*Wave Propagation*

This subsection presents the results of the analysis of changes in the propagating elastic waves in the modelled active periodical beam. For modelling the discussed structure the shape functions based on the FDSFEM [13] was used and the node distribution the same as in the TDSFEM [15] was adopted. Such a combination of the two totally different methods enabled a thorough analysis of the influence of frequency-dependent changes in the Young's modulus of piezoelectric material—Equation (1). The amplitude at both ends of the beam was determined. The aim of a such calculation programme was to analyse the changes of the propagating wave. The analysed periodic beam with active piezoelectric elements was excited to transverse vibrations at one end with the sinusoidal signal (eight pulses) modulated by the Hanning window, by the force of a *F* = 1 N amplitude. Two excitation frequencies *f* = 50 kHz and *f* = 100 kHz have been chosen; the first one from the frequency range of normal behaviour of the periodic beam, and the second within the passive band gap of the periodic structure.

Next Figure 8 shows the dispersion curves determined for the analysed periodical beam in the first Brillouin zone [16,17]. These curves have been determined by the use of the Bloch reduction method [18] by taking into account the relation from Equation (1). The graph in Figure 8a shows the first Brillouin zone of the passive system. One can notice the red lines meaning the vibrations of the propagating wave. The frequency ranges, marked with grey areas, at which there is no wave propagation are visible, i.e., there are no corresponding wave vectors. On the second chart in Figure 8b there is the first Brillouin zone of the system with each RLC circuit tuned to 100 kHz (frequency from the range of the band gap). RLC circuits create an area of anti-resonant vibrations independent of the wave vector that effectively widens the area of the grey fields of blocked wave propagation.

**Figure 8.** Dispersion curves for the analysed beam, (**a**) RLC circuits off, (**b**) RLC circuits tuned to 100 kHz.

Flexural wave propagation in an electromechanical periodic structure has been presented in Figure 9. The presented results represent six cases—the open RLC circuit (Figure 9a,b), the RLC tuned to the frequency of *fR* = 50 kHz (Figure 9c,d) and the RLC tuned to the frequency of *fR* = 100 kHz (Figure 9e,f) for both 50 kHz and 100 kHz excitation signals—left and right column, respectively. It may be concluded that the active RLC circuits with the resonant frequency equal to the excitation carrier frequency had the features of an active vibration damper.

To illustrate the damping character of an active periodic beam with the piezoelectric RLC circuits the following set of results was gathered (Figure 10). Here the vibration spectra before and after passing through the structure have been shown. The green colours show the spectra of vibration measured before passing through the structure (*P*1), the blue after passing through the structure (*P*2). The red line represents the tuned resonant frequency (*fR*) of the RLC circuit. The left hand side column represents the data calculated for 50 kHz excitation signal, the right hand side column for 100 kHz respectively.

In the case of the passive system (Figure 10a,b), the wave of the carrier frequency of *f* = 50 kHz was free to propagate itself, there was no remarkable change in the amplitude magnitude (Figure 10a). In case of the wave of the carrier frequency of *f* = 100 kHz it may be noticed that some amount of the energy was blocked due to the presence of band gaps for this spectrum range as the band gap was a barrier for propagation of waves of these frequencies. However, the natural band gap was of relatively small width, therefore a certain amount of wave energy could propagate through the band gap.

The diagram below (Figure 10c,d) shows the changes in the excitation spectrum in the active periodic beam with the RLC circuits tuned to *fR* = 50 kHz. As it can be noticed, the excitation wave (of the carrier frequency of *f* = 50 kHz) at that frequency was unable to propagate freely due to the dissipation of energy on the electrical resistance band gap that appears. Although the band gap was very narrow the amplitude of the wave decreased in a significant manner. On the other hand for this *fR* = 50 kHz no significant changes were observed in the amplitude of the excitation signal of the carrier frequency of *f* = 100 kHz in comparison to the amplitude registered for the passive system.

Finally, tuning of the RLC circuit to the frequency of *fR* = 100 kHz (Figure 10e,f) did not cause any changes in the wave propagation of the excitation wave of the frequency of *f* = 50 kHz (also comparing to the passive structure). However, this value of the *fR* = 100 kHz caused a significant widening of the band gap. The amplitude of the wave after passing through the structure with band gap decreased tenfold. It was caused by the synergy of the natural periodic structure band gap with energy dissipation caused by the active RLC circuit and its electrical resistance.

**Figure 9.** Patterns of flexural wave propagation in an electromechanical periodic structure.

**Figure 10.** Spectra of flexural oscillations measured before passing through the structure (green) and after passing through the structure (blue). The excitation frequency equal to 50 kHz (**left**) and 100 kHz (**right**) marked with red line.
