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.

Sound produced by an aerodynamic source adjacent to a partly coated, finite elastic plate

1992 ◽  
Vol 436 (1897) ◽  
pp. 351-372 ◽  
Keyword(s):  
Elastic Plate ◽  
Loss Factor ◽  
Dipole Source ◽  
Thin Elastic Plate ◽  
Fluid Loading ◽  
Direct Radiation ◽  
Vortex Interactions ◽  
Bending Waves ◽  
Ducted Rotor ◽  
Clamped Edges

An analysis is made of the scattering of bending waves at the edge of an unbafiled, thin elastic plate in the presence of arbitrary fluid loading. Detailed predictions are made of the sound scattered from free and clamped edges, and empirical formulae given for the radiation loss factor over a range of frequencies and fluid loadings. Application is made to the generation of sound by an aerodynamic dipole source adjacent to a finite plate, a finite length of which has been treated with damping material. The dipole models the production of sound by blade-vortex interactions occurring when turbulence or discrete vortices are ingested by a ducted rotor, in which the plate assumes the role of a neighbouring duct wall. In typical underwater applications, when the influence of fluid loading is important, sound produced by the source at frequencies below the coincidence frequency of the bending waves can propagate directly to the far field, essentially as if the plate were absent. However, flexural plate-motions are also generated by the source. These contribute to the radiation by scattering at the edges and, in the absence of dissipation in the plate, the intensity of the edge-scattered sound can dominate the direct radiation from the source. When the edges can vibrate freely, it is shown that a relatively modest amount of damping is sufficient to reduce the edge generated sound to levels below those of the direct radiation. The efficiency with which bending wave energy is converted into sound is much larger for clamped edges, and larger values of coating loss factor and length are necessary to achieve significant reductions in the structural component of the radiated sound.

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.

Acoustic scattering by a finite elastic plate connecting two fluid-loaded parallel plates

1993 ◽  
Vol 443 (1918) ◽  
pp. 429-444 ◽  
Keyword(s):  
Elastic Plate ◽  
Acoustic Scattering ◽  
Elastic Plates ◽  
Parallel Plates ◽  
Fluid Loading ◽  
Heavy Fluid ◽  
Low Frequencies ◽  
Plane Sound ◽  
Two Fluid ◽  
Good Agreement

A plane sound wave is incident upon two infinite parallel elastic plates which are connected by a finite elastic plate. All three plates support compressional and bending motion, and interact with any compressible fluid with which they are in contact. A method, which can be applied to obtain numerical results, for calculating the sound scattered by the connecting plate is presented. In the absence of fluid between the plates an approximate solution, valid for low frequencies and heavy fluid-loading on the upper plate, has been derived which exhibits good agreement with results obtained numerically.

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)’.

Solitary and Travelling Wave Solutions of Certain Nonlinear Diffusion-Reaction Equations

2020 ◽  
Author(s):  
Miftachul Hadi
Keyword(s):  
Nonlinear Diffusion ◽  
Travelling Wave ◽  
Diffusion Reaction ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Reaction Equations

We review the work of Ranjit Kumar, R S Kaushal, Awadhesh Prasad. The work is still in progress.

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