## 5 Conclusions  
Filtering properties of systems having a continuous and discrete character and arranged in series and parallel have been considered. The analysis has been done by implementing a transfer matrix approach; the analytical model simplifies the identification of the contributions of mechanical properties, phase character and arrangement. The model has been given for out-of-plane shear waves, which are governed by Helmholtz type equations of motion, so that the results can be easily extended to other physical applications governed by the same type of equation, such as longitudinal or torsional waves in a rod (for sufficiently low frequencies), as well as acoustics and optics. Several properties have been detected, which can be implemented in different applications. In the laminate system, it has been shown that dispersion is induced by the elastic impedance contrast, namely for \(\xi_{c}\
eq 1\), whereas the frequency contrast does not necessarily introduce any dispersion. For oscillators arranged in parallel, localised effects are obtained, where most of the energy is confined in the resonators. This effect is implemented when tuned mass dampers are added to a structure in order to damp dangerous vibrations. The simple model developed in this work shows that energy localisation is higher for the discrete oscillator, but the continuous one gives the possibility to obtain this effect at multiple frequencies. The analysis also shows that this type of localisation, where the dispersive behaviour of the system in series is strongly perturbed only in the neighbourhood of the resonance frequencies of the oscillator, is retrieved when the elastic impedance of the resonator is sufficiently small. If systems having discrete and continuous character are arranged in series, the dispersion properties drastically change, combining the effect of creating tiny pass/stop bands typical of discrete systems with the frequency periodicity or quasi-periodicity induced by the continuous ones. Such behaviour, addressed as continuous-discrete like, can have direct application in delay-line components, where there is the necessity to slow down  
Figure 15: Energy distribution between different phases as a function of the radian frequency \(\omega\). Dashed lines correspond to the bi-layered system, while the continuous line corresponds to the bi-layered system with the oscillator disposed in series (indicated in black). Details of the second and third pass-bands are given on the right. Material and geometrical parameters are the same as Figures 6a and 11. Pass band frequency intervals are indicated at the top and at the bottom of the plot. waves and energy propagation, which is associated with the group velocity (the slope of the dispersion curves), with a small amount of reflection. A similar continuous discrete-like behaviour is obtained for a phase arranged in parallel, when the elastic impedance of the resonator is sufficiently high. In both cases, it is possible to predict the central frequencies of the tiny pass bands by evaluating the eigenfrequencies of an isolated single unit cell with ad hoc boundary conditions. These types of systems can be implemented in order to propagate selected frequencies at a constant interval. For the systems in parallel, this behaviour is obtained when the elastic impedance of the resonator is sufficiently high, while for the system in series, this is obtained when the frequency of the incoming wave is above the resonance frequency of the discrete oscillator, a condition that is reasonably easier to reach in technological applications. The transition regions between the continuous-discrete and the continuous-like behaviours can be explored to enlarge the frequency intervals, where the localisation induced by the resonators is still effective. The results presented in this work can find applications that range from the control of mechanical vibrations in large scale structures, such as slender bridges, to frequency filtering in Micro and Nano Electro-Mechanical Systems. Conceptualization, M.B.