Analysis of the flow-induced vibrations of viscoelastically supported rectangular plates

The purpose of this study was to develop a theoretical model for the flow-induced vibration of viscoelastically supported rectangular plates. In particular, the influence of the dynamic mechanical properties of the elements supporting the plate was investigated. The case of a homogeneous rectangular plate supported along all four edges by a complex viscoelastic element was treated. The Rayleigh-Ritz method was used applying beam functions as the trial functions. This approach ensured a fast convergence rate, which is advantageous for vibration analysis of high order modes. The flow-induced vibration of the plate was calculated using the Corcos model for the surface pressure loading. The results suggest that there is an optimal support stiffness that minimizes the flow-induced vibration response of the plate.     

Analytical modeling of squeeze film damping for rectangular elastic plates using Green's functions

An analytical method for the solution of squeeze film damping based on Green's function to the nonlinear Reynolds equation considering an elastic plate is presented. This allows calculating the stiffness and damping forces rapidly for various boundary conditions. The elastic plate velocity is applied to the nonlinear Reynolds equation as a forcing term. The nonlinear Reynolds equation is divided into multiple linear nonhomogeneous Helmholtz equations, which then can be solvable using the presented approach. Approximate mode shapes of a rectangular elastic plate are used, enabling the calculation of the damping ratio and frequency shift for the linear case, as well as the complex resistant pressure, for both linear and nonlinear cases.     

Vibrations of heated plates with two opposite edges simply supported

Vibration analysis of the family of rectangular plates with two opposite sides simply supported can be simplified by assuming mode shapes. In the present paper a vibration analysis of such plates which are heated so as to have a temperature varying in the direction parallel to these sides is presented. A steady state temperature which satisfies the Laplace equation is considered. Due to the assumption of mode shapes the governing plate differential equation, which in general is a function of the x and y co-ordinates, becomes a function of one co-ordinate. This equation is analyzed by a finite difference method and solved by a standard simultaneous iteration technique. The accuracy of the method is ascertained by comparing the results for some well known boundary conditions when there is no temperature effect with the standard solutions available in the literature. From the results an attempt has been made to correlate the natural frequency with the temperature. Plates of uniform thickness with different length to breadth ratios have been analyzed. The assumed linear temperature distribution satisfies the Laplace equation and the plate is free to expand in its plane at its edges so that no thermal stresses will be induced.     

Free vibration of a point-supported membrane stretched by inextensible strings

An analysis is presented for the free vibration of a point-supported rectangular membrane with uniform tension stretched by inextensible strings along the edges. The membrane is transformed into a square membrane of unit length by a transformation of variables. The transverse deflection of the square membrane is expressed in a series of the products of the deflection functions of strings parallel to the edges, and the frequency equation is derived by the Ritz method. This method is applied to point-supported membranes symmetrical with respect to the center lines, and the natural frequencies and the mode shapes are calculated numerically up to higher modes.     

Free vibration analysis of stiffened plates using finite difference method

Free vibration characteristics of rectangular stiffened plates having a single stiffener have been examined by using the finite difference method. A variational technique has been used to minimize the total energy of the stiffened plate and the derivatives appearing in the energy functional are replaced by finite difference equations. The energy functional is minimized with respect to discretized displacement components and natural frequencies and mode shapes of the stiffened plate have been determined as the solutions of a linear algebraic eigenvalue problem. The analysis takes into consideration inplane deformation of the plate and the stiffener and the effect of inplane inertia on the natural frequencies and mode shapes. The effect of the ratio of stiffener depth to plate thickness on the natural frequencies of the stiffened plate has also been examined.     

Determination of dynamic characteristics of rectangular plates with cutouts using a finite difference formulation

The paper describes a method for the prediction of dynamic characteristics of rectangular plates with cutouts. The method is based on the use of variational principles in conjunction with finite difference technique. A concept of interlacing grids has been developed to express the strain energy of nodal subdomains into which the plate is divided. The use of this idea has been demonstrated in relation to internal and boundary nodes. Natural frequencies and corresponding mode shapes of rectangular plates with one and two cutouts have been predicted and experimentally verified.     

Damage detection of a rectangular plate by spatial wavelet based approach

This paper presents a technique for structure damage detection based on spatial wavelet analysis. Many damage detection methods require modal properties before and after damage. This method only needs the spatially distributed signals (e.g. the displacements or mode shapes) of the rectangular plate after damage. First, spatially distributed signals of the rectangular plate with damage are obtained by finite element method. The damaged region is represented as the elements with reduced stiffness. Then these spatially distributed signals are analyzed by wavelet transformation. It is observed that distributions of the wavelet coefficients can identify the damage position of rectangular plate by showing a peak at the position of the damage. It is also demonstrated that this method is very sensitivity to the damage size.     

EXPERIMENTAL AND THEORETICAL INVESTIGATION OF THE LINEAR AND NON-LINEAR DYNAMIC BEHAVIOUR OF A GLARE 3 HYBRID COMPOSITE PANEL

A theoretical model based on Hamilton's principle and spectral analysis is used to study the non-linear free vibration of hybrid composite plates made of Glare 3, a new aircraft structural material. It consists of alternating layers of metal- and fibre-reinforced composites. In previous work, the theoretical model has been used to calculate the first non-linear mode of fully clamped rectangular composite fibre-reinforced plastic (CFRP) laminated plates. This study concerns determination of the linear dynamic properties of the Glare 3 hybrid composite rectangular panel (G3HCRP) such as natural frequencies and mode shapes. The theoretical model is used to calculate the fundamental non-linear mode shape and associated flexural behaviour of the fully clamped G3HCRP. A series of experimental investigations have been conducted using a G3HCRP in order to determine linear dynamic properties. The response due to random excitation was investigated and the experimental measurements are analyzed and discussed. Comparisons are made with finite element predictions and response estimates given by the ESDU method, the latter being a “design guide” approach used by industry. Concerning the non-linear analysis, the results are given for various plate aspect ratios and vibration amplitudes, showing a higher increase of the induced bending stress near the clamps at large deflections. Comparisons between the dynamic behaviour of an isotropic plate and G3HCRP at large vibration amplitudes are presented and good results are obtained.     

Vibrations of a polygonal plate having orthogonal straight edges by an extended rayleigh-ritz method

An extended Rayleigh-Ritz method is presented for solving vibration problems of a polygonal plate having orthogonal straight edges. The polygonal plate is considered as an assemblage of several rectangular plates. For each element rectangular plate, the transverse displacement is approximated by interpolation functions corresponding to unknown displacements and slopes at the discrete points which are chosen along the edges, and series of trial functions which satisfy homogeneous artificial boundary conditions. By minimizing the energy functional corresponding to the assumed displacement function, the dynamic stiffness matrix of the element rectangular plate, which is similar to that obtained in the finite element method, is derived. The dynamic stiffness matrix of the whole system is obtained by summing up those of the element rectangular plates. Numerical results are presented for the natural frequencies and mode shapes of cantilever L-shaped and T-shaped plates.     

Shock response of viscoelastically damped sandwich plates

Analysis for the transient response of a simply supported three layer viscoelastically damped sandwich plate, subjected to a half sine shock pulse, has been carried out, with account taken of the transverse inertia effects only. The properties of the viscoelastic core material have been represented by those of a four element viscoelastic model. The influences of the variation of various geometrical and physical parameters of the damped sandwich plate on the shock response are investigated. The decay rate of the transverse vibrations of the plate is evaluated in terms of the logarithmic decrement.     

A note on the damping of large amplitude beam vibrations

This paper presents a new series-type method for solving the eigenvalue problems of irregularly shaped plates clamped at all edges. An irregularly shaped plate is formed on a simply supported rectangular plate by rigidly fixing several segments. With the reaction forces and moments acting on all edges of an actual plate of irregular shape regarded as unknown harmonic loads, the stationary response of the plate to these loads is expressed by the use of the Green function. The force and moment distributions along the edges are expanded into Fourier series with unknown coefficients, and the homogeneous equations for the coefficients are derived by restraint conditions on the edges. The natural frequencies and the mode shapes of the actual plate are determined by calculating the eigenvalues and eigenvectors of the equations. The method is applied to a cross-shaped, an I-shaped and an L-shaped plate clamped at all edges, the natural frequencies and the mode shapes of the plates are calculated numerically and the effect of the shape is discussed.     

Semi-analytical solution for three-dimensional transient response of functionally graded annular plate on a two parameter viscoelastic foundation

The three-dimensional transient analysis of functionally graded annular plates with arbitrary boundary conditions is carried out in this paper. The material properties of the FGM plate are assumed to vary smoothly in an exponential law along the thickness direction. The plate is assumed to rest on a two parameter viscoelastic foundation. A semi-analytical method, which integrates the state space method (SSM), Laplace transform and its inversion, as well as the one-dimensional differential quadrature method (DQM), is proposed to obtain the transient response of the plate. The state space method is used to obtain the analytical solution in the thickness direction. The differential quadrature method is employed to approximate the solution in the radial direction. The Laplace transform and the numerical inversion are used to obtain the solution in time domain. Numerical results show a good agreement between the response histories obtained by the present method and finite element method. The effects of the boundary conditions at the edges, the material graded index, the Winkler and shearing layer elastic coefficients, and the damping coefficient are studied. Numerical examples show that the peak response decreases as the material graded index, the Winkler and shearing layer elastic coefficients, and the damping coefficient increase. The results obtained in this paper can serve as benchmark data in further research.     

Transient vibration analysis of a completely free plate using modes obtained by Gorman's superposition method

This paper shows that the transient response of a plate undergoing flexural vibration can be calculated accurately and efficiently using the natural frequencies and modes obtained from the superposition method. The response of a completely free plate is used to demonstrate this. The case considered is one where all supports of a simply supported thin rectangular plate under self weight are suddenly removed. The resulting motion consists of a combination of the natural modes of a completely free plate. The modal superposition method is used for determining the transient response, and the natural frequencies and mode shapes of the plates used are obtained by Gorman's superposition method. These are compared with corresponding results based on the modes using the Rayleigh-Ritz method using the ordinary and degenerated free-free beam functions. There is an excellent agreement between the results from both approaches but the superposition method has shown faster convergence and the results may serve as benchmarks for the transient response of completely free plates.     

基于类表面等离子体激元的矩形金属光栅色散特性的研究

等离子体激元(surface plasmon polaritons, SPP)因其独特的光学和物理特性, 使其具有诸如透射增强和局域共振等一系列新颖现象, 已成为当前国内外学者研究的热点. 本文对基于类表面等离子体激元(Spoof Surface Plasmons, SSP)的矩形金属光栅色散特性和模式分布进行了研究. 利用本征函数法并结合场匹配条件, 获得了矩形栅表面SSP的场表达式、色散关系和模式分布, 并通过电磁仿真进行了验证. 在此基础上分析了矩形栅各参数对SSP色散及模式分布的影响, 研究结果表明: 由本征函数法获得的SSP色散特性与仿真结果基本符合; 增大金属栅高度或减小排列周期能减小SSP的相速度; 而增大金属栅周期占空比能在一定程度上拓展SSP与电子束互作用的带宽; 改变金属盖板高度对慢波SSP色散模式基本没有影响; 减小金属栅侧面宽度能增大模式之间的间隔, 从而能有效避免模式竞争的发生. 本文对基于SSP的矩形金属光栅色散特性的研究将为进一步研究SSP与电子束的相互作用, 形成高效、宽带的新型太赫兹源奠定良好的理论基础.     

Optimum thickness distribution of unconstrained viscoelastic damping layer treatments for plates

This paper concerns the optimum thickness distribution of unconstrained viscoelastic damping layer treatments for plates. The system loss factor is expressed in terms of the mechanical properties of the plate and damping layer and the layer/plate thickness ratio. Optimum distributions of the thickness ratio that maximize the system loss factor are obtained through sequential unconstrained minimization techniques. Results are presented for both simply-supported and edge-fixed rectangular plates with aspect ratios of 1·0 to 4·0. These results indicate that the system loss factor can be increased by as much as 100%, or more, by optimizing the thickness distribution of the damping treatment. Also revealed are the regions of the plate where added damping treatments are most effective.     

Dynamic stability of a sandwich plate with a constraining layer and electrorheological fluid core

The dynamic stability problems of a sandwich plate with a constraining layer and an electrorheological (ER) fluid core subjected to an axial dynamic force are investigated. The rectangular plate is covered in an ER fluid core and a constraining layer to improve the stability of the system. Effects of the natural frequencies, static buckling loads, and loss factors on the dynamic stability behavior of the sandwich plate are studied in the paper. Rheological property of an ER material, such as viscosity, plasticity, and elasticity may be changed when applying an electric field. The modal damper and the natural frequencies for the sandwich plate are calculated for various electric fields. When an electric field is applied, the damping of the system is more effective. In this study, finite element method and the harmonic balance method are used to calculate the instability regions of the sandwich plate. The ER fluid core is found to have a significant effect on the dynamic stability regions.     

Smearing technique for vibration analysis of simply supported cross-stiffened and doubly curved thin rectangular shells

Plates stiffened with ribs can be modeled as equivalent homogeneous isotropic or orthotropic plates. Modeling such an equivalent smeared plate numerically, say, with the finite element method requires far less computer resources than modeling the complete stiffened plate. This may be important when a number of stiffened plates are combined in a complicated assembly composed of many plate panels. However, whereas the equivalent smeared plate technique is well established and recently improved for flat panels, there is no similar established technique for doubly curved stiffened shells. In this paper the improved smeared plate technique is combined with the equation of motion for a doubly curved thin rectangular shell, and a solution is offered for using the smearing technique for stiffened shell structures. The developed prediction technique is validated by comparing natural frequencies and mode shapes as well as forced responses from simulations based on the smeared theory with results from experiments with a doubly curved cross-stiffened shell. Moreover, natural frequencies of cross-stiffened panels determined by finite element simulations that include the exact cross-sectional geometries of panels with cross-stiffeners are compared with predictions based on the smeared theory for a range of different panel curvatures. Good agreement is found.     

Dynamics of a rectangular plate resting on an elastic foundation and partially in contact with a quiescent fluid

In this study, a method of analysis is presented for investigating the effects of elastic foundation and fluid on the dynamic response characteristics (natural frequencies and associated mode shapes) of rectangular Kirchhoff plates. For the interaction of the Kirchhoff plate–Pasternak foundation, a mixed-type finite element formulation is employed by using the Gâteaux differential. The plate finite element adopted in this study is quadrilateral and isoparametric having four corner nodes, and at each node four degrees of freedom are present (one transverse displacement, two bending moments and one torsional moment). Therefore, a total number of 16 degrees-of-freedom are assigned to each element. A consistent mass formulation is used for the eigenvalue solution in the mixed finite element analysis. The plate structure considered is assumed clamped or simply supported along its edges and resting on a Pasternak foundation. Furthermore, the plate is fully or partially in contact with fresh water on its one side. For the calculation of the fluid–structure interaction effects (generalized fluid–structure interaction forces), a boundary element method is adopted together with the method of images in order to impose an appropriate boundary condition on the fluid's free surface. It is assumed that the fluid is ideal, i.e., inviscid, incompressible, and its motion is irrotational. It is also assumed that the plate–elastic foundation system vibrates in its in vacuo eigenmodes when it is in contact with fluid, and that each mode gives rise to a corresponding surface pressure distribution on the wetted surface of the structure. At the fluid–structure interface, continuity considerations require that the normal velocity of the fluid is equal to that of the structure. The normal velocities on the wetted surface of the structure are expressed in terms of the modal structural displacements, obtained from the finite element analysis. By using the boundary integral equation method the fluid pressure is eliminated from the problem, and the fluid–structure interaction forces are calculated in terms of the generalized hydrodynamic added mass coefficients (due to the inertial effect of fluid). To asses the influences of the elastic foundation and fluid on the dynamic behavior of the plate structure, the natural frequencies and associated mode shapes are presented. Furthermore, the influence of the submerging depth on the dynamic behavior is also investigated.     

Numerical and experimental analysis of uncertainty on modal parameters estimated with the stochastic subspace method

Modal parameters of structures are often used as inputs for finite element model updating, vibration control, structural design or structural health monitoring (SHM). In order to test the robustness of these methods, it is a common practice to introduce uncertainty on the eigenfrequencies and modal damping coefficients under the form of a Gaussian perturbation, while the uncertainty on the mode shapes is modeled in the form of independent Gaussian noise at each measured location. A more rigorous approach consists however in adding uncorrelated noise on the time domain responses at each sensor before proceeding to an operational modal analysis. In this paper, we study in detail the resulting uncertainty when modal analysis is performed using the stochastic subspace identification method. A Monte-Carlo simulation is performed on a simply supported beam, and the uncertainty on a set of 5000 modal parameters identified with the stochastic subspace identification method is discussed. Next, 4000 experimental modal identifications of a small clamped–free steel plate equipped with 8 piezoelectric patches are performed in order to confirm the conclusions drawn in the numerical case study. In particular, the results point out that the uncertainty on eigenfrequencies and modal damping coefficients may exhibit a non-normal distribution, and that there is a non-negligible spatial correlation between the uncertainty on mode shapes at sensors of different locations.     

Vibrations of non-uniform rectangular plates: A spline technique method of solution

The fourth order differential equation governing the transverse motion of an elastic rectangular plate of variable thickness has been solved, by using the quintic spline interpolation technique. An algorithm for computing the solution of this differential equation is presented, for the case of equal intervals. Frequencies, mode shapes and moments for doubly symmetric, antisymmetric-symmetric and second symmetric-symmetric modes of vibration are presented for various cases of boundary conditions.