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Highlights: [•] Vibrating membranes composed of long strips with local enlargement may lead to trapped modes. [•] Often there exists a much lowered fundamental frequency which dominates the vibration mode. [•] Results are applicable to vibrating plates, acoustic and EM waveguides, and quantum waveguides. [Copyright &y& Elsevier]

Abstract: Vibration of a circular plate internally concentrically supported is basic in structural mechanics. The complete first six frequencies and mode shapes are determined for clamped, simply-supported, sliding, and ring supports using exact characteristic frequency equations. Complex mode switches occur as the support radius is varied. Asymptotic formulas for large support radii are derived. Different types of singularities as the support radius shrinks to zero are distinguished and classified. [Copyright &y& Elsevier]

Presented herein are exact solutions for the free vibration problems of a class of annular plates with parabolically varying rigidity and for a sandwiched non-uniform annular plate. These exact solutions are useful for verifying numerical methods and algorithms. [ABSTRACT FROM AUTHOR]

Abstract: The Stokes flow through a wavy or corrugated channel with surface slip is studied. The correct Navier''s partial slip condition is applied and perturbation solutions about the small amplitude to channel width ratio are obtained. As in Stokes slip flow over a sphere, the resistance is not zero even when slip is infinite. The resistance (due to the interaction of waviness and slip) is larger when the corrugations of the two plates are out of phase than that when they are in phase. [Copyright &y& Elsevier]

Abstract: Buckling of a standing plate subjected to self weight and top load is studied. The vertical sides are simply supported and Levy separation yields an ordinary differential equation that is integrated by an accurate, efficient initial value method. Exact and asymptotic formulas are also found. The effects of self weight and top load both contribute to the buckling. Their contributions are not linear. For narrow plates, if top load dominates it does not matter whether an end is clamped or simply supported. If self weight dominates, the buckling of a narrow plate only depends on the bottom condition. [Copyright &y& Elsevier]

Abstract: Fundamental frequencies of vibrating circularly periodic plates with a circular core have been determined analytically. A boundary perturbation method is developed to extract the fundamental eigenvalue of the governing biharmonic boundary value problem. The method is then applied to wavy and polygonal plates with clamped and simply supported outer boundary conditions. Clamped, simply supported, and free circular cores are considered. Approximate analytical formulations of the fundamental frequency for such plates with core are obtained. [Copyright &y& Elsevier]

The vibration of a rectangular plate under self-weight (or acceleration) is studied. The vertical edges are simply supported and a semi-analytic Levy-integration method is used. The fundamental frequencies are determined for various top and bottom boundary conditions. It is found that self-weight has considerable effect on both the frequency and the mode shape. [Copyright &y& Elsevier]

Abstract: The three dimensional problem of steady fluid deposition on an inclined rotating disk is solved by similarity transform. For a given spraying rate there may be one, two or no steady state solution. The inclination causes a downward draining flow and a lateral flow. Perturbation solutions compare well with exact similarity solutions when the fluid film is thin. [Copyright &y& Elsevier]

Abstract: Vibrations of thin annular plates with at least one sliding or movable edge are considered. The fundamental frequencies for the seven cases may correspond to axisymmetric or non-axisymmetric modes. [Copyright &y& Elsevier]

Discusses the importance of buckling of plates caused by uniform in-plane radial edge compression in elastic stability. Investigation of a rectangular plate with an interior hinge; Importance of exact solutions in checking nonexact numerical methods such as the Ritz method; Square root of the normalized buckling load for the weakened simply supported plates.

Abstract: For an isotropic circular plate with constant thickness and free edges, its fundamental vibration mode takes the form of a twisting mode with two nodal diameters, i.e. n=2. In certain applications, it may be necessary to have an axisymmetric shape for the fundamental vibration mode (i.e. mode shape with no nodal diameter, n=0). In this paper, we show that such an axisymmetric vibration mode can be realized by increasing the bending rigidity of the outer rim of the circular plate by using a larger thickness or by using a material with a larger Young''s modulus or both. We also determine the critical flexural rigidity of the outer rim that will trigger this vibration mode change from n=2 type to n=0 type for a given rim width. The ability to alter the mode shape of circular plates with free edges has useful applications in sensing and actuating devices and large pontoon-type floating circular structures. [Copyright &y& Elsevier]