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This paper presents a study of wave propagation through an infinite periodic structure that consists of elastic Timoshenko beams interconnected with rigid bodies. This is a generalized approach in which the beams are not coaxial and the centre of mass of each rigid body is placed away from the intersection of their neutral axes. An analytical approach is used by applying the transfer matrix method (TMM), along with the Floquet–Bloch theorem for elastic wave propagation. Subsequent parametric analysis is performed with visualization of resulting band diagrams of a representative structure. These results are verified through comparison with solutions obtained using the finite-element method (FEM). In this manner, a comprehensive dynamical analysis of tailored metastructures is provided. This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 2)'. [ABSTRACT FROM AUTHOR]

The paper proposes a topology optimization method for 1D phononic structures to minimize the tangential component of particle velocity at the objective boundary. The tangential derivative of the boundary integral equation (TDBIE) is employed for the evaluation of the objective function involves the tangential derivative quantities. The topological derivative is derived through the adjoint method and the Neumann boundary condition of the adjoint field is obtained using the TDBIE. The average topological derivative which is obtained by calculating the average value of the topological derivative in each layer of design domains, is employed for the updating of the level set function. Numerical implementations demonstrate the proposed method is effective for the design of the 1D phononic crystals with the objective function involving tangential derivative quantities. [ABSTRACT FROM AUTHOR]

In this work, the impact of the defect on the transmission of a mechanical wave in a periodic quasi-one-dimensional structure was investigated. The multilayer structure was made of PLA and air, while the defect layer was PNM-0.38PT with a significantly higher value of acoustic impedance in relation to the materials of the base structure. The influence of the position of the defect in the structure and its thickness was analysed. Transmission as a function of frequency was determined using the Transfer Matrix Method algorithm. The work showed the presence of band gaps in the analyzed structures. The influence of the symmetry of structures and substructures on the transmission of a mechanical wave was investigated. The influence of the number of layers with very low acoustic impedance (air) on the number of high transmission peaks with a small half-width was also demonstrated.

The work compares the design of phononic structures using two types of optimization algorithms. Using the genetic algorithm and the simulated annealing algorithm, optimal layer distributions were obtained in which the phononic band gap phenomenon occurs. The mechanical wave propagating in the obtained structure, for the given frequency ranges, significantly reduces the transmitted energy, thanks to which the building facade or monument located behind the obtained barrier is exposed to much smaller vibrations, which significantly reduces damage related to long-term fatigue load. The mechanical wave propagation was modeled using the Transfer Matrix Method algorithm and the proprietary objective function allows for the reduction of wave transmission with the simultaneous reduction of high transmission peaks with small half-widths. [ABSTRACT FROM AUTHOR]

In this work, the impact of the defect on the transmission of a mechanical wave in a periodic quasi-one-dimensional structure was investigated. The multilayer structure was made of PLA and air, while the defect layer was PNM-0.38PT with a significantly higher value of acoustic impedance in relation to the materials of the base structure. The influence of the position of the defect in the structure and its thickness was analysed. Transmission as a function of frequency was determined using the Transfer Matrix Method algorithm. The work showed the presence of band gaps in the analyzed structures. The influence of the symmetry of structures and substructures on the transmission of a mechanical wave was investigated. The influence of the number of layers with very low acoustic impedance (air) on the number of high transmission peaks with a small half-width was also demonstrated. [ABSTRACT FROM AUTHOR]

In this work, the influence of different cross-sections of meta-atoms and their distribution on the occurrence of local resonance regions in inter-meta-atomic spaces of finite phononic structures was investigated. Software based on the Mathematica package was designed and implemented using the finite difference algorithm in the time domain to simulate mechanical wave propagation in phononic structures. Then, for the recorded time series from the inter-meta-atomic spaces, resonant frequency distributions were determined using Fourier transforms, and an analysis of the differences in frequency distributions depending on the location of the inter-meta-atomic space was carried out. [ABSTRACT FROM AUTHOR]

The concept of frequency gaps in phononic crystals is widely used in physics. The feasibility and efficiency of applying this principle in damping out seismic and mechanical induced vibrations in real scale of civil engineering constructions are presented in this article through the results obtained from numerical modeling and analysis of a concrete substratum embedding steel elements (pillars) coated in a polyvinyl chloride polymer (PVC). The first configuration having the elements fully embedded into the substrate resulted in two narrow band gaps at relatively high frequencies; and when only the metallic pillars are emerging from the substrate, the band gaps shift towards the low frequencies. The results are improved and show the existence of three band gaps at medium frequencies ranging from 80 to 200 m/s when both the pillars and the polymer are emerging from the foundation. Exploring other metal-polymer pairs of materials such as "steel-rubber", "steel-silicone"," lead-rubber" and "lead-silicone", shows that a range of band gaps has shifted again towards the lower frequencies which cover part of the seismic frequency domain. Further improvement is obtained by notching the ends of the substrate in order to widen and lower the band gaps especially for "metal-rubber" pairs. These results show the potential of using periodic networks to mitigate seismic and mechanical vibration effects on large scale structures and components.

In the paper, all possible transmissions of quasi one-dimensional structures permutations made of glass and epoxy resin surrounded by water for the frequency range of acoustic waves was determined using the Transfer Matrix Method (TMM) algorithm. On their basis, the phase space of solutions for minimizing the objective function for the genetic algorithm was determined. This space consisted of the transmission integral (energy reduction of propagating wave) and the integral of the absolute value of the transmission functions derivative (elimination of high transmission peaks with a small half width). The paper demonstrates the existence of bandgaps for the analyzed multilayer structures (waves with characteristic frequencies do not propagate in the structure) and demonstrates the existence of local minima of solution spaces with low probability of finding optimal structure by the genetic algorithm. State space search for various initial values of the optimization algorithm was also analyzed. [ABSTRACT FROM AUTHOR]

A novel design of origami phononic structures with tunable waveguiding features are studied in this research. The structure is formed by attaching cylindrical inclusions on top of origami sheet; in this kind of architecture, folding of the underlying origami sheet can change the periodicity of inclusions. Being periodic in nature, the origami structure exhibits bandgaps features and have the potential to steer wave energy through a path of defects (also known as waveguide) created in the lattice of inclusions. Since the spectral content transmitted through the waveguide depends on the periodicity surrounding it, the folding induced lattice transformation in such origami structures can lead to adaptation in waveguide frequency. In fact, in origami structures that can transform between different Bravais-lattice types, for example between a square and hexagon lattice, the waveguide frequency can be drastically tuned. Such phenomenal adaptation is studied in this research by extracting dispersion diagrams of defective origami structures using the plane wave expansion method. Further, numerical and experimental investigations are performed on finite origami structures with waveguides and the transmission results of different folding configurations demonstrate that waveguide frequency can be significantly altered. Overall, the low effort one-degree-of-freedom (1DOF) folding mechanism combined with the scalable nature of the origami architecture makes the origami phononic structure a novel and effective device for waveguiding applications across a range of frequencies. [ABSTRACT FROM AUTHOR]

There are two classes of phononic structures that can support elastic waves with non-conventional topology, namely intrinsic and extrinsic systems. The non-conventional topology of elastic wave results from breaking time reversal symmetry (T-symmetry) of wave propagation. In extrinsic systems, energy is injected into the phononic structure to break T-symmetry. In intrinsic systems symmetry is broken through the medium microstructure that may lead to internal resonances. Mass-spring composite structures are introduced as metaphors for more complex phononic crystals with non-conventional topology. The elastic wave equation of motion of an intrinsic phononic structure composed of two coupled one-dimensional (1D) harmonic chains can be factored into a Dirac-like equation, leading to antisymmetric modes that have spinor character and therefore non-conventional topology in wave number space. The topology of the elastic waves can be further modified by subjecting phononic structures to externally-induced spatio-temporal modulation of their elastic properties. Such modulations can be actuated through photo-elastic effects, magneto-elastic effects, piezo-electric effects or external mechanical effects. We also uncover an analogy between a combined intrinsic-extrinsic systems composed of a simple one-dimensional harmonic chain coupled to a rigid substrate subjected to a spatio-temporal modulation of the side spring stiffness and the Dirac equation in the presence of an electromagnetic field. The modulation is shown to be able to tune the spinor part of the elastic wave function and therefore its topology. This analogy between classical mechanics and quantum phenomena offers new modalities for developing more complex functions of phononic crystals and acoustic metamaterials.

This article presents methods for modeling, analysis, and design of metamaterial beams for broadband vibration absorption/isolation. The proposed metamaterial beam consists of a uniform isotropic beam with many small spring-mass-damper subsystems integrated at separated locations along the beam to act as vibration absorbers. For a unit cell of an infinite metamaterial beam, governing equations are derived using the extended Hamilton principle. The existence of stopband is demonstrated using a model based on averaging material properties over a cell length and a model based on finite element modeling and the Bloch-Floquet theory for periodic structures. However, these two idealized models cannot be used for finite beams and/or elastic waves having short wavelengths. For finite metamaterial beams, a linear finite element method is used for detailed modeling and analysis. Both translational and rotational absorbers are considered. Because results show that rotational absorbers are not efficient, only translational absorbers are recommended for practical designs. The concepts of negative effective mass and stiffness and how the spring-massdamper subsystems create a stopband (i.e., no elastic waves in this frequency range can propagate forward) are explained in detail. Numerical simulations reveal that the actual working mechanism of the proposed metamaterial beam is based on the concept of conventional mechanical vibration absorbers. It uses the incoming elastic wave in the beam to resonate the integrated spring-mass-damper absorbers to vibrate in their optical mode at frequencies close to but above their local resonance frequencies to create shear forces and bending moments to straighten the beam and stop the wave propagation. This concept can be easily extended to design a broadband absorber that works for elastic waves of short and long wavelengths. Numerical examples validate the concept and show that, for high-frequency waves, the structure's boundary conditions do not have significant influence on the absorbers' function. However, for absorption of low-frequency waves, the boundary conditions and resonant modes of the structure need to be considered in the design. With appropriate design calculations, finite discrete spring-mass-damper absorbers can be used, and hence expensive micro- or nanomanufacturing techniques are not needed for design and manufacturing of such metamaterial beams for broadband vibration absorption/isolation. [ABSTRACT FROM AUTHOR]

The paper presents a level set based topology optimization method for unidirectional phononic structures with finite layers of lattice cells. Boundary element method(BEM) is employed as the numerical approach to solve the acoustic problems governed by Helmholtz equation. A sized reduced coefficient matrix is derived due to the iteration forms for input and output quantities on the periodic boundary of unit cells. Topological derivatives are formulated by boundary integral equation combined with adjoint variable method and computed for each layer. An average topological sensitivity of a single design domain is proposed for the updating of the level set function(LSF) governed by an evolution equation. Numerical models with different number of layers are considered and several optimized structures of unit cells are obtained in concerned frequencies. A further investigation into the transmission of acoustic waves is carried out by employing more layers of the periodic structures between the input and output domains. The results demonstrate the effectiveness of the proposed optimization method for the finite unidirectional phononic structures. • Average topological derivative is specified for each design domain. • Original and adjoint fields are both computed through a size reduced system matrix. • Resluts from different numbers of design domains are obtained. • The band gap properties are enhanced with more layers. [ABSTRACT FROM AUTHOR]