Effect of thermal gradient on vibration of non-homogeneous visco-elastic elliptic plate of variable thickness

An analysis and numerical results are presented for free transverse vibrations of non-homogeneous visco-elastic elliptic plate whose temperature and thickness spatial variations both are parabolic along a line through plate centre. The variation in density is assumed as parabolic along a line through plate centre, which raise non-homogeneity of the plate materials and make problem interesting as introducing variation in non-homogeneity of the material mass density reduce the problem practical importance in comparison to homogenous plate. For visco-elastic, the basic elastic and viscous elements are combined. We have taken Kelvin model for visco-elasticity that is the combination of the elastic and viscous elements in parallel. Here the elastic element means the spring and the viscous element means the dashpot. The governing differential equation of motion has been solved by Galerkin’s technique. Deflection, time period and logarithmic decrement corresponding to the first two modes of vibrations of a clamped non-homogeneous visco-elastic elliptic plate for various values of taper constant, thermal constants, non-homogeneity constant and aspect ratio are obtained and shown graphically.     

Study of the effect of thermal gradient on free vibration of clamped visco-elastic rectangular plates with linearly thickness variation in both directions

The effect of thermal gradient on the free vibration of clamped visco-elastic rectangular plate with linearly thickness variations in both the directions has been studied here. The governing differential equation has been solved using Rayleigh-Ritz technique. The frequency equation is derived for the clamped boundary condition on all the four edges. The effect of linear temperature variation has been considered. Deflection and time period corresponding to the first two modes of vibrations of a clamped plate have been computed for various values of aspect ratio, thermal constants, and taper constants.     

A general analytical solution for elastic vibration of rectangular thin plates

A general solution of differential equation for lateral displacement lunction of rectangular elastic thin plates in free vibration is established in this paper. It can be used to solve the vibration problem of rectangular plate with arbitrary boundaries. As an example, the frequency and its vibration mode of a rectangular plate with four edges free have been solved.     

Study of thermal gradient effect on vibrations of a non-homogeneous orthotropic rectangular plate having bi-direction linearly thickness variations

Here, thermal gradient effect on vibration of non-homogeneous orthotropic rectangular plate having bi-direction linearly thickness variations is studied. The non-homogeneity is assumed to arise due to the variation in the density of the plate material. Rayleigh Ritz method is used to evaluate the fundamental frequencies and deflection function. The two-dimensional thickness variation is taken as the Cartesian product of linear variations along the two concurrent edges of the plate. Yet the method used for the solution of present problem provides an approximate solution but it is quite convenient and authentic one. A two-term deflection function is used. Frequency corresponding to first mode of vibration is calculated for clamped plate for different values of temperature gradient, taper constants and non-homogeneity constant. In special cases, comparisons have been made with results that are available in the literature.     

Vibrations of nonhomogeneous orthotropic rectangular plates with bilinear thickness variation resting on Winkler foundation

Free transverse vibrations of nonhomogeneous orthotropic rectangular plates with bilinear thickness variation resting on Winkler foundation are presented here using two dimensional boundary characteristic orthogonal polynomials in the Rayleigh-Ritz method on the basis of classical plate theory. Gram-Schmidt process has been used to generate orthogonal polynomials. The nonhomogeneity of the plate is assumed to arise due to linear variations in elastic properties and density of the plate material with the in-plane coordinates. The two dimensional thickness variation is taken as the Cartesian product of linear variations along the two concurrent edges of the plate. Effect of nonhomogeneity parameters, aspect ratio and thickness variation together with foundation parameter on the natural frequencies has been illustrated for the first three modes of vibration for four different combinations of clamped, simply supported and free edges correct to four decimal places. Three dimensional mode shapes for specified plate for all the four boundary conditions have been plotted. A comparison of results in special cases with published one has been presented.     

Winkler地基上四边自由正交各向异性矩形薄板弯曲问题的辛叠加解

研究Winkler地基上正交各向异性矩形薄板弯曲方程所对应的Hamilton正则方程, 计算出其对边滑支条件下相应Hamilton算子的本征值和本征函数系, 证明该本征函数系的辛正交性以及在Cauchy主值意义下的完备性, 进而给出对边滑支边界条件下Hamilton正则方程的通解, 之后利用辛叠加方法求出Winkler地基上四边自由正交各向异性矩形薄板弯曲问题的解析解. 最后通过两个具体算例验证了所得解析解的正确性.     

Buckling strength analysis of adhesively bonded aluminum hat sections

This paper provides an analytical solution for the critical buckling stress of adhesively bonded aluminum hat sections under static axial compression. The governing rectangular plate member of the structure is treated based on the differential equation for out-of-plane deflections of thin plates. Finite element eigenvalue buckling analysis is performed to verify the assumed simply supported boundary conditions for common edges between adjacent plate elements. Elastic restraint is applied to the two loaded edges of the rectangular plate, and the relative critical buckling stress is computed according to the transcendental equations. It is found from experiments that there is no adhesive bonding failure in the elastic buckling stage. The analytical solution yields buckling stress predictions which are in reasonable agreement with measured values.     

DSC-Ritz element method for vibration analysis of rectangular Mindlin plates with mixed edge supports

Based on Mindlin's first-order shear deformable plate theory, a DSC-Ritz element method is developed for the free vibration analysis of moderately thick rectangular plates with mixed supporting edges. The rationale of the present approach is not only to apply the discrete singular convolution (DSC) delta type wavelet kernel as a trial function with the Ritz method, but also to incorporate the method in finite elements in order to handle the mixed boundary constraints. The approach is novel and flexible as it passes through a bottleneck of the global DSC-Ritz method in treating the kinematic supporting edges with assorted discontinuities. A series of numerical simulations for rectangular Mindlin plates with various edge support discontinuities, plate thicknesses and aspect ratios are presented. For verification, the vibration frequencies thus established are directly compared with those reported in the open literature. New sets of numerical results for several other cases of moderately thick plates with mixed simply supported, clamped and free edges are presented and discussed in detail.     

Buckling and vibration of non-ideal simply supported rectangular isotropic plates

Buckling and vibration of a rectangular isotropic plate which has non-ideal simply supported boundary conditions along one of its edges are investigated. It is assumed that one of the edges of the plate allows a small non-zero deflection and a small non-zero moment. Externally applied in-plane loads are considered to be parallel and perpendicular to the edge where non-ideal boundary conditions are present in the buckling problem. Analytical solutions of the buckling and vibration problems are obtained by using the Linshtead–Poincare perturbation technique. Improved buckling loads and natural frequencies are determined for various values of the aspect ratio of the plate.     

薄板小波有限元理论及其应用

利用样条小波尺度函数构造了常用的三角形和矩形薄板单元的位移函数,得到了利用小波函数表示的形函数。采用合理的局部坐标,对单元进行压缩,使单元在局部坐标区间上有其值,成功地推导出了分域的三角形和矩形薄板小波有限元列式。在此基础上,提出了弹性地基薄板的小波有限元求解方法。通过两个算例对薄板的挠度和弯矩进行了计算,数值结果表明,求解结果具有收敛快、精度高的特点。     

The calculation of bending and stability of a thin rectangular plate for variable rigidity

The subject discussed in this paper is the rectangular plate. Its wwo opposite edges are simply supported, while the other two are arbitrary, and the rigidity of the plate is variable along the direction parallel to the simply supported edges. In order to solve the problem, the author adopts the finite plate-strip element method^[1], which is different from the usual finite element method or the finite strip method. The steps of the above method is no longer to establish a rigidity matrix for elements or strips and gather them into a total matrix for solution. Now the relation of transfer between the strain and inner force of every plate-strip is shown. Finally a practical example is given and this method is found to be easier and more effective.     

Experimental and theoretical study on large amplitude vibrations of clamped rubber plates

In this paper, the large amplitude forced vibrations of thin rectangular plates made of different types of rubbers are investigated both experimentally and theoretically. The excitation is provided by a concentrated transversal harmonic load. Clamped boundary conditions at the edges are considered, while rotary inertia, geometric imperfections and shear deformation are neglected since they are negligible for the studied cases. The von Kármán nonlinear strain-displacement relationships are used in the theoretical study; the viscoelastic behaviour of the material is modelled using the Kelvin-Voigt model, which introduces nonlinear damping. An equivalent viscous damping model has also been created for comparison. In-plane pre-loads applied during the assembly of the plate to the frame are taken into account. In the experimental study, two rubber plates with different material and thicknesses have been considered; a silicone plate and a neoprene plate. The plates have been fixed to a heavy rectangular metal frame with an initial stretching. The large amplitude vibrations of the plates in the spectral neighbourhood of the first resonance have been measured at various harmonic force levels. A laser Doppler vibrometer has been used to measure the plate response. Maximum vibration amplitude larger than three times the thickness of the plate has been achieved, corresponding to a hardening type nonlinear response. Experimental frequency-response curves have been very satisfactorily compared to numerical results. Results show that the identified retardation time increases when the excitation level is increased, similar to the equivalent viscous damping but to a lesser extent due to its nonlinear nature. The nonlinearity introduced by the Kelvin-Voigt viscoelasticity model is found to be not sufficient to capture the dissipation present in the rubber plates during large amplitude vibrations.     

Numerical investigation of the effect of boundary conditions on hydroelastic stability of two parallel plates interacting with a layer of ideal flowing fluid

The paper studies the hydroelastic stability of two parallel identical rectangular plates interacting with a flowing fluid confined between them. General equations describing the behavior of ideal compressible liquid in the case of small perturbations are written in terms of the perturbation velocity potential and transformed using the Bubnov–Galerkin method. The small deformations of elastic plates are defined using the first-order shear deformation plate theory. A mathematical formulation of the dynamic problem for elastic structures is developed using the variational principle of virtual displacements, which takes into account the work done by the inertial forces and hydrodynamic pressure. The numerical solution of the problem is carried out in three-dimensional formulation by means of the finite element method. A stability criterion is based on the analysis of complex eigenvalues of the coupled system of equations obtained for different values of flow velocity. The existence of different types of instability has been shown depending on the combinations of the kinematic boundary conditions defined at the edges of both plates. We considered both the symmetric and asymmetric types of clamping. It has been found that the dependence of the lowest eigenfrequency of two parallel plates on the height of quiescent fluid is nonmonotonic with a pronounced peak. At the same time, critical velocities of instability change insignificantly if the distance between plates is greater than half of the maximum linear dimensions of the structure. It should be noted that the critical velocities of divergence increase monotonically with growth of the height of the fluid layer, but critical velocities for the onset of flutter instability have sharp jumps. The cause of these jumps is a change in the mode shapes at which the system loses stability.     

Investigation of the interaction between a viscous incompressible fluid layer and walls of a channel formed by coaxial vibrating discs

The interaction between a viscous incompressible fluid layer and walls of a channel formed by two concentric discs moving perpendicularly to their planes due to vibration of the base on which the channel is mounted is investigated. The case of two absolutely rigid discs with elastic suspension and the case in which one of the discs is an elastic plate with the rigid restrain on the edges are considered. The velocity and pressure distributions over the fluid and the laws of motion of the walls and their frequency characteristics which make it possible to determine the resonance vibration frequencies of the mechanical system considered are found.     

无网格法在点弹性支承矩形薄板横向振动中的应用

基于薄板理论和弹性动力学Hamilton原理的推广,采用无网格伽辽金法,建立了具有有限多个点弹性支承的弹性矩形薄板横向振动的无量纲量运动微分方程,给出了其特征方程。通过求解特征方程,得出了四边简支板的无量纲固有频率随点弹性支承的刚性系数和支承位置的变化曲线,分析了点弹性支承的刚性系数和支承位置对矩形薄板横向振动特性的影响。数值计算结果表明,无网格法求解点弹性支承板横向振动问题是切实可行的。     

Characteristic orthogonal polynomials in the study of transverse vibrations of nonhomogeneous rectangular orthotropic plates of bilinearly varying thickness

Effect of nonhomogeneity on the vibrational characteristics of thin orthotropic rectangular plates of bilinearly varying thickness has been studied using boundary characteristic orthogonal polynomials in the Rayleigh-Ritz method. The thickness variation is taken as the Cartesian product of linear variations along two concurrent edges of the plate. The orthogonal polynomials in two variables are generated using the Gram-Schmidt process. The nonhomogeneity of the plate material is assumed to arise due to linear variations in Young’s moduli, shear modulus and density of the plate with the in-plane coordinates. Numerical results have been computed for four different combinations of clamped, simply supported and free edges. Effect of thickness variation together with varying values of aspect ratio and nonhomogeneity on the natural frequencies is illustrated for the first three modes of vibration. Three dimensional mode shapes have been presented. Comparison has been made with the known results.     

NONLINEAR DYNAMICS OF A FGM PLATE WITH TWO CLAMPED OPPOSITE EDGES AND TWO FREE EDGES

This paper deals with the nonlinear forced vibration of FGM rectangular plate with a boundary of two edges clamped opposite and the other two free. The plate is subjected to transversal and in-plane excitations. The present research treats the material properties of the FGM plates as temperature-dependent and graded continuously throughout the thickness direction, following the volume fraction of the constituent materials according to the power law. The temperature is assumed to be constant in the plane and varied only in the thickness direction of the plate. In the framework of geometrical nonlinearity the plate is modeled and the equations of motion are obtained on Hamilton＇s principle. With the help of Galerkin discretization, the nonlinear ordinary differential equations describing transverse vibration of the plate are proposed. By the numerical method, the nonlinear dynamical responses of the FGM plate with two clamped opposite and two free edges are analyzed.     

非均匀Winkler弹性地基上变厚度矩形板自由振动的DTM求解

针对非均匀Winkler弹性地基上变厚度矩形板的自由振动问题,通过一种有效的数值求解方法——微分变换法（DTM）,研究其无量纲固有频率特性。已知变厚度矩形板对边为简支边界条件,其他两边的边界条件为简支、固定或自由任意组合。采用DTM将非均匀Winkler弹性地基上变厚度矩形板无量纲化的自由振动控制微分方程及其边界条件变换为等价的代数方程,得到含有无量纲固有频率的特征方程。数值结果退化为均匀Winker弹性地基上矩形板以及变厚度矩形板的情形,并与已有文献采用的不同求解方法进行比较,结果表明,DTM具有非常高的精度和很强的适用性。最后,在不同边界条件下分析地基变化参数、厚度变化参数和长宽比对矩形板无量纲固有频率的影响,并给出了非均匀Winkler弹性地基上对边简支对边固定变厚度矩形板的前六阶振型。     

Finite Difference Method for the Acoustic Radiation of an Elastic Plate Excited by a Turbulent Boundary Layer: A Spectral Domain Solution

A finite difference method is developed to study, on a two-dimensional model, the acoustic pressure radiated when a thin elastic plate, clamped at its boundaries, is excited by a turbulent boundary layer. Consider a homogeneous thin elastic plate clamped at its boundaries and extended to infinity by a plane, perfectly rigid, baffle. This plate closes a rectangular cavity. Both the cavity and the outside domain contain a perfect fluid. The fluid in the cavity is at rest. The fluid in the outside domain moves in the direction parallel to the system plate/baffle with a constant speed. A turbulent boundary layer develops at the interface baffle/plate. The wall pressure fluctuations in this boundary layer generates a vibration of the plate and an acoustic radiation in the two fluid domains. Modeling the wall pressure fluctuations spectrum in a turbulent boundary layer developed over a vibrating surface is a very complex and unresolved task. Ducan and Sirkis [1] proposed a model for the two-way interactions between a membrane and a turbulent flow of fluid. The excitation of the membrane is modeled by a potential flow randomly perturbed. This potential flow is modified by the displacement of the membrane. Howe [2] proposed a model for the turbulent wall pressure fluctuations power spectrum over an elastomeric material. The model presented in this article is based on a hypothesis of one-way interaction between the flow and the structure: the flow generates wall pressure fluctuations which are at the origin of the vibration of the plate, but the vibration of the plate does not modify the characteristics of the flow. A finite difference scheme that incorporates the vibration of the plate and the acoustic pressure inside the fluid cavity has been developed and coupled with a boundary element method that ensures the outside domain coupling. In this paper, we focus on the resolution of the coupled vibration/interior acoustic problem. We compare the results obtained with three numerical methods: (a) a finite difference representation for both the plate displacement and the acoustic pressure inside the cavity; (b) a coupled method involving a finite difference representation for the displacement of the plate and a boundary element method for the interior acoustic pressure; (c) a boundary element method for both the vibration of the plate and the interior acoustic pressure. A comparison of the numerical results obtained with two models of turbulent wall pressure fluctuations spectrums - the Corcos model [3] and the Chase model [4] - is proposed. A difference of 20 dB is found in the vibro-acoustic response of the structure. In [3], this difference is explained by calculating a wavenumber transfer function of the plate. In [6], coupled beam-cavity modes for similar geometry are calculated by the finite difference method. This revised version was published online in July 2006 with corrections to the Cover Date.     

AN ANALYTICAL SOLUTION OF TRANSVERSE VIBRATION OF RECTANGULAR PLATES SIMPLY SUPPORTED AT TWO OPPOSITE EDGES WITH ARBITRARY NUMBER OF ELASTIC LINE SUPPORTS IN ONE WAY

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