scholarly journals Abnormal long wave dispersion phenomena in a slightly compressible elastic plate with non-classical boundary conditions

2007 ◽  
Vol 42 (2) ◽  
pp. 298-309 ◽  
Author(s):  
Graham A. Rogerson ◽  
Kevin J. Sandiford ◽  
Ludmila A. Prikazchikova
Keyword(s):  
Boundary Conditions ◽  
Elastic Plate ◽  
Wave Dispersion ◽  
Slightly Compressible ◽  
Long Wave ◽  
Classical Boundary

scholarly journals On the applicability limits of refined theories in describing of the flexural edge wave in plates

2020 ◽  
pp. 206-213
Author(s):  
Мария Владимировна Вильде ◽  
Янина Александровна Парфенова ◽  
Мария Юрьевна Сурова
Keyword(s):  
Boundary Conditions ◽  
Asymptotic Theory ◽  
Wave Dispersion ◽  
Excitation Amplitude ◽  
Edge Wave ◽  
Numerical Comparison ◽  
Plate Bending ◽  
Classical Boundary ◽  
Kirchhoff Theory ◽  
3D Problem

Исследуются пределы применимости уточненных теорий изгиба пластины при описании дисперсии изгибной краевой волны и амплитуды её возбуждения парой сосредоточенных скручивающих моментов, приложенных на торце. Методом численного сравнения с решением трехмерной задачи показано, что теория типа Тимошенко пригодна для описания краевой волны на частотах, не превосходящих 30% от первой частоты запирания. Уточненная теория изгиба пластин с приведенной инерцией в сочетании с классическими граничными условиями позволяет уточнить скорость волны по сравнению с теорией Кирхгофа, но значительно искажает амплитуду. The applicability limits of refined plate bending theories in describing of the flexural edge wave dispersion and its excitation amplitude are investigated. The wave is excited by a pair of twisting couples applied to the edge of the plate. Numerical comparison with the solution of 3D problem shows that Uflyand-Mindlin theory is applicable at the frequencies up to 30% of the first cut-off. The higher order asymptotic theory of plate bending with modified inertia and classical boundary conditions allows to improve the describing of the velocity comparing to Kirchhoff theory, but leads to a considerable error in describing of the amplitude.

Natural Frequencies and Normal Modes of a Spinning Timoshenko Beam With General Boundary Conditions

1992 ◽  
Vol 59 (2S) ◽  
pp. S197-S204 ◽  
Author(s):  
Jean Wu-Zheng Zu ◽  
Ray P. S. Han
Keyword(s):  
Boundary Conditions ◽  
Mode Shape ◽  
Timoshenko Beam ◽  
Natural Frequencies ◽  
Normal Modes ◽  
Single Mode ◽  
Mode Shapes ◽  
Simulation Studies ◽  
Classical Boundary ◽  
First Time

A free flexural vibrations of a spinning, finite Timoshenko beam for the six classical boundary conditions are analytically solved and presented for the first time. Expressions for computing natural frequencies and mode shapes are given. Numerical simulation studies show that the simply-supported beam possesses very peculiar free vibration characteristics: There exist two sets of natural frequencies corresponding to each mode shape, and the forward and backward precession mode shapes of each set coincide identically. These phenomena are not observed in beams with the other five types of boundary conditions. In these cases, the forward and backward precessions are different, implying that each natural frequency corresponds to a single mode shape.

Effects of inertia and stratification in incompressible ideal fluids: pressure imbalances by rigid confinement

2013 ◽  
Vol 726 ◽  
pp. 404-438 ◽  
Author(s):  
R. Camassa ◽  
S. Chen ◽  
G. Falqui ◽  
G. Ortenzi ◽  
M. Pedroni
Keyword(s):  
Euler Equations ◽  
Initial Conditions ◽  
Numerical Algorithms ◽  
Wave Dispersion ◽  
Variable Density ◽  
Density Variation ◽  
Fluid Inertia ◽  
Time Dynamics ◽  
Long Wave ◽  
Set Up

AbstractConsequences of density stratification are studied for an ideal (Euler) incompressible fluid, confined to move under gravity between rigid lids but otherwise free to move along horizontal directions. Initial conditions that generate horizontal pressure imbalances in a laterally unbounded domain are examined. The aim is to show analytically the existence of classes of initial data for which total horizontal momentum evolves in time, even though only vertical forces act on the fluid in this set-up. A simple class of such initial conditions, leading to momentum evolution, is identified by systematic asymptotic expansions of the governing inhomogeneous Euler equations in the small-density-variation limit. These results for Euler equations are compared and confirmed with long-wave asymptotic models, which can handle arbitrary density variations and provide closed-form mathematical expressions for limiting cases. In particular, the role of wave dispersion arising from the fluid inertia is captured by the long-wave models, even for short-time dynamics emanating from initial conditions outside the models’ asymptotic range of validity. These results are compared with direct numerical simulations for variable-density Euler fluids, which further validate the numerical algorithms and the analysis.

Pull-In Instability of Functionally Graded Cantilever Nanoactuators Incorporating Effects of Microstructure, Surface Energy and Intermolecular Forces

2018 ◽  
Vol 10 (08) ◽  
pp. 1850091 ◽  
Author(s):  
Mohamed A. Attia ◽  
Salwa A. Mohamed
Keyword(s):  
Residual Stress ◽  
Surface Energy ◽  
Boundary Conditions ◽  
Van Der Waals ◽  
Surface Elasticity ◽  
Van Der Waals Forces ◽  
Functionally Graded ◽  
Surface Residual Stress ◽  
Electrically Actuated ◽  
Classical Boundary

In this paper, an integrated non-classical continuum model is developed to investigate the pull-in instability of electrostatically actuated functionally graded nanocantilevers. The model accounts for the simultaneous effects of local-microstructure, surface elasticity and surface residual in the presence of fringing field as well as Casimir and van der Waals forces. The modified couple stress and Gurtin–Murdoch surface elasticity theories are employed to conduct the scaling effects of microstructure and surface energy, respectively, in the context of Euler–Bernoulli beam hypothesis. Bulk and surface material properties are varied according to the power-law distribution through the beam thickness. The physical neutral axis position for mentioned FG nanobeams is considered. Hamilton principle is employed to derive the nonlinear size-dependent governing equations and the non-classical boundary conditions. The resulting nonlinear differential equations are solved utilizing the generalized differential quadrature method (GDQM). In addition, the non-classical boundary conditions of nanocantilever beams due to surface residual stress are exactly implemented. After validation of the obtained results by previously available data in the literature, the influences of different geometrical and material parameters on the pull-in instability of the FG nanocantilevers are examined in detail. It is concluded that the pull-in behavior of electrically actuated FG micro/nanocantilevers is significantly influenced by the material distribution, material length scale parameter, surface elasticity constant, surface residual stress, initial gap, slenderness ratio, Casimir, and van der Waals forces. The obtained results can be considered for modeling and analysis of electrically actuated FG nanocantilevers.

A model study of elastic waves in a layered sphere

1967 ◽  
Vol 57 (5) ◽  
pp. 959-981
Author(s):  
Victor Gregson
Keyword(s):  
Surface Wave ◽  
Surface Waves ◽  
Elastic Waves ◽  
Wave Dispersion ◽  
Flat Space ◽  
Dispersion Curves ◽  
Body Waves ◽  
Surface Wave Dispersion ◽  
Steel Sphere ◽  
Long Wave

abstract Elastic waves produced by an impact were recorded at the surface of a solid 12.0 inch diameter steel sphere coated with a 0.3 inch copper layer. Conventional modeling techniques employing both compressional and shear piezoelectric transducers were used to record elastic waves for one millisecond at various points around the great circle of the sphere. Body, PL, and surface waves were observed. Density, layer thickness, compressional and shear-wave velocities were measured so that accurate surface-wave dispersion curves could be computed. Surface-wave dispersion was measured as well as computed. Measured PL mode dispersion compared favorably with theoretical computations. In addition, dispersion curves for Rayleigh, Stoneley, and Love modes were computed. Measured surface-wave dispersion showed Rayleigh and Love modes were observed but not Stoneley modes. Measured dispersion compared favorably with theoretical computations. The curvature correction applied to dispersion calculations in a flat space has been estimated to correct dispersion values at long-wave lengths to about one per cent of correct dispersion in a spherical model. Measured dispersion compared with such flat space dispersion corrected for curvature proved accurate within one per cent at long wave lengths. Two sets of surface waves were observed. One set was associated with body waves radiating outward from impact. The other set was associated with body waves reflecting at the pole opposite impact. For each set of surface waves, measured dispersion compared favorably with computed dispersion.

An Experimental Approach to the Design of PVDF Modal Sensors

2001 ◽  
Author(s):  
Daniel Cuhat ◽  
Patricia Davies
Keyword(s):  
Boundary Conditions ◽  
Experimental Approach ◽  
Experimental Methodology ◽  
Laser Doppler Velocimeter ◽  
Piezoelectric Film ◽  
Bernoulli Beam ◽  
Beam Structure ◽  
Arbitrary Boundary ◽  
Classical Boundary ◽  
Euler Bernoulli Beam

Abstract The principle of modal sensing is based on the use of a shaped PVDF piezoelectric film measuring strains on the surface of a bending beam and acting as a modal filter. So far, the use of this type of sensors has remained confined to studies involving uniform structures with classical boundary conditions. The goal of this paper is to present an experimental methodology for the design of a shaped modal sensor applicable to an non-uniform Euler-Bernoulli beam with arbitrary boundary conditions. This approach is illustrated with test data collected on a cantilever beam structure with a laser Doppler velocimeter.

Comments on the use of the Korteweg-de Vries equation in the study of anharmonic lattices

1974 ◽  
Vol 339 (1616) ◽  
pp. 119-126 ◽  
Keyword(s):  
Differential Equation ◽  
Lattice Dynamics ◽  
Continuum Limit ◽  
Vries Equation ◽  
Wave Dispersion ◽  
Order Partial Differential Equation ◽  
Lennard Jones ◽  
Long Wave ◽  
De Vries ◽  
The Continuum

The use of the Korteweg-de Vries equation as the continuum limit of the equations describing the anharmonic motion of atoms in a lattice is examined in the light of the periodic solutions recently constructed by Askar (1973). It is shown that the Korteweg-de Vries equation does not exactly represent the behaviour of the nonlinear lattice in the limit of long waves and that a sixth-order partial differential equation gives a more accurate description of the lattice dynamics. The relative accuracies of two long wave dispersion relations derived from the Korteweg-de Vries equation are discussed and numerical results are presented for Morse, Born-Meyer and Lennard-Jones potentials. Askar’s paper contains several misprints and errors and the corrected forms of his equations are presented in the appendix.

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