On natural frequencies of transversely isotropic circular plates

The paper discusses the impact of the material properties of transversely isotropic circular plates on its natural frequencies. Two refined theories of plates have been used to analyze the free vibration behavior of homogeneous plates. Both theories take into account normal and rotary inertias. Fundamental frequencies for plates with radial inhomogeneity have been obtained with the help of finite element package Comsol Multiphysics 5.0. It has been shown that the inhomogeneity of the plate have a profound impact on the first (lowest) frequency of the plate, while the plate orthotropy has a greater influence on the second and higher vibration mode [2] (Fig. 1, Table 1).     

Stability of plastic plates and shells

Recent research on the stability of plastic plates and shells, mostly by workers in the Soviet Union, is briefly reviewed.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, Vol. 5, No. 1, pp. 150–156, January–February, 1969.     

Design of circular plates for minimum weight

Zusammenfassung Diese Arbeit befasst sich mit dem Problem der Bemessung einer elastischplastischen Kreisplatte veränderlicher Dicke von minimalem Gewicht. Die Platte ist am Rand frei drebhar gelagert oder eingespannt und trägt eine rotationssymmetrische, sonst beliebig verteilte Last. Es wird das vonDrucker undShield entwickelte Verfahren zur Dimensionierung verwendet, welches ein absolutes Minimum für das Gewicht einer Sandwichplatte und ein relatives Minimum für dasjenige einer homogenen Platte liefert.

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The results presented in this paper were obtained in the course of research work sponsored by Watertown Arsenal Laboratory under Contract DA-19-020-ORD-2598-DA-19-020-501-ORD-3984.     

Dynamic response of circular and annular circular plates using spectral element method

This paper presents development of the spectral element method (SEM) for analysis of circular and annular circular plates vibration under impact load. A novel formulation is proposed in frequency domain for conducting spectral element matrix. Several numerical examples are presented in order to demonstrate performance of the presented method compared to other numerical methods in literature. The presented formulation was applied in order to analyze vibration of annular circular plates with various thicknesses and various internal-to-external radius ratios. In addition, annular circular plate displacements subjected to an impact load were calculated using the presented SEM.     

Cylindrically orthotropic circular plates with large deflections

Zusammenfassung Es werden die Verschiebungs- und Spannungszustände in einer Kreisplatte mit grossen Durchbiegungen und zylindrischer Orthotropie mit Hilfe zweier Methoden untersucht, nämlich der Methode des kleinen Parameters vonPoincaré und einer modifizierten Methode vonGalerkin.Ein numerisches Beispiel für gleichmässige Belastung der Platte, die Poissonsche Zahlv =1/4 und die Beziehungv :v _r =4 wird gegeben.

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This work has been sponsored by the United States Army under contract No. DA-11-002-ORD-2059     

The minimum weight design of circular sandwich plates

Résumé Dans ce mémoire nous étudions le problème statique du choix des dimensions suivant: Comment doit varier l'épaisseur de point en point dans les plaques circulaires fléchies du type sandwich, pourque leur poids soit minimum, mais sans qu'il existe du fluage illimité? Comme hypothèse de base, nous avons pris la condition de plasticité devon Mises et la règle de fluage associée. Les plaques sont simplement posées ou encastrées le long de leur contour et portent une charge répartie symétriquement par rapport à l'axe. Les résultats obtenus sont comparés à ceux du problème analogue, où l'on utilise la condition de plasticité deTresca. Nous avons présenté finalement quelques résultats numériques.     

Axisymmetric vibrations of circular plates of linearly varying thickness

Summary Free axisymmetric vibrations of circular plates of linearly varying thickness have been studied on the basis of classical theory of plates. The differential equation governing the motion of such plates have been solved by Frobenius method. The transverse displacement has been expressed as an infinite power series in the radial coordinate. Frequency parameters have been computed for the first three modes of vibration for clamped as well as simply supported circular plates corresponding to different values of taper constant.

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Zusammenfassung Auf Grund der klassischen Scheibentheorie werden die freien axialsymmetrischen Schwingungen der Kreisscheibe mit linear veränderlicher Dicke studiert. Die Differentialgleichung der Bewegung wird mit Hilfe der Methode von Frobenius gelöst. Die transversale Verschiebung wird als eine Reihe der radialen Koordinaten ausgedrückt. Die Schwingungsparameter werden für die ersten drei Eigenschwingungen sowohl für die eingespannte als auch für die aufgelegte Kreisscheibe für verschiedene Werte des Anzugs berechnet.

     

A semi-analytic approach for the nonlinear dynamic response of circular plates

This paper presents a new semi-analytic perturbation differential quadrature method for geometrically nonlinear vibration analysis of circular plates. The nonlinear governing equations are converted into a linear differential equation system by using Linstedt–Poincaré perturbation method. The solutions of nonlinear dynamic response and the nonlinear free vibration are then sought through the use of differential quadrature approximation in space domain and analytical series expansion in time domain. The present method is validated against analytical results using elliptic function in several examples for both clamped and simply supported circular plates, showing that it has excellent accuracy and convergence. Compared with numerical methods involving iterative time integration, the present method does not suffer from error accumulation and is able to give very accurate results over a long time interval.     

Capacity of a condenser whose plates are circular arcs

We find an asymptotic formula for the conformal capacity of a plane condenser both plates of which are concentric circular arcs as the distance between them vanishes. This result generalizes the formula for the capacity of parallel linear plate condenser found by Simonenko and Chekulaeva (1972). On a capacity of a condenser consisting of infinite bands, Izv. Vyssh. Uchebn. Zaved. Elektromekh., 4, 362–370 (in Russian) and sheds light on the problem of finding an asymptotic formula for the capacity of condenser whose plates are arbitrary parallel curves. This problem was posed and partially solved by R. Kühnau (1998). Randeffekten beim elektostatischen Kondensator, Zap. Nauchn. Semin. POMI, 254, 132–154.     

Nonlinear vibrations of elastic circular plates exhibiting rectilinear orthotropy

Zusammenfassung Es werden mittels einer Verallgemeinerung der von-Kármánschen Gleichungen auf den dynamischen und orthotropen Fall die nichtlinearen Schwingungen elastisch orthotroper Kreisplatten untersucht. Die Bewegungsgleichung wird mittels des Galerkin-Verfahrens von den Ortsvariablen befreit. Als Beispiel wird Einspannung des Randes unter Erhaltung der radialen Verschieblichkeit angenommen, und es wird unter diesen Randbedingungen der Einfluss der Anisotropie und der relativen Amplitude auf die Eigenschwingungen sowie den Spannungszustand studiert. Als Sonderfall wird der nichtlineare Gleichgewichtszustand diskutiert.

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The preparation of this paper has been sponsored in part by the National Defense Education Act.     

A note on finite deflections of circular ring plates

We solve the problem of the symmetrically deforming edge loaded polar-orthotropic circular ring plate for the case of large deformations of a radially rigid plate, and compare the results with the corresponding solutions in accordance with the small finite deformation equations of von Karman.

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Zusammenfassung Wir lösen das Problem der rotationssymmetrischen Biegung von polar-orthotropen Kreisringplatten für den Fall von grossen Durchbiegungen von Platten welche starr in radialer Richtung sind. Wir vergleichen unsere Resultate mit den entsprechenden Lösungen welche aus den von Karman'schen Gleichungen für kleine endliche Durchbiegungen folgen.

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A report on work supported by the Office of Naval Research.     

Minimum weight design of circular plates under arbitrary loading

Zussamenfassung Das Dimensionierungsverfahren vonDrucker undShield [1, 2] wurde kürzlich auf die Dimensionierung rotationssymmetrisch belasteter Kreisplatten für Mindestgewicht angewandt [3], wobei sowohl gelenkig gestützte als auch eingespannte Sandwichplatten betrachtet wurden. In der vorliegenden Arbeit werden diese Untersuchungen auf kreisförmige Sandwichplatten unter nicht rotationssymmetrischer Belastung ausgedehnt. Es zeigt sich, dass auch für solche Lasten der Dimensionierung für Mindestgewicht diejenigen Verformungszustände zugrunde gelegt werden könne, welche schon zur Lösung der entsprechenden Aufgabe bei rotationssymmetrischer Belastung konstruiert wurden. Für Belastung einer gelenkig gestützten oder eingespannten Kreisplatte durch eine Einzellast mit beliebigem Angriffspunkt werden explizite Dimensionierungsformeln gewonnen, für beliebig verteilte Belastung werden Integraldarstellungen der optimalen Dimensionierung gegeben.

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To Professor F. K. G. Odqvist on the occasion of his 60th anniversary

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The results presented in this paper were obtained in the course of research sponsored by the Office of Ordnance Research, Department of the Army, under Contract No. DA-19-020-ORD-4564.     

On the circular chromatic number of circular partitionable graphs

This article studies the circular chromatic number of a class of circular partitionable graphs. We prove that an infinite family of circular partitionable graphs G has . A consequence of this result is that we obtain an infinite family of graphs G with the rare property that the deletion of each vertex decreases its circular chromatic number by exactly 1. © 2006 Wiley Periodicals, Inc. J Graph Theory     

Analysis of the states of stress and strain of circular oriented-plastic plates

Retaining the fundamental assumption _z=0, the author investigates two-dimensional problems of the bending of cylindrically orthotropic circular plates in relation to the anisotropy parameter k=E/E_r and a generalized parameter depending both on the transverse shears and the dimensions of the plates h²/a²×E_r/G_rz. Equations are given for calculating certain quantities from the classical variant of the solutions. There are shown to be important differences in the solutions for boundary conditions of the clamped-edge type (when a horizontal or vertical element of the edge is fixed). The effect of the stress _z is estimated from the standpoint of the Ambartsumyan theory [2]. Two types of loading, a distributed load q=const and a concentrated load P applied at the center of the plate, are considered.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 5, pp. 899–908, September–October, 1969.     

On the effects of coupling between in-plane and out-of-plane vibrating modes of smart functionally graded circular/annular plates

In recent years many articles concerned with the mechanics of functionally graded plates have been published. The variation in material properties through the thickness of the plate introduces a coupling between in-plane and transverse displacements, the coupling is important in the vibration of functionally graded plates (FGPs), but none have produced an exact closed-form solution for the in-plane as well as transverse vibrations of smart circular/annular FGPs. Therefore, this paper develops an exact closed-form solution for the free vibration of piezoelectric coupled thick circular/annular FGPs subjected to different boundary conditions on the basis of the Mindlin’s first-order shear deformation theory. Through the comparison of present results with those available, the accuracy of the present method was verified. The effects of coupling between in-plane and transverse displacements on the frequency parameters are proved to be significant. It is concluded that the developed model can describe vibrational behavior of smart FGM plates more realistic. Due to the inherent features of the present solution, all findings will be a useful benchmark for evaluating other analytical and numerical methods developed by researchers in the future.     

Basic studies on the dynamics of rotating circular structures

In this paper the dynamic effects due to rotation of spinning circular structures will be discussed by analytical, experimental and numerical investigations. Basic phenomena of spinning are studied on the analytical solution of a spinning ring. The observed effects will be underlined by rather simple experimental investigations on a rotating wine glass. Based on this fundamental knowledge special attention has to be payed to the finite element modeling and implementation of the eigenvalue analysis of gyroscopic systems, which requires the numerical solution in complex numbers. Besides the influence of rotational speed, the disturbances due to rolling contact will be discussed on rotating tubes as well as detailed finite element models of rolling tires. The application of these techniques with emphasis to the prediction of tire noise will be outlined. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Free vibration analysis of circular plates by differential transformation method

This study analyses the free vibrations of circular thin plates for simply supported, clamped and free boundary conditions. The solution method used is differential transform method (DTM), which is a semi-numerical-analytical solution technique that can be applied to various types of differential equations. By using DTM, the governing differential equations are reduced to recurrence relations and its related boundary/regularity conditions are transformed into a set of algebraic equations. The frequency equations are obtained for the possible combinations of the outer edge boundary conditions and the regularity conditions at the center of the circular plate. Numerical results for the dimensionless natural frequencies are presented and then compared to the Bessel function solution and the numerical solutions that appear in literature. It is observed that DTM is a robust and powerful tool for eigenvalue analysis of circular thin plates.     

On the deformation of elastic plates

Summary The purpose of the paper is to indicate some elementary applications of a method for the resolution of elliptic problems. The method consists in fitting solutions obtained in several subdomains and depending on arbitrary constants. One of the interests of the method is that it seems particularly efficient in the case of singularities. An application to the bending of triangular and parallelogramic plates is given.

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Résumé Les auteurs se proposent d'exposer sur quelques cas simples une méthode de résolution pour des problèmes elliptiques. La méthode, qui consiste à raccorder en certains points choisis diverses solutions obtenues dans des sous-domaines, semble particulièrement efficace pour traiter des problèmes présentant des singularités. La méthode est appliquée à titre d'exemple à la flexion de plaques triangulaires ou en forme de paralléllogrammes.

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Dédié au Professeur E. Stiefel, avec notre entière reconnaissance