scholarly journals (Bi)-orthogonality relation for eigenfunctions of self-adjoint operators

2019 ◽  
Vol 377 (2156) ◽  
pp. 20190112 ◽  
Author(s):  
L. S. Ledet ◽  
S. V. Sorokin
Keyword(s):  
Energy Flow ◽  
Orthogonality Relation ◽  
Thin Elastic Plate ◽  
Fluid Loading ◽  
Heavy Fluid ◽  
Theme Issue ◽  
Structured Media ◽  
Flow Components ◽  
Adjoint Operators ◽  
Dynamic Phenomena

The bi-orthogonality relation for eigenfunctions of self-adjoint operators is derived. Its composition is explained in view of the structure of a characteristic equation and of the energy flow components. Application of the bi-orthogonality relation for solving forcing problems is generalized and the connection between the bi-orthogonality relation and the virtual wave method is highlighted. Technicalities are illustrated in a non-trivial example of propagation of free/forced cylindrical waves in a thin elastic plate under heavy fluid loading. This article is part of the theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 1)’.

scholarly journals Frequency-dependent impedance and surface waves on the boundary of a stratified dielectric medium

2019 ◽  
Vol 377 (2156) ◽  
pp. 20190218
Author(s):  
Kirill Cherednichenko ◽  
William Graham
Keyword(s):  
Dispersion Relation ◽  
Surface Waves ◽  
Half Space ◽  
Dielectric Medium ◽  
Interface Condition ◽  
Theme Issue ◽  
Structured Media ◽  
Loss Parameter ◽  
The Relationship ◽  
Dynamic Phenomena

We analyse waves propagating along the interface between half-spaces filled with a perfect dielectric and a Lorentz material. We show that the corresponding interface condition leads to a generalization of the classical Leontovich condition on the boundary of a dielectric half-space. We study when this condition supports propagation of (dispersive) surface waves. We derive the related dispersion relation for waves along the boundary of a stratified half-space and determine the relationship between the loss parameter, frequency and wavenumber for which interfacial waves exist. This article is part of the theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 1)’.

From statics of composites to acoustic metamaterials

2019 ◽  
Vol 377 (2156) ◽  
pp. 20190099 ◽  
Author(s):  
J. R. Willis
Keyword(s):  
Simple Model ◽  
Momentum Density ◽  
Acoustic Metamaterials ◽  
Static Response ◽  
Weighted Mean ◽  
Theme Issue ◽  
Structured Media ◽  
Explicit Analysis ◽  
Random Composite ◽  
Dynamic Phenomena

The Hashin–Shtrikman formalism for the static response of a composite is first summarized, in a form applicable to anisotropic media and arbitrary two-point correlations. Its extension to dynamic response is then explained and in particular the extra coupling, for stress with velocity as well as strain, and momentum density with strain as well as velocity that inevitably follows, is highlighted. The more recent dynamical formulations which incorporate extra flexibility through the use of a weighted mean displacement also receive mention. The article is concluded with an explicit analysis of a simple model random composite and some lessons are drawn from the solution, which exposes at least one clear gap in existing knowledge and requires further research. This article is part of the theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 1)’.

scholarly journals Flexural vibration systems with gyroscopic spinners

2019 ◽  
Vol 377 (2156) ◽  
pp. 20190154 ◽  
Author(s):  
G. Carta ◽  
M. J. Nieves ◽  
I. S. Jones ◽  
N. V. Movchan ◽  
A. B. Movchan
Keyword(s):  
Numerical Simulation ◽  
Periodic System ◽  
Spectral Properties ◽  
Flexural Vibration ◽  
Finite System ◽  
Band Gaps ◽  
Theme Issue ◽  
Structured Media ◽  
Transient Numerical Simulation ◽  
Dynamic Phenomena

In this paper, we study the spectral properties of a finite system of flexural elements connected by gyroscopic spinners. We determine how the eigenfrequencies and eigenmodes of the system depend on the gyricity of the spinners. In addition, we present a transient numerical simulation that shows how a gyroscopic spinner attached to the end of a hinged beam can be used as a ‘stabilizer’, reducing the displacements of the beam. We also discuss the dispersive properties of an infinite periodic system of beams with gyroscopic spinners at the junctions. In particular, we investigate how the band-gaps of the structure can be tuned by varying the gyricity of the spinners. This article is part of the theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 1)’.

A nonlinear model of the dynamics of a large elastic plate with heavy fluid loading

2006 ◽  
Vol 462 (2072) ◽  
pp. 2205-2224 ◽  
Author(s):  
N Peake ◽  
S.V Sorokin
Keyword(s):  
Elastic Plate ◽  
Plane Waves ◽  
Travelling Wave ◽  
Plane Motion ◽  
Transverse Motion ◽  
Thin Elastic Plate ◽  
Transverse Displacement ◽  
Fluid Loading ◽  
Heavy Fluid ◽  
Wave Solutions

In this paper, we derive weakly nonlinear equations for the dynamics of a thin elastic plate of large extent under conditions of heavy fluid loading. Two situations are then considered. First, we consider the case in which transverse motion of the plate generates a weaker in-plane motion, which is in turn coupled back to the evolution of the transverse motion. This results in the familiar nonlinear Schrödinger equation for the amplitude of a transverse plane wave, and we show that solitary-wave solutions are possible over the range of (non-dimensional) frequencies ω > ω c , which depends on the material properties. Dimensional values of ω c are physically realizable for a typical composite material underwater. Second, we consider the case in which the amplitudes of the transverse and in-plane motion are of the same order of magnitude, possible at a single resonant frequency, which leads to an evolution equation of rather novel type. We find a range of travelling-wave solutions, including cases in which incident in-plane waves can generate localized regions of transverse displacement.

About some projects with attendant circumstances

2019 ◽  
Vol 377 (2156) ◽  
pp. 20190097 ◽  
Author(s):  
Leonid I. Slepyan
Keyword(s):  
Theme Issue ◽  
Structured Media ◽  
Dynamic Phenomena

In this brief article (written at the suggestion of Prof. Gennady Mishuris), I discuss some projects, the circumstances under which the associated problems appeared on my desk, the results and what helped me to obtain them. This article is part of the theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 1)’.

The modes, resonances and forced response of elastic structures under heavy fluid loading

1984 ◽  
Vol 312 (1521) ◽  
pp. 295-341 ◽  
Keyword(s):  
Order Approximation ◽  
Low Frequency ◽  
Forced Response ◽  
Mode Shapes ◽  
Thin Elastic Plate ◽  
Fluid Loading ◽  
Heavy Fluid ◽  
Elastic Structures ◽  
Strip Plate ◽  
First Time

This paper reports analytical studies of problems that involve the motion of plane elastic structures under conditions of heavy fluid loading. The main aspect concerns the description of the vibration response of a thin elastic plate (or membrane), of finite extent in at least one dimension, when the structure is excited by concentrated mechanical drive along a line or at a point; and as part of this the possibility of resonant response is discussed, and the resonance conditions and free modes of oscillation are obtained. There is also some discussion of the acoustic fields radiated by the structures under localized mechanical excitation. The analysis makes extensive use of results for the reflection of a structural wave (subject to heavy fluid loading) at an edge, and the paper gives results for that reflection process covering waves incident normally on eight different edge configurations and waves incident obliquely on two edge configurations. These results include the reflection coefficient (whose magnitude is unity in the leading-order approximation of low-frequency heavy fluid loading), and the amplitude and directivity of the edge-scattered sound. By using the argument that edge reflection is a local process, the response is then calculated for a strip plate, under both line and point forcing, and the response is, for the first time, obtained for structures finite in both dimensions and subject to heavy fluid loading. Specifically, solutions are given here for a circular plate with eccentric drive, and for a membrane model of a rectangular panel, with central point drive. For some conditions and geometries expressions in simple form are found for the natural frequencies and mode shapes, and for the off-resonance forced response. Expressions for the drive admittances are found which display a variety of interesting features.

scholarly journals Waves in elastic bodies with discrete and continuous dynamic microstructure

2019 ◽  
Vol 378 (2162) ◽  
pp. 20190313 ◽  
Author(s):  
Gennady S. Mishuris ◽  
Alexander B. Movchan ◽  
Leonid I. Slepyan
Keyword(s):  
Comparative Analysis ◽  
Energy Distribution ◽  
Unified Approach ◽  
Elastic Solids ◽  
Theme Issue ◽  
Elastic Bodies ◽  
Structured Media ◽  
Continuous Dynamic ◽  
Dynamic Phenomena ◽  
Non Locality

This paper presents a unified approach to the modelling of elastic solids with embedded dynamic microstructures. General dependences are derived based on Green's kernel formulations. Specifically, we consider systems consisting of a master structure and continuously or discretely distributed oscillators. Several classes of connections between oscillators are studied. We examine how the microstructure affects the dispersion relations and determine the energy distribution between the master structure and microstructures, including the vibration shield phenomenon. Special attention is given to the comparative analysis of discrete and continuous distributions of the oscillators, and to the effects of non-locality and trapped vibrations. This article is part of the theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 2)’.

scholarly journals Resonant waves and localization phenomena in lattices

2019 ◽  
Vol 377 (2156) ◽  
pp. 20190110
Author(s):  
Sagdulla A. Abdukadirov ◽  
Mark V. Ayzenberg-Stepanenko ◽  
Gregory G. Osharovich
Keyword(s):  
Three Dimensional ◽  
Wave Processes ◽  
Theme Issue ◽  
Hexagonal Cell ◽  
Transient Wave ◽  
Structured Media ◽  
Different Types ◽  
Mass Spring ◽  
Square Cell ◽  
Dynamic Phenomena

Transient wave processes in mass-spring lattices excited by point oscillating sources are studied. Dispersion properties of uniform periodic three-dimensional (3D) square-cell and two-dimensional (2D) hexagonal-cell lattices including revealed star-shaped localization phenomena are analysed. The resonant-like waves and localization-like patterns in non-uniform lattices possessing predetermined and randomly distributed defects are numerically examined in order to identify the sensitivity of star-shape forms to different types of defects. This article is part of the theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 1)’.

scholarly journals Nested Bloch waves in elastic structures with configurational forces

2019 ◽  
Vol 377 (2156) ◽  
pp. 20190101 ◽  
Author(s):  
F. Dal Corso ◽  
D. Tallarico ◽  
N. V. Movchan ◽  
A. B. Movchan ◽  
D. Bigoni
Keyword(s):  
Longitudinal Vibration ◽  
Configurational Forces ◽  
Longitudinal Vibrations ◽  
Theme Issue ◽  
Structured Media ◽  
Flexural Oscillation ◽  
Mechanical Devices ◽  
Band Gap Structure ◽  
Dynamic Phenomena ◽  
Axial Resonance

Small axial and flexural oscillations are analysed for a periodic and infinite structure, constrained by sliding sleeves and composed of elastic beams. A nested Bloch–Floquet technique is introduced to treat the nonlinear coupling between longitudinal and transverse displacements induced by the configurational forces generated at the sliding sleeve ends. The action of configurational forces is shown to play an important role from two perspectives. First, the band gap structure for purely longitudinal vibration is broken so that axial propagation may occur at frequencies that are forbidden in the absence of a transverse oscillation and, second, a flexural oscillation may induce axial resonance, a situation in which the longitudinal vibrations tend to become unbounded. The presented results disclose the possibility of exploiting configurational forces in the design of mechanical devices towards longitudinal actuation from flexural vibrations of small amplitude at given frequency. This article is part of the theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 1)’.

scholarly journals Dynamic fracture of a dissimilar chain

2019 ◽  
Vol 377 (2156) ◽  
pp. 20190103 ◽  
Author(s):  
N. Gorbushin ◽  
G. Mishuris
Keyword(s):  
Steady State ◽  
Dynamic Fracture ◽  
Model Parameters ◽  
Static Case ◽  
Theme Issue ◽  
Double Chain ◽  
Structured Media ◽  
Applied Forces ◽  
Mass Spring ◽  
Dynamic Phenomena

In this paper, we study the dynamic fracture of a dissimilar chain composed of two different mass-spring chains and connected with other springs. The propagation of the fault (crack) is realized under externally applied moving forces. In comparison with a homogeneous double chain, the considered structure displays some new essential features of steady-state crack propagation. Specifically, the externally applied forces are of a different strength, unlike a static case, and should be appropriately chosen to satisfy the equilibrium of the structure. Moreover, there exists a gap in the range of crack speeds where the steady-state fracture cannot occur. We analyse the admissibility of solutions for different model parameters and crack speeds. We complement analytical findings with numerical simulations to validate our results. This article is part of the theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 1)’.

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