**4. Conclusions**

In this paper numerical investigations of a beam structural element built out of periodically arranged elementary cells with active piezoelectric elements has been performed. For this analysis the authors propose numerical models based on the use of the finite element method (FEM) and the spectral finite element methods defined in the frequency domain (FDSFEM) and the time domain (TDSFEM). The application of different modelling methods allow the authors to formulate conclusions that result from the calculations performed.

The FEM is the most common method and generally gives correct results. However, for high-frequency analysis problems it is necessary to use either a heavily dense grid or higher-order approximation polynomials. This leads to a correspondingly large sizes of the problems to be solved or a Runge effect. There is another reason why the FEM can be disadvantageous in the applications related to periodic structures—numerical models themselves show periodic characteristics [6,15]. Thus, it is easy to predict that the results obtained may have features typical to periodic structures resulting not only from the geometry of finite elements, but also from the features of the numerical models. The consequent misinterpretation of results may be simply dangerous.

Due to the above mentioned characteristics of the FEM, the study proposes to utilise the TDSFEM method. The unquestionable advantage of the TDSFEM method is its ability to employ higher order approximation polynomials, which results in higher calculation accuracy. Additionally, an un-uniform distribution of nodes in single finite elements enables one to obtain a diagonal form of the inertia matrix, which significantly reduces the time of numerical calculations. All presented amplitude-frequency characteristics of the analysed periodic beam structural element, taking into account the dependence of the PZT Young's modulus on the frequency, have been determined by this method.

While in the case of determination of the amplitude-frequency characteristics of the active periodic beam, the use of the TDSFEM allowed the authors to obtain results at a satisfactory level, the calculations for changes in the propagation of elastic wave required another modification of the modelling method. In order to precisely map changes in Young's modulus for the value of PZT material for each analysed frequency, it was necessary to use the FDSFEM method. Modification of the method involved the use of non-uniform mesh of nodes in the finite elements known from the TDSFEM and shape functions from the FDSFEM. In this way the changes in propagation of elastic waves in the active periodic beam structural element have been modelled.

After all numerical tests performed it may be concluded that periodic structures with active piezoelectric elements incorporated into the RLC resonance circuit can be successfully used to attenuate vibrations in a controlled manner. A resonant circuit with piezoelectric elements causes the appearance of an additional band gap in the spectrum of mechanical vibrations in the vicinity of the natural frequency of the RLC resonance system. Adjusting the resonance frequencies of the RLC systems to the frequency of the naturally present band gap in the spectrum of mechanical vibrations results in a significant widening of the band gap, which leads to effective vibration damping in this frequency range.

Although the proposed approach clearly demonstrates that there is a possibility of active control of band gaps, it should be added that the problem still requires a number of analyses and will definitely be the subject of further scientific considerations of the authors.

**Author Contributions:** Conceptualization, W.W., M.P. and M.K.; methodology, W.W., M.P. and M.K.; software, W.W.; validation, W.W., M.P. and M.K.; formal analysis, W.W., M.P. and M.K.; investigation, W.W., M.P. and M.K.; resources, W.W., M.P. and M.K.; data curation, W.W. and M.P.; writing—original draft preparation, M.P.; writing—review and editing, M.P.; visualization, W.W. and M.P.; supervision, M.K.; project administration, M.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** The authors would like to gratefully acknowledge the support of the Academic Computer Centre in Gda ´nsk, the provider of the software used for the research done by the author and described in this paper.

**Conflicts of Interest:** The authors declare no conflict of interest.
