Rigid Rectangular Plate Research Articles

Models for resonant acoustic metasurfaces with application to moth wing ultrasound absorption.

Taking as bioinspiration the remarkable acoustic absorption properties of moth wings, we develop a simple analytical model that describes the interaction between acoustic pressure fields, and thin elastic plates incorporating resonant sub-structures. The moth wing is an exemplar of a natural acoustic metamaterial; the wings are deeply subwavelength in thickness at the frequencies of interest, the absorption is broadband and the tiny scales resonate on the moth wing acting in concert. The simplified model incorporates only the essential physics and the scales are idealized to flat rigid rectangular plates coupled via a spring to an elastic plate that forms the wing; all the components are deep-subwavelength at desired frequencies. Based on Fourier analysis, complemented by phenomenological modelling, our theory shows excellent agreement with simulation mimicking the moth-wing structure. Moth wings operate as broadband sound absorbers employing a range of scale sizes. We demonstrate that a random distribution of scale sizes generates a broadband absorption spectrum. To further illustrate the potential of the model, we design a deeply sub-wavelength acoustic counterpart of electromagnetically induced reflectance.This article is part of the theme issue ‘Wave generation and transmission in multi-scale complex media and structured metamaterials (part 2)’.

Non-Linear Oscillations of the Channel Wall Filled with a Viscous Liquid Induced by the Vibrating Foundation

The paper considers the issues of mathematical modelling nonlinear vibrations for the wall of a narrow parallel walled channel filled with a viscous incompressible liquid. The upper channel wall is a rigid rectangular plate supported by nonlinear spring with a cubic restoring force, while the bottom one is a fixed rigid rectangular plate. The case of nonlinear oscillations of the upper channel wall due to channel’s foundation vibration has been investigated in the frame of hydroelasticity problem. The main attention is paid to the consideration of steady-state nonlinear oscillations for the upper channel wall and the creeping motion of the liquid in the channel. The liquid layer reaction acting on the upper channel wall is found and the channel’s wall nonlinear oscillation equation is obtained taking into account the energy dissipation due to the liquid viscosity. It was shown that this equation coincided with the Duffing one. The solution of the nonlinear oscillation equation was carried out by the harmonic balance method. Based on the found solution, the hydroelastic response of the channel wall on a nonlinear elastic suspension to the channel foundation vibration was determined.

Analytical solution to vertical and rocking vibration of a rigid rectangular plate on a layered transversely isotropic half-space

Analytical solution to vertical and rocking vibration of a rigid rectangular plate on a layered transversely isotropic half-space

Flow induced oscillation of two rigid rectangular plates in a side-by-side configuration

The present work is an experimental study of two oscillating rigid plates placed in side-by-side configuration, hinged at their leading edges, subjected to low subsonic flow. This problem is investigated using smoke-wire flow visualization, hot-wire anemometry, and time resolved particle image velocimetry. It is found that beyond a critical Reynolds number, the plates set into oscillatory motion. This critical Reynolds number depends on the gap between the plates. It is also seen that this value of Reynolds number, at lower values of gap to thickness ratio (<7) is significantly higher than that of the single plate configuration value. The frequency and amplitude of the oscillating plates at various gaps and Reynolds numbers have been studied and compared with the characteristics of an oscillating single plate. It is also found that depending on the gap and acceleration of the free-stream, there exist two modes of oscillation - (i) in-phase and (ii) out-of-phase. For gap to thickness ratio less than 10, only in-phase oscillations take place for all values of free-stream velocity considered in the present work, whereas, when this ratio is greater than 10, the mode of oscillation depends on the initial conditions up to a certain free-stream velocity, beyond which the plates switch to in-phase mode. Smoke wire flow visualization technique along with time resolved particle image velocimetry reveal that the vorticity distributions around the plates are responsible for the initiation of the two modes of oscillations.

Models for analyzing squeeze film air damping depending on oscillation modes of micro/nano beam resonators

Models for analyzing squeeze film air damping depending on the oscillation mode of cantilever and double-clamped micro/nanobeam resonators are presented. The models are obtained from the damping theory of a rigid rectangular plate moving parallelly to a nearby object in viscous and molecular flow regime and electrostatic theory. The analysis results based on the models show that the influence of oscillation mode on squeeze film air damping in viscous regime is remarkably large compared to that in molecular flow regime. The damping in the first-order mode is larger than that in the high-order modes, while there is little deference in damping among the high-order modes. The damping of oscillation modes depends strongly on the oscillation amplitude and electrostatic spring softening. The obtained damping analysis models are useful for the optimal design of micro/nanobeam resonators and verifying the discrepancy between the theoretically calculated results and experiment data.

Sound diffraction prediction of a rectangular rigid plate using the physical theory of diffraction

For the purpose of numerical efficiency in room acoustic modeling, the physical theory of diffraction (PTD) is applied to approximate solutions of a finite-sized, rigid rectangular plate. The application of the PTD enables sound diffraction contributions from the two edge pairs of the rigid plate to be predicted separately while ignoring the edge waves around the corners in far-field. This paper discusses numerical implementations of the theoretical predictions to show the efficiency of the application of the PTD in finite-sized objects. The numerical results implemented will also be compared with preliminary experimental results carried out using an acoustical goniometer.

Diffraction simulation from wedges to finite-sized plates based on the physical theory of diffraction

Efficient predictions of sound diffraction around objects are of critical significance in room-acoustic simulations. An advanced diffraction theory has recently been investigated for potential applications in room acoustics for some semi-infinite, canonical wedges and for finite-sized rectangular plates [Rozynova and Xiang, J. Acoust. Soc. Am. 144 (to be published)]. The physical theory of diffraction (PTD) still relies on both geometrical and physical principles, yet emphasizes the physical one. Important features of the PTD approach are its computational efficiency and the high degree of accuracy for the diffracted sound field. This paper reviews the fringe field predictions of canonical semi-infinite wedges and further discusses solutions of diffraction problems on finite, rigid rectangular plates. The PTD is applied to approximate the solutions of a finite-sized, rigid rectangular plate that achieves high numerical efficiency. The PTD simulation allows sound diffraction contributions to be determined independently from two pairs of edges of the rigid plate, while ignoring the edge waves around the corner in far-field. This paper uses numerical implementations of the PTD predictions to demonstrate the simulation efficiency of the PTD in finite-sized objects. The numerical simulations are also validated by some preliminary experimental results carried out using an acoustic goniometer.

Flow-induced oscillation of a rigid rectangular plate hinged at its leading edge

In the present study, the flow-induced oscillation of an elongated rectangular cylinder (a rigid plate with flat leading and trailing edges), hinged at its leading edge, is studied experimentally. Effects of various parameters such as the Reynolds number, chord, thickness, and moment of inertia on oscillation characteristics of the plate were investigated. Flow visualization and hotwire measurements were carried out to study the nature of the flow field around the plate. The angular deflection and angular velocity of the oscillating plate were obtained using a high-speed camera. There exists a critical chord to thickness ratio, below which the rectangular plate hinged at the leading edge starts oscillating beyond a critical wind speed. The flow-induced oscillation of the plate begins with a small amplitude oscillation which increases with time, and the plate finally attains a limit cycle oscillation. Impinging shear layer vortices, originating at the leading edge, play a key role in the initiation of flow-induced oscillation of the plate. The frequency of the limit cycle oscillation increases almost linearly with the wind speed and decreases with an increase in the chord to thickness ratio of the plate. A dimensional analysis is carried out to identify the important dimensionless parameters, and the same has been used for extracting the scaling for oscillation frequency. Good agreement between the present experimental results and the derived analytical expressions is obtained.

Vertical Vibration of Two Axle Railway Vehicle

The vertical vibration of the two axle vehicle has been solved analytically using the model of rigid rectangular plate with asymmetrically load distribution related to geometrical symmetry axes of central plane. Vertical displacements are solved for various cases of asymmetry, under kinematic excitation of the system with three degrees of freedom. The displacement of the centre of gravity and rotations to central axes are solved. The elastic support of the plate is combined with parallel viscous dumper. The task of vibration analysis is defined generally (and solved analytically) and for further solution (or extended) SYMPACT, ADAMS or MEDINA etc. software are possible to use.

Direct drive surge wave energy converter with grid integration functionality

Summary This paper addresses the design and simulation validation of a direct drive surge wave energy converter (DDSWEC) as an efficient device for harnessing marine wave energy in near-shore locations. The proposed DDSWEC is composed of a rigid rectangular plate producing linear motions in response to surge forces of ocean waves. A translator is devised to transfer the wave energy to a linear permanent magnet synchronous generator, which all together form the power take-off (PTO) system. In the proposed scheme, the mechanical simplicity of PTO system is a superior feature in reducing power losses in the wave-electric power conversion chain. Additionally, suitable mechanical degrees of freedom lead to an impressive increase in the power conversion efficiency. Besides, mathematical modeling of the mechanical and electrical systems governing DDSWEC is interrogated in depth. In addition, through a comprehensive frequency contents analysis, the capture width of DDSWEC and the device efficiency are scrutinized to validate its well performance. Because the stochastic behavior of waves induces voltages with variable frequency and magnitude, which are undesirable features for grid integration, a full-scale back-to-back converter with both generator and grid-side controls is accordingly implemented. To validate the overall performance of the proposed DDSWEC, a test system is designed in Matlab/Simulink platform by which several time-domain simulations for both the monochromatic and irregular waves are conducted. Copyright © 2015 John Wiley & Sons, Ltd.

The analysis of a rigid rectangular plate on a transversely isotropic multilayered medium

A semi-analytical method is developed for the analysis of a rigid rectangular plate on a transversely isotropic multilayered medium. Based on the governing equations of elasticity in Cartesian coordinates, transfer matrices for a single layer are deduced through a double Fourier transform and a Laplace transform. The transfer matrix solution of a multilayered medium is subsequently obtained with the consideration of the continuity conditions between two adjacent layers. By the application of the mixed boundary conditions of the contact problem, a pair of 2-D dual integral equations are derived, which are further converted to linear equations by means of the Jacobi orthogonal polynomials. The results of numerical calculation carried out by the corresponding computer program agree well with those obtained from the FEM software ABAQUS. Other numerical examples are presented to elucidate the influence of transversely isotropy, thickness and stratification of the medium.

Numerical Simulation of a Three-Dimensional Fish-like Body Swimming with Finlets

AbstractThe swimming of a 3D fish-like body with finlets is numerically investigated at Re = 1000 (the Reynolds number is based on the uniform upstream flow and the length of the fish-like body). The finlets are simply modeled as thin rigid rectangular plates that undulate with the body. The wake structures and the flow around the caudal peduncle are studied. The finlets redirect the local flow across the caudal peduncle but the vortical structures in the wake are almost not affected by the finlets. Improvement of hydrodynamic performance has not been found in the simulation based on this simple model. The present numerical result is in agreement with that of the work of Nauen and Lauder [J. Exp. Biol., 204 (2001), pp. 2251-2263] and partially supports the hypothesis of Webb [Bull. Fish. Res. Bd. Can., 190 (1975), pp. 1-159].

Analytical and numerical modeling of squeeze-film damping in perforated microstructures

The literature includes a variety of analytical and semi-analytical models to describe squeeze-film damping in MEMS perforated structures. Even if many of them have been validated by means of numerical simulations, nobody seems to have discussed about the accuracy of numerical approaches in this field. In the present paper, we apply both the main analytical models and a commercial finite element software, COMSOL Multiphysics, to solve a good number of squeeze-film problems. They refer to some cases, which were experimentally investigated during the past by different authors. The tested structures are rigid rectangular plates fabricated with different material, different perforation ratio (i.e., the ratio of the hole side to the holes pitch) and different number of perforations. We compare both the analytical and the numerical results with the available experimental data, in order to have an overview about their effectiveness. Numerical simulations offer in all the considered cases valuable agreement with experiments.

The linear-wave response of a periodic array of floating elastic plates

The problem of an infinite periodic array of identical floating elastic plates subject to forcing from plane incident waves is considered. This study is motivated by the problem of trying to model wave propagation in the marginal ice zone, a region of ocean consisting of an arbitrary packing of floating ice sheets. It is shown that the problem considered can be formulated exactly in terms of the solution to an integral equation in a manner similar to that used for the problem of wave scattering by a single elastic floating plate, the key difference here being the use of a modified periodic Green function. The convergence of this Green function in its original form is poor, but can be accelerated by a transformation. It is shown that the results from the method satisfy energy conservation and that in the particular case of a fixed rigid rectangular plate which spans the periodicity uniformly the solution reduces to that for a two-dimensional rigid dock. Solutions for a range of elastic-plate geometries are also presented.

Diffraction of an acoustic plane wave by a rectangular plate

A scalar field diffracted by a rigid rectangular plate can be computed with the formula that Otsuki derived by applying the properties of Weber–Schafheitlin’s discontinuous integral (also referred to as the Kobayashi potential). This method may be regarded as giving an eigenfunction expansion of the wave functions of the present configuration, although these functions are expressed by double infinite integrals. Expansion coefficients are determined from the matrix equations, and the matrix elements are given by double infinite integrals. An algorithm effective for computing these integrals is developed here by using the asymptotic expansions for the integrands. As a result, the original integral reduces to a finite double integral plus two finite single integrals. The numerical results for some diffraction patterns are presented. With relatively large plates, the results agree well with those obtained from the Kirchhoff diffraction integral.

Trefftz's Method for Vibration of Rigid Plate

Trefftz’s method is applied to study the vertical vibration of a rigid rectangular plate of infinite length resting on an elastic half-space. The problem is formulated as one of diffraction of elastic waves. The method consists in using c-complete or T-complete (T after Trefftz) families of solutions of the governing equations in the medium to construct the diffracted displacement fields as linear combinations of such solutions. Coefficients are obtained from the treatment of boundary conditions using a collocation least-square scheme. The relationship between the vertical contact force applied to the plate and vertical displacement is given in terms of a complex stiffness that depends on the frequency. Numerical values are given for different Poisson’s ratios. They are compared with those obtained by an analytical method.

Approximations for the low frequency response of a rigid plate embedded in an infinite space

This paper is concerned with the low frequency harmonic oscillation of rigid rectangular foundations deeply embedded in an isotropic elastic soil media. It is shown that a solution based on matched asymptotic expansion techniques can be combined with certain approximate solutions developed for the static compliance of a rigid rectangular plate embedded in an elastic infinite space, to yield a set of bounds for the dynamic response of an embedded rigid rectangular foundation.

Squeeze film lubrication of a viscoelastic fluid with reference to human joints

Squeeze film lubrication between two approaching surfaces is considered with reference to normally loaded human joints. The fluid between the surfaces is viscoelastic. The upper surface is a rigid rectangular plate and the lower surface is a porous rectangular plate composed of three thin layers with differing porosities. Analytical closed-form solutions are obtained for the pressure distribution and the load-carrying capacity. The effects of the viscoelastic behaviour of the synovial fluid and the effects of the porosity variation of the cartilage on the pressure and on the loadcarrying capacity are presented graphically.

Diffraction of an Acoustic Wave by a Rigid Rectangular Plate

The diffraction of a plane acoustic wave by an infinitely thin and rigid rectangular plate is treated theoretically for the case of normal incidence. The rigorous solution satisfying the wave equation and the boundary conditions is obtained by the use of modified Weber-Schafheitlin integrals and hypergeometric polynomials. The far field, the diffraction coefficient, the total pressure on the plate and its time average are expressed as functions of k a and k b , where 2 a and 2 b are the side lengths of the plate and k is the wave number.