Three-Dimensional Vibration Analysis of Rectangular Plates With Mixed Boundary Conditions

The determination of natural frequencies of rectangular plates with mixed boundary conditions by discrete singular convolution

A nine-node isoparametric plate element in conjunction with first-order shear deformation theory is used for free vibration analysis of rectangular plates with central cutouts. Both thick and thin plate problems are solved for various aspect ratios and boundary conditions. In this article, primary focus is given to the effect of rotary inertia on natural frequencies of perforated rectangular plates. It is found that rotary inertia has significant effect on thick plates, while for thin plates the rotary inertia term can be ignored. It is seen that the numerical convergence is very rapid and based on comparison with experimental and analytical data from literature, it is proposed that the present formulation is capable of yielding highly accurate results. Finally, some new numerical solutions are provided here, which may serve as benchmark for future research on similar problems.

This article presents a simple and efficient procedure for the natural vibration analysis of rectangular plates and stiffened panels in contact with fluid on one side. The assumed mode method is applied, where the natural frequencies and mode shapes are obtained by solving an eigenvalue problem of a multi-degree-of-freedom system matrix equation derived by using Lagrange’s equation of motion. The Mindlin thick plate theory is applied for a plate, and in the case of stiffened panels, the effect of framing is taken into account by adding its strain and kinetic energies to the corresponding plate energies. Potential flow theory assumptions are adopted for the fluid, and free surface waves are ignored. The fluid velocity potential is derived from the boundary conditions for the fluid and structure and is utilized for the calculation of added mass using the assumed modes. The applicability and accuracy of the developed procedure are illustrated with several numerical examples using a developed in-house code. A comparison of the results with those obtained by general purpose finite element analysis software is provided, where very good agreement is achieved.

In this paper an examination of methods for vibration analysis of moderately thick rectangular plates has been presented. First, the state-of-the art in thick plate vibration theories and analysis methods is described and it is followed with basic equations of the original Mindlin theory, which represents a starting point for the development of all other mathematical models. Then, the problem of analytical solving of equilibrium equations is considered based on the modified Mindlin theory of thick plate vibrations, which has been published in the literature recently. Further, energy methods that can be applied to arbitrary boundary conditions are discussed and outline of the assumed mode method is presented. Finally, in the context of numerical methods a new quadrilateral finite element, based on the above mentioned advanced thick plate theory, is included. It should be emphasized that it doesn’t suffers of shear locking problem associated with finite elements, due to natural relation among bending and shear polynomials, and moreover, it gives very accurate results. Application of the presented methods is illustrated by several numerical examples which include natural vibration analyses of rectangular plates with various thicknesses and different combinations of boundary conditions (simply supported, clamped, free and elastically restrained). Comparisons of natural frequencies and mode shapes with results available in the relevant literature and with those obtained by the commercial finite element software are also provided.

The model problem of a plane angle for a second-order elliptic system subject to Dirichlet, mixed, and Neumann boundary conditions is analyzed. For each boundary condition, the existence of solutions of the form $r^λv$ is reduced to spectral analysis of a particular matrix. Focusing on Dirichlet and mixed boundary conditions, optimal bounds on $|\Re λ|$ are derived, employing tools from numerical range analysis and accretive operator theory. The developed framework is novel and recovers known bounds for Dirichlet boundary conditions. The results for mixed boundary conditions are new and represent the central contribution of this work. Immediate applications of these findings are new regularity results for linear second-order elliptic systems subject to mixed boundary conditions.

Acoustic power radiation from a rectangular plate with mixed boundary conditions

The title problem is investigated using the differential quadrature method based on the state‐space formalism. The plates, with mixed boundary conditions, may cross over one‐way internal rigid line supports that impose zero transverse displacement constraints. Differential quadrature procedure is applied in the direction of line supports, while exact solution is sought in the transfer domain perpendicular to the line supports using the state space method. To avoid numerical instability in the transfer matrix method, joint coupling matrices are introduced, mainly according to the continuity conditions at line joints. Natural frequencies of rectangular Kirchhoff plates with general boundary conditions are calculated and compared with the results from other methods. Effects of location of internal line supports and the mixed boundary conditions on the frequency parameters are discussed. Copyright © 2006 John Wiley & Sons, Ltd.

Stress intensity factors and T-stress inedge-cracked rectangular plates under mixed boundary conditions

Heat sources on rectangular plates with double-sided convective cooling are a common application of heat transfer in electronic packaging. However, it is considerably difficult to obtain analytical solutions for improving thermal performance because of the mixed boundary conditions on a plate’s upper surface. In this study, mixed boundary conditions were adjusted twice. Based on the method of separation of variables, a new analytical model was established and used to calculate the thermal performance of a random application case, including temperature distribution, thermal spreading resistance, and total thermal resistance. The results indicated that accurate analytical solutions can be obtained with few computing resources when more than 30 terms are used in each of single and double summations with a total mesh number of 196 in the heat source region. Subsequently, a series of thermal performances were calculated for random application cases and simulations were also conducted using the commercial software Icepak to offer a reference. The data comparisons revealed that the analytical model is able to calculate the temperature values at arbitrary point coordinates accurately over wide ranges of a plate’s thermal conductivity and heat transfer coefficient.

The flexibility function approach to vibration analysis of rectangular plates with arbitrary multiple point supports on the edges

Performance of similitude methods for structural vibration analyses of rectangular plates

Several sets of static beam functions have been established by the authors, which are the static solutions of beams under arbitrary static loads. The beam is a strip taken from the rectangular plate to be considered, and the static load acting on the beam is expanded into a series which may be a series of point loads or a series of sinusoid loads or a series of polynomial loads. These sets of static beam functions have been successfully applied to the vibration analysis of rectangular plates with varying complexity by the Rayleigh-Ritz method and finite strip method. Some numerical results are given. By comparing the results with those available from literature it can be shown that the present method is extremely effective for solving the aforementioned problems and usually a few static beam functions will be sufficient to give acceptable results.

. In this article, the interval expansion of the structure of solving basic types of boundary value problems for partial differential equations of the second order of making the basic operations that compose interval arithmetic is developed. For the differential equation (1) of the type, when constructing the interval expansion of the structure of the formula, structural formulas were used to construct with the Rfunction method and 4 problems were studied — the Dirichlet problem, the Neumann problem, the third type problem, the mixed boundary conditions problem. For the Dirichlet problem, the solution is an interval expansion of the structure in the form (5), where 𝑃 = {𝜔𝛷 , 𝜔𝛷̅, 𝜔̅𝛷, 𝜔̅𝛷̅} и [ 𝛷, 𝛷̅]is an indefinite interval function. For the Neumann problem, a solution is solved in the interval extension of the structure, [ 𝛷1, 𝛷1̅̅̅̅], [ 𝛷2, 𝛷2̅̅̅̅] is an indefinite interval function and 𝐷1 is a differential operator of the form. For the problem of the third type, the solution is solved in the interval extension of the structure, [ 𝛷1, 𝛷1̅̅̅̅], [𝛷2, 𝛷2̅̅̅̅] -indefinite, interval function, 𝐷1 - differential operator of the form (9). For the problem, mixed boundary conditions are treated. The solution In the interval extension of the structure,[ 𝛷1, 𝛷1], [ 𝛷2, 𝛷2̅̅̅̅] is an indefinite interval function and 𝐷1 is a differential operator of the form.

Maximum frequency design of laminated plates with mixed boundary conditions

Nonexistence of Positive Solutions for Singular Hyperbolic Differential Inequalities