Static Analysis for Exact Vibration Analysis of Clamped Plates

2015 ◽  
Vol 15 (08) ◽  
pp. 1540030 ◽  
Author(s):  
Moshe Eisenberger ◽  
Aharon Deutsch
Keyword(s):  
Static Analysis ◽  
Elastic Foundation ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Point Load ◽  
Vibration Frequencies ◽  
Concentrated Load ◽  
Simply Supported ◽  
Winkler Elastic Foundation ◽  
Isotropic Rectangular Plate

Presented herein is a new method for the analysis of plates with clamped edges. The solutions for the natural frequencies of the plates are found using static analysis. The starting are the equations of motion of an isotropic rectangular plate supported on Winkler elastic foundation, with a positive or negative value. In either case, one can solve the displacements of such a plate under a given concentrated load. This deflection will be infinite if the plate losses its stiffness, or in other words, the generalized foundation is causing the plate to be unstable. The solution for the vibration frequencies of the plate is equivalent to finding the values of the negative elastic foundation that will yield infinite deflection under a point load on the plate. The solution for a clamped plate is decomposed as the sum of three cases of plates resting on elastic foundation: simply supported plate with a concentrated load, and two cases of distributed moments along opposite edges. The solution for simply supported plates with elastic foundation is found using Navier's method. For zero force, the vibration frequencies are found up to the desired accuracy by careful calculations at the neighborhood of the roots.

Benchmark Vibration Frequencies of Square Thin Plates with all Possible Combinations of Classical Boundary Conditions

2019 ◽  
Vol 19 (11) ◽  
pp. 1950131
Author(s):  
Aharon Deutsch ◽  
Joseph Tenenbaum ◽  
Moshe Eisenberger
Keyword(s):  
Boundary Conditions ◽  
Elastic Foundation ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Point Load ◽  
Thin Plates ◽  
Vibration Frequencies ◽  
Winkler Elastic Foundation ◽  
Edge Conditions ◽  
Classical Boundary

In this work, a new method is used for the exact vibration analysis of plates with classical boundary conditions. Four classical edge conditions are included: C — clamped, S — Simply supported, F — free, and G — guided. For square plates, all the possibilities add up to 55 cases. The solutions for the natural frequencies of the plates are found in this paper using static analysis. Starting from the equations of motion of an isotropic rectangular thin plate supported on Winkler elastic foundation, with a positive or negative value, the solution for the vibration frequencies of the plate is equivalent to finding the values of the negative elastic foundation that will yield infinite deflection under a point load on the plate. The solution is composed of three parts, the sum of which satisfies exactly both the field equation and the boundary conditions. For zero force, the vibration frequencies are found up to the desired accuracy. Benchmark results of the first six normalized natural frequencies, of isotropic square plates, for all possible 55 combinations of classical boundary conditions are given, many for the first time.

Vibration behavior of a micro cylindrical sandwich panel reinforced by graphene platelet

2020 ◽  
Vol 26 (13-14) ◽  
pp. 1311-1343 ◽  
Author(s):  
Mohammadreza Anvari ◽  
Mehdi Mohammadimehr ◽  
Ali Amiri
Keyword(s):  
Elastic Foundation ◽  
Sandwich Panel ◽  
Natural Frequencies ◽  
Volume Fraction ◽  
Couple Stress Theory ◽  
Equations Of Motion ◽  
Modified Couple Stress Theory ◽  
Theoretical Background ◽  
Graphene Platelet ◽  
Vibration Behavior

In this article, vibration behavior of a micro cylindrical sandwich panel with foam core and reinforced graphene platelet composite layers on the top and bottom resting on elastic foundation based on modified couple stress theory is investigated. Hamilton’s principle is used to determine the governing equations of motion. These equations are solved by Navier’s method to obtain the natural frequencies. The results are compared with the extracted results by the other literatures. The effects of different parameters such as temperature change, volume fraction of graphene platelet, length to radius ratio, and the elastic foundation on the natural frequencies have been carried out. Also, the effects of reinforced materials for layers is discussed and compared with unreinforced composites layers. Sandwich structures are wildly used in different applications such as spacecraft, aeronautical, pressurized gas tanks, boilers, aircraft fuselage, marines, and civil structures, and these cases need high strength and low weight. The present work is a theoretical background for more explorations and further experimental researchers in the field of cylindrical reinforced panels.

scholarly journals Peridynamic Higher-Order Beam Formulation

2020 ◽  
Author(s):  
Zhenghao Yang ◽  
Erkan Oterkus ◽  
Selda Oterkus
Keyword(s):  
Finite Element Method ◽  
Cantilever Beam ◽  
Higher Order ◽  
Point Load ◽  
Transverse Displacement ◽  
Lagrange Equations ◽  
Concentrated Load ◽  
Simply Supported Beam ◽  
Simply Supported ◽  
Good Agreement

Abstract In this study, a novel higher-order peridynamic beam formulation is presented. The formulation is obtained by using Euler-Lagrange equations and Taylor’s expansion. To demonstrate the capability of the presented approach, several different beam configurations are considered including simply supported beam subjected to distributed loading, simply supported beam with concentrated load, clamped-clamped beam subjected to distributed loading, cantilever beam subjected to a point load at its free end and cantilever beam subjected to a moment at its free end. Transverse displacement results along the beam obtained from peridynamics and finite element method are compared with each other and very good agreement is obtained between the two approaches.

Effect of Winkler and Pasternak elastic foundation on the vibration of rotating functionally graded material cylindrical shell

2018 ◽  
Vol 232 (24) ◽  
pp. 4564-4577 ◽  
Author(s):  
Muzamal Hussain ◽  
Muhammad Nawaz Naeem ◽  
Mohammad Reza Isvandzibaei
Keyword(s):  
Cylindrical Shell ◽  
Boundary Conditions ◽  
Elastic Foundation ◽  
Natural Frequencies ◽  
Cylindrical Shells ◽  
Functionally Graded ◽  
Radius Ratio ◽  
Wave Number ◽  
Simply Supported ◽  
Elastic Foundations

In this paper, vibration characteristics of rotating functionally graded cylindrical shell resting on Winkler and Pasternak elastic foundations have been investigated. These shells are fabricated from functionally graded materials. Shell dynamical equations are derived by using the Hamilton variational principle and the Langrangian functional framed from the shell strain and kinetic energy expressions. Elastic foundations, namely Winkler and Pasternak moduli are inducted in the tangential direction of the shell. The rotational motions of the shells are due to the Coriolis and centrifugal acceleration as well as the hoop tension produced in the rotating case. The wave propagation approach in standard eigenvalue form has been employed in order to derive the characteristic frequency equation describing the natural frequencies of vibration in rotating functionally graded cylindrical shell. The complex exponential functions, with the axial modal numbers that depend on the boundary conditions stated at edges of a cylindrical shell, have been used to compute the axial modal dependence. In our new investigation, frequency spectra are obtained for circumferential wave number, length-to-radius ratio, height-to-radius ratio with simply supported–simply supported and clamped–clamped boundary conditions without elastic foundation. Also, the effect of elastic foundation on the rotating cylindrical shells is examined with the simply supported–simply supported edge. To check the validity of the present method, the fundamental natural frequencies of non-rotating isotropic and functionally graded cylindrical shells are compared with the open literature. Also, a comparison is made for infinitely long rotating with the earlier published paper.

Transverse Vibration of Two Axially Moving Beams Connected by an Elastic Foundation

2005 ◽  
Author(s):  
Mohamed Gaith ◽  
Sinan Mu¨ftu¨
Keyword(s):  
Elastic Foundation ◽  
Transverse Vibration ◽  
Natural Frequencies ◽  
Flexural Rigidity ◽  
Beam Theory ◽  
Mode Shapes ◽  
Winkler Elastic Foundation ◽  
Canonical State ◽  
Axially Moving Beams ◽  
Axially Moving

Transverse vibration of two axially moving beams connected by a Winkler elastic foundation is analyzed analytically. The system is a model of paper and paper-cloth (wire-screen) used in paper making. The two beams are tensioned, translating axially with a common constant velocity, simply supported at their ends, and of different materials and geometry. Due to the effect of translation, the dynamics of the system displays gyroscopic motion. The Euler-Bernoulli beam theory is used to model the deflections, and the governing equations are expressed in the canonical state form. The natural frequencies and associated mode shapes are obtained. It is found that the natural frequencies of the system are composed of two infinite sets describing in-phase and out-of-phase vibrations. In case the beams are identical, these modes become synchronous and asynchronous, respectively. Divergence instability occurs at the critical velocity; and, the frequency-velocity relationship is similar to that of a single traveling beam. The effects of the mass, flexural rigidity, and axial tension ratios of the two beams, as well as the effects of the elastic foundation stiffness are investigated.

scholarly journals Natural Frequencies of FG Plates with Two New Distribution of Porosity

2021 ◽  
Vol 26 (2) ◽  
pp. 128-142
Author(s):  
Slimane Merdaci ◽  
Adda Hadj Mostefa ◽  
Osama M.E.S. Khayal
Keyword(s):  
Natural Frequencies ◽  
Equations Of Motion ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Functionally Graded Plates ◽  
Simply Supported ◽  
Fg Plates ◽  
Porosity Coefficient ◽  
The Impact ◽  
New Distribution

Abstract The functionally graded plates (FGP) with two new porosity distributions are examined in this paper. In this work the plate is modeled using the higher-order shear deformation plate principle. The shear correction variables are neglected. To evaluate the equations of motion, the Hamilton method will be used herein. Therefore, the free vibration analysis of FG plate is developed in this work. For porous smart plates with simply-supported sides, natural frequencies are obtained and verified with the established findings in the literature. The impact of the porosity coefficient on the normal frequencies of the plate for various thickness ratios, geometric ratios, and material properties was investigated in a thorough numerical analysis.

scholarly journals Free vibration analysis of sandwich beams with FG porous core and FGM faces resting on Winkler elastic foundation by various shear deformation theories

2018 ◽  
Vol 12 (3) ◽  
pp. 23-33 ◽  
Author(s):  
Dang Xuan Hung ◽  
Huong Quy Truong
Keyword(s):  
Free Vibration ◽  
Elastic Foundation ◽  
Equations Of Motion ◽  
Sandwich Beam ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Porous Core ◽  
Winkler Elastic Foundation ◽  
Beam Theories ◽  
Good Agreement

This paper studies the free vibration behavior of a sandwich beam resting on Winkler elastic foundation. The sandwich beam is composed of two FGM face layers and a functionally graded (FG) porous core. A common general form of different beam theories is proposed and the equations of motion are formulated using Hamilton's principle. The result of the general form is validated against those of a particular case and shows a good agreement. The effect of different parameters on the fundamental natural frequency of the sandwich beam is investigated. Article history: Received 02 March 2018, Revised 26 March 2018, Accepted 27 April 2018

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.

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