\sin\varphi+mgh\sin \varphi+F_{mag}sin(2\varphi+1.5)=0, \tag{4}\]  
where: \(I\)--moment of inertia of pendulum, \(c\)--damping coefficient, \(h\)--distance from center of gravity to pivot axis, \(m\)--mass of pendulum and rotational additional masses, \(g\)--gravitation, \(\omega\)--angular velocity of pendulum, \(A\)--amplitude of vertical excitation, \(F_{mag}\)--magnetic force amplitude acting on pendulum, \(\varphi\)--tilt angle of pendulum. The phase shift of 1.5 rad present in the magnetic force term denotes the tilting of the pendulum (Figure 2). Here, we simplified the angular dependence of the magnetic force using sinusoidal dependence. Some selected features of the Hamiltonian version of the considered model together with the separatrices which defines the regions of two basic solutions: oscillations and rotations are presented in Figure 9a-c. Note that the double well potential (Figure 9c) is generated by the interaction between the pendulum tip permanent magnet and additional permanent magnets distributed in the system as presented in Figure 2. In the studied system the magnetic forces are much stronger than gravity, which leads to a small splitting of the separatrices (see the differences of red and black lines in Figure 9b). The numerical results of the frequency sweeps are presented in Figure 10.  
Figure 9: Selected model features for the mechanical resonator: (**a**) The total energy for the Hamiltonian system (without damping and denoted in colors \(\left|\right|\) (see the color bar); (**b**) Possible solutions: oscillation (blue), rotation (green) and the separatrices—orbital lines crossing the saddle points (black and red); (**c**) Double well potential defined with in the pendulum angular variable \([-\pi,\pi]\). The system parameters are indicated in Table 4.  
Figure 10 shows the frequency sweep results for pendulum response in its angular velocity. Our interest was focused on the angular velocity and the corresponding variability that are directly proportional to the electromotive forces in the electrical circuit.