Draft effect on wave action with a semi-infinite elastic plate

A method for analyzing reflection and transmission of ocean waves from a semi-infinlte elastic plate with a draft is developed. The relation of energy conservation for plates with three different edge conditions （ free, simply supported and built-in） is also derived. It is found that the present method satisfies the energy relation very well. The effects of draft on wave reflection and transmission coefficients as well as on the vertical vibration of the plates are examined through numerical tests. It is demonstrated that the zero draft assumption works well for low wave frequencies, but the effect of plate draft becomes significant for high wave frequencies.

Oblique wave scattering by a semi-infinite elastic plate with finite draft floating on a step topography

On the basis of a potential theory and Euler-Bernoulli beam theory, an analytical solution for oblique wave scattering by a semi-infinite elastic plate with finite draft floating on a step topography is developed using matched eigenfunction expansions. Different from previous studies, the effects of a wave incident angle, a plate draft, three different plate edge conditions （free, simply supported and built-in） and a sea-bottom topography are all taken into account. Moreover, the plate edge conditions are directly incorporated into linear algebraic equations for determining unknown expansion coefficients in velocity potentials, which leads to a simple and efficient solving procedure. Numerical results show that the convergence of the present solution is good, and an energy conservation relation is well satisfied. Also, the present predictions are in good agreement with known results for special cases. The effects of the wave incident angle, the plate draft, the plate edge conditions and the sea-bottom topography on various hydrodynamic quantities are analyzed. Some useful results are presented for engineering designs.

On the Interaction of Surface Waves with An Elastic Plate of Finite Length in Head Seas

An eigen-function expansion method based on a new orthogonal inner product is proposed by Sahoo et al. (2000) for the study of the hydroelastic response of mat-type VLFS in head seas. However, their main emphasis is on the effect of edge conditions and they assume that the plate is of a semi-infinite length. In reality, the plate is of finite length. For consideration of the finite length effect, the reflection and transmission from the other end must be considered. The effect of this reflection and transmission on the hydroelastic response of VLFS is of interest for practical application. Furthermore, the physical meaning of the new inner product was not given in their paper. In this paper, it is shown that the new inner product can he derived from the governing equation and the bottom boundary conditions. Then the same eigen-function expansion method is adopted for the study of the hydroelastic response of an elastic plate of finite length in surface waves. Detailed comparisons are made between the present finite length model and the semi-infinite model and between the present model predictions and the experimental results. It is found that that the finite length effect is significant and the accuracy of present model is higher than the semi-infinite model. Furthermore, a new phenomenon, which is not mentioned in Sahoo et al. (2000), is found. Taht is, for larger L/h ratios, the reflection and transmission coefficients will oscillate with the non-dimensional parameter k(0) h. Further study is needed for full understanding of this phenomenon.

ON THE ORIENTATION OF BUCKLING DIRECTION OF ANISOTROPIC ELASTIC PLATE UNDER UNIAXIAL COMPRESSION

The theory of small deformation superimposed on a largedeformation of an elastic solid is used to investigate the bucklingof anisotropic elastic plate under uniaxial compression. The bucklingdirec- tion (th direction of buckling direction is obtained. It isfound that the out-of-plane buckling of anisotropic elastic plate ispossible and both buckling conditions for flexural and extensionalmodes are presented.

Scattering of Surface Water Waves Involving Semi-infinite Floating Elastic Plates on Water of Finite Depth

Two problems of scattering of surface water waves involving a semi-infinite elastic plate and a pair of semi-infinite elastic plates,separated by a gap of finite width,floating horizontally on water of finite depth,are investigated in the present work for a two-dimensional time-harmonic case.Within the frame of linear water wave theory,the solutions of the two boundary value problems under consideration have been represented in the forms of eigenfunction expansions.Approximate values of the reflection and transmission coefficients are obtained by solving an over-determined system of linear algebraic equations in each problem.In both the problems,the method of least squares as well as the singular value decomposition have been employed and tables of numerical values of the reflection and transmission coefficients are presented for specific choices of the parameters for modelling the elastic plates.Our main aim is to check the energy balance relation in each problem which plays a very important role in the present approach of solutions of mixed boundary value problems involving Laplace equations.The main advantage of the present approach of solutions is that the results for the values of reflection and transmission coefficients obtained by using both the methods are found to satisfy the energy-balance relations associated with the respective scattering problems under consideration.The absolute values of the reflection and transmission coefficients are presented graphically against different values of the wave numbers.

EQUIVALENCE OF REFINED THEORY AND DECOMPOSED THEOREM OF AN ELASTIC PLATE

A connection between Cheng's refined theory and Gregory's decomposed theorem is analyzed. The equivalence of the refined theory and the decomposed theorem is given. Using operator matrix determinant of partial differential equation, Cheng gained one equation, and he substituted the sum of the general integrals of three differential equations for the solution of the equation. But he did not prove the rationality of substitute. There, a whole proof for the refined theory from Papkovich?_Neuber solution was given. At first expressions were obtained for all the displacements and stress components in term of the mid_plane displacement and its derivatives. Using Lur'e method and the theorem of appendix, the refined theory was given. At last, using basic mathematic method, the equivalence between Cheng's refined theory and Gregory's decomposed theorem was proved, i.e., Cheng's bi_harmonic equation, shear equation and transcendental equation are equivalent to Gregory's interior state, shear state and Papkovich_Fadle state, respectively.

RAYLEIGH LAMB WAVES IN MICROPOLAR ISOTROPIC ELASTIC PLATE

The propagation of waves in a homogeneous isotropic micropolar elastic cylindrical plate subjected to stress free conditions is investigated. The secular equations for symmetric and skew symmetric wave mode propagation are derived. At short wave limit, the secular equations for symmetric and skew symmetric waves in a stress free circular plate reduces to Rayleigh surface wave frequency equation. Thin plate results are also obtained. The amplitudes of displacements and microrotation components are obtained and depicted graphically. Some special cases are also deduced from the present investigations. The secular equations for symmetric and skew symmetric modes are also presented graphically.

NONLINEAR ANALYSIS ON THE VIBRATION OF ELASTIC PLATES

We consider the vibration of elastic thin plates under certain reasonable assumptions. We derive the nonlinear equations for this model by the Hamilton Principle. Under the conditions on the hyperbolicity for the initial data, we establish the local time wellposedness for the initial and boundary value problem by Picard iteration scheme, and obtain the estimates for the solutions.

ON OUT-OF-PLANE BUCKLING OF ANISOTROPIC ELASTIC PLATE SUBJECTED TO A SIMPLE SHEAR

Out-of-plane buckling of anisotropic elastic plate subjected to asimple shear is investigated. From exact 3-D equilibrium conditionsof anisotropic elastic body with a plane of elastic symmetry atcritical Configuration, the equation for buckling direction (bucklingwave direction) parameter is derived and the Shape functions ofpossible buckling modes are obtained. The traction free boundaryconditions which must Hold on the upper and lower surfaces of platelead to a linear eigenvalue problem whose nontrivial solutions Arejust the possible buckling modes for the plate.

Analytic Solution of Interaction Between Surface Waves and Floating Elastic Plates with Elastic End-Restraints

Based on the dynamic theories of water waves and Mindlin plates, the analytic solution of interaction between smface waves and two-dimensional floating elastic plates with edge-restraint is constructed by use of the Wiener-Hopf technique. Firstly, without regard for elastic edge restraint, the wave-induced responses of elastic floating plate analyzed by the present method are in good agreement with the results from literature and experimental results. Therefore, it can be shown that the present method is valid. Secondly, three end-restraint cases （i.e., the left-end elastic restraints, the both-end elastic restraints, and the right-end elastic restraints） are proposed to reduce the vibration of floating plates, in which the spring is used to connect the sea bottom and the floating plate＇ s left （ or right ） edge. The relations between the spring stiffness and the parameters of wave-induced responses of floating plates are discussed. Moreover, the effective method to reduce the vibration of floating elastic plates can be obtained through comparison.

Vibration reduction of floating elastic plates in surface waves

Based on dynamical theories of water waves and dynamics of Mindlin thick plates, the investigation of the wave-induced responses and the vibration reduction of an elastic floating plate are presented using the Wiener-Hopf technique. Without regard to the case of elastic connector, the calculated results obtained by the present method are in good agreement with those from the literature and the experiment. It can be shown that the present method is valid. Relations between the spring stiffness to be used to connect the sea bottom and the floating plate and the parameters of wave-induced responses of floating plates are investigated using the present method. Therefore, these results can be used as theoretical bases for the design stage of super floating platform systems.

Sound Transmission Loss of Helmholtz Resonators with Elastic Bottom Plate

Helmholtz resonators are widely used to control low frequency noise propagating in pipes.In this paper,the elastic bottom plate of Helmholtz resonator is simplified as a single degree of freedom(SDOF)vibration system with acoustic excitation,and a one-dimensional lumped-parameter analytical model was developed to accurately characterize the structure-acoustic coupling and sound transmission loss(STL)of a Helmholtz resonator with an elastic bottom plate.The effect of dynamical parameters of elastic bottom plate on STL is analyzed by utilizing the model.A design criterion to circumvent the effect of wall elasticity of Helmholtz resonators is proposed,i.e.,the structural natural frequency of the wall should be greater than three times the resonant frequency of the resonator to avoid the adverse effects of wall elasticity.This study can provide guidance for the rapid and effective design of Helmholtz resonators.

Unified Solution Method of Rectangular Plate Elastic Bending

The bending of rectangular plate is divided into the generalized statically determinate bending and the generalized statically indeterminate bending based on the analysis of the completeness of calculating condition at the corner point. The former can be solved directly by the equilibrium differential equation and the boundary conditions of four edges of the plate. The latter can be solved by using the superposition principle. Making use of the recommended method, the bending of the plate with all kinds of...

Hydroelastic Response of A Circular Plate in Waves on A Two-Layer Fluid of Finite Depth

The hydroelastic response of a circular, very large floating structure(VLFS), idealized as a floating circular elastic thin plate, is investigated for the case of time-harmonic incident waves of the surface and interfacial wave modes, of a given wave frequency, on a two-layer fluid of finite and constant depth. In linear potential-flow theory, with the aid of angular eigenfunction expansions, the diffraction potentials can be expressed by the Bessel functions. A system of simultaneous equations is derived by matching the velocity and the pressure between the open-water and the platecovered regions, while incorporating the edge conditions of the plate. Then the complex nested series are simplified by utilizing the orthogonality of the vertical eigenfunctions in the open-water region. Numerical computations are presented to investigate the effects of different physical quantities, such as the thickness of the plate, Young’s modulus, the ratios of the densities and of the layer depths, on the dispersion relations of the flexural-gravity waves for the two-layer fluid. Rapid convergence of the method is observed, but is slower at higher wave frequency. At high frequency, it is found that there is some energy transferred from the interfacial mode to the surface mode.

Analytic approximation to nonlinear hydroelastic waves traveling in a thin elastic plate floating on a fluid

An analytic approximation method known as the homotopy analysis method(HAM)is applied to study the nonlinear hydroelastic progressive waves traveling in an infinite elastic plate such as an ice sheet or a very large floating structure(VLFS)on the surface of deep water.A convergent analytical series solution for the plate deflection is derived by choosing the optimal convergencecontrol parameter.Based on the analytical solution the efects of diferent parameters are considered.We find that the plate deflection becomes lower with an increasing Young’s modulus of the plate.The displacement tends to be flattened at the crest and be sharpened at the trough as the thickness of the plate increases,and the larger density of the plate also causes analogous results.Furthermore,it is shown that the hydroelastic response of the plate is greatly afected by the high-amplitude incident wave.The results obtained can help enrich our understanding of the nonlinear hydroelastic response of an ice sheet or a VLFS on the water surface.

Hydroelastic interaction between water waves and a thin elastic plate of arbitrary geometry

An analytical method is developed for the hydroelastic interaction between surface incident waves and a thin elastic plate of arbitrary geometry floating on an inviscid fluid of finite depth in the framework of linear potential flow.Three kinds of edge conditions are considered and the corresponding analytical representations are derived in the polar coordinate system.According to the surface boundary conditions,the fluid domain is divided into two regions,namely,an open water region and a plate-covered region.With the assumption that all the motion is time-harmonic,the series solutions for the spatial velocity potentials are derived by the method of eigenfunction expansion.The matching conditions for the continuities of the velocity and pressure are transformed by taking the inner products successively with respect to the vertical eigenfunction for the free surface and the angular eigenfunction.A system of simultaneous equations,including two edge conditions and two matching conditions,is set up for deriving the expansion coefficients.As an example,numerical computation for the expansion coefficients of truncated series is performed for an elliptic plate.The results show that the method suggested here is useful to revealing the physical features of the gravity wave scattering in the open water and the hydroelastic response in the plate.

AN ACCURATE TWO-DIMENSIONAL THEORY OF VIBRATIONS OF ISOTROPIC,ELASTIC PLATES

An infinite system of two-dimensional equations of motion of isotropic elastic plates with edge and corner conditions are deduced from the three-dimensional equations of elasticity by expansion of displacements in a series of trigonometrical functions and a linear function of the thickness coordinate of the plate. The linear term in the expansion is to accommodate the in-plane displacements induced by the rotation of the plate normal in low-frequency flexural motions. A system of first-order equations of flexural motions and accompanying boundary conditions are extracted from the infinite system. It is shown that the present system of equations is equivalent to the Mindlin＇s first-order equations, and the dispersion relation of straight-crested waves of the present theory is identical to that of the Mindlin＇s without introducing any corrections. Reduction of present equations and boundary conditions to those of classical plate theories of flexural motions is also presented.

UNILATERAL EIGENVALUE PROBLEMS FOR NONLINEARLY ELASTIC PLATES: AN APPROACH VIA PSEUDO-MONOTONE OPERATORS

This paper considers a class of variational inequalities that model the buckling of a von Karman plate clamped on a part of its boundary and lying on a fiat rigid support. The existence and bifurcation results of D. Goeleven, V. H. Nguyen and M. Thera[6] rely on the Leray- Schauder degree. Using the topological degree for pseudo-monotone operators of type (S+), the author establishes a more general existence result for such unilateral eigenvalue problems.

On a Kind of Biharmonic Boundary Value Problems Related to Clamped Elastic Thin Plates

By using complex variable methods, the boundary value problem for biharmonic functions arisen from the theory of clamped elastic thin plate is shown to be equivalent to the first fundamental problem in plane elasticity which, as well-known, may be easily solved by reduction to a Fredholm integral equation. The case of circular plate is illustrated in detail, the solution of which is obtained in closed form.

USING FREDHOLM INTEGRAL EQUATION OF THE SECOND KINDTO SOLVE THE VERTICAL VIBRATION OF ELASTICPLATE ON AN ELASTIC HALF SPACE

The dual integral equations of vertical forced vibration of elastic plate on an elastic half space subject to harmonic uniform distribution loading are established according to the mixed boundary-value condition. By applying Abel transformation the dual integral equations are reduced to Fredholm integral equation of the second kind which is solved numerically.