In the work, multi-criteria optimization of phononic structures was performed to minimize the transmission in the frequency range of acoustic waves, eliminate high transmission peaks with a small half-width inside of the band gap, and what was the most important part of the work – to minimize the number of layers in the structure. Two types of the genetic algorithm were compared in the study. The first one was characterized by a constant number of layers (GACL) of the phononic structure of each individual in each population. Then, the algorithm was run for a different number of layers, as a result of which the structures with the best value of the objective function were determined. In the second version of the algorithm, individuals in populations had a variable number of layers (GAVL) which required a different type of target function and crossover procedure. The transmission for quasi-one-dimensional phononic structures was determined with the use of the transfer matrix method algorithm. Based on the research, it can be concluded that the developed GAVL algorithm with an appropriately selected objective function achieved optimal solutions in a much smaller number of iterations than the GACL algorithm, and the value of the k parameter below 1 leads to faster achievement of the optimal structure.

The study investigated the effect of the fill factor, lattice constant, and the shape and type of meta-atom material on the reduction of mechanical wave transmission in quasi-two-dimensional phononic structures. A finite difference algorithm in the time domain was used for the analysis, and the obtained time series were converted into the frequency domain using the discrete Fourier transform. The use of materials with large differences in acoustic impedance allowed to determine the influence of the meta-atom material on the propagation of the mechanical wave.

The study analyzed the influence of materials and different types of damping on the dynamic stability of the Bernoulli-Euler beam. Using the mode summation method and applying an orthogonal condition of eigenfunctions and describing the analyzed system with the Mathieu equation, the problem of dynamic stability was solved. By examining the influence of internal and external damping and damping in the beam supports, their influence on the regions of stability and instability of the solution to the Mathieu equation was determined.

Although the study of oscillatory motion has a long history, going back four centuries, it is still an active subject of scientificr esearch. In this review paper prospective research directions in the field of mechanical vibrations were pointed out. Four groups of important issues in which advanced research is conducted were discussed. The first are energy harvester devices, thanks to which we can obtain or save significant amounts of energy, and thus reduce the amount of greenhouse gases. The next discussed issue helps in the design of structures using vibrations and describes the algorithms that allow to identify and search for optimal parameters for the devices being developed. The next section describes vibration in multi-body systems and modal analysis, which are key to understanding the phenomena in vibrating machines. The last part describes the properties of granulated materials from which modern, intelligent vacuum-packed particles are made. They are used, for example, as intelligent vibration damping devices.