An analogy between free vibration of a plate and of a particle of mass

In this paper, the free flexural vibration of an elastic circular thin plate with an initial imperfection is investigated. Approximate solution of the problem for the fundamental frequency of vibration, of large amplitude and with the plate imperfection, leads to a non-linear ordinary differential equation with time as the independent variable. It is shown that this equation also represents the free vibration of a particle of mass on a shallow curve of fourth degree.. With this similarity in view, it is possible to draw an analogy between these two vibrations. A numerical analysis is made with particular reference to this analogy and the results are given in various figures which represent the vibratory motion and the period of vibration versus the initial amplitude of the plate or of the particle of mass.     

Large amplitude vibration of an initially stressed cross ply laminated plates

In this paper, nonlinear equations of large amplitude vibration for a laminated plate in a general state of nonuniform initial stress are derived. The equations include the effects of transverse shear and rotary inertia. Using these derived governing equations, the large amplitude vibration behaviour of an initially stressed cross-ply laminated plate is studied. The initial stress is taken to be a combination of pure bending stress plus an extensional stress in the plane of the plate. The Galerkin method is used to reduce the governing nonlinear partial differential equations to ordinary nonlinear differential equations and the Runge-Kutta method is used to obtain the nonlinear to linear frequencies. The frequency responses of nonlinear vibration are sensitive of the vibration amplitude, aspect ratio, thickness ratio, modulus ratio, stack sequence, layer number and state of initial stresses. The effects of various parameters on the large amplitude free vibrations are presented.     

Transverse Vibration and Stability Analysis of Circular Plate Subjected to Follower Force and Thermal Load

Transverse vibration and stability analysis of circular plate subjected to follower force and thermal load are analyzed . B ased on the thin plate theory in involving the variable temperature, the differential equation of transverse vibration for the axisymmetric circular plate subjected to follower force and thermal load is established. Then, the differential equation of vibration and corresponding boundary conditions are discretized by the differential quadrature method. Meanwhile, the generalized eigenvalue under three different boundary conditions are calculated. In this case, the change curve of the first order dimensionless complex frequency of the circular plate subjected to the follower force in the different conditions with the variable temperature coefficient and temperature load is analyzed. The stability and corresponding critical loads of the circular plate subjected to follower force and thermal load with simply supported edge, clamped edge and free edge are discussed. The results provide theoretical basis for improving the dynamic stability of the circular plate.     

The buckling and frequency of flexural vibration of rectangular isotropic and orthotropic plates using Rayleigh's method

A simple approximate formula for the natural frequencies of flexural vibration of isotropic plates, originally developed by Warburton using characteristic beam functions in Rayleigh's method, is modified to apply to specially orthotropic plates and extended to include the effect of uniform, direct inplane forces. The initial buckling problem is treated simply by equating the frequency expression to zero. The approach permits the ready determination of reasonably accurate natural frequencies and/or buckling loads for a given plate involving any combination of free, simply supported or clamped edges, without requiring the aid of a sophisticated calculating device or a knowledge of plate, vibration or buckling theory. To illustrate the applicability and accuracy of the approach, numerical results for a number of specific plate problems are presented.     

Free vibration analysis of the completely free rectangular plate by the method of superposition

While the subject of free vibration analysis of the completely free rectangular plate has a history which goes back nearly two centuries it remains a fact that most theoretical solutions to this classical problem are considered to be at best approximate in nature. This is because of the difficulties which have been encountered in trying to obtain solutions which satisfy the free edge conditions as well as the governing differential equation. In a new approach to this problem, by using the method of superposition, it is shown that solutions which satisfy identically the differential equation and which satisfy the boundary conditions with any desired degree of accuracy are obtained. Eigenvalues of four digit accuracy are provided for a wide range of plate aspect ratios and modal shapes. Exact delineation is made between the three families of modes which are characteristic of this plate vibration problem. Accurate modal shapes are provided for the response of completely free square plates.     

Evolution of the vibrational behavior of a guitar soundboard along successive construction phases by means of the modal analysis technique

The crucial piece of a craft guitar is the soundboard since it determines the quality of the sound given by the instrument. From the initial phase (the gross plate with no hole) until the last construction phase (the plate with the full structure) several stages are followed to modify the dynamic behavior of the structure and hence the acoustic response of the final guitar. The aim of this work is to analyze the effect of each modification on the vibrational response of the plate. With this in mind, an experimental modal analysis of all the construction phases has been performed in the low frequency range under free conditions. The response of the plate in each stage is defined by vibration patterns, resonance frequencies, quality factors, and admittance curves.     

Analysis of in-plane vibration of shear walls by a finite element method

The purpose of this paper is to present the application of a finite element technique in the free vibration analysis of shear-wall-type structures. An in-plane plate element with six degrees of freedom per node is developed and the results are compared with those for other plate elements.     

Thermal postbuckling and vibration analyses of functionally graded plates

Thermal postbuckling and vibration behaviors of the functionally graded (FG) plate are investigated. The material properties of the FG plate are assumed to vary continuously through the thickness of the plate and as temperature with the nonlinearity. The nonlinear finite element equations based on the first-order shear deformation plate theory are formulated for the FG plate. The von Karman nonlinear strain–displacement relationship is used to account for the large deflection of the plate. The incremental form considering the initial displacement and initial stress is adopted for the nonlinear temperature-dependent material properties of the functionally graded material. The numerical result shows the characteristics of the thermal postbuckling and vibration of the FG plate in the pre- and post-buckled regions.     

Large amplitude vibration of an initially stressed moderately thick plate

Non-linear equations of motion for a transversely isotropic moderately thick plate in a general state of non-uniform initial stress where the effects of transverse shear and rotary inertia are included are derived. The large amplitude flexural vibration of a simply supported rectangular moderately thick plate subjected to initial stress is investigated. The initial stress is taken to be a combination of a pure bending stress plus an extensional stress in the plane of the plate. These equations are used to solve the vibrations problem by the Galerkin method. The effects of various parameters on the non-linear vibration frequencies are studied.     

FREE VIBRATION OF ORTHOTROPIC SQUARE PLATES WITH A SQUARE HOLE

An approximate method is extended for analyzing the free vibration problem of orthotropic square plate with a square hole. In this paper, a square plate with a square hole is transformed into an equivalent square plate with non-uniform thickness by considering the hole as an extremely thin part of the equivalent plate. Therefore, the dynamic characteristics of a plate with a hole can be obtained by analyzing the equivalent plate. The Green function, which is the discrete solution for the deflection of the equivalent plate, is used to obtain the characteristic equation of the free vibration. The effects of the side to thickness ratio, hole side to plate side ratio and the variation of the thickness on the frequencies are considered. Some numerical analyses are carried out for the simply supported orthotropic square plate with a square hole. The efficiency and accuracy of the numerical solutions by the present method are investigated.     

Free vibration of non-circular cylindrical shells with longitudinal interior partitions

An analysis is presented for the free vibration of a simply supported non-circular cylindrical shell with longitudinal interior partitions. For this purpose, the governing equations of vibration of a non-circular cylindrical shell including a plate as special case are written in a matrix differential equation by using the transfer matrix in the circumferential direction. Once the matrix has been determined, the entire structure matrix is obtained by the product of the transfer matrix of the shell and the point matrix at the joint of the structure matrix. The method is applied to approximately elliptical cylindrical shells with an interior plate, and the natural frequencies and the mode shapes of vibration are calculated numerically, giving the results.     

Flexural vibration insulation with an elastic layer for a plate under nonuniform vibratory action

A method is proposed for calculating the vibration insulation of an elastic layer positioned between a finite-size plate with free edges and a vibrationally active base. The elastic layer consists of a discrete set of springs. In the case of an out-of-phase vibration of the base surface, the mean dynamic action on the plate decreases, which leads to an increase in vibration insulation. By contrast, the resonances of flexural plate vibrations reduce the vibration insulation at resonance frequencies. For different cases, the system parameters that correspond to a positive effect of vibration insulation are estimated. The simplest mass-spring system possessing a single degree of freedom is discussed.     

Forced vibrations of a non-uniform thickness rectangular plate with two free sides

A forced vibration of a rectangular plate with thickness varying linearly in one direction is studied. The plate is simply supported along two opposite sides, and is free along the other sides. Approximate formulae are proposed for estimating the maximum deflection and surface stresses of a rectangular plate subject to a uniformly distributed harmonic lateral load. The accuracy of the formulae is discussed. The approximate formulae show the possibility that one can obtain solutions for the forced vibration of a plate with non-uniform thickness from those for the static bending of the same plate when the plate is subjected to a uniformly distributed load.     

Free vibrations of laminated composite cylindrical shells with an interior rectangular plate

The free vibration analysis of a laminated composite cylindrical shell with an interior rectangular plate is performed by the analytical and experimental methods. The frequency equations of vibration of the shell including the plate are formulated by using the receptance method. To obtain the free vibration characteristics before the combination of two structures, the energy principle based on the classical plate theory and Love's thin shell theory is adopted. The numerical results are compared with the results from an experiment, as well as a finite element analysis, to validate the current formulation. The influences of the length-to-radius ratio (L_S/a) and radius-to-thickness ratio (a/h_S) of the shell and fiber orientation angles (Θ) of symmetric cross- and angle-ply composite materials on the natural frequencies of a cylindrical laminated combined shell are also discussed in details.     

Free vibration of size-dependent Mindlin microplates based on the modified couple stress theory

This paper develops a Mindlin microplate model based on the modified couple stress theory for the free vibration analysis of microplates. This non-classical plate model contains an internal material length scale parameter related to the material microstructures and is capable of interpreting the size effect that the classical Mindlin plate model is unable to describe. The higher-order governing equations of motion and boundary conditions are derived using the Hamilton principle. The p-version Ritz method is employed to determine the natural frequencies of the microplate with different boundary conditions. A detailed parametric study is conducted to study the influences of the length scale parameter, side-to-thickness ratio and aspect ratio on the free vibration characteristics of the microplate. It is found that the size effect is significant when the thickness of microplate is close to the material length scale parameter.     

Dynamic behaviour of a simply supported circular cylindrical panel subjected to an initial bending moment

The vibration of cylindrical panels with simply supported boundary conditions in the presence of an initial bending moment is investigated. The analysis is based on the theory of Herrmann and Armenàkas. The results for the stability and free vibration of the panels are obtained as special cases.     

An exact frequency analysis of annular plates with small core having elastically restrained outer edge and sliding inner edge

The present study deals with an exact analysis of free transverse vibrations of annular plates having small core and sliding inner edge and the outer edge being elastically restrained based on classical plate theory. This study focuses mainly on the influence of variations in the elastic restraint parameters on the fundamental frequencies of plate vibration. The natural frequencies for the first six modes of annular plate vibrations are computed for different materials and varying values of the radius parameter and these natural frequencies may correspond to either axisymmetric and/or non-axisymmetric modes of plate vibration. The extensive data of values of fundamental frequency parameter presented in this paper is believed to be of use in the design of acoustic underwater transducers, ocean and naval structures, compressor and pump elements, offshore platforms. These results may serve as bench mark values for researchers to validate their results obtained using approximate numerical methods.     

阻尼受击板的时域建模及声音合成

为满足声源辨识中对合成冲击声的迫切需求,建立了球-板撞击的时域模型,提出一种时域快速求解方法,并进行了实验验证.首先给出一种将时域有限差分法(FDTD)和模态展开法(MEM)相结合的时域混合方法,求解板的振动方程,并解决了混合方法中MEM的模态截断和初值问题,及两种方法中阻尼的一致性问题;随后,给出了简支矩形板的冲击声计算结果,通过与FDTD方法的运算量进行对比,验证了混合方法的高效性;最后,针对自由边界下的L形板进行了实验验证.结果表明,与传统FDTD方法相比,时域混合方法在保证合成冲击声精度的前提下可将计算效率提高100至1200倍。      

Vibration and stability of a free circular plate subjected to non-conservative loading

This paper is concerned with the stability and vibration of a completely free circular plate subjected to a non-conservative edge loading. The eigencurves and mode shapes of the plate are obtained for various values of the non-conservativeness parameter. Numerical results are presented for the asymmetrical mode shapes of the plate. Interesting conclusions are drawn from these results some of which are verified analytically.     

Axially symmetric stability of a completely free circular plate subjected to a non-conservative edge load

This paper presents a study on the behaviour of the vibration and stability of a two dimensional structure: i.e., a completely free circular plate subjected to non-conservative radial loading. The eigencurves and mode shapes of the circular plate are presented for various values of the non-conservativeness parameter. Some interesting conclusions concerning the behaviour of a completely free plate are drawn from the analytical investigation of the solution of the problem and from the numerical calculations as well.