Exact Solutions for Vibration of Multi-Span Rectangular Mindlin Plates

2002 ◽  
Vol 124 (4) ◽  
pp. 545-551 ◽  
Author(s):  
Y. Xiang ◽  
G. W. Wei
Keyword(s):  
Boundary Conditions ◽  
Exact Solutions ◽  
Analytical Approach ◽  
Solution Method ◽  
Plate Boundary ◽  
Vibration Frequencies ◽  
Type Solution ◽  
Simply Supported ◽  
Space Technique ◽  
Benchmark Solutions

This paper presents the first-known exact solutions for the vibration of multi-span rectangular Mindlin plates with two opposite edges simply supported. The Levy type solution method and the state-space technique are employed to develop an analytical approach to deal with the vibration of rectangular Mindlin plates of multiple spans. Exact vibration frequencies are obtained for two-span square Mindlin plates with varying span ratios and two-, three- and four-equal-span rectangular Mindlin plates. The influence of the span ratios, the number of spans and plate boundary conditions on the vibration behavior of square and rectangular Mindlin plates is examined. The presented exact vibration results may serve as benchmark solutions for such plates.

A Levy-Type Solution for a Rectangular Plate of Variable Thickness

1958 ◽  
Vol 25 (2) ◽  
pp. 297-298
Author(s):  
H. D. Conway
Keyword(s):  
Boundary Conditions ◽  
Rectangular Plate ◽  
Variable Thickness ◽  
Thickness Variation ◽  
Type Solution ◽  
Arbitrary Boundary ◽  
Simply Supported ◽  
Arbitrary Boundary Conditions ◽  
Plate Of Variable Thickness

Abstract A solution is given for the bending of a uniformly loaded rectangular plate, simply supported on two opposite edges and having arbitrary boundary conditions on the others. The thickness variation is taken as exponential in order to make the solution tractable, and thus closely approximates to uniform taper if the latter is small.

Three alternative versions of the theory for a Timoshenko–Ehrenfest beam on a Winkler–Pasternak foundation

2020 ◽  
pp. 108128652094777
Author(s):  
Giulio Maria Tonzani ◽  
Isaac Elishakoff
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Detailed Comparison ◽  
Vibration Frequencies ◽  
Pasternak Foundation ◽  
New Model ◽  
Simply Supported ◽  
Model Based

This paper analyzes the free vibration frequencies of a beam on a Winkler–Pasternak foundation via the original Timoshenko–Ehrenfest theory, a truncated version of the Timoshenko–Ehrenfest equation, and a new model based on slope inertia. We give a detailed comparison between the three models in the context of six different sets of boundary conditions. In particular, we analyze the most common combinations of boundary conditions deriving from three typical end constraints, namely the simply supported end, clamped end, and free end. An interesting intermingling phenomenon is presented for a simply-supported (S-S) beam together with proof of the ‘non-existence’ of zero frequencies for free-free (F-F) and simply supported-free (S-F) beams on a Winkler–Pasternak foundation.

EXACT EIGEN-RELATIONS OF CLAMPED-CLAMPED AND SIMPLY SUPPORTED PIPES CONVEYING FLUIDS

2012 ◽  
Vol 04 (03) ◽  
pp. 1250035 ◽  
Author(s):  
PIN LU ◽  
HONGYU SHENG
Keyword(s):  
Boundary Conditions ◽  
Dynamic Stability ◽  
Dynamic Properties ◽  
Pipe Conveying Fluid ◽  
Simply Supported ◽  
Pipeline Systems ◽  
Benchmark Solutions ◽  
Conveying Fluid ◽  
Flow Velocities

The exact eigen-equations of pipe conveying fluid with clamped-clamped and simply supported boundary conditions are derived. The simplified forms of the general eigen-equations for some specific cases are determined, and the corresponding dynamic properties are calculated and discussed. These properties provide a better understanding on the relationships between the dynamic stability and the flow velocities of the fluid-conveying components and help to design stable pipeline systems. In addition, the dynamic properties obtained by the exact eigen-equations can also serve as benchmark solutions for verifying results obtained by other approximate approaches.

scholarly journals Frequency Equations and Modes of Free Vibrations of Rectangular Plates with Various Edge Conditions

1994 ◽  
Vol 208 (5) ◽  
pp. 307-319 ◽  
Author(s):  
C W Bert ◽  
M Malik
Keyword(s):  
Boundary Conditions ◽  
Exact Solutions ◽  
Free Vibrations ◽  
Edge Condition ◽  
Rectangular Plates ◽  
Simply Supported ◽  
Aspect Ratios ◽  
Edge Conditions ◽  
Classical Boundary ◽  
Free Edges

This paper considers linear free vibrations of thin isotropic rectangular plates with combinations of the classical boundary conditions of simply supported, clamped and free edges and the mathematically possible condition of guided edges. The total number of plate configurations with the classical boundary conditions are known to be twenty-one. The inclusion of the guided edge condition gives rise to an additional thirty-four plate configurations. Of these additional cases, twenty-one cases have exact solutions for which frequency equations in explicit or transcendental form may be obtained. The frequency equations of these cases are given and, for each case, results of the first nine mode frequencies are tabulated for a range of the plate aspect ratios.

scholarly journals An exact solution method for Fredholm integro-differential equations

2019 ◽  
pp. 2-8 ◽  
Author(s):  
Kyriaki Tsilika
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Exact Solutions ◽  
Solution Method ◽  
Initial Conditions ◽  
Integral Boundary Conditions ◽  
Integral Boundary ◽  
Abstract Operator ◽  
Nonlocal Integral Boundary Conditions

Introduction: Linear boundary value problems for Fredholm ordinary integro-differential equations are seldom consideredwith integral boundary conditions in the literature. In our case, integro-differential equations are subject to multipoint or nonlocalintegral boundary conditions. It should be noted that finding exact solutions even for multipoint problems or problems with nonlocalintegral boundary conditions with a differential equation is a difficult task. Purpose: Finding the uniqueness and existencecriterion of solutions for Fredholm ordinary integro-differential equations with multipoint or nonlocal integral boundary conditionsand obtaining exact solutions in closed form of such problems. Results: Within the class of abstract operator equations, for thespecial case of Fredholm integro-differential equations with multipoint or nonlocal integral boundary conditions, a criterion for theexistence and uniqueness of an exact solution is proved and the analytical representation of the solution is given. A direct methodanalytically solving such problems is proposed, in which all calculations are reproducible in any program of symbolic calculations.If the user sets the input parameters and the initial conditions of the problem, the computer codes check the conditions of existenceand uniqueness and of solution generate the analytical solution. The stages of the solution method are illustrated by twoexamples. The article uses computer algebra system Mathematica to demonstrate the results.

scholarly journals Stability plate with longitudinal constructive discontinuity

2011 ◽  
Vol 9 (1) ◽  
pp. 23-33
Author(s):  
Snezana Mitic ◽  
Ratko Pavlovic
Keyword(s):  
Exact Solutions ◽  
Analytical Approach ◽  
Plate Theory ◽  
Elastic Stability ◽  
The Other ◽  
Uniform Thickness ◽  
Simply Supported ◽  
Thin Plate Theory ◽  
The Stability ◽  
Different Thickness

The influence of longitudinal constructive discontinuity on the stability of the plate in the domain of elastic stability is solved based on the classical thin plate theory. The constructive discontinuities divide the plate into fields of different thickness. The plate has two opposite edges simply supported while the other two edges can take any combination of free, simply supported and clamped conditions. The Levy method is used for the solution of the problem of stability, with the aim of developing an analytical approach when researching the stability of plates with longitudinal constructive discontinuities and also with the aim of obtaining exact solutions for plates with non-uniform thickness. The exact solutions for stability presented herein are very valuable as they may serve as benchmark results for researches in this area.

BUCKLING AND VIBRATION OF STEPPED, SYMMETRIC CROSS-PLY LAMINATED RECTANGULAR PLATES

2001 ◽  
Vol 01 (03) ◽  
pp. 385-408 ◽  
Author(s):  
Y. XIANG ◽  
J. N. REDDY
Keyword(s):  
Solution Method ◽  
Natural Frequencies ◽  
Laminated Plates ◽  
The Other ◽  
Rectangular Plates ◽  
Vibration Frequencies ◽  
Buckling Loads ◽  
Simply Supported ◽  
Step Thickness ◽  
Thickness Variations

This paper presents the exact buckling loads and vibration frequencies of multi-stepped symmetric cross-ply laminated rectangular plates having two opposite edges simply supported while the other two edges may have any combination of free, simply supported, and clamped conditions. An analytical method that uses the Lévy solution method and the domain decomposition technique is proposed to determine the buckling loads and natural frequencies of stepped laminated plates. Buckling and vibration solutions are obtained for symmetric cross-ply laminated rectangular plates having two-, three- and four-step thickness variations.

A New Solution Method for Vibration Analysis of Circular Cylindrical Thin Shells with a Circumferential Surface Crack

2013 ◽  
Vol 639-640 ◽  
pp. 1003-1009 ◽  
Author(s):  
Tao Yin ◽  
Dian Qing Li ◽  
Hong Ping Zhu
Keyword(s):  
Boundary Conditions ◽  
Natural Frequency ◽  
Thin Shell ◽  
Solution Method ◽  
Cylindrical Shells ◽  
Axial Direction ◽  
Mode Shapes ◽  
Governing Equations ◽  
Simply Supported ◽  
Initial Trial

In this paper, a new solution method is proposed for determining the natural frequency of a given mode for a finite-length circular cylindrical thin shell with a circumferential part-through crack. The governing equation of the cracked cylindrical shell is derived by integrating the line-spring model with the classical thin shell theory. The proposed method calculates the natural frequency from an initial trial to satisfy both the governing equations and appropriate boundary conditions through an optimization process. The initial trial is proposed to satisfy the governing equations by using the beam modal function to determine the modal wavenumbers and mode shapes of cylindrical shells in the axial direction, assuming the flexural mode shapes of cylindrical shells in the axial direction to be of the same form as that of a flexural vibration beam with the same boundary conditions. Four representative sets of boundary conditions are considered: simply supported (SS-SS), clamped-clamped (C-C), clamped-simply supported (C-SS), and clamped-free (C-F). Compared with the finite element (FE) method, the proposed solution method is verified to provide an accurate and efficient way to calculate the dynamic characteristics of both intact and cracked cylindrical shells.

Double Fourier series and boundary value problems

1944 ◽  
Vol 40 (3) ◽  
pp. 222-228 ◽  
Author(s):  
A. E. Green
Keyword(s):  
Fourier Series ◽  
Boundary Conditions ◽  
Exact Solutions ◽  
Rectangular Plate ◽  
Strain Energy ◽  
Double Fourier Series ◽  
Energy Methods ◽  
Rectangular Plates ◽  
Simply Supported ◽  
Points Of View

1. Problems in elasticity which are concerned with isotropic rectangular plates have attracted the attention of many writers both from the theoretical and practical points of view. When the boundary conditions are of the simply supported type the solution of the problems is usually simple, although when double Fourier series are used the validity of such solutions is not very clearly shown in most cases. Satisfactory exact solutions for many classical problems in which the edges of the rectangular plate are clamped have only been obtained in recent years, but approximate strain energy methods often gave results which were useful for practical purposes.

Natural Frequencies Formula for Simply Supported Mindlin Plates

1994 ◽  
Vol 116 (4) ◽  
pp. 536-540 ◽  
Author(s):  
C. M. Wang
Keyword(s):  
Exact Solutions ◽  
Explicit Formula ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Kirchhoff Plate ◽  
Vibration Frequencies ◽  
Simply Supported ◽  
Plate Shape

This paper presents an explicit formula for the vibration frequencies of simply supported Mindlin plates in terms of the corresponding thin (Kirchhoff) plate frequencies. The formula has been obtained from an exact vibration analysis of simply supported rectangular Mindlin plates. When the formula was applied to other simply supported plate shapes such as skew plates, circular and annular sectorial plates, it was found to give almost exact solutions. It appears that the formula can be used to predict the frequencies accurately for any simply supported plate shape and thus should be valuable to designers as Mindlin vibration solutions are scarce.

Sign in / Sign up

Export Citation Format

Share Document